Identifier
-
Mp00302:
Parking functions
—insertion tableau⟶
Semistandard tableaux
Mp00075: Semistandard tableaux —reading word permutation⟶ Permutations
Mp00065: Permutations —permutation poset⟶ Posets
St001880: Posets ⟶ ℤ
Values
[1,1,1] => [[1,1,1]] => [1,2,3] => ([(0,2),(2,1)],3) => 3
[1,1,2] => [[1,1,2]] => [1,2,3] => ([(0,2),(2,1)],3) => 3
[1,1,3] => [[1,1,3]] => [1,2,3] => ([(0,2),(2,1)],3) => 3
[1,2,2] => [[1,2,2]] => [1,2,3] => ([(0,2),(2,1)],3) => 3
[1,2,3] => [[1,2,3]] => [1,2,3] => ([(0,2),(2,1)],3) => 3
[1,1,1,1] => [[1,1,1,1]] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4) => 4
[1,1,1,2] => [[1,1,1,2]] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4) => 4
[1,1,1,3] => [[1,1,1,3]] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4) => 4
[1,1,1,4] => [[1,1,1,4]] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4) => 4
[1,1,2,2] => [[1,1,2,2]] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4) => 4
[1,1,2,3] => [[1,1,2,3]] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4) => 4
[1,1,2,4] => [[1,1,2,4]] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4) => 4
[1,1,3,3] => [[1,1,3,3]] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4) => 4
[1,1,3,4] => [[1,1,3,4]] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4) => 4
[1,2,2,2] => [[1,2,2,2]] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4) => 4
[1,2,2,3] => [[1,2,2,3]] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4) => 4
[1,2,2,4] => [[1,2,2,4]] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4) => 4
[1,2,3,3] => [[1,2,3,3]] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4) => 4
[1,2,3,4] => [[1,2,3,4]] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4) => 4
[1,1,1,1,1] => [[1,1,1,1,1]] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5) => 5
[1,1,1,1,2] => [[1,1,1,1,2]] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5) => 5
[1,1,1,1,3] => [[1,1,1,1,3]] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5) => 5
[1,1,1,1,4] => [[1,1,1,1,4]] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5) => 5
[1,1,1,1,5] => [[1,1,1,1,5]] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5) => 5
[1,1,1,2,2] => [[1,1,1,2,2]] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5) => 5
[1,1,1,2,3] => [[1,1,1,2,3]] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5) => 5
[1,1,1,2,4] => [[1,1,1,2,4]] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5) => 5
[1,1,1,2,5] => [[1,1,1,2,5]] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5) => 5
[1,1,1,3,3] => [[1,1,1,3,3]] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5) => 5
[1,1,1,3,4] => [[1,1,1,3,4]] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5) => 5
[1,1,1,3,5] => [[1,1,1,3,5]] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5) => 5
[1,1,1,4,4] => [[1,1,1,4,4]] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5) => 5
[1,1,1,4,5] => [[1,1,1,4,5]] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5) => 5
[1,1,2,2,2] => [[1,1,2,2,2]] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5) => 5
[1,1,2,2,3] => [[1,1,2,2,3]] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5) => 5
[1,1,2,2,4] => [[1,1,2,2,4]] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5) => 5
[1,1,2,2,5] => [[1,1,2,2,5]] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5) => 5
[1,1,2,3,3] => [[1,1,2,3,3]] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5) => 5
[1,1,2,3,4] => [[1,1,2,3,4]] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5) => 5
[1,1,2,3,5] => [[1,1,2,3,5]] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5) => 5
[1,1,2,4,4] => [[1,1,2,4,4]] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5) => 5
[1,1,2,4,5] => [[1,1,2,4,5]] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5) => 5
[1,1,3,3,3] => [[1,1,3,3,3]] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5) => 5
[1,1,3,3,4] => [[1,1,3,3,4]] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5) => 5
[1,1,3,3,5] => [[1,1,3,3,5]] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5) => 5
[1,1,3,4,4] => [[1,1,3,4,4]] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5) => 5
[1,1,3,4,5] => [[1,1,3,4,5]] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5) => 5
[1,2,2,2,2] => [[1,2,2,2,2]] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5) => 5
[1,2,2,2,3] => [[1,2,2,2,3]] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5) => 5
[1,2,2,2,4] => [[1,2,2,2,4]] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5) => 5
[1,2,2,2,5] => [[1,2,2,2,5]] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5) => 5
[1,2,2,3,3] => [[1,2,2,3,3]] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5) => 5
[1,2,2,3,4] => [[1,2,2,3,4]] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5) => 5
[1,2,2,3,5] => [[1,2,2,3,5]] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5) => 5
[1,2,2,4,4] => [[1,2,2,4,4]] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5) => 5
[1,2,2,4,5] => [[1,2,2,4,5]] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5) => 5
[1,2,3,3,3] => [[1,2,3,3,3]] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5) => 5
[1,2,3,3,4] => [[1,2,3,3,4]] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5) => 5
[1,2,3,3,5] => [[1,2,3,3,5]] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5) => 5
[1,2,3,4,4] => [[1,2,3,4,4]] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5) => 5
[1,2,3,4,5] => [[1,2,3,4,5]] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5) => 5
[1,1,1,1,1,1] => [[1,1,1,1,1,1]] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6) => 6
[1,1,1,1,1,2] => [[1,1,1,1,1,2]] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6) => 6
[1,1,1,1,1,3] => [[1,1,1,1,1,3]] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6) => 6
[1,1,1,1,1,4] => [[1,1,1,1,1,4]] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6) => 6
[1,1,1,1,2,2] => [[1,1,1,1,2,2]] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6) => 6
[1,1,1,1,2,3] => [[1,1,1,1,2,3]] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6) => 6
[1,1,1,1,2,4] => [[1,1,1,1,2,4]] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6) => 6
[1,1,1,1,3,3] => [[1,1,1,1,3,3]] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6) => 6
[1,1,1,1,3,4] => [[1,1,1,1,3,4]] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6) => 6
[1,1,1,1,4,4] => [[1,1,1,1,4,4]] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6) => 6
[1,1,1,2,2,2] => [[1,1,1,2,2,2]] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6) => 6
[1,1,1,2,2,3] => [[1,1,1,2,2,3]] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6) => 6
[1,1,1,2,2,4] => [[1,1,1,2,2,4]] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6) => 6
[1,1,1,2,3,3] => [[1,1,1,2,3,3]] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6) => 6
[1,1,1,2,3,4] => [[1,1,1,2,3,4]] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6) => 6
[1,1,1,2,4,4] => [[1,1,1,2,4,4]] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6) => 6
[1,1,1,3,3,3] => [[1,1,1,3,3,3]] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6) => 6
[1,1,1,3,3,4] => [[1,1,1,3,3,4]] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6) => 6
[1,1,1,3,4,4] => [[1,1,1,3,4,4]] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6) => 6
[1,1,1,4,4,4] => [[1,1,1,4,4,4]] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6) => 6
[1,1,2,2,2,2] => [[1,1,2,2,2,2]] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6) => 6
[1,1,2,2,2,3] => [[1,1,2,2,2,3]] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6) => 6
[1,1,2,2,2,4] => [[1,1,2,2,2,4]] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6) => 6
[1,1,2,2,3,3] => [[1,1,2,2,3,3]] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6) => 6
[1,1,2,2,3,4] => [[1,1,2,2,3,4]] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6) => 6
[1,1,2,2,4,4] => [[1,1,2,2,4,4]] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6) => 6
[1,1,2,3,3,3] => [[1,1,2,3,3,3]] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6) => 6
[1,1,2,3,3,4] => [[1,1,2,3,3,4]] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6) => 6
[1,1,2,3,4,4] => [[1,1,2,3,4,4]] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6) => 6
[1,1,2,4,4,4] => [[1,1,2,4,4,4]] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6) => 6
[1,1,3,3,3,3] => [[1,1,3,3,3,3]] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6) => 6
[1,1,3,3,3,4] => [[1,1,3,3,3,4]] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6) => 6
[1,1,3,3,4,4] => [[1,1,3,3,4,4]] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6) => 6
[1,1,3,4,4,4] => [[1,1,3,4,4,4]] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6) => 6
[1,2,2,2,2,2] => [[1,2,2,2,2,2]] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6) => 6
[1,2,2,2,2,3] => [[1,2,2,2,2,3]] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6) => 6
[1,2,2,2,2,4] => [[1,2,2,2,2,4]] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6) => 6
[1,2,2,2,3,3] => [[1,2,2,2,3,3]] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6) => 6
[1,2,2,2,3,4] => [[1,2,2,2,3,4]] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6) => 6
[1,2,2,2,4,4] => [[1,2,2,2,4,4]] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6) => 6
>>> Load all 110 entries. <<<
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
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
insertion tableau
Description
The insertion tableau obtained by applying RSK to the parking function regarded as a word.
searching the database
Sorry, this statistic was not found in the database
or
add this statistic to the database – it's very simple and we need your support!