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

David Broadhurst, Xavier Roulleau

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.

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

Abstract

For integers , we give a formula for the number of order subsets of the ring whose sum of elements is modulo . To do so, we describe explicitly a sequence of matrices , for positive integers , such that the size of is the number of divisors of , and for two coprime integers , the matrix is the Kronecker product of and . For , and for when is even, the sequences are related to the number of necklaces with black beads and white beads, and to Lyndon words. This work begins with empirical determinations of up to , 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.

Paper Structure

This paper contains 22 sections, 19 theorems, 72 equations.

Table of Contents

  1. Introduction
  2. A sequence of matrices with remarkable properties
  3. Definitions of the matrices $M(k)$
  4. Matrices indexed by finite sets
  5. Divisor indexed matrices
  6. The matrices $M(k)$
  7. Main properties of the matrices $M(k)$
  8. The sequences $(T(n,k,s))_{n\geq k}$
  9. Definitions and properties of the integers $T(n,k,s)$
  10. Generating functions
  11. Route to proof
  12. Last row of $M(k)$ and generating function of $T(n,k,0)$
  13. A route to the sequence of matrices $M(k)$
  14. Proof of the matrix construction using Ramanujan sums
  15. Further remarks and open questions
  16. ...and 7 more sections

Key Result

Theorem 1

For integers $n$, $k$ and $s$, with $n\ge k>0$, one has where $M(k)_{t,d}$ with $t\mid k$ and $d\mid k$ is the integer computed by factorizing $k$ into prime powers $q=p^{e}$, with $e$ the valuation of $k$ at prime $p|k$, to give

Theorems & Definitions (37)

  • Theorem 1
  • Theorem 2
  • Definition 3
  • Remark 4
  • Definition 5
  • Theorem 6
  • proof
  • Proposition 7
  • proof
  • Proposition 8
  • ...and 27 more