q-Narayana numbers and the flag h-vector of J(2×n)

The Narayana numbers are N(n,k)=(1/n)nknk+1. There are several natural statistics on Dyck paths with a distribution given by N(n,k). We show the equidistribution of Narayana statistics by computing the flag h-vector of J(2×n) in different ways. In the process we discover new Narayana statistics and provide co-statistics for the Narayana statistics so that the bi-statistics have a distribution given by Fürlinger and Hofbauerʹs q-Narayana numbers. We interpret the flag h-vector in terms of semi-standard Young tableaux, which enables us to express the q-Narayana numbers in terms of Schur functions. We also introduce what we call pre-shellings of simplicial complexes.