12,812 research outputs found
The Euler and Springer numbers as moment sequences
I study the sequences of Euler and Springer numbers from the point of view of
the classical moment problem.Comment: LaTeX2e, 30 pages. Version 2 contains some small clarifications
suggested by a referee. Version 3 contains new footnotes 9 and 10. To appear
in Expositiones Mathematica
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
The symmetric and unimodal expansion of Eulerian polynomials via continued fractions
This paper was motivated by a conjecture of Br\"{a}nd\'{e}n (European J.
Combin. \textbf{29} (2008), no.~2, 514--531) about the divisibility of the
coefficients in an expansion of generalized Eulerian polynomials, which implies
the symmetric and unimodal property of the Eulerian numbers. We show that such
a formula with the conjectured property can be derived from the combinatorial
theory of continued fractions. We also discuss an analogous expansion for the
corresponding formula for derangements and prove a -analogue of the fact
that the (-1)-evaluation of the enumerator polynomials of permutations (resp.
derangements) by the number of excedances gives rise to tangent numbers (resp.
secant numbers). The -analogue unifies and generalizes our recent
results (European J. Combin. \textbf{31} (2010), no.~7, 1689--1705.) and that
of Josuat-Verg\`es (European J. Combin. \textbf{31} (2010), no.~7, 1892--1906).Comment: 19 pages, 2 figure
The Rayleigh-Schr\"odinger perturbation series of quasi-degenerate systems
We present the first representation of the general term of the
Rayleigh-Schr\"odinger series for quasidegenerate systems. Each term of the
series is represented by a tree and there is a straightforward relation between
the tree and the analytical expression of the corresponding term. The
combinatorial and graphical techniques used in the proof of the series
expansion allow us to derive various resummation formulas of the series. The
relation with several combinatorial objects used for special cases (degenerate
or non-degenerate systems) is established.Comment: 16 pages, 3 figure
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
- …