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A211350 Refined triangle A124323: T(n,k) is the number of partitions of an n-set that are of type k (k-th integer partition, defined by A194602).

%I A211350 #11 Apr 18 2012 15:46:59
%S A211350 1,1,1,1,3,1,1,6,4,3,1,1,10,10,15,5,10,1,1,15,20,45,15,60,6,15,15,10,
%T A211350 1,1,21,35,105,35,210,21,105,105,70,7,105,21,35,1,1,28,56,210,70,560,
%U A211350 56,420,420,280,28,840,168,280,8,105,210,280,28
%N A211350 Refined triangle A124323: T(n,k) is the number of partitions of an n-set that are of type k (k-th integer partition, defined by A194602).
%C A211350 Name could also be "Triangle of multinomial coefficients, read by rows (version 4)", compare A036040, A080575, A178867. The latter and this one differ only in the order of columns.
%C A211350 The rows are counted from 1, the columns from 0.
%C A211350 Row lengths: 1,2,3,5,7,11... (partition numbers A000041)
%C A211350 Row sums: 1,2,5,15,52,203... (Bell numbers A000110)
%C A211350 Row maxima: 1,1,3,6,15,60,210,840,3780,12600,69300,415800... (A102356)
%C A211350 Distinct entries per row: 1,1,2,4,4,7,7,13,17,23,26,40... (A102465)
%C A211350 Rightmost columns are those from Pascal's triangle A007318 without the second one (i.e. triangle A184049). The other columns - (always?) without a 1 at the top - are multiples of these columns from Pascal's triangle; so actually only the top elements of each column are needed to calculate the other entries; these top elements are in A211360. (The top elements of the related triangle A178867 are in A178866.)
%H A211350 Tilman Piesk, <a href="/A211350/b211350.txt">Rows n=1..12 of triangle, flattened</a>
%H A211350 Tilman Piesk, <a href="http://en.wikiversity.org/wiki/Partition_related_number_triangles#all1">Partition related number triangles</a>
%Y A211350 Cf. A124323, A194602, A178867, A000041, A000110, A102356, A102465, A184049, A211360.
%K A211350 tabf,nonn
%O A211350 1,5
%A A211350 _Tilman Piesk_, Apr 09 2012