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A generating tree with a single label for permutations avoiding the vincular pattern 1−32−4

A generating tree with a single label for permutations avoiding the vincular pattern 1−32−4 AbstractWe continue the study of permutations avoiding the vincular pattern 1−32−4 by constructing a generating tree with a single label for these permutations. This construction finally provides a clearer explanation of why a certain recursive formula found by Callan actually counts these permutations, since this formula was originally obtained as a consequence of a very intricate bijection with a certain class of ordered rooted trees. This responds to a theoretical issue already raised by Duchi, Guerrini and Rinaldi. As a byproduct, we also obtain an algorithm to generate all these permutations and we refine their enumeration according to a simple statistic, which is the number of right-to-left maxima to the right of 1. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Pure Mathematics and Applications de Gruyter

A generating tree with a single label for permutations avoiding the vincular pattern 1−32−4

Cervetti, Matteo
Pure Mathematics and Applications , Volume 30 (1): 7 – Jun 1, 2022
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Publisher
de Gruyter
Copyright
© 2022 Matteo Cervetti, published by Sciendo
eISSN
1788-800X
DOI
10.2478/puma-2022-0009
Publisher site
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Abstract

AbstractWe continue the study of permutations avoiding the vincular pattern 1−32−4 by constructing a generating tree with a single label for these permutations. This construction finally provides a clearer explanation of why a certain recursive formula found by Callan actually counts these permutations, since this formula was originally obtained as a consequence of a very intricate bijection with a certain class of ordered rooted trees. This responds to a theoretical issue already raised by Duchi, Guerrini and Rinaldi. As a byproduct, we also obtain an algorithm to generate all these permutations and we refine their enumeration according to a simple statistic, which is the number of right-to-left maxima to the right of 1.

Journal

Pure Mathematics and Applications – de Gruyter

Published: Jun 1, 2022

Keywords: permutation; vincular pattern; generating tree; powered Catalan numbers; 05A15

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