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An $(X,Y)$-descent in a permutation is a pair of adjacent elements such thatthe first element is from $X$, the second element is from $Y$, and the firstelement is greater than the second one. An $(X,Y)$-adjacency in a permutationis a pair of adjacent elements such that the first one is from $X$ and thesecond one is from $Y$. An $(X,Y)$-place-value pair in a permutation is anelement $y$ in position $x$, such that $y$ is in $Y$ and $x$ is in $X$. Itturns out, that for certain choices of $X$ and $Y$ some of the three statisticsabove become equidistributed. Moreover, it is easy to derive the distributionformula for $(X,Y)$-place-value pairs thus providing distribution for otherstatistics under consideration too. This generalizes some results in theliterature. As a result of our considerations, we get combinatorial proofs ofseveral remarkable identities. We also conjecture existence of a bijectionbetween two objects in question preserving a certain statistic.

Equidistribution of (X,Y)-descents, (X,Y)-adjacent pairs, and (X,Y)-place-value pairs on permutations