Abstract
For a subsetS, let the descent statistic β(S) be the number of permutations that have descent setS. 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 (k i,P,k n−i, whereP is a palindrome, whose first entry is larger thank. 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.
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Ehrenborg, R., Mahajan, S. Maximizing the descent statistic. Annals of Combinatorics 2, 111–129 (1998). https://doi.org/10.1007/BF01608482
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DOI: https://doi.org/10.1007/BF01608482