- FindStat

Identifier

Mp00079: Set partitions —shape⟶ Integer partitions
St000567: Integer partitions ⟶ ℤ

Values

{{1,2}} => [2] => 0

{{1},{2}} => [1,1] => 1

{{1,2,3}} => [3] => 0

{{1,2},{3}} => [2,1] => 2

{{1,3},{2}} => [2,1] => 2

{{1},{2,3}} => [2,1] => 2

{{1},{2},{3}} => [1,1,1] => 3

{{1,2,3,4}} => [4] => 0

{{1,2,3},{4}} => [3,1] => 3

{{1,2,4},{3}} => [3,1] => 3

{{1,2},{3,4}} => [2,2] => 4

{{1,2},{3},{4}} => [2,1,1] => 5

{{1,3,4},{2}} => [3,1] => 3

{{1,3},{2,4}} => [2,2] => 4

{{1,3},{2},{4}} => [2,1,1] => 5

{{1,4},{2,3}} => [2,2] => 4

{{1},{2,3,4}} => [3,1] => 3

{{1},{2,3},{4}} => [2,1,1] => 5

{{1,4},{2},{3}} => [2,1,1] => 5

{{1},{2,4},{3}} => [2,1,1] => 5

{{1},{2},{3,4}} => [2,1,1] => 5

{{1},{2},{3},{4}} => [1,1,1,1] => 6

{{1,2,3,4,5}} => [5] => 0

{{1,2,3,4},{5}} => [4,1] => 4

{{1,2,3,5},{4}} => [4,1] => 4

{{1,2,3},{4,5}} => [3,2] => 6

{{1,2,3},{4},{5}} => [3,1,1] => 7

{{1,2,4,5},{3}} => [4,1] => 4

{{1,2,4},{3,5}} => [3,2] => 6

{{1,2,4},{3},{5}} => [3,1,1] => 7

{{1,2,5},{3,4}} => [3,2] => 6

{{1,2},{3,4,5}} => [3,2] => 6

{{1,2},{3,4},{5}} => [2,2,1] => 8

{{1,2,5},{3},{4}} => [3,1,1] => 7

{{1,2},{3,5},{4}} => [2,2,1] => 8

{{1,2},{3},{4,5}} => [2,2,1] => 8

{{1,2},{3},{4},{5}} => [2,1,1,1] => 9

{{1,3,4,5},{2}} => [4,1] => 4

{{1,3,4},{2,5}} => [3,2] => 6

{{1,3,4},{2},{5}} => [3,1,1] => 7

{{1,3,5},{2,4}} => [3,2] => 6

{{1,3},{2,4,5}} => [3,2] => 6

{{1,3},{2,4},{5}} => [2,2,1] => 8

{{1,3,5},{2},{4}} => [3,1,1] => 7

{{1,3},{2,5},{4}} => [2,2,1] => 8

{{1,3},{2},{4,5}} => [2,2,1] => 8

{{1,3},{2},{4},{5}} => [2,1,1,1] => 9

{{1,4,5},{2,3}} => [3,2] => 6

{{1,4},{2,3,5}} => [3,2] => 6

{{1,4},{2,3},{5}} => [2,2,1] => 8

{{1,5},{2,3,4}} => [3,2] => 6

{{1},{2,3,4,5}} => [4,1] => 4

{{1},{2,3,4},{5}} => [3,1,1] => 7

{{1,5},{2,3},{4}} => [2,2,1] => 8

{{1},{2,3,5},{4}} => [3,1,1] => 7

{{1},{2,3},{4,5}} => [2,2,1] => 8

{{1},{2,3},{4},{5}} => [2,1,1,1] => 9

{{1,4,5},{2},{3}} => [3,1,1] => 7

{{1,4},{2,5},{3}} => [2,2,1] => 8

{{1,4},{2},{3,5}} => [2,2,1] => 8

{{1,4},{2},{3},{5}} => [2,1,1,1] => 9

{{1,5},{2,4},{3}} => [2,2,1] => 8

{{1},{2,4,5},{3}} => [3,1,1] => 7

{{1},{2,4},{3,5}} => [2,2,1] => 8

{{1},{2,4},{3},{5}} => [2,1,1,1] => 9

{{1,5},{2},{3,4}} => [2,2,1] => 8

{{1},{2,5},{3,4}} => [2,2,1] => 8

{{1},{2},{3,4,5}} => [3,1,1] => 7

{{1},{2},{3,4},{5}} => [2,1,1,1] => 9

{{1,5},{2},{3},{4}} => [2,1,1,1] => 9

{{1},{2,5},{3},{4}} => [2,1,1,1] => 9

{{1},{2},{3,5},{4}} => [2,1,1,1] => 9

{{1},{2},{3},{4,5}} => [2,1,1,1] => 9

{{1},{2},{3},{4},{5}} => [1,1,1,1,1] => 10

{{1,2,3,4,5,6}} => [6] => 0

{{1,2,3,4,5},{6}} => [5,1] => 5

{{1,2,3,4,6},{5}} => [5,1] => 5

{{1,2,3,4},{5,6}} => [4,2] => 8

{{1,2,3,4},{5},{6}} => [4,1,1] => 9

{{1,2,3,5,6},{4}} => [5,1] => 5

{{1,2,3,5},{4,6}} => [4,2] => 8

{{1,2,3,5},{4},{6}} => [4,1,1] => 9

{{1,2,3,6},{4,5}} => [4,2] => 8

{{1,2,3},{4,5,6}} => [3,3] => 9

{{1,2,3},{4,5},{6}} => [3,2,1] => 11

{{1,2,3,6},{4},{5}} => [4,1,1] => 9

{{1,2,3},{4,6},{5}} => [3,2,1] => 11

{{1,2,3},{4},{5,6}} => [3,2,1] => 11

{{1,2,3},{4},{5},{6}} => [3,1,1,1] => 12

{{1,2,4,5,6},{3}} => [5,1] => 5

{{1,2,4,5},{3,6}} => [4,2] => 8

{{1,2,4,5},{3},{6}} => [4,1,1] => 9

{{1,2,4,6},{3,5}} => [4,2] => 8

{{1,2,4},{3,5,6}} => [3,3] => 9

{{1,2,4},{3,5},{6}} => [3,2,1] => 11

{{1,2,4,6},{3},{5}} => [4,1,1] => 9

{{1,2,4},{3,6},{5}} => [3,2,1] => 11

{{1,2,4},{3},{5,6}} => [3,2,1] => 11

{{1,2,4},{3},{5},{6}} => [3,1,1,1] => 12

{{1,2,5,6},{3,4}} => [4,2] => 8

{{1,2,5},{3,4,6}} => [3,3] => 9

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Description

The sum of the products of all pairs of parts.
This is the evaluation of the second elementary symmetric polynomial which is equal to
$$e_2(\lambda) = \binom{n+1}{2} - \sum_{i=1}^\ell\binom{\lambda_i+1}{2}$$
for a partition $\lambda = (\lambda_1,\dots,\lambda_\ell) \vdash n$, see [1].
This is the maximal number of inversions a permutation with the given shape can have, see [2, cor.2.4].

Map

shape

Description

Sends a set partition to the integer partition obtained by the sizes of the blocks.