Identifier
-
Mp00306:
Posets
—rowmotion cycle type⟶
Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
St001034: Dyck paths ⟶ ℤ
Values
([],1) => [2] => [1,0,1,0] => 2
([],2) => [2,2] => [1,1,1,0,0,0] => 4
([(0,1)],2) => [3] => [1,0,1,0,1,0] => 3
([],3) => [2,2,2,2] => [1,1,1,1,0,1,0,0,0,0] => 8
([(1,2)],3) => [6] => [1,0,1,0,1,0,1,0,1,0,1,0] => 6
([(0,1),(0,2)],3) => [3,2] => [1,0,1,1,1,0,0,0] => 5
([(0,2),(2,1)],3) => [4] => [1,0,1,0,1,0,1,0] => 4
([(0,2),(1,2)],3) => [3,2] => [1,0,1,1,1,0,0,0] => 5
([],4) => [2,2,2,2,2,2,2,2] => [1,1,1,1,0,1,0,1,0,1,0,1,0,1,0,0,0,0] => 16
([(2,3)],4) => [6,6] => [1,1,1,0,1,0,1,0,1,0,1,0,0,0] => 12
([(1,2),(1,3)],4) => [6,2,2] => [1,0,1,0,1,0,1,0,1,1,1,1,0,0,0,0] => 10
([(0,1),(0,2),(0,3)],4) => [3,2,2,2] => [1,0,1,1,1,1,0,1,0,0,0,0] => 9
([(0,2),(0,3),(3,1)],4) => [7] => [1,0,1,0,1,0,1,0,1,0,1,0,1,0] => 7
([(0,1),(0,2),(1,3),(2,3)],4) => [4,2] => [1,0,1,0,1,1,1,0,0,0] => 6
([(1,2),(2,3)],4) => [4,4] => [1,1,1,0,1,0,1,0,0,0] => 8
([(0,3),(3,1),(3,2)],4) => [4,2] => [1,0,1,0,1,1,1,0,0,0] => 6
([(1,3),(2,3)],4) => [6,2,2] => [1,0,1,0,1,0,1,0,1,1,1,1,0,0,0,0] => 10
([(0,3),(1,3),(3,2)],4) => [4,2] => [1,0,1,0,1,1,1,0,0,0] => 6
([(0,3),(1,3),(2,3)],4) => [3,2,2,2] => [1,0,1,1,1,1,0,1,0,0,0,0] => 9
([(0,3),(1,2)],4) => [3,3,3] => [1,1,1,1,1,0,0,0,0,0] => 9
([(0,3),(1,2),(1,3)],4) => [5,3] => [1,0,1,0,1,1,1,0,1,0,0,0] => 8
([(0,2),(0,3),(1,2),(1,3)],4) => [3,2,2] => [1,0,1,1,1,1,0,0,0,0] => 7
([(0,3),(2,1),(3,2)],4) => [5] => [1,0,1,0,1,0,1,0,1,0] => 5
([(0,3),(1,2),(2,3)],4) => [7] => [1,0,1,0,1,0,1,0,1,0,1,0,1,0] => 7
([(3,4)],5) => [6,6,6,6] => [1,1,1,1,1,1,1,0,1,0,1,0,0,0,0,0,0,0] => 24
([(0,1),(0,2),(0,3),(0,4)],5) => [3,2,2,2,2,2,2,2] => [1,0,1,1,1,1,0,1,0,1,0,1,0,1,0,1,0,0,0,0] => 17
([(0,2),(0,3),(0,4),(4,1)],5) => [7,6] => [1,0,1,1,1,0,1,0,1,0,1,0,1,0,0,0] => 13
([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5) => [4,2,2,2] => [1,0,1,0,1,1,1,1,0,1,0,0,0,0] => 10
([(1,2),(1,3),(2,4),(3,4)],5) => [4,4,2,2] => [1,1,1,0,1,0,1,1,0,1,0,0,0,0] => 12
([(0,2),(0,3),(2,4),(3,4),(4,1)],5) => [5,2] => [1,0,1,0,1,0,1,1,1,0,0,0] => 7
([(0,3),(0,4),(3,2),(4,1)],5) => [4,3,3] => [1,0,1,1,1,1,1,0,0,0,0,0] => 10
([(0,2),(0,3),(2,4),(3,1),(3,4)],5) => [5,4] => [1,0,1,1,1,0,1,0,1,0,0,0] => 9
([(0,1),(0,2),(1,3),(1,4),(2,3),(2,4)],5) => [4,2,2] => [1,0,1,0,1,1,1,1,0,0,0,0] => 8
([(2,3),(3,4)],5) => [4,4,4,4] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0] => 16
([(1,4),(4,2),(4,3)],5) => [4,4,2,2] => [1,1,1,0,1,0,1,1,0,1,0,0,0,0] => 12
([(0,4),(4,1),(4,2),(4,3)],5) => [4,2,2,2] => [1,0,1,0,1,1,1,1,0,1,0,0,0,0] => 10
([(1,4),(2,4),(4,3)],5) => [4,4,2,2] => [1,1,1,0,1,0,1,1,0,1,0,0,0,0] => 12
([(0,4),(1,4),(4,2),(4,3)],5) => [4,2,2] => [1,0,1,0,1,1,1,1,0,0,0,0] => 8
([(0,4),(1,4),(2,4),(4,3)],5) => [4,2,2,2] => [1,0,1,0,1,1,1,1,0,1,0,0,0,0] => 10
([(0,4),(1,4),(2,4),(3,4)],5) => [3,2,2,2,2,2,2,2] => [1,0,1,1,1,1,0,1,0,1,0,1,0,1,0,1,0,0,0,0] => 17
([(0,4),(1,3),(1,4),(2,3),(2,4)],5) => [5,3,2,2] => [1,0,1,0,1,1,1,0,1,1,0,1,0,0,0,0] => 12
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5) => [3,2,2,2,2] => [1,0,1,1,1,1,0,1,0,1,0,0,0,0] => 11
([(0,4),(1,4),(2,3),(4,2)],5) => [5,2] => [1,0,1,0,1,0,1,1,1,0,0,0] => 7
([(0,4),(1,4),(2,3),(2,4)],5) => [6,5,3] => [1,0,1,1,1,0,1,0,1,1,1,0,0,0,0,0] => 14
([(0,4),(1,4),(2,3),(3,4)],5) => [7,6] => [1,0,1,1,1,0,1,0,1,0,1,0,1,0,0,0] => 13
([(1,4),(2,3)],5) => [6,6,6] => [1,1,1,1,1,0,1,0,1,0,1,0,0,0,0,0] => 18
([(0,3),(1,2),(1,3),(2,4),(3,4)],5) => [5,4] => [1,0,1,1,1,0,1,0,1,0,0,0] => 9
([(0,3),(0,4),(1,3),(1,4),(4,2)],5) => [7,2] => [1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,0] => 9
([(0,3),(0,4),(1,3),(1,4),(3,2),(4,2)],5) => [4,2,2] => [1,0,1,0,1,1,1,1,0,0,0,0] => 8
([(0,4),(1,2),(1,4),(4,3)],5) => [10] => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0] => 10
([(0,4),(1,2),(1,3),(1,4)],5) => [6,5,3] => [1,0,1,1,1,0,1,0,1,1,1,0,0,0,0,0] => 14
([(0,2),(0,4),(3,1),(4,3)],5) => [5,4] => [1,0,1,1,1,0,1,0,1,0,0,0] => 9
([(0,2),(0,3),(1,4),(2,4),(3,1)],5) => [8] => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0] => 8
([(0,3),(0,4),(1,2),(1,3),(1,4)],5) => [5,3,2,2] => [1,0,1,0,1,1,1,0,1,1,0,1,0,0,0,0] => 12
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4)],5) => [3,2,2,2,2] => [1,0,1,1,1,1,0,1,0,1,0,0,0,0] => 11
([(0,3),(0,4),(1,2),(1,3),(2,4)],5) => [10] => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0] => 10
([(0,3),(0,4),(1,2),(2,3),(2,4)],5) => [7,2] => [1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,0] => 9
([(1,4),(3,2),(4,3)],5) => [10] => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0] => 10
([(0,3),(3,4),(4,1),(4,2)],5) => [5,2] => [1,0,1,0,1,0,1,1,1,0,0,0] => 7
([(0,4),(1,2),(2,4),(4,3)],5) => [8] => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0] => 8
([(0,4),(3,2),(4,1),(4,3)],5) => [8] => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0] => 8
([(0,4),(1,2),(2,3),(2,4)],5) => [10] => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0] => 10
([(0,4),(2,3),(3,1),(4,2)],5) => [6] => [1,0,1,0,1,0,1,0,1,0,1,0] => 6
([(0,3),(1,2),(2,4),(3,4)],5) => [4,3,3] => [1,0,1,1,1,1,1,0,0,0,0,0] => 10
([(0,4),(1,2),(2,3),(3,4)],5) => [5,4] => [1,0,1,1,1,0,1,0,1,0,0,0] => 9
([(0,3),(1,4),(2,4),(3,1),(3,2)],5) => [5,2] => [1,0,1,0,1,0,1,1,1,0,0,0] => 7
([(0,2),(0,3),(0,4),(2,5),(3,5),(4,5),(5,1)],6) => [5,2,2,2] => [1,0,1,0,1,0,1,1,1,1,0,1,0,0,0,0] => 11
([(0,1),(0,2),(0,3),(1,5),(2,4),(2,5),(3,4),(3,5)],6) => [5,4,2,2] => [1,0,1,1,1,0,1,0,1,1,0,1,0,0,0,0] => 13
([(0,1),(0,2),(0,3),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6) => [4,2,2,2,2] => [1,0,1,0,1,1,1,1,0,1,0,1,0,0,0,0] => 12
([(0,2),(0,3),(0,4),(2,5),(3,5),(4,1),(4,5)],6) => [6,5,4] => [1,0,1,1,1,0,1,1,1,0,1,0,0,0,0,0] => 15
([(0,2),(0,3),(0,4),(3,5),(4,5),(5,1)],6) => [5,4,2,2] => [1,0,1,1,1,0,1,0,1,1,0,1,0,0,0,0] => 13
([(0,3),(0,4),(0,5),(4,2),(5,1)],6) => [7,6,6] => [1,0,1,1,1,1,1,0,1,0,1,0,1,0,0,0,0,0] => 19
([(0,3),(0,4),(3,5),(4,5),(5,1),(5,2)],6) => [5,2,2] => [1,0,1,0,1,0,1,1,1,1,0,0,0,0] => 9
([(0,2),(0,3),(2,4),(2,5),(3,4),(3,5),(5,1)],6) => [8,2] => [1,0,1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,0] => 10
([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6) => [5,2,2] => [1,0,1,0,1,0,1,1,1,1,0,0,0,0] => 9
([(0,3),(0,4),(3,5),(4,1),(4,2),(4,5)],6) => [6,5,4] => [1,0,1,1,1,0,1,1,1,0,1,0,0,0,0,0] => 15
([(0,2),(0,3),(2,4),(2,5),(3,1),(3,4),(3,5)],6) => [5,4,2,2] => [1,0,1,1,1,0,1,0,1,1,0,1,0,0,0,0] => 13
([(0,1),(0,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5)],6) => [4,2,2,2,2] => [1,0,1,0,1,1,1,1,0,1,0,1,0,0,0,0] => 12
([(2,3),(3,5),(5,4)],6) => [10,10] => [1,1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0,0] => 20
([(0,4),(4,5),(5,1),(5,2),(5,3)],6) => [5,2,2,2] => [1,0,1,0,1,0,1,1,1,1,0,1,0,0,0,0] => 11
([(0,5),(1,5),(5,2),(5,3),(5,4)],6) => [4,2,2,2,2] => [1,0,1,0,1,1,1,1,0,1,0,1,0,0,0,0] => 12
([(0,5),(1,5),(2,5),(5,3),(5,4)],6) => [4,2,2,2,2] => [1,0,1,0,1,1,1,1,0,1,0,1,0,0,0,0] => 12
([(0,5),(1,5),(2,5),(3,4),(5,3)],6) => [5,2,2,2] => [1,0,1,0,1,0,1,1,1,1,0,1,0,0,0,0] => 11
([(1,5),(2,5),(3,4)],6) => [6,6,6,6,6] => [1,1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0,0] => 30
([(0,5),(1,3),(1,5),(2,3),(2,5),(3,4),(5,4)],6) => [5,4,2,2] => [1,0,1,1,1,0,1,0,1,1,0,1,0,0,0,0] => 13
([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6) => [4,2,2,2,2] => [1,0,1,0,1,1,1,1,0,1,0,1,0,0,0,0] => 12
([(0,5),(1,5),(4,2),(5,3),(5,4)],6) => [8,2] => [1,0,1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,0] => 10
([(0,5),(1,5),(2,3),(2,5),(3,4),(5,4)],6) => [6,5,4] => [1,0,1,1,1,0,1,1,1,0,1,0,0,0,0,0] => 15
([(0,5),(1,5),(4,2),(4,3),(5,4)],6) => [5,2,2] => [1,0,1,0,1,0,1,1,1,1,0,0,0,0] => 9
([(0,3),(0,4),(1,5),(2,5),(4,1),(4,2)],6) => [5,4,2,2] => [1,0,1,1,1,0,1,0,1,1,0,1,0,0,0,0] => 13
([(0,3),(0,4),(1,5),(2,5),(3,2),(4,1)],6) => [5,3,3] => [1,0,1,0,1,1,1,1,1,0,0,0,0,0] => 11
([(0,5),(1,5),(3,2),(4,3),(5,4)],6) => [6,2] => [1,0,1,0,1,0,1,0,1,1,1,0,0,0] => 8
([(0,5),(1,5),(2,3),(3,4),(4,5)],6) => [5,4,4,4] => [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0] => 17
([(0,5),(1,4),(2,5),(3,5),(4,2),(4,3)],6) => [5,4,2,2] => [1,0,1,1,1,0,1,0,1,1,0,1,0,0,0,0] => 13
([(0,4),(1,5),(2,5),(3,5),(4,1),(4,2),(4,3)],6) => [5,2,2,2] => [1,0,1,0,1,0,1,1,1,1,0,1,0,0,0,0] => 11
([(0,5),(1,4),(2,4),(3,5),(4,3)],6) => [5,4,2,2] => [1,0,1,1,1,0,1,0,1,1,0,1,0,0,0,0] => 13
([(0,4),(1,4),(2,5),(3,5),(4,2),(4,3)],6) => [5,2,2] => [1,0,1,0,1,0,1,1,1,1,0,0,0,0] => 9
([(0,4),(1,2),(1,4),(2,3),(2,5),(4,5)],6) => [6,4,3] => [1,0,1,0,1,1,1,0,1,1,1,0,0,0,0,0] => 13
([(0,5),(1,2),(1,5),(2,3),(2,4),(5,3),(5,4)],6) => [5,4,2] => [1,0,1,1,1,0,1,0,1,1,0,0,0,0] => 11
([(0,4),(1,2),(1,4),(2,5),(4,5),(5,3)],6) => [5,5] => [1,1,1,0,1,0,1,0,1,0,0,0] => 10
([(0,4),(0,5),(1,4),(1,5),(4,3),(5,2)],6) => [4,3,3,2] => [1,0,1,1,1,1,1,0,0,1,0,0,0,0] => 12
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Description
The area of the parallelogram polyomino associated with the Dyck path.
The (bivariate) generating function is given in [1].
The (bivariate) generating function is given in [1].
Map
parallelogram polyomino
Description
Return the Dyck path corresponding to the partition interpreted as a parallogram polyomino.
The Ferrers diagram of an integer partition can be interpreted as a parallogram polyomino, such that each part corresponds to a column.
This map returns the corresponding Dyck path.
The Ferrers diagram of an integer partition can be interpreted as a parallogram polyomino, such that each part corresponds to a column.
This map returns the corresponding Dyck path.
Map
rowmotion cycle type
Description
The cycle type of rowmotion on the order ideals of a poset.
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