arXiv subjects

Andrew Sack

Publications and source records attributed to Andrew Sack.

8 recordsLinked to original sources

arXiv2026

Interval hypergraphic polytopes (or deformed associahedra), Tamari interval posets, and weeping willows

For a hypergraph $\mathbb{H}$ on $[n]$, the hypergraphic polytope $\triangle_{\mathbb{H}}$ is the Minkowski sum of the standard simplices $\triangle_H$ for all $H \in \mathbb{H}$. We focus here on interval hypergraphs, where all hyperedges are intervals of $[n]$. They are precisely the deformations of Loday's associahedron. Their vertex posets are Tamari interval posets, and we describe which Tamari interval poset appears as a vertex poset in which interval hypergraphic polytope. We also characterize the interval hypergraphs $\mathbb{I}$ for which the hypergraphic polytope $\triangle_\mathbb{I}$ is simple, and we study their vertex posets, which we call weeping willows.

math.CO↗

arXiv2026

Lattices from Pointed Building Sets: Generalized Ornamentation Lattices

We introduce a novel combinatorial structure called pointed building sets, which can be viewed as families of lattices equipped with compatibility relations. To each pointed building set $\mathsf{B}$, we associate a complete lattice $\mathcal{O}(\mathsf{B})$, called the ornamentation lattice of $\mathsf{B}$. Special cases of this construction have already proven useful in understanding the structure of three families of posets: operahedron lattices, the affine Tamari lattice, and hypergraphic posets of subhypergraphs of the path hypergraph of an increasing tree. The goal of this paper is to establish the theory of these generalized ornamentations. We examine several natural classes of pointed building sets which recover classical lattices such as the Tamari lattice, the lattice of topologies ordered by coarsening, and the lattice of naturally labeled partial orders. Furthermore, several theoretical directions are explored, including inverse limits and group actions. Notably, this leads to a straightforward construction of inverse limits of Tamari lattices, yielding infinite analogs of the Tamari lattice.

math.CO↗

arXiv2025

Ornamentation lattices and intreeval hypergraphic lattices

Given a directed graph $D$ with transitive closure $\operatorname{tc}(D)$ and path hypergraph $\mathbb{P}(D)$, we study the connections between the (acyclic) reorientation poset of $\operatorname{tc}(D)$, the (acyclic) sourcing poset of $\mathbb{P}(D)$, and the (acyclic) ornamentation poset of $D$. Geometrically, the acyclic reorientation poset of $\operatorname{tc}(D)$ (resp. the acyclic sourcing poset of $\mathbb{P}(D)$) is the transitive closure of the skeleton of the graphical zonotope of $\operatorname{tc}(D)$ (resp. of the hypergraphic polytope of $\mathbb{P}(D)$) oriented in a linear direction. When $D$ is a rooted (or even unstarred) increasing tree, we show that the acyclic sourcing poset of $\mathbb{P}(D)$ is isomorphic to the ornamentation lattice of $D$, and that they form a lattice quotient of the acyclic reorientation lattice of $\operatorname{tc}(D)$. As a consequence, we obtain polytopal realizations of the ornamentation lattices of rooted (or even unstarred) increasing trees, answering an open question of C. Defant and A. Sack. When $D$ is an increasing tree, we show that the ornamentation lattice of $D$ is the MacNeille completion of the acyclic sourcing poset of $\mathbb{P}(D)$. Finally, still when $D$ is an increasing tree, we use the ornamentation lattice of $D$ to characterize the subhypergraphs of the path hypergraph $\mathbb{P}(D)$ whose acyclic sourcing poset is a lattice.

math.CO↗

arXiv2024

Operahedron Lattices

Laplante-Anfossi associated to each rooted plane tree a polytope called an operahedron. He also defined a partial order on the vertex set of an operahedron and asked if the resulting poset is a lattice. We answer this question in the affirmative, motivating us to name Laplante-Anfossi's posets operahedron lattices. The operahedron lattice of a chain with $n+1$ vertices is isomorphic to the $n$-th Tamari lattice, while the operahedron lattice of a claw with $n+1$ vertices is isomorphic to $\mathrm{Weak}(\mathfrak S_n)$, the weak order on the symmetric group $\mathfrak S_n$. We characterize semidistributive operahedron lattices and trim operahedron lattices. Let $\Delta_{\mathrm{Weak}(\mathfrak S_n)}(w_\circ(k,n))$ be the principal order ideal of $\mathrm{Weak}(\mathfrak S_n)$ generated by the permutation ${w_\circ(k,n)=k(k-1)\cdots 1(k+1)(k+2)\cdots n}$. Our final result states that the operahedron lattice of a broom with $n+1$ vertices and $k$ leaves is isomorphic to the subposet of $\mathrm{Weak}(\mathfrak S_n)$ consisting of the preimages of $\Delta_{\mathrm{Weak}(\mathfrak S_n)}(w_\circ(k,n))$ under West's stack-sorting map; as a consequence, we deduce that this subposet is a semidistributive lattice.

math.CO↗

arXiv2023

Poset Associahedra and Stack-sorting

For any finite connected poset $P$, Galashin introduced a simple convex $(|P|-2)$-dimensional polytope $\mathscr{A}(P)$ called the poset associahedron. For a certain family of posets, whose poset associahedra interpolate between the classical permutohedron and associahedron, we give a simple combinatorial interpretation of the $h$-vector. Our interpretation relates to the theory of stack-sorting of permutations. It also allows us to prove real-rootedness of some of their $h$-polynomials.

math.CO↗

arXiv2023

The poset associahedron $f$-vector is a comparability invariant

We show that the $f$-vector of Galashin's poset associahedron $\mathscr A(P)$ only depends on the comparability graph of $P$. In particular, this allows us to produce a family of polytopes with the same $f$-vectors as permutohedra, but that are not combinatorially equivalent to permutohedra.

math.CO↗

arXiv2023

A realization of poset associahedra

Given any connected poset $P$, we give a simple realization of Galashin's poset associahedron $\mathscr{A}(P)$ as a convex polytope in $\mathbb{R}^P.$ The realization is inspired by the description of $\mathscr{A}(P)$ as a compactification of the configuration space of order-preserving maps $P \to \mathbb{R}.$ In addition, we give an analogous realization for Galashin's affine poset cyclohedra.

math.CO↗

arXiv2021

On audio enhancement via online non-negative matrix factorization

We propose a method for noise reduction, the task of producing a clean audio signal from a recording corrupted by additive noise. Many common approaches to this problem are based upon applying non-negative matrix factorization to spectrogram measurements. These methods use a noiseless recording, which is believed to be similar in structure to the signal of interest, and a pure-noise recording to learn dictionaries for the true signal and the noise. One may then construct an approximation of the true signal by projecting the corrupted recording on to the clean dictionary. In this work, we build upon these methods by proposing the use of \emph{online} non-negative matrix factorization for this problem. This method is more memory efficient than traditional non-negative matrix factorization and also has potential applications to real-time denoising.

eess.AS↗