

A102232


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


1



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

G.f.: (exp(z)^33*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 132.
For n=3 we have obviously a(3)=6 possible rankings:
231, 321, 123, 213, 312, 132.
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)
3124, 2431, 4132.


MAPLE

series((exp(z)^33*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


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



