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### A note on blockers in posets [article]

Anders Björner, Axel Hultman
2004 arXiv   pre-print
The posets P for which A^**=A for all antichains are characterized. 2. The blocker A^* of a symmetric antichain in the partition lattice is characterized. 3.  ...  The blocker A^* of an antichain A in a finite poset P is the set of elements minimal with the property of having with each member of A a common predecessor. The following is done: 1.  ...  Symmetric blockers in partition lattices Recall that the partition lattice Π n consists of all set partitions of [n] = {1, . . . , n} ordered by refinement.  ...

### Page 79 of Mathematical Reviews Vol. , Issue 97A [page]

1997 Mathematical Reviews
H. (1-CAR; Riverside, CA) Large antichains in the partition lattice. (English summary) Random Structures Algorithms 6 (1995), no. 1, 89-104.  ...  Let d,, be the largest size of an antichain in the lattice of partitions Il, of an n-element set ordered by refinement and let b, be the largest number of partitions of an n-element set all having the  ...

### Partitioning Boolean lattices into antichains

Muktar Elzobi, Zbigniew Lonc
2003 Discrete Mathematics
Let f(n) be the smallest integer t such that a poset obtained from a Boolean lattice with n atoms by deleting both the largest and the smallest elements can be partitioned into t antichains of the same  ...  In this paper, it is shown that f(n) 6 b n 2 =log n. This is an improvement of the best previously known upper bound for f(n).  ...  In this article, we consider a variation of a dual problem, i.e. the problem of partitioning of a Boolean lattice into antichains of equal size.  ...

### Partitions of large Boolean lattices

Zbigniew Lonc
1994 Discrete Mathematics
Definef(n) to be the minimum number k such that there is a partition of 2" into k antichains of the same size except for at most one antichain of a smaller size.  ...  In the paper we examine the asymptotic behavior of f(n) and we show that c1 n < f(n) < c2n2 for some constants c1 and c2 and n sufficiently large.  ...  In this paper we deal with a variation of the dual problem: (Q) Can, for n sufficiently large given m, the ordered set 2" be partitioned into antichains of size m except for at most m-1 elements which  ...

### A splitting property of maximal antichains

Rudolf Ahlswede, Péter L. Erdős, Niall Graham
1995 Combinatorica
In any dense poset P (e.g. in the Boolean lattice) every maximal antichain S may be partitioned into disjoint subsets S 1 and S 2 , such that the union of the downset of S 1 with the upset of S 2 yields  ...  To find a similar splitting of maximal antichains in posets is NP-hard in general.  ...  For convenience, we sometimes say that the antichain S is partitionable when S satisfies the splitting property in the poset P.  ...

### The Size of the Largest Antichain in the Partition Lattice

E.Rodney Canfield
1998 Journal of combinatorial theory. Series A
Consider the poset n of partitions of an n-element set, ordered by re nement. The sizes of the various ranks within this poset are the Stirling numbers of the second kind. Let a = 1 2 ? e log(2)=4.  ...  My thanks to Ed Bender, whose comments on the previous version of this paper led to improved style and clari ed exposition.  ...  Since x no y speci es that x be a re nement of y of a special sort, we see the implication x no y ) x y: 4 Largest Antichain in the Partition Lattice Consequently, any set which is an antichain in (  ...

### 4-Connected Triangulations on Few Lines [article]

Stefan Felsner
2019 arXiv   pre-print
The same holds for all subgraphs of such triangulations. The proof is based on a corresponding result for diagrams of planar lattices which makes use of orthogonal chain and antichain families.  ...  We show that 4-connected plane triangulations can be redrawn such that edges are represented by straight segments and the vertices are covered by a set of at most √(2n) lines each of them horizontal or  ...  Work on this problem began at the 2018 Bertinoro Workshop of Graph Drawing. I thank the organizers of the event for making this possible.  ...

### Book Review: Combinatorics of finite sets

Douglas B. West
1988 Bulletin of the American Mathematical Society
In 1928, Sperner determined the maximum-sized antichains in the subset lattice.  ...  Concerning the Sperner property for special posets, Anderson outlines Shearer's proof of Canfield's result that for large n the lattice of partitions of an n-set is not Sperner, contrary to a conjecture  ...

### Page 666 of Mathematical Reviews Vol. , Issue 89B [page]

1989 Mathematical Reviews
The seminal result in the theory is the result of Sperner (1928) that gives the maximum size of an antichain in the Boolean lattice B, (consisting of all subsets of (1,---,m) ordered by inclusion).  ...  While the Sperner property for ranked posets is treated in detail, the book does not describe any of the counterexamples to the long-standing conjecture of Rota (that the partition lattice is Sperner,  ...

### Making communities show respect for order

Vaiva Vasiliauskaite, Tim S. Evans
2020 Applied Network Science
We study the algorithm's performance and antichain properties in artificial models and in real networks, such as citation graphs and food webs.  ...  We show how well this partitioning algorithm distinguishes and groups together nodes of the same origin (in a citation network, the origin is a topic or a research field).  ...  Authors' contributions Both authors developed the theoretical concepts and designed the experiments discussed in the paper. VV created the software used and performed the data analysis.  ...

### Minimum sized fibres in distributive lattices

Dwight Duffus, Bill Sands
2001 Journal of the Australian Mathematical Society
We find a lower bound on the function / (£>), the minimum fibre size in the distributive lattice D, in terms of the size of D.  ...  This fact depends upon being able to split every maximal antichain of this class of distributive lattices into two parts so that the lattice is the union of the upset of one part and the downset of the  ...  To prove Theorem 1, we play off the dimension of a distributive lattice against its length and invoke a description of partitions induced by maximal antichains in a certain class of distributive lattices  ...

### Enumeration of extensions of the cycle matroid of a complete graph [article]

Peter Nelson, Shayla Redlin, Jorn van der Pol
2021 arXiv   pre-print
We prove that the number of single element extensions of M(K_n+1) is 2^n n/2(1+o(1)). This is done using a characterization of extensions as "linear subclasses".  ...  Since ⌊ n 2 ⌋ + 1 = ⌈ n+1 2 ⌉, the result follows. Antichains in the Boolean lattice are well studied and each intersecting antichain is also an antichain.  ...  The Boolean lattice is used as the auxiliary graph in this paper, so known results about enumerating antichains in a Boolean lattice are used instead of a direct application of the container method.  ...

### Making Communities Show Respect for Order [article]

Vaiva Vasiliauskaite, Tim S. Evans
2019 arXiv   pre-print
We study the algorithm's performance and antichain properties in artificial models and in real networks, such as citation graphs and food webs.  ...  We show how well this partitioning algorithm distinguishes and groups together nodes of the same origin (in a citation network, the origin is a topic or a research field).  ...  We partition the nodes of a DAG into antichains which have large neighbourhood overlaps.  ...

### On the widths of finite distributive lattices

Jeff Kahn, Michael Saks
1987 Discrete Mathematics
In words this says that as one considers ~ increasingly large distributive lattices, the maximum sized antichain contains small proportion of the elements.  ...  The following conjecture of U Faigle and B Sands is proved: For every number R > 0 there exists a number n(R) such that if 2 is a finite distributive lattice whose width w(Z) (size of the largest antichain  ...  Acknowledgments The authors wish to thank Dean Sturtevant for several useful discussions. We also thank the referees and Jim Walker for their comments and corrections.  ...

### Chains, Antichains, and Complements in Infinite Partition Lattices [article]

James Emil Avery, Jean-Yves Moyen, Pavel Ruzicka, Jakob Grue Simonsen
2017 arXiv   pre-print
We consider the partition lattice Π_κ on any set of transfinite cardinality κ and properties of Π_κ whose analogues do not hold for finite cardinalities.  ...  Moreover, we give a direct formula for the number of complements to a given partition; (VI) Under the Generalized Continuum Hypothesis, the cardinalities of maximal chains, maximal antichains, and numbers  ...  James Avery was supported by VILLUM FONDEN through the network for Experimental Mathematics in Number Theory, Operator Algebras, and Topology.  ...
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