Annals of Combinatorics 2 (1998) 111-129

Maximizing the Descent Statistic

Richard Ehrenborg1 and Swapneel Mahajan2

Department of Mathematics, White Hall, Cornell University, Ithaca, NY 14853-7901, USA

Received May 29, 1998

AMS Subject Classification: 05A05, 05A20

Abstract.For a subset S, let the descent statistic β (S) be the number of permutations that have descent set S.We study inequalities between the descent statistics of subsets. Each subset (and its complement) is encoded by a list containing the lengths of the runs. We define two preorders that compare different lists based on the descent statistic. Using these preorders, we obtain a complete order on lists of the form (ki,P,kn-i),where P is a palindrome, whose first entry is larger than k. We prove a conjecture due to Gessel, which determines the list that maximizes the descent statistic, among lists of a given size and given length. We also have a generalization of the boustrophedon transform of Millar, Sloane and Young.

Keywords: permutations, descent set, Euler numbers, Boustrophedon transform


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