On the log-concavity of sequences arising from integer bases.
type: Item
instance of: scholarly article
title: On the log-concavity of sequences arising from integer bases.
zbMATH Open document ID: 1045.05005
DOI: 10.1016/S0166-218X(03)00292-0
author: Philippe Pitteloud
published in: Discrete Applied Mathematics
publication date: 2004-03-29T00:00:00Z
review text: An {integer basis} is a strictly increasing sequence \((b_i)_{i\geq 0}\) of positive integers, if there exists a sequence \((c_i)_{i\geq 0}\) of positive integers, such that every positive integer \(n\) can be represented uniquely in the form \(n=\sum_{i=0}^k a_ib_i\) with \(0\leq a_i \leq c_i\) and \(a_k\not= 0\). (Georg Cantor characterized integer bases.) Let us be given an integer basis \({\mathcal B}= (b_i)_{i\geq 0}\). Let \(\Theta_{\mathcal B}(m,l)\) denote the number of integers in the interval \([0,m-1]\), which has exactly \(l\) nonzero digits in its \({\mathcal B}\)-representation. The main result of the paper is that for every integer basis \({\mathcal B}\) and every \(m\), \(\Theta_{\mathcal B}(m,l)\) is a strongly log-concave sequence for \(l=0,1,2,\dots\). The main result implies the log-concavity of weighted sums of several important combinatorial sequences. Technically, the proof of the theorem boils down to proving inequalites for quotients of elementary symmetric polynomials. The paper proposes a general log-concavity conjecture in terms of compression operators on partially ordered sets, as a natural generalization for the present result.
reviewed by: László A. Székely
Mathematics Subject Classification ID: 05A10
Mathematics Subject Classification ID: 05A15
Mathematics Subject Classification ID: 05A20
Mathematics Subject Classification ID: 06A07
Mathematics Subject Classification ID: 11A67
zbMATH DE Number: 2062821
zbMATH Keywords: unimodality
zbMATH Keywords: log-concavity
zbMATH Keywords: elementary symmetric polynomials
zbMATH Keywords: integer bases
zbMATH Keywords: binomial coefficients
zbMATH Keywords: Stirling numbers of the first kind
zbMATH Keywords: f-vectors
zbMATH Keywords: profiles
zbMATH Keywords: compression
MaRDI profile type: Publication
cites work: Q4320807
cites work: Q4338928
cites work: A \(q\)-analog of the exponential formula
cites work: Q4320535
cites work: A property of colored complexes and their duals
cites work: Log-concavity and compressed ideals in certain Macaulay posets
cites work: Q4294628
cites work: Q4344108
description: scientific article; zbMATH DE number 2062821