Restricted permutations, continued fractions, and Chebyshev polynomials

Abstract

We study generating functions for the number of permutations in S n subject to set of restrictions. One of the restrictions belongs to S 3 , while the others to S k . It turns out that in a large variety of cases the answer can be expressed via continued fractions, and Chebyshev polynomials of the second kind.

Pattern‐avoiding permutations and Brownian excursion part I: Shapes and fluctuations

“…Combining the limits for (25) and the middle term in (26) and picking d arbitrarily close to 1 we find that lim n→∞ n 1/2 n k= cn…”

Restricted Stirling Permutations

“…As a final example, let us count the occurrences of the pattern 122 in Q n (213) (motivated by the study of counting occurrences of the pattern 12 · · · k in a 132-avoiding permutation, for example see [20,23]). To do so, we denote the number occurrences of the pattern 122 in σ by 122(σ).…”

“…Since the reversal operation (σ 1 σ 2 · · · σ 2n → σ 2n · · · σ 2 σ 1 ) preserves the set of Stirling permutations, we only need to consider the three cases, τ = 123, τ = 132, τ = 213. For the latter case, we use the block decompositions technique (for instance, see [22]), while for the former two cases, we use the kernel method (for instance, see [12]). x n p plat (σ) q des (σ) r asc (σ) .…”

Locally Convex Words and Permutations

“…We will now give a method of determining F (a, b) given some fixed p and k using the Transfer Matrix method, which is described in [12], Section 4.7. For a description and examples of the transfer matrix method as used for enumerating permutations, see [7], [8], and [13].…”

“…If L(π) is k-convex, we say that π left descends, and similarly define what it means for a permutation to right descend. 8 Analogously, let W = W 1 W 2 . .…”