Related papers: The poset of posets
The interval poset of a permutation is the set of intervals of a permutation, ordered with respect to inclusion. It has been introduced and studied recently in [B. Tenner, arXiv:2007.06142]. We study this poset from the perspective of the…
This is an introduction to the M\"obius function of a poset. The chief novelty is in the exposition. We show how order-preserving maps from one poset to another can be used to relate their M\"obius functions. We derive the basic results on…
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.…
A poset is called upper homogeneous (or "upho") if every principal order filter of the poset is isomorphic to the whole poset. We observe that the rank and characteristic generating functions of upho posets are multiplicative inverses of…
The set of all permutations, ordered by pattern containment, forms a poset. This paper presents the first explicit major results on the topology of intervals in this poset. We show that almost all (open) intervals in this poset have a…
We determine the M\"obius function of the poset of compositions of an integer. In fact we give two proofs of this formula, one using an involution and one involving discrete Morse theory. The composition poset turns out to be intimately…
This paper introduces and develops M\"obius homology, a homology theory for representations of finite posets into abelian categories. Although the connection between poset topology and M\"obius functions is classical, we go further by…
We study the poset $\mathcal{G}$ of all unlabelled graphs, up to isomorphism, with $H\le G$ if $H$ occurs as an induced subgraph in $G$. We present some general results on the M\"obius function of intervals of $\mathcal{G}$ and some results…
The consecutive pattern poset is the infinite partially ordered set of all permutations where $\sigma\le\tau$ if $\tau$ has a subsequence of adjacent entries in the same relative order as the entries of $\sigma$. We study the structure of…
The interval poset of a permutation catalogues the intervals that appear in its one-line notation, according to set inclusion. We study this poset, describing its structural, characterizing, and enumerative properties.
The absolute order on the hyperoctahedral group $B_n$ is investigated. It is proved that the order ideal of this poset generated by the Coxeter elements is homotopy Cohen-Macaulay and the M\"obius number of this ideal is computed. Moreover,…
We define the notion of connectivity set for elements of any finitely generated Coxeter group. Then we define an order related to this new statistic and show that the poset is graded and each interval is a shellable lattice. This implies…
Given a symmetric monoidal category $C$ with product $\sqcup$, where the neutral element for the product is an initial object, we consider the poset of $\sqcup$-complemented subobjects of a given object $X$. When this poset has finite…
We introduce the poset of mesh patterns, which generalises the permutation pattern poset. We fully classify the mesh patterns for which the interval [1^\emptyset,m] is non-pure, where 1^\emptyset is the unshaded singleton mesh pattern. We…
An occurrence of a consecutive permutation pattern $p$ in a permutation $\pi$ is a segment of consecutive letters of $\pi$ whose values appear in the same order of size as the letters in $p$. The set of all permutations forms a poset with…
A poset $P = (X,\prec)$ has an interval representation if each $x \in X$ can be assigned a real interval $I_x$ so that $x \prec y$ in $P$ if and only if $I_x$ lies completely to the left of $I_y$. Such orders are called \emph{interval…
We present a two term formula for the M\"obius function of intervals in the poset of all permutations, ordered by pattern containment. The first term in this formula is the number of so called normal occurrences of one permutation in…
We introduce a new class of poset edge labelings for locally finite lattices which we call $SB$-labelings. We prove for finite lattices which admit an $SB$-labeling that each open interval has the homotopy type of a ball or of a sphere of…
The classes of sequentially Cohen-Macaulay and sequentially homotopy Cohen-Macaulay complexes and posets are studied. First, some different versions of the definitions are discussed and the homotopy type is determined. Second, it is shown…
The set of all permutations, ordered by pattern containment, is a poset. We give a formula for the M\"obius function of intervals $[1,\pi]$ in this poset, for any permutation $\pi$ with at most one descent. We compute the M\"obius function…