458 research outputs found
What power of two divides a weighted Catalan number?
Given a sequence of integers b = (b_0,b_1,b_2,...) one gives a Dyck path P of
length 2n the weight
wt(P) = b_{h_1} b_{h_2} ... b_{h_n},
where h_i is the height of the ith ascent of P. The corresponding weighted
Catalan number is
C_n^b = sum_P wt(P),
where the sum is over all Dyck paths of length 2n. So, in particular, the
ordinary Catalan numbers C_n correspond to b_i = 1 for all i >= 0. Let xi(n)
stand for the base two exponent of n, i.e., the largest power of 2 dividing n.
We give a condition on b which implies that
xi(C_n^b) = xi(C_n).
In the special case b_i=(2i+1)^2, this settles a conjecture of Postnikov
about the number of plane Morse links. Our proof generalizes the recent
combinatorial proof of Deutsch and Sagan of the classical formula for xi(C_n).Comment: Fixed reference
On the diagram of 132-avoiding permutations
The diagram of a 132-avoiding permutation can easily be characterized: it is
simply the diagram of a partition. Based on this fact, we present a new
bijection between 132-avoiding and 321-avoiding permutations. We will show that
this bijection translates the correspondences between these permutations and
Dyck paths given by Krattenthaler and by Billey-Jockusch-Stanley, respectively,
to each other. Moreover, the diagram approach yields simple proofs for some
enumerative results concerning forbidden patterns in 132-avoiding permutations.Comment: 20 pages; additional reference is adde
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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