Related papers: Pattern avoidance in partial permutations

We show that for any permutation $\pi$ there exists an integer $k_{\pi}$ such that every permutation avoiding $\pi$ as a pattern is a product of at most $k_{\pi}$ separable permutations. In other words, every strict class $\mathcal C$ of…

We give some new Wilf equivalences for signed patterns which allow the complete classification of signed patterns of lengths three and four. The problem is considered for pattern avoidance by general as well as involutive signed…

A sequence $f\colon\{1,\dots,n\}\to\mathbb{R}$ contains a permutation $\pi$ of length $k$ if there exist $i_1<\dots<i_k$ such that, for all $x,y$, $f(i_x)<f(i_y)$ if and only if $\pi(x)<\pi(y)$; otherwise, $f$ is said to be $\pi$-free. In…

We consider the number of passes a permutation needs to take through a stack if we only pop the appropriate output values and start over with the remaining entries in their original order. We define a permutation $\pi$ to be $k$-pass…

A permutation $\pi \in S_n$ is said to {\it avoid} a permutation $\sigma \in S_k$ whenever $\pi$ contains no subsequence with all of the same pairwise comparisons as $\sigma$. For any set $R$ of permutations, we write $S_n(R)$ to denote the…

In the set of all patterns in $S_n$, it is clear that each k-pattern occurs equally often. If we instead restrict to the class of permutations avoiding a specific pattern, the situation quickly becomes more interesting. Mikl\'os B\'ona…

We consider avoidance of permutation patterns with designated gap sizes between pairs of consecutive letters. We call the patterns having such constraints distant patterns (DPs) and we show their relation to other pattern notions…

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,…

For a set of permutations (patterns) $\Pi$ in $S_k$, consider the set of all permutations in $S_n$ that avoid all patterns in $\Pi$. An important problem in current algebraic combinatorics is to find pattern sets $\Pi$ such that the…

The extension of pattern avoidance from ordinary permutations to those on multisets gave birth to several interesting enumerative results. We study permutations on regular multisets, i.e., multisets in which each element occurs the same…

We study questions of even-Wilf-equivalence, the analogue of Wilf-equivalence when attention is restricted to pattern avoidance by permutations in the alternating group. Although some Wilf-equivalence results break when considering…

Permutons are probability measures on the unit square with uniform marginals that provide a natural way to describe limits of permutations. We are interested in the permuton limits for permutations sampled uniformly from certain…

Pattern avoidance in the symmetric group $S_n$ has provided a number of useful connections between seemingly unrelated problems from stack-sorting to Schubert varieties. Recent work has generalized these results to $S_n\wr C_c$, the objects…

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.…

We investigate the notion of almost avoiding a permutation: $\pi$ almost avoids $\beta$ if one can remove a single entry from $\pi$ to obtain a $\beta$-avoiding permutation.

A permutation is called Grassmannian if it has at most one descent. In this paper, we investigate pattern avoidance and parity restrictions for such permutations. As our main result, we derive formulas for the enumeration of Grassmannian…

Given a permutation $\pi$, let $\text{Av}_n(\pi)$ be the number of permutations of length $n$ that avoid $\pi$ as a subpermutation. The celebrated resolution of the Stanley-Wilf conjecture by Marcus and Tardos confirmed that the limit…

Problem 1.5.7 from Pitman's Saint-Flour lecture notes: Does there exist for each n a fragmentation process (\Pi_{n,k}, 1 \leq k \leq n) taking values in the space of partitions of {1,2,...,n} such that \Pi_{n,k} is distributed like the…

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…

$\textit{Partial words}$ are words that contain, in addition to letters, special symbols $\diamondsuit$ called $\textit{holes}$. Two partial words of $a=a_0 \dots a_n$ and $b=b_0 \dots b_n$ are $\textit{compatible}$ if for all $i$, $a_i =…