- FindStat

Identifier

Mp00083: Standard tableaux —shape⟶ Integer partitions
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00093: Dyck paths —to binary word⟶ Binary words
St001371: Binary words ⟶ ℤ

Values

[[1]] => [1] => [1,0,1,0] => 1010 => 0
[[1,2]] => [2] => [1,1,0,0,1,0] => 110010 => 0
[[1],[2]] => [1,1] => [1,0,1,1,0,0] => 101100 => 0
[[1,2,3]] => [3] => [1,1,1,0,0,0,1,0] => 11100010 => 0
[[1,3],[2]] => [2,1] => [1,0,1,0,1,0] => 101010 => 0
[[1,2],[3]] => [2,1] => [1,0,1,0,1,0] => 101010 => 0
[[1],[2],[3]] => [1,1,1] => [1,0,1,1,1,0,0,0] => 10111000 => 0
[[1,3,4],[2]] => [3,1] => [1,1,0,1,0,0,1,0] => 11010010 => 0
[[1,2,4],[3]] => [3,1] => [1,1,0,1,0,0,1,0] => 11010010 => 0
[[1,2,3],[4]] => [3,1] => [1,1,0,1,0,0,1,0] => 11010010 => 0
[[1,3],[2,4]] => [2,2] => [1,1,0,0,1,1,0,0] => 11001100 => 0
[[1,2],[3,4]] => [2,2] => [1,1,0,0,1,1,0,0] => 11001100 => 0
[[1,4],[2],[3]] => [2,1,1] => [1,0,1,1,0,1,0,0] => 10110100 => 0
[[1,3],[2],[4]] => [2,1,1] => [1,0,1,1,0,1,0,0] => 10110100 => 0
[[1,2],[3],[4]] => [2,1,1] => [1,0,1,1,0,1,0,0] => 10110100 => 0
[[1,3,5],[2,4]] => [3,2] => [1,1,0,0,1,0,1,0] => 11001010 => 0
[[1,2,5],[3,4]] => [3,2] => [1,1,0,0,1,0,1,0] => 11001010 => 0
[[1,3,4],[2,5]] => [3,2] => [1,1,0,0,1,0,1,0] => 11001010 => 0
[[1,2,4],[3,5]] => [3,2] => [1,1,0,0,1,0,1,0] => 11001010 => 0
[[1,2,3],[4,5]] => [3,2] => [1,1,0,0,1,0,1,0] => 11001010 => 0
[[1,4,5],[2],[3]] => [3,1,1] => [1,0,1,1,0,0,1,0] => 10110010 => 0
[[1,3,5],[2],[4]] => [3,1,1] => [1,0,1,1,0,0,1,0] => 10110010 => 0
[[1,2,5],[3],[4]] => [3,1,1] => [1,0,1,1,0,0,1,0] => 10110010 => 0
[[1,3,4],[2],[5]] => [3,1,1] => [1,0,1,1,0,0,1,0] => 10110010 => 0
[[1,2,4],[3],[5]] => [3,1,1] => [1,0,1,1,0,0,1,0] => 10110010 => 0
[[1,2,3],[4],[5]] => [3,1,1] => [1,0,1,1,0,0,1,0] => 10110010 => 0
[[1,4],[2,5],[3]] => [2,2,1] => [1,0,1,0,1,1,0,0] => 10101100 => 0
[[1,3],[2,5],[4]] => [2,2,1] => [1,0,1,0,1,1,0,0] => 10101100 => 0
[[1,2],[3,5],[4]] => [2,2,1] => [1,0,1,0,1,1,0,0] => 10101100 => 0
[[1,3],[2,4],[5]] => [2,2,1] => [1,0,1,0,1,1,0,0] => 10101100 => 0
[[1,2],[3,4],[5]] => [2,2,1] => [1,0,1,0,1,1,0,0] => 10101100 => 0
[[1,4,6],[2,5],[3]] => [3,2,1] => [1,0,1,0,1,0,1,0] => 10101010 => 0
[[1,3,6],[2,5],[4]] => [3,2,1] => [1,0,1,0,1,0,1,0] => 10101010 => 0
[[1,2,6],[3,5],[4]] => [3,2,1] => [1,0,1,0,1,0,1,0] => 10101010 => 0
[[1,3,6],[2,4],[5]] => [3,2,1] => [1,0,1,0,1,0,1,0] => 10101010 => 0
[[1,2,6],[3,4],[5]] => [3,2,1] => [1,0,1,0,1,0,1,0] => 10101010 => 0
[[1,4,5],[2,6],[3]] => [3,2,1] => [1,0,1,0,1,0,1,0] => 10101010 => 0
[[1,3,5],[2,6],[4]] => [3,2,1] => [1,0,1,0,1,0,1,0] => 10101010 => 0
[[1,2,5],[3,6],[4]] => [3,2,1] => [1,0,1,0,1,0,1,0] => 10101010 => 0
[[1,3,4],[2,6],[5]] => [3,2,1] => [1,0,1,0,1,0,1,0] => 10101010 => 0
[[1,2,4],[3,6],[5]] => [3,2,1] => [1,0,1,0,1,0,1,0] => 10101010 => 0
[[1,2,3],[4,6],[5]] => [3,2,1] => [1,0,1,0,1,0,1,0] => 10101010 => 0
[[1,3,5],[2,4],[6]] => [3,2,1] => [1,0,1,0,1,0,1,0] => 10101010 => 0
[[1,2,5],[3,4],[6]] => [3,2,1] => [1,0,1,0,1,0,1,0] => 10101010 => 0
[[1,3,4],[2,5],[6]] => [3,2,1] => [1,0,1,0,1,0,1,0] => 10101010 => 0
[[1,2,4],[3,5],[6]] => [3,2,1] => [1,0,1,0,1,0,1,0] => 10101010 => 0
[[1,2,3],[4,5],[6]] => [3,2,1] => [1,0,1,0,1,0,1,0] => 10101010 => 0

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$F_{1} = 1$

$F_{2} = 2$

$F_{3} = 4$

Description

The length of the longest Yamanouchi prefix of a binary word.
This is the largest index

$i$ such that in each of the prefixes

$w_1$ ,

$w_1w_2$ ,

$w_1w_2\dots w_i$ the number of zeros is greater than or equal to the number of ones.

Map

to binary word

Description

Return the Dyck word as binary word.

Map

shape

Description

Sends a tableau to its shape.

Map

to Dyck path

Description

Sends a partition to the shortest Dyck path tracing the shape of its Ferrers diagram.