Identifier
Values
[1] => [1,0,1,0] => [1,1,0,0] => 1100 => 0
[2] => [1,1,0,0,1,0] => [1,0,1,0,1,0] => 101010 => 1
[1,1] => [1,0,1,1,0,0] => [1,1,0,1,0,0] => 110100 => 0
[3] => [1,1,1,0,0,0,1,0] => [1,1,0,1,0,0,1,0] => 11010010 => 0
[2,1] => [1,0,1,0,1,0] => [1,1,1,0,0,0] => 111000 => 2
[1,1,1] => [1,0,1,1,1,0,0,0] => [1,1,1,0,0,1,0,0] => 11100100 => 1
[3,1] => [1,1,0,1,0,0,1,0] => [1,0,1,1,0,1,0,0] => 10110100 => 0
[2,2] => [1,1,0,0,1,1,0,0] => [1,0,1,0,1,0,1,0] => 10101010 => 2
[2,1,1] => [1,0,1,1,0,1,0,0] => [1,1,0,1,1,0,0,0] => 11011000 => 1
[3,2] => [1,1,0,0,1,0,1,0] => [1,0,1,1,0,0,1,0] => 10110010 => 0
[3,1,1] => [1,0,1,1,0,0,1,0] => [1,1,0,1,0,1,0,0] => 11010100 => 1
[2,2,1] => [1,0,1,0,1,1,0,0] => [1,1,1,0,1,0,0,0] => 11101000 => 2
[3,2,1] => [1,0,1,0,1,0,1,0] => [1,1,1,1,0,0,0,0] => 11110000 => 2
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Description
The number of distinct cubes in a binary word.
A factor of a word is a sequence of consecutive letters. This statistic records the number of distinct non-empty words $u$ such that $uuu$ is a factor of the word.
Map
to binary word
Description
Return the Dyck word as binary word.
Map
Knuth-Krattenthaler
Description
The map that sends the Dyck path to a 321-avoiding permutation, then applies the Robinson-Schensted correspondence and finally interprets the first row of the insertion tableau and the second row of the recording tableau as up steps.
Interpreting a pair of two-row standard tableaux of the same shape as a Dyck path is explained by Knuth in [1, pp. 60].
Krattenthaler's bijection between Dyck paths and $321$-avoiding permutations used is Mp00119to 321-avoiding permutation (Krattenthaler), see [2].
This is the inverse of the map Mp00127left-to-right-maxima to Dyck path that interprets the left-to-right maxima of the permutation obtained from Mp00024to 321-avoiding permutation as a Dyck path.
Map
to Dyck path
Description
Sends a partition to the shortest Dyck path tracing the shape of its Ferrers diagram.