Identifier
-
Mp00031:
Dyck paths
—to 312-avoiding permutation⟶
Permutations
Mp00252: Permutations —restriction⟶ Permutations
Mp00065: Permutations —permutation poset⟶ Posets
St001964: Posets ⟶ ℤ
Values
[1,0,1,0] => [1,2] => [1] => ([],1) => 0
[1,1,0,0] => [2,1] => [1] => ([],1) => 0
[1,0,1,0,1,0] => [1,2,3] => [1,2] => ([(0,1)],2) => 0
[1,0,1,1,0,0] => [1,3,2] => [1,2] => ([(0,1)],2) => 0
[1,1,0,0,1,0] => [2,1,3] => [2,1] => ([],2) => 0
[1,1,0,1,0,0] => [2,3,1] => [2,1] => ([],2) => 0
[1,1,1,0,0,0] => [3,2,1] => [2,1] => ([],2) => 0
[1,0,1,0,1,0,1,0] => [1,2,3,4] => [1,2,3] => ([(0,2),(2,1)],3) => 0
[1,0,1,0,1,1,0,0] => [1,2,4,3] => [1,2,3] => ([(0,2),(2,1)],3) => 0
[1,0,1,1,0,0,1,0] => [1,3,2,4] => [1,3,2] => ([(0,1),(0,2)],3) => 0
[1,0,1,1,0,1,0,0] => [1,3,4,2] => [1,3,2] => ([(0,1),(0,2)],3) => 0
[1,0,1,1,1,0,0,0] => [1,4,3,2] => [1,3,2] => ([(0,1),(0,2)],3) => 0
[1,1,0,0,1,0,1,0] => [2,1,3,4] => [2,1,3] => ([(0,2),(1,2)],3) => 0
[1,1,0,0,1,1,0,0] => [2,1,4,3] => [2,1,3] => ([(0,2),(1,2)],3) => 0
[1,1,1,0,0,0,1,0] => [3,2,1,4] => [3,2,1] => ([],3) => 0
[1,1,1,0,0,1,0,0] => [3,2,4,1] => [3,2,1] => ([],3) => 0
[1,1,1,0,1,0,0,0] => [3,4,2,1] => [3,2,1] => ([],3) => 0
[1,1,1,1,0,0,0,0] => [4,3,2,1] => [3,2,1] => ([],3) => 0
[1,0,1,0,1,0,1,0,1,0] => [1,2,3,4,5] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4) => 0
[1,0,1,0,1,0,1,1,0,0] => [1,2,3,5,4] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4) => 0
[1,0,1,0,1,1,0,0,1,0] => [1,2,4,3,5] => [1,2,4,3] => ([(0,3),(3,1),(3,2)],4) => 1
[1,0,1,0,1,1,0,1,0,0] => [1,2,4,5,3] => [1,2,4,3] => ([(0,3),(3,1),(3,2)],4) => 1
[1,0,1,0,1,1,1,0,0,0] => [1,2,5,4,3] => [1,2,4,3] => ([(0,3),(3,1),(3,2)],4) => 1
[1,0,1,1,0,0,1,0,1,0] => [1,3,2,4,5] => [1,3,2,4] => ([(0,1),(0,2),(1,3),(2,3)],4) => 0
[1,0,1,1,0,0,1,1,0,0] => [1,3,2,5,4] => [1,3,2,4] => ([(0,1),(0,2),(1,3),(2,3)],4) => 0
[1,0,1,1,0,1,0,0,1,0] => [1,3,4,2,5] => [1,3,4,2] => ([(0,2),(0,3),(3,1)],4) => 0
[1,0,1,1,0,1,0,1,0,0] => [1,3,4,5,2] => [1,3,4,2] => ([(0,2),(0,3),(3,1)],4) => 0
[1,0,1,1,0,1,1,0,0,0] => [1,3,5,4,2] => [1,3,4,2] => ([(0,2),(0,3),(3,1)],4) => 0
[1,0,1,1,1,0,0,0,1,0] => [1,4,3,2,5] => [1,4,3,2] => ([(0,1),(0,2),(0,3)],4) => 1
[1,0,1,1,1,0,0,1,0,0] => [1,4,3,5,2] => [1,4,3,2] => ([(0,1),(0,2),(0,3)],4) => 1
[1,0,1,1,1,0,1,0,0,0] => [1,4,5,3,2] => [1,4,3,2] => ([(0,1),(0,2),(0,3)],4) => 1
[1,0,1,1,1,1,0,0,0,0] => [1,5,4,3,2] => [1,4,3,2] => ([(0,1),(0,2),(0,3)],4) => 1
[1,1,0,0,1,0,1,0,1,0] => [2,1,3,4,5] => [2,1,3,4] => ([(0,3),(1,3),(3,2)],4) => 1
[1,1,0,0,1,0,1,1,0,0] => [2,1,3,5,4] => [2,1,3,4] => ([(0,3),(1,3),(3,2)],4) => 1
[1,1,0,0,1,1,0,0,1,0] => [2,1,4,3,5] => [2,1,4,3] => ([(0,2),(0,3),(1,2),(1,3)],4) => 2
[1,1,0,0,1,1,0,1,0,0] => [2,1,4,5,3] => [2,1,4,3] => ([(0,2),(0,3),(1,2),(1,3)],4) => 2
[1,1,0,0,1,1,1,0,0,0] => [2,1,5,4,3] => [2,1,4,3] => ([(0,2),(0,3),(1,2),(1,3)],4) => 2
[1,1,0,1,0,0,1,0,1,0] => [2,3,1,4,5] => [2,3,1,4] => ([(0,3),(1,2),(2,3)],4) => 0
[1,1,0,1,0,0,1,1,0,0] => [2,3,1,5,4] => [2,3,1,4] => ([(0,3),(1,2),(2,3)],4) => 0
[1,1,0,1,0,1,0,0,1,0] => [2,3,4,1,5] => [2,3,4,1] => ([(1,2),(2,3)],4) => 0
[1,1,0,1,0,1,0,1,0,0] => [2,3,4,5,1] => [2,3,4,1] => ([(1,2),(2,3)],4) => 0
[1,1,0,1,0,1,1,0,0,0] => [2,3,5,4,1] => [2,3,4,1] => ([(1,2),(2,3)],4) => 0
[1,1,1,0,0,0,1,0,1,0] => [3,2,1,4,5] => [3,2,1,4] => ([(0,3),(1,3),(2,3)],4) => 1
[1,1,1,0,0,0,1,1,0,0] => [3,2,1,5,4] => [3,2,1,4] => ([(0,3),(1,3),(2,3)],4) => 1
[1,1,1,0,0,1,0,0,1,0] => [3,2,4,1,5] => [3,2,4,1] => ([(1,3),(2,3)],4) => 0
[1,1,1,0,0,1,0,1,0,0] => [3,2,4,5,1] => [3,2,4,1] => ([(1,3),(2,3)],4) => 0
[1,1,1,0,0,1,1,0,0,0] => [3,2,5,4,1] => [3,2,4,1] => ([(1,3),(2,3)],4) => 0
[1,1,1,1,0,0,0,0,1,0] => [4,3,2,1,5] => [4,3,2,1] => ([],4) => 0
[1,1,1,1,0,0,0,1,0,0] => [4,3,2,5,1] => [4,3,2,1] => ([],4) => 0
[1,1,1,1,0,0,1,0,0,0] => [4,3,5,2,1] => [4,3,2,1] => ([],4) => 0
[1,1,1,1,0,1,0,0,0,0] => [4,5,3,2,1] => [4,3,2,1] => ([],4) => 0
[1,1,1,1,1,0,0,0,0,0] => [5,4,3,2,1] => [4,3,2,1] => ([],4) => 0
[1,0,1,0,1,0,1,0,1,0,1,0] => [1,2,3,4,5,6] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5) => 0
[1,0,1,0,1,0,1,0,1,1,0,0] => [1,2,3,4,6,5] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5) => 0
[1,0,1,0,1,0,1,1,0,0,1,0] => [1,2,3,5,4,6] => [1,2,3,5,4] => ([(0,3),(3,4),(4,1),(4,2)],5) => 1
[1,0,1,0,1,0,1,1,0,1,0,0] => [1,2,3,5,6,4] => [1,2,3,5,4] => ([(0,3),(3,4),(4,1),(4,2)],5) => 1
[1,0,1,0,1,0,1,1,1,0,0,0] => [1,2,3,6,5,4] => [1,2,3,5,4] => ([(0,3),(3,4),(4,1),(4,2)],5) => 1
[1,0,1,0,1,1,0,0,1,0,1,0] => [1,2,4,3,5,6] => [1,2,4,3,5] => ([(0,3),(1,4),(2,4),(3,1),(3,2)],5) => 1
[1,0,1,0,1,1,0,0,1,1,0,0] => [1,2,4,3,6,5] => [1,2,4,3,5] => ([(0,3),(1,4),(2,4),(3,1),(3,2)],5) => 1
[1,0,1,0,1,1,0,1,0,0,1,0] => [1,2,4,5,3,6] => [1,2,4,5,3] => ([(0,4),(3,2),(4,1),(4,3)],5) => 1
[1,0,1,0,1,1,0,1,0,1,0,0] => [1,2,4,5,6,3] => [1,2,4,5,3] => ([(0,4),(3,2),(4,1),(4,3)],5) => 1
[1,0,1,0,1,1,0,1,1,0,0,0] => [1,2,4,6,5,3] => [1,2,4,5,3] => ([(0,4),(3,2),(4,1),(4,3)],5) => 1
[1,0,1,0,1,1,1,0,0,0,1,0] => [1,2,5,4,3,6] => [1,2,5,4,3] => ([(0,4),(4,1),(4,2),(4,3)],5) => 2
[1,0,1,0,1,1,1,0,0,1,0,0] => [1,2,5,4,6,3] => [1,2,5,4,3] => ([(0,4),(4,1),(4,2),(4,3)],5) => 2
[1,0,1,0,1,1,1,0,1,0,0,0] => [1,2,5,6,4,3] => [1,2,5,4,3] => ([(0,4),(4,1),(4,2),(4,3)],5) => 2
[1,0,1,0,1,1,1,1,0,0,0,0] => [1,2,6,5,4,3] => [1,2,5,4,3] => ([(0,4),(4,1),(4,2),(4,3)],5) => 2
[1,0,1,1,0,0,1,0,1,0,1,0] => [1,3,2,4,5,6] => [1,3,2,4,5] => ([(0,2),(0,3),(2,4),(3,4),(4,1)],5) => 1
[1,0,1,1,0,0,1,0,1,1,0,0] => [1,3,2,4,6,5] => [1,3,2,4,5] => ([(0,2),(0,3),(2,4),(3,4),(4,1)],5) => 1
[1,0,1,1,0,0,1,1,0,0,1,0] => [1,3,2,5,4,6] => [1,3,2,5,4] => ([(0,1),(0,2),(1,3),(1,4),(2,3),(2,4)],5) => 2
[1,0,1,1,0,0,1,1,0,1,0,0] => [1,3,2,5,6,4] => [1,3,2,5,4] => ([(0,1),(0,2),(1,3),(1,4),(2,3),(2,4)],5) => 2
[1,0,1,1,0,0,1,1,1,0,0,0] => [1,3,2,6,5,4] => [1,3,2,5,4] => ([(0,1),(0,2),(1,3),(1,4),(2,3),(2,4)],5) => 2
[1,0,1,1,0,1,0,0,1,0,1,0] => [1,3,4,2,5,6] => [1,3,4,2,5] => ([(0,2),(0,3),(1,4),(2,4),(3,1)],5) => 0
[1,0,1,1,0,1,0,0,1,1,0,0] => [1,3,4,2,6,5] => [1,3,4,2,5] => ([(0,2),(0,3),(1,4),(2,4),(3,1)],5) => 0
[1,0,1,1,0,1,0,1,0,0,1,0] => [1,3,4,5,2,6] => [1,3,4,5,2] => ([(0,2),(0,4),(3,1),(4,3)],5) => 0
[1,0,1,1,0,1,0,1,0,1,0,0] => [1,3,4,5,6,2] => [1,3,4,5,2] => ([(0,2),(0,4),(3,1),(4,3)],5) => 0
[1,0,1,1,0,1,0,1,1,0,0,0] => [1,3,4,6,5,2] => [1,3,4,5,2] => ([(0,2),(0,4),(3,1),(4,3)],5) => 0
[1,0,1,1,0,1,1,0,0,0,1,0] => [1,3,5,4,2,6] => [1,3,5,4,2] => ([(0,3),(0,4),(4,1),(4,2)],5) => 1
[1,0,1,1,0,1,1,0,0,1,0,0] => [1,3,5,4,6,2] => [1,3,5,4,2] => ([(0,3),(0,4),(4,1),(4,2)],5) => 1
[1,0,1,1,0,1,1,0,1,0,0,0] => [1,3,5,6,4,2] => [1,3,5,4,2] => ([(0,3),(0,4),(4,1),(4,2)],5) => 1
[1,0,1,1,0,1,1,1,0,0,0,0] => [1,3,6,5,4,2] => [1,3,5,4,2] => ([(0,3),(0,4),(4,1),(4,2)],5) => 1
[1,0,1,1,1,0,0,0,1,0,1,0] => [1,4,3,2,5,6] => [1,4,3,2,5] => ([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5) => 1
[1,0,1,1,1,0,0,0,1,1,0,0] => [1,4,3,2,6,5] => [1,4,3,2,5] => ([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5) => 1
[1,0,1,1,1,0,0,1,0,0,1,0] => [1,4,3,5,2,6] => [1,4,3,5,2] => ([(0,1),(0,2),(0,3),(2,4),(3,4)],5) => 1
[1,0,1,1,1,0,0,1,0,1,0,0] => [1,4,3,5,6,2] => [1,4,3,5,2] => ([(0,1),(0,2),(0,3),(2,4),(3,4)],5) => 1
[1,0,1,1,1,0,0,1,1,0,0,0] => [1,4,3,6,5,2] => [1,4,3,5,2] => ([(0,1),(0,2),(0,3),(2,4),(3,4)],5) => 1
[1,0,1,1,1,0,1,0,0,0,1,0] => [1,4,5,3,2,6] => [1,4,5,3,2] => ([(0,2),(0,3),(0,4),(4,1)],5) => 1
[1,0,1,1,1,0,1,0,0,1,0,0] => [1,4,5,3,6,2] => [1,4,5,3,2] => ([(0,2),(0,3),(0,4),(4,1)],5) => 1
[1,0,1,1,1,0,1,0,1,0,0,0] => [1,4,5,6,3,2] => [1,4,5,3,2] => ([(0,2),(0,3),(0,4),(4,1)],5) => 1
[1,0,1,1,1,0,1,1,0,0,0,0] => [1,4,6,5,3,2] => [1,4,5,3,2] => ([(0,2),(0,3),(0,4),(4,1)],5) => 1
[1,0,1,1,1,1,0,0,0,0,1,0] => [1,5,4,3,2,6] => [1,5,4,3,2] => ([(0,1),(0,2),(0,3),(0,4)],5) => 2
[1,0,1,1,1,1,0,0,0,1,0,0] => [1,5,4,3,6,2] => [1,5,4,3,2] => ([(0,1),(0,2),(0,3),(0,4)],5) => 2
[1,0,1,1,1,1,0,0,1,0,0,0] => [1,5,4,6,3,2] => [1,5,4,3,2] => ([(0,1),(0,2),(0,3),(0,4)],5) => 2
[1,0,1,1,1,1,0,1,0,0,0,0] => [1,5,6,4,3,2] => [1,5,4,3,2] => ([(0,1),(0,2),(0,3),(0,4)],5) => 2
[1,0,1,1,1,1,1,0,0,0,0,0] => [1,6,5,4,3,2] => [1,5,4,3,2] => ([(0,1),(0,2),(0,3),(0,4)],5) => 2
[1,1,0,0,1,0,1,0,1,0,1,0] => [2,1,3,4,5,6] => [2,1,3,4,5] => ([(0,4),(1,4),(2,3),(4,2)],5) => 1
[1,1,0,0,1,0,1,0,1,1,0,0] => [2,1,3,4,6,5] => [2,1,3,4,5] => ([(0,4),(1,4),(2,3),(4,2)],5) => 1
[1,1,0,0,1,0,1,1,0,0,1,0] => [2,1,3,5,4,6] => [2,1,3,5,4] => ([(0,4),(1,4),(4,2),(4,3)],5) => 2
[1,1,0,0,1,0,1,1,0,1,0,0] => [2,1,3,5,6,4] => [2,1,3,5,4] => ([(0,4),(1,4),(4,2),(4,3)],5) => 2
[1,1,0,0,1,0,1,1,1,0,0,0] => [2,1,3,6,5,4] => [2,1,3,5,4] => ([(0,4),(1,4),(4,2),(4,3)],5) => 2
[1,1,0,0,1,1,0,0,1,0,1,0] => [2,1,4,3,5,6] => [2,1,4,3,5] => ([(0,3),(0,4),(1,3),(1,4),(3,2),(4,2)],5) => 2
[1,1,0,0,1,1,0,0,1,1,0,0] => [2,1,4,3,6,5] => [2,1,4,3,5] => ([(0,3),(0,4),(1,3),(1,4),(3,2),(4,2)],5) => 2
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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
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
$$\{ (2, 1) \prec (5, 2),\ (2, 1) \prec (4, 4),\ (2, 1) \prec (3, 5),\ (1, 3) \prec (4, 4),\ (1, 3) \prec (3, 5) \}.$$
Map
to 312-avoiding permutation
Description
Map
restriction
Description
The permutation obtained by removing the largest letter.
This map is undefined for the empty permutation.
This map is undefined for the empty permutation.
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