Identifier
-
Mp00080:
Set partitions
—to permutation⟶
Permutations
Mp00061: Permutations —to increasing tree⟶ Binary trees
Mp00011: Binary trees —to graph⟶ Graphs
St000311: Graphs ⟶ ℤ
Values
{{1}} => [1] => [.,.] => ([],1) => 0
{{1,2}} => [2,1] => [[.,.],.] => ([(0,1)],2) => 2
{{1},{2}} => [1,2] => [.,[.,.]] => ([(0,1)],2) => 2
{{1,2,3}} => [2,3,1] => [[.,[.,.]],.] => ([(0,2),(1,2)],3) => 2
{{1,2},{3}} => [2,1,3] => [[.,.],[.,.]] => ([(0,2),(1,2)],3) => 2
{{1,3},{2}} => [3,2,1] => [[[.,.],.],.] => ([(0,2),(1,2)],3) => 2
{{1},{2,3}} => [1,3,2] => [.,[[.,.],.]] => ([(0,2),(1,2)],3) => 2
{{1},{2},{3}} => [1,2,3] => [.,[.,[.,.]]] => ([(0,2),(1,2)],3) => 2
{{1,2,3,4}} => [2,3,4,1] => [[.,[.,[.,.]]],.] => ([(0,3),(1,2),(2,3)],4) => 2
{{1,2,3},{4}} => [2,3,1,4] => [[.,[.,.]],[.,.]] => ([(0,3),(1,2),(2,3)],4) => 2
{{1,2,4},{3}} => [2,4,3,1] => [[.,[[.,.],.]],.] => ([(0,3),(1,2),(2,3)],4) => 2
{{1,2},{3,4}} => [2,1,4,3] => [[.,.],[[.,.],.]] => ([(0,3),(1,2),(2,3)],4) => 2
{{1,2},{3},{4}} => [2,1,3,4] => [[.,.],[.,[.,.]]] => ([(0,3),(1,2),(2,3)],4) => 2
{{1,3,4},{2}} => [3,2,4,1] => [[[.,.],[.,.]],.] => ([(0,3),(1,3),(2,3)],4) => 4
{{1,3},{2,4}} => [3,4,1,2] => [[.,[.,.]],[.,.]] => ([(0,3),(1,2),(2,3)],4) => 2
{{1,3},{2},{4}} => [3,2,1,4] => [[[.,.],.],[.,.]] => ([(0,3),(1,2),(2,3)],4) => 2
{{1,4},{2,3}} => [4,3,2,1] => [[[[.,.],.],.],.] => ([(0,3),(1,2),(2,3)],4) => 2
{{1},{2,3,4}} => [1,3,4,2] => [.,[[.,[.,.]],.]] => ([(0,3),(1,2),(2,3)],4) => 2
{{1},{2,3},{4}} => [1,3,2,4] => [.,[[.,.],[.,.]]] => ([(0,3),(1,3),(2,3)],4) => 4
{{1,4},{2},{3}} => [4,2,3,1] => [[[.,.],[.,.]],.] => ([(0,3),(1,3),(2,3)],4) => 4
{{1},{2,4},{3}} => [1,4,3,2] => [.,[[[.,.],.],.]] => ([(0,3),(1,2),(2,3)],4) => 2
{{1},{2},{3,4}} => [1,2,4,3] => [.,[.,[[.,.],.]]] => ([(0,3),(1,2),(2,3)],4) => 2
{{1},{2},{3},{4}} => [1,2,3,4] => [.,[.,[.,[.,.]]]] => ([(0,3),(1,2),(2,3)],4) => 2
{{1,2,3,4,5}} => [2,3,4,5,1] => [[.,[.,[.,[.,.]]]],.] => ([(0,4),(1,3),(2,3),(2,4)],5) => 2
{{1,2,3,4},{5}} => [2,3,4,1,5] => [[.,[.,[.,.]]],[.,.]] => ([(0,4),(1,3),(2,3),(2,4)],5) => 2
{{1,2,3,5},{4}} => [2,3,5,4,1] => [[.,[.,[[.,.],.]]],.] => ([(0,4),(1,3),(2,3),(2,4)],5) => 2
{{1,2,3},{4,5}} => [2,3,1,5,4] => [[.,[.,.]],[[.,.],.]] => ([(0,4),(1,3),(2,3),(2,4)],5) => 2
{{1,2,3},{4},{5}} => [2,3,1,4,5] => [[.,[.,.]],[.,[.,.]]] => ([(0,4),(1,3),(2,3),(2,4)],5) => 2
{{1,2,4,5},{3}} => [2,4,3,5,1] => [[.,[[.,.],[.,.]]],.] => ([(0,4),(1,4),(2,3),(3,4)],5) => 4
{{1,2,4},{3,5}} => [2,4,5,1,3] => [[.,[.,[.,.]]],[.,.]] => ([(0,4),(1,3),(2,3),(2,4)],5) => 2
{{1,2,4},{3},{5}} => [2,4,3,1,5] => [[.,[[.,.],.]],[.,.]] => ([(0,4),(1,3),(2,3),(2,4)],5) => 2
{{1,2,5},{3,4}} => [2,5,4,3,1] => [[.,[[[.,.],.],.]],.] => ([(0,4),(1,3),(2,3),(2,4)],5) => 2
{{1,2},{3,4,5}} => [2,1,4,5,3] => [[.,.],[[.,[.,.]],.]] => ([(0,4),(1,3),(2,3),(2,4)],5) => 2
{{1,2},{3,4},{5}} => [2,1,4,3,5] => [[.,.],[[.,.],[.,.]]] => ([(0,4),(1,4),(2,3),(3,4)],5) => 4
{{1,2,5},{3},{4}} => [2,5,3,4,1] => [[.,[[.,.],[.,.]]],.] => ([(0,4),(1,4),(2,3),(3,4)],5) => 4
{{1,2},{3,5},{4}} => [2,1,5,4,3] => [[.,.],[[[.,.],.],.]] => ([(0,4),(1,3),(2,3),(2,4)],5) => 2
{{1,2},{3},{4,5}} => [2,1,3,5,4] => [[.,.],[.,[[.,.],.]]] => ([(0,4),(1,3),(2,3),(2,4)],5) => 2
{{1,2},{3},{4},{5}} => [2,1,3,4,5] => [[.,.],[.,[.,[.,.]]]] => ([(0,4),(1,3),(2,3),(2,4)],5) => 2
{{1,3,4,5},{2}} => [3,2,4,5,1] => [[[.,.],[.,[.,.]]],.] => ([(0,4),(1,4),(2,3),(3,4)],5) => 4
{{1,3,4},{2,5}} => [3,5,4,1,2] => [[.,[[.,.],.]],[.,.]] => ([(0,4),(1,3),(2,3),(2,4)],5) => 2
{{1,3,4},{2},{5}} => [3,2,4,1,5] => [[[.,.],[.,.]],[.,.]] => ([(0,4),(1,4),(2,3),(3,4)],5) => 4
{{1,3,5},{2,4}} => [3,4,5,2,1] => [[[.,[.,[.,.]]],.],.] => ([(0,4),(1,3),(2,3),(2,4)],5) => 2
{{1,3},{2,4,5}} => [3,4,1,5,2] => [[.,[.,.]],[[.,.],.]] => ([(0,4),(1,3),(2,3),(2,4)],5) => 2
{{1,3},{2,4},{5}} => [3,4,1,2,5] => [[.,[.,.]],[.,[.,.]]] => ([(0,4),(1,3),(2,3),(2,4)],5) => 2
{{1,3,5},{2},{4}} => [3,2,5,4,1] => [[[.,.],[[.,.],.]],.] => ([(0,4),(1,4),(2,3),(3,4)],5) => 4
{{1,3},{2,5},{4}} => [3,5,1,4,2] => [[.,[.,.]],[[.,.],.]] => ([(0,4),(1,3),(2,3),(2,4)],5) => 2
{{1,3},{2},{4,5}} => [3,2,1,5,4] => [[[.,.],.],[[.,.],.]] => ([(0,4),(1,3),(2,3),(2,4)],5) => 2
{{1,3},{2},{4},{5}} => [3,2,1,4,5] => [[[.,.],.],[.,[.,.]]] => ([(0,4),(1,3),(2,3),(2,4)],5) => 2
{{1,4,5},{2,3}} => [4,3,2,5,1] => [[[[.,.],.],[.,.]],.] => ([(0,4),(1,4),(2,3),(3,4)],5) => 4
{{1,4},{2,3,5}} => [4,3,5,1,2] => [[[.,.],[.,.]],[.,.]] => ([(0,4),(1,4),(2,3),(3,4)],5) => 4
{{1,4},{2,3},{5}} => [4,3,2,1,5] => [[[[.,.],.],.],[.,.]] => ([(0,4),(1,3),(2,3),(2,4)],5) => 2
{{1,5},{2,3,4}} => [5,3,4,2,1] => [[[[.,.],[.,.]],.],.] => ([(0,4),(1,4),(2,3),(3,4)],5) => 4
{{1},{2,3,4,5}} => [1,3,4,5,2] => [.,[[.,[.,[.,.]]],.]] => ([(0,4),(1,3),(2,3),(2,4)],5) => 2
{{1},{2,3,4},{5}} => [1,3,4,2,5] => [.,[[.,[.,.]],[.,.]]] => ([(0,4),(1,4),(2,3),(3,4)],5) => 4
{{1,5},{2,3},{4}} => [5,3,2,4,1] => [[[[.,.],.],[.,.]],.] => ([(0,4),(1,4),(2,3),(3,4)],5) => 4
{{1},{2,3,5},{4}} => [1,3,5,4,2] => [.,[[.,[[.,.],.]],.]] => ([(0,4),(1,3),(2,3),(2,4)],5) => 2
{{1},{2,3},{4,5}} => [1,3,2,5,4] => [.,[[.,.],[[.,.],.]]] => ([(0,4),(1,4),(2,3),(3,4)],5) => 4
{{1},{2,3},{4},{5}} => [1,3,2,4,5] => [.,[[.,.],[.,[.,.]]]] => ([(0,4),(1,4),(2,3),(3,4)],5) => 4
{{1,4,5},{2},{3}} => [4,2,3,5,1] => [[[.,.],[.,[.,.]]],.] => ([(0,4),(1,4),(2,3),(3,4)],5) => 4
{{1,4},{2,5},{3}} => [4,5,3,1,2] => [[[.,[.,.]],.],[.,.]] => ([(0,4),(1,3),(2,3),(2,4)],5) => 2
{{1,4},{2},{3,5}} => [4,2,5,1,3] => [[[.,.],[.,.]],[.,.]] => ([(0,4),(1,4),(2,3),(3,4)],5) => 4
{{1,4},{2},{3},{5}} => [4,2,3,1,5] => [[[.,.],[.,.]],[.,.]] => ([(0,4),(1,4),(2,3),(3,4)],5) => 4
{{1,5},{2,4},{3}} => [5,4,3,2,1] => [[[[[.,.],.],.],.],.] => ([(0,4),(1,3),(2,3),(2,4)],5) => 2
{{1},{2,4,5},{3}} => [1,4,3,5,2] => [.,[[[.,.],[.,.]],.]] => ([(0,4),(1,4),(2,3),(3,4)],5) => 4
{{1},{2,4},{3,5}} => [1,4,5,2,3] => [.,[[.,[.,.]],[.,.]]] => ([(0,4),(1,4),(2,3),(3,4)],5) => 4
{{1},{2,4},{3},{5}} => [1,4,3,2,5] => [.,[[[.,.],.],[.,.]]] => ([(0,4),(1,4),(2,3),(3,4)],5) => 4
{{1,5},{2},{3,4}} => [5,2,4,3,1] => [[[.,.],[[.,.],.]],.] => ([(0,4),(1,4),(2,3),(3,4)],5) => 4
{{1},{2,5},{3,4}} => [1,5,4,3,2] => [.,[[[[.,.],.],.],.]] => ([(0,4),(1,3),(2,3),(2,4)],5) => 2
{{1},{2},{3,4,5}} => [1,2,4,5,3] => [.,[.,[[.,[.,.]],.]]] => ([(0,4),(1,3),(2,3),(2,4)],5) => 2
{{1},{2},{3,4},{5}} => [1,2,4,3,5] => [.,[.,[[.,.],[.,.]]]] => ([(0,4),(1,4),(2,3),(3,4)],5) => 4
{{1,5},{2},{3},{4}} => [5,2,3,4,1] => [[[.,.],[.,[.,.]]],.] => ([(0,4),(1,4),(2,3),(3,4)],5) => 4
{{1},{2,5},{3},{4}} => [1,5,3,4,2] => [.,[[[.,.],[.,.]],.]] => ([(0,4),(1,4),(2,3),(3,4)],5) => 4
{{1},{2},{3,5},{4}} => [1,2,5,4,3] => [.,[.,[[[.,.],.],.]]] => ([(0,4),(1,3),(2,3),(2,4)],5) => 2
{{1},{2},{3},{4,5}} => [1,2,3,5,4] => [.,[.,[.,[[.,.],.]]]] => ([(0,4),(1,3),(2,3),(2,4)],5) => 2
{{1},{2},{3},{4},{5}} => [1,2,3,4,5] => [.,[.,[.,[.,[.,.]]]]] => ([(0,4),(1,3),(2,3),(2,4)],5) => 2
{{1,2,3,4,5,6}} => [2,3,4,5,6,1] => [[.,[.,[.,[.,[.,.]]]]],.] => ([(0,5),(1,4),(2,3),(2,4),(3,5)],6) => 2
{{1,2,3,4,5},{6}} => [2,3,4,5,1,6] => [[.,[.,[.,[.,.]]]],[.,.]] => ([(0,5),(1,4),(2,3),(2,4),(3,5)],6) => 2
{{1,2,3,4,6},{5}} => [2,3,4,6,5,1] => [[.,[.,[.,[[.,.],.]]]],.] => ([(0,5),(1,4),(2,3),(2,4),(3,5)],6) => 2
{{1,2,3,4},{5,6}} => [2,3,4,1,6,5] => [[.,[.,[.,.]]],[[.,.],.]] => ([(0,5),(1,4),(2,3),(2,4),(3,5)],6) => 2
{{1,2,3,4},{5},{6}} => [2,3,4,1,5,6] => [[.,[.,[.,.]]],[.,[.,.]]] => ([(0,5),(1,4),(2,3),(2,4),(3,5)],6) => 2
{{1,2,3,5,6},{4}} => [2,3,5,4,6,1] => [[.,[.,[[.,.],[.,.]]]],.] => ([(0,5),(1,5),(2,3),(3,4),(4,5)],6) => 4
{{1,2,3,5},{4,6}} => [2,3,5,6,1,4] => [[.,[.,[.,[.,.]]]],[.,.]] => ([(0,5),(1,4),(2,3),(2,4),(3,5)],6) => 2
{{1,2,3,5},{4},{6}} => [2,3,5,4,1,6] => [[.,[.,[[.,.],.]]],[.,.]] => ([(0,5),(1,4),(2,3),(2,4),(3,5)],6) => 2
{{1,2,3,6},{4,5}} => [2,3,6,5,4,1] => [[.,[.,[[[.,.],.],.]]],.] => ([(0,5),(1,4),(2,3),(2,4),(3,5)],6) => 2
{{1,2,3},{4,5,6}} => [2,3,1,5,6,4] => [[.,[.,.]],[[.,[.,.]],.]] => ([(0,5),(1,4),(2,3),(2,4),(3,5)],6) => 2
{{1,2,3},{4,5},{6}} => [2,3,1,5,4,6] => [[.,[.,.]],[[.,.],[.,.]]] => ([(0,5),(1,5),(2,3),(3,4),(4,5)],6) => 4
{{1,2,3,6},{4},{5}} => [2,3,6,4,5,1] => [[.,[.,[[.,.],[.,.]]]],.] => ([(0,5),(1,5),(2,3),(3,4),(4,5)],6) => 4
{{1,2,3},{4,6},{5}} => [2,3,1,6,5,4] => [[.,[.,.]],[[[.,.],.],.]] => ([(0,5),(1,4),(2,3),(2,4),(3,5)],6) => 2
{{1,2,3},{4},{5,6}} => [2,3,1,4,6,5] => [[.,[.,.]],[.,[[.,.],.]]] => ([(0,5),(1,4),(2,3),(2,4),(3,5)],6) => 2
{{1,2,3},{4},{5},{6}} => [2,3,1,4,5,6] => [[.,[.,.]],[.,[.,[.,.]]]] => ([(0,5),(1,4),(2,3),(2,4),(3,5)],6) => 2
{{1,2,4,5,6},{3}} => [2,4,3,5,6,1] => [[.,[[.,.],[.,[.,.]]]],.] => ([(0,5),(1,4),(2,3),(3,5),(4,5)],6) => 4
{{1,2,4,5},{3,6}} => [2,4,6,5,1,3] => [[.,[.,[[.,.],.]]],[.,.]] => ([(0,5),(1,4),(2,3),(2,4),(3,5)],6) => 2
{{1,2,4,5},{3},{6}} => [2,4,3,5,1,6] => [[.,[[.,.],[.,.]]],[.,.]] => ([(0,5),(1,5),(2,3),(3,4),(4,5)],6) => 4
{{1,2,4,6},{3,5}} => [2,4,5,6,3,1] => [[.,[[.,[.,[.,.]]],.]],.] => ([(0,5),(1,4),(2,3),(2,4),(3,5)],6) => 2
{{1,2,4},{3,5,6}} => [2,4,5,1,6,3] => [[.,[.,[.,.]]],[[.,.],.]] => ([(0,5),(1,4),(2,3),(2,4),(3,5)],6) => 2
{{1,2,4},{3,5},{6}} => [2,4,5,1,3,6] => [[.,[.,[.,.]]],[.,[.,.]]] => ([(0,5),(1,4),(2,3),(2,4),(3,5)],6) => 2
{{1,2,4,6},{3},{5}} => [2,4,3,6,5,1] => [[.,[[.,.],[[.,.],.]]],.] => ([(0,5),(1,4),(2,3),(3,5),(4,5)],6) => 4
{{1,2,4},{3,6},{5}} => [2,4,6,1,5,3] => [[.,[.,[.,.]]],[[.,.],.]] => ([(0,5),(1,4),(2,3),(2,4),(3,5)],6) => 2
{{1,2,4},{3},{5,6}} => [2,4,3,1,6,5] => [[.,[[.,.],.]],[[.,.],.]] => ([(0,5),(1,4),(2,3),(2,4),(3,5)],6) => 2
{{1,2,4},{3},{5},{6}} => [2,4,3,1,5,6] => [[.,[[.,.],.]],[.,[.,.]]] => ([(0,5),(1,4),(2,3),(2,4),(3,5)],6) => 2
{{1,2,5,6},{3,4}} => [2,5,4,3,6,1] => [[.,[[[.,.],.],[.,.]]],.] => ([(0,5),(1,4),(2,3),(3,5),(4,5)],6) => 4
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Description
The number of vertices of odd degree in a graph.
Map
to permutation
Description
Sends the set partition to the permutation obtained by considering the blocks as increasing cycles.
Map
to graph
Description
Return the undirected graph obtained from the tree nodes and edges, with leaves being ignored.
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.
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