- FindStat

Identifier

Mp00327: Dyck paths —inverse Kreweras complement⟶ Dyck paths
St000438: Dyck paths ⟶ ℤ

Values

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

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

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

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

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

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

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

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

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

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

[1,0,1,1,0,1,0,0] => [1,1,1,0,0,1,0,0] => 6

[1,0,1,1,1,0,0,0] => [1,1,0,1,0,1,0,0] => 6

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

[1,1,0,0,1,1,0,0] => [1,0,1,1,0,1,0,0] => 6

[1,1,0,1,0,0,1,0] => [1,1,0,0,1,1,0,0] => 6

[1,1,0,1,0,1,0,0] => [1,1,1,0,0,0,1,0] => 7

[1,1,0,1,1,0,0,0] => [1,1,0,0,1,0,1,0] => 7

[1,1,1,0,0,0,1,0] => [1,0,1,0,1,1,0,0] => 6

[1,1,1,0,0,1,0,0] => [1,1,0,1,0,0,1,0] => 7

[1,1,1,0,1,0,0,0] => [1,0,1,1,0,0,1,0] => 7

[1,1,1,1,0,0,0,0] => [1,0,1,0,1,0,1,0] => 7

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

[1,0,1,0,1,0,1,1,0,0] => [1,1,1,1,0,1,0,0,0,0] => 6

[1,0,1,0,1,1,0,0,1,0] => [1,1,1,0,1,1,0,0,0,0] => 6

[1,0,1,0,1,1,0,1,0,0] => [1,1,1,1,0,0,1,0,0,0] => 7

[1,0,1,0,1,1,1,0,0,0] => [1,1,1,0,1,0,1,0,0,0] => 7

[1,0,1,1,0,0,1,0,1,0] => [1,1,0,1,1,1,0,0,0,0] => 6

[1,0,1,1,0,0,1,1,0,0] => [1,1,0,1,1,0,1,0,0,0] => 7

[1,0,1,1,0,1,0,0,1,0] => [1,1,1,0,0,1,1,0,0,0] => 7

[1,0,1,1,0,1,0,1,0,0] => [1,1,1,1,0,0,0,1,0,0] => 8

[1,0,1,1,0,1,1,0,0,0] => [1,1,1,0,0,1,0,1,0,0] => 8

[1,0,1,1,1,0,0,0,1,0] => [1,1,0,1,0,1,1,0,0,0] => 7

[1,0,1,1,1,0,0,1,0,0] => [1,1,1,0,1,0,0,1,0,0] => 8

[1,0,1,1,1,0,1,0,0,0] => [1,1,0,1,1,0,0,1,0,0] => 8

[1,0,1,1,1,1,0,0,0,0] => [1,1,0,1,0,1,0,1,0,0] => 8

[1,1,0,0,1,0,1,0,1,0] => [1,0,1,1,1,1,0,0,0,0] => 6

[1,1,0,0,1,0,1,1,0,0] => [1,0,1,1,1,0,1,0,0,0] => 7

[1,1,0,0,1,1,0,0,1,0] => [1,0,1,1,0,1,1,0,0,0] => 7

[1,1,0,0,1,1,0,1,0,0] => [1,0,1,1,1,0,0,1,0,0] => 8

[1,1,0,0,1,1,1,0,0,0] => [1,0,1,1,0,1,0,1,0,0] => 8

[1,1,0,1,0,0,1,0,1,0] => [1,1,0,0,1,1,1,0,0,0] => 7

[1,1,0,1,0,0,1,1,0,0] => [1,1,0,0,1,1,0,1,0,0] => 8

[1,1,0,1,0,1,0,0,1,0] => [1,1,1,0,0,0,1,1,0,0] => 8

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

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

[1,1,0,1,1,0,0,0,1,0] => [1,1,0,0,1,0,1,1,0,0] => 8

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

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

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

[1,1,1,0,0,0,1,0,1,0] => [1,0,1,0,1,1,1,0,0,0] => 7

[1,1,1,0,0,0,1,1,0,0] => [1,0,1,0,1,1,0,1,0,0] => 8

[1,1,1,0,0,1,0,0,1,0] => [1,1,0,1,0,0,1,1,0,0] => 8

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

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

[1,1,1,0,1,0,0,0,1,0] => [1,0,1,1,0,0,1,1,0,0] => 8

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

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

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

[1,1,1,1,0,0,0,0,1,0] => [1,0,1,0,1,0,1,1,0,0] => 8

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

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

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

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

[1,0,1,0,1,0,1,0,1,0,1,0] => [1,1,1,1,1,1,0,0,0,0,0,0] => 6

[1,0,1,0,1,0,1,0,1,1,0,0] => [1,1,1,1,1,0,1,0,0,0,0,0] => 7

[1,0,1,0,1,0,1,1,0,0,1,0] => [1,1,1,1,0,1,1,0,0,0,0,0] => 7

[1,0,1,0,1,0,1,1,0,1,0,0] => [1,1,1,1,1,0,0,1,0,0,0,0] => 8

[1,0,1,0,1,0,1,1,1,0,0,0] => [1,1,1,1,0,1,0,1,0,0,0,0] => 8

[1,0,1,0,1,1,0,0,1,0,1,0] => [1,1,1,0,1,1,1,0,0,0,0,0] => 7

[1,0,1,0,1,1,0,0,1,1,0,0] => [1,1,1,0,1,1,0,1,0,0,0,0] => 8

[1,0,1,0,1,1,0,1,0,0,1,0] => [1,1,1,1,0,0,1,1,0,0,0,0] => 8

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

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

[1,0,1,0,1,1,1,0,0,0,1,0] => [1,1,1,0,1,0,1,1,0,0,0,0] => 8

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

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

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

[1,0,1,1,0,0,1,0,1,0,1,0] => [1,1,0,1,1,1,1,0,0,0,0,0] => 7

[1,0,1,1,0,0,1,0,1,1,0,0] => [1,1,0,1,1,1,0,1,0,0,0,0] => 8

[1,0,1,1,0,0,1,1,0,0,1,0] => [1,1,0,1,1,0,1,1,0,0,0,0] => 8

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

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

[1,0,1,1,0,1,0,0,1,0,1,0] => [1,1,1,0,0,1,1,1,0,0,0,0] => 8

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

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

[1,0,1,1,0,1,0,1,0,1,0,0] => [1,1,1,1,1,0,0,0,0,1,0,0] => 10

[1,0,1,1,0,1,0,1,1,0,0,0] => [1,1,1,1,0,0,0,1,0,1,0,0] => 10

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

[1,0,1,1,0,1,1,0,0,1,0,0] => [1,1,1,1,0,1,0,0,0,1,0,0] => 10

[1,0,1,1,0,1,1,0,1,0,0,0] => [1,1,1,0,0,1,1,0,0,1,0,0] => 10

[1,0,1,1,0,1,1,1,0,0,0,0] => [1,1,1,0,0,1,0,1,0,1,0,0] => 10

[1,0,1,1,1,0,0,0,1,0,1,0] => [1,1,0,1,0,1,1,1,0,0,0,0] => 8

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

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

[1,0,1,1,1,0,0,1,0,1,0,0] => [1,1,1,0,1,1,0,0,0,1,0,0] => 10

[1,0,1,1,1,0,0,1,1,0,0,0] => [1,1,1,0,1,0,0,1,0,1,0,0] => 10

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

[1,0,1,1,1,0,1,0,0,1,0,0] => [1,1,1,1,0,0,1,0,0,1,0,0] => 10

[1,0,1,1,1,0,1,0,1,0,0,0] => [1,1,0,1,1,1,0,0,0,1,0,0] => 10

[1,0,1,1,1,0,1,1,0,0,0,0] => [1,1,0,1,1,0,0,1,0,1,0,0] => 10

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

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Description

The position of the last up step in a Dyck path.

Map

inverse Kreweras complement

Description

Return the inverse of the Kreweras complement of a Dyck path, regarded as a noncrossing set partition.
To identify Dyck paths and noncrossing set partitions, this maps uses the following classical bijection. The number of down steps after the $i$-th up step of the Dyck path is the size of the block of the set partition whose maximal element is $i$. If $i$ is not a maximal element of a block, the $(i+1)$-st step is also an up step.