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Register a new userOn scattered posets with finite dimension
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We discuss a possible characterization, by means of forbidden configurations, of posets which are embeddable in a product of finitely many scattered chains.
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We study covering numbers and local covering numbers with respect to difference graphs and complete bipartite graphs. In particular we show that in every cover of a Young diagram with $binom{2k}{k}$ steps with generalized rectangles there is a row or a column in the diagram that is used by at least $k+1$ rectangles, and prove that this is best-possible. This answers two questions by Kim, Martin, Masa{v{r}}{i}k, Shull, Smith, Uzzell, and Wang (Europ. J. Comb. 2020), namely: - What is the local complete bipartite cover number of a difference graph? - Is there a sequence of graphs with constant local difference graph cover number and unbounded local complete bipartite cover number? We add to the study of these local covering numbers with a lower bound construction and some examples. Following Kim emph{et al.}, we use the results on local covering numbers to provide lower and upper bounds for the local dimension of partially ordered sets of height~2. We discuss the local dimension of some posets related to Boolean lattices and show that the poset induced by the first two layers of the Boolean lattice has local dimension $(1 + o(1))log_2log_2 n$. We conclude with some remarks on covering numbers for digraphs and Ferrers dimension.
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 $kgeq 1$, 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 $dim(P)leq d$. We settle their conjecture in the affirmative. We also discuss possibilities of generalizing the result by relaxing the condition that the cover graph is planar.
Motivated by generalizing Khovanovs categorification of the Jones polynomial, we study functors $F$ from thin posets $P$ to abelian categories $mathcal{A}$. Such functors $F$ produce cohomology theories $H^*(P,mathcal{A},F)$. We find that CW posets, that is, face posets of regular CW complexes, satisfy conditions making them particularly suitable for the construction of such cohomology theories. We consider a category of tuples $(P,mathcal{A},F,c)$, where $c$ is a certain ${1,-1}$-coloring of the cover relations in $P$, and show the cohomology arising from a tuple $(P,mathcal{A},F,c)$ is functorial, and independent of the coloring $c$ up to natural isomorphism. Such a construction provides a framework for the categorification of a variety of familiar topological/combinatorial invariants: anything expressible as a rank-alternating sum over a thin poset.
Using a modified version of jeu de taquin, Novelli, Pak and Stoyanovskii gave a bijective proof of the hook-length formula for counting standard Young tableaux of fixed shape. In this paper we consider a natural extension of jeu de taquin to arbitrary posets. Given a poset P, jeu de taquin defines a map from the set of bijective labelings of the poset elements with ${1,2,...,|P|}$ to the set of linear extensions of the poset. One question of particular interest is for which posets this map yields each linear extension equally often. We analyze the double-tailed diamond poset $D_{m,n}$ and show that uniform distribution is obtained if and only if $D_{m,n}$ is d-complete. Furthermore, we observe that the extended hook-length formula for counting linear extensions on d-complete posets provides a combinatorial answer to a seemingly unrelated question, namely: Given a uniformly random standard Young tableau of fixed shape, what is the expected value of the left-most entry in the second row?
In this paper, we show that, for all $ngeq 5$, the maximum number of $2$-chains in a butterfly-free family in the $n$-dimensional Boolean lattice is $leftlceilfrac{n}{2}rightrceilbinom{n}{lfloor n/2rfloor}$. In addition, for the height-2 poset $K_{s,t}$, we show that, for fixed $s$ and $t$, a $K_{s,t}$-free family in the $n$-dimensional Boolean lattice has $Oleft(nbinom{n}{lfloor n/2rfloor}right)$ $2$-chains.
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