Number of partitions of modular integers (with an Appendix by P. Deligne)

Abstract

For integers $n,k,s$, we give a formula for the number $T(n,k,s)$ of order $k$ subsets of the ring $\mathbb{Z}/n\mathbb{Z}$ whose sum of elements is $s$ modulo $n$. To do so, we describe explicitly a sequence of matrices $M(k)$, for positive integers $k$, such that the size of $M(k)$ is the number of divisors of $k$, and for two coprime integers $k_{1},k_{2}$, the matrix $M(k_{1}k_{2})$ is the Kronecker product of $M(k_{1})$ and $M(k_{2})$. For $s=0, 1, 2$, and for $s=k/2$ when $k$ is even, the sequences $T(n,k,s)$ are related to the number of necklaces with $k$ black beads and $n-k$ white beads, and to Lyndon words. This work begins with empirical determinations of $M(k)$ up to $k=10000$, from which we infer a closed formula that encompasses many entries in the Encyclopedia of Integer Sequences. Its proof comes from work on Ramanujan sums, by Ramanathan, with a generalization to wider problems linked to representation theory and recently described by Deligne.