Identifier
-
Mp00081:
Standard tableaux
—reading word permutation⟶
Permutations
Mp00223: Permutations —runsort⟶ Permutations
St001076: Permutations ⟶ ℤ
Values
[[1,2]] => [1,2] => [1,2] => 0
[[1],[2]] => [2,1] => [1,2] => 0
[[1,2,3]] => [1,2,3] => [1,2,3] => 0
[[1,3],[2]] => [2,1,3] => [1,3,2] => 1
[[1,2],[3]] => [3,1,2] => [1,2,3] => 0
[[1],[2],[3]] => [3,2,1] => [1,2,3] => 0
[[1,2,3,4]] => [1,2,3,4] => [1,2,3,4] => 0
[[1,3,4],[2]] => [2,1,3,4] => [1,3,4,2] => 2
[[1,2,4],[3]] => [3,1,2,4] => [1,2,4,3] => 1
[[1,2,3],[4]] => [4,1,2,3] => [1,2,3,4] => 0
[[1,3],[2,4]] => [2,4,1,3] => [1,3,2,4] => 1
[[1,2],[3,4]] => [3,4,1,2] => [1,2,3,4] => 0
[[1,4],[2],[3]] => [3,2,1,4] => [1,4,2,3] => 2
[[1,3],[2],[4]] => [4,2,1,3] => [1,3,2,4] => 1
[[1,2],[3],[4]] => [4,3,1,2] => [1,2,3,4] => 0
[[1],[2],[3],[4]] => [4,3,2,1] => [1,2,3,4] => 0
[[1,2,3,4,5]] => [1,2,3,4,5] => [1,2,3,4,5] => 0
[[1,3,4,5],[2]] => [2,1,3,4,5] => [1,3,4,5,2] => 3
[[1,2,4,5],[3]] => [3,1,2,4,5] => [1,2,4,5,3] => 2
[[1,2,3,5],[4]] => [4,1,2,3,5] => [1,2,3,5,4] => 1
[[1,2,3,4],[5]] => [5,1,2,3,4] => [1,2,3,4,5] => 0
[[1,3,5],[2,4]] => [2,4,1,3,5] => [1,3,5,2,4] => 3
[[1,2,5],[3,4]] => [3,4,1,2,5] => [1,2,5,3,4] => 2
[[1,3,4],[2,5]] => [2,5,1,3,4] => [1,3,4,2,5] => 2
[[1,2,4],[3,5]] => [3,5,1,2,4] => [1,2,4,3,5] => 1
[[1,2,3],[4,5]] => [4,5,1,2,3] => [1,2,3,4,5] => 0
[[1,4,5],[2],[3]] => [3,2,1,4,5] => [1,4,5,2,3] => 4
[[1,3,5],[2],[4]] => [4,2,1,3,5] => [1,3,5,2,4] => 3
[[1,2,5],[3],[4]] => [4,3,1,2,5] => [1,2,5,3,4] => 2
[[1,3,4],[2],[5]] => [5,2,1,3,4] => [1,3,4,2,5] => 2
[[1,2,4],[3],[5]] => [5,3,1,2,4] => [1,2,4,3,5] => 1
[[1,2,3],[4],[5]] => [5,4,1,2,3] => [1,2,3,4,5] => 0
[[1,4],[2,5],[3]] => [3,2,5,1,4] => [1,4,2,5,3] => 3
[[1,3],[2,5],[4]] => [4,2,5,1,3] => [1,3,2,5,4] => 2
[[1,2],[3,5],[4]] => [4,3,5,1,2] => [1,2,3,5,4] => 1
[[1,3],[2,4],[5]] => [5,2,4,1,3] => [1,3,2,4,5] => 1
[[1,2],[3,4],[5]] => [5,3,4,1,2] => [1,2,3,4,5] => 0
[[1,5],[2],[3],[4]] => [4,3,2,1,5] => [1,5,2,3,4] => 3
[[1,4],[2],[3],[5]] => [5,3,2,1,4] => [1,4,2,3,5] => 2
[[1,3],[2],[4],[5]] => [5,4,2,1,3] => [1,3,2,4,5] => 1
[[1,2],[3],[4],[5]] => [5,4,3,1,2] => [1,2,3,4,5] => 0
[[1],[2],[3],[4],[5]] => [5,4,3,2,1] => [1,2,3,4,5] => 0
[[1,2,3,4,5,6]] => [1,2,3,4,5,6] => [1,2,3,4,5,6] => 0
[[1,3,4,5,6],[2]] => [2,1,3,4,5,6] => [1,3,4,5,6,2] => 4
[[1,2,4,5,6],[3]] => [3,1,2,4,5,6] => [1,2,4,5,6,3] => 3
[[1,2,3,5,6],[4]] => [4,1,2,3,5,6] => [1,2,3,5,6,4] => 2
[[1,2,3,4,6],[5]] => [5,1,2,3,4,6] => [1,2,3,4,6,5] => 1
[[1,2,3,4,5],[6]] => [6,1,2,3,4,5] => [1,2,3,4,5,6] => 0
[[1,3,5,6],[2,4]] => [2,4,1,3,5,6] => [1,3,5,6,2,4] => 5
[[1,2,5,6],[3,4]] => [3,4,1,2,5,6] => [1,2,5,6,3,4] => 4
[[1,3,4,6],[2,5]] => [2,5,1,3,4,6] => [1,3,4,6,2,5] => 4
[[1,2,4,6],[3,5]] => [3,5,1,2,4,6] => [1,2,4,6,3,5] => 3
[[1,2,3,6],[4,5]] => [4,5,1,2,3,6] => [1,2,3,6,4,5] => 2
[[1,3,4,5],[2,6]] => [2,6,1,3,4,5] => [1,3,4,5,2,6] => 3
[[1,2,4,5],[3,6]] => [3,6,1,2,4,5] => [1,2,4,5,3,6] => 2
[[1,2,3,5],[4,6]] => [4,6,1,2,3,5] => [1,2,3,5,4,6] => 1
[[1,2,3,4],[5,6]] => [5,6,1,2,3,4] => [1,2,3,4,5,6] => 0
[[1,4,5,6],[2],[3]] => [3,2,1,4,5,6] => [1,4,5,6,2,3] => 6
[[1,3,5,6],[2],[4]] => [4,2,1,3,5,6] => [1,3,5,6,2,4] => 5
[[1,2,5,6],[3],[4]] => [4,3,1,2,5,6] => [1,2,5,6,3,4] => 4
[[1,3,4,6],[2],[5]] => [5,2,1,3,4,6] => [1,3,4,6,2,5] => 4
[[1,2,4,6],[3],[5]] => [5,3,1,2,4,6] => [1,2,4,6,3,5] => 3
[[1,2,3,6],[4],[5]] => [5,4,1,2,3,6] => [1,2,3,6,4,5] => 2
[[1,3,4,5],[2],[6]] => [6,2,1,3,4,5] => [1,3,4,5,2,6] => 3
[[1,2,4,5],[3],[6]] => [6,3,1,2,4,5] => [1,2,4,5,3,6] => 2
[[1,2,3,5],[4],[6]] => [6,4,1,2,3,5] => [1,2,3,5,4,6] => 1
[[1,2,3,4],[5],[6]] => [6,5,1,2,3,4] => [1,2,3,4,5,6] => 0
[[1,3,5],[2,4,6]] => [2,4,6,1,3,5] => [1,3,5,2,4,6] => 3
[[1,2,5],[3,4,6]] => [3,4,6,1,2,5] => [1,2,5,3,4,6] => 2
[[1,3,4],[2,5,6]] => [2,5,6,1,3,4] => [1,3,4,2,5,6] => 2
[[1,2,4],[3,5,6]] => [3,5,6,1,2,4] => [1,2,4,3,5,6] => 1
[[1,2,3],[4,5,6]] => [4,5,6,1,2,3] => [1,2,3,4,5,6] => 0
[[1,4,6],[2,5],[3]] => [3,2,5,1,4,6] => [1,4,6,2,5,3] => 6
[[1,3,6],[2,5],[4]] => [4,2,5,1,3,6] => [1,3,6,2,5,4] => 5
[[1,2,6],[3,5],[4]] => [4,3,5,1,2,6] => [1,2,6,3,5,4] => 4
[[1,3,6],[2,4],[5]] => [5,2,4,1,3,6] => [1,3,6,2,4,5] => 4
[[1,2,6],[3,4],[5]] => [5,3,4,1,2,6] => [1,2,6,3,4,5] => 3
[[1,4,5],[2,6],[3]] => [3,2,6,1,4,5] => [1,4,5,2,6,3] => 5
[[1,3,5],[2,6],[4]] => [4,2,6,1,3,5] => [1,3,5,2,6,4] => 4
[[1,2,5],[3,6],[4]] => [4,3,6,1,2,5] => [1,2,5,3,6,4] => 3
[[1,3,4],[2,6],[5]] => [5,2,6,1,3,4] => [1,3,4,2,6,5] => 3
[[1,2,4],[3,6],[5]] => [5,3,6,1,2,4] => [1,2,4,3,6,5] => 2
[[1,2,3],[4,6],[5]] => [5,4,6,1,2,3] => [1,2,3,4,6,5] => 1
[[1,3,5],[2,4],[6]] => [6,2,4,1,3,5] => [1,3,5,2,4,6] => 3
[[1,2,5],[3,4],[6]] => [6,3,4,1,2,5] => [1,2,5,3,4,6] => 2
[[1,3,4],[2,5],[6]] => [6,2,5,1,3,4] => [1,3,4,2,5,6] => 2
[[1,2,4],[3,5],[6]] => [6,3,5,1,2,4] => [1,2,4,3,5,6] => 1
[[1,2,3],[4,5],[6]] => [6,4,5,1,2,3] => [1,2,3,4,5,6] => 0
[[1,5,6],[2],[3],[4]] => [4,3,2,1,5,6] => [1,5,6,2,3,4] => 6
[[1,4,6],[2],[3],[5]] => [5,3,2,1,4,6] => [1,4,6,2,3,5] => 5
[[1,3,6],[2],[4],[5]] => [5,4,2,1,3,6] => [1,3,6,2,4,5] => 4
[[1,2,6],[3],[4],[5]] => [5,4,3,1,2,6] => [1,2,6,3,4,5] => 3
[[1,4,5],[2],[3],[6]] => [6,3,2,1,4,5] => [1,4,5,2,3,6] => 4
[[1,3,5],[2],[4],[6]] => [6,4,2,1,3,5] => [1,3,5,2,4,6] => 3
[[1,2,5],[3],[4],[6]] => [6,4,3,1,2,5] => [1,2,5,3,4,6] => 2
[[1,3,4],[2],[5],[6]] => [6,5,2,1,3,4] => [1,3,4,2,5,6] => 2
[[1,2,4],[3],[5],[6]] => [6,5,3,1,2,4] => [1,2,4,3,5,6] => 1
[[1,2,3],[4],[5],[6]] => [6,5,4,1,2,3] => [1,2,3,4,5,6] => 0
[[1,4],[2,5],[3,6]] => [3,6,2,5,1,4] => [1,4,2,5,3,6] => 3
[[1,3],[2,5],[4,6]] => [4,6,2,5,1,3] => [1,3,2,5,4,6] => 2
[[1,2],[3,5],[4,6]] => [4,6,3,5,1,2] => [1,2,3,5,4,6] => 1
>>> Load all 331 entries. <<<
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searching the database for the individual values of this statistic
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searching the database for statistics with the same generating function
Description
The minimal length of a factorization of a permutation into transpositions that are cyclic shifts of (12).
In symbols, for a permutation $\pi$ this is
$$\min\{ k \mid \pi = \tau_{i_1} \cdots \tau_{i_k}, 1 \leq i_1,\ldots,i_k \leq n\},$$
where $\tau_a = (a,a+1)$ for $1 \leq a \leq n$ and $n+1$ is identified with $1$.
Put differently, this is the number of cyclically simple transpositions needed to sort a permutation.
In symbols, for a permutation $\pi$ this is
$$\min\{ k \mid \pi = \tau_{i_1} \cdots \tau_{i_k}, 1 \leq i_1,\ldots,i_k \leq n\},$$
where $\tau_a = (a,a+1)$ for $1 \leq a \leq n$ and $n+1$ is identified with $1$.
Put differently, this is the number of cyclically simple transpositions needed to sort a permutation.
Map
runsort
Description
The permutation obtained by sorting the increasing runs lexicographically.
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.
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