Forbidden subsequences and Chebyshev polynomials. (English) Zbl 0932.05007

Summary: In J. West [Discrete Math. 157, No. 1-3, 363-374 (1996; Zbl 0877.05002)] it was shown using transfer matrices that the number \(|S_n(123;3214)|\) of permutations avoiding the patterns 123 and 3214 is the Fibonacci number \(F_{2n}\) (as are also \(|S_n(213; 1234)|\) and \(|S_n(213;4123)|)\). We now find the transfer matrix for \(\bigl|S_n(123;r,r-1, \dots, 2,1,r+1) \bigr|\), \(\bigl|S_n(213;1,2, \dots,r,r+1) \bigr|\), and \(\bigl |S_n(213; r+1,1,2, \dots,r) \bigr |\), determine its characteristic polynomial in terms of the Chebyshev polynomials, and go on to determine the generating function as a quotient of modified Chebyshev polynomials. This leads to an asymptotic result for each \(r\) which collapses to the exact results \(2^n\) when \(r=2\) and \(F_{2n}\) when \(r=3\) and to the Catalan number \(c_n\) as \(r\to\infty\). We observe that our generating function also enumerates certain lattice paths, plane trees, and directed animals, giving hope that these areas of combinatorics can be applied to enumerating permutations with excluded subsequences.


05A15 Exact enumeration problems, generating functions
05A05 Permutations, words, matrices


Zbl 0877.05002
Full Text: DOI


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