Identifier
-
Mp00110:
Posets
—Greene-Kleitman invariant⟶
Integer partitions
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00031: Dyck paths —to 312-avoiding permutation⟶ Permutations
St001583: Permutations ⟶ ℤ (values match St000246The number of non-inversions of a permutation.)
Values
([],1) => [1] => [1,0,1,0] => [1,2] => 1
([],2) => [1,1] => [1,0,1,1,0,0] => [1,3,2] => 2
([(0,1)],2) => [2] => [1,1,0,0,1,0] => [2,1,3] => 2
([],3) => [1,1,1] => [1,0,1,1,1,0,0,0] => [1,4,3,2] => 3
([(1,2)],3) => [2,1] => [1,0,1,0,1,0] => [1,2,3] => 3
([(0,1),(0,2)],3) => [2,1] => [1,0,1,0,1,0] => [1,2,3] => 3
([(0,2),(2,1)],3) => [3] => [1,1,1,0,0,0,1,0] => [3,2,1,4] => 3
([(0,2),(1,2)],3) => [2,1] => [1,0,1,0,1,0] => [1,2,3] => 3
([(2,3)],4) => [2,1,1] => [1,0,1,1,0,1,0,0] => [1,3,4,2] => 4
([(1,2),(1,3)],4) => [2,1,1] => [1,0,1,1,0,1,0,0] => [1,3,4,2] => 4
([(0,1),(0,2),(0,3)],4) => [2,1,1] => [1,0,1,1,0,1,0,0] => [1,3,4,2] => 4
([(0,2),(0,3),(3,1)],4) => [3,1] => [1,1,0,1,0,0,1,0] => [2,3,1,4] => 4
([(0,1),(0,2),(1,3),(2,3)],4) => [3,1] => [1,1,0,1,0,0,1,0] => [2,3,1,4] => 4
([(1,2),(2,3)],4) => [3,1] => [1,1,0,1,0,0,1,0] => [2,3,1,4] => 4
([(0,3),(3,1),(3,2)],4) => [3,1] => [1,1,0,1,0,0,1,0] => [2,3,1,4] => 4
([(1,3),(2,3)],4) => [2,1,1] => [1,0,1,1,0,1,0,0] => [1,3,4,2] => 4
([(0,3),(1,3),(3,2)],4) => [3,1] => [1,1,0,1,0,0,1,0] => [2,3,1,4] => 4
([(0,3),(1,3),(2,3)],4) => [2,1,1] => [1,0,1,1,0,1,0,0] => [1,3,4,2] => 4
([(0,3),(1,2)],4) => [2,2] => [1,1,0,0,1,1,0,0] => [2,1,4,3] => 4
([(0,3),(1,2),(1,3)],4) => [2,2] => [1,1,0,0,1,1,0,0] => [2,1,4,3] => 4
([(0,2),(0,3),(1,2),(1,3)],4) => [2,2] => [1,1,0,0,1,1,0,0] => [2,1,4,3] => 4
([(0,3),(1,2),(2,3)],4) => [3,1] => [1,1,0,1,0,0,1,0] => [2,3,1,4] => 4
([(0,2),(0,3),(0,4),(4,1)],5) => [3,1,1] => [1,0,1,1,0,0,1,0] => [1,3,2,4] => 5
([(0,1),(0,2),(0,3),(2,4),(3,4)],5) => [3,1,1] => [1,0,1,1,0,0,1,0] => [1,3,2,4] => 5
([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5) => [3,1,1] => [1,0,1,1,0,0,1,0] => [1,3,2,4] => 5
([(1,3),(1,4),(4,2)],5) => [3,1,1] => [1,0,1,1,0,0,1,0] => [1,3,2,4] => 5
([(0,3),(0,4),(4,1),(4,2)],5) => [3,1,1] => [1,0,1,1,0,0,1,0] => [1,3,2,4] => 5
([(1,2),(1,3),(2,4),(3,4)],5) => [3,1,1] => [1,0,1,1,0,0,1,0] => [1,3,2,4] => 5
([(0,3),(0,4),(3,2),(4,1)],5) => [3,2] => [1,1,0,0,1,0,1,0] => [2,1,3,4] => 5
([(0,2),(0,3),(2,4),(3,1),(3,4)],5) => [3,2] => [1,1,0,0,1,0,1,0] => [2,1,3,4] => 5
([(0,1),(0,2),(1,3),(1,4),(2,3),(2,4)],5) => [3,2] => [1,1,0,0,1,0,1,0] => [2,1,3,4] => 5
([(2,3),(3,4)],5) => [3,1,1] => [1,0,1,1,0,0,1,0] => [1,3,2,4] => 5
([(1,4),(4,2),(4,3)],5) => [3,1,1] => [1,0,1,1,0,0,1,0] => [1,3,2,4] => 5
([(0,4),(4,1),(4,2),(4,3)],5) => [3,1,1] => [1,0,1,1,0,0,1,0] => [1,3,2,4] => 5
([(1,4),(2,4),(4,3)],5) => [3,1,1] => [1,0,1,1,0,0,1,0] => [1,3,2,4] => 5
([(0,4),(1,4),(4,2),(4,3)],5) => [3,2] => [1,1,0,0,1,0,1,0] => [2,1,3,4] => 5
([(0,4),(1,4),(2,4),(4,3)],5) => [3,1,1] => [1,0,1,1,0,0,1,0] => [1,3,2,4] => 5
([(0,4),(1,4),(2,3)],5) => [2,2,1] => [1,0,1,0,1,1,0,0] => [1,2,4,3] => 5
([(0,4),(1,3),(2,3),(2,4)],5) => [2,2,1] => [1,0,1,0,1,1,0,0] => [1,2,4,3] => 5
([(0,4),(1,3),(1,4),(2,3),(2,4)],5) => [2,2,1] => [1,0,1,0,1,1,0,0] => [1,2,4,3] => 5
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5) => [2,2,1] => [1,0,1,0,1,1,0,0] => [1,2,4,3] => 5
([(0,4),(1,3),(2,3),(3,4)],5) => [3,1,1] => [1,0,1,1,0,0,1,0] => [1,3,2,4] => 5
([(0,4),(1,4),(2,3),(2,4)],5) => [2,2,1] => [1,0,1,0,1,1,0,0] => [1,2,4,3] => 5
([(0,4),(1,4),(2,3),(3,4)],5) => [3,1,1] => [1,0,1,1,0,0,1,0] => [1,3,2,4] => 5
([(1,4),(2,3)],5) => [2,2,1] => [1,0,1,0,1,1,0,0] => [1,2,4,3] => 5
([(1,4),(2,3),(2,4)],5) => [2,2,1] => [1,0,1,0,1,1,0,0] => [1,2,4,3] => 5
([(0,4),(1,2),(1,4),(2,3)],5) => [3,2] => [1,1,0,0,1,0,1,0] => [2,1,3,4] => 5
([(0,3),(1,2),(1,3),(2,4),(3,4)],5) => [3,2] => [1,1,0,0,1,0,1,0] => [2,1,3,4] => 5
([(1,3),(1,4),(2,3),(2,4)],5) => [2,2,1] => [1,0,1,0,1,1,0,0] => [1,2,4,3] => 5
([(0,3),(0,4),(1,3),(1,4),(4,2)],5) => [3,2] => [1,1,0,0,1,0,1,0] => [2,1,3,4] => 5
([(0,3),(0,4),(1,3),(1,4),(3,2),(4,2)],5) => [3,2] => [1,1,0,0,1,0,1,0] => [2,1,3,4] => 5
([(0,4),(1,2),(1,4),(4,3)],5) => [3,2] => [1,1,0,0,1,0,1,0] => [2,1,3,4] => 5
([(0,4),(1,2),(1,3)],5) => [2,2,1] => [1,0,1,0,1,1,0,0] => [1,2,4,3] => 5
([(0,4),(1,2),(1,3),(1,4)],5) => [2,2,1] => [1,0,1,0,1,1,0,0] => [1,2,4,3] => 5
([(0,4),(1,2),(1,3),(3,4)],5) => [3,1,1] => [1,0,1,1,0,0,1,0] => [1,3,2,4] => 5
([(0,4),(1,2),(1,3),(2,4),(3,4)],5) => [3,1,1] => [1,0,1,1,0,0,1,0] => [1,3,2,4] => 5
([(0,3),(0,4),(1,2),(1,4)],5) => [2,2,1] => [1,0,1,0,1,1,0,0] => [1,2,4,3] => 5
([(0,3),(0,4),(1,2),(1,3),(1,4)],5) => [2,2,1] => [1,0,1,0,1,1,0,0] => [1,2,4,3] => 5
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4)],5) => [2,2,1] => [1,0,1,0,1,1,0,0] => [1,2,4,3] => 5
([(0,3),(0,4),(1,2),(1,3),(2,4)],5) => [3,2] => [1,1,0,0,1,0,1,0] => [2,1,3,4] => 5
([(0,3),(1,2),(1,4),(3,4)],5) => [3,2] => [1,1,0,0,1,0,1,0] => [2,1,3,4] => 5
([(0,3),(0,4),(1,2),(2,3),(2,4)],5) => [3,2] => [1,1,0,0,1,0,1,0] => [2,1,3,4] => 5
([(1,4),(2,3),(3,4)],5) => [3,1,1] => [1,0,1,1,0,0,1,0] => [1,3,2,4] => 5
([(0,3),(1,4),(4,2)],5) => [3,2] => [1,1,0,0,1,0,1,0] => [2,1,3,4] => 5
([(0,4),(1,2),(2,3),(2,4)],5) => [3,2] => [1,1,0,0,1,0,1,0] => [2,1,3,4] => 5
([(0,3),(1,2),(2,4),(3,4)],5) => [3,2] => [1,1,0,0,1,0,1,0] => [2,1,3,4] => 5
([(0,2),(0,3),(0,4),(2,5),(3,5),(4,1)],6) => [3,2,1] => [1,0,1,0,1,0,1,0] => [1,2,3,4] => 6
([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(3,5)],6) => [3,2,1] => [1,0,1,0,1,0,1,0] => [1,2,3,4] => 6
([(0,1),(0,2),(0,3),(1,5),(2,4),(2,5),(3,4),(3,5)],6) => [3,2,1] => [1,0,1,0,1,0,1,0] => [1,2,3,4] => 6
([(0,1),(0,2),(0,3),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6) => [3,2,1] => [1,0,1,0,1,0,1,0] => [1,2,3,4] => 6
([(0,2),(0,3),(0,4),(2,5),(3,5),(4,1),(4,5)],6) => [3,2,1] => [1,0,1,0,1,0,1,0] => [1,2,3,4] => 6
([(0,3),(0,4),(0,5),(4,2),(5,1)],6) => [3,2,1] => [1,0,1,0,1,0,1,0] => [1,2,3,4] => 6
([(0,2),(0,3),(0,4),(3,5),(4,1),(4,5)],6) => [3,2,1] => [1,0,1,0,1,0,1,0] => [1,2,3,4] => 6
([(0,1),(0,2),(0,3),(2,4),(2,5),(3,4),(3,5)],6) => [3,2,1] => [1,0,1,0,1,0,1,0] => [1,2,3,4] => 6
([(1,4),(1,5),(4,3),(5,2)],6) => [3,2,1] => [1,0,1,0,1,0,1,0] => [1,2,3,4] => 6
([(1,3),(1,4),(3,5),(4,2),(4,5)],6) => [3,2,1] => [1,0,1,0,1,0,1,0] => [1,2,3,4] => 6
([(1,2),(1,3),(2,4),(2,5),(3,4),(3,5)],6) => [3,2,1] => [1,0,1,0,1,0,1,0] => [1,2,3,4] => 6
([(0,4),(0,5),(4,3),(5,1),(5,2)],6) => [3,2,1] => [1,0,1,0,1,0,1,0] => [1,2,3,4] => 6
([(0,3),(0,4),(3,5),(4,1),(4,2),(4,5)],6) => [3,2,1] => [1,0,1,0,1,0,1,0] => [1,2,3,4] => 6
([(0,3),(0,4),(3,2),(3,5),(4,1),(4,5)],6) => [3,2,1] => [1,0,1,0,1,0,1,0] => [1,2,3,4] => 6
([(0,2),(0,3),(2,4),(2,5),(3,1),(3,4),(3,5)],6) => [3,2,1] => [1,0,1,0,1,0,1,0] => [1,2,3,4] => 6
([(0,1),(0,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5)],6) => [3,2,1] => [1,0,1,0,1,0,1,0] => [1,2,3,4] => 6
([(1,5),(2,5),(5,3),(5,4)],6) => [3,2,1] => [1,0,1,0,1,0,1,0] => [1,2,3,4] => 6
([(0,5),(1,5),(5,2),(5,3),(5,4)],6) => [3,2,1] => [1,0,1,0,1,0,1,0] => [1,2,3,4] => 6
([(0,5),(1,5),(2,5),(5,3),(5,4)],6) => [3,2,1] => [1,0,1,0,1,0,1,0] => [1,2,3,4] => 6
([(0,5),(1,4),(2,4),(2,5),(5,3)],6) => [3,2,1] => [1,0,1,0,1,0,1,0] => [1,2,3,4] => 6
([(0,4),(1,3),(2,3),(2,4),(3,5),(4,5)],6) => [3,2,1] => [1,0,1,0,1,0,1,0] => [1,2,3,4] => 6
([(0,5),(1,4),(1,5),(2,4),(2,5),(4,3)],6) => [3,2,1] => [1,0,1,0,1,0,1,0] => [1,2,3,4] => 6
([(0,5),(1,3),(1,5),(2,3),(2,5),(3,4),(5,4)],6) => [3,2,1] => [1,0,1,0,1,0,1,0] => [1,2,3,4] => 6
([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(5,3)],6) => [3,2,1] => [1,0,1,0,1,0,1,0] => [1,2,3,4] => 6
([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6) => [3,2,1] => [1,0,1,0,1,0,1,0] => [1,2,3,4] => 6
([(0,5),(1,4),(1,5),(2,4),(2,5),(5,3)],6) => [3,2,1] => [1,0,1,0,1,0,1,0] => [1,2,3,4] => 6
([(0,5),(1,5),(2,3),(5,4)],6) => [3,2,1] => [1,0,1,0,1,0,1,0] => [1,2,3,4] => 6
([(0,5),(1,5),(2,4),(5,3),(5,4)],6) => [3,2,1] => [1,0,1,0,1,0,1,0] => [1,2,3,4] => 6
([(0,5),(1,5),(2,3),(2,5),(5,4)],6) => [3,2,1] => [1,0,1,0,1,0,1,0] => [1,2,3,4] => 6
([(0,5),(1,5),(2,3),(2,5),(3,4)],6) => [3,2,1] => [1,0,1,0,1,0,1,0] => [1,2,3,4] => 6
([(0,5),(1,5),(2,3),(2,5),(3,4),(5,4)],6) => [3,2,1] => [1,0,1,0,1,0,1,0] => [1,2,3,4] => 6
([(0,4),(1,4),(2,3),(2,5),(4,5)],6) => [3,2,1] => [1,0,1,0,1,0,1,0] => [1,2,3,4] => 6
([(0,3),(1,3),(2,4),(2,5),(3,4),(3,5)],6) => [3,2,1] => [1,0,1,0,1,0,1,0] => [1,2,3,4] => 6
([(0,5),(1,5),(2,3),(3,4)],6) => [3,2,1] => [1,0,1,0,1,0,1,0] => [1,2,3,4] => 6
([(0,4),(1,4),(2,3),(3,5),(4,5)],6) => [3,2,1] => [1,0,1,0,1,0,1,0] => [1,2,3,4] => 6
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Description
The projective dimension of the simple module corresponding to the point in the poset of the symmetric group under bruhat order.
Map
Greene-Kleitman invariant
Description
The Greene-Kleitman invariant of a poset.
This is the partition $(c_1 - c_0, c_2 - c_1, c_3 - c_2, \ldots)$, where $c_k$ is the maximum cardinality of a union of $k$ chains of the poset. Equivalently, this is the conjugate of the partition $(a_1 - a_0, a_2 - a_1, a_3 - a_2, \ldots)$, where $a_k$ is the maximum cardinality of a union of $k$ antichains of the poset.
This is the partition $(c_1 - c_0, c_2 - c_1, c_3 - c_2, \ldots)$, where $c_k$ is the maximum cardinality of a union of $k$ chains of the poset. Equivalently, this is the conjugate of the partition $(a_1 - a_0, a_2 - a_1, a_3 - a_2, \ldots)$, where $a_k$ is the maximum cardinality of a union of $k$ antichains of the poset.
Map
to 312-avoiding permutation
Description
Map
to Dyck path
Description
Sends a partition to the shortest Dyck path tracing the shape of its Ferrers diagram.
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