4,399 research outputs found
Determinants of (generalised) Catalan numbers
We show that recent determinant evaluations involving Catalan numbers and
generalisations thereof have most convenient explanations by combining the
Lindstr\"om-Gessel-Viennot theorem on non-intersecting lattice paths with a
simple determinant lemma from [Manuscripta Math. 69 (1990), 173-202]. This
approach leads also naturally to extensions and generalisations.Comment: AmS-TeX, 16 pages; minor correction
Smith Normal Form of a Multivariate Matrix Associated with Partitions
Consideration of a question of E. R. Berlekamp led Carlitz, Roselle, and
Scoville to give a combinatorial interpretation of the entries of certain
matrices of determinant~1 in terms of lattice paths. Here we generalize this
result by refining the matrix entries to be multivariate polynomials, and by
determining not only the determinant but also the Smith normal form of these
matrices. A priori the Smith form need not exist but its existence follows from
the explicit computation. It will be more convenient for us to state our
results in terms of partitions rather than lattice paths.Comment: 12 pages; revised version (minor changes on first version); to appear
in J. Algebraic Combinatoric
The Geometry of some Fibonacci Identities in the Hosoya Triangle
In this paper we explore some Fibonacci identities that are interpreted
geometrically in the Hosoya triangle. Specifically we explore a generalization
of the Cassini and Catalan identities from a geometric point of view. We also
extend some properties present in the Pascal triangle to the Hosoya triangle.Comment: There are 16 figures representing Fibonacci identities in the Hosoya
triangl
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
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