Enumeration of zero-sum subsequences: A proof of Bialostocki-Lotspeich conjecture

Let $C_n$ be a cyclic group of order $n\ge 2$, and let $S=g_1\cdot\ldots\cdot g_{\ell}$ be a sequence over $C_n$, with repetitions allowed. We denote by $\mathsf N_0^n(S)$ the number of subsets $I\subseteq[1,\ell]$ with $|I|=n$ and $\sum_{i\in I}g_i=0$. A conjecture proposed by Bialostocki and Lotspeich in 1992 states that $$\mathsf N_0^n(S)\ge \binom{\lfloor |S|/2\rfloor}{n}+ \binom{\lceil |S|/2\rceil}{n}.$$ Several partial results toward this conjecture are known. In this paper, we prove this conjecture completely.