130 research outputs found
Pairs of Noncrossing Free Dyck Paths and Noncrossing Partitions
Using the bijection between partitions and vacillating tableaux, we establish
a correspondence between pairs of noncrossing free Dyck paths of length
and noncrossing partitions of with blocks. In terms of the
number of up steps at odd positions, we find a characterization of Dyck paths
constructed from pairs of noncrossing free Dyck paths by using the Labelle
merging algorithm.Comment: 9 pages, 5 figures, revised version, to appear in Discrete
Mathematic
Reduction of -Regular Noncrossing Partitions
In this paper, we present a reduction algorithm which transforms -regular
partitions of to -regular partitions of .
We show that this algorithm preserves the noncrossing property. This yields a
simple explanation of an identity due to Simion-Ullman and Klazar in connection
with enumeration problems on noncrossing partitions and RNA secondary
structures. For ordinary noncrossing partitions, the reduction algorithm leads
to a representation of noncrossing partitions in terms of independent arcs and
loops, as well as an identity of Simion and Ullman which expresses the Narayana
numbers in terms of the Catalan numbers
Enumeration of saturated chains in Dyck lattices
We determine a general formula to compute the number of saturated chains in
Dyck lattices, and we apply it to find the number of saturated chains of length
2 and 3. We also compute what we call the Hasse index (of order 2 and 3) of
Dyck lattices, which is the ratio between the total number of saturated chains
(of length 2 and 3) and the cardinality of the underlying poset.Comment: 9 page
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