A001206 Number of self-dual monotone Boolean functions of n variables.

0, 1, 2, 4, 12, 81, 2646, 1422564, 229809982112, 423295099074735261880

Offset: 0

Author

N. J. A. Sloane

Keywords

• nonn
• hard
• nice
• more

Comments

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

Rivière gives a(n) for n <= 5. - N. J. A. Sloane, May 12 2012

Examples

			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.
Cf. A059090.
From _Gus Wiseman_, Jul 03 2019: (Start)
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)
		

References

• 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).

Links

• Taras Banakh, Volodymyr Gavrylkiv, Automorphism groups of superextensions of groups, arXiv:1802.05804 [math.GR], 2018.
• Jan C. Bioch and Toshihide Ibaraki, Generating and approximating nondominated coteries, IEEE Transactions on parallel and distributed systems 6 (1995), 905-914.
• A. E. Brouwer and A. Verbeek, Counting families of mutually intersecting sets, Report ZN 41, March 1972, Math. Centr., Amsterdam. Gives a(n) for n <= 7.
• A. E. Brouwer and A. Verbeek, Counting families of mutually intersecting sets, Electronic Journal of Combinatorics, Volume 20, Issue 2 (2013), Paper #P8.
• 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.
• Jesús A. De Loera, Serkan Hoşten, Robert Krone, Lily Silverstein, Average Behavior of Minimal Free Resolutions of Monomial Ideals, arXiv:1802.06537 [math.AC], 2018.
• Serkan Hosten and Walter D. Morris, Jr., The order dimension of the complete graph, Discrete Math. 201 (1999), pp. 133-139.
• D. E. Loeb, Challenges in playing multiplayer games, in Levy and Beal, editors, Heuristic Programming in Artificial Intelligence, vol. 4, Ellis Horwood, 1994. [broken link]
• D. E. Loeb and A. Meyerowitz, The maximal intersecting family of sets graph, in H. Barcelo and G. Kalai, editors, Proceedings of the Conference on Jerusalem Combinatorics 1993. AMS series Contemporary Mathematics, 1994.
• Bartłomiej Pawelski, Counting and generating monotone Boolean functions, Doctoral Diss., Univ. Gdańsk, (Poland, 2024). See pp. 12, 15, 38, 49, 58, 73.
• Bartlomiej Pawelski and Andrzej Szepietowski, Divisibility properties of Dedekind numbers, arXiv:2302.04615 [math.CO], 2023.
• N. M. Rivière, Recursive formulas on free distributive lattices, J. Combinatorial Theory 5 1968 229--234. MR0231764 (38 #92).
• Tom Trotter, An Application of the Erdős/Stone Theorem, Slides, Sept. 13, 2001.
• Index entries for sequences related to Boolean functions

Crossrefs

Cf. A000372, A059090.

The case with empty edges allowed is A326372.

The maximal case is A007363, or A326363 with empty edges allowed.

The case with empty intersection is A326366.

The inverse binomial transform is the covering case A305844.

Cf. A006126, A014466, A051185, A326361, A326362.

Programs

Mathematica

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 *)

Formula

a(n+1) = Sum_{m=0..A037952(n)} A059090(n, m).

For n > 0, a(n) = A326372(n - 1) - 1. - Gus Wiseman, Jul 03 2019

Extensions

a(8) due to C. F. Mills & W. H. Mills, 1979

a(8) from Daniel E. Loeb, Jan 04 1996

a(8) confirmed by Don Knuth, Feb 08 2008

a(9) from Andries E. Brouwer, Aug 25 2012