Identifier
-
Mp00025:
Dyck paths
—to 132-avoiding permutation⟶
Permutations
Mp00065: Permutations —permutation poset⟶ Posets
St001964: Posets ⟶ ℤ
Values
[1,0] => [1] => ([],1) => 0
[1,0,1,0] => [2,1] => ([],2) => 0
[1,1,0,0] => [1,2] => ([(0,1)],2) => 0
[1,0,1,0,1,0] => [3,2,1] => ([],3) => 0
[1,1,0,1,0,0] => [2,1,3] => ([(0,2),(1,2)],3) => 0
[1,1,1,0,0,0] => [1,2,3] => ([(0,2),(2,1)],3) => 0
[1,0,1,0,1,0,1,0] => [4,3,2,1] => ([],4) => 0
[1,0,1,1,0,1,0,0] => [3,2,4,1] => ([(1,3),(2,3)],4) => 0
[1,0,1,1,1,0,0,0] => [2,3,4,1] => ([(1,2),(2,3)],4) => 0
[1,1,0,1,0,0,1,0] => [4,2,1,3] => ([(1,3),(2,3)],4) => 0
[1,1,0,1,0,1,0,0] => [3,2,1,4] => ([(0,3),(1,3),(2,3)],4) => 1
[1,1,0,1,1,0,0,0] => [2,3,1,4] => ([(0,3),(1,2),(2,3)],4) => 0
[1,1,1,0,0,0,1,0] => [4,1,2,3] => ([(1,2),(2,3)],4) => 0
[1,1,1,0,0,1,0,0] => [3,1,2,4] => ([(0,3),(1,2),(2,3)],4) => 0
[1,1,1,0,1,0,0,0] => [2,1,3,4] => ([(0,3),(1,3),(3,2)],4) => 1
[1,1,1,1,0,0,0,0] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4) => 0
[1,0,1,0,1,0,1,0,1,0] => [5,4,3,2,1] => ([],5) => 0
[1,0,1,0,1,1,0,1,0,0] => [4,3,5,2,1] => ([(2,4),(3,4)],5) => 0
[1,0,1,0,1,1,1,0,0,0] => [3,4,5,2,1] => ([(2,3),(3,4)],5) => 0
[1,0,1,1,0,1,0,0,1,0] => [5,3,2,4,1] => ([(2,4),(3,4)],5) => 0
[1,0,1,1,0,1,1,0,0,0] => [3,4,2,5,1] => ([(1,4),(2,3),(3,4)],5) => 0
[1,0,1,1,1,0,0,0,1,0] => [5,2,3,4,1] => ([(2,3),(3,4)],5) => 0
[1,0,1,1,1,0,0,1,0,0] => [4,2,3,5,1] => ([(1,4),(2,3),(3,4)],5) => 0
[1,0,1,1,1,0,1,0,0,0] => [3,2,4,5,1] => ([(1,4),(2,4),(4,3)],5) => 1
[1,0,1,1,1,1,0,0,0,0] => [2,3,4,5,1] => ([(1,4),(3,2),(4,3)],5) => 0
[1,1,0,0,1,1,0,1,0,0] => [4,3,5,1,2] => ([(0,4),(1,4),(2,3)],5) => 0
[1,1,0,0,1,1,1,0,0,0] => [3,4,5,1,2] => ([(0,3),(1,4),(4,2)],5) => 0
[1,1,0,1,0,0,1,0,1,0] => [5,4,2,1,3] => ([(2,4),(3,4)],5) => 0
[1,1,0,1,0,0,1,1,0,0] => [4,5,2,1,3] => ([(0,4),(1,4),(2,3)],5) => 0
[1,1,0,1,0,1,0,1,0,0] => [4,3,2,1,5] => ([(0,4),(1,4),(2,4),(3,4)],5) => 2
[1,1,0,1,0,1,1,0,0,0] => [3,4,2,1,5] => ([(0,4),(1,4),(2,3),(3,4)],5) => 1
[1,1,0,1,1,0,0,0,1,0] => [5,2,3,1,4] => ([(1,4),(2,3),(3,4)],5) => 0
[1,1,0,1,1,0,0,1,0,0] => [4,2,3,1,5] => ([(0,4),(1,4),(2,3),(3,4)],5) => 1
[1,1,0,1,1,0,1,0,0,0] => [3,2,4,1,5] => ([(0,4),(1,3),(2,3),(3,4)],5) => 1
[1,1,0,1,1,1,0,0,0,0] => [2,3,4,1,5] => ([(0,4),(1,2),(2,3),(3,4)],5) => 0
[1,1,1,0,0,0,1,0,1,0] => [5,4,1,2,3] => ([(2,3),(3,4)],5) => 0
[1,1,1,0,0,0,1,1,0,0] => [4,5,1,2,3] => ([(0,3),(1,4),(4,2)],5) => 0
[1,1,1,0,0,1,0,0,1,0] => [5,3,1,2,4] => ([(1,4),(2,3),(3,4)],5) => 0
[1,1,1,0,0,1,0,1,0,0] => [4,3,1,2,5] => ([(0,4),(1,4),(2,3),(3,4)],5) => 1
[1,1,1,0,0,1,1,0,0,0] => [3,4,1,2,5] => ([(0,3),(1,2),(2,4),(3,4)],5) => 0
[1,1,1,0,1,0,0,0,1,0] => [5,2,1,3,4] => ([(1,4),(2,4),(4,3)],5) => 1
[1,1,1,0,1,0,0,1,0,0] => [4,2,1,3,5] => ([(0,4),(1,3),(2,3),(3,4)],5) => 1
[1,1,1,0,1,0,1,0,0,0] => [3,2,1,4,5] => ([(0,4),(1,4),(2,4),(4,3)],5) => 2
[1,1,1,0,1,1,0,0,0,0] => [2,3,1,4,5] => ([(0,4),(1,2),(2,4),(4,3)],5) => 1
[1,1,1,1,0,0,0,0,1,0] => [5,1,2,3,4] => ([(1,4),(3,2),(4,3)],5) => 0
[1,1,1,1,0,0,0,1,0,0] => [4,1,2,3,5] => ([(0,4),(1,2),(2,3),(3,4)],5) => 0
[1,1,1,1,0,0,1,0,0,0] => [3,1,2,4,5] => ([(0,4),(1,2),(2,4),(4,3)],5) => 1
[1,1,1,1,0,1,0,0,0,0] => [2,1,3,4,5] => ([(0,4),(1,4),(2,3),(4,2)],5) => 1
[1,1,1,1,1,0,0,0,0,0] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5) => 0
[1,0,1,0,1,0,1,0,1,0,1,0] => [6,5,4,3,2,1] => ([],6) => 0
[1,0,1,0,1,0,1,1,0,1,0,0] => [5,4,6,3,2,1] => ([(3,5),(4,5)],6) => 0
[1,0,1,0,1,0,1,1,1,0,0,0] => [4,5,6,3,2,1] => ([(3,4),(4,5)],6) => 0
[1,0,1,0,1,1,0,1,0,0,1,0] => [6,4,3,5,2,1] => ([(3,5),(4,5)],6) => 0
[1,0,1,0,1,1,0,1,1,0,0,0] => [4,5,3,6,2,1] => ([(2,5),(3,4),(4,5)],6) => 0
[1,0,1,0,1,1,1,0,0,0,1,0] => [6,3,4,5,2,1] => ([(3,4),(4,5)],6) => 0
[1,0,1,0,1,1,1,0,0,1,0,0] => [5,3,4,6,2,1] => ([(2,5),(3,4),(4,5)],6) => 0
[1,0,1,0,1,1,1,0,1,0,0,0] => [4,3,5,6,2,1] => ([(2,5),(3,5),(5,4)],6) => 1
[1,0,1,0,1,1,1,1,0,0,0,0] => [3,4,5,6,2,1] => ([(2,3),(3,5),(5,4)],6) => 0
[1,0,1,1,0,0,1,1,0,1,0,0] => [5,4,6,2,3,1] => ([(1,5),(2,5),(3,4)],6) => 0
[1,0,1,1,0,0,1,1,1,0,0,0] => [4,5,6,2,3,1] => ([(1,3),(2,4),(4,5)],6) => 0
[1,0,1,1,0,1,0,0,1,0,1,0] => [6,5,3,2,4,1] => ([(3,5),(4,5)],6) => 0
[1,0,1,1,0,1,0,0,1,1,0,0] => [5,6,3,2,4,1] => ([(1,5),(2,5),(3,4)],6) => 0
[1,0,1,1,0,1,0,1,1,0,0,0] => [4,5,3,2,6,1] => ([(1,5),(2,5),(3,4),(4,5)],6) => 1
[1,0,1,1,0,1,1,0,0,0,1,0] => [6,3,4,2,5,1] => ([(2,5),(3,4),(4,5)],6) => 0
[1,0,1,1,0,1,1,0,0,1,0,0] => [5,3,4,2,6,1] => ([(1,5),(2,5),(3,4),(4,5)],6) => 1
[1,0,1,1,0,1,1,0,1,0,0,0] => [4,3,5,2,6,1] => ([(1,5),(2,4),(3,4),(4,5)],6) => 1
[1,0,1,1,0,1,1,1,0,0,0,0] => [3,4,5,2,6,1] => ([(1,5),(2,3),(3,4),(4,5)],6) => 0
[1,0,1,1,1,0,0,0,1,0,1,0] => [6,5,2,3,4,1] => ([(3,4),(4,5)],6) => 0
[1,0,1,1,1,0,0,0,1,1,0,0] => [5,6,2,3,4,1] => ([(1,3),(2,4),(4,5)],6) => 0
[1,0,1,1,1,0,0,1,0,0,1,0] => [6,4,2,3,5,1] => ([(2,5),(3,4),(4,5)],6) => 0
[1,0,1,1,1,0,0,1,0,1,0,0] => [5,4,2,3,6,1] => ([(1,5),(2,5),(3,4),(4,5)],6) => 1
[1,0,1,1,1,0,0,1,1,0,0,0] => [4,5,2,3,6,1] => ([(1,4),(2,3),(3,5),(4,5)],6) => 0
[1,0,1,1,1,0,1,0,0,0,1,0] => [6,3,2,4,5,1] => ([(2,5),(3,5),(5,4)],6) => 1
[1,0,1,1,1,0,1,0,0,1,0,0] => [5,3,2,4,6,1] => ([(1,5),(2,4),(3,4),(4,5)],6) => 1
[1,0,1,1,1,0,1,1,0,0,0,0] => [3,4,2,5,6,1] => ([(1,5),(2,3),(3,5),(5,4)],6) => 1
[1,0,1,1,1,1,0,0,0,0,1,0] => [6,2,3,4,5,1] => ([(2,3),(3,5),(5,4)],6) => 0
[1,0,1,1,1,1,0,0,0,1,0,0] => [5,2,3,4,6,1] => ([(1,5),(2,3),(3,4),(4,5)],6) => 0
[1,0,1,1,1,1,0,0,1,0,0,0] => [4,2,3,5,6,1] => ([(1,5),(2,3),(3,5),(5,4)],6) => 1
[1,0,1,1,1,1,0,1,0,0,0,0] => [3,2,4,5,6,1] => ([(1,5),(2,5),(3,4),(5,3)],6) => 1
[1,0,1,1,1,1,1,0,0,0,0,0] => [2,3,4,5,6,1] => ([(1,5),(3,4),(4,2),(5,3)],6) => 0
[1,1,0,0,1,0,1,1,0,1,0,0] => [5,4,6,3,1,2] => ([(1,5),(2,5),(3,4)],6) => 0
[1,1,0,0,1,0,1,1,1,0,0,0] => [4,5,6,3,1,2] => ([(1,3),(2,4),(4,5)],6) => 0
[1,1,0,0,1,1,0,1,0,0,1,0] => [6,4,3,5,1,2] => ([(1,5),(2,5),(3,4)],6) => 0
[1,1,0,0,1,1,0,1,1,0,0,0] => [4,5,3,6,1,2] => ([(0,5),(1,3),(2,4),(4,5)],6) => 0
[1,1,0,0,1,1,1,0,0,0,1,0] => [6,3,4,5,1,2] => ([(1,3),(2,4),(4,5)],6) => 0
[1,1,0,0,1,1,1,0,0,1,0,0] => [5,3,4,6,1,2] => ([(0,5),(1,3),(2,4),(4,5)],6) => 0
[1,1,0,0,1,1,1,0,1,0,0,0] => [4,3,5,6,1,2] => ([(0,5),(1,5),(2,3),(5,4)],6) => 1
[1,1,0,0,1,1,1,1,0,0,0,0] => [3,4,5,6,1,2] => ([(0,5),(1,3),(4,2),(5,4)],6) => 0
[1,1,0,1,0,0,1,0,1,0,1,0] => [6,5,4,2,1,3] => ([(3,5),(4,5)],6) => 0
[1,1,0,1,0,0,1,0,1,1,0,0] => [5,6,4,2,1,3] => ([(1,5),(2,5),(3,4)],6) => 0
[1,1,0,1,0,0,1,1,0,0,1,0] => [6,4,5,2,1,3] => ([(1,5),(2,5),(3,4)],6) => 0
[1,1,0,1,0,0,1,1,0,1,0,0] => [5,4,6,2,1,3] => ([(0,5),(1,5),(2,4),(3,4)],6) => 0
[1,1,0,1,0,0,1,1,1,0,0,0] => [4,5,6,2,1,3] => ([(0,5),(1,5),(2,3),(3,4)],6) => 0
[1,1,0,1,0,1,0,1,0,1,0,0] => [5,4,3,2,1,6] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6) => 3
[1,1,0,1,0,1,0,1,1,0,0,0] => [4,5,3,2,1,6] => ([(0,5),(1,5),(2,5),(3,4),(4,5)],6) => 2
[1,1,0,1,0,1,1,0,0,0,1,0] => [6,3,4,2,1,5] => ([(1,5),(2,5),(3,4),(4,5)],6) => 1
[1,1,0,1,0,1,1,0,0,1,0,0] => [5,3,4,2,1,6] => ([(0,5),(1,5),(2,5),(3,4),(4,5)],6) => 2
[1,1,0,1,0,1,1,0,1,0,0,0] => [4,3,5,2,1,6] => ([(0,5),(1,5),(2,4),(3,4),(4,5)],6) => 2
[1,1,0,1,0,1,1,1,0,0,0,0] => [3,4,5,2,1,6] => ([(0,5),(1,5),(2,3),(3,4),(4,5)],6) => 1
[1,1,0,1,1,0,0,0,1,0,1,0] => [6,5,2,3,1,4] => ([(2,5),(3,4),(4,5)],6) => 0
[1,1,0,1,1,0,0,0,1,1,0,0] => [5,6,2,3,1,4] => ([(0,5),(1,3),(2,4),(4,5)],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 132-avoiding permutation
Description
Sends a Dyck path to a 132-avoiding permutation.
This bijection is defined in [1, Section 2].
This bijection is defined in [1, Section 2].
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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