A102232 - OEIS

A102232

Number of preferential arrangements of n labeled elements when at least k=three ranks are required.

0, 0, 0, 6, 60, 510, 4620, 47166, 545580, 7086750, 102246540, 1622630526, 28091563500, 526858340190, 10641342954060, 230283190945086, 5315654681915820, 130370767029004830, 3385534663256583180, 92801587319327886846, 2677687796244383154540, 81124824998504071784670

(list; graph; refs; listen; history; text; internal format)

OFFSET

0,4

COMMENTS

The labeled case for k=2 is given by A052875. The unlabeled case for k=3 is given by A000295 = Eulerian numbers 2^n - n - 1. The unlabeled case for k=2 is given by A000225 = 2^n - 1.

LINKS

Table of n, a(n) for n=0..21.

FORMULA

E.g.f.: -(exp(z)^3-3*exp(z)^2+3*exp(z)-1)/(-2+exp(z)).

EXAMPLE

Let 1,2,3 denote three labeled elements. Let | denote a separation between two ranks. E.g. if element 1 is on rank (also called level) one, element 3 is on rank two and element 2 is on rank three, then we have the ranking 1|3|2.

For n=3 we have obviously a(3)=6 possible rankings:

2|3|1, 3|2|1, 1|2|3, 2|1|3, 3|1|2, 1|3|2.

For n=4 we have a(4) = 60 possible rankings, e.g. (elements 1 and 3 are on the same rank in the first two examples)

31|2|4, 2|4|31, 4|1|3|2.

MAPLE

series(-(exp(z)^3-3*exp(z)^2+3*exp(z)-1)/(-2+exp(z)), z=0, 30);

spec := [S,

{

B = Set(Z, 1 <= card),

C = Sequence(B, 2 <= card),

S = Prod(B, C)

}, labeled]:

struct := n -> combstruct[count](spec, size = n);

seq(struct(n), n = 0..21); # Peter Luschny, Jul 22 2014

MATHEMATICA

m = 22; CoefficientList[-(E^(3z) - 3*E^(2z) + 3*E^z - 1)/(-2 + E^z) + O[z]^m, z] Range[0, m-1]! (* Jean-François Alcover, Jun 11 2019 *)

CROSSREFS

Cf. A000670, A025875, A000295.

Sequence in context: A061495 A220411 A248217 * A121113 A213269 A091710

Adjacent sequences: A102229 A102230 A102231 * A102233 A102234 A102235

KEYWORD

nonn

AUTHOR

Thomas Wieder, Jan 01 2005

EXTENSIONS

More terms from Peter Luschny, Jul 22 2014

STATUS

approved