37 research outputs found
M-partitions: Optimal partitions of weight for one scale pan
An M-partition of a positive integer m is a partition with as few parts as
possible such that any positive integer less than m has a partition made up of
parts taken from that partition of m. This is equivalent to partitioning a
weight m so as to be able to weigh any integer weight l < m with as few weights
as possible and only one scale pan.
We show that the number of parts of an M-partition is a log-linear function
of m and the M-partitions of m correspond to lattice points in a polytope. We
exhibit a recurrence relation for counting the number of M-partitions of m and,
for ``half'' of the positive integers, this recurrence relation will have a
generating function. The generating function will be, in some sense, the same
as the generating function for counting the number of distinct binary
partitions for a given integer.Comment: 11 page
Ehrhart clutters: Regularity and Max-Flow Min-Cut
If C is a clutter with n vertices and q edges whose clutter matrix has column
vectors V={v1,...,vq}, we call C an Ehrhart clutter if {(v1,1),...,(vq,1)} is a
Hilbert basis. Letting A(P) be the Ehrhart ring of P=conv(V), we are able to
show that if A is the clutter matrix of a uniform, unmixed MFMC clutter C, then
C is an Ehrhart clutter and in this case we provide sharp bounds on the
Castelnuovo-Mumford regularity of A(P). Motivated by the Conforti-Cornuejols
conjecture on packing problems, we conjecture that if C is both ideal and the
clique clutter of a perfect graph, then C has the MFMC property. We prove this
conjecture for Meyniel graphs, by showing that the clique clutters of Meyniel
graphs are Ehrhart clutters. In much the same spirit, we provide a simple proof
of our conjecture when C is a uniform clique clutter of a perfect graph. We
close with a generalization of Ehrhart clutters as it relates to total dual
integrality.Comment: Electronic Journal of Combinatorics, to appea