Identifier
Values
[[1]] => [1] => [1,0] => [2,1] => 2
[[1,2]] => [2] => [1,1,0,0] => [2,3,1] => 2
[[1],[2]] => [1,1] => [1,0,1,0] => [3,1,2] => 2
[[1,2,3]] => [3] => [1,1,1,0,0,0] => [2,3,4,1] => 3
[[1,3],[2]] => [1,2] => [1,0,1,1,0,0] => [3,1,4,2] => 2
[[1,2],[3]] => [2,1] => [1,1,0,0,1,0] => [2,4,1,3] => 2
[[1],[2],[3]] => [1,1,1] => [1,0,1,0,1,0] => [4,1,2,3] => 3
[[1,2,3,4]] => [4] => [1,1,1,1,0,0,0,0] => [2,3,4,5,1] => 4
[[1,3,4],[2]] => [1,3] => [1,0,1,1,1,0,0,0] => [3,1,4,5,2] => 3
[[1,2,4],[3]] => [2,2] => [1,1,0,0,1,1,0,0] => [2,4,1,5,3] => 2
[[1,2,3],[4]] => [3,1] => [1,1,1,0,0,0,1,0] => [2,3,5,1,4] => 3
[[1,3],[2,4]] => [1,2,1] => [1,0,1,1,0,0,1,0] => [3,1,5,2,4] => 2
[[1,2],[3,4]] => [2,2] => [1,1,0,0,1,1,0,0] => [2,4,1,5,3] => 2
[[1,4],[2],[3]] => [1,1,2] => [1,0,1,0,1,1,0,0] => [4,1,2,5,3] => 3
[[1,3],[2],[4]] => [1,2,1] => [1,0,1,1,0,0,1,0] => [3,1,5,2,4] => 2
[[1,2],[3],[4]] => [2,1,1] => [1,1,0,0,1,0,1,0] => [2,5,1,3,4] => 3
[[1],[2],[3],[4]] => [1,1,1,1] => [1,0,1,0,1,0,1,0] => [5,1,2,3,4] => 4
[[1,2,3,4,5]] => [5] => [1,1,1,1,1,0,0,0,0,0] => [2,3,4,5,6,1] => 5
[[1,3,4,5],[2]] => [1,4] => [1,0,1,1,1,1,0,0,0,0] => [3,1,4,5,6,2] => 4
[[1,2,4,5],[3]] => [2,3] => [1,1,0,0,1,1,1,0,0,0] => [2,4,1,5,6,3] => 3
[[1,2,3,5],[4]] => [3,2] => [1,1,1,0,0,0,1,1,0,0] => [2,3,5,1,6,4] => 3
[[1,2,3,4],[5]] => [4,1] => [1,1,1,1,0,0,0,0,1,0] => [2,3,4,6,1,5] => 4
[[1,3,5],[2,4]] => [1,2,2] => [1,0,1,1,0,0,1,1,0,0] => [3,1,5,2,6,4] => 2
[[1,2,5],[3,4]] => [2,3] => [1,1,0,0,1,1,1,0,0,0] => [2,4,1,5,6,3] => 3
[[1,3,4],[2,5]] => [1,3,1] => [1,0,1,1,1,0,0,0,1,0] => [3,1,4,6,2,5] => 3
[[1,2,4],[3,5]] => [2,2,1] => [1,1,0,0,1,1,0,0,1,0] => [2,4,1,6,3,5] => 3
[[1,2,3],[4,5]] => [3,2] => [1,1,1,0,0,0,1,1,0,0] => [2,3,5,1,6,4] => 3
[[1,4,5],[2],[3]] => [1,1,3] => [1,0,1,0,1,1,1,0,0,0] => [4,1,2,5,6,3] => 3
[[1,3,5],[2],[4]] => [1,2,2] => [1,0,1,1,0,0,1,1,0,0] => [3,1,5,2,6,4] => 2
[[1,2,5],[3],[4]] => [2,1,2] => [1,1,0,0,1,0,1,1,0,0] => [2,5,1,3,6,4] => 3
[[1,3,4],[2],[5]] => [1,3,1] => [1,0,1,1,1,0,0,0,1,0] => [3,1,4,6,2,5] => 3
[[1,2,4],[3],[5]] => [2,2,1] => [1,1,0,0,1,1,0,0,1,0] => [2,4,1,6,3,5] => 3
[[1,2,3],[4],[5]] => [3,1,1] => [1,1,1,0,0,0,1,0,1,0] => [2,3,6,1,4,5] => 4
[[1,4],[2,5],[3]] => [1,1,2,1] => [1,0,1,0,1,1,0,0,1,0] => [4,1,2,6,3,5] => 3
[[1,3],[2,5],[4]] => [1,2,2] => [1,0,1,1,0,0,1,1,0,0] => [3,1,5,2,6,4] => 2
[[1,2],[3,5],[4]] => [2,1,2] => [1,1,0,0,1,0,1,1,0,0] => [2,5,1,3,6,4] => 3
[[1,3],[2,4],[5]] => [1,2,1,1] => [1,0,1,1,0,0,1,0,1,0] => [3,1,6,2,4,5] => 3
[[1,2],[3,4],[5]] => [2,2,1] => [1,1,0,0,1,1,0,0,1,0] => [2,4,1,6,3,5] => 3
[[1,5],[2],[3],[4]] => [1,1,1,2] => [1,0,1,0,1,0,1,1,0,0] => [5,1,2,3,6,4] => 4
[[1,4],[2],[3],[5]] => [1,1,2,1] => [1,0,1,0,1,1,0,0,1,0] => [4,1,2,6,3,5] => 3
[[1,3],[2],[4],[5]] => [1,2,1,1] => [1,0,1,1,0,0,1,0,1,0] => [3,1,6,2,4,5] => 3
[[1,2],[3],[4],[5]] => [2,1,1,1] => [1,1,0,0,1,0,1,0,1,0] => [2,6,1,3,4,5] => 4
[[1],[2],[3],[4],[5]] => [1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0] => [6,1,2,3,4,5] => 5
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Description
The length of the longest arithmetic progression in a permutation.
For a permutation $\pi$ of length $n$, this is the biggest $k$ such that there exist $1 \leq i_1 < \dots < i_k \leq n$ with
$$\pi(i_2) - \pi(i_1) = \pi(i_3) - \pi(i_2) = \dots = \pi(i_k) - \pi(i_{k-1}).$$
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
Ringel
Description
The Ringel permutation of the LNakayama algebra corresponding to a Dyck path.
Map
bounce path
Description
The bounce path determined by an integer composition.