Your data matches 1 statistic following compositions of up to 3 maps.
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Matching statistic: St001263
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Mapping
St001263: Integer compositions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => 0
[1,1] => 0
[2] => 0
[1,1,1] => 1
[1,2] => 0
[2,1] => 0
[3] => 1
[1,1,1,1] => 1
[1,1,2] => 0
[1,2,1] => 1
[1,3] => 0
[2,1,1] => 0
[2,2] => 1
[3,1] => 0
[4] => 1
[1,1,1,1,1] => 2
[1,1,1,2] => 1
[1,1,2,1] => 1
[1,1,3] => 1
[1,2,1,1] => 1
[1,2,2] => 0
[1,3,1] => 2
[1,4] => 0
[2,1,1,1] => 1
[2,1,2] => 2
[2,2,1] => 0
[2,3] => 0
[3,1,1] => 1
[3,2] => 0
[4,1] => 0
[5] => 2
[1,1,1,1,1,1] => 2
[1,1,1,1,2] => 1
[1,1,1,2,1] => 1
[1,1,1,3] => 1
[1,1,2,1,1] => 2
[1,1,2,2] => 1
[1,1,3,1] => 1
[1,1,4] => 0
[1,2,1,1,1] => 1
[1,2,1,2] => 0
[1,2,2,1] => 2
[1,2,3] => 0
[1,3,1,1] => 1
[1,3,2] => 0
[1,4,1] => 2
[1,5] => 0
[2,1,1,1,1] => 1
[2,1,1,2] => 2
[2,1,2,1] => 0
Description
The index of the maximal parabolic seaweed algebra associated with the composition. Let $a_1,\dots,a_m$ and $b_1,\dots,b_t$ be a pair of compositions of $n$. The meander associated to this pair is obtained as follows: * place $n$ dots on a horizontal line * subdivide the dots into $m$ blocks of sizes $a_1, a_2,\dots$ * within each block, connect the first and the last dot, the second and the next to last, and so on, with an arc above the line * subdivide the dots into $t$ blocks of sizes $b_1, b_2,\dots$ * within each block, connect the first and the last dot, the second and the next to last, and so on, with an arc below the line By [1, thm.5.1], the index of the seaweed algebra associated to the pair of compositions is $$ \operatorname{ind}\displaystyle\frac{b_1|b_2|...|b_t}{a_1|a_2|...|a_m} = 2C+P-1, $$ where $C$ is the number of cycles (of length at least $2$) and P is the number of paths in the meander. This statistic is $\operatorname{ind}\displaystyle\frac{b_1|b_2|...|b_t}{n}$.