Identifier
-
Mp00081:
Standard tableaux
—reading word permutation⟶
Permutations
Mp00064: Permutations —reverse⟶ Permutations
Mp00325: Permutations —ones to leading⟶ Permutations
St000314: Permutations ⟶ ℤ
Values
[[1]] => [1] => [1] => [1] => 1
[[1,2]] => [1,2] => [2,1] => [2,1] => 1
[[1],[2]] => [2,1] => [1,2] => [1,2] => 2
[[1,2,3]] => [1,2,3] => [3,2,1] => [3,2,1] => 1
[[1,3],[2]] => [2,1,3] => [3,1,2] => [3,1,2] => 1
[[1,2],[3]] => [3,1,2] => [2,1,3] => [1,3,2] => 2
[[1],[2],[3]] => [3,2,1] => [1,2,3] => [1,2,3] => 3
[[1,2,3,4]] => [1,2,3,4] => [4,3,2,1] => [4,3,1,2] => 1
[[1,3,4],[2]] => [2,1,3,4] => [4,3,1,2] => [4,2,1,3] => 1
[[1,2,4],[3]] => [3,1,2,4] => [4,2,1,3] => [4,2,3,1] => 1
[[1,2,3],[4]] => [4,1,2,3] => [3,2,1,4] => [1,4,3,2] => 2
[[1,3],[2,4]] => [2,4,1,3] => [3,1,4,2] => [2,4,1,3] => 2
[[1,2],[3,4]] => [3,4,1,2] => [2,1,4,3] => [2,4,3,1] => 2
[[1,4],[2],[3]] => [3,2,1,4] => [4,1,2,3] => [4,1,3,2] => 1
[[1,3],[2],[4]] => [4,2,1,3] => [3,1,2,4] => [1,3,4,2] => 3
[[1,2],[3],[4]] => [4,3,1,2] => [2,1,3,4] => [1,3,2,4] => 3
[[1],[2],[3],[4]] => [4,3,2,1] => [1,2,3,4] => [1,2,3,4] => 4
[[1,2,3,4,5]] => [1,2,3,4,5] => [5,4,3,2,1] => [5,4,1,2,3] => 1
[[1,3,4,5],[2]] => [2,1,3,4,5] => [5,4,3,1,2] => [5,3,1,2,4] => 1
[[1,2,4,5],[3]] => [3,1,2,4,5] => [5,4,2,1,3] => [5,3,1,4,2] => 1
[[1,2,3,5],[4]] => [4,1,2,3,5] => [5,3,2,1,4] => [5,3,4,2,1] => 1
[[1,2,3,4],[5]] => [5,1,2,3,4] => [4,3,2,1,5] => [1,5,4,3,2] => 2
[[1,3,5],[2,4]] => [2,4,1,3,5] => [5,3,1,4,2] => [5,2,3,4,1] => 1
[[1,2,5],[3,4]] => [3,4,1,2,5] => [5,2,1,4,3] => [5,2,3,1,4] => 1
[[1,3,4],[2,5]] => [2,5,1,3,4] => [4,3,1,5,2] => [2,5,1,3,4] => 2
[[1,2,4],[3,5]] => [3,5,1,2,4] => [4,2,1,5,3] => [2,5,4,1,3] => 2
[[1,2,3],[4,5]] => [4,5,1,2,3] => [3,2,1,5,4] => [2,5,4,3,1] => 2
[[1,4,5],[2],[3]] => [3,2,1,4,5] => [5,4,1,2,3] => [5,2,1,4,3] => 1
[[1,3,5],[2],[4]] => [4,2,1,3,5] => [5,3,1,2,4] => [5,2,4,3,1] => 1
[[1,2,5],[3],[4]] => [4,3,1,2,5] => [5,2,1,3,4] => [5,2,4,1,3] => 1
[[1,3,4],[2],[5]] => [5,2,1,3,4] => [4,3,1,2,5] => [1,4,5,3,2] => 3
[[1,2,4],[3],[5]] => [5,3,1,2,4] => [4,2,1,3,5] => [1,4,3,5,2] => 3
[[1,2,3],[4],[5]] => [5,4,1,2,3] => [3,2,1,4,5] => [1,4,3,2,5] => 3
[[1,4],[2,5],[3]] => [3,2,5,1,4] => [4,1,5,2,3] => [3,5,1,4,2] => 2
[[1,3],[2,5],[4]] => [4,2,5,1,3] => [3,1,5,2,4] => [3,5,1,2,4] => 2
[[1,2],[3,5],[4]] => [4,3,5,1,2] => [2,1,5,3,4] => [3,5,4,1,2] => 2
[[1,3],[2,4],[5]] => [5,2,4,1,3] => [3,1,4,2,5] => [1,3,5,2,4] => 3
[[1,2],[3,4],[5]] => [5,3,4,1,2] => [2,1,4,3,5] => [1,3,2,5,4] => 3
[[1,5],[2],[3],[4]] => [4,3,2,1,5] => [5,1,2,3,4] => [5,1,4,2,3] => 1
[[1,4],[2],[3],[5]] => [5,3,2,1,4] => [4,1,2,3,5] => [1,3,4,5,2] => 4
[[1,3],[2],[4],[5]] => [5,4,2,1,3] => [3,1,2,4,5] => [1,3,4,2,5] => 4
[[1,2],[3],[4],[5]] => [5,4,3,1,2] => [2,1,3,4,5] => [1,3,2,4,5] => 4
[[1],[2],[3],[4],[5]] => [5,4,3,2,1] => [1,2,3,4,5] => [1,2,3,4,5] => 5
[[1,2,3,4,5,6]] => [1,2,3,4,5,6] => [6,5,4,3,2,1] => [6,5,1,2,3,4] => 1
[[1,3,4,5,6],[2]] => [2,1,3,4,5,6] => [6,5,4,3,1,2] => [6,4,1,2,3,5] => 1
[[1,2,4,5,6],[3]] => [3,1,2,4,5,6] => [6,5,4,2,1,3] => [6,4,1,2,5,3] => 1
[[1,2,3,5,6],[4]] => [4,1,2,3,5,6] => [6,5,3,2,1,4] => [6,4,1,5,3,2] => 1
[[1,2,3,4,6],[5]] => [5,1,2,3,4,6] => [6,4,3,2,1,5] => [6,4,5,3,2,1] => 1
[[1,2,3,4,5],[6]] => [6,1,2,3,4,5] => [5,4,3,2,1,6] => [1,6,5,4,3,2] => 2
[[1,3,5,6],[2,4]] => [2,4,1,3,5,6] => [6,5,3,1,4,2] => [6,3,1,4,5,2] => 1
[[1,2,5,6],[3,4]] => [3,4,1,2,5,6] => [6,5,2,1,4,3] => [6,3,1,4,2,5] => 1
[[1,3,4,6],[2,5]] => [2,5,1,3,4,6] => [6,4,3,1,5,2] => [6,3,4,5,1,2] => 1
[[1,2,4,6],[3,5]] => [3,5,1,2,4,6] => [6,4,2,1,5,3] => [6,3,4,2,5,1] => 1
[[1,2,3,6],[4,5]] => [4,5,1,2,3,6] => [6,3,2,1,5,4] => [6,3,4,2,1,5] => 1
[[1,3,4,5],[2,6]] => [2,6,1,3,4,5] => [5,4,3,1,6,2] => [2,6,1,3,4,5] => 2
[[1,2,4,5],[3,6]] => [3,6,1,2,4,5] => [5,4,2,1,6,3] => [2,6,5,1,3,4] => 2
[[1,2,3,5],[4,6]] => [4,6,1,2,3,5] => [5,3,2,1,6,4] => [2,6,5,4,1,3] => 2
[[1,2,3,4],[5,6]] => [5,6,1,2,3,4] => [4,3,2,1,6,5] => [2,6,5,4,3,1] => 2
[[1,4,5,6],[2],[3]] => [3,2,1,4,5,6] => [6,5,4,1,2,3] => [6,3,1,2,5,4] => 1
[[1,3,5,6],[2],[4]] => [4,2,1,3,5,6] => [6,5,3,1,2,4] => [6,3,1,5,4,2] => 1
[[1,2,5,6],[3],[4]] => [4,3,1,2,5,6] => [6,5,2,1,3,4] => [6,3,1,5,2,4] => 1
[[1,3,4,6],[2],[5]] => [5,2,1,3,4,6] => [6,4,3,1,2,5] => [6,3,5,4,2,1] => 1
[[1,2,4,6],[3],[5]] => [5,3,1,2,4,6] => [6,4,2,1,3,5] => [6,3,5,2,4,1] => 1
[[1,2,3,6],[4],[5]] => [5,4,1,2,3,6] => [6,3,2,1,4,5] => [6,3,5,2,1,4] => 1
[[1,3,4,5],[2],[6]] => [6,2,1,3,4,5] => [5,4,3,1,2,6] => [1,5,6,4,3,2] => 3
[[1,2,4,5],[3],[6]] => [6,3,1,2,4,5] => [5,4,2,1,3,6] => [1,5,4,6,3,2] => 3
[[1,2,3,5],[4],[6]] => [6,4,1,2,3,5] => [5,3,2,1,4,6] => [1,5,4,3,6,2] => 3
[[1,2,3,4],[5],[6]] => [6,5,1,2,3,4] => [4,3,2,1,5,6] => [1,5,4,3,2,6] => 3
[[1,3,5],[2,4,6]] => [2,4,6,1,3,5] => [5,3,1,6,4,2] => [3,6,2,4,1,5] => 2
[[1,2,5],[3,4,6]] => [3,4,6,1,2,5] => [5,2,1,6,4,3] => [3,6,5,2,4,1] => 2
[[1,3,4],[2,5,6]] => [2,5,6,1,3,4] => [4,3,1,6,5,2] => [3,6,2,1,4,5] => 2
[[1,2,4],[3,5,6]] => [3,5,6,1,2,4] => [4,2,1,6,5,3] => [3,6,5,2,1,4] => 2
[[1,2,3],[4,5,6]] => [4,5,6,1,2,3] => [3,2,1,6,5,4] => [3,6,5,4,2,1] => 2
[[1,4,6],[2,5],[3]] => [3,2,5,1,4,6] => [6,4,1,5,2,3] => [6,2,3,4,1,5] => 1
[[1,3,6],[2,5],[4]] => [4,2,5,1,3,6] => [6,3,1,5,2,4] => [6,2,3,4,5,1] => 1
[[1,2,6],[3,5],[4]] => [4,3,5,1,2,6] => [6,2,1,5,3,4] => [6,2,3,1,4,5] => 1
[[1,3,6],[2,4],[5]] => [5,2,4,1,3,6] => [6,3,1,4,2,5] => [6,2,5,4,1,3] => 1
[[1,2,6],[3,4],[5]] => [5,3,4,1,2,6] => [6,2,1,4,3,5] => [6,2,5,1,4,3] => 1
[[1,4,5],[2,6],[3]] => [3,2,6,1,4,5] => [5,4,1,6,2,3] => [3,6,1,4,5,2] => 2
[[1,3,5],[2,6],[4]] => [4,2,6,1,3,5] => [5,3,1,6,2,4] => [3,6,1,4,2,5] => 2
[[1,2,5],[3,6],[4]] => [4,3,6,1,2,5] => [5,2,1,6,3,4] => [3,6,5,1,4,2] => 2
[[1,3,4],[2,6],[5]] => [5,2,6,1,3,4] => [4,3,1,6,2,5] => [3,6,1,2,5,4] => 2
[[1,2,4],[3,6],[5]] => [5,3,6,1,2,4] => [4,2,1,6,3,5] => [3,6,5,1,2,4] => 2
[[1,2,3],[4,6],[5]] => [5,4,6,1,2,3] => [3,2,1,6,4,5] => [3,6,5,4,1,2] => 2
[[1,3,5],[2,4],[6]] => [6,2,4,1,3,5] => [5,3,1,4,2,6] => [1,4,6,3,5,2] => 3
[[1,2,5],[3,4],[6]] => [6,3,4,1,2,5] => [5,2,1,4,3,6] => [1,4,3,6,5,2] => 3
[[1,3,4],[2,5],[6]] => [6,2,5,1,3,4] => [4,3,1,5,2,6] => [1,4,6,3,2,5] => 3
[[1,2,4],[3,5],[6]] => [6,3,5,1,2,4] => [4,2,1,5,3,6] => [1,4,3,6,2,5] => 3
[[1,2,3],[4,5],[6]] => [6,4,5,1,2,3] => [3,2,1,5,4,6] => [1,4,3,2,6,5] => 3
[[1,5,6],[2],[3],[4]] => [4,3,2,1,5,6] => [6,5,1,2,3,4] => [6,2,1,5,3,4] => 1
[[1,4,6],[2],[3],[5]] => [5,3,2,1,4,6] => [6,4,1,2,3,5] => [6,2,5,3,4,1] => 1
[[1,3,6],[2],[4],[5]] => [5,4,2,1,3,6] => [6,3,1,2,4,5] => [6,2,5,3,1,4] => 1
[[1,2,6],[3],[4],[5]] => [5,4,3,1,2,6] => [6,2,1,3,4,5] => [6,2,5,1,3,4] => 1
[[1,4,5],[2],[3],[6]] => [6,3,2,1,4,5] => [5,4,1,2,3,6] => [1,4,5,6,3,2] => 4
[[1,3,5],[2],[4],[6]] => [6,4,2,1,3,5] => [5,3,1,2,4,6] => [1,4,5,3,6,2] => 4
[[1,2,5],[3],[4],[6]] => [6,4,3,1,2,5] => [5,2,1,3,4,6] => [1,4,3,5,6,2] => 4
[[1,3,4],[2],[5],[6]] => [6,5,2,1,3,4] => [4,3,1,2,5,6] => [1,4,5,3,2,6] => 4
[[1,2,4],[3],[5],[6]] => [6,5,3,1,2,4] => [4,2,1,3,5,6] => [1,4,3,5,2,6] => 4
[[1,2,3],[4],[5],[6]] => [6,5,4,1,2,3] => [3,2,1,4,5,6] => [1,4,3,2,5,6] => 4
[[1,4],[2,5],[3,6]] => [3,6,2,5,1,4] => [4,1,5,2,6,3] => [2,4,6,1,5,3] => 3
[[1,3],[2,5],[4,6]] => [4,6,2,5,1,3] => [3,1,5,2,6,4] => [2,4,6,3,1,5] => 3
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Description
The number of left-to-right-maxima of a permutation.
An integer $\sigma_i$ in the one-line notation of a permutation $\sigma$ is a left-to-right-maximum if there does not exist a $j < i$ such that $\sigma_j > \sigma_i$.
This is also the number of weak exceedences of a permutation that are not mid-points of a decreasing subsequence of length 3, see [1] for more on the later description.
An integer $\sigma_i$ in the one-line notation of a permutation $\sigma$ is a left-to-right-maximum if there does not exist a $j < i$ such that $\sigma_j > \sigma_i$.
This is also the number of weak exceedences of a permutation that are not mid-points of a decreasing subsequence of length 3, see [1] for more on the later description.
Map
ones to leading
Description
The unique permutation obtained by applying the Foata-Riordan map to obtain a Prüfer code, then prepending zero and cyclically shifting.
Let $c_1,\dots, c_{n-1}$ be the Prüfer code obtained via the Foata-Riordan map described in [1, eq (1.2)] and let $c_0 = 0$.
This map returns the a unique permutation $q_1,\dots, q_n$ such that $q_i - c_{i-1}$ is constant modulo $n+1$.
This map is Mp00299ones to leading restricted to permutations.
Let $c_1,\dots, c_{n-1}$ be the Prüfer code obtained via the Foata-Riordan map described in [1, eq (1.2)] and let $c_0 = 0$.
This map returns the a unique permutation $q_1,\dots, q_n$ such that $q_i - c_{i-1}$ is constant modulo $n+1$.
This map is Mp00299ones to leading restricted to permutations.
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.
Map
reverse
Description
Sends a permutation to its reverse.
The reverse of a permutation $\sigma$ of length $n$ is given by $\tau$ with $\tau(i) = \sigma(n+1-i)$.
The reverse of a permutation $\sigma$ of length $n$ is given by $\tau$ with $\tau(i) = \sigma(n+1-i)$.
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