- FindStat

Identifier

Mp00033: Dyck paths —to two-row standard tableau⟶ Standard tableaux
Mp00207: Standard tableaux —horizontal strip sizes⟶ Integer compositions
Mp00094: Integer compositions —to binary word⟶ Binary words
St000629: Binary words ⟶ ℤ

Values

[1,0] => [[1],[2]] => [1,1] => 11 => 0
[1,0,1,0] => [[1,3],[2,4]] => [1,2,1] => 1101 => 0
[1,1,0,0] => [[1,2],[3,4]] => [2,2] => 1010 => 0
[1,0,1,0,1,0] => [[1,3,5],[2,4,6]] => [1,2,2,1] => 110101 => 0
[1,0,1,1,0,0] => [[1,3,4],[2,5,6]] => [1,3,2] => 110010 => 0
[1,1,0,0,1,0] => [[1,2,5],[3,4,6]] => [2,3,1] => 101001 => 0
[1,1,0,1,0,0] => [[1,2,4],[3,5,6]] => [2,2,2] => 101010 => 0
[1,1,1,0,0,0] => [[1,2,3],[4,5,6]] => [3,3] => 100100 => 0
[1,0,1,0,1,0,1,0] => [[1,3,5,7],[2,4,6,8]] => [1,2,2,2,1] => 11010101 => 0
[1,0,1,0,1,1,0,0] => [[1,3,5,6],[2,4,7,8]] => [1,2,3,2] => 11010010 => 0
[1,0,1,1,0,0,1,0] => [[1,3,4,7],[2,5,6,8]] => [1,3,3,1] => 11001001 => 0
[1,0,1,1,0,1,0,0] => [[1,3,4,6],[2,5,7,8]] => [1,3,2,2] => 11001010 => 0
[1,0,1,1,1,0,0,0] => [[1,3,4,5],[2,6,7,8]] => [1,4,3] => 11000100 => 0
[1,1,0,0,1,0,1,0] => [[1,2,5,7],[3,4,6,8]] => [2,3,2,1] => 10100101 => 0
[1,1,0,0,1,1,0,0] => [[1,2,5,6],[3,4,7,8]] => [2,4,2] => 10100010 => 0
[1,1,0,1,0,0,1,0] => [[1,2,4,7],[3,5,6,8]] => [2,2,3,1] => 10101001 => 0
[1,1,0,1,0,1,0,0] => [[1,2,4,6],[3,5,7,8]] => [2,2,2,2] => 10101010 => 0
[1,1,0,1,1,0,0,0] => [[1,2,4,5],[3,6,7,8]] => [2,3,3] => 10100100 => 0
[1,1,1,0,0,0,1,0] => [[1,2,3,7],[4,5,6,8]] => [3,4,1] => 10010001 => 0
[1,1,1,0,0,1,0,0] => [[1,2,3,6],[4,5,7,8]] => [3,3,2] => 10010010 => 0
[1,1,1,0,1,0,0,0] => [[1,2,3,5],[4,6,7,8]] => [3,2,3] => 10010100 => 0
[1,1,1,1,0,0,0,0] => [[1,2,3,4],[5,6,7,8]] => [4,4] => 10001000 => 0
[1,0,1,0,1,0,1,1,0,0] => [[1,3,5,7,8],[2,4,6,9,10]] => [1,2,2,3,2] => 1101010010 => 0
[1,0,1,0,1,1,0,0,1,0] => [[1,3,5,6,9],[2,4,7,8,10]] => [1,2,3,3,1] => 1101001001 => 0
[1,0,1,0,1,1,0,1,0,0] => [[1,3,5,6,8],[2,4,7,9,10]] => [1,2,3,2,2] => 1101001010 => 0
[1,0,1,1,0,0,1,0,1,0] => [[1,3,4,7,9],[2,5,6,8,10]] => [1,3,3,2,1] => 1100100101 => 0
[1,0,1,1,0,1,0,0,1,0] => [[1,3,4,6,9],[2,5,7,8,10]] => [1,3,2,3,1] => 1100101001 => 0
[1,0,1,1,0,1,0,1,0,0] => [[1,3,4,6,8],[2,5,7,9,10]] => [1,3,2,2,2] => 1100101010 => 0
[1,0,1,1,1,0,0,1,0,0] => [[1,3,4,5,8],[2,6,7,9,10]] => [1,4,3,2] => 1100010010 => 0
[1,0,1,1,1,1,0,0,0,0] => [[1,3,4,5,6],[2,7,8,9,10]] => [1,5,4] => 1100001000 => 0
[1,1,0,0,1,0,1,0,1,0] => [[1,2,5,7,9],[3,4,6,8,10]] => [2,3,2,2,1] => 1010010101 => 0
[1,1,0,1,0,0,1,0,1,0] => [[1,2,4,7,9],[3,5,6,8,10]] => [2,2,3,2,1] => 1010100101 => 0
[1,1,0,1,0,1,0,0,1,0] => [[1,2,4,6,9],[3,5,7,8,10]] => [2,2,2,3,1] => 1010101001 => 0
[1,1,0,1,0,1,0,1,0,0] => [[1,2,4,6,8],[3,5,7,9,10]] => [2,2,2,2,2] => 1010101010 => 0
[1,0,1,0,1,0,1,0,1,1,0,0] => [[1,3,5,7,9,10],[2,4,6,8,11,12]] => [1,2,2,2,3,2] => 110101010010 => 0
[1,0,1,0,1,0,1,1,0,0,1,0] => [[1,3,5,7,8,11],[2,4,6,9,10,12]] => [1,2,2,3,3,1] => 110101001001 => 0
[1,0,1,0,1,0,1,1,0,1,0,0] => [[1,3,5,7,8,10],[2,4,6,9,11,12]] => [1,2,2,3,2,2] => 110101001010 => 0
[1,0,1,0,1,1,0,0,1,0,1,0] => [[1,3,5,6,9,11],[2,4,7,8,10,12]] => [1,2,3,3,2,1] => 110100100101 => 0
[1,0,1,0,1,1,0,1,0,0,1,0] => [[1,3,5,6,8,11],[2,4,7,9,10,12]] => [1,2,3,2,3,1] => 110100101001 => 0
[1,0,1,0,1,1,0,1,0,1,0,0] => [[1,3,5,6,8,10],[2,4,7,9,11,12]] => [1,2,3,2,2,2] => 110100101010 => 0
[1,0,1,1,0,0,1,0,1,0,1,0] => [[1,3,4,7,9,11],[2,5,6,8,10,12]] => [1,3,3,2,2,1] => 110010010101 => 0
[1,0,1,1,0,1,0,0,1,0,1,0] => [[1,3,4,6,9,11],[2,5,7,8,10,12]] => [1,3,2,3,2,1] => 110010100101 => 0
[1,0,1,1,0,1,0,1,0,0,1,0] => [[1,3,4,6,8,11],[2,5,7,9,10,12]] => [1,3,2,2,3,1] => 110010101001 => 0
[1,0,1,1,0,1,0,1,0,1,0,0] => [[1,3,4,6,8,10],[2,5,7,9,11,12]] => [1,3,2,2,2,2] => 110010101010 => 0
[1,1,0,0,1,0,1,0,1,0,1,0] => [[1,2,5,7,9,11],[3,4,6,8,10,12]] => [2,3,2,2,2,1] => 101001010101 => 0
[1,1,0,1,0,0,1,0,1,0,1,0] => [[1,2,4,7,9,11],[3,5,6,8,10,12]] => [2,2,3,2,2,1] => 101010010101 => 0
[1,1,0,1,0,1,0,0,1,0,1,0] => [[1,2,4,6,9,11],[3,5,7,8,10,12]] => [2,2,2,3,2,1] => 101010100101 => 0
[1,1,0,1,0,1,0,1,0,0,1,0] => [[1,2,4,6,8,11],[3,5,7,9,10,12]] => [2,2,2,2,3,1] => 101010101001 => 0
[1,1,0,1,0,1,0,1,0,1,0,0] => [[1,2,4,6,8,10],[3,5,7,9,11,12]] => [2,2,2,2,2,2] => 101010101010 => 0

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$F_{1} = 1$

$F_{2} = 2$

$F_{3} = 5$

$F_{4} = 14$

Description

The defect of a binary word.
The defect of a finite word

$w$ is given by the difference between the maximum possible number and the actual number of palindromic factors contained in

$w$ . The maximum possible number of palindromic factors in a word

$w$ is

$|w|+1$ .

Map

to binary word

Description

Return the composition as a binary word, treating ones as separators.
Encoding a positive integer

$i$ as the word

$10\dots 0$ consisting of a one followed by

$i-1$ zeros, the binary word of a composition

$(i_1,\dots,i_k)$ is the concatenation of of words for

$i_1,\dots,i_k$ .
The image of this map contains precisely the words which do not begin with a

$0$ .

Map

horizontal strip sizes

Description

The composition of horizontal strip sizes.
We associate to a standard Young tableau

$T$ the composition

$(c_1,\dots,c_k)$ , such that

$k$ is minimal and the numbers

$c_1+\dots+c_i + 1,\dots,c_1+\dots+c_{i+1}$ form a horizontal strip in

$T$ for all

$i$ .

Map

to two-row standard tableau

Description

Return a standard tableau of shape

$(n,n)$ where

$n$ is the semilength of the Dyck path.
Given a Dyck path

$D$ , its image is given by recording the positions of the up-steps in the first row and the positions of the down-steps in the second row.