2009/03/14 by Emeric Deutsch, Deutsch, Emeric, Sergey Kitaev +3
Mathematics · #05A15 #Combinatorics (math.CO) #FOS: Mathematics #math.CO #msc:05A15
paper · pdf · doi:10.48550/arxiv.0903.2555
19 pages
arxiv created 2009/03/14 · arxiv updated 2009/12/01
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.