Related papers: A matrix approach to the structure, enumeration, a…
We generalize the concept of combinatorial nested set complexes to posets and exhibit the topological relationship between the arising nested set complexes and the order complex of the underlying poset. In particular, a sufficient condition…
We prove that any lower unitriangular and totally nonnegative matrix gives rise to a family of polynomials with only real zeros. This has consequences for problems in several areas of mathematics. We use it to develop a general theory for…
Partially ordered patterns (POPs) generalize the notion of classical patterns studied in the literature in the context of permutations, words, compositions and partitions. In this paper, we give a number of general, and specific enumerative…
A finite poset X carries a natural structure of a topological space. Fix a field k, and denote by D(X) the bounded derived category of sheaves of finite dimensional k-vector spaces over X. Two posets X and Y are said to be derived…
We decompose a matrix Y into a sum of bilinear terms in a stepwise manner, by considering Y as a mapping from a finite dimensional Banach space into another finite dimensional Banach space. We provide transition formulas, and represent them…
In this manuscript we study inclusion posets of Borel orbit closures on (symmetric) matrices. In particular, we show that the Bruhat poset of partial involutions is a lexicographiically shellable poset. Also, studying the embeddings of…
In certain finite posets, the expected down-degree of their elements is the same whether computed with respect to either the uniform distribution or the distribution weighting an element by the number of maximal chains passing through it.…
Matrix completion is a problem that arises in many data-analysis settings where the input consists of a partially-observed matrix (e.g., recommender systems, traffic matrix analysis etc.). Classical approaches to matrix completion assume…
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 poset is representable if it can be embedded in a field of sets in such a way that existing finite meets and joins become intersections and unions respectively (we say finite meets and joins are preserved). More generally, for cardinals…
The proposed article aims at offering a comprehensive tutorial for the computational aspects of structured matrix and tensor factorization. Unlike existing tutorials that mainly focus on {\it algorithmic procedures} for a small set of…
We define toric partial orders, corresponding to regions of graphic toric hyperplane arrangements, just as ordinary partial orders correspond to regions of graphic hyperplane arrangements. Combinatorially, toric posets correspond to finite…
In section 1 we consider a 3-tuple $S=(|S|,\preccurlyeq,E)$ where $|S|$ is a finite set, $\preccurlyeq$ a partial ordering on $|S|,$ and $E$ a set of unordered pairs of distinct members of $|S|,$ and study, as a function of $n\geq 0,$ the…
Mathematica offers, by way of the package Combinatorics, many useful functions to work on graphs and ordered structures, but none of these functions was specific enough to meet the needs of our research group. Moreover, the existing…
The classical linear ordering problem seeks a single ranking representing a given preference matrix. While suitable for homogeneous populations, it fails when observed preferences arise from several latent groups with distinct ranking…
In this thesis, we consider the problem of characterizing and enumerating sets of polyominoes described in terms of some constraints, defined either by convexity or by pattern containment. We are interested in a well known subclass of…
We introduce a new combinatorial invariant, which we call crosscut poset, that is finer than the crosscut complex. We exhibit many applications of the crosscut poset which include a generalization of Bj\"orner's crosscut theorem and two…
Many matching, tracking, sorting, and ranking problems require probabilistic reasoning about possible permutations, a set that grows factorially with dimension. Combinatorial optimization algorithms may enable efficient point estimation,…
We introduce a formal definition of a pattern poset which encompasses several previously studied posets in the literature. Using this definition we present some general results on the M\"obius function and topology of such pattern posets.…
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…