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

Petter Brändén
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Abstract

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 Fürlinger 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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Petter Brändén. 2002-11-20. q-Narayana numbers and the flag h-vector of $J({\bf 2} \times {\bf n})$. https://arxiv.org/abs/math/0211318

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