On the matchings-Jack conjecture for Jack connection coefficients indexed by two single part partitions

Andrei L. Kanunnikov¹ ; Ekaterina A. Vassilieva²

¹Moscow State University, Russia
²Ecole Polytechnique, Palaiseau, France

The electronic journal of combinatorics, Tome 23 (2016) no. 1

Résumé

This article is devoted to the study of Jack connection coefficients, a generalization of the connection coefficients of the classical commutative subalgebras of the group algebra of the symmetric group closely related to the theory of Jack symmetric functions. First introduced by Goulden and Jackson (1996) these numbers indexed by three partitions of a given integer $n$ and the Jack parameter $\alpha$ are defined as the coefficients in the power sum expansion of some Cauchy sum for Jack symmetric functions. Goulden and Jackson conjectured that they are polynomials in $\beta = \alpha-1$ with non negative integer coefficients of combinatorial significance, the Matchings-Jack conjecture.In this paper we look at the case when two of the integer partitions are equal to the single part $(n)$. We use an algebraic framework of Lasalle (2008) for Jack symmetric functions and a bijective construction in order to show that the coefficients satisfy a simple recurrence formula and prove the Matchings-Jack conjecture in this case. Furthermore we exhibit the polynomial properties of more general coefficients where the two single part partitions are replaced by an arbitrary number of integer partitions either equal to $(n)$ or $[1^{n-2}2]$.

Détail

DOI : 10.37236/5085

Classification : 05E05, 05A19
Mots-clés : Jack connection coefficients, Jack symmetric functions, matchings-Jack conjecture