A well-known theorem of Sperner describes the largest collections of subsets of an $n$-element set none of which contains another set from the collection. Generalising this result, Erdős characterised the largest families of subsets of an $n$-element set that do not contain a chain of sets $A_1 \subset \dotsc \subset A_k$ of an arbitrary length $k$. The extremal families contain all subsets whose cardinalities belong to an interval of length $k-1$ centred at $n/2$. In a far-reaching extension of Sperner's theorem, Kleitman determined the smallest number of chains of length two that have to appear in a collection of a given number $a$ of subsets of an $n$-element set. For every $a$, this minimum is achieved by the collection comprising $a$ sets whose cardinalities are as close to $n/2+1/4$ as possible. We show that the same is true about chains of an arbitrary length $k$, for all $a$ and $n$, confirming the prediction Kleitman made fifty years ago. We also characterise all families of $a$ subsets with the smallest number of chains of length $k$ for all $a$ for which this smallest number is positive. Our argument is inspired by an elegant probabilistic lemma from a recent paper of Noel, Scott, and Sudakov, which in turn can be traced back to Lubell's proof of Sperner's theorem.
For two posets $P$ and $Q$, we say $Q$ is $P$-free if there does not exist any order-preserving injection from $P$ to $Q$. The speical case for $Q$ being the Boolean lattice $B_n$ is well-studied, and the optiamal value is denoted as $\lanp$. Let us define $\La(Q,P)$ to be the largest size of any $P$-free subposet of $Q$. In this paper, we give an upper bound for $\La(Q,P)$ when $Q$ is a double chain and $P$ is any graded poset, which is better than the previous known upper bound, by means of finding the indpendence number of an auxiliary graph related to $P$. For the auxiliary graph, we can f…
Suppose $k \ge 2$ is an integer. Let $Y_k$ be the poset with elements $x_1, x_2, y_1, y_2, \ldots, y_{k-1}$ such that $y_1 < y_2 < \cdots < y_{k-1} < x_1, x_2$ and let $Y_k'$ be the same poset but all relations reversed. We say that a family of subsets of $[n]$ contains a copy of $Y_k$ on consecutive levels if it contains $k+1$ subsets $F_1, F_2, G_1, G_2, \ldots, G_{k-1}$ such that $G_1\subset G_2 \subset \cdots \subset G_{k-1} \subset F_1, F_2$ and $|F_1| = |F_2| = |G_{k-1}|+1 =|G_{k-2}|+ 2= \cdots = |G_{1}|+k-1$. If both $Y_k$ and $Y'_k$ on consecutive levels are forbidden, the size of the …
Let $B_n$ be the poset generated by the subsets of $[n]$ with the inclusion as relation and let $P$ be a finite poset. We want to embed $P$ into $B_n$ as many times as possible such that the subsets in different copies are incomparable. The maximum number of such embeddings is asymptotically determined for all finite posets $P$ as $\frac{n \choose \lfloor n/2\rfloor}{M(P)}$, where $M(P)$ denotes the minimal size of the convex hull of a copy of $P$. We discuss both weak and strong (induced) embeddings.
Let $\F\subset 2^{[n]}$ be a family of subsets of $\{1,2,..., n\}$. For any poset $H$, we say $\F$ is $H$-free if $\F$ does not contain any subposet isomorphic to $H$. Katona and others have investigated the behavior of $\La(n,H)$, which denotes the maximum size of $H$-free families $\F\subset 2^{[n]}$. Here we use a new approach, which is to apply methods from extremal graph theory and probability theory to identify new classes of posets $H$, for which $\La(n,H)$ can be determined asymptotically as $n\to\infty$ for various posets $H$, including two-end-forks, up-down trees, and cycles $C_{4k}…
The Scarf complex for lattices is well understood and utilized. In 2014, the author expanded the use of the term Scarf complex to encompass an infinite set in $\mathbb{Z}^n$ that was generated via an action of a lattice $Λ\in\mathbb{Z}^n$ upon a finite subset $A\subset\mathbb{N}^n$. This paper aims to further generalize the use of Scarf complex by removing the integer lattice completely while maintaining the essence of the Scarf complex.
Given a finite poset $P$, we consider the largest size $\lanp$ of a family $\F$ of subsets of $[n]:=\{1,...,n\}$ that contains no subposet $P$. This continues the study of the asymptotic growth of $\lanp$; it has been conjectured that for all $P$, $π(P):= \lim_{n\rightarrow\infty} \lanp/\nchn$ exists and equals a certain integer, $e(P)$. While this is known to be true for paths, and several more general families of posets, for the simple diamond poset $\D_2$, the existence of $π$ frustratingly remains open. Here we develop theory to show that $π(P)$ exists and equals the conjectured value $e(P…
The maximum size, $La(n,P)$, of a family of subsets of $[n]=\{1,2,...,n\}$ without containing a copy of $P$ as a subposet, has been intensively studied. Let $P$ be a graded poset. We say that a family $\mathcal{F}$ of subsets of $[n]=\{1,2,...,n\}$ contains a \emph{rank-preserving} copy of $P$ if it contains a copy of $P$ such that elements of $P$ having the same rank are mapped to sets of same size in $\mathcal{F}$. The largest size of a family of subsets of $[n]=\{1,2,...,n\}$ without containing a rank-preserving copy of $P$ as a subposet is denoted by $La_{rp}(n,P)$. Clearly, $La(n,P) \le L…
We consider the problem of determining the maximum number of pairs $F\subseteq F'$ in a family $\mathcal{F}\subseteq 2^{[n]}$ that avoids certain posets $P$ of height 2. We show that for any such $P$ the number of pairs is $O(n\binom{n}{\lfloor n/2\rfloor})$ and we find the exact value for the butterfly poset and the $N$ poset. Also, we determine the asymptotics of the maximum number of pairs in containment for some posets of which the Hasse diagram is a path.
We introduce two partially ordered sets, $P^A_n$ and $P^B_n$, of the same cardinalities as the type-A and type-B noncrossing partition lattices. The ground sets of $P^A_n$ and $P^B_n$ are subsets of the symmetric and the hyperoctahedral groups, consisting of permutations which avoid certain patterns. The order relation is given by (strict) containment of the descent sets. In each case, by means of an explicit order-preserving bijection, we show that the poset of restricted permutations is an extension of the refinement order on noncrossing partitions. Several structural properties of these per…
Motivated by the theory of correspondence functors, we introduce the notion of {\em germ} in a finite poset, and the notion of {\em germ extension} of a poset. We show that any finite poset admits a largest germ extension called its {\em germ closure}. We say that a subset $U$ of a finite lattice $T$ is {\em germ extensible} in $T$ if the germ closure of $U$ naturally embeds in $T$. We show that any for any subset $S$ of a finite lattice $T$, there is a unique germ extensible subset $U$ of $T$ such that $U\subseteq S\subseteq \overline{G}(U)$, where $\overline{G}(U)\subseteq T$ is the embeddin…
Let $\cX$ be a family of subsets of a finite set $E$. A matroid on $E$ is called an $\cX$-matroid if each set in $\cX$ is a circuit. We consider the problem of determining when there exists a unique maximal $\cX$-matroid in the weak order poset of all $\cX$-matroids on $E$, and characterizing its rank function when it exists.
The paper "Grothendieck Topologies on Posets" by A.J. Lindenhovius shows that when $\mathbf{P}$ is an Artinian poset and $\mathbf{E}$ is the topos $\mathbf{Set}^\mathbf{P}$ then there are bijections between the set of subsets of $\mathbf{P}$, the set of Grothendieck topologies on $\mathbf{E}$, and the set of nuclei on the Heyting Algebra $\mathrm{Sub}(1_\mathbf{E})$. It also shows that there are nice formulas for converting between subsets, Grothendieck topologies, and nuclei, but the formula for converting a nucleus to a subset is not spelled out explicitly. These notes fix that gap.
We consider 'supersaturation' problems in partially ordered sets (posets) of the following form. Given a finite poset $P$ and an integer $m$ greater than the cardinality of the largest antichain in $P$, what is the minimum number of comparable pairs in a subset of $P$ of cardinality $m$? We provide a framework for obtaining lower bounds on this quantity based on counting comparable pairs relative to a random chain and apply this framework to obtain supersaturation results for three classical posets: the boolean lattice, the collection of subspaces of $\mathbb{F}_q^n$ ordered by set inclusion a…
We are interested in maximizing the number of pairwise unrelated copies of a poset $P$ in the family of all subsets of $[n]$. We prove that for any $P$ the maximum number of unrelated copies of $P$ is asymptotic to a constant times the largest binomial coefficient. Moreover, the constant has the form $\frac{1}{c(P)}$, where $c(P)$ is the size of the smallest convex closure over all embeddings of $P$ into the Boolean lattice.
We identify two key conditions that a subset $A$ of a poset $P$ may satisfy to guarantee the transfer of continuity properties from $P$ to $A$. We then highlight practical cases where these key conditions are fulfilled. Along the way we are led to consider subsets of a given poset $P$ whose way-below relation is the restriction of the way-below relation of $P$, which we call way-below preserving subposets. As an application, we show that every conditionally complete poset with the interpolation property contains a largest continuous way-below preserving subposet. Most of our results are expres…
A subfamily $\{F_1,F_2,\dots,F_{|P|}\}\subseteq {\cal F}$ of sets is a copy of a poset $P$ in ${\cal F}$ if there exists a bijection $ϕ:P\rightarrow \{F_1,F_2,\dots,F_{|P|}\}$ such that whenever $x \le_P x'$ holds, then so does $ϕ(x)\subseteq ϕ(x')$. For a family ${\cal F}$ of sets, let $c(P,{\cal F})$ denote the number of copies of $P$ in ${\cal F}$, and we say that ${\cal F}$ is $P$-free if $c(P,{\cal F})=0$ holds. For any two posets $P,Q$ let us denote by $La(n,P,Q)$ the maximum number of copies of $Q$ over all $P$-free families ${\cal F} \subseteq 2^{[n]}$, i.e. $\max\{c(Q,{\cal F}): {\cal…
Let $\mathcal{G}$ be the set of all connected graphs on vertex set $[n]$. Define the partial ordering $<$ on $\mathcal{G}$ as follows: for $G,H\in \mathcal{G}$ let $G<H$ if $E(G)\subset E(H)$. The poset $(\mathcal{G},<)$ is graded, each level containing the connected graphs with the same number of edges. We prove that $(\mathcal{G},<)$ has the Sperner property, namely that the largest antichain of $(\mathcal{G},<)$ is equal to its largest sized level.
A set of necessary and sufficient conditions under which an isotone mapping from a subset of a poset X to a poset Y has an extension to an isotone mapping from X to Y are found.
We consider the problem of determining the maximum order of an induced vertex-disjoint union of cliques in a graph. More specifically, given some family of graphs $\mathcal{G}$ of equal order, we are interested in the parameter $a(\mathcal{G}) = \min_{G \in \mathcal{G}} \max \{ |U| : U \subseteq V, G[U] \text{ is a vertex-disjoint union of cliques} \}$. We determine the value of this parameter precisely when $\mathcal{G}$ is the family of comparability graphs of $n$-element posets with acyclic cover graph. In particular, we show that $a(\mathcal{G}) = (n+o(n))/\log_2 (n)$ in this class.
Let $X$ be a finite set, $Z \subseteq X$ and $y \notin X$. Marcel Erné showed in 1981, that the number of posets on $X$ containing $Z$ as an antichain equals the number of posets $R$ on $X \cup \{ y \}$ in which the points of $Z \cup \{ y \}$ are exactly the maximal points of $R$. We prove the following generalization: For every poset $Q$ with carrier $Z$, the number of posets on $X$ containing $Q$ as an induced sub-poset equals the number of posets $R$ on $X \cup \{ y \}$ which contain $Q^d + A_y$ as an induced sub-poset and in which the maximal points of $Q^d + A_y$ are exactly the maximal p…
Given a graded poset $P$, consider a chain decomposition $\mathcal{C}$ of $P$. If $|C_1|\le |C_2|$ implies that the set of the ranks of elements in $C_1$ is a subset of the ranks of elements in $C_2$ for any chains $C_1,C_2\in \mathcal{C}$, then we say $\mathcal{C}$ is a nested chain decomposition (or nesting, for short) of $P$, and $P$ is said to be nested. In 1970s, Griggs conjectured that every normalized matching rank-unimodal poset is nested. This conjecture is proved to be true only for all posets of rank 2 [W:05], some posets of rank 3 [HLS:09,ENSST:11], and the very special cases for h…
For a given finite poset $P$, $La(n,P)$ denotes the largest size of a family $\mathcal{F}$ of subsets of $[n]$ not containing $P$ as a weak subposet. We exactly determine $La(n,P)$ for infinitely many $P$ posets. These posets are built from seven base posets using two operations. For arbitrary posets, an upper bound is given for $La(n,P)$ depending on $|P|$ and the size of the longest chain in $P$. To prove these theorems we introduce a new method, counting the intersections of $\mathcal{F}$ with double chains, rather than chains.
Given a poset $P$ we say a family $\mathcal{F}\subseteq P$ is centered if it is obtained by `taking sets as close to the middle layer as possible'. A poset $P$ is said to have the centeredness property if for any $M$, among all families of size $M$ in $P$, centered families contain the minimum number of comparable pairs. Kleitman showed that the Boolean lattice $\{0,1\}^n$ has the centeredness property. It was conjectured by Noel, Scott, and Sudakov, and by Balogh and Wagner, that the poset $\{0,1,\ldots,k\}^n$ also has the centeredness property, provided $n$ is sufficiently large compared to …
We introduce a method of decomposing the family of intervals along a cyclic permutation into chains to determine the size of the largest family of subsets of $[n]:= \{1,2,...,n\}$ not containing one or more given posets as a subposet. De Bonis, Katona and Swanepoel determined the size of the largest butterfly-free family. We strengthen this result by showing that, for certain posets containing the butterfly poset as a subposet, the same bound holds. We also obtain the corresponding LYM-type inequalities.
In this paper, we introduce the notion of the containment graph of a family of sets and containment classes of graphs and posets. Let $Z$ be a family of nonempty sets. We call a (simple, finite) graph G = (V, E) a $Z$-containment graph provided one can assign to each vertex $v_i \in V $ a set $S_i \in Z$ such that $v_i v_j \in E$ if and only if $S_i \subset S_j$ or $S_j \subset S_i$ . Similarly, we call a (strict) partially ordered set $P = (V, <)$ a $Z$-containment poset if to each $v_i \in V $ we can assign a set $S_i \in Z$ such that $v_i < v_j$ if and only if $S_i \subset S_j$. Obviously, …
The purpose of the present paper is to give a realization of a cylindric diagram as a subset of root systems of type $A_{κ-1}^{(1)}$ and several characterization of its poset structure. Furthermore, the set of order ideals of a cylindric diagram is described as a weak Bruhat interval of the Weyl group.
The circular peak set of a permutation $σ$ is the set $\{σ(i)\mid σ(i-1)<σ(i)>σ(i+1)\}$. Let $\mathcal{P}_n$ be the set of all the subset $S\subseteq [n]$ such that there exists a permutation $σ$ which has the circular set $S$. We can make the set $\mathcal{P}_n$ into a poset $\mathscr{P}_n$ by defining $S\preceq T$ if $S\subseteq T$ as sets. In this paper, we prove that the poset $\mathscr{P}_n$ is a simplicial complex on the vertex set $[3,n]$. We study the $f$-vector, the $f$-polynomial, the reduced Euler characteristic, the M$\ddot{o}$bius function, the $h$-vector and the $h$-polynomial of…
There is much recent interest in excluded subposets. Given a fixed poset $P$, how many subsets of $[n]$ can found without a copy of $P$ realized by the subset relation? The hardest and most intensely investigated problem of this kind is when $P$ is a diamond, i.e. the power set of a 2 element set. In this paper, we show infinitely many asymptotically tight constructions using random set families defined from posets based on Abelian groups. They are provided by the convergence of Markov chains on groups. Such constructions suggest that the diamond problem is hard.
We show the classical $q$-Stirling numbers of the second kind can be expressed compactly as a pair of statistics on a subset of restricted growth words. The resulting expressions are polynomials in $q$ and $1+q$. We extend this enumerative result via a decomposition of a new poset $Π(n,k)$ which we call the Stirling poset of the second kind. Its rank generating function is the $q$-Stirling number $S_q[n,k]$. The Stirling poset of the second kind supports an algebraic complex and a basis for integer homology is determined. A parallel enumerative, poset theoretic and homological study for the $q…
For given posets $P$ and $Q$ and an integer $n$, the generalized Turán problem for posets, asks for the maximum number of copies of $Q$ in a $P$-free subset of the $n$-dimensional Boolean lattice, $2^{[n]}$. In this paper, among other results, we show the following: (i) For every $n\geq 5$, the maximum number of $2$-chains in a butterfly-free subfamily of $2^{[n]}$ is $\left\lceil\frac{n}{2}\right\rceil\binom{n}{\lfloor n/2\rfloor}$. (ii) For every fixed $s$, $t$ and $k$, a $K_{s,t}$-free family in $2^{[n]}$ has $O\left(n\binom{n}{\lfloor n/2\rfloor}\right)$ $k$-chains. (iii) For every $n\geq …
An unlabeled poset is said to be (2+2)-free if it does not contain an induced subposet that is isomorphic to 2+2, the union of two disjoint 2-element chains. Let $p_n$ denote the number of (2+2)-free posets of size $n$. In a recent paper, Bousquet-Mélou et al.\cite{BCDK} found, using so called ascent sequences, the generating function for the number of (2+2)-free posets of size $n$: $P(t)=\sum_{n \geq 0} p_n t^n = \sum_{n\geq 0} \prod_{i=1}^{n} (1-(1-t)^i)$. We extend this result in two ways. First, we find the generating function for (2+2)-free posets when four statistics are taken into accou…
The totally nonnegative Grassmannian $\mathrm{Gr}(k,n)_{\geq0}$ is the subset of the real Grassmannian $\mathrm{Gr}(k,n)$ consisting of points with all nonnegative Plücker coordinates. The circular Bruhat order is a poset isomorphic to the face poset of A. Postnikov's (2005) positroid cell decomposition of $\mathrm{Gr}(k,n)_{\geq0}$. We provide a closed formula for the sum of its weighted chains in the spirit of J. Stembridge (2002).
University of East Anglia（东英吉利大学）；Merseburg University of Applied Sciences（默瑟堡应用科学大学）；Interdisciplinary Center for Scientific Computing, Heidelberg University（海德堡大学跨学科科学计算中心）
For a closure space (P,f) with f(\emptyset)=\emptyset, the closures of open subsets of P, called the regular closed subsets, form an ortholattice Reg(P,f), extending the poset Clop(P,f) of all clopen subsets. If (P,f) is a finite convex geometry, then Reg(P,f) is pseudocomplemented. The Dedekind-MacNeille completion of the poset of regions of any central hyperplane arrangement can be obtained in this way, hence it is pseudocomplemented. The lattice Reg(P,f) carries a particularly interesting structure for special types of convex geometries, that we call closure spaces of semilattice type. For …
A subfamily $\{F_1,F_2,\dots,F_{|P|}\}\subseteq \mathcal F$ is a copy of the poset $P$ if there exists a bijection $i:P\rightarrow \{F_1,F_2,\dots,F_{|P|}\}$ such that $p\le_P q$ implies $i(p)\subseteq i(q)$. A family $\mathcal F$ is $P$-free, if it does not contain a copy of $P$. In this paper we establish basic results on the maximum possible number of $k$-chains in a $P$-free family $\mathcal F\subseteq 2^{[n]}$. We prove that if the height of $P$, $h(P) > k$, then this number is of the order $Θ(\prod_{i=1}^{k+1}\binom{l_{i-1}}{l_i})$, where $l_0=n$ and $l_1\ge l_2\ge \dots \ge l_{k+1}$ are…
Let $Δ$ be a root system with a subset of positive roots, $Δ^+$. We consider edges of the Hasse diagrams of some posets associated with $Δ^+$. For each edge one naturally defines its type, and we study the partition of the set of edges into types. For $Δ^+$, the type is a simple root, and for the posets of ad-nilpotent and Abelian ideals the type is an affine simple roots. We give several descriptions of the set of edges of given type and uniform expressions for the number of edges. By a result of Peterson, the number of Abelian ideals is $2^n$, where $n$ is the rank of $Δ$. We prove that the …
