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Counting partitions by genus: a compendium of results

Robert Coquereaux, Jean-Bernard Zuber

Abstract

We study the enumeration of set partitions, according to their length, number of parts, cyclic type, and genus. We introduce genus-dependent Bell, Stirling numbers, and Faà di Bruno coefficients. Besides attempting to summarize what is already known on the subject, we obtain new generic results (in particular for partitions into two parts, for arbitrary genus), and present computer generated new data extending the number of terms known for sequences or families of such coefficients; this also leads to new conjectures.

Counting partitions by genus: a compendium of results

Abstract

We study the enumeration of set partitions, according to their length, number of parts, cyclic type, and genus. We introduce genus-dependent Bell, Stirling numbers, and Faà di Bruno coefficients. Besides attempting to summarize what is already known on the subject, we obtain new generic results (in particular for partitions into two parts, for arbitrary genus), and present computer generated new data extending the number of terms known for sequences or families of such coefficients; this also leads to new conjectures.
Paper Structure (47 sections, 3 theorems, 123 equations, 2 figures, 3 tables)

This paper contains 47 sections, 3 theorems, 123 equations, 2 figures, 3 tables.

Table of Contents

  1. Introduction
  2. Bell, Stirling, and Faà di Bruno numbers
  3. Equivalence relations on a set with n elements
  4. Equivalence relations on a set with n elements, with k equivalence classes
  5. Set with $n$ elements, with $k$ equivalence classes of specified cardinalities
  6. Genus of partitions on a cyclically ordered set
  7. Partitions with no singletons
  8. Associated Bell number $\widehat{B}_n$
  9. Associated Stirling numbers of the second kind $\widehat{S}_{n,k}$
  10. Genus-dependent Bell numbers $B_{n}^{(g)}$
  11. Unconstrained partitions: basic numbers $B_{n}^{(g)}$
  12. Genus $0$
  13. Genus $1$
  14. Genus $2$
  15. Genus $3$
  16. ...and 32 more sections

Key Result

Proposition 11

The generating function of the $C_{2k, [2^k]}^{(g)}$ for $g>0$ is of the form where $Q^{(g)}(u)$ is a polynomial of degree $g-1$ in $u$ satisfying

Figures (2)

  • Figure 1: The partition $(\{1,3,4,6,7\},\{2,5,9\},\{8\},\{10\})$ of $\{1,\ldots,10\}$. (a) the four $\ell$-vertices; (b) and (c): two equivalent representations of the special 10-vertex; (d) a contribution to $C^{(g)}_{10, [1^2\,3 \, 5]}$; (e) the double line (fat graph) version of (d), with three faces and thus genus $g=2$.
  • Figure 2: (a) Partition with $n=12$, $p=5$, $f=5$, $s_1=1$, $s_2=3$, $g=3$; (b) removing shaded faces, i.e., singletons of $\sigma\circ \tau^{-1}$, relabeling the points and doubling the edges to make the one cycle (or face) $(1,8,7,6,5,4,3,2)$ more visible, $f'=1$.

Theorems & Definitions (18)

  • Remark 1
  • Remark 2
  • Remark 3
  • Definition 4
  • Conjecture 8: Genus $g=3$ conjecture (weak form)
  • Conjecture 9: Genus $g$ conjecture (strong form)
  • Remark 10
  • Proposition 11
  • proof
  • Example 12
  • ...and 8 more