41 research outputs found
Increasing and Decreasing Sequences of Length Two in 01-Fillings of Moon Polyominoes
We put recent results on the symmetry of the joint distribution of the
numbers of crossings and nestings of two edges over matchings, set partitions
and linked partitions, in the larger context of the enumeration of increasing
and decreasing chains of length 2 in fillings of moon polyominoes.Comment: It is a updated version of a preprint entitled "On the symmetry of
ascents and descents over 01-fillings of moon polyominoes". 19 page
On k-crossings and k-nestings of permutations
We introduce k-crossings and k-nestings of permutations. We show that the
crossing number and the nesting number of permutations have a symmetric joint
distribution. As a corollary, the number of k-noncrossing permutations is equal
to the number of k-nonnesting permutations. We also provide some enumerative
results for k-noncrossing permutations for some values of k
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
Patterns in matchings and rook placements
Extending the notion of pattern avoidance in permutations, we study matchings and set partitions whose arc diagram representation avoids a given configuration of three arcs. These configurations, which generalize 3-crossings and 3-nestings, have an interpretation, in the case of matchings, in terms of patterns in full rook placements on Ferrers boards. We enumerate 312-avoiding matchings and partitions, obtaining algebraic generating functions, unlike in the 321-avoiding (i.e., 3-noncrossing) case. Our approach also provides a more direct proof of a formula of BĂłna for the number of 1342-avoiding permutations. Additionally, we give a bijection proving the shape-Wilf-equivalence of the patterns 321 and 213 which simplifies existing proofs by Backelin–West–Xin and JelĂnek
Crossings and nestings in set partitions of classical types
In this article, we investigate bijections on various classes of set
partitions of classical types that preserve openers and closers. On the one
hand we present bijections that interchange crossings and nestings. For types B
and C, they generalize a construction by Kasraoui and Zeng for type A, whereas
for type D, we were only able to construct a bijection between non-crossing and
non-nesting set partitions. On the other hand we generalize a bijection to type
B and C that interchanges the cardinality of the maximal crossing with the
cardinality of the maximal nesting, as given by Chen, Deng, Du, Stanley and Yan
for type A. Using a variant of this bijection, we also settle a conjecture by
Soll and Welker concerning generalized type B triangulations and symmetric fans
of Dyck paths.Comment: 22 pages, 7 Figures, removed erroneous commen