On the distribution of some Euler-Mahonian statistics

We give a direct combinatorial proof of the equidistribution of two pairs of permutation statistics, $\texttt{(des, aid)}$ and $\texttt{(lec, inv)}$, which have been previously shown to have the same joint distribution as $\texttt{(exc, maj)}$, the major index and the number of excedances of a permutation. Moreover, the triple $\texttt{(pix, lec, inv)}$ was shown to have the same distribution as $\texttt{(fix, exc, maj)}$, where fix is the number of fixed points of a permutation. We define a new statistic $\texttt{aix}$ so that our bijection maps $\texttt{(pix, lec, inv)}$ to $\texttt{(aix, des, aid)}$. We also find an Eulerian partner das for a Mahonian statistic mix defined using mesh patterns, so that $\texttt{(das, mix)}$ is equidistributed with $\texttt{(des, inv)}$.

Keywords

permutation statistic, Eulerian, Mahonian, admissible inversion, descent, hook factorization, pattern

2010 Mathematics Subject Classification

Primary 05A05. Secondary 05A15.

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Published 4 June 2015