منابع مشابه
Minimal permutations with d descents
Recently, Bouvel and Pergola initiated the study of a special class of permutations, minimal permutations with a given number of descents, which arise from the whole genome duplication-random loss model of genome rearrangement. In this paper, we show that the number of minimal permutations of length 2d− 1 with d descents is given by 2d−3(d− 1)cd, where cd is the d-th Catalan number. We also der...
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We find the exponential generating function for permutations with all valleys even and all peaks odd, and use it to determine the asymptotics for its coefficients, answering a question posed by Liviu Nicolaescu. The generating function can be expressed as the reciprocal of a sum involving Euler numbers: ( 1− E1x + E3 x3 3! − E4 x4 4! + E6 x6 6! − E7 x7 7! + · · · )−1 , (∗) where ∑∞ n=0Enx n/n! ...
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We study new statistics on permutations that are variations on the descent and the inversion statistics. In particular, we consider the alternating descent set of a permutation σ = σ1σ2 · · · σn defined as the set of indices i such that either i is odd and σi > σi+1, or i is even and σi < σi+1. We show that this statistic is equidistributed with the odd 3-factor set statistic on permutations σ̃ ...
متن کاملDescents of Permutations in a Ferrers Board
The classical Eulerian polynomials are defined by setting An(t) = ∑ σ∈Sn t = n ∑ k=1 An,kt k where An,k is the number of permutations of length n with k − 1 descents. Let An(t, q) = ∑ π∈Sn t 1+des(π)qinv(π) be the inv q-analogue of the classical Eulerian polynomials whose generating function is well known: ∑ n>0 uAn(t, q) [n]q! = 1 1− t ∑ k>1 (1− t)kuk [k]q! . (0.1) In this paper we consider pe...
متن کاملExcedances and Descents of Bi-increasing Permutations
Motivated by the relations between certain difference statistics and the classical permutation statistics we study permutations whose inversion number and excedance difference coincide. It turns out that these (socalled bi-increasing) permutations are just the 321-avoiding ones. The paper investigates their excedance and descent structure. In particular, we give some combinatorial interpretatio...
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ژورنال
عنوان ژورنال: Journal of Combinatorics
سال: 2020
ISSN: 2156-3527,2150-959X
DOI: 10.4310/joc.2020.v11.n3.a5