Permutation Pattern Avoidance and the Catalan Triangle

DeSantis, Derek; Field, Rebecca; Hough, Wesley K.; Jones, Brant; Meissen, Rebecca; Ziefle, Jacob

doi:10.35834/mjms/1369746397

Cited by 4 publications

(3 citation statements)

References 10 publications

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“…is the set of length n permutations avoiding 213 and starting with n, the initial conditions are given by a 1 n = 1 and a n n = c n−1 where c n is the nth Catalan number c n = 1 n+1 2n n . Therefore, a k n generates the well known Catalan's triangle (see Table 2 and [7,10,16]), which implies that a n = n k=1 a k n corresponds to the nth Catalan number c n (see A000108 and A009766 in [18]).…”

Section: The (123 132)-machinementioning

confidence: 97%

Catalan and Schröder permutations sortable by two restricted stacks

Baril,

Cerbai,

Khalil

et al. 2020

Preprint

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We investigate the permutation sorting problem with two restricted stacks in series by applying a right greedy process. Instead of σ-machine introduced recently by Cerbai et al., we focus our study on (σ, τ )-machine where the first stack avoids two patterns of length three, σ and τ , and the second stack avoids the pattern 21. For (σ, τ ) = (123, 132), we prove that sortable permutations are those avoiding some generalized patterns, which provides a new set enumerated by the Catalan numbers. When (σ, τ ) = (132, 231), we prove that sortable permutations are counted by the Schröder numbers. Finally, we suspect that other pairs of length three patterns lead to sets enumerated by the Catalan numbers and leave them as open problems.

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Section: The (123 132)-machinementioning

confidence: 97%

Catalan and Schröder permutations sortable by two restricted stacks

Baril,

Cerbai,

Khalil

et al. 2020

Preprint

View full text Add to dashboard Cite

show abstract

“…, where c n is the nth Catalan number. Therefore, a k n generates the well-known Catalan triangle (see Table 1 and [6,11,14]). [6,11,14]).…”

Section: Finally Setting a Kmentioning

confidence: 99%

Catalan and Schröder permutations sortable by two restricted stacks

Baril

Cerbai

Khalil

et al. 2021

Information Processing Letters

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Stack sorting/two stacks in series Pattern avoiding permutations/machines Catalan and the Schröder numbers Combinatorial problems Pattern avoiding machines were introduced recently by Claesson, Cerbai and Ferrari as a particular case of the two-stacks in series sorting device. They consist of two restricted stacks in series, ruled by a right-greedy procedure and the stacks avoid some specified patterns. Some of the obtained results have been further generalized to Cayley permutations by Cerbai, specialized to particular patterns by Defant and Zheng, or considered in the context of functions over the symmetric group by Berlow. In this work we study pattern avoiding machines where the first stack avoids a pair of patterns of length 3 and investigate those pairs for which sortable permutations are counted by the (binomial transform of the) Catalan numbers and the Schröder numbers.

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“…A prominent area in which the application of this sequence is seen is in permutation. According to [4] permutations of length n that avoids any fixed pattern of size three are counted by the Catalan number.…”

Section: Introductionmentioning

confidence: 99%