id: 05960194 dt: j an: 05960194 au: Ábrego, Bernardo M.; Fernández-Merchant, Silvia; Llano, Bernardo ti: An inequality for Macaulay functions. so: J. Integer Seq. 14, No. 7, Article 11.7.4, 11 p., electronic only (2011). py: 2011 pu: School of Computer Science, University of Waterloo, Waterloo, ON la: EN cc: ut: Macaulay function; Macaulay’s theorem; binomial representation of a positive integer; shadow of a set ci: li: emis:journals/JIS/VOL14/Abrego/abrego2.html ab: Summary: Given integers \$k\ge 1\$ and \$n\ge 0\$, there is a unique way of writing \$n\$ as \$\$n={n_k\choose k }+ {n_{k-1}\choose k-1}+\cdots+ {n_1\choose 1}\$\$ so that \$0\le n_1<\cdots< n_{k-1}< n_k\$. Using this representation, the \$k\$th Macaulay function of \$n\$ is defined as \$\$ tial^k(n)= {n_k-1\choose k-1}+ {n_{k-1}-1\choose k-2}+\cdots+ {n_1-1\choose 0}.\$\$ Whow that if \$a\ge 0\$ and \$a< tial^{k+1}(n)\$, then \$ tial^k(a)= tial^{k+1}(n- a)\ge tial^{k+1}(n)\$. As a corollary, we obtain a short proof of Macaulay’s theorem. Other previously known results are obtained as direct consequences. rv: