Identifier
Values
[1,0] => [[1],[2]] => [2] => 10 => 1
[1,0,1,0] => [[1,3],[2,4]] => [2,2] => 1010 => 1
[1,1,0,0] => [[1,2],[3,4]] => [3,1] => 1001 => 2
[1,0,1,0,1,0] => [[1,3,5],[2,4,6]] => [2,2,2] => 101010 => 1
[1,0,1,1,0,0] => [[1,3,4],[2,5,6]] => [2,3,1] => 101001 => 2
[1,1,0,0,1,0] => [[1,2,5],[3,4,6]] => [3,3] => 100100 => 2
[1,1,0,1,0,0] => [[1,2,4],[3,5,6]] => [3,2,1] => 100101 => 2
[1,1,1,0,0,0] => [[1,2,3],[4,5,6]] => [4,2] => 100010 => 3
[1,0,1,0,1,0,1,0] => [[1,3,5,7],[2,4,6,8]] => [2,2,2,2] => 10101010 => 1
[1,0,1,0,1,1,0,0] => [[1,3,5,6],[2,4,7,8]] => [2,2,3,1] => 10101001 => 2
[1,0,1,1,0,0,1,0] => [[1,3,4,7],[2,5,6,8]] => [2,3,3] => 10100100 => 2
[1,0,1,1,0,1,0,0] => [[1,3,4,6],[2,5,7,8]] => [2,3,2,1] => 10100101 => 2
[1,0,1,1,1,0,0,0] => [[1,3,4,5],[2,6,7,8]] => [2,4,2] => 10100010 => 3
[1,1,0,0,1,0,1,0] => [[1,2,5,7],[3,4,6,8]] => [3,3,2] => 10010010 => 2
[1,1,0,0,1,1,0,0] => [[1,2,5,6],[3,4,7,8]] => [3,4,1] => 10010001 => 3
[1,1,0,1,0,0,1,0] => [[1,2,4,7],[3,5,6,8]] => [3,2,3] => 10010100 => 2
[1,1,0,1,0,1,0,0] => [[1,2,4,6],[3,5,7,8]] => [3,2,2,1] => 10010101 => 2
[1,1,0,1,1,0,0,0] => [[1,2,4,5],[3,6,7,8]] => [3,3,2] => 10010010 => 2
[1,1,1,0,0,0,1,0] => [[1,2,3,7],[4,5,6,8]] => [4,4] => 10001000 => 3
[1,1,1,0,0,1,0,0] => [[1,2,3,6],[4,5,7,8]] => [4,3,1] => 10001001 => 3
[1,1,1,0,1,0,0,0] => [[1,2,3,5],[4,6,7,8]] => [4,2,2] => 10001010 => 3
[1,1,1,1,0,0,0,0] => [[1,2,3,4],[5,6,7,8]] => [5,3] => 10000100 => 4
[1,0,1,0,1,0,1,0,1,0] => [[1,3,5,7,9],[2,4,6,8,10]] => [2,2,2,2,2] => 1010101010 => 1
[1,0,1,0,1,0,1,1,0,0] => [[1,3,5,7,8],[2,4,6,9,10]] => [2,2,2,3,1] => 1010101001 => 2
[1,0,1,0,1,1,0,1,0,0] => [[1,3,5,6,8],[2,4,7,9,10]] => [2,2,3,2,1] => 1010100101 => 2
[1,0,1,1,0,1,0,1,0,0] => [[1,3,4,6,8],[2,5,7,9,10]] => [2,3,2,2,1] => 1010010101 => 2
[1,1,0,1,0,1,0,1,0,0] => [[1,2,4,6,8],[3,5,7,9,10]] => [3,2,2,2,1] => 1001010101 => 2
[1,0,1,0,1,0,1,0,1,0,1,0] => [[1,3,5,7,9,11],[2,4,6,8,10,12]] => [2,2,2,2,2,2] => 101010101010 => 1
[1,0,1,0,1,0,1,0,1,1,0,0] => [[1,3,5,7,9,10],[2,4,6,8,11,12]] => [2,2,2,2,3,1] => 101010101001 => 2
[1,0,1,0,1,0,1,1,0,1,0,0] => [[1,3,5,7,8,10],[2,4,6,9,11,12]] => [2,2,2,3,2,1] => 101010100101 => 2
[1,0,1,0,1,1,0,1,0,1,0,0] => [[1,3,5,6,8,10],[2,4,7,9,11,12]] => [2,2,3,2,2,1] => 101010010101 => 2
[1,0,1,1,0,1,0,1,0,1,0,0] => [[1,3,4,6,8,10],[2,5,7,9,11,12]] => [2,3,2,2,2,1] => 101001010101 => 2
[1,1,0,1,0,1,0,1,0,1,0,0] => [[1,2,4,6,8,10],[3,5,7,9,11,12]] => [3,2,2,2,2,1] => 100101010101 => 2
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Description
The length of the longest constant subword.
Map
valley composition
Description
The composition corresponding to the valley set of a standard tableau.
Let $T$ be a standard tableau of size $n$.
An entry $i$ of $T$ is a descent if $i+1$ is in a lower row (in English notation), otherwise $i$ is an ascent.
An entry $2 \leq i \leq n-1$ is a valley if $i-1$ is a descent and $i$ is an ascent.
This map returns the composition $c_1,\dots,c_k$ of $n$ such that $\{c_1, c_1+c_2,\dots, c_1+\dots+c_k\}$ is the valley set of $T$.
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
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.