q-Narayana numbers and the flag h-vector of $J({\bf 2} \times {\bf n})$

Details q-Narayana numbers and the flag h-vector of $J({\bf 2} \times {\bf n})$ q-Narayana numbers and the flag h-vector of $J({\bf 2} \times {\bf n})$

2002-11-20

arxiv.org/abs/math/0211318v1

Discrete Math. 281 (2004), no. 1-3, 67--81

Mathematics

Combinatorics

Scientific paper

The Narayana numbers are $N(n,k) = {1 \over n}{n \choose k}{n \choose {k+1}}$. 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 $J({\bf 2} \times {\bf n})$ in different ways. In the process we discover new Narayana statistics and provide co-statistics for which the Narayana statistics in question have a distribution given by F\"urlinger and Hofbauers q-Narayana numbers. We also interpret the h-vector in terms of semi-standard Young tableaux, which enables us to express the q-Narayana numbers in terms of Schur functions.

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