A note on the map expansion of Jack polynomials

Abstract

In a recent work, Maciej Do\l{}e\k{}ga and the author have given a formula of the expansion of the Jack polynomial $J^{(\alpha)}_\lambda$ in the power-sum basis as a non-orientability generating series of bipartite maps whose edges are decorated with the boxes of the partition $\lambda$. We conjecture here a variant of this expansion in which we restrict the sum on maps whose edges are injectively decorated by the boxes of $\lambda$. We prove this conjecture for Jack polynomials indexed by 2-column partitions. The proof uses a mix of combinatorial methods and differential operator computations.