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A001206
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Number of self-dual monotone Boolean functions of n variables.
(Formerly M1267 N0486)
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25
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OFFSET
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0,3
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COMMENTS
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Sometimes called Hosten-Morris numbers (or HM numbers).
Also the number of simplicial complexes on the set {1, ..., n-1} such that no pair of faces covers all of {1, ..., n-1}. [Miller-Sturmfels] - N. J. A. Sloane, Feb 18 2008
Also the maximal number of generators of a neighborly monomial ideal in n variables. [Miller-Sturmfels]. - N. J. A. Sloane, Feb 18 2008
Also the number of intersecting antichains on a labeled (n-1)-set or (n-1)-variable Boolean functions in the Post class F(7,2). Cf. A059090. - Vladeta Jovovic, Goran Kilibarda, Dec 28 2000
Also the number of nondominated coteries on n members. - Don Knuth, Sep 01 2005
The number of maximal families of intersecting subsets of an n-element set. - Bridget Tenner, Nov 16 2006
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REFERENCES
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Martin Aigner and Günter M. Ziegler, Proofs from THE BOOK, Third Edition, Springer-Verlag, 2004. See chapter 22.
V. Jovovic and G. Kilibarda, The number of n-variable Boolean functions in the Post class F(7,2), Belgrade, 2001, in preparation.
D. E. Knuth, The Art of Computer Programming, Vol. 4A, Section 7.1.1, p. 79.
W. F. Lunnon, The IU function: the size of a free distributive lattice, pp. 173-181 of D. J. A. Welsh, editor, Combinatorial Mathematics and Its Applications. Academic Press, NY, 1971.
Charles F. Mills and W. M. Mills, The calculation of λ(8), preprint, 1979. Gives a(8).
E. Miller and B. Sturmfels, Combinatorial Commutative Algebra, Springer, 2005.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
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LINKS
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Gábor Damásdi, Stefan Felsner, António Girão, Balázs Keszegh, David Lewis, Dániel T. Nagy, Torsten Ueckerdt, On Covering Numbers, Young Diagrams, and the Local Dimension of Posets, arXiv:2001.06367 [math.CO], 2020.
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FORMULA
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EXAMPLE
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a(2) = 1 + 1 = 2;
a(3) = 1 + 3 = 4;
a(4) = 1 + 7 + 3 + 1 = 12;
a(5) = 1 + 15 + 30 + 30 + 5 = 81;
a(6) = 1 + 31 + 195 + 605 + 780 + 543 + 300 + 135 + 45 + 10 + 1 = 2646;
a(7) = 1 + 63 + 1050 + 9030 + 41545 + 118629 + 233821 + 329205 + 327915 + 224280 + 100716 + 29337 + 5950 + 910 + 105 + 1 = 1422564.
The a(1) = 1 through a(4) = 12 intersecting antichains of nonempty sets (see Jovovic and Kilibarda's comment):
{} {} {} {}
{{1}} {{1}} {{1}}
{{2}} {{2}}
{{1,2}} {{3}}
{{1,2}}
{{1,3}}
{{2,3}}
{{1,2,3}}
{{1,2},{1,3}}
{{1,2},{2,3}}
{{1,3},{2,3}}
{{1,2},{1,3},{2,3}}
(End)
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MATHEMATICA
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stableSets[u_, Q_]:=If[Length[u]==0, {{}}, With[{w=First[u]}, Join[stableSets[DeleteCases[u, w], Q], Prepend[#, w]&/@stableSets[DeleteCases[u, r_/; r==w||Q[r, w]||Q[w, r]], Q]]]];
Table[Length[stableSets[Subsets[Range[n], {1, n}], Or[Intersection[#1, #2]=={}, SubsetQ[#1, #2]]&]], {n, 0, 5}] (* Gus Wiseman, Jul 03 2019 *)
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CROSSREFS
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The case with empty edges allowed is A326372.
The case with empty intersection is A326366.
The inverse binomial transform is the covering case A305844.
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KEYWORD
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nonn,hard,nice,more
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AUTHOR
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EXTENSIONS
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a(8) due to C. F. Mills & W. H. Mills, 1979
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STATUS
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approved
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