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

1,4

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).

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

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 *)

Prog

(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
(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

Crossrefs

Cf. A000453.

Sequence in context: A177727 A132413 A344907 * A003771 A121100 A203366

Adjacent sequences: A032260 A032261 A032262 * A032264 A032265 A032266

Keyword

nonn,easy,nice

Author

Christian G. Bower

Extensions

Alternative description from Vladeta Jovovic, Goran Kilibarda, Zoran Maksimovic

More terms from Vincenzo Librandi, Oct 19 2013

Status

approved