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A statistic-swapping involution on the Cartesian product of the symmetric group $S_{kn}$ and the generalized symmetric group $S(k,n)$

Peter Kagey, Kai Mawhinney

Abstract

We construct a statistic-swapping involution on the Cartesian product of the generalized symmetric group $S(k,n)$ with the symmetric group $S_{kn}$, which swaps the number of fixed points in the generalized symmetric group element with the number of $k$-cycles in the symmetric group element. This gives a combinatorial proof for a probabilistic observation: the distribution of fixed points on $S(k,n)$ matches the distribution of $k$-cycles on $S_{kn}$.

A statistic-swapping involution on the Cartesian product of the symmetric group $S_{kn}$ and the generalized symmetric group $S(k,n)$

Abstract

We construct a statistic-swapping involution on the Cartesian product of the generalized symmetric group with the symmetric group , which swaps the number of fixed points in the generalized symmetric group element with the number of -cycles in the symmetric group element. This gives a combinatorial proof for a probabilistic observation: the distribution of fixed points on matches the distribution of -cycles on .
Paper Structure (9 sections, 10 theorems, 32 equations, 1 figure, 1 table)

This paper contains 9 sections, 10 theorems, 32 equations, 1 figure, 1 table.

Table of Contents

  1. Introduction
  2. Decomposing the involution
  3. The auxilliary bijection
  4. k-cycles map to fixed points
  5. Inverting the bijection
  6. Example of inverting the map
  7. Shifting blocks
  8. The involution
  9. Further considerations

Key Result

Theorem 1.2

Let $\operatorname{Cyc}_m(S_{kn})$ denote the set of permutations in $S_{kn}$ with exactly $m$$k$-cycles, and let $\operatorname{Fxpt}_m(S(k,n))$ denote the set of elements of $S(k,n)$ with exactly $m$ fixed points. Then

Figures (1)

  • Figure 1: In the $k=2$ and $n=3$ case, $S(2,3)=B_3$ acts on an octahedron, and $S_6$ acts on the $6$-simplex. Here, the fixed points of $S(2,3)$ fix pairs of antipodal vertices of the octahedron, and $k$-cycles reflect edges of the $6$-simplex. In this example, there is one fixed point and two $k$-cycles in the first pair, which map to a pair with two fixed points and one $k$-cycle.

Theorems & Definitions (40)

  • Definition 1.1
  • Example 1
  • Example 2
  • Theorem 1.2
  • Definition 1.3
  • Theorem 1.4
  • proof
  • Definition 1.5: Stanley's fundamental bijection Stanley2011EC1
  • Definition 1.6
  • Example 3
  • ...and 30 more