Related papers: Numerical semigroups, polyhedra, and posets I: the…
Several recent papers have examined a rational polyhedron $P_m$ whose integer points are in bijection with the numerical semigroups (cofinite, additively closed subsets of the non-negative integers) containing $m$. A combinatorial…
This paper is the third in a series of manuscripts that examine the combinatorics of the Kunz polyhedron $P_m$, whose positive integer points are in bijection with numerical semigroups (cofinite subsemigroups of $\mathbb Z_{\ge 0}$) whose…
A common tool in the theory of numerical semigroups is to interpret a desired class of semigroups as the integer lattice points in a rational polyhedron in order to leverage computational and enumerative techniques from polyhedral geometry.…
We study the question whether the affine semigroup of integer points in a convex cone can be finitely generated up to symmetries of the cone. We establish general properties of finite generation up to symmetry, and then concentrate on the…
To every poset P, Stanley (1986) associated two polytopes, the order polytope and the chain polytope, whose geometric properties reflect the combinatorial qualities of P. This construction allows for deep insights into combinatorics by way…
A numerical semigroup is a cofinite subset of $\mathbb Z_{\ge 0}$ containing $0$ and closed under addition. Each numerical semigroup $S$ with smallest positive element $m$ corresponds to an integer point in the Kunz cone $\mathcal C_m…
We investigate the semigroup of integer points inside a convex cone. We extend classical results in integer linear programming to integer conic programming. We show that the semigroup associated with nonpolyhedral cones can sometimes have a…
We examine two natural operations to create numerical semigroups. We say that a numerical semigroup $\mathcal{S}$ is $k$-normalescent if it is the projection of the set of integer points in a $k$-dimensional polyhedral cone, and we say that…
We introduce Cayley posets as posets arising naturally from pairs $S<T$ of semigroups, much in the same way that Cayley graph arises from a (semi)group and a subset. We show that Cayley posets are a common generalization of several known…
We study a class of polyhedra associated to marked posets. Examples of these polyhedra are Gelfand-Tsetlin polytopes and cones, as well as Berenstein-Zelevinsky polytopes, all of which have appeared in the representation theory of…
We introduce a new family of finite posets which we call 2-chains. These first arose in the study of 0-Hecke algebras, but they admit a variety of different characterisations. We give these characterisations, prove that they are equivalent…
A numerical semigroup is a cofinite subset of the non-negative integers that is closed under addition and contains 0. Each numerical semigroup $S$ with fixed smallest positive element $m$ corresponds to an integer point in a rational…
In this paper we present a new approach to construct the set of numerical semigroups with a fixed genus. Our methodology is based on the construction of the set of numerical semigroups with fixed Frobenius number and genus. An equivalence…
We show that double cosets of the infinite symmetric group with respect to some special subgroups admit natural structures of semigroups. We interpret elements of such semigroups in combinatorial terms (chips, colored graphs,…
Finite quasi semimetrics on $n$ can be thought of as nonnegative valuations on the edges of a complete directed graph on $n$ vertices satisfying all possible triangle inequalities. They comprise a polyhedral cone whose symmetry groups were…
Let p<q be coprime integers. Kunz-Waldi semigroups are numerical semigroups containing p and q and contained in <p,q,r>, where 2r=p,q,p+q whichever is even. In this paper, we prove a conjecture on the Betti numbers of the semigroup rings of…
This paper proposes a new, visual method to study numerical semigroups and the Frobenius problem. The method is based on building a so-called reduction graph, whose nodes usually correspond to monogenic semigroups, and whose edges can have…
The recent proof of the Boij-Soederberg conjectures reveals new structure about Betti diagrams of modules, giving a complete description of the cone of Betti diagrams. We begin to expand on this new structure by investigating the semigroup…
We study the structure of the family of numerical semigroups with fixed multiplicity and Frobenius number. We give an algorithmic method to compute all the semigroups in this family. As an application we compute the set of all numerical…
We adapt here the computation of characters on incidence Hopf algebras introduced by W. Schmitt in the 1990s to a family mixing bounded and unbounded posets. We then apply our results to the family of hypertree posets and partition posets.…