Given integers $k\geq1$ and $n\geq0$, there is a unique way of writing $n$ as $n=\binom{ n_{ k}}{ k}+\binom{ n_{ k-1}}{ k-1}+...+\binom{ n_{ 1}}{ 1}$ so that $0\leq n_{ 1}<...<n_{ k-1}<n_{ k}$. Using this representation, the \emph{ Kruskal-Macaulay function of}$n$ is defined as $\partial^{ k}(n) =\binom{ n_{ k}-1}{ k-1}+\binom{ n_{ k-1}-1}{ k-2}+...+\binom{ n_{ 1}-1}% { 0}.$ 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.
Categories: math.CO