Type

Article

Keywords

Restricted permutation, restricted involution, pattern-avoiding permutation, pattern-avoiding involution, forbidden subsequence, Chebyshev polynomial, colored permutation

Abstract

Several authors have examined connections between restricted permutations and Chebyshev polynomials of the second kind. In this paper we prove analogues of these results for colored permutations. First we define a distinguished set of length two and length three patterns, which contains only 312 when just one color is used. Then we give a recursive procedure for computing the generating function for the colored permutations which avoid this distinguished set and any set of additional patterns, which we use to find a new set of signed permutations counted by the Catalan numbers and a new set of signed permutations counted by the large Schr¨oder numbers. We go on to use this result to compute the generating functions for colored permutations which avoid our distinguished set and any layered permutation with three or fewer layers. We express these generating functions in terms of Chebyshev polynomials of the second kind and we show that they are special cases of generating functions for involutions which avoid 3412 and a layered permutation.

Language

English

Department(s)

Mathematics and Statistics

Journal or Book Title

Discrete Mathematics

Publication Year

2007

DOI

10.1016/j.disc.2006.09.027

Publisher

Elsevier

Rights Management

Carleton College does not own the copyright to this work and the work is available through the Carleton College Library following the original publisher policies regarding self-archiving. For more information on the copyright status of this work, refer to the current copyright holder.

RoMEO Color

Green

Preprint Archiving

Yes (with link to journal home page)

Postprint Archiving

Yes

Publisher PDF Archiving

No

Contributing Organization

Carleton College

Format

application/pdf

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Included in

Mathematics Commons

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