The fundamental bijection is a bijection $\theta:\mathcal{S}_n\to\mathcal{S}_n$ in which one uses the standard cycle form of one permutation to obtain another permutation in one-line form. In this paper, we enumerate the set of permutations $\pi\in\mathcal{S}_n$ that avoids a pattern $\sigma\in\mathcal{S}_3$, whose image $\theta(\pi)$ also avoids $\sigma$. We additionally consider what happens under repeated iterations of $\theta$; in particular, we enumerate permutations $\pi\in\mathcal{S}_n$ that have the property that $\pi$ and its first $k$ iterations under $\theta$ all avoid a pattern $\sigma$. Finally, we consider permutations with the property that $\pi=\theta^2(\pi)$ that avoid a given pattern $\sigma$, and end the paper with some directions for future study.
@article{10_37236_13748,
author = {Kassie Archer and Robert P. Laudone},
title = {Pattern avoidance and the fundamental bijection},
journal = {The electronic journal of combinatorics},
year = {2025},
volume = {32},
number = {4},
doi = {10.37236/13748},
zbl = {8120105},
url = {http://geodesic.mathdoc.fr/articles/10.37236/13748/}
}
TY - JOUR
AU - Kassie Archer
AU - Robert P. Laudone
TI - Pattern avoidance and the fundamental bijection
JO - The electronic journal of combinatorics
PY - 2025
VL - 32
IS - 4
UR - http://geodesic.mathdoc.fr/articles/10.37236/13748/
DO - 10.37236/13748
ID - 10_37236_13748
ER -
%0 Journal Article
%A Kassie Archer
%A Robert P. Laudone
%T Pattern avoidance and the fundamental bijection
%J The electronic journal of combinatorics
%D 2025
%V 32
%N 4
%U http://geodesic.mathdoc.fr/articles/10.37236/13748/
%R 10.37236/13748
%F 10_37236_13748
Kassie Archer; Robert P. Laudone. Pattern avoidance and the fundamental bijection. The electronic journal of combinatorics, Tome 32 (2025) no. 4. doi: 10.37236/13748
