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 $\sigma =[2,3,\dots ,n,1]$ 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 ${\mathrm{NC}}_W^{(k)}$, we also obtain the interlacing symmetric decomposition property of the $h$-polynomial $h(\Delta ({\mathrm{NC}}_W^{(k)}),t)$.