arXiv subjects
Publications and source records attributed to T. Mansour.
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.
In this paper, we find explicit formulas or generating functions for the cardinalities of the sets $S_n(T,τ)$ of all permutations in $S_n$ that avoid a pattern $τ\in S_k$ and a set $T$, $|T|\geq 2$, of patterns from $S_3$. The main body of the paper is divided into three sections corresponding to the cases $|T|=2,3$ and $|T|\geq 4$. As an example, in the fifth section, we obtain the complete classification of all cardinalities of the sets $S_n(T,τ)$ for $k=4$.
Let f_n^r(k) be the number of 132-avoiding permutations on n letters that contain exactly r occurrences of 12... k, and let F_r(x;k) and F(x,y;k) be the generating functions defined by $F_r(x;k)=\sum_{n\gs0} f_n^r(k)x^n$ and $F(x,y;k)=\sum_{r\gs0}F_r(x;k)y^r$. We find an explcit expression for F(x,y;k) in the form of a continued fraction. This allows us to express F_r(x;k) for $1\ls r\ls k$ via Chebyshev polynomials of the second kind.
Let T_k^m={σ\in S_k | σ_1=m}. We prove that the number of permutations which avoid all patterns in T_k^m equals (k-2)!(k-1)^{n+1-k} for k <= n. We then prove that for any τin T_k^1 (or any τin T_k^k), the number of permutations which avoid all patterns in T_k^1 (or in T_k^k) except for τand contain τexactly once equals (n+1-k)(k-1)^{n-k} for k <= n. Finally, for any τin T_k^m, 2 <= m <= k-1, this number equals (k-1)^{n-k} for k <= n. These results generalize recent results due to Robertson concerning permutations avoiding 123-pattern and containing 132-pattern exactly once.
In this paper we calculate the cardinality of the set S_n(T,tau) of all permutations in S_n that avoid one pattern from S_4 and a nonempty set of patterns from S_3.