A133789 Let P(A) denote 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 disjoint and for which x is not a subset of y and y is not a subset of x, 1) x and y are disjoint and for which either x is a subset of y or y is a subset of x, or 2) x and y intersect but for which x is not a subset of y and y is not a subset of x.
0, 1, 4, 16, 70, 316, 1414, 6196, 26590, 112156, 466774, 1923076, 7863310, 31972396, 129459334, 522571156, 2104535230, 8460991036, 33972711094, 136277478436, 546270602350, 2188566048076, 8764718254054, 35090241492916, 140455083984670, 562102715143516
Offset: 0
Keywords
Examples
a(3) = 16 because for P(A) = {{},{1},{2},{3},{1,2},{1,3},{2,3},{1,2,3}} we see that
{1} and {2},
{1} and {3},
{2} and {3},
{1} and {2,3},
{2} and {1,3},
{3} and {1,2}
are disjoint, while
{} and {1},
{} and {2},
{} and {3},
{} and {1,2},
{} and {1,3},
{} and {2,3},
{} and {1,2,3}
are disjoint and one is a subset of the other and
{1,2} and {1,3},
{1,2} and {2,3},
{1,3} and {2,3}
are intersecting, but neither is a subset of the other.
Also, through row 8 of Pascal's triangle the a(3)=16 even entries are 2 (so a(0)=0 and a(1)=1) then 4,6,4 (so a(2)=4) then 10,10 then 6,20,6 then 8,28,56,70,56,28,8. [_Aaron Meyerowitz_, Oct 29 2013]
Links
- Ross La Haye, Binary Relations on the Power Set of an n-Element Set, Journal of Integer Sequences, Vol. 12 (2009), Article 09.2.6. [_Ross La Haye_, Feb 22 2009]
Formula
a(n) = (1/2)(4^n - 2*3^n + 3*2^n - 2).
O.g.f.: x*(1-6*x+11*x^2)/[(-1+x)*(-1+2*x)*(-1+3*x)*(-1+4*x)]. - R. J. Mathar, Jan 11 2008
a(n) = A084869(n)-1 = A016269(n-2)+2^n-1. - Vladeta Jovovic, Jan 04 2008, corrected by Eric Rowland, May 15 2017
a(n) = 3*StirlingS2(n+1,4) + StirlingS2(n+1,3) + StirlingS2(n+1,2). - Ross La Haye, Jan 11 2008
a(n) = 3*StirlingS2(n+1,4) + StirlingS2(n+1,3) + StirlingS2(n+1,2). - Ross La Haye, Jan 11 2008
a(n) = 10*a(n-1)-35*a(n-2)+50*a(n-3)-24*a(n-4). [Aaron Meyerowitz, Oct 29 2013]
Extensions
Edited by N. J. A. Sloane, Jan 20 2008 to incorporate suggestions from several contributors.
Comments