Related papers: Quadrant marked mesh patterns in 123-avoiding perm…
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…
Given a permutation $\sg = \sg_1...\sg_n$ 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…
This paper is a continuation of the systematic study of the distributions of quadrant marked mesh patterns initiated in [6]. Given a permutation $\sg = \sg_1 ... \sg_n$ in the symmetric group $S_n$, we say that $\sg_i$ matches the quadrant…
Classical pattern avoidance and occurrence are well studied in the symmetric group $\mathcal{S}_{n}$. In this paper, we provide explicit recurrence relations to the generating functions counting the number of classical pattern occurrence in…
In this paper, we study the distribution of consecutive patterns in the set of 123-avoiding permutations and the set of 132-avoiding permutations, that is, in $\mathcal{S}_n(123)$ and $\mathcal{S}_n(132)$. We first study the distribution of…
We study pattern avoidance by combinatorial objects other than permutations, namely by ordered partitions of an integer and by permutations of a multiset. In the former case we determine the generating function explicitly, for integer…
A notable problem within permutation patterns that has attracted considerable attention in literature since 1973 is the search for a bijective proof demonstrating that 123-avoiding and 132-avoiding permutations are equinumerous, both…
Dokos et. al. studied the distribution of two statistics over permutations $\mathfrak{S}_n$ of $\{1,2,\dots, n\}$ that avoid one or more length three patterns. A permutation $\sigma\in\mathfrak{S}_n$ contains a pattern…
We consider a random permutation drawn from the set of 132-avoiding permutations of length $n$ and show that the number of occurrences of another pattern $\sigma$ has a limit distribution, after scaling by $n^{\lambda(\sigma)/2}$ where…
Extending the notion of pattern avoidance in permutations, we study matchings and set partitions whose arc diagram representation avoids a given configuration of three arcs. These configurations, which generalize 3-crossings and 3-nestings,…
We present a new approach to the problem of enumerating permutations of length n that avoid a fixed consecutive pattern of length m. We use this idea to give explicit upper and lower bounds on the number of permutations avoiding a pattern…
We consider a random permutation drawn from the set of permutations of length $n$ that avoid some given set of patterns of length 3. We show that the number of occurrences of another pattern $\sigma$ has a limit distribution, after suitable…
A set of permutations is called sign-balanced if the set contains the same number of even permutations as odd permutations. Let $S_n(\sigma_1, \sigma_2, \ldots, \sigma_r)$ be the set of permutations in the symmetric group $S_n$ which avoids…
This paper is continuation of the systematic study of distribution of quadrant marked mesh patterns. We study quadrant marked mesh patterns on up-down and down-up permutations, also known as alternating and reverse alternating permutations,…
The simple permutations in two permutation classes --- the 321-avoiding permutations and the skew-merged permutations --- are enumerated using a uniform method. In both cases, these enumerations were known implicitly, by working backwards…
This paper is continuation of the systematic study of distribution of quadrant marked mesh patterns initiated in "S. Kitaev and J. Remmel, Quadrant marked mesh patterns, J. Integer Sequences 12, Issue 4 (2012), Article 12.4.7.". We study…
For a set of permutations $S\subseteq S_n$, consider the quasisymmetric generating function $$Q(S): = \sum_{w\in S}F_{n, \mathrm{Des}(w)},$$ where $\mathrm{Des}(w) := \{i\mid w(i)> w(i+1)\}$ is the descent set of $w$ and $F_{n,…
We extend the concept of pattern avoidance in permutations on a totally ordered set to pattern avoidance in permutations on partially ordered sets. The number of permutations on $P$ that avoid the pattern $\pi$ is denoted $Av_P(\pi)$. We…
We initiate the study of limit shapes for random permutations avoiding a given pattern. Specifically, for patterns of length 3, we obtain delicate results on the asymptotics of distributions of positions of numbers in the permutations. We…
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…