332 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
A q-analogue of Catalan Hankel determinants
In this paper we shall survey the various methods of evaluating Hankel
determinants and as an illustration we evaluate some Hankel determinants of a
q-analogue of Catalan numbers. Here we consider
as a q-analogue of Catalan numbers
, which is known as the moments of the little
q-Jacobi polynomials. We also give several proofs of this q-analogue, in which
we use lattice paths, the orthogonal polynomials, or the basic hypergeometric
series. We also consider a q-analogue of Schr\"oder Hankel determinants, and
give a new proof of Moztkin Hankel determinants using an addition formula for
.Comment: 17 page
Cumulants, lattice paths, and orthogonal polynomials
A formula expressing free cumulants in terms of the Jacobi parameters of the
corresponding orthogonal polynomials is derived. It combines Flajolet's theory
of continued fractions and Lagrange inversion. For the converse we discuss
Gessel-Viennot theory to express Hankel determinants in terms of various
cumulants.Comment: 11 pages, AMS LaTeX, uses pstricks; revised according to referee's
suggestions, in particular cut down last section and corrected some wrong
attribution
Some determinants of path generating functions
We evaluate four families of determinants of matrices, where the entries are
sums or differences of generating functions for paths consisting of up-steps,
down-steps and level steps. By specialisation, these determinant evaluations
have numerous corollaries. In particular, they cover numerous determinant
evaluations of combinatorial numbers - most notably of Catalan, ballot, and of
Motzkin numbers - that appeared previously in the literature.Comment: 35 pages, AmS-TeX; minor corrections; final version to appear in Adv.
Appl. Mat
Constellations and multicontinued fractions: application to Eulerian triangulations
We consider the problem of enumerating planar constellations with two points
at a prescribed distance. Our approach relies on a combinatorial correspondence
between this family of constellations and the simpler family of rooted
constellations, which we may formulate algebraically in terms of multicontinued
fractions and generalized Hankel determinants. As an application, we provide a
combinatorial derivation of the generating function of Eulerian triangulations
with two points at a prescribed distance.Comment: 12 pages, 4 figure
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