Matchings and Partial Patterns

Abstract

A matching of size $2n$ is a partition of the set $[2n]=\{1,2,\dotsc,2n\}$ into $n$ disjoint pairs. A matching may be identified with a canonical sequence, which is a sequence of integers in which each integer $i\in[n]$ occurs exactly twice, and the first occurrence of $i$ precedes the first occurrence of $i+1$. A partial pattern with $k$ symbols is a sequence of integers from the set $[k]$, in which each $i\in[k]$ appears at least once and at most twice, and the first occurrence of $i$ always precedes the first occurrence of $i+1$. Given a partial pattern $\sigma$ and a matching $\mu$, we say that $\mu$ avoids $\sigma$ if the canonical sequence of $\mu$ has no subsequence order-isomorphic to $\sigma$. Two partial patterns $\tau$ and $\sigma$ are equivalent if there is a size-preserving bijection between $\tau$-avoiding and $\sigma$-avoiding matchings. In this paper, we describe several families of equivalent pairs of patterns, most of them involving infinitely many equivalent pairs. We verify by computer enumeration that these families contain all the equivalences among patterns of length at most six. Many of our arguments exploit a close connection between partial patterns and fillings of diagrams.

On multiple pattern avoiding set partitions

“…Intuitively, strong partition equivalence means that we can bijectively map σ-avoiders to τ -avoiders by just permuting the letters of their standard representation. The concept of strong partition equivalence is first explicitly used in [7], although most pairs of strongly partition equivalent patterns follow from the bijections constructed in an earlier paper [6]. Let us list the known facts about strong partition equivalence (references point to the corresponding statement in [6]).…”

“…Known results on pattern equivalences. In the paper on partial patterns in matchings [7], the authors introduce the notion of strong partition equivalence. We say that two patterns σ and τ are strongly partition equivalent, if there exists a bijection f between the sets of σ-avoiding and τ -avoiding partitions with the property that for any σ-avoiding partition ρ, the number of blocks of ρ is equal to the number of blocks of f (ρ), and moreover for any i, the i-th block of ρ has the same size as the i-th block of f (ρ).…”

“…Such a partition is known as a partial matching. Pattern avoidance in partial matchings has already been addressed in a previous paper [7]. We therefore focus on the remaining patterns of size three, that is, we assume σ ∈ {112, 121, 122, 123}.…”

“…First, in Section 2 we consider set partitions avoiding a pair of patterns {σ, τ }, where σ is a pattern of size three and τ is an arbitrary pattern not containing σ. The situation when σ = 111 corresponds to single-pattern avoidance in partial matchings, which has been previously addressed [7]. We therefore restrict our attention to the cases when σ = 111.…”

Enumeration of Some Classes of Pattern Avoiding Matchings, with a Glimpse into the Matching Pattern Poset

“…Given e ∈ τ , the integers min(e) and max(e) will be called the left vertex and the right vertex of e, respectively. Given a subset S of τ and e ∈ S, we will say that e is the leftmost (respectively, rightmost) edge of S when min(e) ≤ min(f ) (respectively, max(e) ≥ max(f )) for every f ∈ S. Following [23], we will represent τ by means of the unique integer sequence τ ∈ [n] 2n such that τmin(e) = τmax(e) and τmin(e) < τmin(f) for every e, f ∈ τ such that min(e) < min(f ). Using this encoding, the vertices of τ are represented by the elements of τ and two vertices of τ are connected by an edge when the corresponding components of τ are equal (see Fig.…”