Identifier
-
Mp00043:
Integer partitions
—to Dyck path⟶
Dyck paths
Mp00129: Dyck paths —to 321-avoiding permutation (Billey-Jockusch-Stanley)⟶ Permutations
St000040: Permutations ⟶ ℤ
Values
[1] => [1,0,1,0] => [2,1] => 2
[2] => [1,1,0,0,1,0] => [1,3,2] => 2
[1,1] => [1,0,1,1,0,0] => [2,1,3] => 2
[3] => [1,1,1,0,0,0,1,0] => [1,2,4,3] => 2
[2,1] => [1,0,1,0,1,0] => [2,3,1] => 4
[1,1,1] => [1,0,1,1,1,0,0,0] => [2,1,3,4] => 2
[4] => [1,1,1,1,0,0,0,0,1,0] => [1,2,3,5,4] => 2
[3,1] => [1,1,0,1,0,0,1,0] => [3,1,4,2] => 8
[2,2] => [1,1,0,0,1,1,0,0] => [1,3,2,4] => 2
[2,1,1] => [1,0,1,1,0,1,0,0] => [2,4,1,3] => 8
[1,1,1,1] => [1,0,1,1,1,1,0,0,0,0] => [2,1,3,4,5] => 2
[5] => [1,1,1,1,1,0,0,0,0,0,1,0] => [1,2,3,4,6,5] => 2
[4,1] => [1,1,1,0,1,0,0,0,1,0] => [4,1,2,5,3] => 16
[3,2] => [1,1,0,0,1,0,1,0] => [1,3,4,2] => 4
[3,1,1] => [1,0,1,1,0,0,1,0] => [2,1,4,3] => 4
[2,2,1] => [1,0,1,0,1,1,0,0] => [2,3,1,4] => 4
[2,1,1,1] => [1,0,1,1,1,0,1,0,0,0] => [2,5,1,3,4] => 16
[1,1,1,1,1] => [1,0,1,1,1,1,1,0,0,0,0,0] => [2,1,3,4,5,6] => 2
[6] => [1,1,1,1,1,1,0,0,0,0,0,0,1,0] => [1,2,3,4,5,7,6] => 2
[5,1] => [1,1,1,1,0,1,0,0,0,0,1,0] => [5,1,2,3,6,4] => 32
[4,2] => [1,1,1,0,0,1,0,0,1,0] => [1,4,2,5,3] => 8
[4,1,1] => [1,1,0,1,1,0,0,0,1,0] => [3,1,2,5,4] => 8
[3,3] => [1,1,1,0,0,0,1,1,0,0] => [1,2,4,3,5] => 2
[3,2,1] => [1,0,1,0,1,0,1,0] => [2,3,4,1] => 8
[3,1,1,1] => [1,0,1,1,1,0,0,1,0,0] => [2,1,5,3,4] => 8
[2,2,2] => [1,1,0,0,1,1,1,0,0,0] => [1,3,2,4,5] => 2
[2,2,1,1] => [1,0,1,1,0,1,1,0,0,0] => [2,4,1,3,5] => 8
[2,1,1,1,1] => [1,0,1,1,1,1,0,1,0,0,0,0] => [2,6,1,3,4,5] => 32
[1,1,1,1,1,1] => [1,0,1,1,1,1,1,1,0,0,0,0,0,0] => [2,1,3,4,5,6,7] => 2
[6,1] => [1,1,1,1,1,0,1,0,0,0,0,0,1,0] => [6,1,2,3,4,7,5] => 64
[5,2] => [1,1,1,1,0,0,1,0,0,0,1,0] => [1,5,2,3,6,4] => 16
[5,1,1] => [1,1,1,0,1,1,0,0,0,0,1,0] => [4,1,2,3,6,5] => 16
[4,3] => [1,1,1,0,0,0,1,0,1,0] => [1,2,4,5,3] => 4
[4,2,1] => [1,1,0,1,0,1,0,0,1,0] => [3,4,1,5,2] => 28
[4,1,1,1] => [1,0,1,1,1,0,0,0,1,0] => [2,1,3,5,4] => 4
[3,3,1] => [1,1,0,1,0,0,1,1,0,0] => [3,1,4,2,5] => 8
[3,2,2] => [1,1,0,0,1,1,0,1,0,0] => [1,3,5,2,4] => 8
[3,2,1,1] => [1,0,1,1,0,1,0,1,0,0] => [2,4,5,1,3] => 28
[3,1,1,1,1] => [1,0,1,1,1,1,0,0,1,0,0,0] => [2,1,6,3,4,5] => 16
[2,2,2,1] => [1,0,1,0,1,1,1,0,0,0] => [2,3,1,4,5] => 4
[2,2,1,1,1] => [1,0,1,1,1,0,1,1,0,0,0,0] => [2,5,1,3,4,6] => 16
[2,1,1,1,1,1] => [1,0,1,1,1,1,1,0,1,0,0,0,0,0] => [2,7,1,3,4,5,6] => 64
[6,2] => [1,1,1,1,1,0,0,1,0,0,0,0,1,0] => [1,6,2,3,4,7,5] => 32
[6,1,1] => [1,1,1,1,0,1,1,0,0,0,0,0,1,0] => [5,1,2,3,4,7,6] => 32
[5,3] => [1,1,1,1,0,0,0,1,0,0,1,0] => [1,2,5,3,6,4] => 8
[5,2,1] => [1,1,1,0,1,0,1,0,0,0,1,0] => [4,5,1,2,6,3] => 92
[5,1,1,1] => [1,1,0,1,1,1,0,0,0,0,1,0] => [3,1,2,4,6,5] => 8
[4,4] => [1,1,1,1,0,0,0,0,1,1,0,0] => [1,2,3,5,4,6] => 2
[4,3,1] => [1,1,0,1,0,0,1,0,1,0] => [3,1,4,5,2] => 16
[4,2,2] => [1,1,0,0,1,1,0,0,1,0] => [1,3,2,5,4] => 4
[4,2,1,1] => [1,0,1,1,0,1,0,0,1,0] => [2,4,1,5,3] => 16
[4,1,1,1,1] => [1,0,1,1,1,1,0,0,0,1,0,0] => [2,1,3,6,4,5] => 8
[3,3,2] => [1,1,0,0,1,0,1,1,0,0] => [1,3,4,2,5] => 4
[3,3,1,1] => [1,0,1,1,0,0,1,1,0,0] => [2,1,4,3,5] => 4
[3,2,2,1] => [1,0,1,0,1,1,0,1,0,0] => [2,3,5,1,4] => 16
[3,2,1,1,1] => [1,0,1,1,1,0,1,0,1,0,0,0] => [2,5,6,1,3,4] => 92
[3,1,1,1,1,1] => [1,0,1,1,1,1,1,0,0,1,0,0,0,0] => [2,1,7,3,4,5,6] => 32
[2,2,2,2] => [1,1,0,0,1,1,1,1,0,0,0,0] => [1,3,2,4,5,6] => 2
[2,2,2,1,1] => [1,0,1,1,0,1,1,1,0,0,0,0] => [2,4,1,3,5,6] => 8
[2,2,1,1,1,1] => [1,0,1,1,1,1,0,1,1,0,0,0,0,0] => [2,6,1,3,4,5,7] => 32
[6,3] => [1,1,1,1,1,0,0,0,1,0,0,0,1,0] => [1,2,6,3,4,7,5] => 16
[6,2,1] => [1,1,1,1,0,1,0,1,0,0,0,0,1,0] => [5,6,1,2,3,7,4] => 292
[6,1,1,1] => [1,1,1,0,1,1,1,0,0,0,0,0,1,0] => [4,1,2,3,5,7,6] => 16
[5,4] => [1,1,1,1,0,0,0,0,1,0,1,0] => [1,2,3,5,6,4] => 4
[5,3,1] => [1,1,1,0,1,0,0,1,0,0,1,0] => [4,1,5,2,6,3] => 56
[5,2,2] => [1,1,1,0,0,1,1,0,0,0,1,0] => [1,4,2,3,6,5] => 8
[5,2,1,1] => [1,1,0,1,1,0,1,0,0,0,1,0] => [3,5,1,2,6,4] => 56
[5,1,1,1,1] => [1,0,1,1,1,1,0,0,0,0,1,0] => [2,1,3,4,6,5] => 4
[4,4,1] => [1,1,1,0,1,0,0,0,1,1,0,0] => [4,1,2,5,3,6] => 16
[4,3,2] => [1,1,0,0,1,0,1,0,1,0] => [1,3,4,5,2] => 8
[4,3,1,1] => [1,0,1,1,0,0,1,0,1,0] => [2,1,4,5,3] => 8
[4,2,2,1] => [1,0,1,0,1,1,0,0,1,0] => [2,3,1,5,4] => 8
[4,2,1,1,1] => [1,0,1,1,1,0,1,0,0,1,0,0] => [2,5,1,6,3,4] => 56
[4,1,1,1,1,1] => [1,0,1,1,1,1,1,0,0,0,1,0,0,0] => [2,1,3,7,4,5,6] => 16
[3,3,3] => [1,1,1,0,0,0,1,1,1,0,0,0] => [1,2,4,3,5,6] => 2
[3,3,2,1] => [1,0,1,0,1,0,1,1,0,0] => [2,3,4,1,5] => 8
[3,3,1,1,1] => [1,0,1,1,1,0,0,1,1,0,0,0] => [2,1,5,3,4,6] => 8
[3,2,2,2] => [1,1,0,0,1,1,1,0,1,0,0,0] => [1,3,6,2,4,5] => 16
[3,2,2,1,1] => [1,0,1,1,0,1,1,0,1,0,0,0] => [2,4,6,1,3,5] => 56
[3,2,1,1,1,1] => [1,0,1,1,1,1,0,1,0,1,0,0,0,0] => [2,6,7,1,3,4,5] => 292
[2,2,2,2,1] => [1,0,1,0,1,1,1,1,0,0,0,0] => [2,3,1,4,5,6] => 4
[2,2,2,1,1,1] => [1,0,1,1,1,0,1,1,1,0,0,0,0,0] => [2,5,1,3,4,6,7] => 16
[6,4] => [1,1,1,1,1,0,0,0,0,1,0,0,1,0] => [1,2,3,6,4,7,5] => 8
[6,3,1] => [1,1,1,1,0,1,0,0,1,0,0,0,1,0] => [5,1,6,2,3,7,4] => 184
[6,2,2] => [1,1,1,1,0,0,1,1,0,0,0,0,1,0] => [1,5,2,3,4,7,6] => 16
[6,2,1,1] => [1,1,1,0,1,1,0,1,0,0,0,0,1,0] => [4,6,1,2,3,7,5] => 184
[6,1,1,1,1] => [1,1,0,1,1,1,1,0,0,0,0,0,1,0] => [3,1,2,4,5,7,6] => 8
[5,5] => [1,1,1,1,1,0,0,0,0,0,1,1,0,0] => [1,2,3,4,6,5,7] => 2
[5,4,1] => [1,1,1,0,1,0,0,0,1,0,1,0] => [4,1,2,5,6,3] => 32
[5,3,2] => [1,1,1,0,0,1,0,1,0,0,1,0] => [1,4,5,2,6,3] => 28
[5,3,1,1] => [1,1,0,1,1,0,0,1,0,0,1,0] => [3,1,5,2,6,4] => 32
[5,2,2,1] => [1,1,0,1,0,1,1,0,0,0,1,0] => [3,4,1,2,6,5] => 28
[5,2,1,1,1] => [1,0,1,1,1,0,1,0,0,0,1,0] => [2,5,1,3,6,4] => 32
[5,1,1,1,1,1] => [1,0,1,1,1,1,1,0,0,0,0,1,0,0] => [2,1,3,4,7,5,6] => 8
[4,4,2] => [1,1,1,0,0,1,0,0,1,1,0,0] => [1,4,2,5,3,6] => 8
[4,4,1,1] => [1,1,0,1,1,0,0,0,1,1,0,0] => [3,1,2,5,4,6] => 8
[4,3,3] => [1,1,1,0,0,0,1,1,0,1,0,0] => [1,2,4,6,3,5] => 8
[4,3,2,1] => [1,0,1,0,1,0,1,0,1,0] => [2,3,4,5,1] => 16
[4,3,1,1,1] => [1,0,1,1,1,0,0,1,0,1,0,0] => [2,1,5,6,3,4] => 28
[4,2,2,2] => [1,1,0,0,1,1,1,0,0,1,0,0] => [1,3,2,6,4,5] => 8
[4,2,2,1,1] => [1,0,1,1,0,1,1,0,0,1,0,0] => [2,4,1,6,3,5] => 32
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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
to Dyck path
Description
Sends a partition to the shortest Dyck path tracing the shape of its Ferrers diagram.
Map
to 321-avoiding permutation (Billey-Jockusch-Stanley)
Description
The Billey-Jockusch-Stanley bijection to 321-avoiding permutations.
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