Counting Descent Pairs with Prescribed Tops and Bottoms

Details Counting Descent Pairs with Prescribed Tops and Bottoms Counting Descent Pairs with Prescribed Tops and Bottoms

2006-10-20

arxiv.org/abs/math/0610608v1

Mathematics

Combinatorics

27 pages, 18 figures

Scientific paper

Given sets X and Y of positive integers and a permutation sigma = sigma_1, sigma_2, ..., sigma_n in S_n, an X,Y-descent of sigma is a descent pair sigma_i > sigma_{i+1} whose "top" sigma_i is in X and whose "bottom" sigma_{i+1} is in Y. We give two formulas for the number P_{n,s}^{X,Y} of sigma in S_n with s X,Y-descents. P_{n,s}^{X,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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