Finite primitive groups of small rank: symmetric and sporadic groups

Abstract

Finite primitive groups of small rank have been studied for a long time because they are a natural source of important examples and because of their applications to various parts of mathematics. This interest has brought to the classification of finite primitive groups of small rank. Historically, in this context, the word “small” means absolute constant. This paper is part of a series aiming to classify finite primitive groups of small rank, where the word “small” is not an absolute constant. We have various motivations for embarking in this classification, ranging from representation theory to combinatorics. We start our work considering almost simple groups whose socle is either an alternating or a sporadic simple group.

1 Introduction

One of the first major applications of the classification of the finite simple groups is the work of Cameron [7], determining the finite 2-transitive groups. This is the first systematic investigation of finite primitive groups based on the rank: 2-transitive groups are groups having rank 2. Besides its intrinsic interest, Cameron’s paper establishes the fact that the CFSGs is a tool of paramount importance for the modern investigation of an old subject as permutation group theory. Indeed, Cameron’s classification gives a very explicit list of groups, an achievement possibly beyond the expectations of Manning and Wielandt (which were also interested in 2-transitive groups), who did not have the CFSGs available in their toolkit.

Next, finite group theorists have got interested in permutation groups having rank 3 because of their application to graph theory and geometry. For instance, rank 3 groups give rise to strongly regular graphs with a rich group of symmetries, and rank 3 groups can be used to construct interesting finite geometries. The classification of the finite primitive groups of rank 3 counts the contribution of many researchers, which culminated again with a list of groups. The finite primitive groups of rank 3 of affine type (according to the O’Nan–Scott partition into types) were classified by Liebeck [18]. Kantor and Liebler [17] have classified the rank 3 primitive permutation groups having socle a finite classical group, and Liebeck and Saxl [19] have classified the rank 3 primitive permutation groups having socle a finite exceptional group of Lie type or a sporadic simple group. The papers [17,18,19] contain all the relevant information for the complete classification of rank 3 primitive permutation groups. Indeed, the classification of the rank 3 primitive permutation groups having socle an alternating group has been determined much earlier, see, for example, [3], and the remaining O’Nan–Scott types are easy to deal with, as remarked by Cameron [7].

Devillers et al. [10, 11] have classified the rank 3 permutation groups which are quasiprimitive but not primitive. When combined with the earlier work on primitive groups, this result yields a classification of all quasiprimitive rank 3 permutation groups. (A permutation group is quasiprimitive if each non-identity normal subgroup of transitive.)

Before continuing our introduction, there is a remark that we need to make in order to explain some of our motivations and some choices we made in this work. Based on a beautiful remark by Seitz [25], Kantor [16, Corollary] has shown that for each $r\in \mathbb {N}$, the classification of the almost simple groups of rank at most r is a finite problem, and this was conjectured by Peter Neumann in 1973. We refer to [16] for a precise definition of “finite problem” in this context. However, although the work in [16] is theoretically very interesting, in practice it is not very effective. For instance [16, Section 5], when $r=4$ and $G=\mathrm {P}\Omega _n^{\pm }(q)$, every permutation representation of G having rank at most 4 is either parabolic, or $n\le 63$ and $q<8.905\cdot 10^{65}$. This shows that already when $r=4$ the bounds in [16] cannot be used to deal with the “small” groups with a computer, for producing the list of permutation representations of rank at most 4 of $\mathrm {P}\Omega _n^{\pm }(q)$. Since 1973, our knowledge on finite simple groups has considerably grown; hence, in our work, we aim to classify the finite primitive groups of small rank, where the word “small” has to be understood as a slow growing function depending on the O’Nan–Scott–Aschbacher class of the groups under consideration. For instance, when G is an almost simple group with socle an alternating group ${\mathrm {Alt}}(n)$ where n is a positive integer with $n\ge 5$, we classify the primitive faithful actions of G of rank at most $n^2$. To some extent, the choice of $n^2$ is purely aesthetic: first, it makes our classification not too cumbersome to use; second, in our opinion, it is interesting to classify the primitive actions of ${\mathrm {Sym}}(n)$ of small rank where small has to be understood as a “not too fast growing function” of the Coxeter rank of ${\mathrm {Sym}}(n)$; third, $n^2$ already highlights some of the deep number-theoretic difficulties (related to the Anand–Dumir–Gupta conjecture on magic squares [2]) arising from this task.

When the classification of rank 3 primitive permutation groups was completed, finite geometers found very useful having in their toolkit also the classification of the rank 4 primitive permutation groups. Except for the primitive groups of affine type, Cuypers in his D. Phil.  thesis [9] has classified the primitive groups having rank 4. Applications of this classification appear in the second part of Cuypers’ thesis. Cuypers’ work has never been published and it is difficult to access, although it has been quoted and used a few times in the literature. Thus, another reason for embarking into our current project is to have a classification including Cuypers’. Therefore, our choices of slow growing functions will always make sure that except for the affine groups, they include finite primitive groups of rank at most 4.

Cuypers’ work heavily relies on the fact that the permutation character of a permutation group of rank at most 4 is multiplicity-free. Character theoretic methods pivoting on the multiplicity-freeness have been extensively used in this context, for example, for the classification of permutation groups of rank 2 and 3. Elementary number-theoretic considerations show that the permutation character of a rank 5 group is also multiplicity-free. Therefore, it is natural asking:

What is the smallest positive integer r such that, there exists a primitive permutation group G having rank r and whose permutation character is not multiplicity free?

The group ${\mathrm {Aut}}(^{2}B_2(8))$ has a primitive permutation representation of degree 560, rank 7, and the permutation character is not multiplicity-free. Thus, $r\le 7$. This and other examples were first found by Jason Williford, after a conversation with the first author, with an exhaustive computer search in the database of primitive groups available in GAP [27].

Our work aims to show that $r=7$ and aims to determine the primitive permutation groups of rank at most 7 whose permutation character is not multiplicity-free. Actually, at the time of this writing, we have only a finite number of such examples. We find this rather surprising and we wonder whether this behaviour holds for larger ranks.

We stress the fact that currently, we have no example with $r=6$. This case is rather interesting on its own; indeed, any rank 6 non-multiplicity-free primitive group can be used to construct a non-commutative primitive association scheme which is schurian. However, to the best of our knowledge, no non-commutative primitive association scheme is known at the moment (regardless of whether the scheme is schurian or non-schurian).

Theorem 1.1

Let G be a finite almost simple primitive group of rank at most 8 with socle a sporadic simple group. Then, G appears in Table 1.

Table 1 Primitive permutation representations of rank at most 8: sporadic socle, see Notation 3.1

Table 2 Primitive permutation representations of small rank: alternating socle

See Notation 3.1 for a detailed description for reading Table 1. From Table 1, we deduce that for the groups in Theorem 1.1, 8 is the smallest rank for a non-multiplicity-free primitive action: arising from the actions of $M_{11}$ of degree 165, $M_{23}$ of degree 1771 and $Co_3$ of degree 37950.

Theorem 1.2

Let G be a finite almost simple primitive group of rank at most $n^2$ with socle an alternating group $\mathrm {Alt}(n)$. Then, G appears in Table 2.

From Table 2, we deduce that for the groups in Theorem 1.2, 12 is the smallest rank for a non-multiplicity-free primitive action.

Theorem 1.2 is only a summary of our results on finite almost simple primitive groups with socle an alternating group. In Sects. 4–6, we obtain much more information on these actions. Let M be the stabiliser of a point in G, where G is a finite almost simple primitive group with socle an alternating group ${\mathrm {Alt}}(n)$. For the proof of Theorem 1.2, our analysis depends on the action of M on $\{1,\ldots ,n\}$. In Sect. 4, we consider the case that M is either intransitive or primitive on $\{1,\ldots ,n\}$. Next, we consider the case that M is imprimitive and hence, G is acting on the set of uniform partitions of $\{1,\ldots ,n\}$. In particular, $n=rs$ for some $r,s\ge 2$. In Sect. 5 and 6, we investigate in detail the rank of G in its action on the set of uniform partitions of $\{1,\ldots ,sr\}$ into r parts of cardinality s. In Sect. 5, we obtain two lower bounds for this rank of independent interest: the first bound gives good estimates when r is large compared to s and the second bound gives good estimates when s is large compared to r. See Theorems 5.8 and 5.9 for details. Finally, in Sects. 5 and 6, we obtain explicit formulae for the rank of ${\mathrm {Sym}}(rs)$ acting on uniform partitions of $\{1,\ldots ,rs\}$ into r parts of cardinality s when $r\le 5$ and when $s=2$; in the process, we show that computing explicitly this rank is related to the Anand–Dumir–Gupta conjecture on magic squares.

Inspired by our work, we pose the following:

Conjecture 1.3

There exists a function $f:\mathbb {N}\rightarrow \mathbb {N}$ with $\lim _{n\rightarrow \infty }f(n)=\infty$ such that if G is a finite primitive group of rank r, then the number of distinct irreducible constituents of the permutation character of G is at least f(r).

The number of irreducible constituents in the decomposition of the permutation character is the dimension of the centre of the centraliser ring of the permutation representation. Here, we coin the term central rank of the permutation representation for this number. With this terminology, our conjecture claims that the central rank of a finite primitive group can be bounded from below with an unbounded function depending on the rank.

2 Basic lemmas and some notation

Given a transitive permutation group G on $\Omega$, we denote by ${{{\text {rk}}}}_\Omega (G)$ the rank of G, that is, the number of orbits of G on the Cartesian product $\Omega \times \Omega$. Equivalently, given $\omega \in \Omega$, ${{{\text {rk}}}}_\Omega (G)$ equals the number of orbits of the stabiliser $G_\omega :=\{g\in G\mid \omega ^g=\omega \}$ of the point $\omega$ in its action on $\Omega$. If the domain $\Omega$ of G is clear from the context, we drop the cumbersome label $\Omega$ from ${{{\text {rk}}}}_\Omega (G)$. We recall that if $\pi$ is the permutation character of G, then ${{{\text {rk}}}}_{\Omega }(G)=\langle \pi ,\pi \rangle$, where $\langle \cdot ,\cdot \rangle$ is the Hermitian product in the space of $\mathbb {C}$-class functions of G.

Given a group G and a subgroup M, we denote by $M{\backslash } G$ the set of right cosets of M in G and we view $M{\backslash } G$ as a G-set, with the group G acting on $M{\backslash } G$ via right multiplication. Moreover, we let $\mathrm {Out}(G)$ denote the outer automorphism of G.

Lemma 2.1

Let G be a transitive permutation group on $\Omega$, let $\omega \in \Omega$ and let N be a normal transitive subgroup of G. Then,

$$\begin{aligned} |\Omega |\le 1+(\mathrm {rk}_\Omega (G)-1)|G_\omega | \end{aligned}$$

(1)

and ${{{\text {rk}}}}_\Omega (N)\le 1+({{{\text {rk}}}}_\Omega (G)-1)[G:N]$.

Proof

The group $G_\omega$ has at least one orbit, namely $\{\omega \}$, of cardinality 1 and all other orbits have cardinality at most $|G_\omega |$; hence, Eq. (1) follows. Since N is transitive on $\Omega$, we have $G=NG_\omega$ and $[G:N]=[G_\omega :N_\omega ]$. Therefore, each $G_\omega$-orbit on $\Omega {\setminus }\{\omega \}$ consists of at most [G : N] $N_\omega$-orbits. Therefore, ${{{\text {rk}}}}_\Omega (N)\le 1+({{{\text {rk}}}}_\Omega (G)-1)[G:N]$. $\square$

Lemma 2.1 is one of our main basic tools; however, sometimes we use it in disguised forms. For instance, when G is an almost simple group with socle $G_0$ and M is a maximal core-free subgroup of G, we deduce from Lemma 2.1 that $[G_0:M\cap G_0]=[G:M]=|M{\backslash } G|\le 1+({{{\text {rk}}}}_{M{\backslash } G}(G)-1)|M|$, that is,

$$\begin{aligned} |G_0|< & {} {{{\text {rk}}}}_{M{\backslash } G}(G)|M||M\cap G_0|\\= & {} {{{\text {rk}}}}_{M{\backslash } G}(G)[G:G_0]|M\cap G_0|^2\le {{{\text {rk}}}}_{M{\backslash } G}(G_0)|\mathrm {Out}(G_0)||M\cap G_0|^2. \end{aligned}$$

Thus,

$$\begin{aligned} |G_0|\le \mathrm {rk}_\Omega (G)|\mathrm {Out}(G_0)||M\cap G_0|^2; \end{aligned}$$

(2)

this upper bound will be very useful combined with information on the simple group $G_0$; in fact, this inequality only uses abstract information on the simple group $G_0$. A similar inequality was used by Cuypers [9, Lemma 3.2] for his classification of the primitive groups of rank at most 4.

Using Eq. (2), for studying the finite primitive almost simple groups with socle a simple group of Lie type, it is important a result of Alavi and Burness [1]. These authors consider the problem of classifying the finite almost simple groups G admitting a triple factorisation, that is, the triples (G, A, B) with G an almost simple group and with A and B proper subgroups of G with $G=ABA$. If $G=ABA$ is a triple factorisation, then $\max \{|A|,|B|\}^3\ge |G|$. Therefore, Alavi and Burness as a first step towards a complete classification of the triple factorisations are interested in classifying first the maximal subgroups M of G with $|M|^3\ge |G|$. In the light of Eq. (2), the cardinality of $G_0$ is bounded above by a multiple of $|M\cap G_0|^2$ and hence, the work of Alavi and Burness is very useful for our project.

Lemma 2.2

Let G be a transitive permutation group on $\Omega$, let $\omega \in \Omega$, let $\pi$ be the permutation character of G, and let $\kappa$ be the maximum degree of a complex irreducible character of G. Then,

$$\begin{aligned} |\Omega |\le 1+(\mathrm {rk}_\Omega (G)-1)\kappa . \end{aligned}$$

(3)

Proof

Write $\pi :=\chi _0+a_1\chi _1+\cdots +a_\ell \chi _\ell$ where $\chi _0$ is the principal character of G, $\chi _0,\ldots ,\chi _\ell$ are the irreducible constituents of $\pi$ and $a_1,\ldots ,a_\ell$ are positive integers. Clearly, ${{{\text {rk}}}}_\Omega (G)=1+a_1^2+\cdots +a_\ell ^2$ and $|\Omega |=\pi (1)=\chi _0(1)+a_1\chi _1(1)+\cdots +a_\ell \chi _\ell (1)\le 1+(a_1+\cdots +a_\ell )\kappa \le 1+(a_1^2+\cdots +a_\ell ^2)\kappa =1+(\langle \pi ,\pi \rangle -1)\kappa =1+({{{\text {rk}}}}_\Omega (G)-1)\kappa$. $\square$

3 Sporadic simple groups

We use the notation of [8] to denote the almost simple groups with socle a sporadic simple group and their maximal subgroups. Let G be a primitive permutation group with socle a sporadic simple group. Whenever the permutation character of G is available in a computer algebra system, as GAP [27] or magma [4], we can check directly whether the rank of G is at most eight and whether its permutation character is multiplicity-free. Apart from

• (a) the Monster,

• (b) the action of the Baby Monster on the cosets of a maximal subgroup of type $(2^2\times F_4(2)):2$,

each permutation character of each primitive permutation representation of an almost simple group with socle a sporadic simple group is available in GAP via the package “The GAP character Table Library”. Our application of this package is not a novelty; indeed, very recently the methods developed in [28] have been used in [6, 12] to solve some open problems concerning the sporadic simple groups.

Notation 3.1

In Table 1, we have listed the primitive permutation representations of the almost simple groups with socle a sporadic group having rank at most 8: in the first column we have reported the almost simple group, in the second column the degree of the action, in the third column the structure of the stabiliser of a point, in the fourth column the rank of the action, and in the fifth column we have reported “Y” if the permutation character under consideration is multiplicity-free and “N” otherwise.

From the previous paragraphs, except for Cases $\mathbf{(a)}$ and $\mathbf{(b)}$ mentioned above, the veracity of Table 1 can be tested using a computer. The permutation character of the Baby Monster G in its action on the cosets of a maximal subgroup M of type $(2^2\times F_4(2)):2$ is missing from the GAP library because the conjugacy fusion of some of the elements of M in G remains a mystery: this information is vital for computing the permutation character. We claim ${{{\text {rk}}}}_{M{\backslash } G}(G)\ge 11$ and hence this action does not appear in Table 1. This action has degree $|G:M|=156849238149120000$, and the largest degree of an irreducible complex character of G is 16547812226400000. From Lemma 2.2, we have

$$\begin{aligned} \mathrm {rk}_{M{\backslash } G}(G)\ge \frac{156849238149120000}{16547812226400000}+1\ge 10.47855. \end{aligned}$$

It remains to deal with the case that G is the Monster group, in all of its primitive faithful actions. To ensure that our proof covers all primitive actions, we will make use of a recent account of the classification of the maximal subgroups of G in [29]. From [29, Section 3.6], we see that the classification of the maximal subgroups of the Monster is complete except for a few small (but rather intriguing) open cases. In particular, if M is a maximal subgroup of G, then either

• (a) M is in [29, Section 3.6], or

• (b) M is almost simple with socle isomorphic to $L_2(8)$, $L_2(13)$, $L_2 (16)$, $U_3 (4)$ or $U_3 (8)$.

Using this information, we can argue analogously to the Baby Monster. Let M be a maximal subgroup of G with ${{{\text {rk}}}}_{M{\backslash } G}(G)\le 8$. From Lemma 2.2, $|\Omega |=|G:M|\le 1+7\kappa$ and hence $|M|\ge |G|/(1+7\kappa )$, where $\kappa$ is the largest degree of an irreducible complex character of G. Using the information from [8], we get

$$\begin{aligned} |M|\ge & {} \frac{|G|}{1+7\kappa }\\= & {} \frac{808017424794512875886459904961710757005754368000000000}{1811764342717385448316640626}\\\ge & {} 4.4\cdot 10^{26}. \end{aligned}$$

The only maximal subgroup of G having order at least $4.4\cdot 10^{26}$ is the twofold cover of the Baby Monster; therefore, $M=2.B$. Let $\pi$ be the permutation character of G in its action on $M{\backslash } G$. Using the character table of G (and the notation in [8]), it is a long computer-aided computation to prove that

$$\begin{aligned} \pi =\chi _1+\chi _2+\chi _4+\chi _5+\chi _9+\chi _{14}+\chi _{21}+\chi _{34}+\chi _{35} \end{aligned}$$

and hence, ${{{\text {rk}}}}_{M{\backslash } G}(G)=9$. This is an integer linear programming problem. Namely, $\pi$ is a linear combination $\chi _1+a_2\chi _2+\cdots +a_{194}\chi _{194}$ of the complex irreducible characters of G with non-negative coefficients. This linear combination satisfies various constraints. First, $[G:M]=\pi (1)=1+a_1\chi _1(1)+\cdots +a_{194}\chi _{194}(1)$. Second, the vector $\pi$ is non-negative in each of its 194 coordinates. Third, $\pi$ is zero on every element not G-conjugate to an element of M, because these elements fix no point of $M{\backslash } G$. Forth, for each $g\in G$ and for each integer $\ell$, $\pi (g)\le \pi (g^\ell )$ because $\pi (g)$ and $\pi (g^\ell )$ are the fixed point numbers of g and $g^\ell$, respectively. Incidentally, this shows that the smallest rank of a faithful permutation representation of the Monster group has rank 9 and that this action is the permutation representation of G on the right cosets of a maximal subgroup of type 2.B.

From Table 1, we deduce the following result.

Corollary 3.2

Let G be an almost simple group with socle a sporadic simple group. If G has rank at most 8, then either the permutation character of G is multiplicity-free, or $G=M_{11}$ in its action of degree 165, or $G=M_{23}$ in its action of degree 1771, or $G=Co_3$ in its action of degree 37950.

4 Alternating groups

Let $\Omega$ be a G-set, with G acting faithfully and primitively on $\Omega$. Fix $\omega \in \Omega$ and consider $G_\omega$, the stabiliser in G of the point $\omega$. We assume that ${{{\text {rk}}}}_\Omega (G)\le n^2$.

Because of its peculiar behaviour, there is one case that we aim to discuss first: $n=6$ and $G\nleq {\mathrm {Sym}}(6)$, that is, G is isomorphic to either $M_{10}$, or ${\mathrm {PGL}}_2(9)$, or $\mathrm {P}\Gamma \mathrm {L}_2(9)$ (recall that ${\mathrm {Sym}}(6)\cong \mathrm {P}\Sigma \mathrm {L}_2(9)$). For each $G\in \{{\mathrm {PGL}}_2(9),M_{10},\mathrm {P}\Gamma \mathrm {L}_2(9)\}$, we compute the maximal core-free subgroups M of G. Then, for each pair (G, M), we construct the permutation character of G in its action on $M{\backslash } G$ and we select the cases in which the rank of this action is at most $6^2=36$: these cases are listed in Table 2.

We now subdivide our argument depending on the action of $G_\omega$ on the underlying set $\{1,\ldots ,n\}$: in Sect. 4.1 we consider $G_\omega$ intransitive on $\{1,\ldots ,n\}$, in Sect. 4.2 we consider $G_\omega$ primitive on $\{1,\ldots ,n\}$, and in Sect. 5 we consider $G_\omega$ imprimitive on $\{1,\ldots ,n\}$. For all of these sections, our results are summarised in Table 2.

4.1 The subgroup $G_\omega$ is intransitive in its action on $\{1,\ldots ,n\}$

The maximality of $G_\omega$ in G yields that $G_\omega$ is the stabiliser in G of a subset of $\{1,\ldots ,n\}$ of cardinality $\kappa$, for some $\kappa \in \mathbb {N}$ with $1\le \kappa <n/2$. Now, the action of G on $\Omega$ is permutation isomorphic to the natural action of G on the $\kappa$-subsets of $\{1,\ldots ,n\}$, that is, on the subsets of $\{1,\ldots ,n\}$ of cardinality $\kappa$. Using this identification of the G-set $\Omega$, it is immediate to see that G has rank $\kappa +1$. Moreover, $G_\omega$ is a Young subgroup of G. Therefore, using the character theory of the alternating and the symmetric group (see [15, Section 2 and Corollary 2.2.22] or [23, Section 2.11 and Theorem 2.11.2]), we get that the permutation character of G acting on $\Omega$ is multiplicity-free. This example is listed in Table 2.

4.2 The subgroup $G_\omega$ is primitive in its action on $\{1,\ldots ,n\}$

For the reader’s convenience, we report a very useful result of Maróti [21, Theorem 1.1] phrased in terms of our current notation:

Theorem 4.1

One of the following holds:

• (i) there exist three natural numbers m, k, r with $m\ge 5$, $m/2>k\ge 1$, $r\ge 1$ and with $r>1$ when $k=1$, such that $G_\omega$ is a subgroup of the wreath product ${\mathrm {Sym}}(m){\mathrm {wr}}{\mathrm {Sym}}(r)$ containing $({\mathrm {Alt}}(m))^r$, where the action of ${\mathrm {Sym}}(m)$ is on k-subsets of $\{1,\ldots ,n\}$ and the wreath product has the product action of degree $n={m\atopwithdelims ()k}^r$;

• (ii) $G_\omega$ equals $M_{11}$, $M_{12}$, $M_{23}$ or $M_{24}$ in their 4-transitive actions;

• (iii) $|G_\omega |\le n\cdot \prod _{i=0}^{\lfloor \log _2(n)\rfloor -1}(n-2^i)<n^{1+\lfloor \log _2(n)\rfloor }$.

We now consider in turn the three possibilities described by Maróti. Assume that Case (i) holds. From Lemma 2.1, we get $|G:G_\omega |=|\Omega |\le 1+(n^2-1)|G_\omega |$; hence, $|G|\le n^2|G_\omega |^2$ and

$$\begin{aligned} \frac{n!}{2}\le n^2\left( m!^rr!\right) ^2, \end{aligned}$$

with $n={m\atopwithdelims ()k}^r$. Now elementary computations involving Stirling’s formula show that this inequality is never satisfied; therefore, in our context Case (i) does not arise.

Assume that Case (ii) holds. We have $G_\omega \le {\mathrm {Alt}}(n)$ and hence $G={\mathrm {Alt}}(n)$. From Lemma 2.1, a quick computation shows that ${{{\text {rk}}}}_\Omega (G)\le n^2$ only when $G_\omega$ is either $M_{11}$ or $M_{12}$ (that is, $|G_\omega |$ is too small compared to $|\Omega |$ for satisfying Eq. (1) with ${{{\text {rk}}}}_\Omega (G)\le n^2$); therefore, we may discard $M_{23}$ and $M_{24}$ form our attention. For each of the two possibilities $M_{11}$ and $M_{12}$ for $G_\omega$, we construct with the invaluable help of magma the permutation representation of ${\mathrm {Alt}}(n)$ on the right cosets of $G_\omega$. For each of these actions, the permutation character is multiplicity-free and the actions have rank 5 and 4, respectively, see Table 2.

Assume, finally, that Case (iii) holds. Arguing as above, from Lemma 2.1 we get

$$\begin{aligned} \frac{n!}{2}<n^2\cdot \left( n^{1+\lfloor \log _2(n)\rfloor }\right) ^2. \end{aligned}$$

Another elementary computation involving Stirling’s formula shows that this inequality is satisfied only when $n\le 17$. To deal with the remaining cases, we invoke once again the help of a computer. Namely, for each $n\in \{5,\ldots ,17\}$, we consider $G\in \{{\mathrm {Alt}}(n),{\mathrm {Sym}}(n)\}$ and we compute the maximal core-free subgroups M of G that are primitive on $\{1,\ldots ,n\}$. Then, for each pair (G, M), we construct the permutation character of G in its action on the right cosets of M and we select the cases in which the rank of this action is at most $n^2$: these cases are listed in Table 2.

The remaining case, that is, “$G_\omega$ is imprimitive in its action on $\{1,\ldots ,n\}$”, is very involved and we deal with it in its own section.

5 The subgroup $G_\omega$ is imprimitive in its action on $\{1,\ldots ,n\}$

The maximality of $G_\omega$ yields that $G_\omega$ is the stabiliser in G of a uniform partition of $\{1,\ldots ,n\}$. In particular, $n=sr$, for some integers r and s with $r,s\ge 2$, and the action of G on $\Omega$ is permutation isomorphic to the natural action of G on the set of all partitions of n into r parts each having size s. Therefore, $G_\omega \cong G\cap ({\mathrm {Sym}}(s){\mathrm {wr}}{\mathrm {Sym}}(r))$, because the imprimitive wreath product ${\mathrm {Sym}}(s){\mathrm {wr}}{\mathrm {Sym}}(r)$ is the stabiliser in ${\mathrm {Sym}}(n)$ of a partition of $\{1,\ldots ,n\}$ having r parts each of cardinality s. In what follows, we identify $\Omega$ with the set of all uniform partitions of $\{1,\ldots ,n\}$ into r parts of cardinality s and we write $\Omega _{r,s}:=\Omega$ to have a notation that help us to remember the action under consideration. In our preliminary discussion, we first restrict our attention to $G={\mathrm {Sym}}(n)$.

Fix a uniform partition $\omega _0\in \Omega _{r,s}$, thus, $\omega _0=\{X_1,\ldots ,X_r\}$ is a partition of $\{1,\ldots ,n\}$ and $|X_i|=s$ for each $i\in \{1,\ldots ,r\}$. For each $\omega =\{Y_1,\ldots ,Y_r\}\in \Omega _{r,s}$, we associate an $(r\times r)$-matrix $M_{\omega }$: the entry of $M_{\omega }$ in row i and column j is the cardinality of $|X_i\cap Y_j|$. Observe that each entry of $M_{\omega }$ is a non-negative integer. Moreover, each row and each column of $M_{\omega }$ adds up to s. In recreational mathematics, the matrix $M_\omega$ is said to be a magic square. (The matrix $M_\omega$ is also called a doubly stochastic non-negative integral square matrix of order r with row and column sum s.) Finally, with a moment’s thought we see that given two elements $\omega$ and $\omega '$ in $\Omega _{r,s}$, $\omega$ and $\omega '$ are in the same $G_{\omega _0}$-orbit if and only if $M_{\omega '}$ can be obtained from $M_{\omega }$ by some permutation of its rows and of its columns. In the light of this paragraph, we introduce some useful notation: let $\Sigma _{r,s}$ be the set of all $(r\times r)$-matrices having non-negative integer coefficients and having constant row and column sum s. Now, the group ${\mathrm {Sym}}(r)\times {\mathrm {Sym}}(r)$ acts on $\Sigma _{r,s}$: the element $(\sigma ,\tau )\in {\mathrm {Sym}}(r)\times {\mathrm {Sym}}(r)$ acts on the matrices in $\Sigma _{r,s}$ by permuting the rows via the permutation $\sigma$ and the columns via the permutation $\tau$. We summarise this paragraph in the following.

Lemma 5.1

The rank of ${\mathrm {Sym}}(sr)$ in its action on the set of uniform partitions of $\{1,\ldots ,sr\}$ into r parts of cardinality s equals the number of orbits of ${\mathrm {Sym}}(r)\times {\mathrm {Sym}}(r)$ in its action on the set of $(r\times r)$-matrices having non-negative integer coefficients and having constant row and column sum s.

Using Lemma 5.1 and the orbit-counting lemma, we can in some cases compute ${{{\text {rk}}}}_{\Omega _{r,s}}(G)$. Since some of the computations that follow are quite involved, we start with two easy remarks. Given a permutation $g\in {\mathrm {Sym}}(\Delta )$, we write ${\mathrm {Fix}}_\Delta (g):=\{\delta \in \Delta \mid \delta ^g=\delta \}$ and ${\mathrm {fix}}_\Delta (g):=|{\mathrm {Fix}}_\Delta (g)|$.

Proposition 5.2

The rank ${\mathrm {Sym}}(2s)$ in its action on the set of uniform partitions of $\{1,\ldots ,2s\}$ into 2 parts of cardinality s equals $\lceil (s+1)/2\rceil$. Moreover, the permutation character of this action is multiplicity-free. When, $s>2$, ${\mathrm {Alt}}(2s)$ has also rank $\lceil (s+1)/2\rceil$ in its action on uniform partitions with 2 parts.

Proof

We use Lemma 5.1 and the orbit-counting lemma. Clearly, $\Sigma _{2,s}$ consists of the $s+1$ matrices

$$\begin{aligned} \begin{pmatrix} i&{}s-i\\ s-i&{}i\\ \end{pmatrix}, \end{aligned}$$

with $i\in \{0,\ldots ,s\}$. The identity and the element ((1, 2), (1, 2)) of ${\mathrm {Sym}}(2)\times {\mathrm {Sym}}(2)$ fix all elements of $\Sigma _{2,s}$, whereas ((1, 2), 1) and (1, (1, 2)) both fix only the matrix

$$\begin{aligned} \begin{pmatrix} s/2&{}s/2\\ s/2&{}s/2\\ \end{pmatrix}, \end{aligned}$$

when s is even and fix no matrix if s is odd. Therefore,

$$\begin{aligned} \mathrm {rk}_{\Omega _{2,s}}({\mathrm {Sym}}(2s))&=\frac{1}{4}\bigg ( \mathrm {fix}_{\Sigma _{2,s}}((1,1))+ \mathrm {fix}_{\Sigma _{2,s}}(((1,2),(1,2)))\\&\quad + \mathrm {fix}_{\Sigma _{2,s}}((1,(1,2)))+ \mathrm {fix}_{\Sigma _{2,s}}(((1,2),1))\bigg )\\&={\left\{ \begin{array}{ll} \frac{1}{4}(2(s+1)+2)=\frac{s+2}{2}&{}\text {if }s\text { is even},\\ \frac{1}{4}(2(s+1))=\frac{s+1}{2}&{}\text {if }s\text { is odd}. \end{array}\right. } \end{aligned}$$

It is straightforward to show that the permutation character of ${\mathrm {Sym}}(2s)$ in its action on $\Omega _{2,s}$ is multiplicity-free. Indeed, write $G:={\mathrm {Sym}}(2s)$, let $G_{\omega _0}\cong {\mathrm {Sym}}(s){\mathrm {wr}}{\mathrm {Sym}}(2)$ be the stabiliser of $\omega _0=\{X_1,X_2\}$ and let $H:=G_{X_1}\cap G_{X_2}={\mathrm {Sym}}(X_1)\times {\mathrm {Sym}}(X_2)$ be the setwise stabiliser in G of both $X_1$ and $X_2$. Now, H is a Young subgroup of G. Therefore, from classical results on the parabolic actions of the symmetric group [15], we see that the permutation character $1_H^G$ is multiplicity-free. As $1_H^G=(1_H^{G_{\omega _0}})^{G}$, we deduce that $1_{G_{\omega _0}}^G$ is a constituent of $1_H^G$ and hence, $1_{G_\omega }^G$ is multiplicity-free too. (This was also observed by Saxl [24].)

Finally, it is not hard to show that when $s>2$, ${{{\text {rk}}}}_{\Omega _{2,s}}({\mathrm {Alt}}(2s))={{{\text {rk}}}}_{\Omega _{2,s}}({\mathrm {Sym}}(2s))$ because $({\mathrm {Alt}}(2s))_{\omega _0}$ acts transitively on each orbit of $({\mathrm {Sym}}(2s))_{\omega _0}$ on $\Omega _{2,s}$. $\square$

The orbits of ${\mathrm {Alt}}(2s)$ or ${\mathrm {Sym}}(2s)$ on the set $\Sigma _{2,s}$ form a well-known metric association scheme generated by the folded Johnson graph. The rank of this scheme and hence Proposition 5.2 is known, see, for instance, [5, p. 259]. We have included a proof of Proposition 5.2 to show in detail how to use Lemma 5.1.

We now take a closer look at the case $r=3$. For the proof of Proposition 5.3, we use the fact that the number of non-negative integer solutions of the equation

$$\begin{aligned} x_1+x_2+\cdots +x_k=n \end{aligned}$$

is the binomial coefficient ${n+k-1\atopwithdelims ()k-1}$.

Proposition 5.3

We have

$$\begin{aligned} \mathrm {rk}_{\Omega _{3,s}}({\mathrm {Alt}}(3s))= & {} \mathrm {rk}_{\Omega _{3,s}}({\mathrm {Sym}}(3s))\\= & {} {\left\{ \begin{array}{ll} \frac{1}{288}(s^4+6s^3+64s^2+192s+160)&{}\text {if }s\equiv 2,4\pmod 6,\\ \frac{1}{288}(s^4+6s^3+64s^2+192s+288)&{}\text {if }s\equiv 0\pmod 6,\\ \frac{1}{288}(s^4+6s^3+64s^2+138s+79)&{}\text {if }s\equiv 1,5\pmod 6,\\ \frac{1}{288}(s^4+6s^3+64s^2+138s+207)&{}\text {if }s\equiv 3\pmod 6. \end{array}\right. } \end{aligned}$$

Moreover, the permutation character of this action is multiplicity-free if and only if $s\le 5$.

Proof

From [14] (see also the work of Anand–Dumir–Gupta [2] or the original proof of MacMahon [20]), we have

$$\begin{aligned}&|\Sigma _{3,s}|=\frac{(s+1)(s+2)(s^2+3s+4)}{8}. \end{aligned}$$

(4)

Actually, in both papers this number is written in the form ${s+4\atopwithdelims ()4}+{s+3\atopwithdelims ()4}+{s+2\atopwithdelims ()4}$.

Table 3 Information on ${\mathrm {Sym}}(3)\times {\mathrm {Sym}}(3)$

In Table 3, we have listed a complete set of representatives for the action of ${\mathrm {Sym}}(3){\mathrm {wr}}{\mathrm {Sym}}(2)=({\mathrm {Sym}}(3)\times {\mathrm {Sym}}(3))\rtimes {\mathrm {Sym}}(2)$ by conjugation on ${\mathrm {Sym}}(3)\times {\mathrm {Sym}}(3)$. In particular, in view of Lemma 5.1 and the orbit-counting lemma, to compute ${{{\text {rk}}}}_{\Omega _{3,s}}({\mathrm {Sym}}(3s))$, it suffices to determine for each of these elements the number of fixed points on $\Sigma _{3,s}$.

The only element of $\Sigma _{3,s}$ fixed by ((1, 2, 3), 1) or by ((1, 2, 3), (1, 2)) is

$$\begin{aligned} \begin{pmatrix} s/3&{}s/3&{}s/3\\ s/3&{}s/3&{}s/3\\ s/3&{}s/3&{}s/3 \end{pmatrix}, \end{aligned}$$

and this element arises only if 3 divides s. Therefore,

$$\begin{aligned} \mathrm {fix}_{\Sigma _{3,s}}((1,2,3),1)=\mathrm {fix}_{\Sigma _{3,s}}((1,2,3),(12))={\left\{ \begin{array}{ll}1&{}\text {if }3\text { divides }s,\\ 0&{}\text {otherwise}.\end{array}\right. } \end{aligned}$$

(5)

An element x of $\Sigma _{3,s}$ is fixed by the permutation ((1, 2, 3), (1, 2, 3)) if an only if

$$\begin{aligned} x=\begin{pmatrix} a&{}b&{}s-a-b\\ s-a-b&{}a&{}b\\ b&{}s-a-b&{}a \end{pmatrix}, \end{aligned}$$

for some $a,b\in \mathbb {N}$ with $s-a-b\ge 0$. In particular, $0\le b\le s-a$ and

$$\begin{aligned} \mathrm {fix}_{\Sigma _{3,s}}((1,2,3),(1,2,3))=\sum _{a=0}^{s}\sum _{b=0}^{s-a}1=\sum _{a=0}^s(s-a+1)=\frac{(s+1)(s+2)}{2}. \end{aligned}$$

(6)

An element x of $\Sigma _{3,s}$ is fixed by the permutation ((1, 2), 1) if an only if

$$\begin{aligned} x=\begin{pmatrix} a&{}b&{}c\\ a&{}b&{}c\\ s-2a&{}s-2b&{}2-2c \end{pmatrix}, \end{aligned}$$

for some $a,b,c\in \mathbb {N}$ with $a+b+c=s$ and $0\le a,b,c\le \lfloor s/2\rfloor$. Without the restriction “$a,b,c\le \lfloor s/2\rfloor$”, we get ${s+2\atopwithdelims ()2}$ solutions. Next, the number of solutions with one of a, b, c greater or equal to $\lfloor s/2\rfloor +1$ is

$$\begin{aligned} {s-\lfloor s/2\rfloor -1+2\atopwithdelims ()2}={\lceil s/2\rceil +1\atopwithdelims ()2}. \end{aligned}$$

Now by the inclusion–exclusion principle, we obtain

$$\begin{aligned} \mathrm {fix}_{\Sigma _{3,s}}(((1,2),1)) = {s+2\atopwithdelims ()2}-3{\lceil s/2\rceil +1\atopwithdelims ()2}= {\left\{ \begin{array}{ll} \frac{(s+2)(s+4)}{8}&{}\text {if }s \text { is even},\\ \frac{s^2-1}{8}&{}\text {if }s \text { is odd}. \end{array}\right. } \end{aligned}$$

(7)

An element x of $\Sigma _{3,s}$ is fixed by the permutation ((1, 2), (1, 2)) if an only if

$$\begin{aligned} x=\begin{pmatrix} a&{}b&{}c\\ b&{}a&{}c\\ s-a-b&{}s-a-b&{}s-2c \end{pmatrix}, \end{aligned}$$

for some $a,b,c\in \mathbb {N}$ with $a+b+c=s$, $0\le a,b,c$ and $c\le \lfloor s/2\rfloor$. Without the latter restriction on c, we have ${s+2\atopwithdelims ()2}$ solutions. The number of solutions with c greater or equal to $\lfloor s/2\rfloor +1$ is ${s-\lfloor s/2\rfloor -1+2\atopwithdelims ()2}={\lceil s/2\rceil +1\atopwithdelims ()2}$. In particular, the number of choices for x is

$$\begin{aligned} \mathrm {fix}_{\Sigma _{3,s}}((1,2),(1,2))= {s+2\atopwithdelims ()2}-{\lceil s/2\rceil +1\atopwithdelims ()2}= {\left\{ \begin{array}{ll} \frac{3s^2+10s+8}{8}&{}\text {if }s \text { is even},\\ \frac{(s+1)(3s+5)}{8}&{}\text {if }s \text { is odd}. \end{array}\right. } \end{aligned}$$

(8)

The formula for ${{{\text {rk}}}}_{\Omega _{3,s}}({\mathrm {Sym}}(3s))$ follows from the orbit-counting lemma, Eqs. (4)–(8) and Table 3. Moreover, it is not hard to show that ${{{\text {rk}}}}_{\Omega _{3,s}}({\mathrm {Alt}}(3s)){=}{{{\text {rk}}}}_{\Omega _{3,s}}({\mathrm {Sym}}(3s))$. Finally, it follows from the results in [13, 24] that the permutation character of ${\mathrm {Sym}}(3s)$ in its action on $\Omega _{3,s}$ is not multiplicity-free when $s\ge 6$. $\square$

There is another case we wish to address: $s=2$.

Proposition 5.4

The rank of ${\mathrm {Sym}}(2r)$ in its natural action on the set $\Omega _{r,2}$ of partitions of $\{1,\ldots ,2r\}$ into r parts of cardinality 2 equals p(r), where p(r) is the number of partitions of r. The permutation character of this action is multiplicity-free. Moreover, the rank of ${\mathrm {Alt}}(2r)$ on $\Omega _{r,2}$ equals p(r) when r is odd and $p(r)+p(r/2)$ when r is even.

Proof

The first statement follows from Lemma 5.1 applied with $s=2$. Both Saxl [24] and Godsil and Meagher [13] have proved that the permutation characters of ${\mathrm {Sym}}(2s)$ and ${\mathrm {Alt}}(2s)$ are multiplicity-free. Finally, observe that given $\omega _0,\omega \in \Omega _{r,2}$, the suborbit $\omega ^{{\mathrm {Sym}}(2r)_{\omega _0}}$ splits into two distinct ${\mathrm {Alt}}(2r)$-suborbits if and only if ${\mathrm {Sym}}(2r)_{\omega _0}\cap {\mathrm {Sym}}(2r)_\omega$ consists of even permutations. From this and Lemma 5.1, it follows, from standard arguments, that $={{{\text {rk}}}}_{\Omega _{r,2}}({\mathrm {Alt}}(2r))={{{\text {rk}}}}_{\Omega _{r,2}}({\mathrm {Sym}}(2r))=p(r)$ when r is odd and ${{{\text {rk}}}}_{\Omega _{r,2}}({\mathrm {Alt}}(2r))=p(r)+p(r/2)$ when r is even. $\square$

Observe that when $s=2$ and $r=4$, ${\mathrm {Alt}}(8)\cap ({\mathrm {Sym}}(2){\mathrm {wr}}{\mathrm {Sym}}(8))$ is contained in $\mathrm {AGL}_3(2)$ and hence, it is not maximal in ${\mathrm {Alt}}(8)$. Therefore, in the row of Table 2 concerning these actions, the value of $r=4$ is missing.

5.1 A first lower bound for $\mathrm {rk}_{\Omega _{r,s}}({\mathrm {Sym}}(rs))$

Given $A,B\in \Sigma _{r,s}$, we say that A is equivalent to B and we write

$$\begin{aligned} A\sim B, \end{aligned}$$

if A and B belong to the same $({\mathrm {Sym}}(r)\times {\mathrm {Sym}}(r))$-orbit, i.e. A may be obtained from B by independent row and column permutations. The matrix $A\in \Sigma _{r,s}$ is called irreducible if it is not equivalent to a block diagonal matrix.

Given $a_1,\ldots ,a_k\in \mathbb {N}$ and $P_1,\ldots ,P_k$$(r\times r)$-permutation matrices, the linear combination $a_1 P_1 + \cdots + a_k P_k$ belongs to $\Sigma _{r,s}$ if and only if $a_1 + \cdots + a_k =s$. For a fixed k, the set of all such linear combinations is a union of $({\mathrm {Sym}}(r)\times {\mathrm {Sym}}(r))$-orbits. We estimate the number of such orbits when $k\in \{2,3\}$.

For every $i\in \{1,\ldots ,r\}$, we denote by P(i) the unique $j\in \{1,\ldots ,r\}$ with $P_{i,j}=1$ in the $r\times r$-permutation matrix P.

Proposition 5.5

Let $a,a'$ be integers with $1\le a,a' < s/2$ and let $P, Q,P', Q'$ be $r\times r$-permutation matrices with $P\ne Q$ and $P'\ne Q'$.

$$\begin{aligned} \text {If }aP+(s-a)Q=a'P+(s-a')Q', \text {then } a=a', P = P', Q=Q'. \end{aligned}$$

Proof

Write $M:=aP+(s-a)Q=a'P+(s-a')Q'$. Clearly, each row $M_i$ of M contains one or two nonzero entries. If $M_i$ has a unique nonzero entry, then $P(i)=Q(i)=P'(i)=Q'(i)$. If $M_i$ has two nonzero entries, then by our assumption on a and $a'$ we have $a=a'$ and $P(i)=P'(i)\ne Q(i)=Q'(i)$. Thus, for each i, $P'(i) = P(i)$ and $Q'(i)=Q(i)$. Since $P\ne Q$ and $P'\ne Q'$, there always exists a row with two nonzero elements. Hence, $a=a'$. $\square$

Now we consider linear combinations of three distinct permutation matrices. Let us call a triple (a, b, c) of positive integers good if $1\le a< b < c$. For each $s\in \mathbb {N}$, let g(s) be the number of good triples summing up to s.

Lemma 5.6

For every positive integer s, we have

$$\begin{aligned} g(s)=\frac{\delta (s)}{3} +\left\{ \begin{array}{cc} \frac{(s-2)(s-4)}{12} &{} s \hbox { is even},\\ \frac{(s-1)(s-5)}{12} &{} s \hbox { is odd}, \end{array} \right. \end{aligned}$$

(9)

where $\delta (s)$ is one or zero depending on whether s is divisible by three or not.

Proof

Let (a, b, c) be a good triple and set $x:=a-1,y:=b-1,z:=c-1$. Then, g(s) equals the number of non-negative solutions of $x+y+z=s-3$ satisfying $0\le x< y < z$. The number of ordered triples (x, y, z) satisfying the linear equation $x+y+z=s-3$ is $\left( {\begin{array}{c}s-1\\ 2\end{array}}\right)$. By the inclusion–exclusion principle, the number of such triples with pairwise distinct x, y, z equals

$$\begin{aligned} \left( {\begin{array}{c}s-1\\ 2\end{array}}\right) - 3 \left\lfloor \frac{s-1}{2}\right\rfloor +2\delta (s). \end{aligned}$$

Since each ordered triple $x< y < z$ appears 6 times as a solution of $x+y+z=s-3$, we obtain Eq. (9). $\square$

Proposition 5.7

Let $(a,b,c),(a',b',c')$ be two good triples summing up to s. Then, for any two triples of pairwise distinct permutation matrices (P, Q, R) and $(P',Q',S')$, we have:

$$\begin{aligned} aP+bQ+cR= & {} a'P'+b'Q'+c'R'\iff (a,b,c)\\= & {} (a',b',c') \hbox { and } (P,Q,R)=(P',Q',R'). \end{aligned}$$

Proof

The backward implication is obvious. Therefore, assume $aP+bQ+cR = a'P'+b'Q'+c'R'$. If $P'=P$ or $Q'=Q$ or $R'=R$, then we are done by Proposition 5.5. Therefore, we assume $P'\ne P,Q'\ne Q, R'\ne R$.

Write $M:=aP+bQ+cR$. If M is reducible, then we can argue with an easy induction on the size of the blocks. Therefore, we may assume that M is irreducible. In what follows, it is convenient to observe that the irreducibility of M is equivalent to the connectivity of the graph $\Gamma$ with vertices the rows $M_1,\ldots ,M_n$ of M and where two rows $M_i,M_j$ are declared to be adjacent if and only if they have common nonzero entries.

If a row or a column of M contains a unique nonzero entry, then M is reducible, contrary to our assumption. Therefore, each row and each column of M have at least two nonzero entries, that is, for each i at least two of the elements P(i), Q(i), R(i) are distinct.

Step 1 The minimal nonzero entry of M is a. In particular, $a=a'$.

If this is not the case, then $P(i)\in \{Q(i),R(i)\}$ for all $i\in \{1,\cdots ,n\}$. Let $I := \{i\,|\, P(i)=Q(i)\}$ and $J:=\{j\,|\,P(j)=R(j)\}$. Clearly, $I\cup J=\{1,\ldots ,n\}$. Since for each i at least two of P(i), Q(i), R(i) are distinct, $I\cap J=\emptyset$. Thus $I\cup J$ is a partition of $\{1,\ldots ,n\}$. Since P, Q, R are pairwise distinct, the sets I and J are non-empty. If two rows $M_i,i\in I, M_j,j\in J$ have a nonzero entry, k say, then $k\in \{Q(i),R(i)\}\cap \{Q(j),R(j)\}$. Since $i\ne j$, $Q(i)\ne Q(j)$ and $R(i)\ne R(j)$. So, either $k=Q(i)=R(j)$ or $k=Q(j)=R(i)$. In both cases, we get $P(i)=P(j)$ contrary to $i\ne j$. Therefore, the inner product $\langle M_i, M_j\rangle = 0$ whenever $i\in I, j\in J$ and $\Gamma$ is disconnected, a contradiction. $_\blacksquare$

Step 2 The maximal nonzero entry of M is c. In particular, $c=c'$.

The proof is analogous to the proof of Step 1. $_\blacksquare$

Combining Step 1 and 2 with $a+b+c = a'+b'+c'$, we obtain $(a,b,c)=(a',b',c')$. It remains to show that $(P,Q,R)=(P',Q',R')$.

Step 3 For each $i\in \{1,\ldots ,n\}$, either $P(i)=P'(i),Q(i)=Q'(i),R(i)=R'(i)$, or $P(i)=Q(i)=R'(i)\ne P'(i)=Q'(i)=R(i)$ and $a+b=c$.

Let $i\in \{1,\ldots ,n\}$. Each row of M contains at most 3 nonzero elements. If the ith row contains exactly three nonzero entries, say $(i,k),(i,\ell ),(i,m)$, then, up to reordering $M_{ik} = a, M_{i\ell } = b, M_{im} = c$, implying $P(i) =k = P'(i) , Q(i) = \ell =Q'(i), R(i)=m=R'(i)$.

If the ith row contains two nonzero entries, say (i, k) and $(i,\ell )$, then, up to reordering, there are three possibilities:

$$\begin{aligned} M_{ik}=a,M_{i\ell }=b+c,\,\, \text {or }\,\, M_{ik}=b, M_{i\ell }=a+c, \,\,\text {or }\,\,M_{ik}=c, M_{i\ell }=a+b. \end{aligned}$$

In the first case, the inequality $a < b+c$ implies $P(i)=k=P'(i), Q(i)=R(i)=\ell = Q'(i)=R'(i)$. Analogously, in the second case the inequality $b < a + c$ implies $Q(i)=k=Q'(i), P(i)=R(i)=\ell = P'(i)=R'(i)$. In the third case, if $a+b\ne c$ then $R(i)=k=R'(i)$ and $P(i)=Q(i)=\ell = P'(i)=Q'(i)$. If $a+b=c$, then either $R(i)=k=R'(i), P(i)=Q(i)=\ell = P'(i)=Q'(i)$ or $R(i)=k=P'(i)=Q'(i), R'(i)=\ell =P(i)=Q(i)$. $_\blacksquare$

Write

$$\begin{aligned} I&:=\{i\in \{1,\ldots ,n\}\mid P(i)=P'(i),Q(i)=Q'(i),R(i)=R'(i)\},\\ J&:=\{i\in \{1,\ldots ,n\}\mid P(i)=Q(i)=R'(i)\ne P'(i)=Q'(i)=R(i)\}. \end{aligned}$$

Clearly, $I\cap J = \emptyset$ and $I\cup J=\{1,\ldots ,n\}$ from Step 3. If $I=\emptyset$, then $P=Q$ contrary to our assumption. If $J=\emptyset$, then $P=P',Q=Q',R=R'$ and we are done. Finally, suppose $I\ne \emptyset \ne J$. We show that in this case $\Gamma$ is disconnected.

Pick an arbitrary pair $i\in I$ and $j\in J$. If $\langle M_i,M_j\rangle \ne 0$, then $M_{ik} > 0 < M_{jk}$ for some $k\in \{1,\ldots ,n\}$. Since $M_j$ has exactly two nonzero entries, they coincide with R(j) and $R'(j)$. Up to reordering, we may assume $R(j)=k$. Denote also $\ell :=R'(j)$. Then, $P'(j)=Q'(j)=k$ and $P(j)=Q(j)=\ell$. It follows from $i\ne j$ that $R(i)\ne k$. Together with the inequality $M_{ik} > 0$, we obtain that at least one of P(i), Q(i) equals k. Since $P(i)=P'(i),Q(i)=Q'(i)$, we conclude that at least one of $P'(i),Q'(i)$ equals k. This contradicts $P'(j)=Q'(j)=k$, because $i\ne j$. $\square$

Combining Eq. (9) with Proposition 5.7, we obtain a lower bound for the number of doubly stochastic matrices of size r with row sum s:

$$\begin{aligned} |\Sigma _{r,s}|\ge g(s) r! (r!-1)(r!-2). \end{aligned}$$

From Lemma 5.1, we immediately obtain:

Theorem 5.8

Let $\mathrm {rk}_{\Omega _{r,s}}(\mathrm {Sym}(rs))$ be the rank of the symmetric group ${\mathrm {Sym}}(rs)$ in its natural action on the set $\Omega _{r,s}$ of uniform partitions of $\{1,\ldots ,rs\}$ into r parts of cardinality s. Then

$$\begin{aligned} \mathrm {rk}_{\Omega _{r,s}(\mathrm {Sym}(rs))}\ge g(s)\left( 1-\frac{1}{r!}\right) (r!-2), \end{aligned}$$

where g(s) is defined in (9).

5.2 A second lower bound for $\mathrm {rk}_{\Omega _{r,s}}({\mathrm {Sym}}(rs))$

Now we apply the positive solution of Stanley to the Anand–Dumir conjecture to prove a general upper bound for ${{{\text {rk}}}}_{\Omega _{r,s}}({\mathrm {Sym}}(rs))$. First, we give some historical background. MacMahon [20] and then later Anand–Dumir–Gupta [2] have showed that

$$\begin{aligned} |\Sigma _{3,s}|={s+4\atopwithdelims ()4}+{s+3\atopwithdelims ()4}+{s+2\atopwithdelims ()4}. \end{aligned}$$

In a moment of inspiration, from this pattern, Anand, Dumir and Gupta [2] proposed the following conjecture.

Conjecture

Let $\sigma _r(s):=|\Sigma _{r,s}|$ be the number of $(r\times r)$-magic squares with row and column sums equal to s. As a function of s,

• (1) $\sigma _{r}(s)\in \mathbb {Q}[s]$;

• (2) $\deg (\sigma _{r}(s))=(r-1)^2$;

• (3) as a polynomial in s, $\sigma _{r}(s)$ is 0 for $s\in \{-1,-2,\ldots ,-(r-1)\}$;

• (4) as a polynomial in s, we have the relation $\sigma _{r}(-s-r)=(-1)^{r-1}\sigma _r(s)$.

Once we encode $\sigma _r(s)$ in a generating function, this conjecture can be phrased in the following equality:

$$\begin{aligned} \sum _{s=0}^\infty \sigma _r(s)\lambda ^s=\frac{h_0+h_1\lambda +\cdots +h_{(r-1)^2+1-r}\lambda ^{(r-1)^2+1-r}}{(1-\lambda )^{(r-1)^2+1}}, \end{aligned}$$

where $h_0,h_1,\ldots ,h_{(r-1)^2+1-r}$ are integers, $h_0+h_1+\cdots +h_{(r-1)^2+1-r}\ne 0$ and $h_{(r-1)^2+1-r-i}=h_i$ for every $i\in \{0,\ldots ,(r-1)^2+1-r\}$. Richard Stanley has made two additional conjectures:

• (5) $h_i\ge 0$ for every $i\in \{0,\ldots ,(r-1)^2+1-r\}$;

• (6) $h_0\le h_1\le \cdots \le h_{\lfloor ((r-1)^2+1-r)/2\rfloor }$.

Stanley has settled affirmatively (1)–(5) in [26]; however, to the best of our knowledge, (6) remains open.

From this, Lemma 5.1 and the orbit-counting lemma we obtain the following.

Theorem 5.9

Let $\mathrm {rk}_{\Omega _{r,s}}({\mathrm {Sym}}(rs))$ be the rank of the symmetric group ${\mathrm {Sym}}(rs)$ in its natural action on the set $\Omega _{r,s}$ of uniform partitions of $\{1,\ldots ,rs\}$ into r parts of cardinality s. Then,

$$\begin{aligned} \frac{1}{r!^2}|\Sigma _{r,s}|+(s+1)^{(r-1)(r-2)}\ge \mathrm {rk}_{\Omega _{r,s}}({\mathrm {Sym}}(rs))\ge \frac{1}{r!^2}|\Sigma _{r,s}|\ge \frac{1}{r!^2}{s+r-1\atopwithdelims ()(r-1)^2}. \end{aligned}$$

Proof

We use the notation established above. From Lemma 5.1 and the orbit-counting lemma, we have

$$\begin{aligned} \mathrm {rk}_{\Omega _{r,s}}({\mathrm {Sym}}(rs))=\frac{1}{r!^2}\sum _{(\sigma ,\tau )\in {\mathrm {Sym}}(r)\times {\mathrm {Sym}}(r)}\mathrm {fix}_{\Sigma _{r,s}}((\sigma ,\tau ))\ge |\Sigma _{r,s}|/r!^2=\sigma _r(s)/r!^2. \end{aligned}$$

Now, from above,

$$\begin{aligned} \sigma _r(s)= & {} h_0{s+(r-1)^2\atopwithdelims ()(r-1)^2}+h_1{s+(r-1)^2-1\atopwithdelims ()(r-1)^2}\\&+\cdots +h_{(r-1)^2+1-r}{s+r-1\atopwithdelims ()(r-1)^2}. \end{aligned}$$

As the binomial coefficients ${s+(r-1)^2-i\atopwithdelims ()(r-1)^2}$ are decreasing as a function of $i\in \{0,\ldots ,d\}$ and $h_0+\cdots +h_{(r-1)^2+1-r}\ne 0$, the first part of the result follows.

Let us now think about $\mathrm {rk}_{\Omega _{r,s}}({\mathrm {Sym}}(rs))$ and $|\Sigma _{r,s}|$ as functions of s, with r being fixed, and let us proved the upper bound for $\mathrm {rk}_{\Omega _{r,s}}({\mathrm {Sym}}(rs))$. From the orbit-counting lemma,

$$\begin{aligned} \mathrm {rk}_{\Omega _{r,s}}({\mathrm {Sym}}(rs))-\frac{1}{r!^2}|\Sigma _{r,s}|=\frac{1}{r!^2}\sum _{\begin{array}{c} (\sigma ,\tau )\in {\mathrm {Sym}}(r)\times {\mathrm {Sym}}(r)\\ (\sigma ,\tau )\ne (1,1) \end{array}}\mathrm {fix}_{\Sigma _{r,s}}((\sigma ,\tau )) \end{aligned}$$

and hence, it suffices to show that ${\mathrm {fix}}_{\Sigma _{r,s}}((\sigma ,\tau ))\le s^{(r-1)(r-2)}$. Since ${\mathrm {fix}}_{\Sigma _{r,s}}((\sigma ,\tau ))\le {\mathrm {fix}}_{\Sigma _{r,s}}((\sigma ,\tau )^\ell )$ for every integer $\ell$, we may assume without loss of generality that $(\sigma ,\tau )$ has prime order p. Moreover, since ${\mathrm {fix}}_{\Sigma _{r,s}}((\sigma ,\tau ))={\mathrm {fix}}_{\Sigma _{r,s}}((\sigma ,\tau )^x)$ for every $x\in {\mathrm {Sym}}(r){\mathrm {wr}}{\mathrm {Sym}}(2)$, replacing $(\sigma ,\tau )$ by a suitable ${\mathrm {Sym}}(r){\mathrm {wr}}{\mathrm {Sym}}(2)$-conjugate in we may assume

$$\begin{aligned}&\sigma =(1,\ldots ,p)(p+1,\ldots ,2p)\cdots ((\ell -1)p+1,\ldots ,\ell p)\\&\tau =(1,\ldots ,p)(p+1,\ldots ,2p)\cdots ((\ell '-1)p+1,\ldots ,\ell ' p), \end{aligned}$$

for some $\ell ,\ell '\in \mathbb {N}$ with $\ell '\le \ell$. Observe now that $x\in {\mathrm {Fix}}_{\Omega _{r,s}}((\sigma ,\tau ))$ if and only if the coordinates of the matrix x are constant on the cycles of $(\sigma ,\tau )$ in its action on $\{1,\ldots ,r\}\times \{1,\ldots ,r\}$. Using the explicit representation of $\sigma$ and $\tau$ above and a moment’s thought, we obtain that the largest number of orbits of a non-identity element of ${\mathrm {Sym}}(r)\times {\mathrm {Sym}}(s)$ in its action on $\{1,\ldots ,r\}\times \{1,\ldots ,r\}$ arises when $p=2$, $\ell =1$ and $\ell '=1$, that is, $(\sigma ,\tau )=((1,2),1)$. Now, an element $x\in \Sigma _{r,s}$ is fixed by ((1, 2), 1) if and only if it is of the form:

$$\begin{aligned} x=\begin{pmatrix} a_{1,1}&{}a_{1,2}&{}\cdots &{}a_{1,r-1}&{}s-\sum _{j=1}^{r-1}a_{1,j}\\ a_{1,1}&{}a_{1,2}&{}\cdots &{}a_{1,r-1}&{}s-\sum _{j=1}^{r-1}a_{1,j}\\ a_{2,1}&{}a_{2,2}&{}\cdots &{}a_{2,r-1}&{}s-\sum _{j=1}^{r-1}a_{2,j}\\ a_{3,1}&{}a_{3,2}&{}\cdots &{}a_{3,r-1}&{}s-\sum _{j=1}^{r-1}a_{3,j}\\ \vdots &{}\vdots &{}\ddots &{}\vdots &{}\vdots \\ a_{r-2,1}&{}a_{r-2,2}&{}\cdots &{}a_{r-2,r-1}&{}s-\sum _{j=1}^{r-1}a_{r-2,j}\\ s-\sum _{k=1}^{r-2}a_{k,1}&{} s-\sum _{k=1}^{r-2}a_{k,2}&{}\cdots &{} s-\sum _{k=1}^{r-2}a_{k,r-1}&{}-(r-1)s+\sum _{j=1}^{r-1}\sum _{k=1}^{r-2}a_{k,j} \end{pmatrix}. \end{aligned}$$

From this, we see that the entries of x depend upon $(r-1)(r-2)$ coefficients $a_{k,j}$s and each of these coefficients is an integer in $\{0,\ldots ,s\}$. Thus, the number of choices for x is at most $(s+1)^{(r-1)(r-2)}$. $\square$

5.3 Proof of Theorem 1.2 when $G_\omega$ is imprimitive on $\{1,\ldots ,n\}$

We are now ready to determine when the rank of ${\mathrm {Sym}}(rs)$ on $\Omega _{r,s}$ is at most $(rs)^2$.

Theorem 5.10

Let r and s be positive integers with $r,s\ge 2$. The rank $\mathrm {rk}_{\Omega _{r,s}}({\mathrm {Sym}}(rs))$ of ${\mathrm {Sym}}(rs)$ in its action on the uniform partitions of $\{1,\ldots ,rs\}$ into r parts of cardinality s is at most $(rs)^2$, unless $r=2$, or $s=2$ and $r\le 26$, or $r=3$ and $s\le 47$, or $r=4$ and $s\le 6$, or $r=5$ and $s\le 4$, or $r=6$ and $s\le 3$, or $r=7$ and $s\le 3$. The value of $\mathrm {rk}_{\Omega _{r,s}}({\mathrm {Sym}}(rs))$ for each of these exceptions is listed in Table 2.

Proof

For simplicity, write ${{{\text {rk}}}}_{r,s}:={{{\text {rk}}}}_{\Omega _{r,s}}({\mathrm {Sym}}(rs))$. It is clear that ${{{\text {rk}}}}_{r,s}\ge {{{\text {rk}}}}_{r,s'}$ and ${{{\text {rk}}}}_{r,s}\ge {{{\text {rk}}}}_{r',s}$, for every $s'\le s$ and $r'\le r$, that is, ${{{\text {rk}}}}_{r,s}$ is increasing in both variables r and s.

From Proposition 5.2, ${{{\text {rk}}}}_{2,s}\le (2s)^2$ and hence, $r=2$ is an exception. A direct application of the formula in Proposition 5.3 shows that ${{{\text {rk}}}}_{3,s}\le (3s)^2$ only when $s\le 47$. In particular, we may now assume that $r\ge 4$.

A tedious application of the bound in Proposition 5.4 and some elementary estimates on the number of partitions of r show that ${{{\text {rk}}}}_{r,2}\le (2r)^2$ only when $r\le 26$. In particular, we may now assume that $s\ge 3$.

If the rank is bounded by $(rs)^2$, then from Theorem 5.8 we obtain

$$\begin{aligned} g(s) \left( 1-\frac{1}{r!}\right) \left( r!-2\right) \le r^2 s^2. \end{aligned}$$

(10)

Similarly, from Theorem 5.9, we have

$$\begin{aligned} \frac{1}{r!^2}{s+r-1\atopwithdelims ()(r-1)^2}\le (rs)^2. \end{aligned}$$

(11)

Similarly, ${{{\text {rk}}}}_{r,s}\ge {{{\text {rk}}}}_{r,2}=p(r)$, where in the last equality we have applied Proposition 5.4 and p(r) is the number of partitions of r. Now, from a beautiful result of Maróti [22], we have $p(r)\ge e^{2.5\sqrt{r}}/(13r)$. Therefore,

$$\begin{aligned} \frac{e^{2.5\sqrt{r}}}{13r}\le (rs)^2, \end{aligned}$$

(12)

where p(r) is the number of partitions of r.

The function g(s) is positive starting with $s=6$ and the ratio $g(s)/s^2$ attains its minimal value in $s=6$ when s is even and in $s=7$ when s is odd. The minimum, for $s\ge 6$, of g(s) is $g(7)=1/49$. Using these observations and a computation, we see that when $s\ge 6$, Eq. (10) is satisfied only when $r\le 6$. Similar arguments show that Eqs. (10)–(12) are satisfied only when

• $s=3$ and $r\le 32$, or

• $s=4$ and $r\le 37$, or

• $s=5$ and $r\le 40$, or

• $r=4$ and $6\le s\le 24$, or

• $r=5$ and $6\le s\le 26$, or

• $r=6$ and $6\le s\le 13$.

For the remaining cases, we invoke again the help of magma, using the built-in character theoretic package, we compute ${{{\text {rk}}}}_{r,s}$ calculating the inner product of the permutation character of ${\mathrm {Sym}}(rs)$ acting on $\Omega _{r,s}$ (it is actually more efficient to obtain the permutation character by inducing up to ${\mathrm {Sym}}(rs)$ the principal character of ${\mathrm {Sym}}(s){\mathrm {wr}}{\mathrm {Sym}}(r)$). The only pairs (r, s) with $\mathrm {rk}_{\Omega _{r,s}}({\mathrm {Sym}}(rs))\le (rs)^2$ are listed in the statement of this theorem, and the corresponding value of $\mathrm {rk}_{\Omega _{r,s}}({\mathrm {Sym}}(rs))$ is given in Table 2. $\square$

Proof of Theorem 1.2

We use that notation that we have established above. The proof follows from Sect. 4.1 when $G_\omega$ is intransitive on $\{1,\ldots ,n\}$, from Sect. 4.2 when $G_\omega$ is primitive on $\{1,\ldots ,n\}$. Assume now that $G_\omega$ is imprimitive on $\{1,\ldots ,n\}$. When $G={\mathrm {Sym}}(n)$, the proof follows from Theorem 5.10 and, for the multiplicity-freeness of the permutation character, from Propositions 5.2–5.4, and from a computation with magma for the remaining exceptions. When $G={\mathrm {Alt}}(n)$, the proof follows immediately from the fact that $\mathrm {rk}_{\Omega _{r,s}}({\mathrm {Sym}}(rs))\le {{{\text {rk}}}}_{\Omega _{r,s}}({\mathrm {Alt}}(rs))$ and then arguing as in the previous line. $\square$

6 Some exact values for the rank of the symmetric group acting on uniform partitions

Proposition 5.2 is elementary; observe, however, that there are two ingredients to make the argument elementary. First, the size of $\Sigma _{2,s}$ is easy to compute, and second, it is straightforward to determine the fixed points of the elements of ${\mathrm {Sym}}(r)\times {\mathrm {Sym}}(r)$ on $\Sigma _{r,s}$. In general, the exact cardinality of $\Sigma _{r,s}$ is not known: many details on its asymptotic behaviour are related to a conjecture of Anand–Dumir–Gupta [2] which is now solved by Stanley [26]. To the best of our knowledge, $|\Sigma _{r,s}|$ has been computed only when $2\le r\le 6$, see [14]. Therefore, it seems plausible to be able to compute the rank of ${\mathrm {Sym}}(rs)$ on $\Omega _{r,s}$ whenever $2\le r\le 6$. Indeed, we have already computed in Proposition 5.3 this rank when $r=3$. In practice, we will see that already when $r\in \{4,5\}$ this is a difficult task because it is difficult to establish the number of fixed points of certain elements of ${\mathrm {Sym}}(r)\times {\mathrm {Sym}}(r)$ on $\Sigma _{r,s}$.

We now briefly deal with the case that $r=4$. The proof is computationally much harder than the previous two cases; however, it uses the ideas already described in the case $r=2$ and $r=3$. For instance, the cardinality of $\Sigma _{4,s}$ is given in [14]. The problem of computing ${\mathrm {fix}}_{\Sigma _{4,s}}((\sigma ,\tau ))$, for each $(\sigma ,\tau )\in {\mathrm {Sym}}(4)\times {\mathrm {Sym}}(4)$, is related in counting the internal lattice points in a certain polyhedral in $\mathbb {R}^9$, which is a classical problem in integer linear programming.

Proposition 6.1

The rank ${\mathrm {Sym}}(4s)$ in its action on the set $\Omega _{4,s}$ of uniform partitions of $\{1,\ldots ,4s\}$ into 4 parts of cardinality s is

$$\begin{aligned} f(s)+ \frac{36330}{6531840}s^5+\frac{7938}{6531840}s^6+\frac{1596}{6531840}s^7+\frac{198}{6531840}s^8+\frac{11}{6531840}s^9, \end{aligned}$$

where f(s) is a polynomial of degree 4 in s, depending on the congruence of s modulo 12, and is tabulated in Table 5. Moreover, the permutation character of this action is multiplicity-free if and only if $s\le 3$. When, $s\notin \{2,4\}$, ${\mathrm {Alt}}(4s)$ has the same rank as ${\mathrm {Sym}}(4s)$ in its action on $\Omega _{4,s}$.

Proof

In Table 4, we have listed all the information needed to compute ${{{\text {rk}}}}_{\Omega _{4,s}}({\mathrm {Sym}}(4s))$ via Lemma 5.1 and the orbit-counting lemma. In the first column of Table 4, we have a complete set of representatives for the action of ${\mathrm {Sym}}(4){\mathrm {wr}}{\mathrm {Sym}}(2)=({\mathrm {Sym}}(4)\times {\mathrm {Sym}}(4))\rtimes {\mathrm {Sym}}(2)$ on ${\mathrm {Sym}}(4)\times {\mathrm {Sym}}(4)$ by conjugation. In the second column, we have the size of the corresponding ${\mathrm {Sym}}(4){\mathrm {wr}}{\mathrm {Sym}}(2)$-conjugacy class. Then, in the third column we have the number of fixed points. In particular, the first and the second columns are straightforward and the third column is the bulk of the argument for proving this proposition.

Table 4 Information on ${\mathrm {Sym}}(4)\times {\mathrm {Sym}}(4)$ acting on $\Sigma _{4,s}$

Some of the entries in the third column of Table 4 are very difficult to compute and require careful computations (for instance, in the first row we have $|\Sigma _{4,s}|$, which is given in [14]); however, the computations are similar in spirit to the computations in Propositions 5.2 and 5.3 (Table 5).

Here, we omit the computations, but to give an idea of the proof we compute ${\mathrm {fix}}_{\Sigma _{4,s}}((1,2)(3,4),1)$. An element x of $\Sigma _{4,s}$ is fixed by the permutation ((1, 2)(3, 4), 1) if an only if

$$\begin{aligned} x=\begin{pmatrix} a&{}b&{}c&{}d\\ a&{}b&{}c&{}d\\ \frac{s}{2}-a&{}\frac{s}{2}-b&{}\frac{s}{2}-c&{}\frac{s}{2}-c\\ \frac{s}{2}-a&{}\frac{s}{2}-b&{}\frac{s}{2}-c&{}\frac{s}{2}-c \end{pmatrix}, \end{aligned}$$

for some $a,b,c,d\in \mathbb {N}$ subject to the conditions $a+b+c+d=s$ and $0\le a,b,c,d\le s/2$. The equation $a+b+c+d=s$ has ${s+3\atopwithdelims ()3}$ non-negative integer solutions. Here, s must be even. The number of solutions with one of a, b, c, d greater than $s/2+1$ is ${s-(s/2+1)+3\atopwithdelims ()3}={s/2+2\atopwithdelims ()3}$. Therefore, by the inclusion–exclusion principle,

$$\begin{aligned} \mathrm {fix}_{\Sigma _{4,s}}((1,2)(3,4),1)= {s+3\atopwithdelims ()3}-4{s/2+2\atopwithdelims ()2}=\frac{s^3}{12}+\frac{s^2}{2}+\frac{7s}{6}+1. \end{aligned}$$

(13)

It follows from the work of Saxl [24] and Godsil and Meagher [13] that the permutation character of ${\mathrm {Sym}}(4s)$ acting on $\Omega _{4,s}$ is multiplicity-free if and only if $s\le 6$.

Finally, standard arguments show that ${\mathrm {Alt}}(4s)$ and ${\mathrm {Sym}}(4s)$ have the same orbitals on $\Omega _{4,s}$ whenever $s>4$. A direct inspection in the case $s\in \{2,3,4\}$ reveals that ${{{\text {rk}}}}_{\Omega _{4,s}}({\mathrm {Alt}}(4s))={{{\text {rk}}}}_{\Omega _{4,s}}({\mathrm {Sym}}(4s))$ also when $s=3$. $\square$

Table 5 The polynomials f(s) for computing the rank of ${\mathrm {Sym}}(4s)$ in its action on $\Omega _{4,s}$

Proposition 6.2

The rank ${\mathrm {Sym}}(5s)$ in its action on the set $\Omega _{5,s}$ of uniform partitions of $\{1,\ldots ,5s\}$ into 5 parts of cardinality s is

$$\begin{aligned}&f(s)+\frac{132029851}{4213820620800}s^{10}+\frac{1374677}{206928691200}s^{11} + \frac{72980263}{66217181184000}s^{12}+\frac{22531}{165542952960}s^{13}\\&\quad +\frac{1008757}{86082335539200}s^{14}+\frac{188723}{301288174387200}s^{15}+\frac{188723}{12051526975488000}s^{16}, \end{aligned}$$

where f(s) is a polynomial of degree 9 in s, depending on the congruence of s modulo 60, and is tabulated in Table 6. Moreover, the permutation character of this action is multiplicity-free if and only if $s= 2$. The group ${\mathrm {Alt}}(5s)$ has the same rank as ${\mathrm {Sym}}(5s)$ in its action on $\Omega _{5,s}$.

Proof

We omit the proof and it is similar to the proofs in the previous propositions; computations are just harder. $\square$

Table 6 The polynomials f(s) for computing the rank of ${\mathrm {Sym}}(5s)$ in its action on $\Omega _{5,s}$ (see Proposition 6.2)

Remark 6.3

Recall that Jackson and Van Rees [14] have computed the value of $|\Sigma _{6,s}|$. Indeed, from [14, Section 6], we have

$$\begin{aligned} |\Sigma _{6,s}|&=1+ \frac{3899}{600}s+ \frac{46584105377}{2141691552}s^2+ \frac{12246206617138789}{247365374256000}s^3\\&\quad + \frac{382955230861099213}{4517106834240000}s^4+ \frac{155498465793777230567}{1355132050272000000}s^5\\&\quad + \frac{14226886368398551}{112634352230400}s^6+ \frac{243245111626317349}{2111894104320000}s^7\\&\quad + \frac{232132948167689}{2634721689600}s^8+ \frac{253578194011961479}{4446092851200000}s^9+ \frac{736591080322991}{23433524674560}s^{10}\\&\quad + \frac{16265048187290869}{1098446469120000}s^{11}+ \frac{2000221303490489}{334764638208000}s^{12}\\&\quad + \frac{570713692223620411}{276180826521600000}s^{13}+ \frac{8346012436199}{13638559334400}s^{14}\\&\quad + \frac{1424745952102609}{9206027550720000}s^{15}+ \frac{77984295979769}{2343352467456000}s^{16}+ \frac{1062348478211833}{175751435059200000}s^{17}\\&\quad \frac{18674864899}{20324995891200}s^{18}+ \frac{2462417656967}{21341245685760000}s^{19}+ \frac{141248912237}{12014330904576000}s^{20}\\&\quad + \frac{853529939221}{901074817843200000}s^{21}+ \frac{4394656999}{75690284698828800}s^{22}\\&\quad + \frac{158824242127}{62444484876533760000}s^{23}+ \frac{9700106723}{136783157348597760000}s^{24}\\&\quad + \frac{9700106723}{10258736801144832000000}s^{25}. \end{aligned}$$

In particular, this explicit value for $|\Sigma _{6,s}|$ and Theorem 5.9 have be used to obtain good estimates on ${{{\text {rk}}}}_{\Omega _{6,s}}({\mathrm {Sym}}(6s))$.