Crossings and Nestings in Colored Set Partitions

Marberg, Eric

doi:10.37236/3163

Cited by 10 publications

(25 citation statements)

References 36 publications

(106 reference statements)

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“…This is equivalent to counting the number of walks on the Hasse diagram of the r-rim hook lattice. Marberg [10] noted isomorphism between his walks on the Hasse diagram of the r-fold product of the Young lattice of integer partitions (produced from oscillating r-partite tableau) and those of Chen and Guo [3]. Arc-coloured permutations have two types of r-partite tableau simultaneously accounting for both the upper and the lower arc diagrams.…”

Section: Discussionmentioning

confidence: 99%

“…The result is shown in Figure 6. The adjacency matrix of Figure 6 As noted by Marberg [10], these are every second Fibonacci numbers.…”

Section: Examples Of G 221 and G 222 For Set Partitionsmentioning

confidence: 94%

“…Next, Chen and Guo [3] generalized symmetric equidistribution of crossing and nesting statistics to coloured complete matchings. Most recently, Marberg [10] extended the result to coloured set partitions with a novel way of proving that the ordinary generating functions of j-noncrossing, knonnesting, r-coloured partitions according to size n are rational functions. We extend their results to r-arc-coloured permutations, or r-coloured permutations in short.…”

Section: Definitions and Terminologymentioning

confidence: 99%

“…Since crossing and nesting statistics involves arcs, we define an r-coloured permutation parallel to [10] as a pair, (S, φ) consisting of a permutation of [n] and an arc-colour assigning map φ : Arc(S) → [r], and use a capital Greek letter, Σ, to denote these objects. We say Σ has a k-crossing (resp.…”

Section: Main Theoremmentioning

confidence: 99%

“…In all cases, bijective proofs were given; and for some, generating functions were found. Inspired by recent works of Chen and Guo [3] on coloured matchings and Marberg [10] on coloured set partitions, we give a bijection to establish symmetric joint distribution of crossing and nesting statistics for arc-coloured permutations. We also show that the ordinary generating functions for jnoncrossing, k-nonnesting, r-coloured permutations according to size n are rational functions.…”

Section: Introductionmentioning

confidence: 99%

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Crossings and Nestings for Arc-Coloured Permutations and Automation

Yen

2015

Electron. J. Combin.

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The equidistribution of many crossing and nesting statistics exists in several combinatorial objects like matchings, set partitions, permutations, and embedded labelled graphs. The involutions switching nesting and crossing numbers for set partitions given by Krattenthaler, also by Chen, Deng, Du, Stanley, and Yan, and for permutations given by Burrill, Mishna, and Post involved passing through tableau-like objects. Recently, Chen and Guo for matchings, and Marberg for set partitions extended the result to coloured arc annotated diagrams. We prove that symmetric joint distribution continues to hold for arc-coloured permutations. As in Marberg's recent work, but through a different interpretation, we also conclude that the ordinary generating functions for all j-noncrossing, k-nonnesting, r-coloured permutations according to size n are rational functions. We use the interpretation to automate the generation of these rational series for both noncrossing and nonnesting coloured set partitions and permutations.

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Section: Discussionmentioning

confidence: 99%

“…The result is shown in Figure 6. The adjacency matrix of Figure 6 As noted by Marberg [10], these are every second Fibonacci numbers.…”

Section: Examples Of G 221 and G 222 For Set Partitionsmentioning

confidence: 94%

Section: Definitions and Terminologymentioning

confidence: 99%

Section: Main Theoremmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Crossings and Nestings for Arc-Coloured Permutations and Automation

Yen

2015

Electron. J. Combin.

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A Generating Tree Approach to k-Nonnesting Partitions and Permutations

et al. 2016

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We describe a generating tree approach to the enumeration and exhaustive generation of knonnesting set partitions and permutations. Unlike previous work in the literature using the connections of these objects to Young tableaux and restricted lattice walks, our approach deals directly with partition and permutation diagrams. We provide explicit functional equations for the generating functions, with k as a parameter.The heart of the argument maps the objects to a sequence of Young tableaux, and the bijection is achieved by taking the transpose of the tableaux.From the enumerative point of view, results have been far less forthcoming. The enumeration of k-nonnesting matchings was completed by Chen et al. in their master work by using a bijection to collections of noncrossing Dyck paths whose enumeration was known. Consequently, the results are elegant. Also, the class of 2-noncrossing (or simply noncrossing) partitions is counted by Catalan numbers, and so one could hope for a similarly beautiful enumeration scheme. So far, only exact formulas exist for 3-noncrossing set partitions, and beyond that, a far more complicated structure is conjectured. Specifically, Bousquet-Mélou and Xin [4] did a complete functional equation analysis, and determined explicit exact and asymptotic enumeration formulas.Although their functional equations can be adapted to enumerate set partitions with higher noncrossing numbers, they conjecture that the structure of the generating function becomes more complicated. Conjecture 1.1 (Bousquet-Mélou,Xin 2005 [4]). For every k > 3, the generating function of k-noncrossing set partitions is not D-finite. Mishna and Yen [26] determined functional equations for k-nonnesting set partitions, and described a process for isolating coefficients, giving additional evidence for Conjecture 1.1. A more recent development considers set partitions where the maximum nesting size and the maximum crossing size are controlled simultaneously. The resulting generating functions are rational [25], and (in theory) can be determined explicitly. They can be summed together for a different picture of generating functions for k-nonnesting partitions. 1.2. Permutations. The arc diagram is a convenient way to view permutations, as it simultaneously highlights both the cycle structure, and the line notation structure. Corteel [14] used arc diagrams to directly connect various permutation statistics, such as exceedences and permutation patterns to occurrences of nestings and crossings. Burill, Mishna and Post [8] extended this definition, and directly adapted the results of Chen et al. to prove that k-nonnesting permutations are in bijection with k-noncrossing permutations. They computed enumerative data by brute force, by finding the nesting and crossing numbers of each permutation. Consequently, the data is restricted to small sizes only. The generating function for permutations in which both the maximum nesting and maximum crossing numbers are controlled is rational [31], similarly to the case of set partitions. Exten...

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A simple bijection for enhanced, classical, and 2-distant k-noncrossing partitions

Gil

Tirrell

2020

Discrete Mathematics

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Crossings and Nestings in Colored Set Partitions

Cited by 10 publications

References 36 publications

Crossings and Nestings for Arc-Coloured Permutations and Automation

Crossings and Nestings for Arc-Coloured Permutations and Automation

A Generating Tree Approach to k-Nonnesting Partitions and Permutations

A simple bijection for enhanced, classical, and 2-distant k-noncrossing partitions

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