Identifier
-
Mp00045:
Integer partitions
—reading tableau⟶
Standard tableaux
Mp00081: Standard tableaux —reading word permutation⟶ Permutations
Mp00310: Permutations —toric promotion⟶ Permutations
St000842: Permutations ⟶ ℤ
Values
[2] => [[1,2]] => [1,2] => [1,2] => 2
[1,1] => [[1],[2]] => [2,1] => [2,1] => 2
[3] => [[1,2,3]] => [1,2,3] => [3,2,1] => 2
[2,1] => [[1,3],[2]] => [2,1,3] => [3,1,2] => 2
[1,1,1] => [[1],[2],[3]] => [3,2,1] => [1,2,3] => 2
[4] => [[1,2,3,4]] => [1,2,3,4] => [4,2,3,1] => 2
[3,1] => [[1,3,4],[2]] => [2,1,3,4] => [4,1,2,3] => 2
[2,2] => [[1,2],[3,4]] => [3,4,1,2] => [2,3,1,4] => 2
[2,1,1] => [[1,4],[2],[3]] => [3,2,1,4] => [1,2,4,3] => 2
[1,1,1,1] => [[1],[2],[3],[4]] => [4,3,2,1] => [1,3,2,4] => 2
[5] => [[1,2,3,4,5]] => [1,2,3,4,5] => [5,2,3,4,1] => 2
[4,1] => [[1,3,4,5],[2]] => [2,1,3,4,5] => [5,1,2,3,4] => 2
[3,2] => [[1,2,5],[3,4]] => [3,4,1,2,5] => [2,3,5,4,1] => 2
[3,1,1] => [[1,4,5],[2],[3]] => [3,2,1,4,5] => [1,2,5,3,4] => 2
[2,2,1] => [[1,3],[2,5],[4]] => [4,2,5,1,3] => [3,5,4,2,1] => 2
[2,1,1,1] => [[1,5],[2],[3],[4]] => [4,3,2,1,5] => [1,3,2,5,4] => 2
[1,1,1,1,1] => [[1],[2],[3],[4],[5]] => [5,4,3,2,1] => [1,4,3,2,5] => 2
[6] => [[1,2,3,4,5,6]] => [1,2,3,4,5,6] => [6,2,3,4,5,1] => 2
[5,1] => [[1,3,4,5,6],[2]] => [2,1,3,4,5,6] => [6,1,2,3,4,5] => 2
[4,2] => [[1,2,5,6],[3,4]] => [3,4,1,2,5,6] => [2,3,6,4,5,1] => 2
[4,1,1] => [[1,4,5,6],[2],[3]] => [3,2,1,4,5,6] => [1,2,6,3,4,5] => 2
[3,3] => [[1,2,3],[4,5,6]] => [4,5,6,1,2,3] => [3,4,5,6,2,1] => 2
[3,2,1] => [[1,3,6],[2,5],[4]] => [4,2,5,1,3,6] => [3,6,4,2,5,1] => 3
[3,1,1,1] => [[1,5,6],[2],[3],[4]] => [4,3,2,1,5,6] => [1,3,2,6,4,5] => 2
[2,2,2] => [[1,2],[3,4],[5,6]] => [5,6,3,4,1,2] => [4,5,2,3,1,6] => 2
[2,2,1,1] => [[1,4],[2,6],[3],[5]] => [5,3,2,6,1,4] => [4,2,6,5,3,1] => 2
[2,1,1,1,1] => [[1,6],[2],[3],[4],[5]] => [5,4,3,2,1,6] => [1,4,3,2,6,5] => 2
[1,1,1,1,1,1] => [[1],[2],[3],[4],[5],[6]] => [6,5,4,3,2,1] => [1,5,4,3,2,6] => 2
[6,1] => [[1,3,4,5,6,7],[2]] => [2,1,3,4,5,6,7] => [7,1,2,3,4,5,6] => 2
[5,1,1] => [[1,4,5,6,7],[2],[3]] => [3,2,1,4,5,6,7] => [1,2,7,3,4,5,6] => 2
[4,1,1,1] => [[1,5,6,7],[2],[3],[4]] => [4,3,2,1,5,6,7] => [1,3,2,7,4,5,6] => 2
[3,1,1,1,1] => [[1,6,7],[2],[3],[4],[5]] => [5,4,3,2,1,6,7] => [1,4,3,2,7,5,6] => 2
[7,1] => [[1,3,4,5,6,7,8],[2]] => [2,1,3,4,5,6,7,8] => [8,1,2,3,4,5,6,7] => 2
[5,1,1,1] => [[1,5,6,7,8],[2],[3],[4]] => [4,3,2,1,5,6,7,8] => [1,3,2,8,4,5,6,7] => 2
[8,1] => [[1,3,4,5,6,7,8,9],[2]] => [2,1,3,4,5,6,7,8,9] => [9,1,2,3,4,5,6,7,8] => 2
[9,1] => [[1,3,4,5,6,7,8,9,10],[2]] => [2,1,3,4,5,6,7,8,9,10] => [10,1,2,3,4,5,6,7,8,9] => 2
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Description
The breadth of a permutation.
According to [1, Def.1.6], this is the minimal Manhattan distance between two ones in the permutation matrix of $\pi$: $$\min\{|i-j|+|\pi(i)-\pi(j)|: i\neq j\}.$$
According to [1, Def.1.3], a permutation $\pi$ is $k$-prolific, if the set of permutations obtained from $\pi$ by deleting any $k$ elements and standardising has maximal cardinality, i.e., $\binom{n}{k}$.
By [1, Thm.2.22], a permutation is $k$-prolific if and only if its breath is at least $k+2$.
By [1, Cor.4.3], the smallest permutations that are $k$-prolific have size $\lceil k^2+2k+1\rceil$, and by [1, Thm.4.4], there are $k$-prolific permutations of any size larger than this.
According to [2] the proportion of $k$-prolific permutations in the set of all permutations is asymptotically equal to $\exp(-k^2-k)$.
According to [1, Def.1.6], this is the minimal Manhattan distance between two ones in the permutation matrix of $\pi$: $$\min\{|i-j|+|\pi(i)-\pi(j)|: i\neq j\}.$$
According to [1, Def.1.3], a permutation $\pi$ is $k$-prolific, if the set of permutations obtained from $\pi$ by deleting any $k$ elements and standardising has maximal cardinality, i.e., $\binom{n}{k}$.
By [1, Thm.2.22], a permutation is $k$-prolific if and only if its breath is at least $k+2$.
By [1, Cor.4.3], the smallest permutations that are $k$-prolific have size $\lceil k^2+2k+1\rceil$, and by [1, Thm.4.4], there are $k$-prolific permutations of any size larger than this.
According to [2] the proportion of $k$-prolific permutations in the set of all permutations is asymptotically equal to $\exp(-k^2-k)$.
Map
toric promotion
Description
Toric promotion of a permutation.
Let $\sigma\in\mathfrak S_n$ be a permutation and let
$ \tau_{i, j}(\sigma) = \begin{cases} \sigma & \text{if $|\sigma^{-1}(i) - \sigma^{-1}(j)| = 1$}\\ (i, j)\circ\sigma & \text{otherwise}. \end{cases} $
The toric promotion operator is the product $\tau_{n,1}\tau_{n-1,n}\dots\tau_{1,2}$.
This is the special case of toric promotion on graphs for the path graph. Its order is $n-1$.
Let $\sigma\in\mathfrak S_n$ be a permutation and let
$ \tau_{i, j}(\sigma) = \begin{cases} \sigma & \text{if $|\sigma^{-1}(i) - \sigma^{-1}(j)| = 1$}\\ (i, j)\circ\sigma & \text{otherwise}. \end{cases} $
The toric promotion operator is the product $\tau_{n,1}\tau_{n-1,n}\dots\tau_{1,2}$.
This is the special case of toric promotion on graphs for the path graph. Its order is $n-1$.
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
reading tableau
Description
Return the RSK recording tableau of the reading word of the (standard) tableau $T$ labeled down (in English convention) each column to the shape of a partition.
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