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Constant Term Solution for an Arbitrary Number of Osculating Lattice Paths

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Abstract

Osculating paths are sets of directed lattice paths which are not allowed to cross each other or have common edges, but are allowed to have common vertices. In this work we derive a constant term formula for the number of such lattice paths by solving a set of simultaneous difference equations.

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References

  1. Andrews G.E.: Plane partitions v: the tsscpp conjecture. J. Combin. Theory Ser. A 66, 28–39 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  2. Baxter R.J.: Exactly Solved Models in Statistical Mechanics. Academic Press, London (1982)

    MATH  Google Scholar 

  3. Bethe H.A.: Zur Theorie der Metalle. I. Eigenwerte und Eigenfunktionen der linearen Atomkette. Z. Phys. 71, 205 (1931)

    Article  ADS  Google Scholar 

  4. Bousquet-Mélou M.: Three osculating walkers. J. Phys. Conf. Ser. 42, 35–46 (2006)

    Article  ADS  Google Scholar 

  5. Brak, R.: Osculating lattice paths and alternating sign matrices. In: Formal Power Series and Alegebraic Combinatorics, 9th Conference (1997)

  6. Brak R., Essam J.W., Owczarek A.L.: Partial difference equation method for lattice path problems. Ann. Comb. 3(2–4), 265–275 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  7. Brenti F.: Determinants of super-Schur functions, lattice paths, and dotted plane partitions. Adv. Math. 98(1), 27–64 (1993)

    Article  MathSciNet  MATH  Google Scholar 

  8. Essam J.W.: Three attractive osculating walkers and a polymer collapse transition. J. Stat. Phys. 110(3-6), 1191–1207 (2003)

    Article  MATH  Google Scholar 

  9. Fischer I.: The number of monotone triangles with prescribed bottom row. Adv. Appl. Math. 37(2), 249–267 (2006)

    Article  MATH  Google Scholar 

  10. Fisher M.E.: Walks, walls, wetting, and melting. J. Stat. Phys. 34, 667–730 (1984)

    Article  ADS  MATH  Google Scholar 

  11. Gessel, I., Viennot, G.: Determinants, paths and plane partitions. (Unpublished) (1989)

  12. Guttmann A.J., Owczarek A.L., Viennot X.G.: Vicious walkers and Young tableaux I: without walls. J. Phys. A-Math. Gen. 31(40), 8123–8135 (1998)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  13. Izergin A.G.: Statistical sum of the 6-vertex model in a finite lattice. Sov. Phys. Dokl. 32, 878 (1987)

    ADS  Google Scholar 

  14. Korepin V.E.: Calculation of norms of Bethe wave functions. Commun. Math. Phys. 86, 391–418 (1982)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  15. Korff, C.: Quantum cohomology via vicious and osculating walkers. ArXiv e-prints (2012)

  16. Krattenthaler C., Guttmann A.J., Viennot X.G.: Vicious walkers, friendly walkers and Young tableaux: II. With a wall. J. Phys. A-Math. Gen. 33(48), 8835–8866 (2000)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  17. Krattenthaler C., Guttmann A.J., Viennot X.G.: Vicious walkers, friendly walkers, and young tableaux. III. Between two walls. J. Stat. Phys. 110(3-6), 1069–1086 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  18. Kuperberg G.: Another proof of the alternating sign matrix conjecture. Int. Math. Res. Notes 1996, 139–150 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  19. Lieb E.H.: Residual entropy of square lattice. Phys. Rev. 162, 162 (1967)

    Article  ADS  Google Scholar 

  20. Mills W.H., Robbins D.P., Rumsey H. Jr: Alternating sign matrices and descending plane partitions. J. Comb. Theory Ser. A 34, 340–359 (1983)

    Article  MathSciNet  MATH  Google Scholar 

  21. Robbins D.P.: The story of 1, 2, 7, 42, 492, 7436,.... Math. Intell. 13, 12–19 (1991)

    Article  MathSciNet  MATH  Google Scholar 

  22. Stembridge, J.: On the fully commutative elements of coxeter groups. J. Algebr. Combin. 353–385 (1995)

  23. Stembridge J.R.: Nonintersection paths, Pfaffians, and plane partitions. Adv. Math. 83(1), 96–131 (1990)

    Article  MathSciNet  MATH  Google Scholar 

  24. Stembridge J.R.: The enurameration of totally symmetrical plane partitons. Adv. Math. 111(2), 227–243 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  25. Tracy, C.A., Widom, H.: Integral formulas for the asymmetric simple exclusion process. Commun. Math. Phys. 279 (2008)

  26. Zeilberger D.: Proof of the alternating sign matrix conjecture. Elect. J. Comb 3, 84 (1996)

    Google Scholar 

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Correspondence to Richard Brak.

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Brak, R., Galleas, W. Constant Term Solution for an Arbitrary Number of Osculating Lattice Paths. Lett Math Phys 103, 1261–1272 (2013). https://doi.org/10.1007/s11005-013-0646-1

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  • DOI: https://doi.org/10.1007/s11005-013-0646-1

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