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

dmtcs:3210 - Discrete Mathematics & Theoretical Computer Science, March 23, 2017, Vol 19 no. 1
A class of symmetric difference-closed sets related to commuting involutions

Authors: Campbell, John

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.

Source : oai:HAL:hal-01345066v4
Volume: Vol 19 no. 1
Section: Combinatorics
Published on: March 23, 2017
Submitted on: March 22, 2017
Keywords: symmetric difference-closed set,commuting involution,Klein four-group,permutation group,combinatorics of finite sets,[MATH.MATH-CO] Mathematics [math]/Combinatorics [math.CO]