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Matching statistic: St000913
Mapping
St000913: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1]
 => 1
[2]
 => 1
[1,1]
 => 1
[3]
 => 1
[2,1]
 => 1
[1,1,1]
 => 1
[4]
 => 2
[3,1]
 => 1
[2,2]
 => 1
[2,1,1]
 => 1
[1,1,1,1]
 => 1
[5]
 => 4
[4,1]
 => 2
[3,2]
 => 2
[3,1,1]
 => 1
[2,2,1]
 => 1
[2,1,1,1]
 => 1
[1,1,1,1,1]
 => 1
[6]
 => 11
[5,1]
 => 4
[4,2]
 => 5
[4,1,1]
 => 2
[3,3]
 => 2
[3,2,1]
 => 2
[3,1,1,1]
 => 1
[2,2,2]
 => 1
[2,2,1,1]
 => 1
[2,1,1,1,1]
 => 1
[1,1,1,1,1,1]
 => 1
[7]
 => 33
[6,1]
 => 11
[5,2]
 => 12
[5,1,1]
 => 4
[4,3]
 => 10
[4,2,1]
 => 5
[4,1,1,1]
 => 2
[3,3,1]
 => 2
[3,2,2]
 => 3
[3,2,1,1]
 => 2
[3,1,1,1,1]
 => 1
[2,2,2,1]
 => 1
[2,2,1,1,1]
 => 1
[2,1,1,1,1,1]
 => 1
[1,1,1,1,1,1,1]
 => 1
[8]
 => 116
[7,1]
 => 33
[6,2]
 => 37
[6,1,1]
 => 11
[5,3]
 => 27
[5,2,1]
 => 12
Description
The number of ways to refine the partition into singletons. For example there is only one way to refine $[2,2]$: $[2,2] > [2,1,1] > [1,1,1,1]$. However, there are two ways to refine $[3,2]$: $[3,2] > [2,2,1] > [2,1,1,1] > [1,1,1,1,1$ and $[3,2] > [3,1,1] > [2,1,1,1] > [1,1,1,1,1]$. In other words, this is the number of saturated chains in the refinement order from the bottom element to the given partition. The sequence of values on the partitions with only one part is [[A002846]].