Related papers: 321-polygon-avoiding permutations and Chebyshev po…

We say that a permutation $\pi$ is a Motzkin permutation if it avoids 132 and there do not exist $a<b$ such that $\pi_a<\pi_b<\pi_{b+1}$. We study the distribution of several statistics in Motzkin permutations, including the length of the…

Shallow permutations were defined in 1977 to be those that satisfy the lower bound of the Diaconis-Graham inequality. Recently, there has been renewed interest in these permutations. In particular, Berman and Tenner showed they satisfy…

Motivated by the concept of partial words, we introduce an analogous concept of partial permutations. A partial permutation of length n with k holes is a sequence of symbols $\pi = \pi_1\pi_2 ... \pi_n$ in which each of the symbols from the…

We define a map between the set of permutations that avoid either the four patterns $3214,3241,4213,4231$ or $3124,3142,4123,4132$, and the set of Dyck prefixes. This map, when restricted to either of the two classes, turns out to be a…

Babson and Steingr\`imsson introduced generalized permutation patterns that allow the requirement that two adjacent letters in a pattern must be adjacent in the permutation. Subsequently, Claesson presented a complete solution for the…

Chen and collaborators give a recursively defined bijection from 021-avoiding ascent sequences to 021-avoiding (aka 132-avoiding) permutations. Here we give an algorithmic bijection from 021-avoiding ascent sequences to Dyck paths. Our…

The diagram of a 132-avoiding permutation can easily be characterized: it is simply the diagram of a partition. Based on this fact, we present a new bijection between 132-avoiding and 321-avoiding permutations. We will show that this…

Two well-known polytopes whose vertices are indexed by permutations in the symmetric group $\mathfrak{S}_n$ are the permutohedron $P_n$ and the Birkhoff polytope $B_n$. We consider polytopes $P_n(\Pi)$ and $B_n(\Pi)$, whose vertices…

Let I_n(\pi) denote the number of involutions in the symmetric group S_n which avoid the permutation \pi. We say that two permutations \alpha,\beta\in\S{j} may be exchanged if for every n, k, and ordering \tau of j+1,...,k, we have…

We study the longest increasing subsequence problem for random permutations avoiding the pattern $312$ and another pattern $\tau$ under the uniform probability distribution. We determine the exact and asymptotic formulas for the average…

Starting from some considerations we make about the relations between certain difference statistics and the classical permutation statistics we study permutations whose inversion number and excedance difference coincide. It turns out that…

Partially ordered patterns (POPs) play an important role in the study of permutation patterns, providing a convenient framework for describing large families of classical patterns. The problem of enumerating permutations that avoid POPs has…

A bijection between $(31245,32145,31254,32154)$-avoiding permutations and $(31425,32415,31524,32514)$-avoiding permutations is constructed, which preserves five classical set-valued statistics. Combining with two codings of permutations due…

We use catalytic variables to derive generating functions for the permutation classes $Av(\textbf{4123},\textbf{1324})$, $Av(\textbf{4123},\textbf{1243})$, and $Av(\textbf{4123},\textbf{1342})$. Each generating function is algebraic of…

Motivated by Kitaev and Zhang's recent work on non-overlapping ascents in stack-sortable permutations and Dumont's permutation interpretation of the Jacobi elliptic functions, we investigate some parity statistics on restricted…

We extend earlier work of the same author to enumerate alternating permutations avoiding the permutation pattern 2143. We use a generating tree approach to construct a recursive bijection between the set A_{2n}(2143) of alternating…

The discrete cosine transforms of types V--VIII are generalized to the antisymmetric and symmetric multivariate discrete cosine transforms. Four families of discretely and continuously orthogonal Chebyshev-like polynomials corresponding to…

In this paper we study pattern avoidance for affine permutations. In particular, we show that for a given pattern p, there are only finitely many affine permutations in $\widetilde{S}_n$ that avoid p if and only if p avoids the pattern 321.…

In 1990 West conjectured that there are $2(3n)!/((n+1)!(2n+1)!)$ two-stack sortable permutations on $n$ letters. This conjecture was proved analytically by Zeilberger in 1992. Later, Dulucq, Gire, and Guibert gave a combinatorial proof of…

We describe the distribution of the number and location of the fixed points of permu- tations that avoid the pattern 321 via a bijection with rooted plane trees on n + 1 vertices. Using the local limit theorem for Galton-Watson trees, we…