15 research outputs found
The Maximal Denumerant of a Numerical Semigroup
Given a numerical semigroup S = and n in S, we
consider the factorization n = c_0 a_0 + c_1 a_1 + ... + c_t a_t where c_i >=
0. Such a factorization is maximal if c_0 + c_1 + ... + c_t is a maximum over
all such factorizations of n. We provide an algorithm for computing the maximum
number of maximal factorizations possible for an element in S, which is called
the maximal denumerant of S. We also consider various cases that have
connections to the Cohen-Macualay and Gorenstein properties of associated
graded rings for which this algorithm simplifies.Comment: 13 Page
Numerical semigroups with large embedding dimension satisfy Wilf's conjecture
We give an affirmative answer to Wilf's conjecture for numerical semigroups
satisfying 2 \nu \geq m, where \nu and m are respectively the embedding
dimension and the multiplicity of a semigroup. The conjecture is also proved
when m \leq 8 and when the semigroup is generated by a generalized arithmetic
sequence.Comment: 13 page
Parametric Polyhedra with at least Lattice Points: Their Semigroup Structure and the k-Frobenius Problem
Given an integral matrix , the well-studied affine semigroup
\mbox{ Sg} (A)=\{ b : Ax=b, \ x \in {\mathbb Z}^n, x \geq 0\} can be
stratified by the number of lattice points inside the parametric polyhedra
. Such families of parametric polyhedra appear in
many areas of combinatorics, convex geometry, algebra and number theory. The
key themes of this paper are: (1) A structure theory that characterizes
precisely the subset \mbox{ Sg}_{\geq k}(A) of all vectors b \in \mbox{
Sg}(A) such that has at least solutions. We
demonstrate that this set is finitely generated, it is a union of translated
copies of a semigroup which can be computed explicitly via Hilbert bases
computations. Related results can be derived for those right-hand-side vectors
for which has exactly solutions or fewer
than solutions. (2) A computational complexity theory. We show that, when
, are fixed natural numbers, one can compute in polynomial time an
encoding of \mbox{ Sg}_{\geq k}(A) as a multivariate generating function,
using a short sum of rational functions. As a consequence, one can identify all
right-hand-side vectors of bounded norm that have at least solutions. (3)
Applications and computation for the -Frobenius numbers. Using Generating
functions we prove that for fixed the -Frobenius number can be
computed in polynomial time. This generalizes a well-known result for by
R. Kannan. Using some adaptation of dynamic programming we show some practical
computations of -Frobenius numbers and their relatives
Linear Extensions and Comparable Pairs in Partial Orders
We study the number of linear extensions of a partial order with a given proportion of comparable pairs of elements, and estimate the maximum and minimum possible numbers. We also show that a random interval partial order on elements has close to a third of the pairs comparable with high probability, and the number of linear extensions is with high probability