On the log-concavity of sequences arising from integer bases.
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.
scientific article; zbMATH DE number 2062821