Identifier
-
Mp00276:
Graphs
—to edge-partition of biconnected components⟶
Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00120: Dyck paths —Lalanne-Kreweras involution⟶ Dyck paths
St000015: Dyck paths ⟶ ℤ (values match St000053The number of valleys of the Dyck path., St001068Number of torsionless simple modules in the corresponding Nakayama algebra., St001169Number of simple modules with projective dimension at least two in the corresponding Nakayama algebra.)
Values
([(0,1)],2) => [1] => [1,0] => [1,0] => 1
([(1,2)],3) => [1] => [1,0] => [1,0] => 1
([(0,2),(1,2)],3) => [1,1] => [1,1,0,0] => [1,0,1,0] => 2
([(0,1),(0,2),(1,2)],3) => [3] => [1,0,1,0,1,0] => [1,1,1,0,0,0] => 1
([(2,3)],4) => [1] => [1,0] => [1,0] => 1
([(1,3),(2,3)],4) => [1,1] => [1,1,0,0] => [1,0,1,0] => 2
([(0,3),(1,3),(2,3)],4) => [1,1,1] => [1,1,0,1,0,0] => [1,1,0,1,0,0] => 2
([(0,3),(1,2)],4) => [1,1] => [1,1,0,0] => [1,0,1,0] => 2
([(0,3),(1,2),(2,3)],4) => [1,1,1] => [1,1,0,1,0,0] => [1,1,0,1,0,0] => 2
([(1,2),(1,3),(2,3)],4) => [3] => [1,0,1,0,1,0] => [1,1,1,0,0,0] => 1
([(0,3),(1,2),(1,3),(2,3)],4) => [3,1] => [1,0,1,0,1,1,0,0] => [1,1,1,0,0,0,1,0] => 2
([(0,2),(0,3),(1,2),(1,3)],4) => [4] => [1,0,1,0,1,0,1,0] => [1,1,1,1,0,0,0,0] => 1
([(0,2),(0,3),(1,2),(1,3),(2,3)],4) => [5] => [1,0,1,0,1,0,1,0,1,0] => [1,1,1,1,1,0,0,0,0,0] => 1
([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4) => [6] => [1,0,1,0,1,0,1,0,1,0,1,0] => [1,1,1,1,1,1,0,0,0,0,0,0] => 1
([(3,4)],5) => [1] => [1,0] => [1,0] => 1
([(2,4),(3,4)],5) => [1,1] => [1,1,0,0] => [1,0,1,0] => 2
([(1,4),(2,4),(3,4)],5) => [1,1,1] => [1,1,0,1,0,0] => [1,1,0,1,0,0] => 2
([(0,4),(1,4),(2,4),(3,4)],5) => [1,1,1,1] => [1,1,0,1,0,1,0,0] => [1,1,1,0,1,0,0,0] => 2
([(1,4),(2,3)],5) => [1,1] => [1,1,0,0] => [1,0,1,0] => 2
([(1,4),(2,3),(3,4)],5) => [1,1,1] => [1,1,0,1,0,0] => [1,1,0,1,0,0] => 2
([(0,1),(2,4),(3,4)],5) => [1,1,1] => [1,1,0,1,0,0] => [1,1,0,1,0,0] => 2
([(2,3),(2,4),(3,4)],5) => [3] => [1,0,1,0,1,0] => [1,1,1,0,0,0] => 1
([(0,4),(1,4),(2,3),(3,4)],5) => [1,1,1,1] => [1,1,0,1,0,1,0,0] => [1,1,1,0,1,0,0,0] => 2
([(1,4),(2,3),(2,4),(3,4)],5) => [3,1] => [1,0,1,0,1,1,0,0] => [1,1,1,0,0,0,1,0] => 2
([(0,4),(1,4),(2,3),(2,4),(3,4)],5) => [3,1,1] => [1,0,1,0,1,1,0,1,0,0] => [1,1,1,1,0,0,0,1,0,0] => 2
([(1,3),(1,4),(2,3),(2,4)],5) => [4] => [1,0,1,0,1,0,1,0] => [1,1,1,1,0,0,0,0] => 1
([(0,4),(1,2),(1,3),(2,4),(3,4)],5) => [4,1] => [1,0,1,0,1,0,1,1,0,0] => [1,1,1,1,0,0,0,0,1,0] => 2
([(1,3),(1,4),(2,3),(2,4),(3,4)],5) => [5] => [1,0,1,0,1,0,1,0,1,0] => [1,1,1,1,1,0,0,0,0,0] => 1
([(0,4),(1,3),(2,3),(2,4),(3,4)],5) => [3,1,1] => [1,0,1,0,1,1,0,1,0,0] => [1,1,1,1,0,0,0,1,0,0] => 2
([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5) => [5,1] => [1,0,1,0,1,0,1,0,1,1,0,0] => [1,1,1,1,1,0,0,0,0,0,1,0] => 2
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5) => [6] => [1,0,1,0,1,0,1,0,1,0,1,0] => [1,1,1,1,1,1,0,0,0,0,0,0] => 1
([(0,4),(1,3),(2,3),(2,4)],5) => [1,1,1,1] => [1,1,0,1,0,1,0,0] => [1,1,1,0,1,0,0,0] => 2
([(0,1),(2,3),(2,4),(3,4)],5) => [3,1] => [1,0,1,0,1,1,0,0] => [1,1,1,0,0,0,1,0] => 2
([(0,3),(1,2),(1,4),(2,4),(3,4)],5) => [3,1,1] => [1,0,1,0,1,1,0,1,0,0] => [1,1,1,1,0,0,0,1,0,0] => 2
([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5) => [3,3] => [1,1,1,0,1,0,0,0] => [1,1,0,1,0,1,0,0] => 3
([(0,3),(0,4),(1,2),(1,4),(2,3)],5) => [5] => [1,0,1,0,1,0,1,0,1,0] => [1,1,1,1,1,0,0,0,0,0] => 1
([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5) => [6] => [1,0,1,0,1,0,1,0,1,0,1,0] => [1,1,1,1,1,1,0,0,0,0,0,0] => 1
([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5) => [5,1] => [1,0,1,0,1,0,1,0,1,1,0,0] => [1,1,1,1,1,0,0,0,0,0,1,0] => 2
([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5) => [6] => [1,0,1,0,1,0,1,0,1,0,1,0] => [1,1,1,1,1,1,0,0,0,0,0,0] => 1
([(4,5)],6) => [1] => [1,0] => [1,0] => 1
([(3,5),(4,5)],6) => [1,1] => [1,1,0,0] => [1,0,1,0] => 2
([(2,5),(3,5),(4,5)],6) => [1,1,1] => [1,1,0,1,0,0] => [1,1,0,1,0,0] => 2
([(1,5),(2,5),(3,5),(4,5)],6) => [1,1,1,1] => [1,1,0,1,0,1,0,0] => [1,1,1,0,1,0,0,0] => 2
([(0,5),(1,5),(2,5),(3,5),(4,5)],6) => [1,1,1,1,1] => [1,1,0,1,0,1,0,1,0,0] => [1,1,1,1,0,1,0,0,0,0] => 2
([(2,5),(3,4)],6) => [1,1] => [1,1,0,0] => [1,0,1,0] => 2
([(2,5),(3,4),(4,5)],6) => [1,1,1] => [1,1,0,1,0,0] => [1,1,0,1,0,0] => 2
([(1,2),(3,5),(4,5)],6) => [1,1,1] => [1,1,0,1,0,0] => [1,1,0,1,0,0] => 2
([(3,4),(3,5),(4,5)],6) => [3] => [1,0,1,0,1,0] => [1,1,1,0,0,0] => 1
([(1,5),(2,5),(3,4),(4,5)],6) => [1,1,1,1] => [1,1,0,1,0,1,0,0] => [1,1,1,0,1,0,0,0] => 2
([(0,1),(2,5),(3,5),(4,5)],6) => [1,1,1,1] => [1,1,0,1,0,1,0,0] => [1,1,1,0,1,0,0,0] => 2
([(2,5),(3,4),(3,5),(4,5)],6) => [3,1] => [1,0,1,0,1,1,0,0] => [1,1,1,0,0,0,1,0] => 2
([(0,5),(1,5),(2,5),(3,4),(4,5)],6) => [1,1,1,1,1] => [1,1,0,1,0,1,0,1,0,0] => [1,1,1,1,0,1,0,0,0,0] => 2
([(1,5),(2,5),(3,4),(3,5),(4,5)],6) => [3,1,1] => [1,0,1,0,1,1,0,1,0,0] => [1,1,1,1,0,0,0,1,0,0] => 2
([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6) => [3,1,1,1] => [1,0,1,0,1,1,0,1,0,1,0,0] => [1,1,1,1,1,0,0,0,1,0,0,0] => 2
([(2,4),(2,5),(3,4),(3,5)],6) => [4] => [1,0,1,0,1,0,1,0] => [1,1,1,1,0,0,0,0] => 1
([(0,5),(1,5),(2,4),(3,4)],6) => [1,1,1,1] => [1,1,0,1,0,1,0,0] => [1,1,1,0,1,0,0,0] => 2
([(1,5),(2,3),(2,4),(3,5),(4,5)],6) => [4,1] => [1,0,1,0,1,0,1,1,0,0] => [1,1,1,1,0,0,0,0,1,0] => 2
([(0,5),(1,5),(2,3),(3,4),(4,5)],6) => [1,1,1,1,1] => [1,1,0,1,0,1,0,1,0,0] => [1,1,1,1,0,1,0,0,0,0] => 2
([(2,4),(2,5),(3,4),(3,5),(4,5)],6) => [5] => [1,0,1,0,1,0,1,0,1,0] => [1,1,1,1,1,0,0,0,0,0] => 1
([(1,5),(2,4),(3,4),(3,5),(4,5)],6) => [3,1,1] => [1,0,1,0,1,1,0,1,0,0] => [1,1,1,1,0,0,0,1,0,0] => 2
([(0,5),(1,5),(2,4),(3,4),(4,5)],6) => [1,1,1,1,1] => [1,1,0,1,0,1,0,1,0,0] => [1,1,1,1,0,1,0,0,0,0] => 2
([(0,5),(1,5),(2,3),(2,4),(3,5),(4,5)],6) => [4,1,1] => [1,0,1,0,1,0,1,1,0,1,0,0] => [1,1,1,1,1,0,0,0,0,1,0,0] => 2
([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => [5,1] => [1,0,1,0,1,0,1,0,1,1,0,0] => [1,1,1,1,1,0,0,0,0,0,1,0] => 2
([(0,5),(1,5),(2,4),(3,4),(3,5),(4,5)],6) => [3,1,1,1] => [1,0,1,0,1,1,0,1,0,1,0,0] => [1,1,1,1,1,0,0,0,1,0,0,0] => 2
([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6) => [6] => [1,0,1,0,1,0,1,0,1,0,1,0] => [1,1,1,1,1,1,0,0,0,0,0,0] => 1
([(0,5),(1,4),(2,4),(2,5),(3,4),(3,5)],6) => [4,1,1] => [1,0,1,0,1,0,1,1,0,1,0,0] => [1,1,1,1,1,0,0,0,0,1,0,0] => 2
([(0,5),(1,4),(2,3)],6) => [1,1,1] => [1,1,0,1,0,0] => [1,1,0,1,0,0] => 2
([(1,5),(2,4),(3,4),(3,5)],6) => [1,1,1,1] => [1,1,0,1,0,1,0,0] => [1,1,1,0,1,0,0,0] => 2
([(0,1),(2,5),(3,4),(4,5)],6) => [1,1,1,1] => [1,1,0,1,0,1,0,0] => [1,1,1,0,1,0,0,0] => 2
([(1,2),(3,4),(3,5),(4,5)],6) => [3,1] => [1,0,1,0,1,1,0,0] => [1,1,1,0,0,0,1,0] => 2
([(0,5),(1,4),(2,3),(3,5),(4,5)],6) => [1,1,1,1,1] => [1,1,0,1,0,1,0,1,0,0] => [1,1,1,1,0,1,0,0,0,0] => 2
([(1,4),(2,3),(2,5),(3,5),(4,5)],6) => [3,1,1] => [1,0,1,0,1,1,0,1,0,0] => [1,1,1,1,0,0,0,1,0,0] => 2
([(0,1),(2,5),(3,4),(3,5),(4,5)],6) => [3,1,1] => [1,0,1,0,1,1,0,1,0,0] => [1,1,1,1,0,0,0,1,0,0] => 2
([(0,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6) => [3,1,1,1] => [1,0,1,0,1,1,0,1,0,1,0,0] => [1,1,1,1,1,0,0,0,1,0,0,0] => 2
([(1,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6) => [3,3] => [1,1,1,0,1,0,0,0] => [1,1,0,1,0,1,0,0] => 3
([(0,5),(1,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6) => [3,3,1] => [1,1,1,0,1,0,0,1,0,0] => [1,1,0,1,1,0,1,0,0,0] => 3
([(1,4),(1,5),(2,3),(2,5),(3,4)],6) => [5] => [1,0,1,0,1,0,1,0,1,0] => [1,1,1,1,1,0,0,0,0,0] => 1
([(0,5),(1,4),(2,3),(2,4),(3,5),(4,5)],6) => [4,1,1] => [1,0,1,0,1,0,1,1,0,1,0,0] => [1,1,1,1,1,0,0,0,0,1,0,0] => 2
([(1,2),(1,5),(2,4),(3,4),(3,5),(4,5)],6) => [6] => [1,0,1,0,1,0,1,0,1,0,1,0] => [1,1,1,1,1,1,0,0,0,0,0,0] => 1
([(0,5),(1,2),(1,4),(2,3),(3,5),(4,5)],6) => [5,1] => [1,0,1,0,1,0,1,0,1,1,0,0] => [1,1,1,1,1,0,0,0,0,0,1,0] => 2
([(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6) => [5,1] => [1,0,1,0,1,0,1,0,1,1,0,0] => [1,1,1,1,1,0,0,0,0,0,1,0] => 2
([(0,5),(1,4),(2,3),(3,4),(3,5),(4,5)],6) => [3,1,1,1] => [1,0,1,0,1,1,0,1,0,1,0,0] => [1,1,1,1,1,0,0,0,1,0,0,0] => 2
([(0,5),(1,4),(2,3),(2,4),(3,5)],6) => [1,1,1,1,1] => [1,1,0,1,0,1,0,1,0,0] => [1,1,1,1,0,1,0,0,0,0] => 2
([(0,1),(2,4),(2,5),(3,4),(3,5)],6) => [4,1] => [1,0,1,0,1,0,1,1,0,0] => [1,1,1,1,0,0,0,0,1,0] => 2
([(0,5),(1,5),(2,3),(2,4),(3,4)],6) => [3,1,1] => [1,0,1,0,1,1,0,1,0,0] => [1,1,1,1,0,0,0,1,0,0] => 2
([(0,4),(1,2),(1,3),(2,5),(3,5),(4,5)],6) => [4,1,1] => [1,0,1,0,1,0,1,1,0,1,0,0] => [1,1,1,1,1,0,0,0,0,1,0,0] => 2
([(0,4),(1,2),(1,5),(2,5),(3,4),(3,5)],6) => [3,1,1,1] => [1,0,1,0,1,1,0,1,0,1,0,0] => [1,1,1,1,1,0,0,0,1,0,0,0] => 2
([(0,4),(1,2),(2,5),(3,4),(3,5),(4,5)],6) => [3,1,1,1] => [1,0,1,0,1,1,0,1,0,1,0,0] => [1,1,1,1,1,0,0,0,1,0,0,0] => 2
([(0,1),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => [5,1] => [1,0,1,0,1,0,1,0,1,1,0,0] => [1,1,1,1,1,0,0,0,0,0,1,0] => 2
([(0,4),(1,4),(2,3),(2,5),(3,5),(4,5)],6) => [3,1,1,1] => [1,0,1,0,1,1,0,1,0,1,0,0] => [1,1,1,1,1,0,0,0,1,0,0,0] => 2
([(0,3),(0,4),(1,2),(1,5),(2,5),(3,5),(4,5)],6) => [4,3] => [1,0,1,1,1,0,1,0,0,0] => [1,1,1,0,0,1,0,1,0,0] => 3
([(0,4),(1,2),(1,5),(2,5),(3,4),(3,5),(4,5)],6) => [3,3,1] => [1,1,1,0,1,0,0,1,0,0] => [1,1,0,1,1,0,1,0,0,0] => 3
([(0,1),(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => [5,3] => [1,0,1,0,1,1,1,0,1,0,0,0] => [1,1,1,1,0,0,0,1,0,1,0,0] => 3
([(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => [6] => [1,0,1,0,1,0,1,0,1,0,1,0] => [1,1,1,1,1,1,0,0,0,0,0,0] => 1
([(0,4),(0,5),(1,2),(1,3),(2,5),(3,4)],6) => [6] => [1,0,1,0,1,0,1,0,1,0,1,0] => [1,1,1,1,1,1,0,0,0,0,0,0] => 1
([(0,4),(0,5),(1,2),(1,3),(2,3),(4,5)],6) => [3,3] => [1,1,1,0,1,0,0,0] => [1,1,0,1,0,1,0,0] => 3
([(0,1),(0,5),(1,5),(2,3),(2,4),(3,4),(4,5)],6) => [3,3,1] => [1,1,1,0,1,0,0,1,0,0] => [1,1,0,1,1,0,1,0,0,0] => 3
([(0,1),(0,5),(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6) => [5,3] => [1,0,1,0,1,1,1,0,1,0,0,0] => [1,1,1,1,0,0,0,1,0,1,0,0] => 3
([(5,6)],7) => [1] => [1,0] => [1,0] => 1
([(4,6),(5,6)],7) => [1,1] => [1,1,0,0] => [1,0,1,0] => 2
([(3,6),(4,6),(5,6)],7) => [1,1,1] => [1,1,0,1,0,0] => [1,1,0,1,0,0] => 2
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search for individual values
searching the database for the individual values of this statistic
Description
The number of peaks of a Dyck path.
Map
Lalanne-Kreweras involution
Description
The Lalanne-Kreweras involution on Dyck paths.
Label the upsteps from left to right and record the labels on the first up step of each double rise. Do the same for the downsteps. Then form the Dyck path whose ascent lengths and descent lengths are the consecutives differences of the labels.
Label the upsteps from left to right and record the labels on the first up step of each double rise. Do the same for the downsteps. Then form the Dyck path whose ascent lengths and descent lengths are the consecutives differences of the labels.
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
to edge-partition of biconnected components
Description
Sends a graph to the partition recording the number of edges in its biconnected components.
The biconnected components are also known as blocks of a graph.
The biconnected components are also known as blocks of a graph.
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