Séminaire Lotharingien de Combinatoire, 80B.45 (2018), 12 pp.
Andrei L. Kanunnikov, Valentin V. Promyslov and Ekaterina A.
Vassilieva
A Labelled Variant of the Matchings-Jack and Hypermap-Jack Conjectures
Abstract.
Introduced by Goulden and Jackson in their 1996 paper, the
matchings-Jack and hypermap-Jack conjectures are two major open
questions relating symmetric functions, representation theory and
combinatorial maps.
These conjectures state an important combinatorial interpretation of
the coefficients in the power sum expansion of two related formal
power series involving Jack symmetric functions.
These coefficients are indexed by three partitions of a given integer
n. This paper is devoted to the case when one of them is equal to
(n). We exhibit some of their polynomial properties and prove a
variant of the two conjectures involving labelled hypermaps and
matchings in some important cases.
Received: November 14, 2017.
Accepted: February 17, 2018.
Final version: April 1, 2018.
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