Matchings Avoiding Partial Patterns and Lattice Paths

Jelínek, Vít; Li, Nelson Y.; Mansour, Toufik; Yan, Sherry H. F.

doi:10.37236/1115

Cited by 3 publications

(4 citation statements)

References 13 publications

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“…For example, let π = {{1, 5}, {2, 3, 4, 7}, {6, 8}} ∈ S (8,3). There are 5 arcs in π, so we must add 4 vertices in order to get a partial matching in Q (12,5).…”

Section: Neighbor Alignments and Left Nestingsmentioning

confidence: 99%

“…There are 5 arcs in π, so we must add 4 vertices in order to get a partial matching in Q (12,5). We first add a singleton before the arc (2, 3) and a singleton before the arc (6,8). Then change the two 2-paths into left crossings from left to right.…”

Section: Neighbor Alignments and Left Nestingsmentioning

confidence: 99%

“…Perfect matchings avoiding certain patterns have been studied in [3,4,5,7,8,9,11,12,16]. Bousquet-Mélou, Claesson, Dukes and Kitaev [1] investigated perfect matchings avoiding left nestings and right nestings, and found bijections with other combinatorial objects such as (2 + 2)-free posets.…”

Section: Introductionmentioning

confidence: 99%

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Partitions and partial matchings avoiding neighbor patterns

Chen

Fan

Zhao

2012

European Journal of Combinatorics

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We obtain the generating functions for partial matchings avoiding neighbor alignments and for partial matchings avoiding neighbor alignments and left nestings. We show that there is a bijection between partial matchings avoiding three neighbor patterns (neighbor alignments, left nestings and right nestings) and set partitions avoiding right nestings via an intermediate structure of integer compositions. Such integer compositions are known to be in one-to-one correspondence with self-modified ascent sequences or 31524avoiding permutations, as shown by Bousquet-Mélou, Claesson, Dukes and Kitaev.

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“…For example, let π = {{1, 5}, {2, 3, 4, 7}, {6, 8}} ∈ S (8,3). There are 5 arcs in π, so we must add 4 vertices in order to get a partial matching in Q (12,5).…”

Section: Neighbor Alignments and Left Nestingsmentioning

confidence: 99%

Section: Neighbor Alignments and Left Nestingsmentioning

confidence: 99%

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Partitions and partial matchings avoiding neighbor patterns

Chen

Fan

Zhao

2012

European Journal of Combinatorics

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“…Apart from these general results, various authors have studied classes of matchings avoiding a particular fixed pattern of small size. Jelínek, Li, Mansour and Yan [8] have shown that the matchings avoiding 1123 correspond bijectively to a certain class of lattice paths, and used this fact to obtain the formula…”

Section: Previous Workmentioning

confidence: 99%

Matchings and Partial Patterns

Jelínek

Mansour

2010

Electron. J. Combin.

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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.

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