0,4

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.

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Thomas Wieder, (Old) Home Page.

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

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.

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

Cf. A000670, A025875, A000295.

Adjacent sequences: A102229 A102230 A102231 this_sequence A102233 A102234 A102235

Sequence in context: A076100 A043033 A061495 this_sequence A121113 A091710 A054880

nonn

Thomas Wieder (wieder.thomas(AT)t-online.de), Jan 01 2005