Layered Restrictions and Chebyshev Polynomials
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A permutation is called layered if it consists of the disjoint union of substrings (layers) so that the entries decrease within each layer, and increase between the layers. We find the generating function for the number of permutations on $n$ letters avoiding $(1,2,3)$ and a layered permutation on $k$ letters. In the most interesting case of two layers, the generating function depends only on $k$ and is expressed via Chebyshev polynomials of the second kind.
8 pages
Permutations, words, matrices, 42C05, permutations, 30B70, Exact enumeration problems, generating functions, enumeration, 05A05; 05A15; 30B70; 42C05, pattern avoidance, 05A05, generating functions, FOS: Mathematics, Mathematics - Combinatorics, Combinatorics (math.CO), Chebyshev polynomials, 05A15, Orthogonal functions and polynomials, general theory of nontrigonometric harmonic analysis, Continued fractions; complex-analytic aspects
Permutations, words, matrices, 42C05, permutations, 30B70, Exact enumeration problems, generating functions, enumeration, 05A05; 05A15; 30B70; 42C05, pattern avoidance, 05A05, generating functions, FOS: Mathematics, Mathematics - Combinatorics, Combinatorics (math.CO), Chebyshev polynomials, 05A15, Orthogonal functions and polynomials, general theory of nontrigonometric harmonic analysis, Continued fractions; complex-analytic aspects
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