Séminaire Lotharingien de Combinatoire, 86B.24 (2022), 12 pp.
Houcine Ben Dali
Integrality in the Matching-Jack Conjecture and the Farahat-Higman Algebra
Abstract.
Using Jack polynomials, Goulden and Jackson have introduced a one parameter deformation τb of the generating series of bipartite maps. The Matching-Jack conjecture suggests that the coefficients
cλμ,ν of the function τb in the power-sum basis are non-negative integer polynomials in the deformation parameter b.
Do\l{}{\k{e}}ga and F\'eray have proved in 2016 the "polynomiality" part in the Matching-Jack conjecture.
In this paper, we prove the "integrality" part.
The proof is based on a recent work of the author that deduces the Matching-Jack conjecture for marginal sums cλμ,l from an analog result for the b-conjecture, established in 2020 by Chapuy and Dołęga.
A key step in the proof involves a new connection with the graded Farahat-Higman algebra.
Received: November 25, 2021.
Accepted: March 4, 2022.
Final version: April 1, 2022.
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