- FindStat

Identifier

Mp00060: Permutations —Robinson-Schensted tableau shape⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00093: Dyck paths —to binary word⟶ Binary words
St001371: Binary words ⟶ ℤ

Values

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

[1,2] => [2] => [1,0,1,0] => 1010 => 0

[2,1] => [1,1] => [1,1,0,0] => 1100 => 0

[1,2,3] => [3] => [1,0,1,0,1,0] => 101010 => 0

[1,3,2] => [2,1] => [1,0,1,1,0,0] => 101100 => 0

[2,1,3] => [2,1] => [1,0,1,1,0,0] => 101100 => 0

[2,3,1] => [2,1] => [1,0,1,1,0,0] => 101100 => 0

[3,1,2] => [2,1] => [1,0,1,1,0,0] => 101100 => 0

[3,2,1] => [1,1,1] => [1,1,0,1,0,0] => 110100 => 0

[1,2,3,4] => [4] => [1,0,1,0,1,0,1,0] => 10101010 => 0

[1,2,4,3] => [3,1] => [1,0,1,0,1,1,0,0] => 10101100 => 0

[1,3,2,4] => [3,1] => [1,0,1,0,1,1,0,0] => 10101100 => 0

[1,3,4,2] => [3,1] => [1,0,1,0,1,1,0,0] => 10101100 => 0

[1,4,2,3] => [3,1] => [1,0,1,0,1,1,0,0] => 10101100 => 0

[1,4,3,2] => [2,1,1] => [1,0,1,1,0,1,0,0] => 10110100 => 0

[2,1,3,4] => [3,1] => [1,0,1,0,1,1,0,0] => 10101100 => 0

[2,1,4,3] => [2,2] => [1,1,1,0,0,0] => 111000 => 0

[2,3,1,4] => [3,1] => [1,0,1,0,1,1,0,0] => 10101100 => 0

[2,3,4,1] => [3,1] => [1,0,1,0,1,1,0,0] => 10101100 => 0

[2,4,1,3] => [2,2] => [1,1,1,0,0,0] => 111000 => 0

[2,4,3,1] => [2,1,1] => [1,0,1,1,0,1,0,0] => 10110100 => 0

[3,1,2,4] => [3,1] => [1,0,1,0,1,1,0,0] => 10101100 => 0

[3,1,4,2] => [2,2] => [1,1,1,0,0,0] => 111000 => 0

[3,2,1,4] => [2,1,1] => [1,0,1,1,0,1,0,0] => 10110100 => 0

[3,2,4,1] => [2,1,1] => [1,0,1,1,0,1,0,0] => 10110100 => 0

[3,4,1,2] => [2,2] => [1,1,1,0,0,0] => 111000 => 0

[3,4,2,1] => [2,1,1] => [1,0,1,1,0,1,0,0] => 10110100 => 0

[4,1,2,3] => [3,1] => [1,0,1,0,1,1,0,0] => 10101100 => 0

[4,1,3,2] => [2,1,1] => [1,0,1,1,0,1,0,0] => 10110100 => 0

[4,2,1,3] => [2,1,1] => [1,0,1,1,0,1,0,0] => 10110100 => 0

[4,2,3,1] => [2,1,1] => [1,0,1,1,0,1,0,0] => 10110100 => 0

[4,3,1,2] => [2,1,1] => [1,0,1,1,0,1,0,0] => 10110100 => 0

[4,3,2,1] => [1,1,1,1] => [1,1,0,1,0,1,0,0] => 11010100 => 0

[1,3,2,5,4] => [3,2] => [1,0,1,1,1,0,0,0] => 10111000 => 0

[1,3,5,2,4] => [3,2] => [1,0,1,1,1,0,0,0] => 10111000 => 0

[1,4,2,5,3] => [3,2] => [1,0,1,1,1,0,0,0] => 10111000 => 0

[1,4,5,2,3] => [3,2] => [1,0,1,1,1,0,0,0] => 10111000 => 0

[2,1,3,5,4] => [3,2] => [1,0,1,1,1,0,0,0] => 10111000 => 0

[2,1,4,3,5] => [3,2] => [1,0,1,1,1,0,0,0] => 10111000 => 0

[2,1,4,5,3] => [3,2] => [1,0,1,1,1,0,0,0] => 10111000 => 0

[2,1,5,3,4] => [3,2] => [1,0,1,1,1,0,0,0] => 10111000 => 0

[2,1,5,4,3] => [2,2,1] => [1,1,1,0,0,1,0,0] => 11100100 => 0

[2,3,1,5,4] => [3,2] => [1,0,1,1,1,0,0,0] => 10111000 => 0

[2,3,5,1,4] => [3,2] => [1,0,1,1,1,0,0,0] => 10111000 => 0

[2,4,1,3,5] => [3,2] => [1,0,1,1,1,0,0,0] => 10111000 => 0

[2,4,1,5,3] => [3,2] => [1,0,1,1,1,0,0,0] => 10111000 => 0

[2,4,5,1,3] => [3,2] => [1,0,1,1,1,0,0,0] => 10111000 => 0

[2,5,1,3,4] => [3,2] => [1,0,1,1,1,0,0,0] => 10111000 => 0

[2,5,1,4,3] => [2,2,1] => [1,1,1,0,0,1,0,0] => 11100100 => 0

[2,5,4,1,3] => [2,2,1] => [1,1,1,0,0,1,0,0] => 11100100 => 0

[3,1,2,5,4] => [3,2] => [1,0,1,1,1,0,0,0] => 10111000 => 0

[3,1,4,2,5] => [3,2] => [1,0,1,1,1,0,0,0] => 10111000 => 0

[3,1,4,5,2] => [3,2] => [1,0,1,1,1,0,0,0] => 10111000 => 0

[3,1,5,2,4] => [3,2] => [1,0,1,1,1,0,0,0] => 10111000 => 0

[3,1,5,4,2] => [2,2,1] => [1,1,1,0,0,1,0,0] => 11100100 => 0

[3,2,1,5,4] => [2,2,1] => [1,1,1,0,0,1,0,0] => 11100100 => 0

[3,2,5,1,4] => [2,2,1] => [1,1,1,0,0,1,0,0] => 11100100 => 0

[3,2,5,4,1] => [2,2,1] => [1,1,1,0,0,1,0,0] => 11100100 => 0

[3,4,1,2,5] => [3,2] => [1,0,1,1,1,0,0,0] => 10111000 => 0

[3,4,1,5,2] => [3,2] => [1,0,1,1,1,0,0,0] => 10111000 => 0

[3,4,5,1,2] => [3,2] => [1,0,1,1,1,0,0,0] => 10111000 => 0

[3,5,1,2,4] => [3,2] => [1,0,1,1,1,0,0,0] => 10111000 => 0

[3,5,1,4,2] => [2,2,1] => [1,1,1,0,0,1,0,0] => 11100100 => 0

[3,5,2,1,4] => [2,2,1] => [1,1,1,0,0,1,0,0] => 11100100 => 0

[3,5,2,4,1] => [2,2,1] => [1,1,1,0,0,1,0,0] => 11100100 => 0

[3,5,4,1,2] => [2,2,1] => [1,1,1,0,0,1,0,0] => 11100100 => 0

[4,1,2,5,3] => [3,2] => [1,0,1,1,1,0,0,0] => 10111000 => 0

[4,1,5,2,3] => [3,2] => [1,0,1,1,1,0,0,0] => 10111000 => 0

[4,1,5,3,2] => [2,2,1] => [1,1,1,0,0,1,0,0] => 11100100 => 0

[4,2,1,5,3] => [2,2,1] => [1,1,1,0,0,1,0,0] => 11100100 => 0

[4,2,5,1,3] => [2,2,1] => [1,1,1,0,0,1,0,0] => 11100100 => 0

[4,2,5,3,1] => [2,2,1] => [1,1,1,0,0,1,0,0] => 11100100 => 0

[4,3,1,5,2] => [2,2,1] => [1,1,1,0,0,1,0,0] => 11100100 => 0

[4,3,5,1,2] => [2,2,1] => [1,1,1,0,0,1,0,0] => 11100100 => 0

[4,5,1,2,3] => [3,2] => [1,0,1,1,1,0,0,0] => 10111000 => 0

[4,5,1,3,2] => [2,2,1] => [1,1,1,0,0,1,0,0] => 11100100 => 0

[4,5,2,1,3] => [2,2,1] => [1,1,1,0,0,1,0,0] => 11100100 => 0

[4,5,2,3,1] => [2,2,1] => [1,1,1,0,0,1,0,0] => 11100100 => 0

[4,5,3,1,2] => [2,2,1] => [1,1,1,0,0,1,0,0] => 11100100 => 0

[5,2,1,4,3] => [2,2,1] => [1,1,1,0,0,1,0,0] => 11100100 => 0

[5,2,4,1,3] => [2,2,1] => [1,1,1,0,0,1,0,0] => 11100100 => 0

[5,3,1,4,2] => [2,2,1] => [1,1,1,0,0,1,0,0] => 11100100 => 0

[5,3,4,1,2] => [2,2,1] => [1,1,1,0,0,1,0,0] => 11100100 => 0

[2,1,4,3,6,5] => [3,3] => [1,1,1,0,1,0,0,0] => 11101000 => 0

[2,1,4,6,3,5] => [3,3] => [1,1,1,0,1,0,0,0] => 11101000 => 0

[2,1,5,3,6,4] => [3,3] => [1,1,1,0,1,0,0,0] => 11101000 => 0

[2,1,5,6,3,4] => [3,3] => [1,1,1,0,1,0,0,0] => 11101000 => 0

[2,4,1,3,6,5] => [3,3] => [1,1,1,0,1,0,0,0] => 11101000 => 0

[2,4,1,6,3,5] => [3,3] => [1,1,1,0,1,0,0,0] => 11101000 => 0

[2,4,6,1,3,5] => [3,3] => [1,1,1,0,1,0,0,0] => 11101000 => 0

[2,5,1,3,6,4] => [3,3] => [1,1,1,0,1,0,0,0] => 11101000 => 0

[2,5,1,6,3,4] => [3,3] => [1,1,1,0,1,0,0,0] => 11101000 => 0

[2,5,6,1,3,4] => [3,3] => [1,1,1,0,1,0,0,0] => 11101000 => 0

[3,1,4,2,6,5] => [3,3] => [1,1,1,0,1,0,0,0] => 11101000 => 0

[3,1,4,6,2,5] => [3,3] => [1,1,1,0,1,0,0,0] => 11101000 => 0

[3,1,5,2,6,4] => [3,3] => [1,1,1,0,1,0,0,0] => 11101000 => 0

[3,1,5,6,2,4] => [3,3] => [1,1,1,0,1,0,0,0] => 11101000 => 0

[3,2,1,6,5,4] => [2,2,2] => [1,1,1,1,0,0,0,0] => 11110000 => 0

[3,2,6,1,5,4] => [2,2,2] => [1,1,1,1,0,0,0,0] => 11110000 => 0

[3,2,6,5,1,4] => [2,2,2] => [1,1,1,1,0,0,0,0] => 11110000 => 0

[3,4,1,2,6,5] => [3,3] => [1,1,1,0,1,0,0,0] => 11101000 => 0

>>> Load all 133 entries. <<<

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$F_{1} = 1$

$F_{2} = 2$

$F_{3} = 6$

$F_{4} = 24$

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

parallelogram polyomino

Description

Return the Dyck path corresponding to the partition interpreted as a parallogram polyomino.
The Ferrers diagram of an integer partition can be interpreted as a parallogram polyomino, such that each part corresponds to a column.
This map returns the corresponding Dyck path.

Map

to binary word

Description

Return the Dyck word as binary word.

Map

Robinson-Schensted tableau shape

Description

Sends a permutation to its Robinson-Schensted tableau shape.
The Robinson-Schensted corrspondence is a bijection between permutations of length

$n$ and pairs of standard Young tableaux of the same shape and of size

$n$ , see [1]. These two tableaux are the insertion tableau and the recording tableau.
This map sends a permutation to the shape of its corresponding insertion and recording tableau.