Restricted 132-avoiding permutations

Annals of Combinatorics, 2003

A permutation is said to be alternating if it starts with rise and then descents and rises come in turn. In this paper we study the generating function for the number of alternating permutations on n letters that avoid or contain exactly once 132 and also avoid or contain exactly once an arbitrary pattern on k letters. In several interesting cases the generating function depends only on k and is expressed via Chebyshev polynomials of the second kind.

2002

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

Advances in Applied Mathematics, 2002

We study the generating function for the number of permutations on n letters containing exactly $r\gs0$ occurences of 132. It is shown that finding this function for a given r amounts to a routine check of all permutations in $S_{2r}$.

2002

A permutation $\pi$ is said to be {\em Dumont permutations of the first kind} if each even integer in $\pi$ must be followed by a smaller integer, and each odd integer is either followed by a larger integer or is the last element of $\pi$ (see, for example, \cite{Z}). In \cite{D} Dumont showed that certain classes of permutations on $n$ letters are counted by the Genocchi numbers. In particular, Dumont showed that the $(n+1)$st Genocchi number is the number of Dummont permutations of the first kind on $2n$ letters. In this paper we study the number of Dumont permutations of the first kind on $n$ letters avoiding the pattern 132 and avoiding (or containing exactly once) an arbitrary pattern on $k$ letters. In several interesting cases the generating function depends only on $k$.

Advances in Applied Mathematics, 2002

We study the generating function for the number of permutations on n letters containing exactly r 0 occurrences of 132. It is shown that finding this function for a given r amounts to a routine check of all permutations in S 2r .

Pure Mathematics and Applications

We enumerate permutations avoiding 1324, 2143, and a third 4-letter pattern τ a step toward the goal of enumerating avoiders for all triples of 4-letter patterns. The enumeration is already known for all but five patterns τ, which are treated in this paper.

2014

We give an improved algorithm for counting the number of $1324$-avoiding permutations, resulting in 5 further terms of the generating function. We analyse the known coefficients and find compelling evidence that unlike other classical length-4 pattern-avoiding permutations, the generating function in this case does not have an algebraic singularity. Rather, the number of 1324-avoiding permutations of length $n$ behaves as $$B\cdot \mu^n \cdot \mu_1^{n^{\sigma}} \cdot n^g.$$ We estimate $\mu=11.60 \pm 0.01,$ $\sigma=1/2,$ $\mu_1 = 0.0398 \pm 0.0010,$ $g = -1.1 \pm 0.2$ and $B =9.5 \pm 1.0.$

2002

"We study generating functions for the number of permutations in $S_n$ subject to two restrictions. One of the restrictions belongs to $S_3$, while the other belongs to $S_k$. It turns out that in a large variety of cases the answer can be expressed via Chebyshev polynomials of the second k"

Discrete Applied Mathematics, 2006

Several authors have examined connections among 132-avoiding permutations, continued fractions, and Chebyshev polynomials of the second kind. In this paper we find analogues for some of these results for permutations π avoiding 132 and 1223 (there is no occurrence π i < π j < π j+1 such that 1 ≤ i ≤ j − 2) and provide a combinatorial interpretation for such permutations in terms of lattice paths. Using tools developed to prove these analogues, we give enumerations and generating functions for permutations which avoid both 132 and 1223, and certain additional patterns. We also give generating functions for permutations avoiding 132 and 1223 and containing certain additional patterns exactly once. In all cases we express these generating functions in terms of Chebyshev polynomials of the second kind.

Advances in Applied Mathematics, 2006

We study generating functions for the number of n-long k-ary words that avoid both 132 and an arbitrary-ary pattern. In several interesting cases the generating function depends only on and is expressed via Chebyshev polynomials of the second kind and continued fractions.