Journal of Integer Sequences, Vol. 14 (2011), Article 11.7.4

An Inequality for Macaulay Functions


Bernardo M. Ábrego and Silvia Fernández-Merchant
Department of Mathematics
California State University, Northridge
18111 Nordhoff Street
Northridge, CA 91330
USA

Bernardo Llano
Departamento de Matemáticas
Universidad Autónoma Metropolitana, Iztapalapa
San Rafael Atlixco 186
Colonia Vicentina, 09340, México, D.F.
México

Abstract:

Given integers $ k\geq1$ and $ n\geq0$, there is a unique way of writing $ n$ as $ n=\binom{n_{k}}{k}+\binom{n_{k-1}}{k-1}+\cdots+\binom{n_{1}}{1}$ so that $ 0\leq n_{1}<\cdots<n_{k-1}<n_{k}$. Using this representation, the k $ ^{\text{th}}$ Macaulay function of $ n$ is defined as $ \partial^{k}( n) =\binom{n_{k}-1}{k-1}+\binom{n_{k-1}-1}{k-2}+\cdots+\binom{n_{1}-1} {0}.$ We show that if $ a\geq0$ and $ a<\partial^{k+1}(n) $, then $ \partial^{k}(a) +\partial^{k+1}( n-a) \geq \partial^{k+1}(n)$. As a corollary, we obtain a short proof of Macaulay's theorem. Other previously known results are obtained as direct consequences.

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(Concerned with sequences A123578 A123579 A123580 A123731.)

Received December 18 2010; revised version received July 8 2011. Published in Journal of Integer Sequences, September 5 2011.

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