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).

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, 1, 1, 21, 35, 105, 35, 210, 21, 105, 105, 70, 7, 105, 21, 35, 1, 1, 28, 56, 210, 70, 560, 56, 420, 420, 280, 28, 840, 168, 280, 8, 105, 210, 280, 28

Offset: 1

Author

Tilman Piesk, Apr 09 2012

Keywords

• tabf
• nonn

Comments

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.

The rows are counted from 1, the columns from 0.

Row lengths: 1,2,3,5,7,11... (partition numbers A000041)

Row sums: 1,2,5,15,52,203... (Bell numbers A000110)

Row maxima: 1,1,3,6,15,60,210,840,3780,12600,69300,415800... (A102356)

Distinct entries per row: 1,1,2,4,4,7,7,13,17,23,26,40... (A102465)

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.)

Links

• Tilman Piesk, Rows n=1..12 of triangle, flattened
• Tilman Piesk, Partition related number triangles

Crossrefs

Cf. A124323, A194602, A178867, A000041, A000110, A102356, A102465, A184049, A211360.