Pattern avoiding permutations and independent sets in graphs

Bean, Christian; Tannock, Murray; Ulfarsson, Henning

doi:10.48550/arxiv.1512.08155

Cited by 2 publications

(19 citation statements)

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“…2: The sum and skew-sum of two permutations. Bean et al (2015) introduced the staircase encoding, a function which maps a permutation to a staircase grid where cells are filled with non-negative integers. The integers represented the size of the monotone sequence in each cell of the grid.…”

Section: ⊕ 21 ⊖ 21mentioning

confidence: 99%

“…We first recover the results for Av(123) and Av(132) from Bean et al (2015) in Section 2, using our more refined encoding and weighted independent sets. Our technique is then extended to describe the structure of Av(2314, 3124) and Av(2413, 3142) in Section 3 and 4.…”

Section: ⊕ 21 ⊖ 21mentioning

confidence: 99%

“…Our technique is then extended to describe the structure of Av(2314, 3124) and Av(2413, 3142) in Section 3 and 4. In Section 5, we recall the updown core graph introduced by Bean et al (2015) and use it to enumerate Av (2314,3124,2413,3142), before introducing a new core graph in Section 6 that is used to give the structure of Av(2314, 3124, 3142). Our notion of weighted independent sets is then generalized to allow labelling.…”

Section: ⊕ 21 ⊖ 21mentioning

confidence: 99%

“…We say that B 0 is the empty staircase grid which corresponds to the staircase encoding of ε, the empty permutation. Bean et al (2015) used the staircase grid to enumerate Av(123) and Av(132). We briefly review these in terms of our redefined staircase encoding.…”

Section: Encoding Permutations On Gridmentioning

confidence: 99%

“…Theorem 2.6 (Lemma 2.2 in Bean et al (2015)). The map SE is a bijection between Av (n) (123) and the weighted independent sets WI(U(B n ), Av + (12)).…”

Section: Encoding Permutations On Gridmentioning

confidence: 99%

See 4 more Smart Citations

Enumeration of Permutation Classes and Weighted Labelled Independent Sets

Bean,

Nadeau,

Ulfarsson

2019

Preprint

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In this paper, we study the staircase encoding of permutations, which maps a permutation to a staircase grid with cells filled with permutations. We study many cases where restricted to a permutation class, the staircase encoding becomes a bijection to its image. We describe the image of those restrictions using independent sets of graphs weighted with permutations. We derive the generating function for the independent sets and then for their weighted counterparts. The bijections we establish provide the enumeration of permutation classes. We use our results to uncover some unbalanced Wilf-equivalences of permutation classes and outline how to do random sampling in the permutation classes. In particular, we cover the classes

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Section: ⊕ 21 ⊖ 21mentioning

confidence: 99%

Section: ⊕ 21 ⊖ 21mentioning

confidence: 99%

Section: ⊕ 21 ⊖ 21mentioning

confidence: 99%

Section: Encoding Permutations On Gridmentioning

confidence: 99%

“…Theorem 2.6 (Lemma 2.2 in Bean et al (2015)). The map SE is a bijection between Av (n) (123) and the weighted independent sets WI(U(B n ), Av + (12)).…”

Section: Encoding Permutations On Gridmentioning

confidence: 99%

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Enumeration of Permutation Classes and Weighted Labelled Independent Sets

Bean,

Nadeau,

Ulfarsson

2019

Preprint

Self Cite

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Resolving Two Conjectures on Staircase Encodings and Boundary Grids of $132$ and $123$-avoiding permutations

Narayanan

2019

Electron. J. Combin.

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This paper analyzes relations between pattern avoidance of certain permutations and graphs on staircase grids and boundary grids, and proves two conjectures posed by Bean, Tannock, and Ulfarsson (2015). More specifically, this paper enumerates a certain family of staircase encodings and proves that the downcore graph, a certain graph established on the boundary grid, is pure if and only if the permutation corresponding to the boundary grid avoids the classical patterns 123 and 2143.

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Pattern avoiding permutations and independent sets in graphs

Cited by 2 publications

References 0 publications

Enumeration of Permutation Classes and Weighted Labelled Independent Sets

Enumeration of Permutation Classes and Weighted Labelled Independent Sets

Resolving Two Conjectures on Staircase Encodings and Boundary Grids of $132$ and $123$-avoiding permutations

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