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Title: Pattern-Functions, Statistics, and Shallow Permutations
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.  more » « less
Award ID(s):
2054436
NSF-PAR ID:
10405453
Author(s) / Creator(s):
;
Date Published:
Journal Name:
The Electronic Journal of Combinatorics
Volume:
29
Issue:
4
ISSN:
1077-8926
Format(s):
Medium: X
Sponsoring Org:
National Science Foundation
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