Summary: 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 $ ^{{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.

05A05, 05A20

Macaulay function, Macaulay's theorem, binomial representation of a positive integer, shadow of a set