2,637 research outputs found
Triangular Recurrences, Generalized Eulerian Numbers, and Related Number Triangles
Many combinatorial and other number triangles are solutions of recurrences of
the Graham-Knuth-Patashnik (GKP) type. Such triangles and their defining
recurrences are investigated analytically. They are acted on by a
transformation group generated by two involutions: a left-right reflection and
an upper binomial transformation, acting row-wise. The group also acts on the
bivariate exponential generating function (EGF) of the triangle. By the method
of characteristics, the EGF of any GKP triangle has an implicit representation
in terms of the Gauss hypergeometric function. There are several parametric
cases when this EGF can be obtained in closed form. One is when the triangle
elements are the generalized Stirling numbers of Hsu and Shiue. Another is when
they are generalized Eulerian numbers of a newly defined kind. These numbers
are related to the Hsu-Shiue ones by an upper binomial transformation, and can
be viewed as coefficients of connection between polynomial bases, in a manner
that generalizes the classical Worpitzky identity. Many identities involving
these generalized Eulerian numbers and related generalized Narayana numbers are
derived, including closed-form evaluations in combinatorially significant
cases.Comment: 62 pages, final version, accepted by Advances in Applied Mathematic
Path counting and random matrix theory
We establish three identities involving Dyck paths and alternating Motzkin
paths, whose proofs are based on variants of the same bijection. We interpret
these identities in terms of closed random walks on the halfline. We explain
how these identities arise from combinatorial interpretations of certain
properties of the -Hermite and -Laguerre ensembles of random
matrix theory. We conclude by presenting two other identities obtained in the
same way, for which finding combinatorial proofs is an open problem.Comment: 14 pages, 13 figures and diagrams; submitted to the Electronic
Journal of Combinatoric
A probabilistic interpretation of a sequence related to Narayana polynomials
A sequence of coefficients appearing in a recurrence for the Narayana
polynomials is generalized. The coefficients are given a probabilistic
interpretation in terms of beta distributed random variables. The recurrence
established by M. Lasalle is then obtained from a classical convolution
identity. Some arithmetical properties of the generalized coefficients are also
established
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
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