# Equidistribution of descents, adjacent pairs, and place-value pairs on permutations

Deutsch, Emeric and Kitaev, Sergey and Remmel, Jeffrey (2009) Equidistribution of descents, adjacent pairs, and place-value pairs on permutations. Journal of Integer Sequences, 12 (5). 09.5.1.

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## Abstract

An \$(X,Y)\$-descent in a permutation is a pair of adjacent elements such that the first element is from \$X\$, the second element is from \$Y\$, and the first element is greater than the second one. An \$(X,Y)\$-adjacency in a permutation is a pair of adjacent elements such that the first one is from \$X\$ and the second one is from \$Y\$. An \$(X,Y)\$-place-value pair in a permutation is an element \$y\$ in position \$x\$, such that \$y\$ is in \$Y\$ and \$x\$ is in \$X\$. It turns out, that for certain choices of \$X\$ and \$Y\$ some of the three statistics above become equidistributed. Moreover, it is easy to derive the distribution formula for \$(X,Y)\$-place-value pairs thus providing distribution for other statistics under consideration too. This generalizes some results in the literature. As a result of our considerations, we get combinatorial proofs of several remarkable identities. We also conjecture existence of a bijection between two objects in question preserving a certain statistic.