We study the relationship between two notions of pattern avoidance for involutions in the symmetric group and their restriction to fixed-point-free involutions. The first is classical, while the second appears in the geometry of certain spherical varieties and generalizes the notion of pattern avoidance for perfect matchings studied by Jelínek. The first notion can always be expressed in terms of the second, and we give an effective algorithm to do so. We also give partial results characterizing the families of involutions where the converse holds. As a consequence, we prove two conjectures of McGovern characterizing (rational) smoothness of certain varieties. We also give new enumerative results, and conclude by proposing several lines of inquiry that extend our current work.
@article{10_37236_10155,
author = {Jonathan J. Fang and Zachary Hamaker and Justin M. Troyka},
title = {On pattern avoidance in matchings and involutions},
journal = {The electronic journal of combinatorics},
year = {2022},
volume = {29},
number = {1},
doi = {10.37236/10155},
zbl = {1486.05004},
url = {http://geodesic.mathdoc.fr/articles/10.37236/10155/}
}
TY - JOUR
AU - Jonathan J. Fang
AU - Zachary Hamaker
AU - Justin M. Troyka
TI - On pattern avoidance in matchings and involutions
JO - The electronic journal of combinatorics
PY - 2022
VL - 29
IS - 1
UR - http://geodesic.mathdoc.fr/articles/10.37236/10155/
DO - 10.37236/10155
ID - 10_37236_10155
ER -
%0 Journal Article
%A Jonathan J. Fang
%A Zachary Hamaker
%A Justin M. Troyka
%T On pattern avoidance in matchings and involutions
%J The electronic journal of combinatorics
%D 2022
%V 29
%N 1
%U http://geodesic.mathdoc.fr/articles/10.37236/10155/
%R 10.37236/10155
%F 10_37236_10155
Jonathan J. Fang; Zachary Hamaker; Justin M. Troyka. On pattern avoidance in matchings and involutions. The electronic journal of combinatorics, Tome 29 (2022) no. 1. doi: 10.37236/10155
