Identifier
Values
[1,0] => 10 => [1,2] => [2,1] => 1
[1,0,1,0] => 1010 => [1,2,2] => [2,2,1] => 2
[1,1,0,0] => 1100 => [1,1,3] => [3,1,1] => 3
[1,0,1,0,1,0] => 101010 => [1,2,2,2] => [2,2,2,1] => 3
[1,0,1,1,0,0] => 101100 => [1,2,1,3] => [3,2,1,1] => 4
[1,1,0,0,1,0] => 110010 => [1,1,3,2] => [3,2,1,1] => 4
[1,1,0,1,0,0] => 110100 => [1,1,2,3] => [3,2,1,1] => 4
[1,1,1,0,0,0] => 111000 => [1,1,1,4] => [4,1,1,1] => 6
[1,0,1,0,1,0,1,0] => 10101010 => [1,2,2,2,2] => [2,2,2,2,1] => 4
[1,0,1,0,1,1,0,0] => 10101100 => [1,2,2,1,3] => [3,2,2,1,1] => 5
[1,0,1,1,0,0,1,0] => 10110010 => [1,2,1,3,2] => [3,2,2,1,1] => 5
[1,0,1,1,0,1,0,0] => 10110100 => [1,2,1,2,3] => [3,2,2,1,1] => 5
[1,0,1,1,1,0,0,0] => 10111000 => [1,2,1,1,4] => [4,2,1,1,1] => 7
[1,1,0,0,1,0,1,0] => 11001010 => [1,1,3,2,2] => [3,2,2,1,1] => 5
[1,1,0,0,1,1,0,0] => 11001100 => [1,1,3,1,3] => [3,3,1,1,1] => 6
[1,1,0,1,0,0,1,0] => 11010010 => [1,1,2,3,2] => [3,2,2,1,1] => 5
[1,1,0,1,0,1,0,0] => 11010100 => [1,1,2,2,3] => [3,2,2,1,1] => 5
[1,1,0,1,1,0,0,0] => 11011000 => [1,1,2,1,4] => [4,2,1,1,1] => 7
[1,1,1,0,0,0,1,0] => 11100010 => [1,1,1,4,2] => [4,2,1,1,1] => 7
[1,1,1,0,0,1,0,0] => 11100100 => [1,1,1,3,3] => [3,3,1,1,1] => 6
[1,1,1,0,1,0,0,0] => 11101000 => [1,1,1,2,4] => [4,2,1,1,1] => 7
[1,1,1,1,0,0,0,0] => 11110000 => [1,1,1,1,5] => [5,1,1,1,1] => 10
[1,0,1,0,1,0,1,0,1,0] => 1010101010 => [1,2,2,2,2,2] => [2,2,2,2,2,1] => 5
[1,0,1,0,1,0,1,1,0,0] => 1010101100 => [1,2,2,2,1,3] => [3,2,2,2,1,1] => 6
[1,0,1,0,1,1,0,0,1,0] => 1010110010 => [1,2,2,1,3,2] => [3,2,2,2,1,1] => 6
[1,0,1,0,1,1,0,1,0,0] => 1010110100 => [1,2,2,1,2,3] => [3,2,2,2,1,1] => 6
[1,0,1,0,1,1,1,0,0,0] => 1010111000 => [1,2,2,1,1,4] => [4,2,2,1,1,1] => 8
[1,0,1,1,0,0,1,0,1,0] => 1011001010 => [1,2,1,3,2,2] => [3,2,2,2,1,1] => 6
[1,0,1,1,0,0,1,1,0,0] => 1011001100 => [1,2,1,3,1,3] => [3,3,2,1,1,1] => 7
[1,0,1,1,0,1,0,0,1,0] => 1011010010 => [1,2,1,2,3,2] => [3,2,2,2,1,1] => 6
[1,0,1,1,0,1,0,1,0,0] => 1011010100 => [1,2,1,2,2,3] => [3,2,2,2,1,1] => 6
[1,0,1,1,0,1,1,0,0,0] => 1011011000 => [1,2,1,2,1,4] => [4,2,2,1,1,1] => 8
[1,0,1,1,1,0,0,0,1,0] => 1011100010 => [1,2,1,1,4,2] => [4,2,2,1,1,1] => 8
[1,0,1,1,1,0,0,1,0,0] => 1011100100 => [1,2,1,1,3,3] => [3,3,2,1,1,1] => 7
[1,0,1,1,1,0,1,0,0,0] => 1011101000 => [1,2,1,1,2,4] => [4,2,2,1,1,1] => 8
[1,0,1,1,1,1,0,0,0,0] => 1011110000 => [1,2,1,1,1,5] => [5,2,1,1,1,1] => 11
[1,1,0,0,1,0,1,0,1,0] => 1100101010 => [1,1,3,2,2,2] => [3,2,2,2,1,1] => 6
[1,1,0,0,1,0,1,1,0,0] => 1100101100 => [1,1,3,2,1,3] => [3,3,2,1,1,1] => 7
[1,1,0,0,1,1,0,0,1,0] => 1100110010 => [1,1,3,1,3,2] => [3,3,2,1,1,1] => 7
[1,1,0,0,1,1,0,1,0,0] => 1100110100 => [1,1,3,1,2,3] => [3,3,2,1,1,1] => 7
[1,1,0,0,1,1,1,0,0,0] => 1100111000 => [1,1,3,1,1,4] => [4,3,1,1,1,1] => 9
[1,1,0,1,0,0,1,0,1,0] => 1101001010 => [1,1,2,3,2,2] => [3,2,2,2,1,1] => 6
[1,1,0,1,0,0,1,1,0,0] => 1101001100 => [1,1,2,3,1,3] => [3,3,2,1,1,1] => 7
[1,1,0,1,0,1,0,0,1,0] => 1101010010 => [1,1,2,2,3,2] => [3,2,2,2,1,1] => 6
[1,1,0,1,0,1,0,1,0,0] => 1101010100 => [1,1,2,2,2,3] => [3,2,2,2,1,1] => 6
[1,1,0,1,0,1,1,0,0,0] => 1101011000 => [1,1,2,2,1,4] => [4,2,2,1,1,1] => 8
[1,1,0,1,1,0,0,0,1,0] => 1101100010 => [1,1,2,1,4,2] => [4,2,2,1,1,1] => 8
[1,1,0,1,1,0,0,1,0,0] => 1101100100 => [1,1,2,1,3,3] => [3,3,2,1,1,1] => 7
[1,1,0,1,1,0,1,0,0,0] => 1101101000 => [1,1,2,1,2,4] => [4,2,2,1,1,1] => 8
[1,1,0,1,1,1,0,0,0,0] => 1101110000 => [1,1,2,1,1,5] => [5,2,1,1,1,1] => 11
[1,1,1,0,0,0,1,0,1,0] => 1110001010 => [1,1,1,4,2,2] => [4,2,2,1,1,1] => 8
[1,1,1,0,0,0,1,1,0,0] => 1110001100 => [1,1,1,4,1,3] => [4,3,1,1,1,1] => 9
[1,1,1,0,0,1,0,0,1,0] => 1110010010 => [1,1,1,3,3,2] => [3,3,2,1,1,1] => 7
[1,1,1,0,0,1,0,1,0,0] => 1110010100 => [1,1,1,3,2,3] => [3,3,2,1,1,1] => 7
[1,1,1,0,0,1,1,0,0,0] => 1110011000 => [1,1,1,3,1,4] => [4,3,1,1,1,1] => 9
[1,1,1,0,1,0,0,0,1,0] => 1110100010 => [1,1,1,2,4,2] => [4,2,2,1,1,1] => 8
[1,1,1,0,1,0,0,1,0,0] => 1110100100 => [1,1,1,2,3,3] => [3,3,2,1,1,1] => 7
[1,1,1,0,1,0,1,0,0,0] => 1110101000 => [1,1,1,2,2,4] => [4,2,2,1,1,1] => 8
[1,1,1,0,1,1,0,0,0,0] => 1110110000 => [1,1,1,2,1,5] => [5,2,1,1,1,1] => 11
[1,1,1,1,0,0,0,0,1,0] => 1111000010 => [1,1,1,1,5,2] => [5,2,1,1,1,1] => 11
[1,1,1,1,0,0,0,1,0,0] => 1111000100 => [1,1,1,1,4,3] => [4,3,1,1,1,1] => 9
[1,1,1,1,0,0,1,0,0,0] => 1111001000 => [1,1,1,1,3,4] => [4,3,1,1,1,1] => 9
[1,1,1,1,0,1,0,0,0,0] => 1111010000 => [1,1,1,1,2,5] => [5,2,1,1,1,1] => 11
[1,1,1,1,1,0,0,0,0,0] => 1111100000 => [1,1,1,1,1,6] => [6,1,1,1,1,1] => 15
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Description
The number of ways to select a row of a Ferrers shape and two cells in this row. Equivalently, if $\lambda = (\lambda_0\geq\lambda_1 \geq \dots\geq\lambda_m)$ is an integer partition, then the statistic is
$$\frac{1}{2} \sum_{i=0}^m \lambda_i(\lambda_i -1).$$
Map
to partition
Description
Sends a composition to the partition obtained by sorting the entries.
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 binary word
Description
Return the Dyck word as binary word.