12,006 research outputs found
A unified approach to polynomial sequences with only real zeros
We give new sufficient conditions for a sequence of polynomials to have only
real zeros based on the method of interlacing zeros. As applications we derive
several well-known facts, including the reality of zeros of orthogonal
polynomials, matching polynomials, Narayana polynomials and Eulerian
polynomials. We also settle certain conjectures of Stahl on genus polynomials
by proving them for certain classes of graphs, while showing that they are
false in general.Comment: 19 pages, Advances in Applied Mathematics, in pres
Planar maps as labeled mobiles
We extend Schaeffer's bijection between rooted quadrangulations and
well-labeled trees to the general case of Eulerian planar maps with prescribed
face valences, to obtain a bijection with a new class of labeled trees, which
we call mobiles. Our bijection covers all the classes of maps previously
enumerated by either the two-matrix model used by physicists or by the
bijection with blossom trees used by combinatorists. Our bijection reduces the
enumeration of maps to that, much simpler, of mobiles and moreover keeps track
of the geodesic distance within the initial maps via the mobiles' labels.
Generating functions for mobiles are shown to obey systems of algebraic
recursion relations.Comment: 31 pages, 17 figures, tex, lanlmac, epsf; improved tex
A Generalization of the Eulerian Numbers
In the present paper we generalize the Eulerian numbers (also of the second
and third orders). The generalization is connected with an autonomous
first-order differential equation, solutions of which are used to obtain
integral representations of some numbers, including the Bernoulli numbers.Comment: 13 pages, preprin
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