Identifier
-
Mp00322:
Integer partitions
—Loehr-Warrington⟶
Integer partitions
St000810: Integer partitions ⟶ ℤ
Values
[1] => [1] => 1
[2] => [1,1] => 3
[1,1] => [2] => 1
[3] => [1,1,1] => 10
[2,1] => [3] => 1
[1,1,1] => [2,1] => 2
[4] => [1,1,1,1] => 47
[3,1] => [2,1,1] => 7
[2,2] => [4] => 1
[2,1,1] => [2,2] => 3
[1,1,1,1] => [3,1] => 2
[5] => [1,1,1,1,1] => 246
[4,1] => [2,1,1,1] => 26
[3,2] => [5] => 1
[3,1,1] => [4,1] => 2
[2,2,1] => [2,2,1] => 6
[2,1,1,1] => [3,1,1] => 6
[1,1,1,1,1] => [3,2] => 2
[6] => [1,1,1,1,1,1] => 1602
[5,1] => [2,1,1,1,1] => 138
[4,2] => [2,2,1,1] => 26
[4,1,1] => [3,1,1,1] => 24
[3,3] => [6] => 1
[3,2,1] => [5,1] => 2
[3,1,1,1] => [3,3] => 3
[2,2,2] => [2,2,2] => 10
[2,2,1,1] => [4,1,1] => 6
[2,1,1,1,1] => [4,2] => 2
[1,1,1,1,1,1] => [3,2,1] => 6
[7] => [1,1,1,1,1,1,1] => 11481
[6,1] => [2,1,1,1,1,1] => 767
[5,2] => [2,2,1,1,1] => 105
[5,1,1] => [3,1,1,1,1] => 114
[4,3] => [7] => 1
[4,2,1] => [5,1,1] => 6
[4,1,1,1] => [3,2,1,1] => 20
[3,3,1] => [6,1] => 2
[3,2,2] => [2,2,2,1] => 23
[3,2,1,1] => [5,2] => 2
[3,1,1,1,1] => [3,2,2] => 6
[2,2,2,1] => [4,1,1,1] => 23
[2,2,1,1,1] => [4,3] => 2
[2,1,1,1,1,1] => [3,3,1] => 6
[1,1,1,1,1,1,1] => [4,2,1] => 5
[8] => [1,1,1,1,1,1,1,1] => 95503
[7,1] => [2,1,1,1,1,1,1] => 5295
[6,2] => [2,2,1,1,1,1] => 599
[6,1,1] => [3,1,1,1,1,1] => 652
[5,3] => [2,2,2,1,1] => 111
[5,2,1] => [4,1,1,1,1] => 111
[5,1,1,1] => [3,2,1,1,1] => 90
[4,4] => [8] => 1
[4,3,1] => [7,1] => 2
[4,2,2] => [6,1,1] => 6
[4,2,1,1] => [4,4] => 3
[4,1,1,1,1] => [5,2,1] => 5
[3,3,2] => [2,2,2,2] => 47
[3,3,1,1] => [6,2] => 2
[3,2,2,1] => [3,2,2,1] => 20
[3,2,1,1,1] => [3,3,1,1] => 22
[3,1,1,1,1,1] => [4,2,1,1] => 19
[2,2,2,2] => [5,1,1,1] => 23
[2,2,2,1,1] => [5,3] => 2
[2,2,1,1,1,1] => [3,3,2] => 6
[2,1,1,1,1,1,1] => [4,2,2] => 7
[1,1,1,1,1,1,1,1] => [4,3,1] => 6
[9] => [1,1,1,1,1,1,1,1,1] => 871030
[8,1] => [2,1,1,1,1,1,1,1] => 39468
[7,2] => [2,2,1,1,1,1,1] => 3586
[7,1,1] => [3,1,1,1,1,1,1] => 4285
[6,3] => [2,2,2,1,1,1] => 512
[6,2,1] => [4,1,1,1,1,1] => 622
[6,1,1,1] => [3,2,1,1,1,1] => 471
[5,4] => [9] => 1
[5,3,1] => [3,2,2,1,1] => 85
[5,2,2] => [6,1,1,1] => 23
[5,2,1,1] => [3,3,1,1,1] => 91
[5,1,1,1,1] => [4,2,1,1,1] => 80
[4,4,1] => [8,1] => 2
[4,3,2] => [7,1,1] => 6
[4,3,1,1] => [7,2] => 2
[4,2,2,1] => [6,3] => 2
[4,2,1,1,1] => [6,2,1] => 5
[4,1,1,1,1,1] => [4,4,1] => 6
[3,3,3] => [2,2,2,2,1] => 110
[3,3,2,1] => [3,2,2,2] => 23
[3,3,1,1,1] => [5,2,1,1] => 18
[3,2,2,2] => [5,1,1,1,1] => 110
[3,2,2,1,1] => [3,3,3] => 10
[3,2,1,1,1,1] => [5,2,2] => 6
[3,1,1,1,1,1,1] => [5,3,1] => 5
[2,2,2,2,1] => [5,4] => 2
[2,2,2,1,1,1] => [3,3,2,1] => 21
[2,2,1,1,1,1,1] => [4,2,2,1] => 18
[2,1,1,1,1,1,1,1] => [4,3,1,1] => 19
[1,1,1,1,1,1,1,1,1] => [4,3,2] => 5
[10] => [1,1,1,1,1,1,1,1,1,1] => 8879558
[9,1] => [2,1,1,1,1,1,1,1,1] => 340198
[8,2] => [2,2,1,1,1,1,1,1] => 26038
[8,1,1] => [3,1,1,1,1,1,1,1] => 32048
[7,3] => [2,2,2,1,1,1,1] => 3134
>>> Load all 272 entries. <<<
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Description
The sum of the entries in the column specified by the partition of the change of basis matrix from powersum symmetric functions to monomial symmetric functions.
For example, $p_{22} = 2m_{22} + m_4$, so the statistic on the partition $22$ is 3.
For example, $p_{22} = 2m_{22} + m_4$, so the statistic on the partition $22$ is 3.
Map
Loehr-Warrington
Description
Return a partition whose diagonal inversion number is the length of the preimage.
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