Some refinements of Stanley's shuffle theorem

Define a permutation to be any sequence of distinct positive integers. Given two permutations π and σ on disjoint underlying sets, we denote by π ⧢ σ the set of shuffles of π and σ (the set of all permutations obtained by interleaving ...

Let r des and r exc denote the permutation statistics r-descent number and r-excedance number, respectively. We prove that the pairs of permutation statistics ( r des , r maj ) and ( r exc , r den ) are equidistributed, where r maj denotes the r-...

We give a new bijective proof of Stanley's Shuffling Theorem using a more elementary approach, mapping a shuffling of two permutations to a pair of stars and bars arrangements. We conclude with a few remarks about consequences of the shuffling theorem.