8,786 research outputs found
On the complexity of the chip-firing reachability problem
In this paper, we study the complexity of the chip-firing reachability
problem. We show that for Eulerian digraphs, the reachability problem can be
decided in strongly polynomial time, even if the digraph has multiple edges. We
also show a special case when the reachability problem can be decided in
polynomial time for general digraphs: if the target distribution is recurrent
restricted to each strongly connected component. As a further positive result,
we show that the chip-firing reachability problem is in co-NP for general
digraphs. We also show that the chip-firing halting problem is in co-NP for
Eulerian digraphs
Powerful sets: a generalisation of binary matroids
A set of binary vectors, with positions indexed by ,
is said to be a \textit{powerful code} if, for all , the number
of vectors in that are zero in the positions indexed by is a power of
2. By treating binary vectors as characteristic vectors of subsets of , we
say that a set of subsets of is a \textit{powerful set} if
the set of characteristic vectors of sets in is a powerful code. Powerful
sets (codes) include cocircuit spaces of binary matroids (equivalently, linear
codes over ), but much more besides. Our motivation is that, to
each powerful set, there is an associated nonnegative-integer-valued rank
function (by a construction of Farr), although it does not in general satisfy
all the matroid rank axioms.
In this paper we investigate the combinatorial properties of powerful sets.
We prove fundamental results on special elements (loops, coloops, frames,
near-frames, and stars), their associated types of single-element extensions,
various ways of combining powerful sets to get new ones, and constructions of
nonlinear powerful sets. We show that every powerful set is determined by its
clutter of minimal nonzero members. Finally, we show that the number of
powerful sets is doubly exponential, and hence that almost all powerful sets
are nonlinear.Comment: 19 pages. This work was presented at the 40th Australasian Conference
on Combinatorial Mathematics and Combinatorial Computing (40ACCMCC),
University of Newcastle, Australia, Dec. 201
Counting subspaces of a finite vector space
In this expository article, we discuss the relation between the Gaussian
binomial and multinomial coefficients and ordinary binomial and multinomial
coefficients from a combinatorial viewpoint, based on expositions by Butler,
Knuth and Stanley.Comment: 15 pages, 3 figures, corrected versio
- …