cp's OEIS Frontend

A036239 Number of 2-element intersecting families of an n-element set; number of 2-way interactions when 2 subsets of power set on {1..n} are chosen at random.

%I A036239 #59 Mar 02 2024 13:27:33
%S A036239 0,2,15,80,375,1652,7035,29360,120975,494252,2007555,8120840,32753175,
%T A036239 131818052,529680075,2125927520,8525298975,34165897052,136857560595,
%U A036239 548011897400,2193792030375,8780400395252,35137296305115,140596265198480
%N A036239 Number of 2-element intersecting families of an n-element set; number of 2-way interactions when 2 subsets of power set on {1..n} are chosen at random.
%C A036239 Let P(A) be the power set of an n-element set A. Then a(n) = the number of pairs of elements {x,y} of P(A) for which either 0) x and y are intersecting but for which x is not a subset of y and y is not a subset of x, or 1) x and y are intersecting and for which either x is a proper subset of y or y is a proper subset of x. - _Ross La Haye_, Jan 10 2008
%C A036239 Graph theory formulation. Let P(A) be the power set of an n-element set A. Then a(n) = the number of edges in the intersection graph G of P(A). The vertices of G are the elements of P(A) and the edges of G are the pairs of elements {x,y} of P(A) such that x and y are intersecting (and x <> y). - _Ross La Haye_, Dec 23 2017
%D A036239 W. W. Kokko, "Interactions", manuscript, 1983.
%H A036239 T. D. Noe, <a href="/A036239/b036239.txt">Table of n, a(n) for n = 1..200</a>
%H A036239 Taylor Brysiewicz, Holger Eble, and Lukas Kühne, <a href="https://arxiv.org/abs/2105.14542">Enumerating chambers of hyperplane arrangements with symmetry</a>, arXiv:2105.14542 [math.CO], 2021.
%H A036239 V. Jovovic and G. Kilibarda, <a href="http://dx.doi.org/10.4213/dm398">On the number of Boolean functions in the Post classes F^{mu}_8</a>, in Russian, Diskretnaya Matematika, 11 (1999), no. 4, 127-138.
%H A036239 V. Jovovic and G. Kilibarda, <a href="http://dx.doi.org/10.1515/dma.1999.9.6.593">On the number of Boolean functions in the Post classes F^{mu}_8</a>, English translation, in Discrete Mathematics and Applications, 9, (1999), no. 6.
%H A036239 Ross La Haye, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL12/LaHaye/lahaye5.html">Binary Relations on the Power Set of an n-Element Set</a>, Journal of Integer Sequences, Vol. 12 (2009), Article 09.2.6.
%H A036239 Thomas Wieder, <a href="http://www.math.nthu.edu.tw/~amen/2008/070301.pdf">The number of certain k-combinations of an n-set</a>, Applied Mathematics Electronic Notes, vol. 8 (2008).
%H A036239 <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (10,-35,50,-24).
%F A036239 a(n) = (1/2) * (4^n - 3^n - 2^n + 1).
%F A036239 a(n) = 3*Stirling2(n+1,4) + 2*Stirling2(n+1,3). - _Ross La Haye_, Jan 10 2008
%F A036239 a(n) = A006516(n) - A003462(n). - _Zerinvary Lajos_, Jun 05 2009
%F A036239 From _Harvey P. Dale_, May 11 2011: (Start)
%F A036239 a(n) = 10*a(n-1) - 35*a(n-2) + 50*a(n-3) - 24*a(n-4); a(0)=0, a(1)=2, a(2)=15, a(3)=80.
%F A036239 G.f.: x^2*(2-5*x)/(1 - 10*x + 35*x^2 - 50*x^3 + 24*x^4). (End)
%F A036239 E.g.f.: exp(x)*(exp(x) - 1)^2*(exp(x) + 1)/2. - _Stefano Spezia_, Jun 26 2022
%t A036239 LinearRecurrence[{10,-35,50,-24},{0,2,15,80},40] (* or *) With[{c=1/2!}, Table[ c(4^n-3^n-2^n+1),{n,40}]] (* _Harvey P. Dale_, May 11 2011 *)
%o A036239 (Sage) [(4^n - 2^n)/2-(3^n - 1)/2 for n in range(1,24)] # _Zerinvary Lajos_, Jun 05 2009
%o A036239 (PARI) a(n)=(4^n-3^n-2^n+1)/2 \\ _Charles R Greathouse IV_, Jul 25 2011
%Y A036239 Cf. A036240, A051180-A051185, A028243, A032263.
%Y A036239 Cf. A003462, A006516, A016127, A016129, A016130, A016311, A016316, A016321, A016325. - _Zerinvary Lajos_, Jun 05 2009
%K A036239 nonn,easy,nice
%O A036239 1,2
%A A036239 _N. J. A. Sloane_