Identifier
-
Mp00023:
Dyck paths
—to non-crossing permutation⟶
Permutations
Mp00064: Permutations —reverse⟶ Permutations
Mp00065: Permutations —permutation poset⟶ Posets
St001880: Posets ⟶ ℤ
Values
[1,1,1,0,0,0] => [3,2,1] => [1,2,3] => ([(0,2),(2,1)],3) => 3
[1,1,1,0,1,0,0,0] => [4,2,3,1] => [1,3,2,4] => ([(0,1),(0,2),(1,3),(2,3)],4) => 4
[1,1,1,1,0,0,0,0] => [4,3,2,1] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4) => 4
[1,1,1,0,1,0,1,0,0,0] => [5,2,3,4,1] => [1,4,3,2,5] => ([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5) => 1
[1,1,1,0,1,1,0,0,0,0] => [5,2,4,3,1] => [1,3,4,2,5] => ([(0,2),(0,3),(1,4),(2,4),(3,1)],5) => 4
[1,1,1,1,0,0,1,0,0,0] => [5,3,2,4,1] => [1,4,2,3,5] => ([(0,2),(0,3),(1,4),(2,4),(3,1)],5) => 4
[1,1,1,1,0,1,0,0,0,0] => [5,3,4,2,1] => [1,2,4,3,5] => ([(0,3),(1,4),(2,4),(3,1),(3,2)],5) => 5
[1,1,1,1,1,0,0,0,0,0] => [5,4,3,2,1] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5) => 5
[1,1,1,0,1,0,1,0,1,0,0,0] => [6,2,3,4,5,1] => [1,5,4,3,2,6] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,5),(3,5),(4,5)],6) => 1
[1,1,1,0,1,0,1,1,0,0,0,0] => [6,2,3,5,4,1] => [1,4,5,3,2,6] => ([(0,2),(0,3),(0,4),(1,5),(2,5),(3,5),(4,1)],6) => 1
[1,1,1,0,1,1,0,0,1,0,0,0] => [6,2,4,3,5,1] => [1,5,3,4,2,6] => ([(0,2),(0,3),(0,4),(1,5),(2,5),(3,5),(4,1)],6) => 1
[1,1,1,0,1,1,0,1,0,0,0,0] => [6,2,4,5,3,1] => [1,3,5,4,2,6] => ([(0,3),(0,4),(1,5),(2,5),(3,5),(4,1),(4,2)],6) => 2
[1,1,1,0,1,1,1,0,0,0,0,0] => [6,2,5,4,3,1] => [1,3,4,5,2,6] => ([(0,2),(0,4),(1,5),(2,5),(3,1),(4,3)],6) => 4
[1,1,1,1,0,0,1,0,1,0,0,0] => [6,3,2,4,5,1] => [1,5,4,2,3,6] => ([(0,2),(0,3),(0,4),(1,5),(2,5),(3,5),(4,1)],6) => 1
[1,1,1,1,0,0,1,1,0,0,0,0] => [6,3,2,5,4,1] => [1,4,5,2,3,6] => ([(0,3),(0,4),(1,5),(2,5),(3,2),(4,1)],6) => 4
[1,1,1,1,0,1,0,0,1,0,0,0] => [6,3,4,2,5,1] => [1,5,2,4,3,6] => ([(0,3),(0,4),(1,5),(2,5),(3,5),(4,1),(4,2)],6) => 2
[1,1,1,1,0,1,0,1,0,0,0,0] => [6,3,4,5,2,1] => [1,2,5,4,3,6] => ([(0,4),(1,5),(2,5),(3,5),(4,1),(4,2),(4,3)],6) => 2
[1,1,1,1,0,1,1,0,0,0,0,0] => [6,3,5,4,2,1] => [1,2,4,5,3,6] => ([(0,4),(1,5),(2,5),(3,2),(4,1),(4,3)],6) => 5
[1,1,1,1,1,0,0,0,1,0,0,0] => [6,4,3,2,5,1] => [1,5,2,3,4,6] => ([(0,2),(0,4),(1,5),(2,5),(3,1),(4,3)],6) => 4
[1,1,1,1,1,0,0,1,0,0,0,0] => [6,4,3,5,2,1] => [1,2,5,3,4,6] => ([(0,4),(1,5),(2,5),(3,2),(4,1),(4,3)],6) => 5
[1,1,1,1,1,0,1,0,0,0,0,0] => [6,5,3,4,2,1] => [1,2,4,3,5,6] => ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6) => 6
[1,1,1,1,1,1,0,0,0,0,0,0] => [6,5,4,3,2,1] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6) => 6
[1,1,1,0,1,0,1,0,1,0,1,0,0,0] => [7,2,3,4,5,6,1] => [1,6,5,4,3,2,7] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,6),(2,6),(3,6),(4,6),(5,6)],7) => 1
[1,1,1,0,1,0,1,0,1,1,0,0,0,0] => [7,2,3,4,6,5,1] => [1,5,6,4,3,2,7] => ([(0,2),(0,3),(0,4),(0,5),(1,6),(2,6),(3,6),(4,6),(5,1)],7) => 1
[1,1,1,0,1,0,1,1,0,0,1,0,0,0] => [7,2,3,5,4,6,1] => [1,6,4,5,3,2,7] => ([(0,2),(0,3),(0,4),(0,5),(1,6),(2,6),(3,6),(4,6),(5,1)],7) => 1
[1,1,1,0,1,0,1,1,0,1,0,0,0,0] => [7,2,3,5,6,4,1] => [1,4,6,5,3,2,7] => ([(0,3),(0,4),(0,5),(1,6),(2,6),(3,6),(4,6),(5,1),(5,2)],7) => 1
[1,1,1,0,1,0,1,1,1,0,0,0,0,0] => [7,2,3,6,5,4,1] => [1,4,5,6,3,2,7] => ([(0,2),(0,3),(0,5),(1,6),(2,6),(3,6),(4,1),(5,4)],7) => 1
[1,1,1,0,1,1,0,0,1,0,1,0,0,0] => [7,2,4,3,5,6,1] => [1,6,5,3,4,2,7] => ([(0,2),(0,3),(0,4),(0,5),(1,6),(2,6),(3,6),(4,6),(5,1)],7) => 1
[1,1,1,0,1,1,0,0,1,1,0,0,0,0] => [7,2,4,3,6,5,1] => [1,5,6,3,4,2,7] => ([(0,3),(0,4),(0,5),(1,6),(2,6),(3,6),(4,2),(5,1)],7) => 1
[1,1,1,0,1,1,0,1,0,0,1,0,0,0] => [7,2,4,5,3,6,1] => [1,6,3,5,4,2,7] => ([(0,3),(0,4),(0,5),(1,6),(2,6),(3,6),(4,6),(5,1),(5,2)],7) => 1
[1,1,1,0,1,1,0,1,0,1,0,0,0,0] => [7,2,4,5,6,3,1] => [1,3,6,5,4,2,7] => ([(0,4),(0,5),(1,6),(2,6),(3,6),(4,6),(5,1),(5,2),(5,3)],7) => 2
[1,1,1,0,1,1,0,1,1,0,0,0,0,0] => [7,2,4,6,5,3,1] => [1,3,5,6,4,2,7] => ([(0,3),(0,5),(1,6),(2,6),(3,6),(4,2),(5,1),(5,4)],7) => 2
[1,1,1,0,1,1,1,0,0,0,1,0,0,0] => [7,2,5,4,3,6,1] => [1,6,3,4,5,2,7] => ([(0,2),(0,3),(0,5),(1,6),(2,6),(3,6),(4,1),(5,4)],7) => 1
[1,1,1,0,1,1,1,0,0,1,0,0,0,0] => [7,2,5,4,6,3,1] => [1,3,6,4,5,2,7] => ([(0,3),(0,5),(1,6),(2,6),(3,6),(4,2),(5,1),(5,4)],7) => 2
[1,1,1,0,1,1,1,0,1,0,0,0,0,0] => [7,2,6,4,5,3,1] => [1,3,5,4,6,2,7] => ([(0,3),(0,4),(1,5),(2,5),(3,6),(4,1),(4,2),(5,6)],7) => 4
[1,1,1,0,1,1,1,1,0,0,0,0,0,0] => [7,2,6,5,4,3,1] => [1,3,4,5,6,2,7] => ([(0,2),(0,5),(1,6),(2,6),(3,4),(4,1),(5,3)],7) => 4
[1,1,1,1,0,0,1,0,1,0,1,0,0,0] => [7,3,2,4,5,6,1] => [1,6,5,4,2,3,7] => ([(0,2),(0,3),(0,4),(0,5),(1,6),(2,6),(3,6),(4,6),(5,1)],7) => 1
[1,1,1,1,0,0,1,0,1,1,0,0,0,0] => [7,3,2,4,6,5,1] => [1,5,6,4,2,3,7] => ([(0,3),(0,4),(0,5),(1,6),(2,6),(3,6),(4,2),(5,1)],7) => 1
[1,1,1,1,0,0,1,1,0,0,1,0,0,0] => [7,3,2,5,4,6,1] => [1,6,4,5,2,3,7] => ([(0,3),(0,4),(0,5),(1,6),(2,6),(3,6),(4,2),(5,1)],7) => 1
[1,1,1,1,0,0,1,1,0,1,0,0,0,0] => [7,3,2,5,6,4,1] => [1,4,6,5,2,3,7] => ([(0,4),(0,5),(1,6),(2,6),(3,6),(4,3),(5,1),(5,2)],7) => 2
[1,1,1,1,0,0,1,1,1,0,0,0,0,0] => [7,3,2,6,5,4,1] => [1,4,5,6,2,3,7] => ([(0,4),(0,5),(1,6),(2,6),(3,2),(4,3),(5,1)],7) => 4
[1,1,1,1,0,1,0,0,1,0,1,0,0,0] => [7,3,4,2,5,6,1] => [1,6,5,2,4,3,7] => ([(0,3),(0,4),(0,5),(1,6),(2,6),(3,6),(4,6),(5,1),(5,2)],7) => 1
[1,1,1,1,0,1,0,0,1,1,0,0,0,0] => [7,3,4,2,6,5,1] => [1,5,6,2,4,3,7] => ([(0,4),(0,5),(1,6),(2,6),(3,6),(4,3),(5,1),(5,2)],7) => 2
[1,1,1,1,0,1,0,1,0,0,1,0,0,0] => [7,3,4,5,2,6,1] => [1,6,2,5,4,3,7] => ([(0,4),(0,5),(1,6),(2,6),(3,6),(4,6),(5,1),(5,2),(5,3)],7) => 2
[1,1,1,1,0,1,0,1,0,1,0,0,0,0] => [7,3,4,5,6,2,1] => [1,2,6,5,4,3,7] => ([(0,5),(1,6),(2,6),(3,6),(4,6),(5,1),(5,2),(5,3),(5,4)],7) => 2
[1,1,1,1,0,1,0,1,1,0,0,0,0,0] => [7,3,4,6,5,2,1] => [1,2,5,6,4,3,7] => ([(0,5),(1,6),(2,6),(3,6),(4,3),(5,1),(5,2),(5,4)],7) => 2
[1,1,1,1,0,1,1,0,0,0,1,0,0,0] => [7,3,5,4,2,6,1] => [1,6,2,4,5,3,7] => ([(0,3),(0,5),(1,6),(2,6),(3,6),(4,2),(5,1),(5,4)],7) => 2
[1,1,1,1,0,1,1,0,0,1,0,0,0,0] => [7,3,5,4,6,2,1] => [1,2,6,4,5,3,7] => ([(0,5),(1,6),(2,6),(3,6),(4,3),(5,1),(5,2),(5,4)],7) => 2
[1,1,1,1,0,1,1,0,1,0,0,0,0,0] => [7,3,6,4,5,2,1] => [1,2,5,4,6,3,7] => ([(0,4),(1,6),(2,5),(3,5),(4,1),(4,2),(4,3),(5,6)],7) => 3
[1,1,1,1,0,1,1,1,0,0,0,0,0,0] => [7,3,6,5,4,2,1] => [1,2,4,5,6,3,7] => ([(0,5),(1,6),(2,6),(3,4),(4,2),(5,1),(5,3)],7) => 5
[1,1,1,1,1,0,0,0,1,0,1,0,0,0] => [7,4,3,2,5,6,1] => [1,6,5,2,3,4,7] => ([(0,2),(0,3),(0,5),(1,6),(2,6),(3,6),(4,1),(5,4)],7) => 1
[1,1,1,1,1,0,0,0,1,1,0,0,0,0] => [7,4,3,2,6,5,1] => [1,5,6,2,3,4,7] => ([(0,4),(0,5),(1,6),(2,6),(3,2),(4,3),(5,1)],7) => 4
[1,1,1,1,1,0,0,1,0,0,1,0,0,0] => [7,4,3,5,2,6,1] => [1,6,2,5,3,4,7] => ([(0,3),(0,5),(1,6),(2,6),(3,6),(4,2),(5,1),(5,4)],7) => 2
[1,1,1,1,1,0,0,1,0,1,0,0,0,0] => [7,4,3,5,6,2,1] => [1,2,6,5,3,4,7] => ([(0,5),(1,6),(2,6),(3,6),(4,3),(5,1),(5,2),(5,4)],7) => 2
[1,1,1,1,1,0,0,1,1,0,0,0,0,0] => [7,4,3,6,5,2,1] => [1,2,5,6,3,4,7] => ([(0,5),(1,6),(2,6),(3,2),(4,1),(5,3),(5,4)],7) => 5
[1,1,1,1,1,0,1,0,0,0,1,0,0,0] => [7,5,3,4,2,6,1] => [1,6,2,4,3,5,7] => ([(0,3),(0,4),(1,5),(2,5),(3,6),(4,1),(4,2),(5,6)],7) => 4
[1,1,1,1,1,0,1,0,0,1,0,0,0,0] => [7,5,3,4,6,2,1] => [1,2,6,4,3,5,7] => ([(0,4),(1,6),(2,5),(3,5),(4,1),(4,2),(4,3),(5,6)],7) => 3
[1,1,1,1,1,0,1,0,1,0,0,0,0,0] => [7,6,3,4,5,2,1] => [1,2,5,4,3,6,7] => ([(0,5),(1,6),(2,6),(3,6),(5,1),(5,2),(5,3),(6,4)],7) => 3
[1,1,1,1,1,0,1,1,0,0,0,0,0,0] => [7,6,3,5,4,2,1] => [1,2,4,5,3,6,7] => ([(0,5),(1,6),(2,6),(4,2),(5,1),(5,4),(6,3)],7) => 6
[1,1,1,1,1,1,0,0,0,0,1,0,0,0] => [7,5,4,3,2,6,1] => [1,6,2,3,4,5,7] => ([(0,2),(0,5),(1,6),(2,6),(3,4),(4,1),(5,3)],7) => 4
[1,1,1,1,1,1,0,0,0,1,0,0,0,0] => [7,5,4,3,6,2,1] => [1,2,6,3,4,5,7] => ([(0,5),(1,6),(2,6),(3,4),(4,2),(5,1),(5,3)],7) => 5
[1,1,1,1,1,1,0,0,1,0,0,0,0,0] => [7,6,4,3,5,2,1] => [1,2,5,3,4,6,7] => ([(0,5),(1,6),(2,6),(4,2),(5,1),(5,4),(6,3)],7) => 6
[1,1,1,1,1,1,0,1,0,0,0,0,0,0] => [7,6,4,5,3,2,1] => [1,2,3,5,4,6,7] => ([(0,4),(1,6),(2,6),(4,5),(5,1),(5,2),(6,3)],7) => 7
[1,1,1,1,1,1,1,0,0,0,0,0,0,0] => [7,6,5,4,3,2,1] => [1,2,3,4,5,6,7] => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7) => 7
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
to non-crossing permutation
Description
Sends a Dyck path $D$ with valley at positions $\{(i_1,j_1),\ldots,(i_k,j_k)\}$ to the unique non-crossing permutation $\pi$ having descents $\{i_1,\ldots,i_k\}$ and whose inverse has descents $\{j_1,\ldots,j_k\}$.
It sends the area St000012The area of a Dyck path. to the number of inversions St000018The number of inversions of a permutation. and the major index St000027The major index of a Dyck path. to $n(n-1)$ minus the sum of the major index St000004The major index of a permutation. and the inverse major index St000305The inverse major index of a permutation..
It sends the area St000012The area of a Dyck path. to the number of inversions St000018The number of inversions of a permutation. and the major index St000027The major index of a Dyck path. to $n(n-1)$ minus the sum of the major index St000004The major index of a permutation. and the inverse major index St000305The inverse major index of a permutation..
Map
reverse
Description
Sends a permutation to its reverse.
The reverse of a permutation $\sigma$ of length $n$ is given by $\tau$ with $\tau(i) = \sigma(n+1-i)$.
The reverse of a permutation $\sigma$ of length $n$ is given by $\tau$ with $\tau(i) = \sigma(n+1-i)$.
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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