37,043 research outputs found
Restricted 132-Dumont permutations
A permutation is said to be {\em Dumont permutations of the first kind}
if each even integer in must be followed by a smaller integer, and each
odd integer is either followed by a larger integer or is the last element of
(see, for example, \cite{Z}). In \cite{D} Dumont showed that certain
classes of permutations on letters are counted by the Genocchi numbers. In
particular, Dumont showed that the st Genocchi number is the number of
Dummont permutations of the first kind on letters.
In this paper we study the number of Dumont permutations of the first kind on
letters avoiding the pattern 132 and avoiding (or containing exactly once)
an arbitrary pattern on letters. In several interesting cases the
generating function depends only on .Comment: 12 page
Counting peaks at height k in a Dyck path
A Dyck path is a lattice path in the plane integer lattice
consisting of steps (1,1) and (1,-1), which never
passes below the x-axis. A peak at height k on a Dyck path is a point on the
path with coordinate y=k that is immediately preceded by a (1,1) step and
immediately followed by a (1,-1) step. In this paper we find an explicit
expression to the generating function for the number of Dyck paths starting at
(0,0) and ending at (2n,0) with exactly r peaks at height k. This allows us to
express this function via Chebyshev polynomials of the second kind and
generating function for the Catalan numbers.Comment: 7 pages, 3 figure
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