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

TL;DR

This paper provides a formula for counting specific subsets in modular rings, linking combinatorial objects like necklaces and Lyndon words with algebraic structures, and introduces matrices with multiplicative properties.

Contribution

It explicitly constructs matrices M(k) that encode the counts of subsets, revealing their multiplicative structure and connecting combinatorics with algebra and number theory.

Findings

Derived a formula for T(n,k,s) involving matrices M(k)

Established the multiplicative property of M(k) matrices via Kronecker products

Connected the counts to necklaces, Lyndon words, and Ramanujan sums

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.