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
search for individual values
searching the database for the individual values of this statistic
Description
The number of 2-Gorenstein indecomposable injective modules in the incidence algebra of the lattice.
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
$$\{ (2, 1) \prec (5, 2),\ (2, 1) \prec (4, 4),\ (2, 1) \prec (3, 5),\ (1, 3) \prec (4, 4),\ (1, 3) \prec (3, 5) \}.$$
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).
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