Sums Involving Multinomial Coefficients

Abstract

Let $g_k(n)=\sum _{\underset{\_}{v}\in C_k(n)}(\genfrac{}{}{0ex}{}{n}{v})2^{v_1{v}_2+v_2{v}_3+v_3{v}_4+\dots +v_{k-1}{v}_k}$ where $C_k(n)$ denote the set of $k$-compositions of $n$. We show that

1. $g_k(n+p-1)\equiv g_k(n)(\mathrm{mod}p)$ for all $k,n\ge 1$, prime $p$;
2. $g_k(n)$ is a polynomial in $k$ of degree $n$ for $k\ge n+1$;

and, moreover, that these properties hold for wider classes of functions which are sums involving multinomial coefficients.