Counting Descent Pairs with Prescribed Tops and Bottoms
Abstract
Given sets X and Y of positive integers and a permutation sigma = sigma1, sigma2, ..., sigman in Sn, an X,Y-descent of sigma is a descent pair sigmai > sigmai+1 whose "top" sigmai is in X and whose "bottom" sigmai+1 is in Y. We give two formulas for the number Pn,sX,Y of sigma in Sn with s X,Y-descents. Pn,sX,Y is also shown to be a hit number of a certain Ferrers board. This work generalizes results of Kitaev and Remmel on counting descent pairs whose top (or bottom) is equal to 0 mod k.
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