We show that the well known {\em homotopy complementation formula} of Björner and Walker admits several closely related generalizations on different classes of topological posets (lattices). The utility of this technique is demonstrated on some classes of topological posets including the Grassmannian and configuration posets, $\widetilde{\mathbf G}_n(R)$ and $\exp_n(X)$ which were introduced and studied by V.~Vassiliev. Among other applications we present a reasonably complete description, in terms of more standard spaces, of homology types of configuration posets $\exp_n(S^m)$ which leads to a negative answer to a question of Vassilev raised at the workshop ``Geometric Combinatorics'' (MSRI, February 1997).
Sectional pseudocomplementation (sp-complementation) on a poset is a partial operation $*$ which associates with every pair $(x,y)$ of elements, where $x \ge y$, the pseudocomplement $x*y$ of $x$ in the upper section $[y)$. Any total extension $\to$ of $*$ is said to be an extended sp-complementation and is considered as an implication-like operation. Extended sp-complementations have already be studied on semilattices and lattices. We describe several naturally arising classes of general posets with extended sp-complementation, present respective elementary properties of this operation, demon…
We initiate the study of the enumerative combinatorics of the intervals in the Dyck pattern poset. More specifically, we find some closed formulas to express the size of some specific intervals, as well as the number of their covering relations. In most of the cases, we are also able to refine our formulas by rank. We also provide the first results on the Möbius function of the Dyck pattern poset, giving for instance a closed expression for the Möbius function of initial intervals whose maximum is a Dyck path having exactly two peaks.
In this paper we study maximal chains in certain lattices constructed from powers of chains by iterated lax colimits in the $2$-category of posets. Such a study is motivated by the fact that in lower dimensions, we get some familiar combinatorial objects such as Dyck paths and Kreweras walks.
Many combinatorial proofs rely on induction. When these proofs are formulated in traditional language, they can be bulky and unmanageable. Coalgebras provide a language which can reduce reduce many inductive proofs in graded poset theory to comprehensible size. As a bonus, the visual form of the resulting recursive proofs suggests combinatorial interpretations for constants appearing in the longer arguments. We use the techniques of coalgebras to compute invariants of toric and affine arrangements as well as of poset products. In additional chapters we prove structure theorems for acyclic orie…
We develop a tighter implementation of basic PL topology, which keeps track of some combinatorial structure beyond PL homeomorphism type. With this technique we clarify some aspects of PL transversality and give combinatorial proofs of a number of known results. New results include a combinatorial characterization of collapsible polyhedra in terms of constructible posets (generalizing face posets of constructible simplicial complexes in the sense of Hochster). The relevant constructible posets are also characterized in terms of Reading's zipping of posets, which is a variation of edge contract…
We study $ν$-Schröder paths, which are Schröder paths which stay weakly above a given lattice path $ν$. Some classical bijective and enumerative results are extended to the $ν$-setting, including the relationship between small and large Schröder paths. We introduce two posets of $ν$-Schröder objects, namely $ν$-Schröder paths and trees, and show that they are isomorphic to the face poset of the $ν$-associahedron $A_ν$ introduced by Ceballos, Padrol and Sarmiento. A consequence of our results is that the $i$-dimensional faces of $A_ν$ are indexed by $ν$-Schröder paths with $i$ diagonal steps, a…
This is both an expository and research paper where we advocate a systematic study of continuous analogues of finite partially ordered sets, convex polytopes, oriented matroids, arrangements of subspaces, finite simplicial complexes, and other combinatorial structures. Among the illustrative examples are an Euler formula for a class of `continuous convex polytopes' (conjectured by Kalai and Wigderson), a duality result for a class of `continuous matroids', a calculation of the Euler characteristic of ideals in the Grassmannian poset (related to a problem of Gian-Carlo Rota), an exposition of t…
Linear extensions of posets are important objects in enumerative and algebraic combinatorics that are difficult to count in general. Families of posets like Young diagrams of straight shapes and $d$-complete posets have hook-length product formulas to count linear extensions, whereas families like Young diagrams of skew shapes have determinant or positive sum formulas like the Naruse hook-length formula from 2014. In 2020, Garver et. al. gave determinant formulas to count linear extensions of a family of posets called mobile posets that refine $d$-complete posets and border strip skew shapes. …
We present a beautiful interplay between combinatorial topology and homological algebra for a class of monoids that arise naturally in algebraic combinatorics. We explore several applications of this interplay. For instance, we provide a new interpretation of the Leray number of a clique complex in terms of non-commutative algebra. Résumé. Nous présentons une magnifique interaction entre la topologie combinatoire et l'algèbre homologique d'une classe de monoïdes qui figurent naturellement dans la combinatoire algébrique. Nous explorons plusieurs applications de cette interaction. Par exemple, …
We study the stratification of the space of monic polynomials with real coefficients according to the number and multiplicities of real zeros. In the first part, for each of these strata we provide a purely combinatorial chain complex calculating (co)homology of its one-point compactification and describe the homotopy type by order complexes of a class of posets of compositions. In the second part, we determine the homotopy type of the one-point compactification of the space of monic polynomials of fixed degree which have only real roots (i.e., hyperbolic polynomials) and at least one root is …
The Rees product of partially ordered sets was introduced by Björner and Welker. Using the theory of lexicographic shellability, Linusson, Shareshian and Wachs proved formulas, of significance in the theory of gamma-positivity, for the dimension of the homology of the Rees product of a graded poset $P$ with a certain $t$-analogue of the chain of the same length as $P$. Equivariant generalizations of these formulas are proven in this paper, when a group of automorphisms acts on $P$, and are applied to establish the Schur gamma-positivity of certain symmetric functions arising in algebraic and g…
División de Ingenierías, Universidad de Guanajuato（瓜纳华托大学工程学院）；Research Data Centre, German Federal Pension Insurance（德国联邦养老金保险研究数据中心）；Yonsei University（延世大学）
The theory of pictures between posets is known to encode much of the combinatorics of symmetric group representations and related topics such as Young diagrams and tableaux. Many reasons, com-binatorial (e.g. since semi-standard tableaux can be viewed as double quasi-posets) and topological (quasi-posets identify with finite topolo-gies) lead to extend the theory to quasi-posets. This is the object of the present article.
In recent years it has been noted that a number of combinatorial structures such as real and complex hyperplane arrangements, interval greedoids, matroids and oriented matroids have the structure of a finite monoid called a left regular band. Random walks on the monoid model a number of interesting Markov chains such as the Tsetlin library and riffle shuffle. The representation theory of left regular bands then comes into play and has had a major influence on both the combinatorics and the probability theory associated to such structures. In a recent paper, the authors established a close conn…
A quasi-hereditary algebra is an Artin algebra together with a partial order on its set of isomorphism classes of simple modules which satisfies certain conditions. In this article we investigate all the possible choices that yield to quasi-hereditary structures on a given algebra, in particular we introduce and study what we call the poset of quasi-hereditary structures. Our techniques involve certain quiver decompositions and idempotent reductions. For a path algebra of Dynkin type $\mathbb{A}$, we provide a full classification of its quasi-hereditary structures. For types $\mathbb{D}$ and $…
This is a survey on the categorification of the poset of generalized non-crossing partitions, using the representation theory of a hereditary artin algebra H, looking at the set P of exceptional subcategories in mod H. This categorification is due to Ingalls and Thomas, and a subsequent paper by Igusa and Schiffler. Starting point is a refinement of the classical tilting theory for mod H, replacing torsion pairs by torsion triples, thus putting it into the realm of the stability theory of King. The torsion pairs in mod H correspond nicely to the perpendicular pairs of exceptional subcategories…
This memoir constitutes the author's PhD thesis at Cornell University. It serves both as an expository work and as a description of new research. At the heart of the memoir, we introduce and study a poset $NC^{(k)}(W)$ for each finite Coxeter group $W$ and for each positive integer $k$. When $k=1$, our definition coincides with the generalized noncrossing partitions introduced by Brady-Watt and Bessis. When $W$ is the symmetric group, we obtain the poset of classical $k$-divisible noncrossing partitions, first studied by Edelman. Along the way, we include a comprehensive introduction to relate…
The face poset of the permutohedron realizes the combinatorics of linearly ordered partitions of the set $[n]=\{1,...,n\}$. Similarly, the cyclopermutohedron is a virtual polytope that realizes the combinatorics of cyclically ordered partitions of the set $[n+1]$. The cyclopermutohedron was introduced by the third author by motivations coming from configuration spaces of polygonal linkages. In the paper we prove two facts: (1) the volume of the cyclopermutohedron equals zero, and (2) the homology groups $H_k$ for $k=0,...,n-2$ of the face poset of the cyclopermutohedron are non-zero free abeli…
We study $r$-differential posets, a class of combinatorial objects introduced in 1988 by the first author, which gathers together a number of remarkable combinatorial and algebraic properties, and generalizes important examples of ranked posets, including the Young lattice. We first provide a simple bijection relating differential posets to a certain class of hypergraphs, including all finite projective planes, which are shown to be naturally embedded in the initial ranks of some differential poset. As a byproduct, we prove the existence, if and only if $r\geq 6$, of $r$-differential posets no…
Matroid bundles, introduced by MacPherson, are combinatorial analogues of real vector bundles. This paper sets up the foundations of matroid bundles, and defines a natural transformation from isomorphism classes of real vector bundles to isomorphism classes of matroid bundles, as well as a transformation from matroid bundles to spherical quasifibrations. The poset of oriented matroids of a fixed rank classifies matroid bundles, and the above transformations give a splitting from topology to combinatorics back to topology. This shows the mod 2 cohomology of the poset of rank k oriented matroids…
The Razumov-Stroganov correspondence, an important link between statistical physics and combinatorics proved in 2011 by L. Cantini and A. Sportiello, relates the ground state eigenvector of the O(1) dense loop model on a semi-infinite cylinder to a refined enumeration of fully-packed loops, which are in bijection with alternating sign matrices. This paper reformulates a key component of this proof in terms of posets, the toggle group, and homomesy, and proves two new homomesy results on general posets which we hope will have broader implications.
