The proof of a conjecture of Simion for certain partitions
type: Item
instance of: scholarly article
title: The proof of a conjecture of Simion for certain partitions
zbMATH Open document ID: 1001.05012
DOI: 10.1016/S0012-365X(00)00111-4
published in: Discrete Mathematics
publication date: 2002-12-11T00:00:00Z
review text: A partition of an integer \(N\) is a non-increasing sequence \(\lambda=\{\lambda_1,\lambda_2,\dots ,\lambda_m\},\) with \(\lambda_1\leq n, \sum_{i=1}^{m}\lambda_i=N\), and let \(\{\lambda'\}\) be its conjugate, i.e. \(\lambda'_i=|\{j:\lambda_j\geq i\}|\), with \(\lambda_1\leq m.\) Consider the \(m\times n\) grid with the Ferrers diagram of partition \(\lambda\) removed. Furthermore denote by \(N(m,n,\lambda)\) the number of paths in the \(m\times n\) grid such that the path starts at \((0,0)\) (at the left-lower corner) and ends at \((n,m)\) (at the right-upper corner) and at each step it goes one unit up (\(+(0,1)\)) or one unit right (\(+(1,0)\)) but not inside the removed \(\lambda\). A sequence \(\{a_1,a_2,\dots, a_n\}\) is said to be unimodal if, for some \(k\), \(a_1\leq \dots \leq a_k\) and \(a_{k+1}\leq \dots\leq a_n\). The author proves a conjecture of Simion in the case when \(\lambda\) is self-conjugate. It reads: if \(\lambda\) is self-conjugate and \(t\) is a non-negative integer then the sequence \(N(\lambda'_1,\lambda_1+t,\lambda),N(\lambda'_1+1,\lambda_1+t-1,\lambda),\dots, N(\lambda'_1+t,\lambda_1,\lambda)\) is unimodal.
reviewed by: Norbert Hegyvári
Mathematics Subject Classification ID: 05A17
Mathematics Subject Classification ID: 05A15
zbMATH DE Number: 1533344
zbMATH Keywords: partitions
zbMATH Keywords: lattice paths
zbMATH Keywords: unimodality
Wikidata QID: Q123331771
author: Martin V. Hildebrand
MaRDI profile type: Publication
full work available at URL: s0012-365x(00)00111-4
OpenAlex ID: W2060777787
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description: scientific article; zbMATH DE number 1533344