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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
mailto:bernardo.abrego@csun.edubernardo.abrego@csun.edu
mailto:silvia.fernandez@csun.edusilvia.fernandez@csun.edu

Bernardo Llano
Departamento de Matemáticas
Universidad Autónoma Metropolitana, Iztapalapa
San Rafael Atlixco 186
Colonia Vicentina, 09340, México, D.F.
México
mailto:llano@xanum.uam.mxllano@xanum.uam.mx

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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.