Identifier
-
Mp00080:
Set partitions
—to permutation⟶
Permutations
Mp00061: Permutations —to increasing tree⟶ Binary trees
Mp00013: Binary trees —to poset⟶ Posets
St001880: Posets ⟶ ℤ
Values
{{1,2,3}} => [2,3,1] => [[.,[.,.]],.] => ([(0,2),(2,1)],3) => 3
{{1,3},{2}} => [3,2,1] => [[[.,.],.],.] => ([(0,2),(2,1)],3) => 3
{{1},{2,3}} => [1,3,2] => [.,[[.,.],.]] => ([(0,2),(2,1)],3) => 3
{{1},{2},{3}} => [1,2,3] => [.,[.,[.,.]]] => ([(0,2),(2,1)],3) => 3
{{1,2,3,4}} => [2,3,4,1] => [[.,[.,[.,.]]],.] => ([(0,3),(2,1),(3,2)],4) => 4
{{1,2,4},{3}} => [2,4,3,1] => [[.,[[.,.],.]],.] => ([(0,3),(2,1),(3,2)],4) => 4
{{1,4},{2,3}} => [4,3,2,1] => [[[[.,.],.],.],.] => ([(0,3),(2,1),(3,2)],4) => 4
{{1},{2,3,4}} => [1,3,4,2] => [.,[[.,[.,.]],.]] => ([(0,3),(2,1),(3,2)],4) => 4
{{1},{2,4},{3}} => [1,4,3,2] => [.,[[[.,.],.],.]] => ([(0,3),(2,1),(3,2)],4) => 4
{{1},{2},{3,4}} => [1,2,4,3] => [.,[.,[[.,.],.]]] => ([(0,3),(2,1),(3,2)],4) => 4
{{1},{2},{3},{4}} => [1,2,3,4] => [.,[.,[.,[.,.]]]] => ([(0,3),(2,1),(3,2)],4) => 4
{{1,2,3,4,5}} => [2,3,4,5,1] => [[.,[.,[.,[.,.]]]],.] => ([(0,4),(2,3),(3,1),(4,2)],5) => 5
{{1,2,3,5},{4}} => [2,3,5,4,1] => [[.,[.,[[.,.],.]]],.] => ([(0,4),(2,3),(3,1),(4,2)],5) => 5
{{1,2,5},{3,4}} => [2,5,4,3,1] => [[.,[[[.,.],.],.]],.] => ([(0,4),(2,3),(3,1),(4,2)],5) => 5
{{1,3,5},{2,4}} => [3,4,5,2,1] => [[[.,[.,[.,.]]],.],.] => ([(0,4),(2,3),(3,1),(4,2)],5) => 5
{{1},{2,3,4,5}} => [1,3,4,5,2] => [.,[[.,[.,[.,.]]],.]] => ([(0,4),(2,3),(3,1),(4,2)],5) => 5
{{1},{2,3,5},{4}} => [1,3,5,4,2] => [.,[[.,[[.,.],.]],.]] => ([(0,4),(2,3),(3,1),(4,2)],5) => 5
{{1,5},{2,4},{3}} => [5,4,3,2,1] => [[[[[.,.],.],.],.],.] => ([(0,4),(2,3),(3,1),(4,2)],5) => 5
{{1},{2,5},{3,4}} => [1,5,4,3,2] => [.,[[[[.,.],.],.],.]] => ([(0,4),(2,3),(3,1),(4,2)],5) => 5
{{1},{2},{3,4,5}} => [1,2,4,5,3] => [.,[.,[[.,[.,.]],.]]] => ([(0,4),(2,3),(3,1),(4,2)],5) => 5
{{1},{2},{3,5},{4}} => [1,2,5,4,3] => [.,[.,[[[.,.],.],.]]] => ([(0,4),(2,3),(3,1),(4,2)],5) => 5
{{1},{2},{3},{4,5}} => [1,2,3,5,4] => [.,[.,[.,[[.,.],.]]]] => ([(0,4),(2,3),(3,1),(4,2)],5) => 5
{{1},{2},{3},{4},{5}} => [1,2,3,4,5] => [.,[.,[.,[.,[.,.]]]]] => ([(0,4),(2,3),(3,1),(4,2)],5) => 5
{{1,2,3,4,5,6}} => [2,3,4,5,6,1] => [[.,[.,[.,[.,[.,.]]]]],.] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6) => 6
{{1,2,3,4,6},{5}} => [2,3,4,6,5,1] => [[.,[.,[.,[[.,.],.]]]],.] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6) => 6
{{1,2,3,6},{4,5}} => [2,3,6,5,4,1] => [[.,[.,[[[.,.],.],.]]],.] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6) => 6
{{1,2,4,6},{3,5}} => [2,4,5,6,3,1] => [[.,[[.,[.,[.,.]]],.]],.] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6) => 6
{{1,2,6},{3,5},{4}} => [2,6,5,4,3,1] => [[.,[[[[.,.],.],.],.]],.] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6) => 6
{{1,3,6},{2,4,5}} => [3,4,6,5,2,1] => [[[.,[.,[[.,.],.]]],.],.] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6) => 6
{{1,3,6},{2,5},{4}} => [3,5,6,4,2,1] => [[[.,[[.,[.,.]],.]],.],.] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6) => 6
{{1},{2,3,4,5,6}} => [1,3,4,5,6,2] => [.,[[.,[.,[.,[.,.]]]],.]] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6) => 6
{{1},{2,3,4,6},{5}} => [1,3,4,6,5,2] => [.,[[.,[.,[[.,.],.]]],.]] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6) => 6
{{1},{2,3,6},{4,5}} => [1,3,6,5,4,2] => [.,[[.,[[[.,.],.],.]],.]] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6) => 6
{{1},{2,4,6},{3,5}} => [1,4,5,6,3,2] => [.,[[[.,[.,[.,.]]],.],.]] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6) => 6
{{1,6},{2,5},{3,4}} => [6,5,4,3,2,1] => [[[[[[.,.],.],.],.],.],.] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6) => 6
{{1},{2},{3,4,5,6}} => [1,2,4,5,6,3] => [.,[.,[[.,[.,[.,.]]],.]]] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6) => 6
{{1},{2},{3,4,6},{5}} => [1,2,4,6,5,3] => [.,[.,[[.,[[.,.],.]],.]]] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6) => 6
{{1},{2,6},{3,5},{4}} => [1,6,5,4,3,2] => [.,[[[[[.,.],.],.],.],.]] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6) => 6
{{1},{2},{3,6},{4,5}} => [1,2,6,5,4,3] => [.,[.,[[[[.,.],.],.],.]]] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6) => 6
{{1},{2},{3},{4,5,6}} => [1,2,3,5,6,4] => [.,[.,[.,[[.,[.,.]],.]]]] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6) => 6
{{1},{2},{3},{4,6},{5}} => [1,2,3,6,5,4] => [.,[.,[.,[[[.,.],.],.]]]] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6) => 6
{{1},{2},{3},{4},{5,6}} => [1,2,3,4,6,5] => [.,[.,[.,[.,[[.,.],.]]]]] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6) => 6
{{1},{2},{3},{4},{5},{6}} => [1,2,3,4,5,6] => [.,[.,[.,[.,[.,[.,.]]]]]] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6) => 6
{{1,2,3,4,5,6,7}} => [2,3,4,5,6,7,1] => [[.,[.,[.,[.,[.,[.,.]]]]]],.] => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7) => 7
{{1,2,3,4,5,7},{6}} => [2,3,4,5,7,6,1] => [[.,[.,[.,[.,[[.,.],.]]]]],.] => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7) => 7
{{1,2,3,4,7},{5,6}} => [2,3,4,7,6,5,1] => [[.,[.,[.,[[[.,.],.],.]]]],.] => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7) => 7
{{1,2,3,5,7},{4,6}} => [2,3,5,6,7,4,1] => [[.,[.,[[.,[.,[.,.]]],.]]],.] => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7) => 7
{{1,2,3,7},{4,6},{5}} => [2,3,7,6,5,4,1] => [[.,[.,[[[[.,.],.],.],.]]],.] => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7) => 7
{{1,2,4,7},{3,5,6}} => [2,4,5,7,6,3,1] => [[.,[[.,[.,[[.,.],.]]],.]],.] => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7) => 7
{{1,2,4,7},{3,6},{5}} => [2,4,6,7,5,3,1] => [[.,[[.,[[.,[.,.]],.]],.]],.] => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7) => 7
{{1,2,7},{3,6},{4,5}} => [2,7,6,5,4,3,1] => [[.,[[[[[.,.],.],.],.],.]],.] => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7) => 7
{{1,3,5,7},{2,4,6}} => [3,4,5,6,7,2,1] => [[[.,[.,[.,[.,[.,.]]]]],.],.] => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7) => 7
{{1,3,7},{2,4,6},{5}} => [3,4,7,6,5,2,1] => [[[.,[.,[[[.,.],.],.]]],.],.] => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7) => 7
{{1,3,7},{2,6},{4,5}} => [3,6,7,5,4,2,1] => [[[.,[[[.,[.,.]],.],.]],.],.] => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7) => 7
{{1},{2,3,4,5,6,7}} => [1,3,4,5,6,7,2] => [.,[[.,[.,[.,[.,[.,.]]]]],.]] => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7) => 7
{{1},{2,3,4,5,7},{6}} => [1,3,4,5,7,6,2] => [.,[[.,[.,[.,[[.,.],.]]]],.]] => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7) => 7
{{1},{2,3,4,7},{5,6}} => [1,3,4,7,6,5,2] => [.,[[.,[.,[[[.,.],.],.]]],.]] => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7) => 7
{{1},{2,3,5,7},{4,6}} => [1,3,5,6,7,4,2] => [.,[[.,[[.,[.,[.,.]]],.]],.]] => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7) => 7
{{1},{2,3,7},{4,6},{5}} => [1,3,7,6,5,4,2] => [.,[[.,[[[[.,.],.],.],.]],.]] => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7) => 7
{{1},{2,4,7},{3,5,6}} => [1,4,5,7,6,3,2] => [.,[[[.,[.,[[.,.],.]]],.],.]] => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7) => 7
{{1},{2,4,7},{3,6},{5}} => [1,4,6,7,5,3,2] => [.,[[[.,[[.,[.,.]],.]],.],.]] => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7) => 7
{{1},{2},{3,4,5,6,7}} => [1,2,4,5,6,7,3] => [.,[.,[[.,[.,[.,[.,.]]]],.]]] => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7) => 7
{{1},{2},{3,4,5,7},{6}} => [1,2,4,5,7,6,3] => [.,[.,[[.,[.,[[.,.],.]]],.]]] => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7) => 7
{{1},{2},{3,4,7},{5,6}} => [1,2,4,7,6,5,3] => [.,[.,[[.,[[[.,.],.],.]],.]]] => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7) => 7
{{1,7},{2,6},{3,5},{4}} => [7,6,5,4,3,2,1] => [[[[[[[.,.],.],.],.],.],.],.] => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7) => 7
{{1},{2},{3,5,7},{4,6}} => [1,2,5,6,7,4,3] => [.,[.,[[[.,[.,[.,.]]],.],.]]] => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7) => 7
{{1},{2,7},{3,6},{4,5}} => [1,7,6,5,4,3,2] => [.,[[[[[[.,.],.],.],.],.],.]] => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7) => 7
{{1},{2},{3},{4,5,6,7}} => [1,2,3,5,6,7,4] => [.,[.,[.,[[.,[.,[.,.]]],.]]]] => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7) => 7
{{1},{2},{3},{4,5,7},{6}} => [1,2,3,5,7,6,4] => [.,[.,[.,[[.,[[.,.],.]],.]]]] => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7) => 7
{{1},{2},{3,7},{4,6},{5}} => [1,2,7,6,5,4,3] => [.,[.,[[[[[.,.],.],.],.],.]]] => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7) => 7
{{1},{2},{3},{4,7},{5,6}} => [1,2,3,7,6,5,4] => [.,[.,[.,[[[[.,.],.],.],.]]]] => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7) => 7
{{1},{2},{3},{4},{5,6,7}} => [1,2,3,4,6,7,5] => [.,[.,[.,[.,[[.,[.,.]],.]]]]] => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7) => 7
{{1},{2},{3},{4},{5,7},{6}} => [1,2,3,4,7,6,5] => [.,[.,[.,[.,[[[.,.],.],.]]]]] => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7) => 7
{{1},{2},{3},{4},{5},{6,7}} => [1,2,3,4,5,7,6] => [.,[.,[.,[.,[.,[[.,.],.]]]]]] => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7) => 7
{{1},{2},{3},{4},{5},{6},{7}} => [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 poset
Description
Return the poset obtained by interpreting the tree as a Hasse diagram.
Map
to increasing tree
Description
Sends a permutation to its associated increasing tree.
This tree is recursively obtained by sending the unique permutation of length $0$ to the empty tree, and sending a permutation $\sigma$ of length $n \geq 1$ to a root node with two subtrees $L$ and $R$ by splitting $\sigma$ at the index $\sigma^{-1}(1)$, normalizing both sides again to permutations and sending the permutations on the left and on the right of $\sigma^{-1}(1)$ to the trees $L$ and $R$, respectively.
This tree is recursively obtained by sending the unique permutation of length $0$ to the empty tree, and sending a permutation $\sigma$ of length $n \geq 1$ to a root node with two subtrees $L$ and $R$ by splitting $\sigma$ at the index $\sigma^{-1}(1)$, normalizing both sides again to permutations and sending the permutations on the left and on the right of $\sigma^{-1}(1)$ to the trees $L$ and $R$, respectively.
Map
to permutation
Description
Sends the set partition to the permutation obtained by considering the blocks as increasing cycles.
searching the database
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