34 research outputs found
The Tutte Polynomial of a Morphism of Matroids 6. A Multi-Faceted Counting Formula for Hyperplane Regions and Acyclic Orientations
We show that the 4-variable generating function of certain orientation
related parameters of an ordered oriented matroid is the evaluation at (x + u,
y+v) of its Tutte polynomial. This evaluation contains as special cases the
counting of regions in hyperplane arrangements and of acyclic orientations in
graphs. Several new 2-variable expansions of the Tutte polynomial of an
oriented matroid follow as corollaries.
This result hold more generally for oriented matroid perspectives, with
specific special cases the counting of bounded regions in hyperplane
arrangements or of bipolar acyclic orientations in graphs.
In corollary, we obtain expressions for the partial derivatives of the Tutte
polynomial as generating functions of the same orientation parameters.Comment: 23 pages, 2 figures, 3 table
Short rewriting, and geometric explanations related to the active bijection, for: Extension-lifting bijections for oriented matroids, by S. Backman, F. Santos, C.H. Yuen, arXiv:1904.03562v2 (October 29, 2023)
For an oriented matroid M, and given a generic single element extension and a
generic single element lifting of M, the main result of [1] provides a
bijection between bases of M and certain reorientations of M induced by the
extension-lifting. This note is intended to somehow clarify and precise the
geometric setting for this paper in terms of oriented matroid arrangements and
oriented matroid programming, to describe and prove the main bijective result
in a short simple way, and to show how it consists of combining two direct
bijections and a central bijection, which is the same as a special case -
practically uniform - of the bounded case of the active bijection [5, 6]. (The
relation with the active bijection is addressed in [1] in an indirect and more
complicated way.