Restricted 132-Avoiding Permutations

Abstract

We study generating functions for the number of permutations on n letters avoiding 132 and an arbitrary permutation τ on k letters, or containing τ exactly once. In several interesting cases the generating function depends only on k and is expressed via Chebyshev polynomials of the second kind.

Restricted Dumont Permutations

“…So far, there has been a single paper on restricted Dumont permutations, namely Mansour [7]. Most of it is devoted to the study of 132-avoiding Dumont permutations of the first kind (as there are no 132-avoiding Dumont permutations of the second kind other than 21 ∈ D 2 2 ).…”

“…to find |D 1 2n (τ )| for τ = 2143, 3421, 4213 and |D 2 2n (τ )| for τ = 2143, 4132. Another is to combine the forbidden patterns of Section 3 with additional restrictions as in [7]. Yet another is to find the complete distribution for the number of occurrences of these patterns possibly combined with other restrictions, or to find equidistributed statistics on some of these restricted sets.…”

Restricted 132-Alternating Permutations and Chebyshev Polynomials

“…This problem was solved completely for τ 1 , τ 2 ∈ S 3 (see [SS]), for τ 1 ∈ S 3 and τ 2 ∈ S 4 (see [W]), and for τ 1 , τ 2 ∈ S 4 (see [Bo, Km] and references therein). Several recent papers [CW,MV1,Kr,MV2,MV3,MV4] deal with the case τ 1 ∈ S 3 , τ 2 ∈ S k for various pairs τ 1 , τ 2 . Another natural question is to study permutations avoiding τ 1 and containing τ 2 exactly t times.…”

“…Apparently, for the first time the relation between restricted permutations and Chebyshev polynomials was discovered by Chow and West in [CW], and later by Mansour and Vainshtein [MV1,MV2,MV3,MV4], Krattenthaler [Kr]. These results related to a rational function…”

Equivalence classes of permutations modulo descents and left-to-right maxima

“…Many classical sequences in combinatorics appear as the cardinality of pattern-avoiding permutation classes. A large number of these results were rstly obtained by West and Knuth [8,12,13,14,15,16] (see books of Kitaev [7] and Mansour [11]). …”