Identifier
-
Mp00276:
Graphs
—to edge-partition of biconnected components⟶
Integer partitions
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00121: Dyck paths —Cori-Le Borgne involution⟶ Dyck paths
St000144: Dyck paths ⟶ ℤ
Values
([(0,1)],2) => [1] => [1,0,1,0] => [1,0,1,0] => 2
([(1,2)],3) => [1] => [1,0,1,0] => [1,0,1,0] => 2
([(0,2),(1,2)],3) => [1,1] => [1,0,1,1,0,0] => [1,1,0,1,0,0] => 2
([(0,1),(0,2),(1,2)],3) => [3] => [1,1,1,0,0,0,1,0] => [1,1,1,0,0,1,0,0] => 3
([(2,3)],4) => [1] => [1,0,1,0] => [1,0,1,0] => 2
([(1,3),(2,3)],4) => [1,1] => [1,0,1,1,0,0] => [1,1,0,1,0,0] => 2
([(0,3),(1,3),(2,3)],4) => [1,1,1] => [1,0,1,1,1,0,0,0] => [1,1,1,0,1,0,0,0] => 2
([(0,3),(1,2)],4) => [1,1] => [1,0,1,1,0,0] => [1,1,0,1,0,0] => 2
([(0,3),(1,2),(2,3)],4) => [1,1,1] => [1,0,1,1,1,0,0,0] => [1,1,1,0,1,0,0,0] => 2
([(1,2),(1,3),(2,3)],4) => [3] => [1,1,1,0,0,0,1,0] => [1,1,1,0,0,1,0,0] => 3
([(0,3),(1,2),(1,3),(2,3)],4) => [3,1] => [1,1,0,1,0,0,1,0] => [1,0,1,1,0,0,1,0] => 4
([(0,2),(0,3),(1,2),(1,3)],4) => [4] => [1,1,1,1,0,0,0,0,1,0] => [1,1,1,1,0,0,1,0,0,0] => 3
([(0,2),(0,3),(1,2),(1,3),(2,3)],4) => [5] => [1,1,1,1,1,0,0,0,0,0,1,0] => [1,1,1,1,1,0,0,1,0,0,0,0] => 3
([(3,4)],5) => [1] => [1,0,1,0] => [1,0,1,0] => 2
([(2,4),(3,4)],5) => [1,1] => [1,0,1,1,0,0] => [1,1,0,1,0,0] => 2
([(1,4),(2,4),(3,4)],5) => [1,1,1] => [1,0,1,1,1,0,0,0] => [1,1,1,0,1,0,0,0] => 2
([(0,4),(1,4),(2,4),(3,4)],5) => [1,1,1,1] => [1,0,1,1,1,1,0,0,0,0] => [1,1,1,1,0,1,0,0,0,0] => 2
([(1,4),(2,3)],5) => [1,1] => [1,0,1,1,0,0] => [1,1,0,1,0,0] => 2
([(1,4),(2,3),(3,4)],5) => [1,1,1] => [1,0,1,1,1,0,0,0] => [1,1,1,0,1,0,0,0] => 2
([(0,1),(2,4),(3,4)],5) => [1,1,1] => [1,0,1,1,1,0,0,0] => [1,1,1,0,1,0,0,0] => 2
([(2,3),(2,4),(3,4)],5) => [3] => [1,1,1,0,0,0,1,0] => [1,1,1,0,0,1,0,0] => 3
([(0,4),(1,4),(2,3),(3,4)],5) => [1,1,1,1] => [1,0,1,1,1,1,0,0,0,0] => [1,1,1,1,0,1,0,0,0,0] => 2
([(1,4),(2,3),(2,4),(3,4)],5) => [3,1] => [1,1,0,1,0,0,1,0] => [1,0,1,1,0,0,1,0] => 4
([(0,4),(1,4),(2,3),(2,4),(3,4)],5) => [3,1,1] => [1,0,1,1,0,0,1,0] => [1,1,0,1,0,0,1,0] => 3
([(1,3),(1,4),(2,3),(2,4)],5) => [4] => [1,1,1,1,0,0,0,0,1,0] => [1,1,1,1,0,0,1,0,0,0] => 3
([(0,4),(1,2),(1,3),(2,4),(3,4)],5) => [4,1] => [1,1,1,0,1,0,0,0,1,0] => [1,0,1,1,1,0,0,1,0,0] => 4
([(1,3),(1,4),(2,3),(2,4),(3,4)],5) => [5] => [1,1,1,1,1,0,0,0,0,0,1,0] => [1,1,1,1,1,0,0,1,0,0,0,0] => 3
([(0,4),(1,3),(2,3),(2,4),(3,4)],5) => [3,1,1] => [1,0,1,1,0,0,1,0] => [1,1,0,1,0,0,1,0] => 3
([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5) => [5,1] => [1,1,1,1,0,1,0,0,0,0,1,0] => [1,0,1,1,1,1,0,0,1,0,0,0] => 4
([(0,4),(1,3),(2,3),(2,4)],5) => [1,1,1,1] => [1,0,1,1,1,1,0,0,0,0] => [1,1,1,1,0,1,0,0,0,0] => 2
([(0,1),(2,3),(2,4),(3,4)],5) => [3,1] => [1,1,0,1,0,0,1,0] => [1,0,1,1,0,0,1,0] => 4
([(0,3),(1,2),(1,4),(2,4),(3,4)],5) => [3,1,1] => [1,0,1,1,0,0,1,0] => [1,1,0,1,0,0,1,0] => 3
([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5) => [3,3] => [1,1,1,0,0,0,1,1,0,0] => [1,1,1,0,0,0,1,1,0,0] => 5
([(0,3),(0,4),(1,2),(1,4),(2,3)],5) => [5] => [1,1,1,1,1,0,0,0,0,0,1,0] => [1,1,1,1,1,0,0,1,0,0,0,0] => 3
([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5) => [5,1] => [1,1,1,1,0,1,0,0,0,0,1,0] => [1,0,1,1,1,1,0,0,1,0,0,0] => 4
([(4,5)],6) => [1] => [1,0,1,0] => [1,0,1,0] => 2
([(3,5),(4,5)],6) => [1,1] => [1,0,1,1,0,0] => [1,1,0,1,0,0] => 2
([(2,5),(3,5),(4,5)],6) => [1,1,1] => [1,0,1,1,1,0,0,0] => [1,1,1,0,1,0,0,0] => 2
([(1,5),(2,5),(3,5),(4,5)],6) => [1,1,1,1] => [1,0,1,1,1,1,0,0,0,0] => [1,1,1,1,0,1,0,0,0,0] => 2
([(0,5),(1,5),(2,5),(3,5),(4,5)],6) => [1,1,1,1,1] => [1,0,1,1,1,1,1,0,0,0,0,0] => [1,1,1,1,1,0,1,0,0,0,0,0] => 2
([(2,5),(3,4)],6) => [1,1] => [1,0,1,1,0,0] => [1,1,0,1,0,0] => 2
([(2,5),(3,4),(4,5)],6) => [1,1,1] => [1,0,1,1,1,0,0,0] => [1,1,1,0,1,0,0,0] => 2
([(1,2),(3,5),(4,5)],6) => [1,1,1] => [1,0,1,1,1,0,0,0] => [1,1,1,0,1,0,0,0] => 2
([(3,4),(3,5),(4,5)],6) => [3] => [1,1,1,0,0,0,1,0] => [1,1,1,0,0,1,0,0] => 3
([(1,5),(2,5),(3,4),(4,5)],6) => [1,1,1,1] => [1,0,1,1,1,1,0,0,0,0] => [1,1,1,1,0,1,0,0,0,0] => 2
([(0,1),(2,5),(3,5),(4,5)],6) => [1,1,1,1] => [1,0,1,1,1,1,0,0,0,0] => [1,1,1,1,0,1,0,0,0,0] => 2
([(2,5),(3,4),(3,5),(4,5)],6) => [3,1] => [1,1,0,1,0,0,1,0] => [1,0,1,1,0,0,1,0] => 4
([(0,5),(1,5),(2,5),(3,4),(4,5)],6) => [1,1,1,1,1] => [1,0,1,1,1,1,1,0,0,0,0,0] => [1,1,1,1,1,0,1,0,0,0,0,0] => 2
([(1,5),(2,5),(3,4),(3,5),(4,5)],6) => [3,1,1] => [1,0,1,1,0,0,1,0] => [1,1,0,1,0,0,1,0] => 3
([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6) => [3,1,1,1] => [1,0,1,1,1,0,0,1,0,0] => [1,1,1,0,1,0,0,0,1,0] => 3
([(2,4),(2,5),(3,4),(3,5)],6) => [4] => [1,1,1,1,0,0,0,0,1,0] => [1,1,1,1,0,0,1,0,0,0] => 3
([(0,5),(1,5),(2,4),(3,4)],6) => [1,1,1,1] => [1,0,1,1,1,1,0,0,0,0] => [1,1,1,1,0,1,0,0,0,0] => 2
([(1,5),(2,3),(2,4),(3,5),(4,5)],6) => [4,1] => [1,1,1,0,1,0,0,0,1,0] => [1,0,1,1,1,0,0,1,0,0] => 4
([(0,5),(1,5),(2,3),(3,4),(4,5)],6) => [1,1,1,1,1] => [1,0,1,1,1,1,1,0,0,0,0,0] => [1,1,1,1,1,0,1,0,0,0,0,0] => 2
([(2,4),(2,5),(3,4),(3,5),(4,5)],6) => [5] => [1,1,1,1,1,0,0,0,0,0,1,0] => [1,1,1,1,1,0,0,1,0,0,0,0] => 3
([(1,5),(2,4),(3,4),(3,5),(4,5)],6) => [3,1,1] => [1,0,1,1,0,0,1,0] => [1,1,0,1,0,0,1,0] => 3
([(0,5),(1,5),(2,4),(3,4),(4,5)],6) => [1,1,1,1,1] => [1,0,1,1,1,1,1,0,0,0,0,0] => [1,1,1,1,1,0,1,0,0,0,0,0] => 2
([(0,5),(1,5),(2,3),(2,4),(3,5),(4,5)],6) => [4,1,1] => [1,1,0,1,1,0,0,0,1,0] => [1,1,0,1,1,0,0,1,0,0] => 4
([(1,5),(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] => [1,0,1,1,1,1,0,0,1,0,0,0] => 4
([(0,5),(1,5),(2,4),(3,4),(3,5),(4,5)],6) => [3,1,1,1] => [1,0,1,1,1,0,0,1,0,0] => [1,1,1,0,1,0,0,0,1,0] => 3
([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => [5,1,1] => [1,1,1,0,1,1,0,0,0,0,1,0] => [1,1,0,1,1,1,0,0,1,0,0,0] => 4
([(0,5),(1,4),(2,4),(2,5),(3,4),(3,5)],6) => [4,1,1] => [1,1,0,1,1,0,0,0,1,0] => [1,1,0,1,1,0,0,1,0,0] => 4
([(0,5),(1,4),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => [5,1,1] => [1,1,1,0,1,1,0,0,0,0,1,0] => [1,1,0,1,1,1,0,0,1,0,0,0] => 4
([(0,5),(1,4),(2,3)],6) => [1,1,1] => [1,0,1,1,1,0,0,0] => [1,1,1,0,1,0,0,0] => 2
([(1,5),(2,4),(3,4),(3,5)],6) => [1,1,1,1] => [1,0,1,1,1,1,0,0,0,0] => [1,1,1,1,0,1,0,0,0,0] => 2
([(0,1),(2,5),(3,4),(4,5)],6) => [1,1,1,1] => [1,0,1,1,1,1,0,0,0,0] => [1,1,1,1,0,1,0,0,0,0] => 2
([(1,2),(3,4),(3,5),(4,5)],6) => [3,1] => [1,1,0,1,0,0,1,0] => [1,0,1,1,0,0,1,0] => 4
([(0,5),(1,4),(2,3),(3,5),(4,5)],6) => [1,1,1,1,1] => [1,0,1,1,1,1,1,0,0,0,0,0] => [1,1,1,1,1,0,1,0,0,0,0,0] => 2
([(1,4),(2,3),(2,5),(3,5),(4,5)],6) => [3,1,1] => [1,0,1,1,0,0,1,0] => [1,1,0,1,0,0,1,0] => 3
([(0,1),(2,5),(3,4),(3,5),(4,5)],6) => [3,1,1] => [1,0,1,1,0,0,1,0] => [1,1,0,1,0,0,1,0] => 3
([(0,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6) => [3,1,1,1] => [1,0,1,1,1,0,0,1,0,0] => [1,1,1,0,1,0,0,0,1,0] => 3
([(1,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6) => [3,3] => [1,1,1,0,0,0,1,1,0,0] => [1,1,1,0,0,0,1,1,0,0] => 5
([(0,5),(1,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6) => [3,3,1] => [1,1,0,1,0,0,1,1,0,0] => [1,1,0,0,1,0,1,1,0,0] => 5
([(1,4),(1,5),(2,3),(2,5),(3,4)],6) => [5] => [1,1,1,1,1,0,0,0,0,0,1,0] => [1,1,1,1,1,0,0,1,0,0,0,0] => 3
([(0,5),(1,4),(2,3),(2,4),(3,5),(4,5)],6) => [4,1,1] => [1,1,0,1,1,0,0,0,1,0] => [1,1,0,1,1,0,0,1,0,0] => 4
([(0,5),(1,2),(1,4),(2,3),(3,5),(4,5)],6) => [5,1] => [1,1,1,1,0,1,0,0,0,0,1,0] => [1,0,1,1,1,1,0,0,1,0,0,0] => 4
([(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6) => [5,1] => [1,1,1,1,0,1,0,0,0,0,1,0] => [1,0,1,1,1,1,0,0,1,0,0,0] => 4
([(0,5),(1,4),(2,3),(3,4),(3,5),(4,5)],6) => [3,1,1,1] => [1,0,1,1,1,0,0,1,0,0] => [1,1,1,0,1,0,0,0,1,0] => 3
([(0,5),(1,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => [5,1,1] => [1,1,1,0,1,1,0,0,0,0,1,0] => [1,1,0,1,1,1,0,0,1,0,0,0] => 4
([(0,5),(1,4),(2,3),(2,4),(3,5)],6) => [1,1,1,1,1] => [1,0,1,1,1,1,1,0,0,0,0,0] => [1,1,1,1,1,0,1,0,0,0,0,0] => 2
([(0,1),(2,4),(2,5),(3,4),(3,5)],6) => [4,1] => [1,1,1,0,1,0,0,0,1,0] => [1,0,1,1,1,0,0,1,0,0] => 4
([(0,5),(1,5),(2,3),(2,4),(3,4)],6) => [3,1,1] => [1,0,1,1,0,0,1,0] => [1,1,0,1,0,0,1,0] => 3
([(0,4),(1,2),(1,3),(2,5),(3,5),(4,5)],6) => [4,1,1] => [1,1,0,1,1,0,0,0,1,0] => [1,1,0,1,1,0,0,1,0,0] => 4
([(0,4),(1,2),(1,5),(2,5),(3,4),(3,5)],6) => [3,1,1,1] => [1,0,1,1,1,0,0,1,0,0] => [1,1,1,0,1,0,0,0,1,0] => 3
([(0,4),(1,2),(2,5),(3,4),(3,5),(4,5)],6) => [3,1,1,1] => [1,0,1,1,1,0,0,1,0,0] => [1,1,1,0,1,0,0,0,1,0] => 3
([(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] => [1,0,1,1,1,1,0,0,1,0,0,0] => 4
([(0,4),(1,4),(2,3),(2,5),(3,5),(4,5)],6) => [3,1,1,1] => [1,0,1,1,1,0,0,1,0,0] => [1,1,1,0,1,0,0,0,1,0] => 3
([(0,3),(0,4),(1,2),(1,5),(2,5),(3,5),(4,5)],6) => [4,3] => [1,1,1,0,0,0,1,0,1,0] => [1,1,1,0,0,1,0,1,0,0] => 4
([(0,3),(1,4),(1,5),(2,4),(2,5),(3,5),(4,5)],6) => [5,1,1] => [1,1,1,0,1,1,0,0,0,0,1,0] => [1,1,0,1,1,1,0,0,1,0,0,0] => 4
([(0,4),(1,2),(1,5),(2,5),(3,4),(3,5),(4,5)],6) => [3,3,1] => [1,1,0,1,0,0,1,1,0,0] => [1,1,0,0,1,0,1,1,0,0] => 5
([(0,1),(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => [5,3] => [1,1,1,1,0,0,0,1,0,0,1,0] => [1,1,1,1,0,0,1,0,0,1,0,0] => 4
([(0,5),(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6) => [5,1,1] => [1,1,1,0,1,1,0,0,0,0,1,0] => [1,1,0,1,1,1,0,0,1,0,0,0] => 4
([(0,5),(1,4),(2,3),(2,4),(2,5),(3,4),(3,5)],6) => [5,1,1] => [1,1,1,0,1,1,0,0,0,0,1,0] => [1,1,0,1,1,1,0,0,1,0,0,0] => 4
([(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] => [1,1,1,0,0,0,1,1,0,0] => 5
([(0,2),(1,4),(1,5),(2,3),(3,4),(3,5),(4,5)],6) => [5,1,1] => [1,1,1,0,1,1,0,0,0,0,1,0] => [1,1,0,1,1,1,0,0,1,0,0,0] => 4
([(0,1),(0,5),(1,5),(2,3),(2,4),(3,4),(4,5)],6) => [3,3,1] => [1,1,0,1,0,0,1,1,0,0] => [1,1,0,0,1,0,1,1,0,0] => 5
([(0,1),(0,5),(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6) => [5,3] => [1,1,1,1,0,0,0,1,0,0,1,0] => [1,1,1,1,0,0,1,0,0,1,0,0] => 4
([(5,6)],7) => [1] => [1,0,1,0] => [1,0,1,0] => 2
([(4,6),(5,6)],7) => [1,1] => [1,0,1,1,0,0] => [1,1,0,1,0,0] => 2
([(3,6),(4,6),(5,6)],7) => [1,1,1] => [1,0,1,1,1,0,0,0] => [1,1,1,0,1,0,0,0] => 2
([(2,6),(3,6),(4,6),(5,6)],7) => [1,1,1,1] => [1,0,1,1,1,1,0,0,0,0] => [1,1,1,1,0,1,0,0,0,0] => 2
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search for individual values
searching the database for the individual values of this statistic
Description
The pyramid weight of the Dyck path.
The pyramid weight of a Dyck path is the sum of the lengths of the maximal pyramids (maximal sequences of the form $1^h0^h$) in the path.
Maximal pyramids are called lower interactions by Le Borgne [2], see St000331The number of upper interactions of a Dyck path. and St000335The difference of lower and upper interactions. for related statistics.
The pyramid weight of a Dyck path is the sum of the lengths of the maximal pyramids (maximal sequences of the form $1^h0^h$) in the path.
Maximal pyramids are called lower interactions by Le Borgne [2], see St000331The number of upper interactions of a Dyck path. and St000335The difference of lower and upper interactions. for related statistics.
Map
to Dyck path
Description
Sends a partition to the shortest Dyck path tracing the shape of its Ferrers diagram.
Map
Cori-Le Borgne involution
Description
The Cori-Le Borgne involution on Dyck paths.
Append an additional down step to the Dyck path and consider its (literal) reversal. The image of the involution is then the unique rotation of this word which is a Dyck word followed by an additional down step. Alternatively, it is the composite $\zeta\circ\mathrm{rev}\circ\zeta^{(-1)}$, where $\zeta$ is Mp00030zeta map.
Append an additional down step to the Dyck path and consider its (literal) reversal. The image of the involution is then the unique rotation of this word which is a Dyck word followed by an additional down step. Alternatively, it is the composite $\zeta\circ\mathrm{rev}\circ\zeta^{(-1)}$, where $\zeta$ is Mp00030zeta map.
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.
searching the database
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