New interpretations for noncrossing partitions of classical types

Kim, Jang Soo

doi:10.1016/j.jcta.2010.12.005

Cited by 7 publications

(6 citation statements)

References 18 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Since, as already mentioned, the symmetry of the Narayana numbers is easy to see in the noncrossing world, a type-preserving bijection between nonnesting and noncrossing partitions would therefore constitute a solution to Problem 1.1. There are a handful of type-preserving bijections between the nonnesting and noncrossing partitions in the literature [13,20]; but these constructions are all ad hoc and thus unsatisfactory in some sense. No uniform type-preserving bijection is known.…”

Section: (For the Appropriate Involution ∈ Depending On C)mentioning

confidence: 99%

Symmetry of Narayana Numbers and Rowvacuation of Root Posets

Defant

Hopkins

2021

Forum of Mathematics, Sigma

View full text Add to dashboard Cite

For a Weyl group W of rank r, the W-Catalan number is the number of antichains of the poset of positive roots, and the W-Narayana numbers refine the W-Catalan number by keeping track of the cardinalities of these antichains. The W-Narayana numbers are symmetric – that is, the number of antichains of cardinality k is the same as the number of cardinality $r-k$ . However, this symmetry is far from obvious. Panyushev posed the problem of defining an involution on root poset antichains that exhibits the symmetry of the W-Narayana numbers. Rowmotion and rowvacuation are two related operators, defined as compositions of toggles, that give a dihedral action on the set of antichains of any ranked poset. Rowmotion acting on root posets has been the subject of a significant amount of research in the recent past. We prove that for the root posets of classical types, rowvacuation is Panyushev’s desired involution.

show abstract

Section: (For the Appropriate Involution ∈ Depending On C)mentioning

confidence: 99%

Symmetry of Narayana Numbers and Rowvacuation of Root Posets

Defant

Hopkins

2021

Forum of Mathematics, Sigma

View full text Add to dashboard Cite

show abstract

“…In Section 2 we recall the definition of NC B (n) and NC D (n). In Section 3 we recall the bijection ψ in [8] between NC B (n) and the set B(n) of pairs (σ , x), where σ ∈ NC(n) and x is either ∅, an edge of σ , or a block of σ . Here an edge of σ is a pair (i, j) of integers with i < j such that i and j are in the same block of σ which does not contain any integer between them.…”

Section: (W ) ∼ = Nc(w ) and Nc (K) (A N−1 ) ∼ = Nc (K) (N)mentioning

confidence: 99%

“…The following theorem is proved by the author in [8]. For the sake of self-containedness, we include a short proof.…”

Section: Interpretation Of Noncrossing Partitions Of Type B Nmentioning

confidence: 99%

Chain enumeration of k-divisible noncrossing partitions of classical types

Kim

2011

Journal of Combinatorial Theory, Series A

Self Cite

View full text Add to dashboard Cite

We give combinatorial proofs of the formulas for the number of multichains in the k-divisible noncrossing partitions of classical types with certain conditions on the rank and the block size due to Krattenthaler and Müller. We also prove Armstrong's conjecture on the zeta polynomial of the poset of k-divisible noncrossing partitions of type A invariant under a 180 • rotation in the cyclic representation.

show abstract

“…[1,2,[11][12][13]17,23,29,30,34,43,45,49], and among other things they have applications to the theory of free probability, see [37,51,52]. In particular, there exist several bijections between noncrossing and nonnesting partitions, see e.g.…”

Section: Introductionmentioning

confidence: 99%

“…In particular, there exist several bijections between noncrossing and nonnesting partitions, see e.g. [12,13,17,23,29,45], so it is very natural to wonder if all results which hold for noncrossing partitions hold for nonnesting partitions as well. Unfortunately, nonnesting partitions are much more mysterious and intricate, and plenty of results valid for noncrossing partitions do not translate to nonnesting partitions.…”

Section: Introductionmentioning

confidence: 99%