102 research outputs found
On Noncrossing and nonnesting partitions of type D
We present an explicit bijection between noncrossing and nonnesting
partitions of Coxeter systems of type D which preserves openers, closers and
transients.Comment: 13 pages, 10 figures. A remark on a reference has been correcte
A bijection between noncrossing and nonnesting partitions of types A and B
The total number of noncrossing partitions of type is the th
Catalan number when , and the
binomial when , and these numbers coincide with the
correspondent number of nonnesting partitions. For type A, there are several
bijective proofs of this equality, being the intuitive map that locally
converts each crossing to a nesting one of them. In this paper we present a
bijection between nonnesting and noncrossing partitions of types A and B that
generalizes the type A bijection that locally converts each crossing to a
nesting.Comment: 11 pages, 11 figures. Inverse map described. Minor changes to correct
typos and clarify conten
New interpretations for noncrossing partitions of classical types
We interpret noncrossing partitions of type and type in terms of
noncrossing partitions of type . As an application, we get type-preserving
bijections between noncrossing and nonnesting partitions of type , type
and type which are different from those in the recent work of Fink and
Giraldo. We also define Catalan tableaux of type and type , and find
bijections between them and noncrossing partitions of type and type
respectively.Comment: 21 pages, 15 figures, final versio
Promotion and Rowmotion
We present an equivariant bijection between two actions--promotion and
rowmotion--on order ideals in certain posets. This bijection simultaneously
generalizes a result of R. Stanley concerning promotion on the linear
extensions of two disjoint chains and recent work of D. Armstrong, C. Stump,
and H. Thomas on root posets and noncrossing partitions. We apply this
bijection to several classes of posets, obtaining equivariant bijections to
various known objects under rotation. We extend the same idea to give an
equivariant bijection between alternating sign matrices under rowmotion and
under B. Wieland's gyration. Finally, we define two actions with related orders
on alternating sign matrices and totally symmetric self-complementary plane
partitions.Comment: 25 pages, 22 figures; final versio
Crossings and nestings in colored set partitions
Chen, Deng, Du, Stanley, and Yan introduced the notion of -crossings and
-nestings for set partitions, and proved that the sizes of the largest
-crossings and -nestings in the partitions of an -set possess a
symmetric joint distribution. This work considers a generalization of these
results to set partitions whose arcs are labeled by an -element set (which
we call \emph{-colored set partitions}). In this context, a -crossing or
-nesting is a sequence of arcs, all with the same color, which form a
-crossing or -nesting in the usual sense. After showing that the sizes of
the largest crossings and nestings in colored set partitions likewise have a
symmetric joint distribution, we consider several related enumeration problems.
We prove that -colored set partitions with no crossing arcs of the same
color are in bijection with certain paths in \NN^r, generalizing the
correspondence between noncrossing (uncolored) set partitions and 2-Motzkin
paths. Combining this with recent work of Bousquet-M\'elou and Mishna affords a
proof that the sequence counting noncrossing 2-colored set partitions is
P-recursive. We also discuss how our methods extend to several variations of
colored set partitions with analogous notions of crossings and nestings.Comment: 25 pages; v2: material revised and condensed; v3 material further
revised, additional section adde
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