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q Statistics on $S_n$ and Pattern Avoidance

Amitai Regev·Yuval Roichman
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Abstract

Natural q analogues of classical statistics on the symmetric groups $S_n$ are introduced; parameters like: the q-length, the q-inversion number, the q-descent number and the q-major index. MacMahon's theorem about the equi-distribution of the inversion number and the reverse major index is generalized to all positive integers q. It is also shown that the q-inversion number and the q-reverse major index are equi-distributed over subsets of permutations avoiding certain patterns. Natural q analogues of the Bell and the Stirling numbers are related to these q statistics -- through the counting of the above pattern-avoiding permutations.

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Amitai Regev, Yuval Roichman. 2003-05-28. q Statistics on $S_n$ and Pattern Avoidance. https://arxiv.org/abs/math/0305393

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