A032263 Number of ways to partition n labeled elements into 4 pie slices allowing the pie to be turned over; number of 2-element proper antichains of an n-element set.

0, 0, 0, 3, 30, 195, 1050, 5103, 23310, 102315, 437250, 1834503, 7597590, 31175235, 127067850, 515396703, 2083011870, 8396420955, 33779000850, 135696347703, 544527210150, 2183335871475, 8749027724250, 35043169903503, 140313869216430, 561679070838795

Offset: 1

Author

Christian G. Bower

Keywords

• nonn
• easy
• nice

Comments

A proper antichain is an antichain iff each two of its members have a nonempty intersection.

Let P(A) be the power set of an n-element set A. Then a(n+1) = the number of pairs of elements {x,y} of P(A) for which x and y are intersecting but for which x is not a subset of y and y is not a subset of x. This is just a different formulation of the alternative sequence description. - Ross La Haye, Jan 09 2008

Links

• Vincenzo Librandi, Table of n, a(n) for n = 1..1000
• C. G. Bower, Transforms (2)
• 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).

Crossrefs

Cf. A000453.

Programs

Magma

I:=[0,0,0,3]; [n le 4 select I[n] else 10*Self(n-1)-35*Self(n-2)+50*Self(n-3)-24*Self(n-4): n in [1..30]]; // Vincenzo Librandi, Oct 19 2013

Maple

A032263 := proc(n) (4^n-4*3^n+6*2^n-4)/8 ; end: seq(A032263(n),n=1..20) ; # R. J. Mathar, Feb 26 2008

Mathematica

CoefficientList[Series[(3x^4)/((1-x)(1-2x)(1-3x)(1-4x)),{x,0,40}],x] (* Harvey P. Dale, Feb 28 2013 *)

PARI

a(n)=([0,1,0,0; 0,0,1,0; 0,0,0,1; -24,50,-35,10]^(n-1)*[0;0;0;3])[1,1] \\ Charles R Greathouse IV, Feb 09 2017

Formula

"DIJ[ 4 ]" (bracelet, indistinct, labeled, 4 parts) transform of 1, 1, 1, 1, ...

3*S(n,4) = (4^n-4*3^n+6*2^n-4)/8. - R. J. Mathar, Feb 26 2008

G.f.: 3*x^4/((1-x)*(1-2*x)*(1-3*x)*(1-4*x)). - Colin Barker, May 29 2012

a(n) = 3*A000453(n). - Alois P. Heinz, Jan 24 2018

E.g.f.: (exp(x) - 1)^4/8. - Stefano Spezia, Apr 06 2022

Extensions

Alternative description from Vladeta Jovovic, Goran Kilibarda, Zoran Maksimovic

More terms from Vincenzo Librandi, Oct 19 2013