Identifier
-
Mp00075:
Semistandard tableaux
—reading word permutation⟶
Permutations
Mp00310: Permutations —toric promotion⟶ Permutations
Mp00065: Permutations —permutation poset⟶ Posets
St001880: Posets ⟶ ℤ
Values
[[1],[2],[3]] => [3,2,1] => [1,2,3] => ([(0,2),(2,1)],3) => 3
[[1],[2],[4]] => [3,2,1] => [1,2,3] => ([(0,2),(2,1)],3) => 3
[[1],[3],[4]] => [3,2,1] => [1,2,3] => ([(0,2),(2,1)],3) => 3
[[2],[3],[4]] => [3,2,1] => [1,2,3] => ([(0,2),(2,1)],3) => 3
[[1],[2],[5]] => [3,2,1] => [1,2,3] => ([(0,2),(2,1)],3) => 3
[[1],[3],[5]] => [3,2,1] => [1,2,3] => ([(0,2),(2,1)],3) => 3
[[1],[4],[5]] => [3,2,1] => [1,2,3] => ([(0,2),(2,1)],3) => 3
[[2],[3],[5]] => [3,2,1] => [1,2,3] => ([(0,2),(2,1)],3) => 3
[[2],[4],[5]] => [3,2,1] => [1,2,3] => ([(0,2),(2,1)],3) => 3
[[3],[4],[5]] => [3,2,1] => [1,2,3] => ([(0,2),(2,1)],3) => 3
[[1],[2],[3],[4]] => [4,3,2,1] => [1,3,2,4] => ([(0,1),(0,2),(1,3),(2,3)],4) => 4
[[1],[2],[6]] => [3,2,1] => [1,2,3] => ([(0,2),(2,1)],3) => 3
[[1],[3],[6]] => [3,2,1] => [1,2,3] => ([(0,2),(2,1)],3) => 3
[[1],[4],[6]] => [3,2,1] => [1,2,3] => ([(0,2),(2,1)],3) => 3
[[1],[5],[6]] => [3,2,1] => [1,2,3] => ([(0,2),(2,1)],3) => 3
[[2],[3],[6]] => [3,2,1] => [1,2,3] => ([(0,2),(2,1)],3) => 3
[[2],[4],[6]] => [3,2,1] => [1,2,3] => ([(0,2),(2,1)],3) => 3
[[2],[5],[6]] => [3,2,1] => [1,2,3] => ([(0,2),(2,1)],3) => 3
[[3],[4],[6]] => [3,2,1] => [1,2,3] => ([(0,2),(2,1)],3) => 3
[[3],[5],[6]] => [3,2,1] => [1,2,3] => ([(0,2),(2,1)],3) => 3
[[4],[5],[6]] => [3,2,1] => [1,2,3] => ([(0,2),(2,1)],3) => 3
[[1],[2],[3],[5]] => [4,3,2,1] => [1,3,2,4] => ([(0,1),(0,2),(1,3),(2,3)],4) => 4
[[1],[2],[4],[5]] => [4,3,2,1] => [1,3,2,4] => ([(0,1),(0,2),(1,3),(2,3)],4) => 4
[[1],[3],[4],[5]] => [4,3,2,1] => [1,3,2,4] => ([(0,1),(0,2),(1,3),(2,3)],4) => 4
[[2],[3],[4],[5]] => [4,3,2,1] => [1,3,2,4] => ([(0,1),(0,2),(1,3),(2,3)],4) => 4
[[1],[2],[7]] => [3,2,1] => [1,2,3] => ([(0,2),(2,1)],3) => 3
[[1],[3],[7]] => [3,2,1] => [1,2,3] => ([(0,2),(2,1)],3) => 3
[[1],[4],[7]] => [3,2,1] => [1,2,3] => ([(0,2),(2,1)],3) => 3
[[1],[5],[7]] => [3,2,1] => [1,2,3] => ([(0,2),(2,1)],3) => 3
[[1],[6],[7]] => [3,2,1] => [1,2,3] => ([(0,2),(2,1)],3) => 3
[[2],[3],[7]] => [3,2,1] => [1,2,3] => ([(0,2),(2,1)],3) => 3
[[2],[4],[7]] => [3,2,1] => [1,2,3] => ([(0,2),(2,1)],3) => 3
[[2],[5],[7]] => [3,2,1] => [1,2,3] => ([(0,2),(2,1)],3) => 3
[[2],[6],[7]] => [3,2,1] => [1,2,3] => ([(0,2),(2,1)],3) => 3
[[3],[4],[7]] => [3,2,1] => [1,2,3] => ([(0,2),(2,1)],3) => 3
[[3],[5],[7]] => [3,2,1] => [1,2,3] => ([(0,2),(2,1)],3) => 3
[[3],[6],[7]] => [3,2,1] => [1,2,3] => ([(0,2),(2,1)],3) => 3
[[4],[5],[7]] => [3,2,1] => [1,2,3] => ([(0,2),(2,1)],3) => 3
[[4],[6],[7]] => [3,2,1] => [1,2,3] => ([(0,2),(2,1)],3) => 3
[[5],[6],[7]] => [3,2,1] => [1,2,3] => ([(0,2),(2,1)],3) => 3
[[1],[2],[3],[6]] => [4,3,2,1] => [1,3,2,4] => ([(0,1),(0,2),(1,3),(2,3)],4) => 4
[[1],[2],[4],[6]] => [4,3,2,1] => [1,3,2,4] => ([(0,1),(0,2),(1,3),(2,3)],4) => 4
[[1],[2],[5],[6]] => [4,3,2,1] => [1,3,2,4] => ([(0,1),(0,2),(1,3),(2,3)],4) => 4
[[1],[3],[4],[6]] => [4,3,2,1] => [1,3,2,4] => ([(0,1),(0,2),(1,3),(2,3)],4) => 4
[[1],[3],[5],[6]] => [4,3,2,1] => [1,3,2,4] => ([(0,1),(0,2),(1,3),(2,3)],4) => 4
[[1],[4],[5],[6]] => [4,3,2,1] => [1,3,2,4] => ([(0,1),(0,2),(1,3),(2,3)],4) => 4
[[2],[3],[4],[6]] => [4,3,2,1] => [1,3,2,4] => ([(0,1),(0,2),(1,3),(2,3)],4) => 4
[[2],[3],[5],[6]] => [4,3,2,1] => [1,3,2,4] => ([(0,1),(0,2),(1,3),(2,3)],4) => 4
[[2],[4],[5],[6]] => [4,3,2,1] => [1,3,2,4] => ([(0,1),(0,2),(1,3),(2,3)],4) => 4
[[3],[4],[5],[6]] => [4,3,2,1] => [1,3,2,4] => ([(0,1),(0,2),(1,3),(2,3)],4) => 4
[[1],[2],[3],[4],[5]] => [5,4,3,2,1] => [1,4,3,2,5] => ([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5) => 1
[[1],[2],[3],[4],[5],[6]] => [6,5,4,3,2,1] => [1,5,4,3,2,6] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,5),(3,5),(4,5)],6) => 1
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Description
The number of 2-Gorenstein indecomposable injective modules in the incidence algebra of the lattice.
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 bottommost row (in English notation).
Map
permutation poset
Description
Sends a permutation to its permutation poset.
For a permutation $\pi$ of length $n$, this poset has vertices
$$\{ (i,\pi(i))\ :\ 1 \leq i \leq n \}$$
and the cover relation is given by $(w, x) \leq (y, z)$ if $w \leq y$ and $x \leq z$.
For example, the permutation $[3,1,5,4,2]$ is mapped to the poset with cover relations
$$\{ (2, 1) \prec (5, 2),\ (2, 1) \prec (4, 4),\ (2, 1) \prec (3, 5),\ (1, 3) \prec (4, 4),\ (1, 3) \prec (3, 5) \}.$$
For a permutation $\pi$ of length $n$, this poset has vertices
$$\{ (i,\pi(i))\ :\ 1 \leq i \leq n \}$$
and the cover relation is given by $(w, x) \leq (y, z)$ if $w \leq y$ and $x \leq z$.
For example, the permutation $[3,1,5,4,2]$ is mapped to the poset with cover relations
$$\{ (2, 1) \prec (5, 2),\ (2, 1) \prec (4, 4),\ (2, 1) \prec (3, 5),\ (1, 3) \prec (4, 4),\ (1, 3) \prec (3, 5) \}.$$
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