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Enumeration of $n$-plexes

Arjun Maniyar

TL;DR

The paper addresses the exact enumeration of $n$-plexes by correcting and providing formulas for the cycle index $Z(S_p^{(r)})$ and the counts $s_p^n$, grounded in Pólya enumeration. It develops a practical method to compute $Z(S_p^{(r)})$ via the induced action of $S_p$ on $r$-subsets, employing the $j_1$-type formula $j_1(eta') = \sum_{(i)} \prod_k {j_k \choose i_k}$ and a rule for how powers $(\beta')^m$ transform cycle indices, to assemble $Z(S_p^{(r)})$. The counting polynomial is then obtained as $s_p^n(x) = Z(S_p^{(n+1)}, 1+x)$, with $s_p^n$ found by evaluating at $x=1$ (substituting $y_i=2$), and results are provided for $p\le 9$, $n\le 3$. The work delivers corrected, explicit formulas for $Z(S_p^{(r)})$ with $p\le 9$ and $r=3,4$, including merged cycle-index expressions and concrete examples, thereby enabling reliable enumeration of $n$-plexes and related hypergraph families. This yields practical benchmarks and a solid foundation for further combinatorial enumeration in this domain.

Abstract

Palmer provides a method of enumerating $n$-plexes, however it has some typographical errors in the formula for the cycle index $Z(S_p^{(r)})$ and the values of $s_p^n$, the number of $n$-plexes on $p$ points. This article is intended to provide the correct formulas.

Enumeration of $n$-plexes

TL;DR

The paper addresses the exact enumeration of -plexes by correcting and providing formulas for the cycle index and the counts , grounded in Pólya enumeration. It develops a practical method to compute via the induced action of on -subsets, employing the -type formula and a rule for how powers transform cycle indices, to assemble . The counting polynomial is then obtained as , with found by evaluating at (substituting ), and results are provided for , . The work delivers corrected, explicit formulas for with and , including merged cycle-index expressions and concrete examples, thereby enabling reliable enumeration of -plexes and related hypergraph families. This yields practical benchmarks and a solid foundation for further combinatorial enumeration in this domain.

Abstract

Palmer provides a method of enumerating -plexes, however it has some typographical errors in the formula for the cycle index and the values of , the number of -plexes on points. This article is intended to provide the correct formulas.
Paper Structure (4 sections, 1 theorem, 19 equations, 1 table)

This paper contains 4 sections, 1 theorem, 19 equations, 1 table.

Table of Contents

  1. Introduction
  2. Computing the cycle index of $S_p^{(r)}$
  3. Number of $n$-plexes
  4. Formulas for $Z(S_p^{(r)})$

Key Result

Theorem 1.1

Let $p\geq n+1$ then the counting polynomial $s_p^n$ for $n$-plexes of order $p$ is given by where $S_p^{(n+1)}$ is the $(n+1)$-group of the symmetric group $S_p$.

Theorems & Definitions (1)

  • Theorem 1.1