Brändén's (p,q)-Eulerian polynomials, André permutations and continued fractions

Restricted Permutations, Continued Fractions, and Chebyshev Polynomials

Let fr n(k) be the number of 132-avoiding permutations on n letters that contain exactly r occurrences of 12 . . . k, and let Fr(x; k) and F (x, y; k) be the generating functions defined by Fr(x; k) = P n>0 f r n(k)x n and F (x, y; k) = P r>0 Fr(x; k)y r. We find an explicit expression for F (x, y; k) in the form of a continued fraction. This allows us to express Fr(x; k) for 1 6 r 6 k via Cheb...

0 Restricted Permutations , Continued Fractions , and Chebyshev Polynomials

Restricted Motzkin permutations, Motzkin paths, continued fractions, and Chebyshev polynomials

We say that a permutation is a Motzkin permutation if it avoids 132 and there do not exist a <b such that a < b < b+1. We study the distribution of several statistics in Motzkin permutations, including the length of the longest increasing and decreasing subsequences and the number of rises and descents. We also enumerate Motzkin permutations with additional restrictions, and study the distribut...

Elliptic Functions, Continued Fractions and Doubled Permutations

The question of the possible combinatorial significance of the integer coefficients appearing in the Taylor expansions of the Jacobian elliptic functions was first raised by Schiitzenberger. Indeed, these integers generalize the Euler numbers, i.e. the coefficients of tan(z), sec(z) whose relation to permutations has been known since the work of Andri: [l]. The first combinatorial interpretatio...

0 Restricted Permutations , Continued Fractions , and Chebyshev