31 pages, 7 figures. Counterexamples to Conjecture 11.2, 11.3, and 11.4 in v2 were found, and are now included in Section 11.5 and 11.10 of v3. To appear in Trans. Amer. Math. Soc
The cross--product conjecture (CPC) of Brightwell, Felsner and Trotter (1995) is a two-parameter quadratic inequality for the number of linear extensions of a poset $P= (X, \prec)$ with given value differences on three distinct elements in $X$. We give two different proofs of this inequality for posets of width two. The first proof is algebraic and generalizes CPC to a four-parameter family. The second proof is combinatorial and extends CPC to a $q$-analogue. Further applications include relationships between CPC and other poset inequalities, including a new $q$-analogue of the Kahn--Saks inequality.
The fixed point property for finite posets of width 3 and 4 is studied in terms of forbidden retracts. The ranked forbidden retracts for width 3 and 4 are determined explicitly. The ranked forbidden retracts for the width 3 case that are linearly indecomposable are examined to see which are minimal automorphic. Part of a problem of Niederle from 1989 is thus solved.
Twin-width is a new parameter informally measuring how diverse are the neighbourhoods of the graph vertices, and it extends also to other binary relational structures, e.g. to digraphs and posets. It was introduced just very recently, in 2020 by Bonnet, Kim, Thomasse and Watrigant. One of the core results of these authors is that FO model checking on graph classes of bounded twin-width is in FPT. With that result, they also claimed that posets of bounded width have bounded twin-width, thus capturing prior result on FO model checking of posets of bounded width in FPT. However, their translation…
Heath and Pemmaraju conjectured that the queue-number of a poset is bounded by its width and if the poset is planar then also by its height. We show that there are planar posets whose queue-number is larger than their height, refuting the second conjecture. On the other hand, we show that any poset of width $2$ has queue-number at most $2$, thus confirming the first conjecture in the first non-trivial case. Moreover, we improve the previously best known bounds and show that planar posets of width $w$ have queue-number at most $3w-2$ while any planar poset with $0$ and $1$ has queue-number at m…
Recently, Kitaev and Remmel posed a conjecture concerning the generating function for the number of unlabeled $(2+2)$-free posets with respect to number of elements and number of minimal elements. In this paper, we present a combinatorial proof of this conjecture.
Extending results of Linial (1984) and Aigner (1985), we prove a uniform lower bound on the balance constant of a poset $P$ of width $2$. This constant is defined as $δ(P) = \max_{(x, y)\in P^2}\min\{\mathbb{P}(x\prec y), \mathbb{P}(y\prec x)\}$, where $\mathbb{P}(x\prec y)$ is the probability $x$ is less than $y$ in a uniformly random linear extension of $P$. In particular, we show that if $P$ is a width $2$ poset that cannot be formed from the singleton poset and the three element poset with one relation using the operation of direct sum, then \[δ(P)\ge\frac{-3 + 5\sqrt{17}}{52}\approx 0.338…
Aharoni and Korman (Order 9 (1992) 245--253) have conjectured that every ordered set without infinite antichains possesses a chain and a partition into antichains so that each part intersects the chain. The conjecture is verified for posets whose incomparability graph is locally finite. It follows that the conjecture is true for $(3 + 1)$-free posets with no infinite antichains.
An $N$-free poset is a poset whose comparability graph does not embed an induced path with four vertices. We use the well-quasi-order property of the class of countable $N$-free posets and some labelled ordered trees to show that a countable $N$-free poset has one or infinitely many siblings, up to isomorphism. This, partially proves a conjecture stated by Thomassé for this class.
We generalize the notion of graded posets to what we call sign-graded (labeled) posets. We prove that the $W$-polynomial of a sign-graded poset is symmetric and unimodal. This extends a recent result of Reiner and Welker who proved it for graded posets by associating a simplicial polytopal sphere to each graded poset $P$. By proving that the $W$-polynomials of sign-graded posets has the right sign at -1, we are able to prove the Charney-Davis Conjecture for these spheres (whenever they are flag).
We consider generalizations of the classical secretary problem, also known as the problem of optimal choice, to posets where the only information we have is the size of the poset and the number of maximal elements. We show that, given this information, there is an algorithm that is successful with probability at least $\frac{1}{e}$. We conjecture that if there are $k$ maximal elements and $k \geq 2$ then this can be improved to $\sqrt[k-1]{\frac{1}{k}}$, and prove this conjecture for posets of width $k$. We also show that no better bound is possible.
We establish strict growth for the rank function of an r-differential poset. We do so by exploiting the representation theoretic techniques developed by Reiner and the author for studying related Smith forms.
Anders Bjorner characterized which finite graded partially ordered sets arise as the posets of closure relations on cells of a finite, regular CW complex. His characterization of these "CW posets" required each open interval $(\hat{0},u)$ to have order complex homeomorphic to a sphere of dimension $rk(u)-2$. Work of Danaraj and Klee showed that sufficient conditions were for the poset to be thin and shellable. The proof of the Poincare Conjecture enables the requirement of shellability to be replaced by the homotopy Cohen-Macaulay property. This expands the range of tools that may be used to p…
The queue number of a poset is the queue number of its cover graph when the vertex order is a linear extension of the poset. Heath and Pemmaraju conjectured that every poset of width $w$ has queue number at most $w$. The conjecture has been confirmed for posets of width $w=2$ and for planar posets with $0$ and $1$. In contrast, the conjecture has been refused by a family of general (non-planar) posets of width $w>2$. In this paper, we study queue layouts of two-dimensional posets. First, we construct a two-dimensional poset of width $w > 2$ with queue number $2(w - 1)$, thereby disproving the …
A famous theorem of Dilworth asserts that any finite poset of width $k$ can be decomposed into $k$ chains. We study the following problem: given a Borel poset $P$ of finite width $k$, is it true that it can be decomposed into $k$ Borel chains? We give a positive answer in a special case of Borel posets embeddable into the real line. We also prove a dual theorem for posets whose comparability graphs are locally countable.
We give a necessary and sufficient condition for a $P_4$-free graph to be a cograph. This allows us to obtain a simple proof of the fact that finite $P_4$-free graphs are finite cographs. We also prove that $N$-free chain complete posets and $N$-free posets with no infinite antichains are series-parallel. As a consequence, we obtain that every $N$-free poset with no infinite antichain has a chain and a partition into antichains so that each part intersects the chain. This answers a conjecture of Aharoni and Korman (Order \textbf{9} (1992) 245--253) in this case.
Kahn and Kim (J. Comput. Sci., 1995) have shown that for a finite poset $P$, the entropy of the incomparability graph of $P$ (normalized by multiplying by the order of $P$) and the base-$2$ logarithm of the number of linear extensions of $P$ are within constant factors from each other. The tight constant for the upper bound was recently shown to be $2$ by Cardinal, Fiorini, Joret, Jungers and Munro (STOC 2010, Combinatorica). Here, we refine this last result in case $P$ has width $2$: we show that the constant can be replaced by $2-\varepsilon$ if one also takes into account the number of conn…
We extend the Main Theorem of Aschbacher and Smith on Quillen's Conjecture from $p>5$ to the remaining odd primes $p = 3,5$. In the process, we develop further combinatorial and homotopical methods for studying the poset of nontrivial elementary abelian $p$-subgroups of a finite group. The techniques lead to a number of further results on the Conjecture, often reducing dependence on the CFSG; in particular, we also provide some partial results toward the case of $p=2$.
The Stern poset $\mathcal{S}$ is a graded infinite poset naturally associated to Stern's triangle, which was defined by Stanley analogously to Pascal's triangle. Let $P_n$ denote the interval of $\mathcal{S}$ from the unique element of row $0$ of Stern's triangle to the $n$-th element of row $r$ for sufficiently large $r$. For $n\geq 1$ let \begin{align*} L_n(q)&=2\cdot\left(\sum_{k=1}^{2^n-1}A_{P_k}(q)\right)+A_{P_{2^n}}(q), \end{align*} where $A_{P}(q)$ represents the corresponding $P$-Eulerian polynomial. For any $n\geq 1$ Stanley conjectured that $L_n(q)$ has only real zeros and $L_{4n+1}(…
We consider 3 (weighted) posets associated with a graph G - the poset P(G) of distinct induced unlabelled subgraphs, the lattice Omega(G) of distinct unlabelled graphs induced by connected partitions, and the poset Q(G) of distinct unlabelled edge-subgraphs. We study these posets given up to isomorphism, and their relation to the reconstruction conjectures. We show that when G is not a star or a disjoint union of edges, P(G) and Omega(G) can be constructed from each other. The result implies that trees are reconstructible from their abstract bond lattice. We present many results on the reconst…
The queue-number of a poset is the queue-number of its cover graph viewed as a directed acyclic graph, i.e., when the vertex order must be a linear extension of the poset. Heath and Pemmaraju conjectured that every poset of width $w$ has queue-number at most $w$. Recently, Alam et al. constructed posets of width $w$ with queue-number $w+1$. Our contribution is a construction of posets with width $w$ with queue-number $Ω(w^2)$. This asymptotically matches the known upper bound.
A certain inequality is shown to hold for the values of the Mobius function of the poset obtained by attaching a maximum element to a lower Eulerian Cohen-Macaulay poset. In two important special cases, this inequality provides partial results supporting Stanley's nonnegativity conjecture for the toric h-vector of a lower Eulerian Cohen-Macaulay meet-semilattice and Adin's nonnegativity conjecture for the cubical h-vector of a Cohen-Macaulay cubical complex.
The original notion of dimension for posets is due to Dushnik and Miller and has been studied extensively in the literature. Quite recently, there has been considerable interest in two variations of dimension known as Boolean dimension and local dimension. For a poset $P$, the Boolean dimension of $P$ and the local dimension of $P$ are both bounded from above by the dimension of $P$ and can be considerably less. Our primary goal will be to study analogies and contrasts among these three parameters. As one example, it is known that the dimension of a poset is bounded as a function of its height…
For any graded poset $P$, we define a new graded poset, $\mathcal E(P)$, whose elements are the edges in the Hasse diagram of P. For any group, $G$, acting on the boolean algebra, $B_n$, we conjecture that $\mathcal E(B_n/G)$ is Peck. We prove that the conjecture holds for "common cover transitive" actions. We give some infinite families of common cover transitive actions and show that the common cover transitive actions are closed under direct and semidirect products.
The gold partition conjecture and the Lexicographic sum of posets
Yonsei Univeristy（延世大学）；Kyungpook National University（庆北国立大学）；Thomas J. Watson Research Center, IBM（IBM 托马斯·J·沃森研究中心）；Pusan National University（釜山国立大学）
Let $S$ be a polynomial ring and let $I \subseteq S$ be a monomial ideal. In this short note, we propose the conjecture that the Betti poset of $I$ determines the Stanley projective dimension of $S/I$ or $I$. Our main result is that this conjecture implies the Stanley conjecture for $I$, and it also implies that \[ \operatorname{sdepth} S/I \geq \operatorname{depth} S/I - 1.\] Recently, Duval et al. found a counterexample to the Stanley conjecture, and their counterexample satisfies $\operatorname{sdepth} S/I = \operatorname{depth} S/I - 1$. So if our conjecture is true, then the conclusion is…
In 1981, Kelly showed that planar posets can have arbitrarily large dimension. However, the posets in Kelly's example have bounded Boolean dimension and bounded local dimension, leading naturally to the questions as to whether either Boolean dimension or local dimension is bounded for the class of planar posets. The question for Boolean dimension was first posed by Nešetřil and Pudlák in 1989 and remains unanswered today. The concept of local dimension is quite new, introduced in 2016 by Ueckerdt. Since that time, researchers have obtained many interesting results concerning Boolean dimension …
In 1971 Trotter conjectured that every finite poset on at least $3$ points has a pair whose removal does not decrease the dimension by more than $1$. In 1992 Brightwell and Scheinerman introduced fractional dimension of posets, and they made a similar conjecture for fractional dimension. This paper settles this latter conjecture.
Over the last 30 years, researchers have investigated connections between dimension for posets and planarity for graphs. Here we extend this line of research to the structural graph theory parameter tree-width by proving that the dimension of a finite poset is bounded in terms of its height and the tree-width of its cover graph.
The notion of Rees product of posets was introduced by Björner and Welker, where they study connections between poset topology and commutative algebra. Björner and Welker conjectured and Jonsson proved that the dimension of the top homology of the Rees product of the truncated Boolean algebra $B_n \setminus \{0\}$ and the $n$-chain $C_n$ is equal to the number of derangements in the symmetric group $\mathfrak S_n$. Here we prove a refinement of this result, which involves the Eulerian numbers, and a $q$-analog of both the refinement and the original conjecture, which comes from replacing the B…
The dimension is a key measure of complexity of partially ordered sets. Small dimension allows succinct encoding. Indeed if $P$ has dimension $d$, then to know whether $x \leq y$ in $P$ it is enough to check whether $x\leq y$ in each of the $d$ linear extensions of a witnessing realizer. Focusing on the encoding aspect Nešetřil and Pudlák defined a more expressive version of dimension. A poset $P$ has boolean dimension at most $d$ if it is possible to decide whether $x \leq y$ in $P$ by looking at the relative position of $x$ and $y$ in only $d$ permutations of the elements of $P$. We prove th…
We conjecture a strong property for the up and down maps U and D in an r-differential poset: DU+tI and UD+tI have Smith normal forms over Z[t]. In particular, this would determine the integral structure of the maps U, D, UD, DU, including their ranks in any characteristic. As evidence, we prove the conjecture for the Young-Fibonacci lattice YF studied by Okada and its r-differential generalizations Z(r), as well as verifying many of its consequences for Young's lattice Y and the r-differential Cartesian products Y^r.
Rowmotion is an invertible operator on the order ideals of a poset which has been extensively studied and is well understood for the rectangle poset. In this paper, we show that rowmotion is equivariant with respect to a bijection of Hamaker, Patrias, Pechenik and Williams between order ideals of rectangle and trapezoid posets, thereby affirming a conjecture of Hopkins that the rectangle and trapezoid posets have the same rowmotion orbit structures. Our main tools in proving this are $K$-jeu-de-taquin and (weak) $K$-Knuth equivalence of increasing tableaux. We define $almost$ $minimal$ $tablea…
It has been known for more than 40 years that there are posets with planar cover graphs and arbitrarily large dimension. Recently, Streib and Trotter proved that such posets must have large height. In fact, all known constructions of such posets have two large disjoint chains with all points in one chain incomparable with all points in the other. Gutowski and Krawczyk conjectured that this feature is necessary. More formally, they conjectured that for every $k\geq 1$, there is a constant $d$ such that if $P$ is a poset with a planar cover graph and $P$ excludes $\mathbf{k}+\mathbf{k}$, then $\…
Resolving a conjecture of Methuku and the first author we determine the size of the largest family of subsets of an $n$-element set avoiding both $Y_k$ and $Y_k'$ as induced subposets. The result follows as a consequence of the analogous result on a cyclical grid poset.
The machinery of noncommutative Schur functions is a general approach to Schur positivity of symmetric functions initiated by Fomin-Greene. Hwang recently adapted this theory to posets to give a new approach to the Stanley-Stembridge conjecture. We further develop this theory to prove that the symmetric function associated to any $P$-Knuth equivalence graph is Schur positive. This settles a conjecture of Kim and the third author, and refines results of Gasharov, Shareshian-Wachs, and Hwang on the Schur positivity of chromatic symmetric functions.
Let $P$ be a graded poset of rank $r$ and let $\mathbf{c}$ be a $c$-element chain. For an order ideal $I$ of $P \times \mathbf{c}$, its rowmotion $ψ(I)$ is the smallest ideal containing the minimal elements of the complementary filter of $I$. The map $ψ$ defines invertible dynamics on the set of ideals. We say that $P$ has NRP ("not relatively prime") rowmotion if no $ψ$-orbit has cardinality relatively prime to $r+c+1$. In work with R. Patrias (2020), we proved a 1995 conjecture of P. Cameron and D. Fon-Der-Flaass by establishing NRP rowmotion for the product $P = \mathbf{a} \times \mathbf{b}…
A poset is (r + s)-free if it does not contain two incomparable chains of size r and s, respectively. We prove that when r and s are at least 2, the First-Fit algorithm partitions every (r + s)-free poset P into at most 8(r-1)(s-1)w chains, where w is the width of P. This solves an open problem of Bosek, Krawczyk, and Szczypka (SIAM J. Discrete Math., 23(4):1992--1999, 2010).
This paper gives a formula for the antichain generating polynomial $\mathcal{N}_{[k]\times Q}$ of the poset $[k]\times Q$, where $[k]$ is an arbitrary chain and $Q$ is any finite graded poset. When $Q$ specializes to be a connected minuscule poet, which was classified by Proctor in 1984, we find that the polynomial $\mathcal{N}_{[k]\times Q}$ bears nice properties. For instance, we will recover the $B_n$-Narayana polynomial and the $D_{2n+2}$-Narayana polynomial. We collect evidence for the conjecture that whenever $\mathcal{N}_{[k]\times P}(x)$ is palindromic, it must be $γ$-positive. Moreove…
Let $(P,\leq)$ be a finite poset (partially ordered set), where $P$ has cardinality $n$. Consider linear extensions of $P$ as permutations $x_1x_2\cdots x_n$ in one-line notation. For distinct elements $x,y\in P$, we define $\mathbb{P}(x\prec y)$ to be the proportion of linear extensions of $P$ in which $x$ comes before $y$. For $0\leq α\leq \frac{1}{2}$, we say $(x,y)$ is an $α$-balanced pair if $α\leq \mathbb{P}(x\prec y) \leq 1-α.$ The $1/3-2/3$ Conjecture states that every finite partially ordered set which is not a chain has a $1/3$-balanced pair. We make progress on this conjecture by sh…
