Identifier
Values
[1,0] => [] => => [1] => 1
[1,0,1,0] => [1] => 10 => [1,2] => 1
[1,1,0,0] => [] => => [1] => 1
[1,0,1,0,1,0] => [2,1] => 1010 => [1,2,2] => 1
[1,0,1,1,0,0] => [1,1] => 110 => [1,1,2] => 1
[1,1,0,0,1,0] => [2] => 100 => [1,3] => 1
[1,1,0,1,0,0] => [1] => 10 => [1,2] => 1
[1,1,1,0,0,0] => [] => => [1] => 1
[1,0,1,0,1,1,0,0] => [2,2,1] => 11010 => [1,1,2,2] => 1
[1,0,1,1,0,1,0,0] => [2,1,1] => 10110 => [1,2,1,2] => 1
[1,0,1,1,1,0,0,0] => [1,1,1] => 1110 => [1,1,1,2] => 1
[1,1,0,0,1,0,1,0] => [3,2] => 10100 => [1,2,3] => 1
[1,1,0,0,1,1,0,0] => [2,2] => 1100 => [1,1,3] => 1
[1,1,0,1,0,0,1,0] => [3,1] => 10010 => [1,3,2] => 1
[1,1,0,1,0,1,0,0] => [2,1] => 1010 => [1,2,2] => 1
[1,1,0,1,1,0,0,0] => [1,1] => 110 => [1,1,2] => 1
[1,1,1,0,0,0,1,0] => [3] => 1000 => [1,4] => 1
[1,1,1,0,0,1,0,0] => [2] => 100 => [1,3] => 1
[1,1,1,0,1,0,0,0] => [1] => 10 => [1,2] => 1
[1,1,1,1,0,0,0,0] => [] => => [1] => 1
[1,0,1,1,1,1,0,0,0,0] => [1,1,1,1] => 11110 => [1,1,1,1,2] => 1
[1,1,0,0,1,1,1,0,0,0] => [2,2,2] => 11100 => [1,1,1,3] => 1
[1,1,0,1,0,1,1,0,0,0] => [2,2,1] => 11010 => [1,1,2,2] => 1
[1,1,0,1,1,0,1,0,0,0] => [2,1,1] => 10110 => [1,2,1,2] => 1
[1,1,0,1,1,1,0,0,0,0] => [1,1,1] => 1110 => [1,1,1,2] => 1
[1,1,1,0,0,0,1,1,0,0] => [3,3] => 11000 => [1,1,4] => 1
[1,1,1,0,0,1,0,1,0,0] => [3,2] => 10100 => [1,2,3] => 1
[1,1,1,0,0,1,1,0,0,0] => [2,2] => 1100 => [1,1,3] => 1
[1,1,1,0,1,0,0,1,0,0] => [3,1] => 10010 => [1,3,2] => 1
[1,1,1,0,1,0,1,0,0,0] => [2,1] => 1010 => [1,2,2] => 1
[1,1,1,0,1,1,0,0,0,0] => [1,1] => 110 => [1,1,2] => 1
[1,1,1,1,0,0,0,0,1,0] => [4] => 10000 => [1,5] => 1
[1,1,1,1,0,0,0,1,0,0] => [3] => 1000 => [1,4] => 1
[1,1,1,1,0,0,1,0,0,0] => [2] => 100 => [1,3] => 1
[1,1,1,1,0,1,0,0,0,0] => [1] => 10 => [1,2] => 1
[1,1,1,1,1,0,0,0,0,0] => [] => => [1] => 1
[1,1,0,1,1,1,1,0,0,0,0,0] => [1,1,1,1] => 11110 => [1,1,1,1,2] => 1
[1,1,1,0,0,1,1,1,0,0,0,0] => [2,2,2] => 11100 => [1,1,1,3] => 1
[1,1,1,0,1,0,1,1,0,0,0,0] => [2,2,1] => 11010 => [1,1,2,2] => 1
[1,1,1,0,1,1,0,1,0,0,0,0] => [2,1,1] => 10110 => [1,2,1,2] => 1
[1,1,1,0,1,1,1,0,0,0,0,0] => [1,1,1] => 1110 => [1,1,1,2] => 1
[1,1,1,1,0,0,0,1,1,0,0,0] => [3,3] => 11000 => [1,1,4] => 1
[1,1,1,1,0,0,1,0,1,0,0,0] => [3,2] => 10100 => [1,2,3] => 1
[1,1,1,1,0,0,1,1,0,0,0,0] => [2,2] => 1100 => [1,1,3] => 1
[1,1,1,1,0,1,0,0,1,0,0,0] => [3,1] => 10010 => [1,3,2] => 1
[1,1,1,1,0,1,0,1,0,0,0,0] => [2,1] => 1010 => [1,2,2] => 1
[1,1,1,1,0,1,1,0,0,0,0,0] => [1,1] => 110 => [1,1,2] => 1
[1,1,1,1,1,0,0,0,0,1,0,0] => [4] => 10000 => [1,5] => 1
[1,1,1,1,1,0,0,0,1,0,0,0] => [3] => 1000 => [1,4] => 1
[1,1,1,1,1,0,0,1,0,0,0,0] => [2] => 100 => [1,3] => 1
[1,1,1,1,1,0,1,0,0,0,0,0] => [1] => 10 => [1,2] => 1
[1,1,1,1,1,1,0,0,0,0,0,0] => [] => => [1] => 1
[1,1,1,0,1,1,1,1,0,0,0,0,0,0] => [1,1,1,1] => 11110 => [1,1,1,1,2] => 1
[1,1,1,1,0,0,1,1,1,0,0,0,0,0] => [2,2,2] => 11100 => [1,1,1,3] => 1
[1,1,1,1,0,1,0,1,1,0,0,0,0,0] => [2,2,1] => 11010 => [1,1,2,2] => 1
[1,1,1,1,0,1,1,0,1,0,0,0,0,0] => [2,1,1] => 10110 => [1,2,1,2] => 1
[1,1,1,1,0,1,1,1,0,0,0,0,0,0] => [1,1,1] => 1110 => [1,1,1,2] => 1
[1,1,1,1,1,0,0,0,1,1,0,0,0,0] => [3,3] => 11000 => [1,1,4] => 1
[1,1,1,1,1,0,0,1,0,1,0,0,0,0] => [3,2] => 10100 => [1,2,3] => 1
[1,1,1,1,1,0,0,1,1,0,0,0,0,0] => [2,2] => 1100 => [1,1,3] => 1
[1,1,1,1,1,0,1,0,0,1,0,0,0,0] => [3,1] => 10010 => [1,3,2] => 1
[1,1,1,1,1,0,1,0,1,0,0,0,0,0] => [2,1] => 1010 => [1,2,2] => 1
[1,1,1,1,1,0,1,1,0,0,0,0,0,0] => [1,1] => 110 => [1,1,2] => 1
[1,1,1,1,1,1,0,0,0,0,1,0,0,0] => [4] => 10000 => [1,5] => 1
[1,1,1,1,1,1,0,0,0,1,0,0,0,0] => [3] => 1000 => [1,4] => 1
[1,1,1,1,1,1,0,0,1,0,0,0,0,0] => [2] => 100 => [1,3] => 1
[1,1,1,1,1,1,0,1,0,0,0,0,0,0] => [1] => 10 => [1,2] => 1
[1,1,1,1,1,1,1,0,0,0,0,0,0,0] => [] => => [1] => 1
[1,1,1,1,0,1,1,1,1,0,0,0,0,0,0,0] => [1,1,1,1] => 11110 => [1,1,1,1,2] => 1
[1,1,1,1,1,0,0,1,1,1,0,0,0,0,0,0] => [2,2,2] => 11100 => [1,1,1,3] => 1
[1,1,1,1,1,0,1,0,1,1,0,0,0,0,0,0] => [2,2,1] => 11010 => [1,1,2,2] => 1
[1,1,1,1,1,0,1,1,0,1,0,0,0,0,0,0] => [2,1,1] => 10110 => [1,2,1,2] => 1
[1,1,1,1,1,0,1,1,1,0,0,0,0,0,0,0] => [1,1,1] => 1110 => [1,1,1,2] => 1
[1,1,1,1,1,1,0,0,0,1,1,0,0,0,0,0] => [3,3] => 11000 => [1,1,4] => 1
[1,1,1,1,1,1,0,0,1,0,1,0,0,0,0,0] => [3,2] => 10100 => [1,2,3] => 1
[1,1,1,1,1,1,0,0,1,1,0,0,0,0,0,0] => [2,2] => 1100 => [1,1,3] => 1
[1,1,1,1,1,1,0,1,0,0,1,0,0,0,0,0] => [3,1] => 10010 => [1,3,2] => 1
[1,1,1,1,1,1,0,1,0,1,0,0,0,0,0,0] => [2,1] => 1010 => [1,2,2] => 1
[1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0] => [1,1] => 110 => [1,1,2] => 1
[1,1,1,1,1,1,1,0,0,0,0,1,0,0,0,0] => [4] => 10000 => [1,5] => 1
[1,1,1,1,1,1,1,0,0,0,1,0,0,0,0,0] => [3] => 1000 => [1,4] => 1
[1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0] => [2] => 100 => [1,3] => 1
[1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0] => [1] => 10 => [1,2] => 1
[1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0] => [] => => [1] => 1
[] => [] => => [1] => 1
[1,1,1,1,1,0,1,1,1,1,0,0,0,0,0,0,0,0] => [1,1,1,1] => 11110 => [1,1,1,1,2] => 1
[1,1,1,1,1,1,0,0,1,1,1,0,0,0,0,0,0,0] => [2,2,2] => 11100 => [1,1,1,3] => 1
[1,1,1,1,1,1,0,1,0,1,1,0,0,0,0,0,0,0] => [2,2,1] => 11010 => [1,1,2,2] => 1
[1,1,1,1,1,1,0,1,1,0,1,0,0,0,0,0,0,0] => [2,1,1] => 10110 => [1,2,1,2] => 1
[1,1,1,1,1,1,0,1,1,1,0,0,0,0,0,0,0,0] => [1,1,1] => 1110 => [1,1,1,2] => 1
[1,1,1,1,1,1,1,0,0,0,1,1,0,0,0,0,0,0] => [3,3] => 11000 => [1,1,4] => 1
[1,1,1,1,1,1,1,0,0,1,0,1,0,0,0,0,0,0] => [3,2] => 10100 => [1,2,3] => 1
[1,1,1,1,1,1,1,0,0,1,1,0,0,0,0,0,0,0] => [2,2] => 1100 => [1,1,3] => 1
[1,1,1,1,1,1,1,0,1,0,0,1,0,0,0,0,0,0] => [3,1] => 10010 => [1,3,2] => 1
[1,1,1,1,1,1,1,0,1,0,1,0,0,0,0,0,0,0] => [2,1] => 1010 => [1,2,2] => 1
[1,1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0,0] => [1,1] => 110 => [1,1,2] => 1
[1,1,1,1,1,1,1,1,0,0,0,0,1,0,0,0,0,0] => [4] => 10000 => [1,5] => 1
[1,1,1,1,1,1,1,1,0,0,0,1,0,0,0,0,0,0] => [3] => 1000 => [1,4] => 1
[1,1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0,0] => [2] => 100 => [1,3] => 1
[1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0] => [1] => 10 => [1,2] => 1
[1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0] => [] => => [1] => 1
>>> Load all 130 entries. <<<
[1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0] => [] => => [1] => 1
[1,1,1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0,0,0] => [1,1] => 110 => [1,1,2] => 1
[1,1,1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0,0,0] => [2] => 100 => [1,3] => 1
[1,1,1,1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0,0,0,0] => [1,1] => 110 => [1,1,2] => 1
[1,1,1,1,1,1,1,0,1,0,1,1,0,0,0,0,0,0,0,0] => [2,2,1] => 11010 => [1,1,2,2] => 1
[1,1,1,1,1,1,1,1,0,0,1,0,1,0,0,0,0,0,0,0] => [3,2] => 10100 => [1,2,3] => 1
[1,1,1,1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0,0,0,0] => [2] => 100 => [1,3] => 1
[1,1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0,0] => [1] => 10 => [1,2] => 1
[1,1,1,1,1,1,1,0,1,1,1,0,0,0,0,0,0,0,0,0] => [1,1,1] => 1110 => [1,1,1,2] => 1
[1,1,1,1,1,1,0,1,1,1,1,0,0,0,0,0,0,0,0,0] => [1,1,1,1] => 11110 => [1,1,1,1,2] => 1
[1,1,1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0,0,0] => [1] => 10 => [1,2] => 1
[1,1,1,1,1,1,1,1,1,0,0,0,0,1,0,0,0,0,0,0] => [4] => 10000 => [1,5] => 1
[1,1,1,1,1,1,1,1,1,0,0,0,1,0,0,0,0,0,0,0] => [3] => 1000 => [1,4] => 1
[1,1,1,1,1,1,1,1,0,0,1,1,0,0,0,0,0,0,0,0] => [2,2] => 1100 => [1,1,3] => 1
[1,1,1,1,1,1,1,1,0,0,0,1,1,0,0,0,0,0,0,0] => [3,3] => 11000 => [1,1,4] => 1
[1,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,0] => [] => => [1] => 1
[1,1,1,1,1,1,1,0,1,1,0,1,0,0,0,0,0,0,0,0] => [2,1,1] => 10110 => [1,2,1,2] => 1
[1,1,1,1,1,1,1,1,0,1,0,0,1,0,0,0,0,0,0,0] => [3,1] => 10010 => [1,3,2] => 1
[1,1,1,1,1,1,1,1,0,1,0,1,0,0,0,0,0,0,0,0] => [2,1] => 1010 => [1,2,2] => 1
[1,1,1,1,1,1,1,1,1,0,1,0,1,0,0,0,0,0,0,0,0,0] => [2,1] => 1010 => [1,2,2] => 1
[1,1,1,1,1,1,1,1,1,1,0,0,0,1,0,0,0,0,0,0,0,0] => [3] => 1000 => [1,4] => 1
[1,1,1,1,1,1,1,1,1,0,1,0,0,1,0,0,0,0,0,0,0,0] => [3,1] => 10010 => [1,3,2] => 1
[1,1,1,1,1,1,1,0,0,1,1,1,0,0,0,0,0,0,0,0] => [2,2,2] => 11100 => [1,1,1,3] => 1
[1,1,1,1,1,1,1,1,0,1,1,0,1,0,0,0,0,0,0,0,0,0] => [2,1,1] => 10110 => [1,2,1,2] => 1
[1,1,1,1,1,1,1,1,0,1,1,1,0,0,0,0,0,0,0,0,0,0] => [1,1,1] => 1110 => [1,1,1,2] => 1
[1,1,1,1,1,1,1,1,1,0,0,1,1,0,0,0,0,0,0,0,0,0] => [2,2] => 1100 => [1,1,3] => 1
[1,1,1,1,1,1,1,1,1,0,0,0,1,1,0,0,0,0,0,0,0,0] => [3,3] => 11000 => [1,1,4] => 1
[1,1,1,1,1,1,1,1,1,1,1,0,0,0,1,0,0,0,0,0,0,0,0,0] => [3] => 1000 => [1,4] => 1
[1,1,1,1,1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0,0,0,0,0] => [1,1] => 110 => [1,1,2] => 1
search for individual values
searching the database for the individual values of this statistic
/ search for generating function
searching the database for statistics with the same generating function
Description
The dominant dimension of the corresponding Comp-Nakayama algebra.
Map
to composition
Description
The composition corresponding to a binary word.
Prepending $1$ to a binary word $w$, the $i$-th part of the composition equals $1$ plus the number of zeros after the $i$-th $1$ in $w$.
This map is not surjective, since the empty composition does not have a preimage.
Map
to partition
Description
The cut-out partition of a Dyck path.
The partition $\lambda$ associated to a Dyck path is defined to be the complementary partition inside the staircase partition $(n-1,\ldots,2,1)$ when cutting out $D$ considered as a path from $(0,0)$ to $(n,n)$.
In other words, $\lambda_{i}$ is the number of down-steps before the $(n+1-i)$-th up-step of $D$.
This map is a bijection between Dyck paths of size $n$ and partitions inside the staircase partition $(n-1,\ldots,2,1)$.
Map
to binary word
Description
Return the partition as binary word, by traversing its shape from the first row to the last row, down steps as 1 and left steps as 0.