Department
Mathematics and Statistics
Journal Title
Discrete Mathematics
Publication Date
2007
First Page
1792
Last Page
1800
Publisher
Elsevier
File Name
030_Egge-Eric_RestrictedColoredPermuationsAndChebyshevPolynomials.pdf
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.
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's 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
Paid OA Option
Yes
Contributing Organization
Carleton College
Type
Article
Format
application/pdf
Language
English
DOI
10.1016/j.disc.2006.09.027
Recommended Citation
Egge, E. S. (2007). Restricted Colored Permutations and Chebyshev Polynomials. Discrete Mathematics, 307 (14), 1792-1800. Available at: https://doi.org/10.1016/j.disc.2006.09.027. Accessed via Faculty Work. Mathematics. Carleton Digital Commons. https://digitalcommons.carleton.edu/math_faculty/2
The definitive version is available at https://doi.org/10.1016/j.disc.2006.09.027
