Related papers: Short, new proofs of Dowker duality
Diagram-chasing arguments frequently lead to "magical" relations between distant points of diagrams: exactness implications, connecting morphisms, etc.. These long connections are usually composites of short "unmagical" connections, but the…
We introduce higher simplicial complexity of a simplicial complex $K$ and higher combinatorial complexity of a finite space $P$ (i.e. $P$ is a finite poset). We relate higher simplicial complexity with higher topological complexity of $|K|$…
We prove a certain duality relation for orthogonal polynomials defined on a finite set. The result is used in a direct proof of the equivalence of two different ways of computing the correlation functions of a discrete orthogonal polynomial…
We review the notion of symplectic duality earlier introduced in the context of topological recursion. We show that the transformation of symplectic duality can be expressed as a composition of $x-y$ dualities in a broader context of log…
We show that every gammoid has special digraph representations, such that a representation of the dual of the gammoid may be easily obtained by reversing all arcs. In an informal sense, the duality notion of a poset applied to the digraph…
We introduce decomposition complexes of posets, which generalize order complexes. The main advantage of our construction is that decomposition complexes are closed under taking products. Other special instances of this theory include nested…
We collect three observations on the homology for Smale spaces defined by Putnam. The definition of such homology groups involves four complexes. It is shown here that a simple convergence theorem for spectral sequences can be used to prove…
The notion of a duality between two derived functors as well as an extension theorem for derived functors to larger categories in which they need not be defined is introduced. These ideas are then applied to extend and study the coext…
We prove analogues for sub-posets of the Dowling lattices of the results of Calderbank, Hanlon, and Robinson on homology of sub-posets of the partition lattices. The technical tool used is the wreath product analogue of the tensor species…
We provide a combinatorial recipe for constructing all posets of height at most two for which the corresponding type-A Lie poset algebra is contact. In the case that such posets are connected, a discrete Morse theory argument establishes…
This paper describes the homology of various simplicial complexes associated to set families from combinatorial number theory, including primitive sets, pairwise coprime sets, product-free sets, and coprime-free sets. We present a condition…
Given two discrete Morse functions on a simplicial complex, we introduce the {\em connectedness homomorphism} between the corresponding discrete Morse complexes. This concept leads to a novel framework for studying the connectedness in…
We introduce several new constructions of finite posets with the number of linear extensions given by generalized continued fractions. We apply our results to the problem of the minimum number of elements needed for a poset with a given…
We show that, for a simplicial complex, the supported cap product operation on Borel-Moore homology coincides with the supported cap product on simplicial homology. For this purpose we introduce the supported cap product for locally finite…
Recently, the author and Yamamoto invented a new proof of the duality for multiple zeta values. The technique is applicable in other series identities. In this article, we exhibit such proofs for some series identities.
This note is devoted to a trick which yields almost trivial proofs that certain complexes associated to topological surfaces are connected or simply connected. Applications include new proofs that the complexes of curves, separating curves,…
We consider sequences of absolute and relative homology and cohomology groups that arise naturally for a filtered cell complex. We establish algebraic relationships between their persistence modules, and show that they contain equivalent…
The theorem of orthogonal-orthogonal duality of Rowe, Repka, and Carvalho is proven by a method based on characters that is very different from theirs and akin to Helmers's half a century earlier proof of the analogous sympletic-symplectic…
We present a comparison map between the uberhomology of a simplicial complex $\mathcal{K}$ and the double homology of its associated moment-angle complex $\mathcal{Z}_{\mathcal{K}}$. We show these two homology theories differ at three…
This paper studies topological properties of the lattices of non-crossing partitions of types A and B and of the poset of injective words. Specifically, it is shown that after the removal of the bottom and top elements (if existent) these…