26 research outputs found
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
From Dyck Paths to Standard Young Tableaux
We present nine bijections between classes of Dyck paths and classes of standard Young tableaux (SYT). In particular, we consider SYT of flag and rectangular shapes, we give Dyck path descriptions for certain SYT of height at most 3, and we introduce a special class of labeled Dyck paths of semilength n that is shown to be in bijection with the set of all SYT with n boxes. In addition, we present bijections from certain classes of Motzkin paths to SYT. As a natural framework for some of our bijections, we introduce a class of set partitions which in some sense is dual to the known class of noncrossing partitions
A construction which relates c-freeness to infinitesimal freeness
We consider two extensions of free probability that have been studied in the
research literature, and are based on the notions of c-freeness and
respectively of infinitesimal freeness for noncommutative random variables. In
a 2012 paper, Belinschi and Shlyakhtenko pointed out a connection between these
two frameworks, at the level of their operations of 1-dimensional free additive
convolution. Motivated by that, we propose a construction which produces a
multi-variate version of the Belinschi-Shlyakhtenko result, together with a
result concerning free products of multi-variate noncommutative distributions.
Our arguments are based on the combinatorics of the specific types of cumulants
used in c-free and in infinitesimal free probability. They work in a rather
general setting, where the initial data consists of a vector space given
together with a linear map . In this setting, all
the needed brands of cumulants live in the guise of families of multilinear
functionals on , and our main result concerns a certain transformation
on such families of multilinear functionals.Comment: Version 2: Minor revision, added reference