- FindStat

Identifier

Mp00102: Dyck paths —rise composition⟶ Integer compositions
Mp00040: Integer compositions —to partition⟶ Integer partitions
St001129: Integer partitions ⟶ ℤ

Values

[1,0] => [1] => [1] => 1

[1,0,1,0] => [1,1] => [1,1] => 1

[1,1,0,0] => [2] => [2] => 4

[1,0,1,0,1,0] => [1,1,1] => [1,1,1] => 1

[1,0,1,1,0,0] => [1,2] => [2,1] => 4

[1,1,0,0,1,0] => [2,1] => [2,1] => 4

[1,1,0,1,0,0] => [2,1] => [2,1] => 4

[1,1,1,0,0,0] => [3] => [3] => 9

[1,0,1,0,1,0,1,0] => [1,1,1,1] => [1,1,1,1] => 1

[1,0,1,0,1,1,0,0] => [1,1,2] => [2,1,1] => 4

[1,0,1,1,0,0,1,0] => [1,2,1] => [2,1,1] => 4

[1,0,1,1,0,1,0,0] => [1,2,1] => [2,1,1] => 4

[1,0,1,1,1,0,0,0] => [1,3] => [3,1] => 9

[1,1,0,0,1,0,1,0] => [2,1,1] => [2,1,1] => 4

[1,1,0,0,1,1,0,0] => [2,2] => [2,2] => 16

[1,1,0,1,0,0,1,0] => [2,1,1] => [2,1,1] => 4

[1,1,0,1,0,1,0,0] => [2,1,1] => [2,1,1] => 4

[1,1,0,1,1,0,0,0] => [2,2] => [2,2] => 16

[1,1,1,0,0,0,1,0] => [3,1] => [3,1] => 9

[1,1,1,0,0,1,0,0] => [3,1] => [3,1] => 9

[1,1,1,0,1,0,0,0] => [3,1] => [3,1] => 9

[1,1,1,1,0,0,0,0] => [4] => [4] => 16

[1,0,1,0,1,0,1,0,1,0] => [1,1,1,1,1] => [1,1,1,1,1] => 1

[1,0,1,0,1,0,1,1,0,0] => [1,1,1,2] => [2,1,1,1] => 4

[1,0,1,0,1,1,0,0,1,0] => [1,1,2,1] => [2,1,1,1] => 4

[1,0,1,0,1,1,0,1,0,0] => [1,1,2,1] => [2,1,1,1] => 4

[1,0,1,0,1,1,1,0,0,0] => [1,1,3] => [3,1,1] => 9

[1,0,1,1,0,0,1,0,1,0] => [1,2,1,1] => [2,1,1,1] => 4

[1,0,1,1,0,0,1,1,0,0] => [1,2,2] => [2,2,1] => 16

[1,0,1,1,0,1,0,0,1,0] => [1,2,1,1] => [2,1,1,1] => 4

[1,0,1,1,0,1,0,1,0,0] => [1,2,1,1] => [2,1,1,1] => 4

[1,0,1,1,0,1,1,0,0,0] => [1,2,2] => [2,2,1] => 16

[1,0,1,1,1,0,0,0,1,0] => [1,3,1] => [3,1,1] => 9

[1,0,1,1,1,0,0,1,0,0] => [1,3,1] => [3,1,1] => 9

[1,0,1,1,1,0,1,0,0,0] => [1,3,1] => [3,1,1] => 9

[1,0,1,1,1,1,0,0,0,0] => [1,4] => [4,1] => 16

[1,1,0,0,1,0,1,0,1,0] => [2,1,1,1] => [2,1,1,1] => 4

[1,1,0,0,1,0,1,1,0,0] => [2,1,2] => [2,2,1] => 16

[1,1,0,0,1,1,0,0,1,0] => [2,2,1] => [2,2,1] => 16

[1,1,0,0,1,1,0,1,0,0] => [2,2,1] => [2,2,1] => 16

[1,1,0,0,1,1,1,0,0,0] => [2,3] => [3,2] => 36

[1,1,0,1,0,0,1,0,1,0] => [2,1,1,1] => [2,1,1,1] => 4

[1,1,0,1,0,0,1,1,0,0] => [2,1,2] => [2,2,1] => 16

[1,1,0,1,0,1,0,0,1,0] => [2,1,1,1] => [2,1,1,1] => 4

[1,1,0,1,0,1,0,1,0,0] => [2,1,1,1] => [2,1,1,1] => 4

[1,1,0,1,0,1,1,0,0,0] => [2,1,2] => [2,2,1] => 16

[1,1,0,1,1,0,0,0,1,0] => [2,2,1] => [2,2,1] => 16

[1,1,0,1,1,0,0,1,0,0] => [2,2,1] => [2,2,1] => 16

[1,1,0,1,1,0,1,0,0,0] => [2,2,1] => [2,2,1] => 16

[1,1,0,1,1,1,0,0,0,0] => [2,3] => [3,2] => 36

[1,1,1,0,0,0,1,0,1,0] => [3,1,1] => [3,1,1] => 9

[1,1,1,0,0,0,1,1,0,0] => [3,2] => [3,2] => 36

[1,1,1,0,0,1,0,0,1,0] => [3,1,1] => [3,1,1] => 9

[1,1,1,0,0,1,0,1,0,0] => [3,1,1] => [3,1,1] => 9

[1,1,1,0,0,1,1,0,0,0] => [3,2] => [3,2] => 36

[1,1,1,0,1,0,0,0,1,0] => [3,1,1] => [3,1,1] => 9

[1,1,1,0,1,0,0,1,0,0] => [3,1,1] => [3,1,1] => 9

[1,1,1,0,1,0,1,0,0,0] => [3,1,1] => [3,1,1] => 9

[1,1,1,0,1,1,0,0,0,0] => [3,2] => [3,2] => 36

[1,1,1,1,0,0,0,0,1,0] => [4,1] => [4,1] => 16

[1,1,1,1,0,0,0,1,0,0] => [4,1] => [4,1] => 16

[1,1,1,1,0,0,1,0,0,0] => [4,1] => [4,1] => 16

[1,1,1,1,0,1,0,0,0,0] => [4,1] => [4,1] => 16

[1,1,1,1,1,0,0,0,0,0] => [5] => [5] => 25

[1,0,1,0,1,0,1,0,1,0,1,0] => [1,1,1,1,1,1] => [1,1,1,1,1,1] => 1

[1,0,1,0,1,0,1,0,1,1,0,0] => [1,1,1,1,2] => [2,1,1,1,1] => 4

[1,0,1,0,1,0,1,1,0,0,1,0] => [1,1,1,2,1] => [2,1,1,1,1] => 4

[1,0,1,0,1,0,1,1,0,1,0,0] => [1,1,1,2,1] => [2,1,1,1,1] => 4

[1,0,1,0,1,0,1,1,1,0,0,0] => [1,1,1,3] => [3,1,1,1] => 9

[1,0,1,0,1,1,0,0,1,0,1,0] => [1,1,2,1,1] => [2,1,1,1,1] => 4

[1,0,1,0,1,1,0,0,1,1,0,0] => [1,1,2,2] => [2,2,1,1] => 16

[1,0,1,0,1,1,0,1,0,0,1,0] => [1,1,2,1,1] => [2,1,1,1,1] => 4

[1,0,1,0,1,1,0,1,0,1,0,0] => [1,1,2,1,1] => [2,1,1,1,1] => 4

[1,0,1,0,1,1,0,1,1,0,0,0] => [1,1,2,2] => [2,2,1,1] => 16

[1,0,1,0,1,1,1,0,0,0,1,0] => [1,1,3,1] => [3,1,1,1] => 9

[1,0,1,0,1,1,1,0,0,1,0,0] => [1,1,3,1] => [3,1,1,1] => 9

[1,0,1,0,1,1,1,0,1,0,0,0] => [1,1,3,1] => [3,1,1,1] => 9

[1,0,1,0,1,1,1,1,0,0,0,0] => [1,1,4] => [4,1,1] => 16

[1,0,1,1,0,0,1,0,1,0,1,0] => [1,2,1,1,1] => [2,1,1,1,1] => 4

[1,0,1,1,0,0,1,0,1,1,0,0] => [1,2,1,2] => [2,2,1,1] => 16

[1,0,1,1,0,0,1,1,0,0,1,0] => [1,2,2,1] => [2,2,1,1] => 16

[1,0,1,1,0,0,1,1,0,1,0,0] => [1,2,2,1] => [2,2,1,1] => 16

[1,0,1,1,0,0,1,1,1,0,0,0] => [1,2,3] => [3,2,1] => 36

[1,0,1,1,0,1,0,0,1,0,1,0] => [1,2,1,1,1] => [2,1,1,1,1] => 4

[1,0,1,1,0,1,0,0,1,1,0,0] => [1,2,1,2] => [2,2,1,1] => 16

[1,0,1,1,0,1,0,1,0,0,1,0] => [1,2,1,1,1] => [2,1,1,1,1] => 4

[1,0,1,1,0,1,0,1,0,1,0,0] => [1,2,1,1,1] => [2,1,1,1,1] => 4

[1,0,1,1,0,1,0,1,1,0,0,0] => [1,2,1,2] => [2,2,1,1] => 16

[1,0,1,1,0,1,1,0,0,0,1,0] => [1,2,2,1] => [2,2,1,1] => 16

[1,0,1,1,0,1,1,0,0,1,0,0] => [1,2,2,1] => [2,2,1,1] => 16

[1,0,1,1,0,1,1,0,1,0,0,0] => [1,2,2,1] => [2,2,1,1] => 16

[1,0,1,1,0,1,1,1,0,0,0,0] => [1,2,3] => [3,2,1] => 36

[1,0,1,1,1,0,0,0,1,0,1,0] => [1,3,1,1] => [3,1,1,1] => 9

[1,0,1,1,1,0,0,0,1,1,0,0] => [1,3,2] => [3,2,1] => 36

[1,0,1,1,1,0,0,1,0,0,1,0] => [1,3,1,1] => [3,1,1,1] => 9

[1,0,1,1,1,0,0,1,0,1,0,0] => [1,3,1,1] => [3,1,1,1] => 9

[1,0,1,1,1,0,0,1,1,0,0,0] => [1,3,2] => [3,2,1] => 36

[1,0,1,1,1,0,1,0,0,0,1,0] => [1,3,1,1] => [3,1,1,1] => 9

[1,0,1,1,1,0,1,0,0,1,0,0] => [1,3,1,1] => [3,1,1,1] => 9

[1,0,1,1,1,0,1,0,1,0,0,0] => [1,3,1,1] => [3,1,1,1] => 9

[1,0,1,1,1,0,1,1,0,0,0,0] => [1,3,2] => [3,2,1] => 36

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Description

The product of the squares of the parts of a partition.

Map

to partition

Description

Sends a composition to the partition obtained by sorting the entries.

Map

rise composition

Description

Send a Dyck path to the composition of sizes of its rises.