0,7

The labeled case for at least k=2 elements per rank is given by A032032 = Partition n labeled elements into sets of sizes of at least 2 and order the sets. The unlabeled case for at least k=3 elements per rank is given by A000930 = A Lam{\'e} sequence of higher order. The unlabeled case for at least k=2 elements per rank is given by A000045 = Fibonacci numbers.

Thomas Wieder, Home Page.

Thomas Wieder, (Old) Home Page.

G.f.: -(z^2-2*exp(z)+2+2*z)/(4-2*exp(z)+2*z+z^2).

Let 1,2,3,4,5,6 denote six labeled elements. Let | denote a separation between two ranks. E.g. if elements 1,2, and 3 are on rank (also called level) one and elements 3,4, and 5 are on rank two, then we have the ranking 123|456.

For n=9 we have a(n)=2101 rankings. The order within a rank does not count. Six examples are:

123|456|789; 123456789; 12345|6789; 129|345678; 1235|46789; 789|123456.

series(-(z^2-2*exp(z)+2+2*z)/(4-2*exp(z)+2*z+z^2), z=0, 30);

with(combstruct): SeqSetL := [S, {S=Sequence(U), U=Set(Z, card >= 3)}, labeled]: seq(count(SeqSetL, size=j), j=1..23); - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Oct 19 2006

Cf. A000640, A102232, A032032.

Sequence in context: A020150 A044159 A044540 this_sequence A109357 A082041 A085027

Adjacent sequences: A102230 A102231 A102232 this_sequence A102234 A102235 A102236

nonn

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

a(0) changed to 1 at the suggestion of Zerinvary Lajos, Oct 26 2006