3,069 research outputs found
Polynomial sequences of binomial-type arising in graph theory
In this paper, we show that the solution to a large class of "tiling"
problems is given by a polynomial sequence of binomial type. More specifically,
we show that the number of ways to place a fixed set of polyominos on an
toroidal chessboard such that no two polyominos overlap is
eventually a polynomial in , and that certain sets of these polynomials
satisfy binomial-type recurrences. We exhibit generalizations of this theorem
to higher dimensions and other lattices. Finally, we apply the techniques
developed in this paper to resolve an open question about the structure of
coefficients of chromatic polynomials of certain grid graphs (namely that they
also satisfy a binomial-type recurrence).Comment: 15 page
An Abstraction of Whitney's Broken Circuit Theorem
We establish a broad generalization of Whitney's broken circuit theorem on
the chromatic polynomial of a graph to sums of type
where is a finite set and is a mapping from the power set of into
an abelian group. We give applications to the domination polynomial and the
subgraph component polynomial of a graph, the chromatic polynomial of a
hypergraph, the characteristic polynomial and Crapo's beta invariant of a
matroid, and the principle of inclusion-exclusion. Thus, we discover several
known and new results in a concise and unified way. As further applications of
our main result, we derive a new generalization of the maximums-minimums
identity and of a theorem due to Blass and Sagan on the M\"obius function of a
finite lattice, which generalizes Rota's crosscut theorem. For the classical
M\"obius function, both Euler's totient function and its Dirichlet inverse, and
the reciprocal of the Riemann zeta function we obtain new expansions involving
the greatest common divisor resp. least common multiple. We finally establish
an even broader generalization of Whitney's broken circuit theorem in the
context of convex geometries (antimatroids).Comment: 18 page
Integer points and their orthogonal lattices
Linnik proved in the late 1950's the equidistribution of integer points on
large spheres under a congruence condition. The congruence condition was lifted
in 1988 by Duke (building on a break-through by Iwaniec) using completely
different techniques. We conjecture that this equidistribution result also
extends to the pairs consisting of a vector on the sphere and the shape of the
lattice in its orthogonal complement. We use a joining result for higher rank
diagonalizable actions to obtain this conjecture under an additional congruence
condition.Comment: 15 pages, and 6 pages of appendix, (Appendix by Ruixiang Zhang
- β¦