Identifier
-
Mp00060:
Permutations
—Robinson-Schensted tableau shape⟶
Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00093: Dyck paths —to binary word⟶ Binary words
St001730: 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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Description
The number of times the path corresponding to a binary word crosses the base line.
Interpret each $0$ as a step $(1,-1)$ and $1$ as a step $(1,1)$. Then this statistic counts the number of times the path crosses the $x$-axis.
Interpret each $0$ as a step $(1,-1)$ and $1$ as a step $(1,1)$. Then this statistic counts the number of times the path crosses the $x$-axis.
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.
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.
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.
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.
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