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A note on blockers in posets
[article]

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. ...

##
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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

2003
*
Discrete Mathematics
*

Let f(n) be

doi:10.1016/s0012-365x(02)00448-x
fatcat:7ojs4dj6orfn7gmb2sn5qveggu
*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. ...##
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Partitions of large Boolean lattices

1994
*
Discrete Mathematics
*

Definef(n) to be

doi:10.1016/0012-365x(94)90382-4
fatcat:x47eskpz3rczdnaw4cfa3u5lnu
*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

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. ...

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

1998
*
Journal of combinatorial theory. Series A
*

Consider

doi:10.1006/jcta.1998.2871
fatcat:22vg2w3nwrcwnm2o2d7a2hylmq
*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]

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. ...

##
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Book Review: Combinatorics of finite sets

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 ...

##
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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

2020
*
Applied Network Science
*

We study

doi:10.1007/s41109-020-00255-5
fatcat:2aaklogywrczvg2wo7rh2t63uu
*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

2001
*
Journal of the Australian Mathematical Society
*

We find a lower bound on

doi:10.1017/s144678870000238x
fatcat:q5ts52laz5cyxp2hv6vef6a5bq
*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]

2021
*
arXiv
*
pre-print

We prove that

arXiv:2111.06585v1
fatcat:shymzumsuvcg5f4xgtpig4oy7e
*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]

2019
*
arXiv
*
pre-print

We study

arXiv:1908.11818v1
fatcat:6652eadrifdgzjgjdefwwdl3la
*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

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. ...

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

2017
*
arXiv
*
pre-print

We consider

arXiv:1501.05284v4
fatcat:rqkjpehqxra63efgpujgsre57i
*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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