Identifier
-
Mp00080:
Set partitions
—to permutation⟶
Permutations
Mp00239: Permutations —Corteel⟶ Permutations
Mp00065: Permutations —permutation poset⟶ Posets
St001964: Posets ⟶ ℤ
Values
{{1}} => [1] => [1] => ([],1) => 0
{{1,2}} => [2,1] => [2,1] => ([],2) => 0
{{1},{2}} => [1,2] => [1,2] => ([(0,1)],2) => 0
{{1,2,3}} => [2,3,1] => [3,2,1] => ([],3) => 0
{{1,2},{3}} => [2,1,3] => [2,1,3] => ([(0,2),(1,2)],3) => 0
{{1},{2,3}} => [1,3,2] => [1,3,2] => ([(0,1),(0,2)],3) => 0
{{1},{2},{3}} => [1,2,3] => [1,2,3] => ([(0,2),(2,1)],3) => 0
{{1,2,3},{4}} => [2,3,1,4] => [3,2,1,4] => ([(0,3),(1,3),(2,3)],4) => 1
{{1,2,4},{3}} => [2,4,3,1] => [3,2,4,1] => ([(1,3),(2,3)],4) => 0
{{1,2},{3,4}} => [2,1,4,3] => [2,1,4,3] => ([(0,2),(0,3),(1,2),(1,3)],4) => 2
{{1,2},{3},{4}} => [2,1,3,4] => [2,1,3,4] => ([(0,3),(1,3),(3,2)],4) => 1
{{1,3},{2,4}} => [3,4,1,2] => [4,3,2,1] => ([],4) => 0
{{1,3},{2},{4}} => [3,2,1,4] => [2,3,1,4] => ([(0,3),(1,2),(2,3)],4) => 0
{{1},{2,3,4}} => [1,3,4,2] => [1,4,3,2] => ([(0,1),(0,2),(0,3)],4) => 1
{{1},{2,3},{4}} => [1,3,2,4] => [1,3,2,4] => ([(0,1),(0,2),(1,3),(2,3)],4) => 0
{{1,4},{2},{3}} => [4,2,3,1] => [2,3,4,1] => ([(1,2),(2,3)],4) => 0
{{1},{2,4},{3}} => [1,4,3,2] => [1,3,4,2] => ([(0,2),(0,3),(3,1)],4) => 0
{{1},{2},{3,4}} => [1,2,4,3] => [1,2,4,3] => ([(0,3),(3,1),(3,2)],4) => 1
{{1},{2},{3},{4}} => [1,2,3,4] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4) => 0
{{1,2,3,4,5}} => [2,3,4,5,1] => [5,2,3,4,1] => ([(2,3),(3,4)],5) => 0
{{1,2,3,4},{5}} => [2,3,4,1,5] => [4,2,3,1,5] => ([(0,4),(1,4),(2,3),(3,4)],5) => 1
{{1,2,3,5},{4}} => [2,3,5,4,1] => [4,2,3,5,1] => ([(1,4),(2,3),(3,4)],5) => 0
{{1,2,3},{4,5}} => [2,3,1,5,4] => [3,2,1,5,4] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5) => 3
{{1,2,3},{4},{5}} => [2,3,1,4,5] => [3,2,1,4,5] => ([(0,4),(1,4),(2,4),(4,3)],5) => 2
{{1,2,4},{3},{5}} => [2,4,3,1,5] => [3,2,4,1,5] => ([(0,4),(1,3),(2,3),(3,4)],5) => 1
{{1,2,5},{3,4}} => [2,5,4,3,1] => [4,2,5,1,3] => ([(0,4),(1,3),(2,3),(2,4)],5) => 0
{{1,2},{3,4,5}} => [2,1,4,5,3] => [2,1,5,4,3] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4)],5) => 3
{{1,2},{3,4},{5}} => [2,1,4,3,5] => [2,1,4,3,5] => ([(0,3),(0,4),(1,3),(1,4),(3,2),(4,2)],5) => 2
{{1,2,5},{3},{4}} => [2,5,3,4,1] => [3,2,4,5,1] => ([(1,4),(2,4),(4,3)],5) => 1
{{1,2},{3,5},{4}} => [2,1,5,4,3] => [2,1,4,5,3] => ([(0,3),(0,4),(1,3),(1,4),(4,2)],5) => 2
{{1,2},{3},{4,5}} => [2,1,3,5,4] => [2,1,3,5,4] => ([(0,4),(1,4),(4,2),(4,3)],5) => 2
{{1,2},{3},{4},{5}} => [2,1,3,4,5] => [2,1,3,4,5] => ([(0,4),(1,4),(2,3),(4,2)],5) => 1
{{1,3,4},{2},{5}} => [3,2,4,1,5] => [2,4,3,1,5] => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5) => 1
{{1,3},{2,4,5}} => [3,4,1,5,2] => [5,3,2,4,1] => ([(2,4),(3,4)],5) => 0
{{1,3},{2,4},{5}} => [3,4,1,2,5] => [4,3,2,1,5] => ([(0,4),(1,4),(2,4),(3,4)],5) => 2
{{1,3},{2},{4,5}} => [3,2,1,5,4] => [2,3,1,5,4] => ([(0,3),(0,4),(1,2),(2,3),(2,4)],5) => 2
{{1,3},{2},{4},{5}} => [3,2,1,4,5] => [2,3,1,4,5] => ([(0,4),(1,2),(2,4),(4,3)],5) => 1
{{1,4,5},{2,3}} => [4,3,2,5,1] => [3,5,1,4,2] => ([(0,3),(0,4),(1,2),(1,4)],5) => 0
{{1,4},{2,3},{5}} => [4,3,2,1,5] => [3,4,1,2,5] => ([(0,3),(1,2),(2,4),(3,4)],5) => 0
{{1},{2,3,4,5}} => [1,3,4,5,2] => [1,5,3,4,2] => ([(0,2),(0,3),(0,4),(4,1)],5) => 1
{{1},{2,3,4},{5}} => [1,3,4,2,5] => [1,4,3,2,5] => ([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5) => 1
{{1,5},{2,3},{4}} => [5,3,2,4,1] => [3,4,1,5,2] => ([(0,3),(1,2),(1,4),(3,4)],5) => 0
{{1},{2,3,5},{4}} => [1,3,5,4,2] => [1,4,3,5,2] => ([(0,1),(0,2),(0,3),(2,4),(3,4)],5) => 1
{{1},{2,3},{4,5}} => [1,3,2,5,4] => [1,3,2,5,4] => ([(0,1),(0,2),(1,3),(1,4),(2,3),(2,4)],5) => 2
{{1},{2,3},{4},{5}} => [1,3,2,4,5] => [1,3,2,4,5] => ([(0,2),(0,3),(2,4),(3,4),(4,1)],5) => 1
{{1,4},{2,5},{3}} => [4,5,3,1,2] => [4,3,5,2,1] => ([(2,4),(3,4)],5) => 0
{{1,4},{2},{3},{5}} => [4,2,3,1,5] => [2,3,4,1,5] => ([(0,4),(1,2),(2,3),(3,4)],5) => 0
{{1,5},{2,4},{3}} => [5,4,3,2,1] => [3,4,5,1,2] => ([(0,3),(1,4),(4,2)],5) => 0
{{1},{2,4,5},{3}} => [1,4,3,5,2] => [1,3,5,4,2] => ([(0,3),(0,4),(4,1),(4,2)],5) => 1
{{1},{2,4},{3,5}} => [1,4,5,2,3] => [1,5,4,3,2] => ([(0,1),(0,2),(0,3),(0,4)],5) => 2
{{1},{2,4},{3},{5}} => [1,4,3,2,5] => [1,3,4,2,5] => ([(0,2),(0,3),(1,4),(2,4),(3,1)],5) => 0
{{1,5},{2},{3,4}} => [5,2,4,3,1] => [2,4,5,1,3] => ([(0,4),(1,2),(1,4),(2,3)],5) => 0
{{1},{2,5},{3,4}} => [1,5,4,3,2] => [1,4,5,2,3] => ([(0,3),(0,4),(3,2),(4,1)],5) => 0
{{1},{2},{3,4,5}} => [1,2,4,5,3] => [1,2,5,4,3] => ([(0,4),(4,1),(4,2),(4,3)],5) => 2
{{1},{2},{3,4},{5}} => [1,2,4,3,5] => [1,2,4,3,5] => ([(0,3),(1,4),(2,4),(3,1),(3,2)],5) => 1
{{1,5},{2},{3},{4}} => [5,2,3,4,1] => [2,3,4,5,1] => ([(1,4),(3,2),(4,3)],5) => 0
{{1},{2,5},{3},{4}} => [1,5,3,4,2] => [1,3,4,5,2] => ([(0,2),(0,4),(3,1),(4,3)],5) => 0
{{1},{2},{3,5},{4}} => [1,2,5,4,3] => [1,2,4,5,3] => ([(0,4),(3,2),(4,1),(4,3)],5) => 1
{{1},{2},{3},{4,5}} => [1,2,3,5,4] => [1,2,3,5,4] => ([(0,3),(3,4),(4,1),(4,2)],5) => 1
{{1},{2},{3},{4},{5}} => [1,2,3,4,5] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5) => 0
{{1,2,3,4,5,6}} => [2,3,4,5,6,1] => [6,2,3,4,5,1] => ([(2,3),(3,5),(5,4)],6) => 0
{{1,2,3,4,5},{6}} => [2,3,4,5,1,6] => [5,2,3,4,1,6] => ([(0,5),(1,5),(2,3),(3,4),(4,5)],6) => 1
{{1,2,3,4,6},{5}} => [2,3,4,6,5,1] => [5,2,3,4,6,1] => ([(1,5),(2,3),(3,4),(4,5)],6) => 0
{{1,2,3,4},{5},{6}} => [2,3,4,1,5,6] => [4,2,3,1,5,6] => ([(0,5),(1,5),(2,3),(3,5),(5,4)],6) => 2
{{1,2,3,5},{4},{6}} => [2,3,5,4,1,6] => [4,2,3,5,1,6] => ([(0,5),(1,4),(2,3),(3,5),(5,4)],6) => 1
{{1,2,3,6},{4,5}} => [2,3,6,5,4,1] => [5,2,3,6,1,4] => ([(0,5),(1,4),(2,3),(3,4),(3,5)],6) => 1
{{1,2,3,6},{4},{5}} => [2,3,6,4,5,1] => [4,2,3,5,6,1] => ([(1,5),(2,3),(3,5),(5,4)],6) => 1
{{1,2,3},{4},{5},{6}} => [2,3,1,4,5,6] => [3,2,1,4,5,6] => ([(0,5),(1,5),(2,5),(3,4),(5,3)],6) => 2
{{1,2,4},{3},{5},{6}} => [2,4,3,1,5,6] => [3,2,4,1,5,6] => ([(0,5),(1,4),(2,4),(4,5),(5,3)],6) => 2
{{1,2,5},{3,4},{6}} => [2,5,4,3,1,6] => [4,2,5,1,3,6] => ([(0,4),(1,3),(2,3),(2,4),(3,5),(4,5)],6) => 2
{{1,2,5},{3},{4},{6}} => [2,5,3,4,1,6] => [3,2,4,5,1,6] => ([(0,5),(1,4),(2,4),(3,5),(4,3)],6) => 1
{{1,2,6},{3},{4},{5}} => [2,6,3,4,5,1] => [3,2,4,5,6,1] => ([(1,5),(2,5),(3,4),(5,3)],6) => 1
{{1,2},{3},{4},{5},{6}} => [2,1,3,4,5,6] => [2,1,3,4,5,6] => ([(0,5),(1,5),(3,2),(4,3),(5,4)],6) => 1
{{1,3,4,5},{2,6}} => [3,6,4,5,1,2] => [6,3,4,5,2,1] => ([(3,4),(4,5)],6) => 0
{{1,3,4},{2,5,6}} => [3,5,4,1,6,2] => [6,3,4,2,5,1] => ([(2,5),(3,4),(4,5)],6) => 0
{{1,3,4},{2,5},{6}} => [3,5,4,1,2,6] => [5,3,4,2,1,6] => ([(0,5),(1,5),(2,5),(3,4),(4,5)],6) => 2
{{1,3,4},{2,6},{5}} => [3,6,4,1,5,2] => [5,3,4,2,6,1] => ([(1,5),(2,5),(3,4),(4,5)],6) => 1
{{1,3,5},{2,4},{6}} => [3,4,5,2,1,6] => [5,4,3,1,2,6] => ([(0,5),(1,5),(2,5),(3,4),(4,5)],6) => 2
{{1,3,6},{2,4,5}} => [3,4,6,5,2,1] => [6,4,3,5,1,2] => ([(1,5),(2,5),(3,4)],6) => 0
{{1,3},{2,4,5,6}} => [3,4,1,5,6,2] => [6,3,2,4,5,1] => ([(2,5),(3,5),(5,4)],6) => 1
{{1,3},{2,4,5},{6}} => [3,4,1,5,2,6] => [5,3,2,4,1,6] => ([(0,5),(1,5),(2,4),(3,4),(4,5)],6) => 2
{{1,3},{2,4,6},{5}} => [3,4,1,6,5,2] => [5,3,2,4,6,1] => ([(1,5),(2,4),(3,4),(4,5)],6) => 1
{{1,3},{2,4},{5},{6}} => [3,4,1,2,5,6] => [4,3,2,1,5,6] => ([(0,5),(1,5),(2,5),(3,5),(5,4)],6) => 3
{{1,3,5},{2,6},{4}} => [3,6,5,4,1,2] => [5,3,4,6,2,1] => ([(2,5),(3,4),(4,5)],6) => 0
{{1,3},{2},{4},{5},{6}} => [3,2,1,4,5,6] => [2,3,1,4,5,6] => ([(0,5),(1,3),(3,5),(4,2),(5,4)],6) => 1
{{1,4},{2,3,5},{6}} => [4,3,5,1,2,6] => [4,5,3,2,1,6] => ([(0,5),(1,5),(2,5),(3,4),(4,5)],6) => 2
{{1,4},{2,3,6},{5}} => [4,3,6,1,5,2] => [4,5,3,2,6,1] => ([(1,5),(2,5),(3,4),(4,5)],6) => 1
{{1,4},{2,3},{5},{6}} => [4,3,2,1,5,6] => [3,4,1,2,5,6] => ([(0,4),(1,3),(3,5),(4,5),(5,2)],6) => 1
{{1,5},{2,3,6},{4}} => [5,3,6,4,1,2] => [4,5,3,6,2,1] => ([(2,5),(3,4),(4,5)],6) => 0
{{1,4},{2,5},{3,6}} => [4,5,6,1,2,3] => [6,5,4,3,2,1] => ([],6) => 0
{{1,4},{2,5},{3},{6}} => [4,5,3,1,2,6] => [4,3,5,2,1,6] => ([(0,5),(1,5),(2,4),(3,4),(4,5)],6) => 2
{{1,4},{2,6},{3},{5}} => [4,6,3,1,5,2] => [4,3,5,2,6,1] => ([(1,5),(2,4),(3,4),(4,5)],6) => 1
{{1,4},{2},{3},{5},{6}} => [4,2,3,1,5,6] => [2,3,4,1,5,6] => ([(0,5),(1,4),(2,5),(4,2),(5,3)],6) => 1
{{1,5},{2,4},{3},{6}} => [5,4,3,2,1,6] => [3,4,5,1,2,6] => ([(0,3),(1,4),(2,5),(3,5),(4,2)],6) => 0
{{1,6},{2,5},{3,4}} => [6,5,4,3,2,1] => [4,5,6,1,2,3] => ([(0,5),(1,4),(4,2),(5,3)],6) => 0
{{1,5},{2,6},{3},{4}} => [5,6,3,4,1,2] => [4,3,5,6,2,1] => ([(2,5),(3,5),(5,4)],6) => 1
{{1,5},{2},{3},{4},{6}} => [5,2,3,4,1,6] => [2,3,4,5,1,6] => ([(0,5),(1,4),(2,5),(3,2),(4,3)],6) => 0
{{1,6},{2,5},{3},{4}} => [6,5,3,4,2,1] => [3,4,5,6,1,2] => ([(0,5),(1,3),(4,2),(5,4)],6) => 0
{{1,6},{2},{3},{4},{5}} => [6,2,3,4,5,1] => [2,3,4,5,6,1] => ([(1,5),(3,4),(4,2),(5,3)],6) => 0
{{1},{2},{3},{4},{5},{6}} => [1,2,3,4,5,6] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6) => 0
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Description
The interval resolution global dimension of a poset.
This is the cardinality of the longest chain of right minimal approximations by interval modules of an indecomposable module over the incidence algebra.
This is the cardinality of the longest chain of right minimal approximations by interval modules of an indecomposable module over the incidence algebra.
Map
to permutation
Description
Sends the set partition to the permutation obtained by considering the blocks as increasing cycles.
Map
Corteel
Description
Corteel's map interchanging the number of crossings and the number of nestings of a permutation.
This involution creates a labelled bicoloured Motzkin path, using the Foata-Zeilberger map. In the corresponding bump diagram, each label records the number of arcs nesting the given arc. Then each label is replaced by its complement, and the inverse of the Foata-Zeilberger map is applied.
This involution creates a labelled bicoloured Motzkin path, using the Foata-Zeilberger map. In the corresponding bump diagram, each label records the number of arcs nesting the given arc. Then each label is replaced by its complement, and the inverse of the Foata-Zeilberger map is applied.
Map
permutation poset
Description
Sends a permutation to its permutation poset.
For a permutation π of length n, this poset has vertices
{(i,π(i)) : 1≤i≤n}
and the cover relation is given by (w,x)≤(y,z) if w≤y and x≤z.
For example, the permutation [3,1,5,4,2] is mapped to the poset with cover relations
{(2,1)≺(5,2), (2,1)≺(4,4), (2,1)≺(3,5), (1,3)≺(4,4), (1,3)≺(3,5)}.
For a permutation π of length n, this poset has vertices
{(i,π(i)) : 1≤i≤n}
and the cover relation is given by (w,x)≤(y,z) if w≤y and x≤z.
For example, the permutation [3,1,5,4,2] is mapped to the poset with cover relations
{(2,1)≺(5,2), (2,1)≺(4,4), (2,1)≺(3,5), (1,3)≺(4,4), (1,3)≺(3,5)}.
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