A134165 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 disjoint and for which either x is a subset of y or y is a subset of x, or 1) x and y are disjoint and for which x is not a subset of y and y is not a subset of x, or 2) x and y are intersecting but for which x is not a subset of y and y is not a subset of x, or 3) x = y.
1, 3, 8, 24, 86, 348, 1478, 6324, 26846, 112668, 467798, 1925124, 7867406, 31980588, 129475718, 522603924, 2104600766, 8461122108, 33972973238, 136278002724, 546271650926
Offset: 0
Examples
a(2) = 8 because for P(A) = {{},{1},{2},{1,2}} we have for case 0 {{},{1}}, {{},{2}}, {{},{1,2}} and we have for case 1 {{1},{2}} and we have for case 3 {{},{}}, {{1},{1}}, {{2},{2}}, {{1,2},{1,2}}. There are 0 {x,y} of P(A) in this example that fall under case 2.
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.
- Index entries for linear recurrences with constant coefficients, signature (10,-35,50,-24).
Programs
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Mathematica
LinearRecurrence[{10,-35,50,-24},{1,3,8,24},30] (* Harvey P. Dale, Feb 29 2020 *)
Formula
a(n) = (1/2)(4^n - 2*3^n + 5*2^n - 2) = 3*StirlingS2(n+1,4) + StirlingS2(n+1,3) + 2*StirlingS2(n+1,2) + 1.
G.f.: (1-7*x+13*x^2-x^3)/((1-x)*(1-2*x)*(1-3*x)*(1-4*x)). [Colin Barker, Jul 30 2012]