Identifier
Values
[1] => [[1]] => [1] => [1] => 0
[2] => [[1,2]] => [1,2] => [1,2] => 1
[1,1] => [[1],[2]] => [2,1] => [2,1] => 0
[3] => [[1,2,3]] => [1,2,3] => [1,3,2] => 0
[2,1] => [[1,2],[3]] => [3,1,2] => [3,1,2] => 1
[1,1,1] => [[1],[2],[3]] => [3,2,1] => [3,2,1] => 0
[4] => [[1,2,3,4]] => [1,2,3,4] => [1,4,3,2] => 0
[3,1] => [[1,2,3],[4]] => [4,1,2,3] => [4,1,3,2] => 0
[2,2] => [[1,2],[3,4]] => [3,4,1,2] => [3,4,1,2] => 2
[2,1,1] => [[1,2],[3],[4]] => [4,3,1,2] => [4,3,1,2] => 1
[1,1,1,1] => [[1],[2],[3],[4]] => [4,3,2,1] => [4,3,2,1] => 0
[5] => [[1,2,3,4,5]] => [1,2,3,4,5] => [1,5,4,3,2] => 0
[4,1] => [[1,2,3,4],[5]] => [5,1,2,3,4] => [5,1,4,3,2] => 0
[3,2] => [[1,2,3],[4,5]] => [4,5,1,2,3] => [4,5,1,3,2] => 1
[3,1,1] => [[1,2,3],[4],[5]] => [5,4,1,2,3] => [5,4,1,3,2] => 0
[2,2,1] => [[1,2],[3,4],[5]] => [5,3,4,1,2] => [5,3,4,1,2] => 2
[2,1,1,1] => [[1,2],[3],[4],[5]] => [5,4,3,1,2] => [5,4,3,1,2] => 1
[1,1,1,1,1] => [[1],[2],[3],[4],[5]] => [5,4,3,2,1] => [5,4,3,2,1] => 0
[6] => [[1,2,3,4,5,6]] => [1,2,3,4,5,6] => [1,6,5,4,3,2] => 0
[5,1] => [[1,2,3,4,5],[6]] => [6,1,2,3,4,5] => [6,1,5,4,3,2] => 0
[4,2] => [[1,2,3,4],[5,6]] => [5,6,1,2,3,4] => [5,6,1,4,3,2] => 1
[4,1,1] => [[1,2,3,4],[5],[6]] => [6,5,1,2,3,4] => [6,5,1,4,3,2] => 0
[3,3] => [[1,2,3],[4,5,6]] => [4,5,6,1,2,3] => [4,6,5,1,3,2] => 0
[3,2,1] => [[1,2,3],[4,5],[6]] => [6,4,5,1,2,3] => [6,4,5,1,3,2] => 1
[3,1,1,1] => [[1,2,3],[4],[5],[6]] => [6,5,4,1,2,3] => [6,5,4,1,3,2] => 0
[2,2,2] => [[1,2],[3,4],[5,6]] => [5,6,3,4,1,2] => [5,6,3,4,1,2] => 3
[2,2,1,1] => [[1,2],[3,4],[5],[6]] => [6,5,3,4,1,2] => [6,5,3,4,1,2] => 2
[2,1,1,1,1] => [[1,2],[3],[4],[5],[6]] => [6,5,4,3,1,2] => [6,5,4,3,1,2] => 1
[1,1,1,1,1,1] => [[1],[2],[3],[4],[5],[6]] => [6,5,4,3,2,1] => [6,5,4,3,2,1] => 0
[7] => [[1,2,3,4,5,6,7]] => [1,2,3,4,5,6,7] => [1,7,6,5,4,3,2] => 0
[2,2,2,1] => [[1,2],[3,4],[5,6],[7]] => [7,5,6,3,4,1,2] => [7,5,6,3,4,1,2] => 3
[2,2,1,1,1] => [[1,2],[3,4],[5],[6],[7]] => [7,6,5,3,4,1,2] => [7,6,5,3,4,1,2] => 2
[2,1,1,1,1,1] => [[1,2],[3],[4],[5],[6],[7]] => [7,6,5,4,3,1,2] => [7,6,5,4,3,1,2] => 1
[1,1,1,1,1,1,1] => [[1],[2],[3],[4],[5],[6],[7]] => [7,6,5,4,3,2,1] => [7,6,5,4,3,2,1] => 0
[8] => [[1,2,3,4,5,6,7,8]] => [1,2,3,4,5,6,7,8] => [1,8,7,6,5,4,3,2] => 0
[4,4] => [[1,2,3,4],[5,6,7,8]] => [5,6,7,8,1,2,3,4] => [5,8,7,6,1,4,3,2] => 0
[2,2,2,2] => [[1,2],[3,4],[5,6],[7,8]] => [7,8,5,6,3,4,1,2] => [7,8,5,6,3,4,1,2] => 4
[2,2,2,1,1] => [[1,2],[3,4],[5,6],[7],[8]] => [8,7,5,6,3,4,1,2] => [8,7,5,6,3,4,1,2] => 3
[2,2,1,1,1,1] => [[1,2],[3,4],[5],[6],[7],[8]] => [8,7,6,5,3,4,1,2] => [8,7,6,5,3,4,1,2] => 2
[2,1,1,1,1,1,1] => [[1,2],[3],[4],[5],[6],[7],[8]] => [8,7,6,5,4,3,1,2] => [8,7,6,5,4,3,1,2] => 1
[1,1,1,1,1,1,1,1] => [[1],[2],[3],[4],[5],[6],[7],[8]] => [8,7,6,5,4,3,2,1] => [8,7,6,5,4,3,2,1] => 0
[9] => [[1,2,3,4,5,6,7,8,9]] => [1,2,3,4,5,6,7,8,9] => [1,9,8,7,6,5,4,3,2] => 0
[2,2,2,2,1] => [[1,2],[3,4],[5,6],[7,8],[9]] => [9,7,8,5,6,3,4,1,2] => [9,7,8,5,6,3,4,1,2] => 4
[2,2,2,1,1,1] => [[1,2],[3,4],[5,6],[7],[8],[9]] => [9,8,7,5,6,3,4,1,2] => [9,8,7,5,6,3,4,1,2] => 3
[2,2,1,1,1,1,1] => [[1,2],[3,4],[5],[6],[7],[8],[9]] => [9,8,7,6,5,3,4,1,2] => [9,8,7,6,5,3,4,1,2] => 2
[2,1,1,1,1,1,1,1] => [[1,2],[3],[4],[5],[6],[7],[8],[9]] => [9,8,7,6,5,4,3,1,2] => [9,8,7,6,5,4,3,1,2] => 1
[1,1,1,1,1,1,1,1,1] => [[1],[2],[3],[4],[5],[6],[7],[8],[9]] => [9,8,7,6,5,4,3,2,1] => [9,8,7,6,5,4,3,2,1] => 0
[10] => [[1,2,3,4,5,6,7,8,9,10]] => [1,2,3,4,5,6,7,8,9,10] => [1,10,9,8,7,6,5,4,3,2] => 0
[2,2,2,2,2] => [[1,2],[3,4],[5,6],[7,8],[9,10]] => [9,10,7,8,5,6,3,4,1,2] => [9,10,7,8,5,6,3,4,1,2] => 5
[2,2,2,2,1,1] => [[1,2],[3,4],[5,6],[7,8],[9],[10]] => [10,9,7,8,5,6,3,4,1,2] => [10,9,7,8,5,6,3,4,1,2] => 4
[2,2,2,1,1,1,1] => [[1,2],[3,4],[5,6],[7],[8],[9],[10]] => [10,9,8,7,5,6,3,4,1,2] => [10,9,8,7,5,6,3,4,1,2] => 3
[2,2,1,1,1,1,1,1] => [[1,2],[3,4],[5],[6],[7],[8],[9],[10]] => [10,9,8,7,6,5,3,4,1,2] => [10,9,8,7,6,5,3,4,1,2] => 2
[2,1,1,1,1,1,1,1,1] => [[1,2],[3],[4],[5],[6],[7],[8],[9],[10]] => [10,9,8,7,6,5,4,3,1,2] => [10,9,8,7,6,5,4,3,1,2] => 1
[1,1,1,1,1,1,1,1,1,1] => [[1],[2],[3],[4],[5],[6],[7],[8],[9],[10]] => [10,9,8,7,6,5,4,3,2,1] => [10,9,8,7,6,5,4,3,2,1] => 0
[1,1,1,1,1,1,1,1,1,1,1,1] => [[1],[2],[3],[4],[5],[6],[7],[8],[9],[10],[11],[12]] => [12,11,10,9,8,7,6,5,4,3,2,1] => [12,11,10,9,8,7,6,5,4,3,2,1] => 0
[] => [] => [] => [] => 0
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Description
The number of rafts of a permutation.
Let $\pi$ be a permutation of length $n$. A small ascent of $\pi$ is an index $i$ such that $\pi(i+1)= \pi(i)+1$, see St000441The number of successions of a permutation., and a raft of $\pi$ is a non-empty maximal sequence of consecutive small ascents.
Map
Simion-Schmidt map
Description
The Simion-Schmidt map sends any permutation to a $123$-avoiding permutation.
Details can be found in [1].
In particular, this is a bijection between $132$-avoiding permutations and $123$-avoiding permutations, see [1, Proposition 19].
Map
initial tableau
Description
Sends an integer partition to the standard tableau obtained by filling the numbers $1$ through $n$ row by row.
Map
reading word permutation
Description
Return the permutation obtained by reading the entries of the tableau row by row, starting with the bottom-most row in English notation.