Log-concavity of the Excedance Enumerators in positive elements of Type\n A and Type B Coxeter Groups

2020/09/22 by Hiranya Kishore Dey, Dey, Hiranya Kishore
Mathematics · Computer Science · #Advanced Combinatorial Mathematics #semigroups and automata theory #Advanced Mathematical Identities

paper · pdf · doi:10.48550/arxiv.2009.10655

Abstract

The classical Eulerian Numbers A_n,k are known to be log-concave. Let\nP_n,k and Q_n,k be the number of even and odd permutations with k\nexcedances. In this paper, we show that P_n,k and Q_n,k are\nlog-concave. For this, we introduce the notion of strong synchronisation and\nratio-alternating which are motivated by the notion of synchronisation and\nratio-dominance, introduced by Gross, Mansour, Tucker and Wang in 2014.\n We show similar results for Type B Coxeter Groups. We finish with some\nconjectures to emphasize the following: though strong synchronisation is\nstronger than log-concavity, many pairs of interesting combinatorial families\nof sequences seem to satisfy this property.\n

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