A note on the γ\gamma-coefficients of the "tree Eulerian polynomial"

We consider the generating polynomial of the number of rooted trees on the set $\{1,2,\dots,n\}$ counted by the number of descending edges (a parent with a greater label than a child). This polynomial is an extension of the descent generating polynomial of the set of permutations of a totally ordered $n$-set, known as the Eulerian polynomial. We show how this extension shares some of the properties of the classical one. B. Drake proved that this polynomial factors completely over the integers. From his product formula it can be concluded that this polynomial has positive coefficients in the $\gamma$-basis and we show that a formula for these coefficients can also be derived. We discuss various combinatorial interpretations of these positive coefficients in terms of leaf-labeled binary trees and in terms of the Stirling permutations introduced by Gessel and Stanley. These interpretations are derived from previous results of the author and Wachs related to the poset of weighted partitions and the free multibracketed Lie algebra.Comment: 13 pages, 6 figures, Interpretations derived from results in arXiv:1309.5527 and arXiv:1408.541

On the (co)homology of the poset of weighted partitions

We consider the poset of weighted partitions $\Pi_n^w$, introduced by Dotsenko and Khoroshkin in their study of a certain pair of dual operads. The maximal intervals of $\Pi_n^w$ provide a generalization of the lattice $\Pi_n$ of partitions, which we show possesses many of the well-known properties of $\Pi_n$. In particular, we prove these intervals are EL-shellable, we show that the M\"obius invariant of each maximal interval is given up to sign by the number of rooted trees on on node set $\{1,2,\dots,n\}$ having a fixed number of descents, we find combinatorial bases for homology and cohomology, and we give an explicit sign twisted $\mathfrak{S}_n$-module isomorphism from cohomology to the multilinear component of the free Lie algebra with two compatible brackets. We also show that the characteristic polynomial of $\Pi_n^w$ has a nice factorization analogous to that of $\Pi_n$.Comment: 50 pages, final version, to appear in Trans. AM

On the half-plane property and the Tutte group of a matroid

A multivariate polynomial is stable if it is non-vanishing whenever all variables have positive imaginary parts. A matroid has the weak half-plane property (WHPP) if there exists a stable polynomial with support equal to the set of bases of the matroid. If the polynomial can be chosen with all of its nonzero coefficients equal to one then the matroid has the half-plane property (HPP). We describe a systematic method that allows us to reduce the WHPP to the HPP for large families of matroids. This method makes use of the Tutte group of a matroid. We prove that no projective geometry has the WHPP and that a binary matroid has the WHPP if and only if it is regular. We also prove that T_8 and R_9 fail to have the WHPP.Comment: 8 pages. To appear in J. Combin. Theory Ser.

Weighted partitions

In this extended abstract we consider the poset of weighted partitions Π _n^w, introduced by Dotsenko and Khoroshkin in their study of a certain pair of dual operads. The maximal intervals of Π _n^w provide a generalization of the lattice Π _n of partitions, which we show possesses many of the well-known properties of Π _n. In particular, we prove these intervals are EL-shellable, we compute the Möbius invariant in terms of rooted trees, we find combinatorial bases for homology and cohomology, and we give an explicit sign twisted S_n-module isomorphism from cohomology to the multilinear component of the free Lie algebra with two compatible brackets. We also show that the characteristic polynomial of Π _n^w has a nice factorization analogous to that of Π _n

Realizing the ss-permutahedron via flow polytopes

Ceballos and Pons introduced the $s$-weak order on $s$-decreasing trees, for any weak composition $s$. They proved that it has a lattice structure and further conjectured that it can be realized as the $1$-skeleton of a polyhedral subdivision of a polytope. We answer their conjecture in the case where $s$ is a strict composition by providing three geometric realizations of the $s$-permutahedron. The first one is the dual graph of a triangulation of a flow polytope of high dimension. The second one, obtained using the Cayley trick, is the dual graph of a fine mixed subdivision of a sum of hypercubes that has the conjectured dimension. The third one, obtained using tropical geometry, is the $1$-skeleton of a polyhedral complex for which we can provide explicit coordinates of the vertices and whose support is a permutahedron as conjectured.Comment: 39 pages, 14 figure

A combinatorial model for computing volumes of flow polytopes

We introduce new families of combinatorial objects whose enumeration computes volumes of flow polytopes. These objects provide an interpretation, based on parking functions, of Baldoni and Vergne's generalization of a volume formula originally due to Lidskii. We recover known flow polytope volume formulas and prove new volume formulas for flow polytopes that were seemingly unapproachable. A highlight of our model is an elegant formula for the flow polytope of a graph we call the caracol graph. As by-products of our work, we uncover a new triangle of numbers that interpolates between Catalan numbers and the number of parking functions, we prove the log-concavity of rows of this triangle along with other sequences derived from volume computations, and we introduce a new Ehrhart-like polynomial for flow polytope volume and conjecture product formulas for the polytopes we consider.Comment: 34 pages, 15 figures. v2: updated after referee reports; includes a proof of Proposition 8.7. Accepted into Transactions of the AM

Weighted partitions

In this extended abstract we consider the poset of weighted partitions Π _n^w, introduced by Dotsenko and Khoroshkin in their study of a certain pair of dual operads. The maximal intervals of Π _n^w provide a generalization of the lattice Π _n of partitions, which we show possesses many of the well-known properties of Π _n. In particular, we prove these intervals are EL-shellable, we compute the Möbius invariant in terms of rooted trees, we find combinatorial bases for homology and cohomology, and we give an explicit sign twisted S_n-module isomorphism from cohomology to the multilinear component of the free Lie algebra with two compatible brackets. We also show that the characteristic polynomial of Π _n^w has a nice factorization analogous to that of Π _n

Conjuntos parcialmente ordenados y retículos

Lecture notes of the minicourse “Introduction to partially ordered sets and lattices”. A great part of these notes is based on the work and lecture notes by Richard Stanley [5, Capítulo 3] and Federico Ardila [1, Sección 4] for the part on partially ordered sets, and on the lecture notes of Michelle Wachs [8] for the poset topology partNotas de clase del minicurso de introducción a los conjuntos parcialmente ordenados y retículos. La mayor parte de estas notas está basada en la obra y notas de clase de Richard Stanley [5, Capítulo 3] y en el artículo de Federico Ardila [1, Sección 4] para la parte de conjuntos parcialmente ordenados; y en las notas de clase de Michelle Wachs [8] para la parte de topología de conjuntos parcialmente ordenados