4 research outputs found
Inclusion Matrices and Chains
Given integers , , and such that , let
be the inclusion matrix of -subsets vs. -subsets of a
-set. We modify slightly the concept of standard tableau to study the notion
of rank of a finite set of positive integers which was introduced by Frankl.
Utilizing this, a decomposition of the poset into symmetric skipless
chains is given. Based on this decomposition, we construct an inclusion matrix,
denoted by , which is row-equivalent to . Its Smith
normal form is determined. As applications, Wilson's diagonal form of
is obtained as well as a new proof of the well known theorem on the
necessary and sufficient conditions for existence of integral solutions of the
system due to Wilson. Finally we present anotherinclusion
matrix with similar properties to those of which is in some
way equivalent to .Comment: Accepted for publication in Journal of Combinatorial Theory, Series
Integer diagonal forms for subset intersection relations
For integers , we give a
description for the Smith group of the incidence matrix with rows (columns)
indexed by the size (, respectively) subsets of an -element set,
where incidence means intersection in a set of size . This generalizes
work of Wilson and Bier from the 1990s which dealt only with the case where
incidence meant inclusion. Our approach also describes the Smith group of any
matrix in the -linear span of these matrices so includes all
integer matrices in the Bose-Mesner algebra of the Johnson association scheme:
for example, the association matrices themselves as well as the Laplacian,
signless Laplacian, Seidel adjacency matrix, etc. of the associated graphs. In
particular, we describe the critical (also known as sandpile) groups of these
graphs. The complexity of our formula grows with the parameters, but is
independent of and , which often leads to an efficient algorithm for
computing these groups. We illustrate our techniques to give diagonal forms of
matrices attached to the Kneser and Johnson graphs for subsets of size ,
whose invariants have never before been described, and recover results from a
variety of papers in the literature in a unified way.Comment: 28 page