Lascoux series, parking functions and noncrossing partitions

Abstract

In the study of the generating series of Demazure characters, Lascoux used isobaric divided differences to define a family of polynomials $\mathcal{E}_{\sigma}(t)$ indexed by permutations $\sigma$, and asked for a satisfactory expression of these polynomials. In this paper we obtain a combinatorial interpretation of $\mathcal{E}_{\sigma}(t)$ for the permutation $\sigma=[2,3,\ldots,n,1]$ or its inverse in terms of the descent statistic of parking functions of length $n-1$. Based on this progress on Lascoux's open problem, we find that the polynomial $\mathcal{E}_{\sigma}(t)$ for this special case coincides with the $h$-polynomial $h(\Delta(\mathrm{NC}_W),t)$ of the order complex of the noncrossing partition lattice associated to the irreducible Coxeter group $W$ of type $A_{n-1}$. We are inspired by this coincidence to give an operator approach to $h(\Delta(\mathrm{NC}_W),t)$ for any finite Coxeter group $W$. As an application, we completely solve an open problem on $h(\Delta(\mathrm{NC}_W),t)$ which was proposed by Athanasiadis, Douvropoulos and Kalampogia-Evangelinou. For any $k$-divisible noncrossing partition poset $\mathrm{NC}^{(k)}_W$, we also obtain the interlacing symmetric decomposition property of the $h$-polynomial $h(\Delta(\mathrm{NC}^{(k)}_W),t)$.