Decision trees, monotone functions, and semimatroids

We define decision trees for monotone functions on a simplicial complex. We define homology decidability of monotone functions, and show that various monotone functions related to semimatroids are homology decidable. Homology decidability is a generalization of semi-nonevasiveness, a notion due to Jonsson. The motivating example is the complex of bipartite graphs, whose Betti numbers are unknown in general. We show that these monotone functions have optimum decision trees, from which we can compute relative Betti numbers of related pairs of simplicial complexes. Moreover, these relative Betti numbers are coefficients of evaluations of the Tutte polynomial, and every semimatroid collapses onto its broken circuit complex.Comment: 16 page

Chromatic Polynomials and Rings in Species

Abstract. We present a generalization of the chromatic polynomial, and chromatic symmetric function, arising in the study of combinatorial species. These invariants are defined for modules over lattice rings in species. The primary examples are graphs and set partitions. For these new invariants, we present analogues of results regarding stable partitions, the bond lattice, the deletion-contraction recurrence, and the subset expansion formula. We also present two detailed examples, one related to enumerating subgraphs by their blocks, and a second example related to enumerating subgraphs of a directed graph by their strongly connected components. Resumé. Nous présentons une généralisation du polynôme chromatique et de la fonction symétrique chromatique, qui apparaissent dans l’étude des espèces de structures. Ces invariants sont définis pour modules sur anneaux réticulés aux espéces de structures. Les exemples principaux sont les graphes et les partitions d’entiers. Pour ces invariants nouveaux, nous présentons d’analogues de rsultats concernants les partitions stables, le treillis de liaisons, la rélation de contraction-suppression, et la formule d’expansion en termes de sous-ensembles. Nous présentons aussi deux exemples détaill´s, l’un lié à l’énumération des sous-graphes par ses blocs, et l’autre lié à l’énumération des sousgraphes d’un graphe dirigé par ses composantes fortement connexes

The Discrete Fundamental Group of the Associahedron, and the Exchange Module

The associahedron is an object that has been well studied and has numerous applications, particularly in the theory of operads, the study of non-crossing partitions, lattice theory and more recently in the study of cluster algebras. We approach the associahedron from the point of view of discrete homotopy theory. We study the abelianization of the discrete fundamental group, and show that it is free abelian of rank $\binom{n+2}{4}$. We also find a combinatorial description for a basis of this rank. We also introduce the exchange module of the type $A_n$ cluster algebra, used to model the relations in the cluster algebra. We use the discrete fundamental group to the study of exchange module, and show that it is also free abelian of rank $\binom{n+2}{3}$.Comment: 16 pages, 4 figure

ALMA and VLA Observations of the HD 141569 System

We present VLA 9 mm (33 GHz) observations of the HD 141569 system from semester 16A. The observations achieve a resolution of 0.25 arcsec ($\sim28$ au) and a sensitivity of $4.7~\mu \rm Jy~beam^{-1}$. We find (1) a $52\pm 5~\mu$Jy point source at the location of HD 141569A that shows potential variability, (2) the detected flux is contained within the SED-inferred central clearing of the disc meaning the spectral index of the dust disc is steeper than previously inferred, and (3) the M dwarf companions are also detected and variable. Previous lower-resolution VLA observations (semester 14A) found a higher flux density, interpreted as solely dust emission. When combined with ALMA observations, the VLA 14A observations suggested the spectral index and grain size distribution of HD 141569's disc was shallow and an outlier among debris systems. Using archival ALMA observations of HD 141569 at 0.87 mm and 2.9 mm we find a dust spectral index of $\alpha_{\rm mm} = 1.81\pm 0.20$. The VLA 16A flux corresponds to a brightness temperature of $\sim5\times10^{6}$ K, suggesting strong non-disc emission is affecting the inferred grain properties. The VLA 16A flux density of the M2V companion HD 141569B is $149\pm9~\mu$Jy, corresponding to a brightness temperature of $\sim2\times10^{8}$ K and suggesting significant stellar variability when compared to the VLA14A observations, which are smaller by a factor of $\sim6$.Comment: Accepted for publication in MNRAS, 8 pages, 6 figure

Coloring complexes and combinatorial Hopf monoids

We generalize the notion of a coloring complex of a graph to linearized combinatorial Hopf monoids. We determine when a linearized combinatorial Hopf monoid has such a construction, and discover some inequalities that are satisfied by the quasisymmetric function invariants associated to the combinatorial Hopf monoid. We show that the collection of all such coloring complexes forms a linearized combinatorial Hopf monoid, which is the terminal object in the category of combinatorial Hopf monoids with convex characters. We also study several examples of combinatorial Hopf monoids

On Equivariant flag f-vectors for balanced relative simplicial complexes

We study the equivariant flag f-vector and equivariant flag h-vector of a balanced relative simplicial complex with respect to a group action. When the complex satisfies Serre\u27s condition (Sℓ), we show that the equivariant flag h-vector, the equivariant h-vector, and the equivariant f-vector satisfy several inequalities.We apply these results to the study of P-partitions of double posets, and weak colorings of mixed graphs