Identifier
Identifier
• St000159: ⟶ ℤ (values match St000318The number of addable cells of the Ferrers diagram of an integer partition., St001124The multiplicity of the standard representation in the Kronecker square corresponding to a partition.)
Values
[] => 0
[1] => 1
[2] => 1
[1,1] => 1
[3] => 1
[2,1] => 2
[1,1,1] => 1
[4] => 1
[3,1] => 2
[2,2] => 1
[2,1,1] => 2
[1,1,1,1] => 1
[5] => 1
[4,1] => 2
[3,2] => 2
[3,1,1] => 2
[2,2,1] => 2
[2,1,1,1] => 2
[1,1,1,1,1] => 1
[6] => 1
[5,1] => 2
[4,2] => 2
[4,1,1] => 2
[3,3] => 1
[3,2,1] => 3
[3,1,1,1] => 2
[2,2,2] => 1
[2,2,1,1] => 2
[2,1,1,1,1] => 2
[1,1,1,1,1,1] => 1
[7] => 1
[6,1] => 2
[5,2] => 2
[5,1,1] => 2
[4,3] => 2
[4,2,1] => 3
[4,1,1,1] => 2
[3,3,1] => 2
[3,2,2] => 2
[3,2,1,1] => 3
[3,1,1,1,1] => 2
[2,2,2,1] => 2
[2,2,1,1,1] => 2
[2,1,1,1,1,1] => 2
[1,1,1,1,1,1,1] => 1
[8] => 1
[7,1] => 2
[6,2] => 2
[6,1,1] => 2
[5,3] => 2
[5,2,1] => 3
[5,1,1,1] => 2
[4,4] => 1
[4,3,1] => 3
[4,2,2] => 2
[4,2,1,1] => 3
[4,1,1,1,1] => 2
[3,3,2] => 2
[3,3,1,1] => 2
[3,2,2,1] => 3
[3,2,1,1,1] => 3
[3,1,1,1,1,1] => 2
[2,2,2,2] => 1
[2,2,2,1,1] => 2
[2,2,1,1,1,1] => 2
[2,1,1,1,1,1,1] => 2
[1,1,1,1,1,1,1,1] => 1
[9] => 1
[8,1] => 2
[7,2] => 2
[7,1,1] => 2
[6,3] => 2
[6,2,1] => 3
[6,1,1,1] => 2
[5,4] => 2
[5,3,1] => 3
[5,2,2] => 2
[5,2,1,1] => 3
[5,1,1,1,1] => 2
[4,4,1] => 2
[4,3,2] => 3
[4,3,1,1] => 3
[4,2,2,1] => 3
[4,2,1,1,1] => 3
[4,1,1,1,1,1] => 2
[3,3,3] => 1
[3,3,2,1] => 3
[3,3,1,1,1] => 2
[3,2,2,2] => 2
[3,2,2,1,1] => 3
[3,2,1,1,1,1] => 3
[3,1,1,1,1,1,1] => 2
[2,2,2,2,1] => 2
[2,2,2,1,1,1] => 2
[2,2,1,1,1,1,1] => 2
[2,1,1,1,1,1,1,1] => 2
[1,1,1,1,1,1,1,1,1] => 1
[10] => 1
[9,1] => 2
[8,2] => 2
[8,1,1] => 2
[7,3] => 2
[7,2,1] => 3
[7,1,1,1] => 2
[6,4] => 2
[6,3,1] => 3
[6,2,2] => 2
[6,2,1,1] => 3
[6,1,1,1,1] => 2
[5,5] => 1
[5,4,1] => 3
[5,3,2] => 3
[5,3,1,1] => 3
[5,2,2,1] => 3
[5,2,1,1,1] => 3
[5,1,1,1,1,1] => 2
[4,4,2] => 2
[4,4,1,1] => 2
[4,3,3] => 2
[4,3,2,1] => 4
[4,3,1,1,1] => 3
[4,2,2,2] => 2
[4,2,2,1,1] => 3
[4,2,1,1,1,1] => 3
[4,1,1,1,1,1,1] => 2
[3,3,3,1] => 2
[3,3,2,2] => 2
[3,3,2,1,1] => 3
[3,3,1,1,1,1] => 2
[3,2,2,2,1] => 3
[3,2,2,1,1,1] => 3
[3,2,1,1,1,1,1] => 3
[3,1,1,1,1,1,1,1] => 2
[2,2,2,2,2] => 1
[2,2,2,2,1,1] => 2
[2,2,2,1,1,1,1] => 2
[2,2,1,1,1,1,1,1] => 2
[2,1,1,1,1,1,1,1,1] => 2
[1,1,1,1,1,1,1,1,1,1] => 1
[11] => 1
[10,1] => 2
[9,2] => 2
[9,1,1] => 2
[8,3] => 2
[8,2,1] => 3
[8,1,1,1] => 2
[7,4] => 2
[7,3,1] => 3
[7,2,2] => 2
[7,2,1,1] => 3
[7,1,1,1,1] => 2
[6,5] => 2
[6,4,1] => 3
[6,3,2] => 3
[6,3,1,1] => 3
[6,2,2,1] => 3
[6,2,1,1,1] => 3
[6,1,1,1,1,1] => 2
[5,5,1] => 2
[5,4,2] => 3
[5,4,1,1] => 3
[5,3,3] => 2
[5,3,2,1] => 4
[5,3,1,1,1] => 3
[5,2,2,2] => 2
[5,2,2,1,1] => 3
[5,2,1,1,1,1] => 3
[5,1,1,1,1,1,1] => 2
[4,4,3] => 2
[4,4,2,1] => 3
[4,4,1,1,1] => 2
[4,3,3,1] => 3
[4,3,2,2] => 3
[4,3,2,1,1] => 4
[4,3,1,1,1,1] => 3
[4,2,2,2,1] => 3
[4,2,2,1,1,1] => 3
[4,2,1,1,1,1,1] => 3
[4,1,1,1,1,1,1,1] => 2
[3,3,3,2] => 2
[3,3,3,1,1] => 2
[3,3,2,2,1] => 3
[3,3,2,1,1,1] => 3
[3,3,1,1,1,1,1] => 2
[3,2,2,2,2] => 2
[3,2,2,2,1,1] => 3
[3,2,2,1,1,1,1] => 3
[3,2,1,1,1,1,1,1] => 3
[3,1,1,1,1,1,1,1,1] => 2
[2,2,2,2,2,1] => 2
[2,2,2,2,1,1,1] => 2
[2,2,2,1,1,1,1,1] => 2
[2,2,1,1,1,1,1,1,1] => 2
[2,1,1,1,1,1,1,1,1,1] => 2
[1,1,1,1,1,1,1,1,1,1,1] => 1
[12] => 1
[11,1] => 2
[10,2] => 2
[10,1,1] => 2
[9,3] => 2
[9,2,1] => 3
[9,1,1,1] => 2
[8,4] => 2
[8,3,1] => 3
[8,2,2] => 2
[8,2,1,1] => 3
[8,1,1,1,1] => 2
[7,5] => 2
[7,4,1] => 3
[7,3,2] => 3
[7,3,1,1] => 3
[7,2,2,1] => 3
[7,2,1,1,1] => 3
[7,1,1,1,1,1] => 2
[6,6] => 1
[6,5,1] => 3
[6,4,2] => 3
[6,4,1,1] => 3
[6,3,3] => 2
[6,3,2,1] => 4
[6,3,1,1,1] => 3
[6,2,2,2] => 2
[6,2,2,1,1] => 3
[6,2,1,1,1,1] => 3
[6,1,1,1,1,1,1] => 2
[5,5,2] => 2
[5,5,1,1] => 2
[5,4,3] => 3
[5,4,2,1] => 4
[5,4,1,1,1] => 3
[5,3,3,1] => 3
[5,3,2,2] => 3
[5,3,2,1,1] => 4
[5,3,1,1,1,1] => 3
[5,2,2,2,1] => 3
[5,2,2,1,1,1] => 3
[5,2,1,1,1,1,1] => 3
[5,1,1,1,1,1,1,1] => 2
[4,4,4] => 1
[4,4,3,1] => 3
[4,4,2,2] => 2
[4,4,2,1,1] => 3
[4,4,1,1,1,1] => 2
[4,3,3,2] => 3
[4,3,3,1,1] => 3
[4,3,2,2,1] => 4
[4,3,2,1,1,1] => 4
[4,3,1,1,1,1,1] => 3
[4,2,2,2,2] => 2
[4,2,2,2,1,1] => 3
[4,2,2,1,1,1,1] => 3
[4,2,1,1,1,1,1,1] => 3
[4,1,1,1,1,1,1,1,1] => 2
[3,3,3,3] => 1
[3,3,3,2,1] => 3
[3,3,3,1,1,1] => 2
[3,3,2,2,2] => 2
[3,3,2,2,1,1] => 3
[3,3,2,1,1,1,1] => 3
[3,3,1,1,1,1,1,1] => 2
[3,2,2,2,2,1] => 3
[3,2,2,2,1,1,1] => 3
[3,2,2,1,1,1,1,1] => 3
[3,2,1,1,1,1,1,1,1] => 3
[3,1,1,1,1,1,1,1,1,1] => 2
[2,2,2,2,2,2] => 1
[2,2,2,2,2,1,1] => 2
[2,2,2,2,1,1,1,1] => 2
[2,2,2,1,1,1,1,1,1] => 2
[2,2,1,1,1,1,1,1,1,1] => 2
[2,1,1,1,1,1,1,1,1,1,1] => 2
[1,1,1,1,1,1,1,1,1,1,1,1] => 1
Description
The number of distinct parts of the integer partition.
This statistic is also the number of removeable cells of the partition, and the number of valleys of the Dyck path tracing the shape of the partition.
References
[1] Tewari, V. V. Kronecker coefficients for some near-rectangular partitions MathSciNet:3320625 arXiv:1403.5327
Code
def statistic(L):
return len(set(L))

Created
Sep 04, 2013 at 14:27 by Christian Stump
Updated
Oct 29, 2017 at 20:23 by Martin Rubey