12,025 research outputs found
The -rational -Catalan polynomials for and their -symmetry
We introduce a new statistic, skip, on rational -Dyck paths and define
a marked rank word for each path when is not a multiple of 3. If a triple
of valid statistics (area,skip,dinv) are given, we have an algorithm to
construct the marked rank word corresponding to the triple. By considering all
valid triples we give an explicit formula for the -rational
-Catalan polynomials when . Then there is a natural bijection on the
triples of statistics (area,skips,dinv) which exchanges the statistics area and
dinv while fixing the skip. Thus we prove the -symmetry of
-rational -Catalan polynomials for .Comment: 11 pages, 4 figure
Counting Dyck paths by area and rank
The set of Dyck paths of length inherits a lattice structure from a
bijection with the set of noncrossing partitions with the usual partial order.
In this paper, we study the joint distribution of two statistics for Dyck
paths: \emph{area} (the area under the path) and \emph{rank} (the rank in the
lattice).
While area for Dyck paths has been studied, pairing it with this rank
function seems new, and we get an interesting -refinement of the Catalan
numbers. We present two decompositions of the corresponding generating
function: one refines an identity of Carlitz and Riordan; the other refines the
notion of -nonnegativity, and is based on a decomposition of the
lattice of noncrossing partitions due to Simion and Ullman.
Further, Biane's correspondence and a result of Stump allow us to conclude
that the joint distribution of area and rank for Dyck paths equals the joint
distribution of length and reflection length for the permutations lying below
the -cycle in the absolute order on the symmetric group.Comment: 24 pages, 7 figures. Connections with work of C. Stump
(arXiv:0808.2822v2) eliminated the need for 5 pages of proof in the first
draf
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