We study relationships between permutation statistics and pattern-functions, counting the number of times particular patterns occur in a permutation. This allows us to write several familiar statistics as linear combinations of pattern counts, both in terms of a permutation and in terms of its image under the fundamental bijection. We use these enumerations to resolve the question of characterizing so-called "shallow" permutations, whose depth (equivalently, disarray/displacement) is minimal with respect to length and reflection length. We present this characterization in several ways, including vincular patterns, mesh patterns, and a new object that we call "arrow patterns." Furthermore, we specialize to characterizing and enumerating shallow involutions and shallow cycles, encountering the Motzkin and large Schröder numbers, respectively.
@article{10_37236_10858,
author = {Yosef Berman and Bridget Eileen Tenner},
title = {Pattern-functions, statistics, and shallow permutations},
journal = {The electronic journal of combinatorics},
year = {2022},
volume = {29},
number = {4},
doi = {10.37236/10858},
zbl = {1506.05001},
url = {http://geodesic.mathdoc.fr/articles/10.37236/10858/}
}
TY - JOUR
AU - Yosef Berman
AU - Bridget Eileen Tenner
TI - Pattern-functions, statistics, and shallow permutations
JO - The electronic journal of combinatorics
PY - 2022
VL - 29
IS - 4
UR - http://geodesic.mathdoc.fr/articles/10.37236/10858/
DO - 10.37236/10858
ID - 10_37236_10858
ER -
