Skip to Main content Skip to Navigation
Journal articles

A class of symmetric difference-closed sets related to commuting involutions

Abstract : Recent research on the combinatorics of finite sets has explored the structure of symmetric difference-closed sets, and recent research in combinatorial group theory has concerned the enumeration of commuting involutions in $S_{n}$ and $A_{n}$. In this article, we consider an interesting combination of these two subjects, by introducing classes of symmetric difference-closed sets of elements which correspond in a natural way to commuting involutions in $S_{n}$ and $A_{n}$. We consider the natural combinatorial problem of enumerating symmetric difference-closed sets consisting of subsets of sets consisting of pairwise disjoint $2$-subsets of $[n]$, and the problem of enumerating symmetric difference-closed sets consisting of elements which correspond to commuting involutions in $A_{n}$. We prove explicit combinatorial formulas for symmetric difference-closed sets of these forms, and we prove a number of conjectured properties related to such sets which had previously been discovered experimentally using the On-Line Encyclopedia of Integer Sequences.
Document type :
Journal articles
Complete list of metadatas

https://hal.archives-ouvertes.fr/hal-01345066
Contributor : John Campbell <>
Submitted on : Saturday, March 18, 2017 - 12:01:10 AM
Last modification on : Thursday, July 4, 2019 - 3:08:01 PM

File

CampbellDMTCSfinal.pdf
Files produced by the author(s)

Identifiers

  • HAL Id : hal-01345066, version 4

Collections

Citation

John Campbell. A class of symmetric difference-closed sets related to commuting involutions. Discrete Mathematics and Theoretical Computer Science, DMTCS, 2017, Vol 19 no. 1. ⟨hal-01345066v4⟩

Share

Metrics

Record views

235

Files downloads

1983