Equidistribution of Descents, Adjacent Pairs, and Place-Value Pairs on Permutations
Emeric Deutsch, Sergey Kitaev, Jeffrey B. Remmel · 2011
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. 1