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
to binary word
Description
Return the Dyck word as binary word.
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.
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