Identifier
-
Mp00042:
Integer partitions
—initial tableau⟶
Standard tableaux
Mp00081: Standard tableaux —reading word permutation⟶ Permutations
Mp00069: Permutations —complement⟶ Permutations
St000040: Permutations ⟶ ℤ
Values
[1] => [[1]] => [1] => [1] => 1
[2] => [[1,2]] => [1,2] => [2,1] => 2
[1,1] => [[1],[2]] => [2,1] => [1,2] => 1
[3] => [[1,2,3]] => [1,2,3] => [3,2,1] => 6
[2,1] => [[1,2],[3]] => [3,1,2] => [1,3,2] => 2
[1,1,1] => [[1],[2],[3]] => [3,2,1] => [1,2,3] => 1
[4] => [[1,2,3,4]] => [1,2,3,4] => [4,3,2,1] => 24
[3,1] => [[1,2,3],[4]] => [4,1,2,3] => [1,4,3,2] => 6
[2,2] => [[1,2],[3,4]] => [3,4,1,2] => [2,1,4,3] => 4
[2,1,1] => [[1,2],[3],[4]] => [4,3,1,2] => [1,2,4,3] => 2
[1,1,1,1] => [[1],[2],[3],[4]] => [4,3,2,1] => [1,2,3,4] => 1
[5] => [[1,2,3,4,5]] => [1,2,3,4,5] => [5,4,3,2,1] => 120
[4,1] => [[1,2,3,4],[5]] => [5,1,2,3,4] => [1,5,4,3,2] => 24
[3,2] => [[1,2,3],[4,5]] => [4,5,1,2,3] => [2,1,5,4,3] => 12
[3,1,1] => [[1,2,3],[4],[5]] => [5,4,1,2,3] => [1,2,5,4,3] => 6
[2,2,1] => [[1,2],[3,4],[5]] => [5,3,4,1,2] => [1,3,2,5,4] => 4
[2,1,1,1] => [[1,2],[3],[4],[5]] => [5,4,3,1,2] => [1,2,3,5,4] => 2
[1,1,1,1,1] => [[1],[2],[3],[4],[5]] => [5,4,3,2,1] => [1,2,3,4,5] => 1
[6] => [[1,2,3,4,5,6]] => [1,2,3,4,5,6] => [6,5,4,3,2,1] => 720
[5,1] => [[1,2,3,4,5],[6]] => [6,1,2,3,4,5] => [1,6,5,4,3,2] => 120
[4,2] => [[1,2,3,4],[5,6]] => [5,6,1,2,3,4] => [2,1,6,5,4,3] => 48
[4,1,1] => [[1,2,3,4],[5],[6]] => [6,5,1,2,3,4] => [1,2,6,5,4,3] => 24
[3,3] => [[1,2,3],[4,5,6]] => [4,5,6,1,2,3] => [3,2,1,6,5,4] => 36
[3,2,1] => [[1,2,3],[4,5],[6]] => [6,4,5,1,2,3] => [1,3,2,6,5,4] => 12
[3,1,1,1] => [[1,2,3],[4],[5],[6]] => [6,5,4,1,2,3] => [1,2,3,6,5,4] => 6
[2,2,2] => [[1,2],[3,4],[5,6]] => [5,6,3,4,1,2] => [2,1,4,3,6,5] => 8
[2,2,1,1] => [[1,2],[3,4],[5],[6]] => [6,5,3,4,1,2] => [1,2,4,3,6,5] => 4
[2,1,1,1,1] => [[1,2],[3],[4],[5],[6]] => [6,5,4,3,1,2] => [1,2,3,4,6,5] => 2
[1,1,1,1,1,1] => [[1],[2],[3],[4],[5],[6]] => [6,5,4,3,2,1] => [1,2,3,4,5,6] => 1
[7] => [[1,2,3,4,5,6,7]] => [1,2,3,4,5,6,7] => [7,6,5,4,3,2,1] => 5040
[6,1] => [[1,2,3,4,5,6],[7]] => [7,1,2,3,4,5,6] => [1,7,6,5,4,3,2] => 720
[5,2] => [[1,2,3,4,5],[6,7]] => [6,7,1,2,3,4,5] => [2,1,7,6,5,4,3] => 240
[5,1,1] => [[1,2,3,4,5],[6],[7]] => [7,6,1,2,3,4,5] => [1,2,7,6,5,4,3] => 120
[4,1,1,1] => [[1,2,3,4],[5],[6],[7]] => [7,6,5,1,2,3,4] => [1,2,3,7,6,5,4] => 24
[3,3,1] => [[1,2,3],[4,5,6],[7]] => [7,4,5,6,1,2,3] => [1,4,3,2,7,6,5] => 36
[3,2,1,1] => [[1,2,3],[4,5],[6],[7]] => [7,6,4,5,1,2,3] => [1,2,4,3,7,6,5] => 12
[3,1,1,1,1] => [[1,2,3],[4],[5],[6],[7]] => [7,6,5,4,1,2,3] => [1,2,3,4,7,6,5] => 6
[2,2,2,1] => [[1,2],[3,4],[5,6],[7]] => [7,5,6,3,4,1,2] => [1,3,2,5,4,7,6] => 8
[2,2,1,1,1] => [[1,2],[3,4],[5],[6],[7]] => [7,6,5,3,4,1,2] => [1,2,3,5,4,7,6] => 4
[2,1,1,1,1,1] => [[1,2],[3],[4],[5],[6],[7]] => [7,6,5,4,3,1,2] => [1,2,3,4,5,7,6] => 2
[1,1,1,1,1,1,1] => [[1],[2],[3],[4],[5],[6],[7]] => [7,6,5,4,3,2,1] => [1,2,3,4,5,6,7] => 1
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Description
The number of regions of the inversion arrangement of a permutation.
The inversion arrangement $\mathcal{A}_w$ consists of the hyperplanes $x_i-x_j=0$ such that $(i,j)$ is an inversion of $w$.
Postnikov [4] conjectured that the number of regions in $\mathcal{A}_w$ equals the number of permutations in the interval $[id,w]$ in the strong Bruhat order if and only if $w$ avoids $4231$, $35142$, $42513$, $351624$. This conjecture was proved by Hultman-Linusson-Shareshian-Sjöstrand [1].
Oh-Postnikov-Yoo [3] showed that the number of regions of $\mathcal{A}_w$ is $|\chi_{G_w}(-1)|$ where $\chi_{G_w}$ is the chromatic polynomial of the inversion graph $G_w$. This is the graph with vertices ${1,2,\ldots,n}$ and edges $(i,j)$ for $i\lneq j$ $w_i\gneq w_j$.
For a permutation $w=w_1\cdots w_n$, Lewis-Morales [2] and Hultman (see appendix in [2]) showed that this number equals the number of placements of $n$ non-attacking rooks on the south-west Rothe diagram of $w$.
The inversion arrangement $\mathcal{A}_w$ consists of the hyperplanes $x_i-x_j=0$ such that $(i,j)$ is an inversion of $w$.
Postnikov [4] conjectured that the number of regions in $\mathcal{A}_w$ equals the number of permutations in the interval $[id,w]$ in the strong Bruhat order if and only if $w$ avoids $4231$, $35142$, $42513$, $351624$. This conjecture was proved by Hultman-Linusson-Shareshian-Sjöstrand [1].
Oh-Postnikov-Yoo [3] showed that the number of regions of $\mathcal{A}_w$ is $|\chi_{G_w}(-1)|$ where $\chi_{G_w}$ is the chromatic polynomial of the inversion graph $G_w$. This is the graph with vertices ${1,2,\ldots,n}$ and edges $(i,j)$ for $i\lneq j$ $w_i\gneq w_j$.
For a permutation $w=w_1\cdots w_n$, Lewis-Morales [2] and Hultman (see appendix in [2]) showed that this number equals the number of placements of $n$ non-attacking rooks on the south-west Rothe diagram of $w$.
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
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.
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