Linial arrangements and local binary search trees

We study the set of NBC sets (no broken circuit sets) of the Linial arrangement and deduce a constructive bijection to the set of local binary search trees. We then generalize this construction to two families of Linial type arrangements for which the bijections are with some $k$-ary labelled trees that we introduce for this purpose.Comment: 13 pages, 1 figure. arXiv admin note: text overlap with arXiv:1403.257

Lattice point counts for the Shi arrangement and other affinographic hyperplane arrangements

Hyperplanes of the form x_j = x_i + c are called affinographic. For an affinographic hyperplane arrangement in R^n, such as the Shi arrangement, we study the function f(M) that counts integral points in [1,M]^n that do not lie in any hyperplane of the arrangement. We show that f(M) is a piecewise polynomial function of positive integers M, composed of terms that appear gradually as M increases. Our approach is to convert the problem to one of counting integral proper colorations of a rooted integral gain graph. An application is to interval coloring in which the interval of available colors for vertex v_i has the form [(h_i)+1,M]. A related problem takes colors modulo M; the number of proper modular colorations is a different piecewise polynomial that for large M becomes the characteristic polynomial of the arrangement (by which means Athanasiadis previously obtained that polynomial). We also study this function for all positive moduli.Comment: 13 p

Lattice Points in Orthotopes and a Huge Polynomial Tutte Invariant of Weighted Gain Graphs

A gain graph is a graph whose edges are orientably labelled from a group. A weighted gain graph is a gain graph with vertex weights from an abelian semigroup, where the gain group is lattice ordered and acts on the weight semigroup. For weighted gain graphs we establish basic properties and we present general dichromatic and forest-expansion polynomials that are Tutte invariants (they satisfy Tutte's deletion-contraction and multiplicative identities). Our dichromatic polynomial includes the classical graph one by Tutte, Zaslavsky's two for gain graphs, Noble and Welsh's for graphs with positive integer weights, and that of rooted integral gain graphs by Forge and Zaslavsky. It is not a universal Tutte invariant of weighted gain graphs; that remains to be found. An evaluation of one example of our polynomial counts proper list colorations of the gain graph from a color set with a gain-group action. When the gain group is Z^d, the lists are order ideals in the integer lattice Z^d, and there are specified upper bounds on the colors, then there is a formula for the number of bounded proper colorations that is a piecewise polynomial function of the upper bounds, of degree nd where n is the order of the graph. This example leads to graph-theoretical formulas for the number of integer lattice points in an orthotope but outside a finite number of affinographic hyperplanes, and for the number of n x d integral matrices that lie between two specified matrices but not in any of certain subspaces defined by simple row equations.Comment: 32 pp. Submitted in 2007, extensive revisions in 2013 (!). V3: Added references, clarified examples. 35 p

An elementary chromatic reduction for gain graphs and special hyperplane arrangements

A gain graph is a graph whose edges are labelled invertibly by "gains" from a group. "Switching" is a transformation of gain graphs that generalizes conjugation in a group. A "weak chromatic function" of gain graphs with gains in a fixed group satisfies three laws: deletion-contraction for links with neutral gain, invariance under switching, and nullity on graphs with a neutral loop. The laws lead to the "weak chromatic group" of gain graphs, which is the universal domain for weak chromatic functions. We find expressions, valid in that group, for a gain graph in terms of minors without neutral-gain edges, or with added complete neutral-gain subgraphs, that generalize the expression of an ordinary chromatic polynomial in terms of monomials or falling factorials. These expressions imply relations for chromatic functions of gain graphs. We apply our relations to some special integral gain graphs including those that correspond to the Shi, Linial, and Catalan arrangements, thereby obtaining new evaluations of and new ways to calculate the zero-free chromatic polynomial and the integral and modular chromatic functions of these gain graphs, hence the characteristic polynomials and hypercubical lattice-point counting functions of the arrangements. We also calculate the total chromatic polynomial of any gain graph and especially of the Catalan, Shi, and Linial gain graphs.Comment: 31 page

How is a Chordal Graph like a Supersolvable Binary Matroid?

Let G be a finite simple graph. From the pioneering work of R. P. Stanley it is known that the cycle matroid of G is supersolvable iff G is chordal (rigid): this is another way to read Dirac's theorem on chordal graphs. Chordal binary matroids are not in general supersolvable. Nevertheless we prove that, for every supersolvable binary matroid M, a maximal chain of modular flats of M canonically determines a chordal graph.Comment: 10 pages, 3 figures, to appear in Discrete Mathematic