1,612 research outputs found
The multivariate arithmetic Tutte polynomial
We introduce an arithmetic version of the multivariate Tutte polynomial, and
(for representable arithmetic matroids) a quasi-polynomial that interpolates
between the two. A generalized Fortuin-Kasteleyn representation with
applications to arithmetic colorings and flows is obtained. We give a new and
more general proof of the positivity of the coefficients of the arithmetic
Tutte polynomial, and (in the representable case) a geometrical interpretation
of them.Comment: 21 page
The multivariate arithmetic Tutte polynomial
We introduce an arithmetic version of the multivariate Tutte polynomial recently studied by Sokal, and a quasi-polynomial that interpolates between the two. We provide a generalized Fortuin-Kasteleyn representation for representable arithmetic matroids, with applications to arithmetic colorings and flows. We give a new proof of the positivity of the coefficients of the arithmetic Tutte polynomial in the more general framework of pseudo-arithmetic matroids. In the case of a representable arithmetic matroid, we provide a geometric interpretation of the coefficients of the arithmetic Tutte polynomial. \ua9 2012 Discrete Mathematics and Theoretical Computer Science (DMTCS), Nancy, France
Arithmetic matroids, Tutte polynomial, and toric arrangements
We introduce the notion of an arithmetic matroid, whose main example is given
by a list of elements of a finitely generated abelian group. In particular we
study the representability of its dual, providing an extension of the Gale
duality to this setting. Guided by the geometry of generalized toric
arrangements, we provide a combinatorial interpretation of the associated
arithmetic Tutte polynomial, which can be seen as a generalization of Crapo's
formula for the classical Tutte polynomial.Comment: 32 pages. Several mistakes correcte
Matroid Chern-Schwartz-MacPherson cycles and Tutte activities
Lop\'ez de Medrano-Rin\'con-Shaw defined Chern-Schwartz-MacPherson cycles for
an arbitrary matroid and proved by an inductive geometric argument that the
unsigned degrees of these cycles agree with the coefficients of ,
where is the Tutte polynomial associated to . Ardila-Denham-Huh
recently utilized this interpretation of these coefficients in order to
demonstrate their log-concavity. In this note we provide a direct calculation
of the degree of a matroid Chern-Schwartz-MacPherson cycle by taking its stable
intersection with a generic tropical linear space of the appropriate
codimension and showing that the weighted point count agrees with the Gioan-Las
Vergnas refined activities expansion of the Tutte polynomial
Lattice path matroids: enumerative aspects and Tutte polynomials
Fix two lattice paths P and Q from (0,0) to (m,r) that use East and North
steps with P never going above Q. We show that the lattice paths that go from
(0,0) to (m,r) and that remain in the region bounded by P and Q can be
identified with the bases of a particular type of transversal matroid, which we
call a lattice path matroid. We consider a variety of enumerative aspects of
these matroids and we study three important matroid invariants, namely the
Tutte polynomial and, for special types of lattice path matroids, the
characteristic polynomial and the beta invariant. In particular, we show that
the Tutte polynomial is the generating function for two basic lattice path
statistics and we show that certain sequences of lattice path matroids give
rise to sequences of Tutte polynomials for which there are relatively simple
generating functions. We show that Tutte polynomials of lattice path matroids
can be computed in polynomial time. Also, we obtain a new result about lattice
paths from an analysis of the beta invariant of certain lattice path matroids.Comment: 28 pages, 11 figure
- …