Identifier
-
Mp00275:
Graphs
—to edge-partition of connected components⟶
Integer partitions
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
St000790: Dyck paths ⟶ ℤ
Values
([(0,1)],2) => [1] => [1,0,1,0] => 0
([(1,2)],3) => [1] => [1,0,1,0] => 0
([(0,2),(1,2)],3) => [2] => [1,1,0,0,1,0] => 0
([(0,1),(0,2),(1,2)],3) => [3] => [1,1,1,0,0,0,1,0] => 0
([(2,3)],4) => [1] => [1,0,1,0] => 0
([(1,3),(2,3)],4) => [2] => [1,1,0,0,1,0] => 0
([(0,3),(1,3),(2,3)],4) => [3] => [1,1,1,0,0,0,1,0] => 0
([(0,3),(1,2)],4) => [1,1] => [1,0,1,1,0,0] => 0
([(0,3),(1,2),(2,3)],4) => [3] => [1,1,1,0,0,0,1,0] => 0
([(1,2),(1,3),(2,3)],4) => [3] => [1,1,1,0,0,0,1,0] => 0
([(0,3),(1,2),(1,3),(2,3)],4) => [4] => [1,1,1,1,0,0,0,0,1,0] => 0
([(0,2),(0,3),(1,2),(1,3)],4) => [4] => [1,1,1,1,0,0,0,0,1,0] => 0
([(0,2),(0,3),(1,2),(1,3),(2,3)],4) => [5] => [1,1,1,1,1,0,0,0,0,0,1,0] => 0
([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4) => [6] => [1,1,1,1,1,1,0,0,0,0,0,0,1,0] => 0
([(3,4)],5) => [1] => [1,0,1,0] => 0
([(2,4),(3,4)],5) => [2] => [1,1,0,0,1,0] => 0
([(1,4),(2,4),(3,4)],5) => [3] => [1,1,1,0,0,0,1,0] => 0
([(0,4),(1,4),(2,4),(3,4)],5) => [4] => [1,1,1,1,0,0,0,0,1,0] => 0
([(1,4),(2,3)],5) => [1,1] => [1,0,1,1,0,0] => 0
([(1,4),(2,3),(3,4)],5) => [3] => [1,1,1,0,0,0,1,0] => 0
([(0,1),(2,4),(3,4)],5) => [2,1] => [1,0,1,0,1,0] => 0
([(2,3),(2,4),(3,4)],5) => [3] => [1,1,1,0,0,0,1,0] => 0
([(0,4),(1,4),(2,3),(3,4)],5) => [4] => [1,1,1,1,0,0,0,0,1,0] => 0
([(1,4),(2,3),(2,4),(3,4)],5) => [4] => [1,1,1,1,0,0,0,0,1,0] => 0
([(0,4),(1,4),(2,3),(2,4),(3,4)],5) => [5] => [1,1,1,1,1,0,0,0,0,0,1,0] => 0
([(1,3),(1,4),(2,3),(2,4)],5) => [4] => [1,1,1,1,0,0,0,0,1,0] => 0
([(0,4),(1,2),(1,3),(2,4),(3,4)],5) => [5] => [1,1,1,1,1,0,0,0,0,0,1,0] => 0
([(1,3),(1,4),(2,3),(2,4),(3,4)],5) => [5] => [1,1,1,1,1,0,0,0,0,0,1,0] => 0
([(0,4),(1,3),(2,3),(2,4),(3,4)],5) => [5] => [1,1,1,1,1,0,0,0,0,0,1,0] => 0
([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5) => [6] => [1,1,1,1,1,1,0,0,0,0,0,0,1,0] => 0
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5) => [6] => [1,1,1,1,1,1,0,0,0,0,0,0,1,0] => 0
([(0,4),(1,3),(2,3),(2,4)],5) => [4] => [1,1,1,1,0,0,0,0,1,0] => 0
([(0,1),(2,3),(2,4),(3,4)],5) => [3,1] => [1,1,0,1,0,0,1,0] => 0
([(0,3),(1,2),(1,4),(2,4),(3,4)],5) => [5] => [1,1,1,1,1,0,0,0,0,0,1,0] => 0
([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5) => [6] => [1,1,1,1,1,1,0,0,0,0,0,0,1,0] => 0
([(0,3),(0,4),(1,2),(1,4),(2,3)],5) => [5] => [1,1,1,1,1,0,0,0,0,0,1,0] => 0
([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5) => [6] => [1,1,1,1,1,1,0,0,0,0,0,0,1,0] => 0
([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5) => [6] => [1,1,1,1,1,1,0,0,0,0,0,0,1,0] => 0
([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5) => [6] => [1,1,1,1,1,1,0,0,0,0,0,0,1,0] => 0
([(4,5)],6) => [1] => [1,0,1,0] => 0
([(3,5),(4,5)],6) => [2] => [1,1,0,0,1,0] => 0
([(2,5),(3,5),(4,5)],6) => [3] => [1,1,1,0,0,0,1,0] => 0
([(1,5),(2,5),(3,5),(4,5)],6) => [4] => [1,1,1,1,0,0,0,0,1,0] => 0
([(0,5),(1,5),(2,5),(3,5),(4,5)],6) => [5] => [1,1,1,1,1,0,0,0,0,0,1,0] => 0
([(2,5),(3,4)],6) => [1,1] => [1,0,1,1,0,0] => 0
([(2,5),(3,4),(4,5)],6) => [3] => [1,1,1,0,0,0,1,0] => 0
([(1,2),(3,5),(4,5)],6) => [2,1] => [1,0,1,0,1,0] => 0
([(3,4),(3,5),(4,5)],6) => [3] => [1,1,1,0,0,0,1,0] => 0
([(1,5),(2,5),(3,4),(4,5)],6) => [4] => [1,1,1,1,0,0,0,0,1,0] => 0
([(0,1),(2,5),(3,5),(4,5)],6) => [3,1] => [1,1,0,1,0,0,1,0] => 0
([(2,5),(3,4),(3,5),(4,5)],6) => [4] => [1,1,1,1,0,0,0,0,1,0] => 0
([(0,5),(1,5),(2,5),(3,4),(4,5)],6) => [5] => [1,1,1,1,1,0,0,0,0,0,1,0] => 0
([(1,5),(2,5),(3,4),(3,5),(4,5)],6) => [5] => [1,1,1,1,1,0,0,0,0,0,1,0] => 0
([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6) => [6] => [1,1,1,1,1,1,0,0,0,0,0,0,1,0] => 0
([(2,4),(2,5),(3,4),(3,5)],6) => [4] => [1,1,1,1,0,0,0,0,1,0] => 0
([(0,5),(1,5),(2,4),(3,4)],6) => [2,2] => [1,1,0,0,1,1,0,0] => 0
([(1,5),(2,3),(2,4),(3,5),(4,5)],6) => [5] => [1,1,1,1,1,0,0,0,0,0,1,0] => 0
([(0,5),(1,5),(2,3),(3,4),(4,5)],6) => [5] => [1,1,1,1,1,0,0,0,0,0,1,0] => 0
([(2,4),(2,5),(3,4),(3,5),(4,5)],6) => [5] => [1,1,1,1,1,0,0,0,0,0,1,0] => 0
([(1,5),(2,4),(3,4),(3,5),(4,5)],6) => [5] => [1,1,1,1,1,0,0,0,0,0,1,0] => 0
([(0,5),(1,5),(2,4),(3,4),(4,5)],6) => [5] => [1,1,1,1,1,0,0,0,0,0,1,0] => 0
([(0,5),(1,5),(2,3),(2,4),(3,5),(4,5)],6) => [6] => [1,1,1,1,1,1,0,0,0,0,0,0,1,0] => 0
([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => [6] => [1,1,1,1,1,1,0,0,0,0,0,0,1,0] => 0
([(0,5),(1,5),(2,4),(3,4),(3,5),(4,5)],6) => [6] => [1,1,1,1,1,1,0,0,0,0,0,0,1,0] => 0
([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6) => [6] => [1,1,1,1,1,1,0,0,0,0,0,0,1,0] => 0
([(0,5),(1,4),(2,4),(2,5),(3,4),(3,5)],6) => [6] => [1,1,1,1,1,1,0,0,0,0,0,0,1,0] => 0
([(0,5),(1,4),(2,3)],6) => [1,1,1] => [1,0,1,1,1,0,0,0] => 0
([(1,5),(2,4),(3,4),(3,5)],6) => [4] => [1,1,1,1,0,0,0,0,1,0] => 0
([(0,1),(2,5),(3,4),(4,5)],6) => [3,1] => [1,1,0,1,0,0,1,0] => 0
([(1,2),(3,4),(3,5),(4,5)],6) => [3,1] => [1,1,0,1,0,0,1,0] => 0
([(0,5),(1,4),(2,3),(3,5),(4,5)],6) => [5] => [1,1,1,1,1,0,0,0,0,0,1,0] => 0
([(1,4),(2,3),(2,5),(3,5),(4,5)],6) => [5] => [1,1,1,1,1,0,0,0,0,0,1,0] => 0
([(0,1),(2,5),(3,4),(3,5),(4,5)],6) => [4,1] => [1,1,1,0,1,0,0,0,1,0] => 0
([(0,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6) => [6] => [1,1,1,1,1,1,0,0,0,0,0,0,1,0] => 0
([(1,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6) => [6] => [1,1,1,1,1,1,0,0,0,0,0,0,1,0] => 0
([(1,4),(1,5),(2,3),(2,5),(3,4)],6) => [5] => [1,1,1,1,1,0,0,0,0,0,1,0] => 0
([(0,5),(1,4),(2,3),(2,4),(3,5),(4,5)],6) => [6] => [1,1,1,1,1,1,0,0,0,0,0,0,1,0] => 0
([(1,2),(1,5),(2,4),(3,4),(3,5),(4,5)],6) => [6] => [1,1,1,1,1,1,0,0,0,0,0,0,1,0] => 0
([(0,5),(1,2),(1,4),(2,3),(3,5),(4,5)],6) => [6] => [1,1,1,1,1,1,0,0,0,0,0,0,1,0] => 0
([(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6) => [6] => [1,1,1,1,1,1,0,0,0,0,0,0,1,0] => 0
([(0,5),(1,4),(2,3),(3,4),(3,5),(4,5)],6) => [6] => [1,1,1,1,1,1,0,0,0,0,0,0,1,0] => 0
([(0,5),(1,4),(2,3),(2,4),(3,5)],6) => [5] => [1,1,1,1,1,0,0,0,0,0,1,0] => 0
([(0,1),(2,4),(2,5),(3,4),(3,5)],6) => [4,1] => [1,1,1,0,1,0,0,0,1,0] => 0
([(0,5),(1,5),(2,3),(2,4),(3,4)],6) => [3,2] => [1,1,0,0,1,0,1,0] => 0
([(0,4),(1,2),(1,3),(2,5),(3,5),(4,5)],6) => [6] => [1,1,1,1,1,1,0,0,0,0,0,0,1,0] => 0
([(0,4),(1,2),(1,5),(2,5),(3,4),(3,5)],6) => [6] => [1,1,1,1,1,1,0,0,0,0,0,0,1,0] => 0
([(0,4),(1,2),(2,5),(3,4),(3,5),(4,5)],6) => [6] => [1,1,1,1,1,1,0,0,0,0,0,0,1,0] => 0
([(0,1),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => [5,1] => [1,1,1,1,0,1,0,0,0,0,1,0] => 0
([(0,4),(1,4),(2,3),(2,5),(3,5),(4,5)],6) => [6] => [1,1,1,1,1,1,0,0,0,0,0,0,1,0] => 0
([(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => [6] => [1,1,1,1,1,1,0,0,0,0,0,0,1,0] => 0
([(0,4),(0,5),(1,2),(1,3),(2,5),(3,4)],6) => [6] => [1,1,1,1,1,1,0,0,0,0,0,0,1,0] => 0
([(0,4),(0,5),(1,2),(1,3),(2,3),(4,5)],6) => [3,3] => [1,1,1,0,0,0,1,1,0,0] => 0
([(0,1),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => [6,1] => [1,1,1,1,1,0,1,0,0,0,0,0,1,0] => 0
([(5,6)],7) => [1] => [1,0,1,0] => 0
([(4,6),(5,6)],7) => [2] => [1,1,0,0,1,0] => 0
([(3,6),(4,6),(5,6)],7) => [3] => [1,1,1,0,0,0,1,0] => 0
([(2,6),(3,6),(4,6),(5,6)],7) => [4] => [1,1,1,1,0,0,0,0,1,0] => 0
([(1,6),(2,6),(3,6),(4,6),(5,6)],7) => [5] => [1,1,1,1,1,0,0,0,0,0,1,0] => 0
([(0,6),(1,6),(2,6),(3,6),(4,6),(5,6)],7) => [6] => [1,1,1,1,1,1,0,0,0,0,0,0,1,0] => 0
([(3,6),(4,5)],7) => [1,1] => [1,0,1,1,0,0] => 0
([(3,6),(4,5),(5,6)],7) => [3] => [1,1,1,0,0,0,1,0] => 0
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searching the database for the individual values of this statistic
Description
The number of pairs of centered tunnels, one strictly containing the other, of a Dyck path.
Apparently, the total number of these is given in [1].
The statistic counting all pairs of distinct tunnels is the area of a Dyck path St000012The area of a Dyck path..
Apparently, the total number of these is given in [1].
The statistic counting all pairs of distinct tunnels is the area of a Dyck path St000012The area of a Dyck path..
Map
to Dyck path
Description
Sends a partition to the shortest Dyck path tracing the shape of its Ferrers diagram.
Map
to edge-partition of connected components
Description
Sends a graph to the partition recording the number of edges in its connected components.
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