Identifier
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Mp00081:
Standard tableaux
—reading word permutation⟶
Permutations
Mp00069: Permutations —complement⟶ Permutations
St000779: Permutations ⟶ ℤ
Values
[[1,2]] => [1,2] => [2,1] => 0
[[1],[2]] => [2,1] => [1,2] => 0
[[1,2,3]] => [1,2,3] => [3,2,1] => 0
[[1,3],[2]] => [2,1,3] => [2,3,1] => 1
[[1,2],[3]] => [3,1,2] => [1,3,2] => 0
[[1],[2],[3]] => [3,2,1] => [1,2,3] => 0
[[1,2,3,4]] => [1,2,3,4] => [4,3,2,1] => 0
[[1,3,4],[2]] => [2,1,3,4] => [3,4,2,1] => 1
[[1,2,4],[3]] => [3,1,2,4] => [2,4,3,1] => 1
[[1,2,3],[4]] => [4,1,2,3] => [1,4,3,2] => 0
[[1,3],[2,4]] => [2,4,1,3] => [3,1,4,2] => 1
[[1,2],[3,4]] => [3,4,1,2] => [2,1,4,3] => 0
[[1,4],[2],[3]] => [3,2,1,4] => [2,3,4,1] => 1
[[1,3],[2],[4]] => [4,2,1,3] => [1,3,4,2] => 1
[[1,2],[3],[4]] => [4,3,1,2] => [1,2,4,3] => 0
[[1],[2],[3],[4]] => [4,3,2,1] => [1,2,3,4] => 0
[[1,2,3,4,5]] => [1,2,3,4,5] => [5,4,3,2,1] => 0
[[1,3,4,5],[2]] => [2,1,3,4,5] => [4,5,3,2,1] => 1
[[1,2,4,5],[3]] => [3,1,2,4,5] => [3,5,4,2,1] => 1
[[1,2,3,5],[4]] => [4,1,2,3,5] => [2,5,4,3,1] => 1
[[1,2,3,4],[5]] => [5,1,2,3,4] => [1,5,4,3,2] => 0
[[1,3,5],[2,4]] => [2,4,1,3,5] => [4,2,5,3,1] => 2
[[1,2,5],[3,4]] => [3,4,1,2,5] => [3,2,5,4,1] => 1
[[1,3,4],[2,5]] => [2,5,1,3,4] => [4,1,5,3,2] => 1
[[1,2,4],[3,5]] => [3,5,1,2,4] => [3,1,5,4,2] => 1
[[1,2,3],[4,5]] => [4,5,1,2,3] => [2,1,5,4,3] => 0
[[1,4,5],[2],[3]] => [3,2,1,4,5] => [3,4,5,2,1] => 1
[[1,3,5],[2],[4]] => [4,2,1,3,5] => [2,4,5,3,1] => 2
[[1,2,5],[3],[4]] => [4,3,1,2,5] => [2,3,5,4,1] => 1
[[1,3,4],[2],[5]] => [5,2,1,3,4] => [1,4,5,3,2] => 1
[[1,2,4],[3],[5]] => [5,3,1,2,4] => [1,3,5,4,2] => 1
[[1,2,3],[4],[5]] => [5,4,1,2,3] => [1,2,5,4,3] => 0
[[1,4],[2,5],[3]] => [3,2,5,1,4] => [3,4,1,5,2] => 1
[[1,3],[2,5],[4]] => [4,2,5,1,3] => [2,4,1,5,3] => 2
[[1,2],[3,5],[4]] => [4,3,5,1,2] => [2,3,1,5,4] => 1
[[1,3],[2,4],[5]] => [5,2,4,1,3] => [1,4,2,5,3] => 1
[[1,2],[3,4],[5]] => [5,3,4,1,2] => [1,3,2,5,4] => 0
[[1,5],[2],[3],[4]] => [4,3,2,1,5] => [2,3,4,5,1] => 1
[[1,4],[2],[3],[5]] => [5,3,2,1,4] => [1,3,4,5,2] => 1
[[1,3],[2],[4],[5]] => [5,4,2,1,3] => [1,2,4,5,3] => 1
[[1,2],[3],[4],[5]] => [5,4,3,1,2] => [1,2,3,5,4] => 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] => [6,5,4,3,2,1] => 0
[[1,3,4,5,6],[2]] => [2,1,3,4,5,6] => [5,6,4,3,2,1] => 1
[[1,2,4,5,6],[3]] => [3,1,2,4,5,6] => [4,6,5,3,2,1] => 1
[[1,2,3,5,6],[4]] => [4,1,2,3,5,6] => [3,6,5,4,2,1] => 1
[[1,2,3,4,6],[5]] => [5,1,2,3,4,6] => [2,6,5,4,3,1] => 1
[[1,2,3,4,5],[6]] => [6,1,2,3,4,5] => [1,6,5,4,3,2] => 0
[[1,3,5,6],[2,4]] => [2,4,1,3,5,6] => [5,3,6,4,2,1] => 2
[[1,2,5,6],[3,4]] => [3,4,1,2,5,6] => [4,3,6,5,2,1] => 1
[[1,3,4,6],[2,5]] => [2,5,1,3,4,6] => [5,2,6,4,3,1] => 2
[[1,2,4,6],[3,5]] => [3,5,1,2,4,6] => [4,2,6,5,3,1] => 2
[[1,2,3,6],[4,5]] => [4,5,1,2,3,6] => [3,2,6,5,4,1] => 1
[[1,3,4,5],[2,6]] => [2,6,1,3,4,5] => [5,1,6,4,3,2] => 1
[[1,2,4,5],[3,6]] => [3,6,1,2,4,5] => [4,1,6,5,3,2] => 1
[[1,2,3,5],[4,6]] => [4,6,1,2,3,5] => [3,1,6,5,4,2] => 1
[[1,2,3,4],[5,6]] => [5,6,1,2,3,4] => [2,1,6,5,4,3] => 0
[[1,4,5,6],[2],[3]] => [3,2,1,4,5,6] => [4,5,6,3,2,1] => 1
[[1,3,5,6],[2],[4]] => [4,2,1,3,5,6] => [3,5,6,4,2,1] => 2
[[1,2,5,6],[3],[4]] => [4,3,1,2,5,6] => [3,4,6,5,2,1] => 1
[[1,3,4,6],[2],[5]] => [5,2,1,3,4,6] => [2,5,6,4,3,1] => 2
[[1,2,4,6],[3],[5]] => [5,3,1,2,4,6] => [2,4,6,5,3,1] => 2
[[1,2,3,6],[4],[5]] => [5,4,1,2,3,6] => [2,3,6,5,4,1] => 1
[[1,3,4,5],[2],[6]] => [6,2,1,3,4,5] => [1,5,6,4,3,2] => 1
[[1,2,4,5],[3],[6]] => [6,3,1,2,4,5] => [1,4,6,5,3,2] => 1
[[1,2,3,5],[4],[6]] => [6,4,1,2,3,5] => [1,3,6,5,4,2] => 1
[[1,2,3,4],[5],[6]] => [6,5,1,2,3,4] => [1,2,6,5,4,3] => 0
[[1,3,5],[2,4,6]] => [2,4,6,1,3,5] => [5,3,1,6,4,2] => 2
[[1,2,5],[3,4,6]] => [3,4,6,1,2,5] => [4,3,1,6,5,2] => 1
[[1,3,4],[2,5,6]] => [2,5,6,1,3,4] => [5,2,1,6,4,3] => 1
[[1,2,4],[3,5,6]] => [3,5,6,1,2,4] => [4,2,1,6,5,3] => 1
[[1,2,3],[4,5,6]] => [4,5,6,1,2,3] => [3,2,1,6,5,4] => 0
[[1,4,6],[2,5],[3]] => [3,2,5,1,4,6] => [4,5,2,6,3,1] => 2
[[1,3,6],[2,5],[4]] => [4,2,5,1,3,6] => [3,5,2,6,4,1] => 3
[[1,2,6],[3,5],[4]] => [4,3,5,1,2,6] => [3,4,2,6,5,1] => 2
[[1,3,6],[2,4],[5]] => [5,2,4,1,3,6] => [2,5,3,6,4,1] => 2
[[1,2,6],[3,4],[5]] => [5,3,4,1,2,6] => [2,4,3,6,5,1] => 1
[[1,4,5],[2,6],[3]] => [3,2,6,1,4,5] => [4,5,1,6,3,2] => 1
[[1,3,5],[2,6],[4]] => [4,2,6,1,3,5] => [3,5,1,6,4,2] => 2
[[1,2,5],[3,6],[4]] => [4,3,6,1,2,5] => [3,4,1,6,5,2] => 1
[[1,3,4],[2,6],[5]] => [5,2,6,1,3,4] => [2,5,1,6,4,3] => 2
[[1,2,4],[3,6],[5]] => [5,3,6,1,2,4] => [2,4,1,6,5,3] => 2
[[1,2,3],[4,6],[5]] => [5,4,6,1,2,3] => [2,3,1,6,5,4] => 1
[[1,3,5],[2,4],[6]] => [6,2,4,1,3,5] => [1,5,3,6,4,2] => 2
[[1,2,5],[3,4],[6]] => [6,3,4,1,2,5] => [1,4,3,6,5,2] => 1
[[1,3,4],[2,5],[6]] => [6,2,5,1,3,4] => [1,5,2,6,4,3] => 1
[[1,2,4],[3,5],[6]] => [6,3,5,1,2,4] => [1,4,2,6,5,3] => 1
[[1,2,3],[4,5],[6]] => [6,4,5,1,2,3] => [1,3,2,6,5,4] => 0
[[1,5,6],[2],[3],[4]] => [4,3,2,1,5,6] => [3,4,5,6,2,1] => 1
[[1,4,6],[2],[3],[5]] => [5,3,2,1,4,6] => [2,4,5,6,3,1] => 2
[[1,3,6],[2],[4],[5]] => [5,4,2,1,3,6] => [2,3,5,6,4,1] => 2
[[1,2,6],[3],[4],[5]] => [5,4,3,1,2,6] => [2,3,4,6,5,1] => 1
[[1,4,5],[2],[3],[6]] => [6,3,2,1,4,5] => [1,4,5,6,3,2] => 1
[[1,3,5],[2],[4],[6]] => [6,4,2,1,3,5] => [1,3,5,6,4,2] => 2
[[1,2,5],[3],[4],[6]] => [6,4,3,1,2,5] => [1,3,4,6,5,2] => 1
[[1,3,4],[2],[5],[6]] => [6,5,2,1,3,4] => [1,2,5,6,4,3] => 1
[[1,2,4],[3],[5],[6]] => [6,5,3,1,2,4] => [1,2,4,6,5,3] => 1
[[1,2,3],[4],[5],[6]] => [6,5,4,1,2,3] => [1,2,3,6,5,4] => 0
[[1,4],[2,5],[3,6]] => [3,6,2,5,1,4] => [4,1,5,2,6,3] => 1
[[1,3],[2,5],[4,6]] => [4,6,2,5,1,3] => [3,1,5,2,6,4] => 2
[[1,2],[3,5],[4,6]] => [4,6,3,5,1,2] => [3,1,4,2,6,5] => 1
>>> Load all 175 entries. <<<
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Description
The tier of a permutation.
This is the number of elements $i$ such that $[i+1,k,i]$ is an occurrence of the pattern $[2,3,1]$. For example, $[3,5,6,1,2,4]$ has tier $2$, with witnesses $[3,5,2]$ (or $[3,6,2]$) and $[5,6,4]$.
According to [1], this is the number of passes minus one needed to sort the permutation using a single stack. The generating function for this statistic appears as OEIS:A122890 and OEIS:A158830 in the form of triangles read by rows, see [sec. 4, 1].
This is the number of elements $i$ such that $[i+1,k,i]$ is an occurrence of the pattern $[2,3,1]$. For example, $[3,5,6,1,2,4]$ has tier $2$, with witnesses $[3,5,2]$ (or $[3,6,2]$) and $[5,6,4]$.
According to [1], this is the number of passes minus one needed to sort the permutation using a single stack. The generating function for this statistic appears as OEIS:A122890 and OEIS:A158830 in the form of triangles read by rows, see [sec. 4, 1].
Map
complement
Description
Sents a permutation to its complement.
The complement of a permutation $\sigma$ of length $n$ is the permutation $\tau$ with $\tau(i) = n+1-\sigma(i)$
The complement of a permutation $\sigma$ of length $n$ is the permutation $\tau$ with $\tau(i) = n+1-\sigma(i)$
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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