Counting Descent Pairs with Prescribed Tops and Bottoms

2006/10/20 by John T. Hall, Hall, John T., Jeffrey B. Remmel +1
Mathematics · #05A05 #05A15 (Primary) #05A19 (Secondary) #Combinatorics (math.CO) #FOS: Mathematics #math.CO #msc:05A05 #msc:05A15 #msc:05A19

paper · pdf · doi:10.48550/arxiv.math/0610608

27 pages, 18 figures

arxiv created 2006/10/20 · arxiv updated 2009/12/01

Abstract

Given sets X and Y of positive integers and a permutation sigma = sigma₁, sigma₂, ..., 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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