23 research outputs found
Counting Humps in Motzkin paths
In this paper we study the number of humps (peaks) in Dyck, Motzkin and
Schr\"{o}der paths. Recently A. Regev noticed that the number of peaks in all
Dyck paths of order is one half of the number of super Dyck paths of order
. He also computed the number of humps in Motzkin paths and found a similar
relation, and asked for bijective proofs. We give a bijection and prove these
results. Using this bijection we also give a new proof that the number of Dyck
paths of order with peaks is the Narayana number. By double counting
super Schr\"{o}der paths, we also get an identity involving products of
binomial coefficients.Comment: 8 pages, 2 Figure
Counting Labelled Trees with Given Indegree Sequence
For a labelled tree on the vertex set , define the
direction of each edge to be if . The indegree sequence of
can be considered as a partition . The enumeration of
trees with a given indegree sequence arises in counting secant planes of curves
in projective spaces. Recently Ethan Cotterill conjectured a formula for the
number of trees on with indegree sequence corresponding to a partition
. In this paper we give two proofs of Cotterill's conjecture: one is
`semi-combinatorial" based on induction, the other is a bijective proof.Comment: 10 page
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