Binomial coefficients and anti-exceedances of even permutations: A combinatorial proof
type: Item
instance of: scholarly article
title: Binomial coefficients and anti-exceedances of even permutations: A combinatorial proof
zbMATH Open document ID: 0777.05002
DOI: 10.1016/0097-3165(93)90064-F
author: R. Mantaci
published in: Journal of Combinatorial Theory. Series A
publication date: 1993-11-28T00:00:00Z
review text: For a permutation \(\pi\in S_ n\), the anti-exceedance \(\text{AX}(\pi)\) is the number of integers \(i\) which are transformed into a smaller integer by \(\pi\). For the alternating group \(A_ n\) define \(P_{n,k}=|\{\pi\in A_ n:\text{AX}(\pi)=k\}|\) and \(D_{n,k}=|\{\pi\in S_ n-A_ n:\text{AX}(\pi)=k\}|\). It is well known that \(P_{n,k}+D_{n,k}=\) the Eulerian number \(A_{n,k}\). The author gives a bijective proof for \(P_{n,k}-D_{n,k}=(-1)^{n- k}{n-1\choose k-1}\).
Mathematics Subject Classification ID: 05A05
Mathematics Subject Classification ID: 05A10
Mathematics Subject Classification ID: 05A15
zbMATH DE Number: 217952
zbMATH Keywords: binomial coefficients
zbMATH Keywords: permutations
zbMATH Keywords: Eulerian number
zbMATH Keywords: anti-exceedance
zbMATH Keywords: alternating group
reviewed by: László A. Székely
MaRDI profile type: Publication
cites work: Théorie géométrique des polynômes eulériens
cites work: Q4529547
description: scientific article; zbMATH DE number 217952