Proof of a conjecture of Mircea Merca

Abstract

We prove that, for any prime $p$ and positive integer $r$ with $p^r>2$, the number of multinomial coefficients such that $$ {k\choose k_1,k_2,\ldots,k_n}=p^r,\quad \text{and}\quad k_1+2k_2+\cdots+nk_n=n, $$ is given by $$ \delta_{p^r,\,k}\left(\left\lfloor\frac{n-1}{p^r-1}\right\rfloor -\delta_{0,\,n\bmod p^r} \right), $$ where $\delta_{i,j}$ is the Kronecker delta and $\lfloor x\rfloor$ stands for the largest integer not exceeding $x$. This confirms a recent conjecture of Mircea Merca.