Mathematics and Statistics
Restricted permutation, restricted involution, pattern-avoiding permutation, pattern-avoiding involution, forbidden subsequence, Chebyshev polynomial, colored permutation
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.
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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