1,222 research outputs found
The growth rate over trees of any family of set defined by a monadic second order formula is semi-computable
Monadic second order logic can be used to express many classical notions of
sets of vertices of a graph as for instance: dominating sets, induced
matchings, perfect codes, independent sets or irredundant sets. Bounds on the
number of sets of any such family of sets are interesting from a combinatorial
point of view and have algorithmic applications. Many such bounds on different
families of sets over different classes of graphs are already provided in the
literature. In particular, Rote recently showed that the number of minimal
dominating sets in trees of order is at most and that
this bound is asymptotically sharp up to a multiplicative constant. We build on
his work to show that what he did for minimal dominating sets can be done for
any family of sets definable by a monadic second order formula.
We first show that, for any monadic second order formula over graphs that
characterizes a given kind of subset of its vertices, the maximal number of
such sets in a tree can be expressed as the \textit{growth rate of a bilinear
system}. This mostly relies on well known links between monadic second order
logic over trees and tree automata and basic tree automata manipulations. Then
we show that this "growth rate" of a bilinear system can be approximated from
above.We then use our implementation of this result to provide bounds on the
number of independent dominating sets, total perfect dominating sets, induced
matchings, maximal induced matchings, minimal perfect dominating sets, perfect
codes and maximal irredundant sets on trees. We also solve a question from D.
Y. Kang et al. regarding -matchings and improve a bound from G\'orska and
Skupie\'n on the number of maximal matchings on trees. Remark that this
approach is easily generalizable to graphs of bounded tree width or clique
width (or any similar class of graphs where tree automata are meaningful)
Betti diagrams from graphs
The emergence of Boij-S\"oderberg theory has given rise to new connections
between combinatorics and commutative algebra. Herzog, Sharifan, and Varbaro
recently showed that every Betti diagram of an ideal with a k-linear minimal
resolution arises from that of the Stanley-Reisner ideal of a simplicial
complex. In this paper, we extend their result for the special case of 2-linear
resolutions using purely combinatorial methods. Specifically, we show bijective
correspondences between Betti diagrams of ideals with 2-linear resolutions,
threshold graphs, and anti-lecture hall compositions. Moreover, we prove that
any Betti diagram of a module with a 2-linear resolution is realized by a
direct sum of Stanley-Reisner rings associated to threshold graphs. Our key
observation is that these objects are the lattice points in a normal reflexive
lattice polytope.Comment: To appear in Algebra and Number Theory, 15 pages, 7 figure
Central limit theorems for Gaussian polytopes
Choose random, independent points in according to the standard
normal distribution. Their convex hull is the {\sl Gaussian random
polytope}. We prove that the volume and the number of faces of satisfy
the central limit theorem, settling a well known conjecture in the field.Comment: to appear in Annals of Probabilit
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