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Coloured Permutations Containing and Avoiding Certain Patterns
Toufik Mansour
Department of Mathematics, Chalmers University of Technology, 412 96 Göteborg, Sweden
Annals of Combinatorics 7 (3) p.349-355 September, 2003
AMS Subject Classification: 05A05, 05A15
Let be the set of all coloured permutations on the symbols 1, 2, ..., n with colours 1, 2, ..., r, which is the analogous of the symmetric group when r = 1, and the hyperoctahedral group when r = 2. Let I {1, 2, ..., r} be a subset of d colours; we define to be the set of all coloured permutations such that where c I. We prove that the number of -avoiding coloured permutations in equals for n k where hj = (r-d)j+(k-1)d. We then prove that for any (or any ), the number of coloured permutations in which avoid all patterns in (or in ) except for and contain exactly once equals for n k. Finally, for any , 2mk-1, this number equals for n k+1. These results generalize recent results due to Mansour, Mansour and West, and Simion.
Keywords: alternating permutations, restricted permutations, generating functions, Chebyshev polynomials


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