Identifier
-
Mp00108:
Permutations
—cycle type⟶
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] => [1,1] => [1,0,1,1,0,0] => 101100 => 0
[2,1] => [2] => [1,1,0,0,1,0] => 110010 => 0
[1,2,3] => [1,1,1] => [1,0,1,1,1,0,0,0] => 10111000 => 0
[1,3,2] => [2,1] => [1,0,1,0,1,0] => 101010 => 0
[2,1,3] => [2,1] => [1,0,1,0,1,0] => 101010 => 0
[2,3,1] => [3] => [1,1,1,0,0,0,1,0] => 11100010 => 0
[3,1,2] => [3] => [1,1,1,0,0,0,1,0] => 11100010 => 0
[3,2,1] => [2,1] => [1,0,1,0,1,0] => 101010 => 0
[1,2,4,3] => [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
[1,3,4,2] => [3,1] => [1,1,0,1,0,0,1,0] => 11010010 => 0
[1,4,2,3] => [3,1] => [1,1,0,1,0,0,1,0] => 11010010 => 0
[1,4,3,2] => [2,1,1] => [1,0,1,1,0,1,0,0] => 10110100 => 0
[2,1,3,4] => [2,1,1] => [1,0,1,1,0,1,0,0] => 10110100 => 0
[2,1,4,3] => [2,2] => [1,1,0,0,1,1,0,0] => 11001100 => 0
[2,3,1,4] => [3,1] => [1,1,0,1,0,0,1,0] => 11010010 => 0
[2,4,3,1] => [3,1] => [1,1,0,1,0,0,1,0] => 11010010 => 0
[3,1,2,4] => [3,1] => [1,1,0,1,0,0,1,0] => 11010010 => 0
[3,2,1,4] => [2,1,1] => [1,0,1,1,0,1,0,0] => 10110100 => 0
[3,2,4,1] => [3,1] => [1,1,0,1,0,0,1,0] => 11010010 => 0
[3,4,1,2] => [2,2] => [1,1,0,0,1,1,0,0] => 11001100 => 0
[4,1,3,2] => [3,1] => [1,1,0,1,0,0,1,0] => 11010010 => 0
[4,2,1,3] => [3,1] => [1,1,0,1,0,0,1,0] => 11010010 => 0
[4,2,3,1] => [2,1,1] => [1,0,1,1,0,1,0,0] => 10110100 => 0
[4,3,2,1] => [2,2] => [1,1,0,0,1,1,0,0] => 11001100 => 0
[1,2,4,5,3] => [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
[1,3,2,5,4] => [2,2,1] => [1,0,1,0,1,1,0,0] => 10101100 => 0
[1,3,4,2,5] => [3,1,1] => [1,0,1,1,0,0,1,0] => 10110010 => 0
[1,3,5,4,2] => [3,1,1] => [1,0,1,1,0,0,1,0] => 10110010 => 0
[1,4,2,3,5] => [3,1,1] => [1,0,1,1,0,0,1,0] => 10110010 => 0
[1,4,3,5,2] => [3,1,1] => [1,0,1,1,0,0,1,0] => 10110010 => 0
[1,4,5,2,3] => [2,2,1] => [1,0,1,0,1,1,0,0] => 10101100 => 0
[1,5,2,4,3] => [3,1,1] => [1,0,1,1,0,0,1,0] => 10110010 => 0
[1,5,3,2,4] => [3,1,1] => [1,0,1,1,0,0,1,0] => 10110010 => 0
[1,5,4,3,2] => [2,2,1] => [1,0,1,0,1,1,0,0] => 10101100 => 0
[2,1,3,5,4] => [2,2,1] => [1,0,1,0,1,1,0,0] => 10101100 => 0
[2,1,4,3,5] => [2,2,1] => [1,0,1,0,1,1,0,0] => 10101100 => 0
[2,1,4,5,3] => [3,2] => [1,1,0,0,1,0,1,0] => 11001010 => 0
[2,1,5,3,4] => [3,2] => [1,1,0,0,1,0,1,0] => 11001010 => 0
[2,1,5,4,3] => [2,2,1] => [1,0,1,0,1,1,0,0] => 10101100 => 0
[2,3,1,4,5] => [3,1,1] => [1,0,1,1,0,0,1,0] => 10110010 => 0
[2,3,1,5,4] => [3,2] => [1,1,0,0,1,0,1,0] => 11001010 => 0
[2,4,3,1,5] => [3,1,1] => [1,0,1,1,0,0,1,0] => 10110010 => 0
[2,4,5,1,3] => [3,2] => [1,1,0,0,1,0,1,0] => 11001010 => 0
[2,5,3,4,1] => [3,1,1] => [1,0,1,1,0,0,1,0] => 10110010 => 0
[2,5,4,3,1] => [3,2] => [1,1,0,0,1,0,1,0] => 11001010 => 0
[3,1,2,4,5] => [3,1,1] => [1,0,1,1,0,0,1,0] => 10110010 => 0
[3,1,2,5,4] => [3,2] => [1,1,0,0,1,0,1,0] => 11001010 => 0
[3,2,1,5,4] => [2,2,1] => [1,0,1,0,1,1,0,0] => 10101100 => 0
[3,2,4,1,5] => [3,1,1] => [1,0,1,1,0,0,1,0] => 10110010 => 0
[3,2,5,4,1] => [3,1,1] => [1,0,1,1,0,0,1,0] => 10110010 => 0
[3,4,1,2,5] => [2,2,1] => [1,0,1,0,1,1,0,0] => 10101100 => 0
[3,4,1,5,2] => [3,2] => [1,1,0,0,1,0,1,0] => 11001010 => 0
[3,4,5,2,1] => [3,2] => [1,1,0,0,1,0,1,0] => 11001010 => 0
[3,5,1,2,4] => [3,2] => [1,1,0,0,1,0,1,0] => 11001010 => 0
[3,5,1,4,2] => [2,2,1] => [1,0,1,0,1,1,0,0] => 10101100 => 0
[3,5,4,1,2] => [3,2] => [1,1,0,0,1,0,1,0] => 11001010 => 0
[4,1,3,2,5] => [3,1,1] => [1,0,1,1,0,0,1,0] => 10110010 => 0
[4,1,5,2,3] => [3,2] => [1,1,0,0,1,0,1,0] => 11001010 => 0
[4,2,1,3,5] => [3,1,1] => [1,0,1,1,0,0,1,0] => 10110010 => 0
[4,2,3,5,1] => [3,1,1] => [1,0,1,1,0,0,1,0] => 10110010 => 0
[4,2,5,1,3] => [2,2,1] => [1,0,1,0,1,1,0,0] => 10101100 => 0
[4,3,2,1,5] => [2,2,1] => [1,0,1,0,1,1,0,0] => 10101100 => 0
[4,3,2,5,1] => [3,2] => [1,1,0,0,1,0,1,0] => 11001010 => 0
[4,3,5,1,2] => [3,2] => [1,1,0,0,1,0,1,0] => 11001010 => 0
[4,5,1,3,2] => [3,2] => [1,1,0,0,1,0,1,0] => 11001010 => 0
[4,5,2,1,3] => [3,2] => [1,1,0,0,1,0,1,0] => 11001010 => 0
[4,5,3,1,2] => [2,2,1] => [1,0,1,0,1,1,0,0] => 10101100 => 0
[5,1,3,4,2] => [3,1,1] => [1,0,1,1,0,0,1,0] => 10110010 => 0
[5,1,4,3,2] => [3,2] => [1,1,0,0,1,0,1,0] => 11001010 => 0
[5,2,1,4,3] => [3,1,1] => [1,0,1,1,0,0,1,0] => 10110010 => 0
[5,2,3,1,4] => [3,1,1] => [1,0,1,1,0,0,1,0] => 10110010 => 0
[5,2,4,3,1] => [2,2,1] => [1,0,1,0,1,1,0,0] => 10101100 => 0
[5,3,2,1,4] => [3,2] => [1,1,0,0,1,0,1,0] => 11001010 => 0
[5,3,2,4,1] => [2,2,1] => [1,0,1,0,1,1,0,0] => 10101100 => 0
[5,3,4,2,1] => [3,2] => [1,1,0,0,1,0,1,0] => 11001010 => 0
[5,4,1,2,3] => [3,2] => [1,1,0,0,1,0,1,0] => 11001010 => 0
[5,4,2,3,1] => [3,2] => [1,1,0,0,1,0,1,0] => 11001010 => 0
[5,4,3,2,1] => [2,2,1] => [1,0,1,0,1,1,0,0] => 10101100 => 0
[1,3,2,5,6,4] => [3,2,1] => [1,0,1,0,1,0,1,0] => 10101010 => 0
[1,3,2,6,4,5] => [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
[1,3,5,6,2,4] => [3,2,1] => [1,0,1,0,1,0,1,0] => 10101010 => 0
[1,3,6,5,4,2] => [3,2,1] => [1,0,1,0,1,0,1,0] => 10101010 => 0
[1,4,2,3,6,5] => [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
[1,4,5,6,3,2] => [3,2,1] => [1,0,1,0,1,0,1,0] => 10101010 => 0
[1,4,6,2,3,5] => [3,2,1] => [1,0,1,0,1,0,1,0] => 10101010 => 0
[1,4,6,5,2,3] => [3,2,1] => [1,0,1,0,1,0,1,0] => 10101010 => 0
[1,5,2,6,3,4] => [3,2,1] => [1,0,1,0,1,0,1,0] => 10101010 => 0
[1,5,4,3,6,2] => [3,2,1] => [1,0,1,0,1,0,1,0] => 10101010 => 0
[1,5,4,6,2,3] => [3,2,1] => [1,0,1,0,1,0,1,0] => 10101010 => 0
[1,5,6,2,4,3] => [3,2,1] => [1,0,1,0,1,0,1,0] => 10101010 => 0
[1,5,6,3,2,4] => [3,2,1] => [1,0,1,0,1,0,1,0] => 10101010 => 0
[1,6,2,5,4,3] => [3,2,1] => [1,0,1,0,1,0,1,0] => 10101010 => 0
[1,6,4,3,2,5] => [3,2,1] => [1,0,1,0,1,0,1,0] => 10101010 => 0
[1,6,4,5,3,2] => [3,2,1] => [1,0,1,0,1,0,1,0] => 10101010 => 0
[1,6,5,2,3,4] => [3,2,1] => [1,0,1,0,1,0,1,0] => 10101010 => 0
[1,6,5,3,4,2] => [3,2,1] => [1,0,1,0,1,0,1,0] => 10101010 => 0
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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.
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
cycle type
Description
The cycle type of a permutation as a partition.
Map
to Dyck path
Description
Sends a partition to the shortest Dyck path tracing the shape of its Ferrers diagram.
Map
to binary word
Description
Return the Dyck word as binary word.
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