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: