Connection coefficients, matchings, maps and combinatorial conjectures for Jack symmetric functions

Goulden, I.; Jackson, D.

doi:10.1090/S0002-9947-96-01503-6

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ISSN 1088-6850(online) ISSN 0002-9947(print)

Connection coefficients, matchings, maps and
combinatorial conjectures for
Jack symmetric functions

Authors: I. P. Goulden and D. M. Jackson
Journal: Trans. Amer. Math. Soc. 348 (1996), 873-892
MSC (1991): Primary 05E05, 05A15, 57M15
MathSciNet review: 1325917
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Abstract: A power series is introduced that is an extension to three sets of variables of the Cauchy sum for Jack symmetric functions in the Jack parameter $\alpha.$ We conjecture that the coefficients of this series with respect to the power sum basis are nonnegative integer polynomials in $b$ , the Jack parameter shifted by $1$ . More strongly, we make the Matchings-Jack Conjecture, that the coefficients are counting series in $b$ for matchings with respect to a parameter of nonbipartiteness. Evidence is presented for these conjectures and they are proved for two infinite families.

The coefficients of a second series, essentially the logarithm of the first, specialize at values $1$ and $2$ of the Jack parameter to the numbers of hypermaps in orientable and locally orientable surfaces, respectively. We conjecture that these coefficients are also nonnegative integer polynomials in $b$ , and we make the Hypermap-Jack Conjecture, that the coefficients are counting series in $b$ for hypermaps in locally orientable surfaces with respect to a parameter of nonorientability.

1. M. Adler and P. van Moerbeke, Birkhoff strata, Backlund transformations, and regularization of isospectral operators, Advances in Math., 108 (1994), 140--204. CMP 95:01
2. G. E. Andrews, I. P. Goulden, and D. M. Jackson, Generalizations of Cauchy’s summation theorem for Schur functions, Trans. Amer. Math. Soc. 310 (1988), no. 2, 805–820. MR 973178 (89m:05009), http://dx.doi.org/10.1090/S0002-9947-1988-0973178-8
3. Nantel Bergeron and Adriano M. Garsia, Zonal polynomials and domino tableaux, Discrete Math. 99 (1992), no. 1-3, 3–15. MR 1158775 (93e:20006), http://dx.doi.org/10.1016/0012-365X(92)90360-R
4. I. P. Goulden and D. M. Jackson, Combinatorial enumeration, A Wiley-Interscience Publication, John Wiley & Sons Inc., New York, 1983. With a foreword by Gian-Carlo Rota; Wiley-Interscience Series in Discrete Mathematics. MR 702512 (84m:05002)
5. I.P. Goulden and D.M. Jackson, Combinatorial constructions for integrals over normally distributed random matrices, Proc. Amer. Math. Soc., 123 (1995), 995--1003. CMP 95:06
6. I.P. Goulden and D.M. Jackson, Maps in locally orientable surfaces, the double coset algebra, and zonal polynomials, Canadian J. Math. (to appear).
7. I.P. Goulden and D.M. Jackson, Symmetric functions and Macdonald's result for top connexion coefficients in the symmetric group, J. Algebra, 166 (1994), 364--378. CMP 94:13
8. Phil Hanlon, Jack symmetric functions and some combinatorial properties of Young symmetrizers, J. Combin. Theory Ser. A 47 (1988), no. 1, 37–70. MR 924451 (90e:05008), http://dx.doi.org/10.1016/0097-3165(88)90042-8
9. Phil Hanlon, A Markov chain on the symmetric group and Jack symmetric functions, Discrete Math. 99 (1992), no. 1-3, 123–140. MR 1158785 (93j:60093), http://dx.doi.org/10.1016/0012-365X(92)90370-U
10. Philip J. Hanlon, Richard P. Stanley, and John R. Stembridge, Some combinatorial aspects of the spectra of normally distributed random matrices, and applications (Tampa, FL, 1991) Contemp. Math., vol. 138, Amer. Math. Soc., Providence, RI, 1992, pp. 151–174. MR 1199126 (93j:05164), http://dx.doi.org/10.1090/conm/138/1199126
11. Henry Jack, A class of symmetric polynomials with a parameter, Proc. Roy. Soc. Edinburgh Sect. A 69 (1970/1971), 1–18. MR 0289462 (44 #6652)
12. D. M. Jackson and T. I. Visentin, A character-theoretic approach to embeddings of rooted maps in an orientable surface of given genus, Trans. Amer. Math. Soc. 322 (1990), no. 1, 343–363. MR 1012517 (91b:05093), http://dx.doi.org/10.1090/S0002-9947-1990-1012517-8
13. D. M. Jackson and T. I. Visentin, Character theory and rooted maps in an orientable surface of given genus: face-colored maps, Trans. Amer. Math. Soc. 322 (1990), no. 1, 365–376. MR 1012516 (91b:05094), http://dx.doi.org/10.1090/S0002-9947-1990-1012516-6
14. K.J. Kadell, The Selberg-Jack symmetric functions, (preprint).
15. I. G. Macdonald, Symmetric functions and Hall polynomials, The Clarendon Press Oxford University Press, New York, 1979. Oxford Mathematical Monographs. MR 553598 (84g:05003)
16. I. G. Macdonald, Commuting differential operators and zonal spherical functions, Algebraic groups Utrecht 1986, Lecture Notes in Math., vol. 1271, Springer, Berlin, 1987, pp. 189–200. MR 911140 (89e:43025), http://dx.doi.org/10.1007/BFb0079238
17. I.G. Macdonald, A new class of symmetric functions, Publ. I.R.M.A. Strasbourg, Actes 20th Seminaire Lotharingien (1988), 131--171.
18. Richard P. Stanley, Some combinatorial properties of Jack symmetric functions, Adv. Math. 77 (1989), no. 1, 76–115. MR 1014073 (90g:05020), http://dx.doi.org/10.1016/0001-8708(89)90015-7
19. J.R. Stembridge, A Maple package for symmetric functions - Version 2, Department of Mathematics, University of Michigan, Ann Arbor, MI 48109, July 1993.

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