The Number of Permutations With A Prescribed Number of 132 and 123 Patterns

In this paper, we characterize and enumerate pattern-avoiding permutations composed of only 3-cycles. In particular, we answer the question for the six patterns of length 3. We find that the number of permutations composed of $n$ 3-cycles that avoid the pattern 231 (equivalently 312) is given by $3^{n-1}$, while the generating function for the number of those that avoid the pattern 132 (equivalently 213) is given by a formula involving the generating functions for the well-kn...

In this paper, we give a formula for the number of permutations that avoid the split patterns $3|12$ and $23|1$ with respect to a position $r$. Such permutations count the number of Schubert varieties for which the projection map from the flag variety to a Grassmannian induces a fiber bundle structure. We also study the corresponding bivariate generating function and show how it is related to modified Bessel functions.

We prove that the total number $S_{n,132}(q)$ of copies of the pattern $q$ in all 132-avoiding permutations of length $n$ is the same for $q=231$, $q=312$, or $q=213$. We provide a combinatorial proof for this unexpected threefold symmetry. We then significantly generalize this result to show an exponential number of different pairs of patterns $q$ and $q'$ of length $k$ for which $S_{n,132}(q)=S_{n,132}(q')$ and the equality is non-trivial.

In this paper, we compute the distributions of the statistic number of crossings over permutations avoiding one of the pairs $\{321,231\}$, $\{123,132\}$ and $\{123,213\}$. The obtained results are new combinatorial interpretations of two known triangles in terms of restricted permutations statistic. For other pairs of patterns of length three, we find relationships between the polynomial distributions of the crossings over permutations that avoid the pairs containing the pat...

We study permutations $p$ such that both $p$ and $p^2$ avoid a given pattern $q$. We obtain a generating function for the case of $q=312$ (equivalently, $q=231$), we prove that if $q$ is monotone increasing, then above a certain length, there are no such permutations, and we prove an upper bound for $q=321$. We also present some intriguing questions in the case of $q=132$.

Babson and Steingr\'{\i}msson introduced generalized permutation patterns that allow the requirement that two adjacent letters in a pattern must be adjacent in the permutation. Claesson presented a complete solution for the number of permutations avoiding any single pattern of type $(1,2)$ or $(2,1)$. For eight of these twelve patterns the answer is given by the Bell numbers. For the remaining four the answer is given by the Catalan numbers. With respect to being equidistribu...

Recently, Babson and Steingrimsson (see \cite{BS}) introduced generalized permutations patterns that allow the requirement that two adjacent letters in a pattern must be adjacent in the permutation. In this paper we study the generating functions for the number of permutations on $n$ letters avoiding a generalized pattern $ab\mn c$ where $(a,b,c)\in S_3$, and containing a prescribed number of occurrences of generalized pattern $cd\mn e$ where $(c,d,e)\in S_3$. As a conseque...

We study generating functions for the number of involutions in $S_n$ avoiding (or containing once) 132, and avoiding (or containing once) an arbitrary permutation $\tau$ on $k$ letters. In several interesting cases the generating function depends only on $k$ and is expressed via Chebyshev polynomials of the second kind. In particular, we establish that involutions avoiding both 132 and $12... k$ have the same enumerative formula according to the length than involutions avoidi...

Given a permutation $\sg = \sg_1 \ldots \sg_n$ in the symmetric group $S_n$, we say that $\sg_i$ matches the marked mesh pattern $MMP(a,b,c,d)$ in $\sg$ if there are at least $a$ points to the right of $\sg_i$ in $\sg$ which are greater than $\sg_i$, at least $b$ points to the left of $\sg_i$ in $\sg$ which are greater than $\sg_i$, at least $c$ points to the left of $\sg_i$ in $\sg$ which are smaller than $\sg_i$, and at least $d$ points to the right of $\sg_i$ in $\sg$ whic...

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