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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...
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Published in: | The Electronic journal of combinatorics 2022-12, Vol.29 (4) |
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Main Authors: | , |
Format: | Article |
Language: | English |
Citations: | Items that cite this one |
Online Access: | Get full text |
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Summary: | 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. |
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ISSN: | 1077-8926 1077-8926 |
DOI: | 10.37236/10858 |