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May 2013 Permutation Pattern Avoidance and the Catalan Triangle
Derek Desantis, Rebecca Field, Wesley Hough, Brant Jones, Rebecca Meissen, Jacob Ziefle
Missouri J. Math. Sci. 25(1): 50-60 (May 2013). DOI: 10.35834/mjms/1369746397

Abstract

In the study of various objects indexed by permutations, a natural notion of minimal excluded structure, now known as a permutation pattern, has emerged and found diverse applications. One of the earliest results from the study of permutation pattern avoidance in enumerative combinatorics is that the Catalan numbers $c_n$ count the permutations of size $n$ that avoid any fixed pattern of size three. We refine this result by enumerating the permutations that avoid a given pattern of size three and have a given letter in the first position of their one-line notation. Since there are two parameters, we obtain triangles of numbers rather than sequences. Our main result is that there are two essentially different triangles for any of the patterns of size three, and each of these triangles generalizes the Catalan sequence in a natural way. All of our proofs are bijective, and relate the permutations being counted to recursive formulas for the triangles.

Citation

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Derek Desantis. Rebecca Field. Wesley Hough. Brant Jones. Rebecca Meissen. Jacob Ziefle. "Permutation Pattern Avoidance and the Catalan Triangle." Missouri J. Math. Sci. 25 (1) 50 - 60, May 2013. https://doi.org/10.35834/mjms/1369746397

Information

Published: May 2013
First available in Project Euclid: 28 May 2013

zbMATH: 1268.05003
MathSciNet: MR3087688
Digital Object Identifier: 10.35834/mjms/1369746397

Subjects:
Primary: 05A05
Secondary: 05A15

Rights: Copyright © 2013 Central Missouri State University, Department of Mathematics and Computer Science

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Vol.25 • No. 1 • May 2013
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