Your data matches 199 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)
Matching statistic: St000774
(load all 4 compositions to match this statistic)
Mapping
St000774: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
([],1)
 => 1
([],2)
 => 2
([(0,1)],2)
 => 1
([],3)
 => 3
([(1,2)],3)
 => 2
([(0,2),(1,2)],3)
 => 1
([(0,1),(0,2),(1,2)],3)
 => 2
([],4)
 => 4
([(2,3)],4)
 => 3
([(1,3),(2,3)],4)
 => 2
([(0,3),(1,3),(2,3)],4)
 => 2
([(0,3),(1,2)],4)
 => 2
([(0,3),(1,2),(2,3)],4)
 => 1
([(1,2),(1,3),(2,3)],4)
 => 2
([(0,3),(1,2),(1,3),(2,3)],4)
 => 1
([(0,2),(0,3),(1,2),(1,3)],4)
 => 2
([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 2
([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 3
([],5)
 => 5
([(3,4)],5)
 => 4
([(2,4),(3,4)],5)
 => 3
([(1,4),(2,4),(3,4)],5)
 => 2
([(0,4),(1,4),(2,4),(3,4)],5)
 => 3
([(1,4),(2,3)],5)
 => 3
([(1,4),(2,3),(3,4)],5)
 => 2
([(0,1),(2,4),(3,4)],5)
 => 2
([(2,3),(2,4),(3,4)],5)
 => 3
([(0,4),(1,4),(2,3),(3,4)],5)
 => 1
([(1,4),(2,3),(2,4),(3,4)],5)
 => 2
([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => 2
([(1,3),(1,4),(2,3),(2,4)],5)
 => 2
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
 => 1
([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 2
([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => 1
([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 1
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
 => 2
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 2
([(0,4),(1,3),(2,3),(2,4)],5)
 => 1
([(0,1),(2,3),(2,4),(3,4)],5)
 => 2
([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
 => 1
([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
 => 2
([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
 => 2
([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => 1
([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
 => 1
([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
 => 1
([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 3
([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 2
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 2
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
 => 1
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
 => 2
Description
The maximal multiplicity of a Laplacian eigenvalue in a graph.
Matching statistic: St000381
(load all 30 compositions to match this statistic)
Mapping
Mp00152: Graphs —Laplacian multiplicities⟶ Integer compositions
St000381: Integer compositions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
([],1)
 => [1] => 1
([],2)
 => [2] => 2
([(0,1)],2)
 => [1,1] => 1
([],3)
 => [3] => 3
([(1,2)],3)
 => [1,2] => 2
([(0,2),(1,2)],3)
 => [1,1,1] => 1
([(0,1),(0,2),(1,2)],3)
 => [2,1] => 2
([],4)
 => [4] => 4
([(2,3)],4)
 => [1,3] => 3
([(1,3),(2,3)],4)
 => [1,1,2] => 2
([(0,3),(1,3),(2,3)],4)
 => [1,2,1] => 2
([(0,3),(1,2)],4)
 => [2,2] => 2
([(0,3),(1,2),(2,3)],4)
 => [1,1,1,1] => 1
([(1,2),(1,3),(2,3)],4)
 => [2,2] => 2
([(0,3),(1,2),(1,3),(2,3)],4)
 => [1,1,1,1] => 1
([(0,2),(0,3),(1,2),(1,3)],4)
 => [1,2,1] => 2
([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => [2,1,1] => 2
([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => [3,1] => 3
([],5)
 => [5] => 5
([(3,4)],5)
 => [1,4] => 4
([(2,4),(3,4)],5)
 => [1,1,3] => 3
([(1,4),(2,4),(3,4)],5)
 => [1,2,2] => 2
([(0,4),(1,4),(2,4),(3,4)],5)
 => [1,3,1] => 3
([(1,4),(2,3)],5)
 => [2,3] => 3
([(1,4),(2,3),(3,4)],5)
 => [1,1,1,2] => 2
([(0,1),(2,4),(3,4)],5)
 => [1,1,1,2] => 2
([(2,3),(2,4),(3,4)],5)
 => [2,3] => 3
([(0,4),(1,4),(2,3),(3,4)],5)
 => [1,1,1,1,1] => 1
([(1,4),(2,3),(2,4),(3,4)],5)
 => [1,1,1,2] => 2
([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => [1,1,2,1] => 2
([(1,3),(1,4),(2,3),(2,4)],5)
 => [1,2,2] => 2
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
 => [1,1,1,1,1] => 1
([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [2,1,2] => 2
([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => [1,1,1,1,1] => 1
([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [1,1,1,1,1] => 1
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
 => [1,1,2,1] => 2
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [2,2,1] => 2
([(0,4),(1,3),(2,3),(2,4)],5)
 => [1,1,1,1,1] => 1
([(0,1),(2,3),(2,4),(3,4)],5)
 => [2,1,2] => 2
([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
 => [1,1,1,1,1] => 1
([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
 => [1,2,1,1] => 2
([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
 => [2,2,1] => 2
([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => [1,1,1,1,1] => 1
([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
 => [1,1,1,1,1] => 1
([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
 => [1,1,1,1,1] => 1
([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [3,2] => 3
([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [1,2,1,1] => 2
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [2,1,1,1] => 2
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
 => [1,1,1,1,1] => 1
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
 => [2,2,1] => 2
Description
The largest part of an integer composition.
Matching statistic: St000771
(load all 18 compositions to match this statistic)
Mapping
Mp00203: Graphs —cone⟶ Graphs
St000771: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
([],1)
 => ([(0,1)],2)
 => 1
([],2)
 => ([(0,2),(1,2)],3)
 => 1
([(0,1)],2)
 => ([(0,1),(0,2),(1,2)],3)
 => 2
([],3)
 => ([(0,3),(1,3),(2,3)],4)
 => 2
([(1,2)],3)
 => ([(0,3),(1,2),(1,3),(2,3)],4)
 => 1
([(0,2),(1,2)],3)
 => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 2
([(0,1),(0,2),(1,2)],3)
 => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 3
([],4)
 => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 3
([(2,3)],4)
 => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => 2
([(1,3),(2,3)],4)
 => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 1
([(0,3),(1,3),(2,3)],4)
 => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 2
([(0,3),(1,2)],4)
 => ([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
 => 2
([(0,3),(1,2),(2,3)],4)
 => ([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
 => 1
([(1,2),(1,3),(2,3)],4)
 => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 2
([(0,3),(1,2),(1,3),(2,3)],4)
 => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 2
([(0,2),(0,3),(1,2),(1,3)],4)
 => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
 => 2
([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 3
([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 4
([],5)
 => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
 => 4
([(3,4)],5)
 => ([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
 => 3
([(2,4),(3,4)],5)
 => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 2
([(1,4),(2,4),(3,4)],5)
 => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 2
([(0,4),(1,4),(2,4),(3,4)],5)
 => ([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 3
([(1,4),(2,3)],5)
 => ([(0,5),(1,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)
 => 2
([(1,4),(2,3),(3,4)],5)
 => ([(0,5),(1,4),(1,5),(2,3),(2,5),(3,4),(3,5),(4,5)],6)
 => 1
([(0,1),(2,4),(3,4)],5)
 => ([(0,1),(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 1
([(2,3),(2,4),(3,4)],5)
 => ([(0,5),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 2
([(0,4),(1,4),(2,3),(3,4)],5)
 => ([(0,4),(0,5),(1,4),(1,5),(2,3),(2,5),(3,4),(3,5),(4,5)],6)
 => 1
([(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 1
([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,4),(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 2
([(1,3),(1,4),(2,3),(2,4)],5)
 => ([(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
 => 2
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
 => ([(0,4),(0,5),(1,2),(1,3),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 1
([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 2
([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => ([(0,4),(0,5),(1,3),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 1
([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 2
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
 => ([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
 => 2
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 3
([(0,4),(1,3),(2,3),(2,4)],5)
 => ([(0,4),(0,5),(1,3),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
 => 1
([(0,1),(2,3),(2,4),(3,4)],5)
 => ([(0,1),(0,5),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 2
([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
 => ([(0,3),(0,5),(1,2),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 1
([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
 => ([(0,3),(0,4),(0,5),(1,2),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 2
([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
 => ([(0,3),(0,4),(0,5),(1,2),(1,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)
 => 2
([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => ([(0,1),(0,4),(0,5),(1,3),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 1
([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,3),(0,4),(0,5),(1,2),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 2
([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
 => ([(0,4),(0,5),(1,2),(1,3),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 1
([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 3
([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 2
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 3
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
 => ([(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
 => 2
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
 => ([(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 3
Description
The largest multiplicity of a distance Laplacian eigenvalue in a connected graph. The distance Laplacian of a graph is the (symmetric) matrix with row and column sums $0$, which has the negative distances between two vertices as its off-diagonal entries. This statistic is the largest multiplicity of an eigenvalue. For example, the cycle on four vertices has distance Laplacian $$ \left(\begin{array}{rrrr} 4 & -1 & -2 & -1 \\ -1 & 4 & -1 & -2 \\ -2 & -1 & 4 & -1 \\ -1 & -2 & -1 & 4 \end{array}\right). $$ Its eigenvalues are $0,4,4,6$, so the statistic is $2$. The path on four vertices has eigenvalues $0, 4.7\dots, 6, 9.2\dots$ and therefore statistic $1$.
Matching statistic: St000013
(load all 8 compositions to match this statistic)
Mapping
Mp00152: Graphs —Laplacian multiplicities⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St000013: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
([],1)
 => [1] => [1,0]
 => 1
([],2)
 => [2] => [1,1,0,0]
 => 2
([(0,1)],2)
 => [1,1] => [1,0,1,0]
 => 1
([],3)
 => [3] => [1,1,1,0,0,0]
 => 3
([(1,2)],3)
 => [1,2] => [1,0,1,1,0,0]
 => 2
([(0,2),(1,2)],3)
 => [1,1,1] => [1,0,1,0,1,0]
 => 1
([(0,1),(0,2),(1,2)],3)
 => [2,1] => [1,1,0,0,1,0]
 => 2
([],4)
 => [4] => [1,1,1,1,0,0,0,0]
 => 4
([(2,3)],4)
 => [1,3] => [1,0,1,1,1,0,0,0]
 => 3
([(1,3),(2,3)],4)
 => [1,1,2] => [1,0,1,0,1,1,0,0]
 => 2
([(0,3),(1,3),(2,3)],4)
 => [1,2,1] => [1,0,1,1,0,0,1,0]
 => 2
([(0,3),(1,2)],4)
 => [2,2] => [1,1,0,0,1,1,0,0]
 => 2
([(0,3),(1,2),(2,3)],4)
 => [1,1,1,1] => [1,0,1,0,1,0,1,0]
 => 1
([(1,2),(1,3),(2,3)],4)
 => [2,2] => [1,1,0,0,1,1,0,0]
 => 2
([(0,3),(1,2),(1,3),(2,3)],4)
 => [1,1,1,1] => [1,0,1,0,1,0,1,0]
 => 1
([(0,2),(0,3),(1,2),(1,3)],4)
 => [1,2,1] => [1,0,1,1,0,0,1,0]
 => 2
([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => [2,1,1] => [1,1,0,0,1,0,1,0]
 => 2
([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => [3,1] => [1,1,1,0,0,0,1,0]
 => 3
([],5)
 => [5] => [1,1,1,1,1,0,0,0,0,0]
 => 5
([(3,4)],5)
 => [1,4] => [1,0,1,1,1,1,0,0,0,0]
 => 4
([(2,4),(3,4)],5)
 => [1,1,3] => [1,0,1,0,1,1,1,0,0,0]
 => 3
([(1,4),(2,4),(3,4)],5)
 => [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
 => 2
([(0,4),(1,4),(2,4),(3,4)],5)
 => [1,3,1] => [1,0,1,1,1,0,0,0,1,0]
 => 3
([(1,4),(2,3)],5)
 => [2,3] => [1,1,0,0,1,1,1,0,0,0]
 => 3
([(1,4),(2,3),(3,4)],5)
 => [1,1,1,2] => [1,0,1,0,1,0,1,1,0,0]
 => 2
([(0,1),(2,4),(3,4)],5)
 => [1,1,1,2] => [1,0,1,0,1,0,1,1,0,0]
 => 2
([(2,3),(2,4),(3,4)],5)
 => [2,3] => [1,1,0,0,1,1,1,0,0,0]
 => 3
([(0,4),(1,4),(2,3),(3,4)],5)
 => [1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
 => 1
([(1,4),(2,3),(2,4),(3,4)],5)
 => [1,1,1,2] => [1,0,1,0,1,0,1,1,0,0]
 => 2
([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => [1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
 => 2
([(1,3),(1,4),(2,3),(2,4)],5)
 => [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
 => 2
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
 => [1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
 => 1
([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
 => 2
([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => [1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
 => 1
([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
 => 1
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
 => [1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
 => 2
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
 => 2
([(0,4),(1,3),(2,3),(2,4)],5)
 => [1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
 => 1
([(0,1),(2,3),(2,4),(3,4)],5)
 => [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
 => 2
([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
 => [1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
 => 1
([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
 => [1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
 => 2
([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
 => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
 => 2
([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => [1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
 => 1
([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
 => [1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
 => 1
([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
 => [1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
 => 1
([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [3,2] => [1,1,1,0,0,0,1,1,0,0]
 => 3
([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
 => 2
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
 => 2
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
 => [1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
 => 1
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
 => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
 => 2
Description
The height of a Dyck path. The height of a Dyck path $D$ of semilength $n$ is defined as the maximal height of a peak of $D$. The height of $D$ at position $i$ is the number of up-steps minus the number of down-steps before position $i$.
Matching statistic: St000147
(load all 2 compositions to match this statistic)
Mapping
Mp00152: Graphs —Laplacian multiplicities⟶ Integer compositions
Mp00040: Integer compositions —to partition⟶ Integer partitions
St000147: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
([],1)
 => [1] => [1]
 => 1
([],2)
 => [2] => [2]
 => 2
([(0,1)],2)
 => [1,1] => [1,1]
 => 1
([],3)
 => [3] => [3]
 => 3
([(1,2)],3)
 => [1,2] => [2,1]
 => 2
([(0,2),(1,2)],3)
 => [1,1,1] => [1,1,1]
 => 1
([(0,1),(0,2),(1,2)],3)
 => [2,1] => [2,1]
 => 2
([],4)
 => [4] => [4]
 => 4
([(2,3)],4)
 => [1,3] => [3,1]
 => 3
([(1,3),(2,3)],4)
 => [1,1,2] => [2,1,1]
 => 2
([(0,3),(1,3),(2,3)],4)
 => [1,2,1] => [2,1,1]
 => 2
([(0,3),(1,2)],4)
 => [2,2] => [2,2]
 => 2
([(0,3),(1,2),(2,3)],4)
 => [1,1,1,1] => [1,1,1,1]
 => 1
([(1,2),(1,3),(2,3)],4)
 => [2,2] => [2,2]
 => 2
([(0,3),(1,2),(1,3),(2,3)],4)
 => [1,1,1,1] => [1,1,1,1]
 => 1
([(0,2),(0,3),(1,2),(1,3)],4)
 => [1,2,1] => [2,1,1]
 => 2
([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => [2,1,1] => [2,1,1]
 => 2
([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => [3,1] => [3,1]
 => 3
([],5)
 => [5] => [5]
 => 5
([(3,4)],5)
 => [1,4] => [4,1]
 => 4
([(2,4),(3,4)],5)
 => [1,1,3] => [3,1,1]
 => 3
([(1,4),(2,4),(3,4)],5)
 => [1,2,2] => [2,2,1]
 => 2
([(0,4),(1,4),(2,4),(3,4)],5)
 => [1,3,1] => [3,1,1]
 => 3
([(1,4),(2,3)],5)
 => [2,3] => [3,2]
 => 3
([(1,4),(2,3),(3,4)],5)
 => [1,1,1,2] => [2,1,1,1]
 => 2
([(0,1),(2,4),(3,4)],5)
 => [1,1,1,2] => [2,1,1,1]
 => 2
([(2,3),(2,4),(3,4)],5)
 => [2,3] => [3,2]
 => 3
([(0,4),(1,4),(2,3),(3,4)],5)
 => [1,1,1,1,1] => [1,1,1,1,1]
 => 1
([(1,4),(2,3),(2,4),(3,4)],5)
 => [1,1,1,2] => [2,1,1,1]
 => 2
([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => [1,1,2,1] => [2,1,1,1]
 => 2
([(1,3),(1,4),(2,3),(2,4)],5)
 => [1,2,2] => [2,2,1]
 => 2
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
 => [1,1,1,1,1] => [1,1,1,1,1]
 => 1
([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [2,1,2] => [2,2,1]
 => 2
([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => [1,1,1,1,1] => [1,1,1,1,1]
 => 1
([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [1,1,1,1,1] => [1,1,1,1,1]
 => 1
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
 => [1,1,2,1] => [2,1,1,1]
 => 2
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [2,2,1] => [2,2,1]
 => 2
([(0,4),(1,3),(2,3),(2,4)],5)
 => [1,1,1,1,1] => [1,1,1,1,1]
 => 1
([(0,1),(2,3),(2,4),(3,4)],5)
 => [2,1,2] => [2,2,1]
 => 2
([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
 => [1,1,1,1,1] => [1,1,1,1,1]
 => 1
([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
 => [1,2,1,1] => [2,1,1,1]
 => 2
([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
 => [2,2,1] => [2,2,1]
 => 2
([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => [1,1,1,1,1] => [1,1,1,1,1]
 => 1
([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
 => [1,1,1,1,1] => [1,1,1,1,1]
 => 1
([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
 => [1,1,1,1,1] => [1,1,1,1,1]
 => 1
([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [3,2] => [3,2]
 => 3
([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [1,2,1,1] => [2,1,1,1]
 => 2
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [2,1,1,1] => [2,1,1,1]
 => 2
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
 => [1,1,1,1,1] => [1,1,1,1,1]
 => 1
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
 => [2,2,1] => [2,2,1]
 => 2
Description
The largest part of an integer partition.
Matching statistic: St001235
(load all 12 compositions to match this statistic)
Mapping
Mp00152: Graphs —Laplacian multiplicities⟶ Integer compositions
Mp00039: Integer compositions —complement⟶ Integer compositions
St001235: Integer compositions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
([],1)
 => [1] => [1] => 1
([],2)
 => [2] => [1,1] => 2
([(0,1)],2)
 => [1,1] => [2] => 1
([],3)
 => [3] => [1,1,1] => 3
([(1,2)],3)
 => [1,2] => [2,1] => 2
([(0,2),(1,2)],3)
 => [1,1,1] => [3] => 1
([(0,1),(0,2),(1,2)],3)
 => [2,1] => [1,2] => 2
([],4)
 => [4] => [1,1,1,1] => 4
([(2,3)],4)
 => [1,3] => [2,1,1] => 3
([(1,3),(2,3)],4)
 => [1,1,2] => [3,1] => 2
([(0,3),(1,3),(2,3)],4)
 => [1,2,1] => [2,2] => 2
([(0,3),(1,2)],4)
 => [2,2] => [1,2,1] => 2
([(0,3),(1,2),(2,3)],4)
 => [1,1,1,1] => [4] => 1
([(1,2),(1,3),(2,3)],4)
 => [2,2] => [1,2,1] => 2
([(0,3),(1,2),(1,3),(2,3)],4)
 => [1,1,1,1] => [4] => 1
([(0,2),(0,3),(1,2),(1,3)],4)
 => [1,2,1] => [2,2] => 2
([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => [2,1,1] => [1,3] => 2
([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => [3,1] => [1,1,2] => 3
([],5)
 => [5] => [1,1,1,1,1] => 5
([(3,4)],5)
 => [1,4] => [2,1,1,1] => 4
([(2,4),(3,4)],5)
 => [1,1,3] => [3,1,1] => 3
([(1,4),(2,4),(3,4)],5)
 => [1,2,2] => [2,2,1] => 2
([(0,4),(1,4),(2,4),(3,4)],5)
 => [1,3,1] => [2,1,2] => 3
([(1,4),(2,3)],5)
 => [2,3] => [1,2,1,1] => 3
([(1,4),(2,3),(3,4)],5)
 => [1,1,1,2] => [4,1] => 2
([(0,1),(2,4),(3,4)],5)
 => [1,1,1,2] => [4,1] => 2
([(2,3),(2,4),(3,4)],5)
 => [2,3] => [1,2,1,1] => 3
([(0,4),(1,4),(2,3),(3,4)],5)
 => [1,1,1,1,1] => [5] => 1
([(1,4),(2,3),(2,4),(3,4)],5)
 => [1,1,1,2] => [4,1] => 2
([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => [1,1,2,1] => [3,2] => 2
([(1,3),(1,4),(2,3),(2,4)],5)
 => [1,2,2] => [2,2,1] => 2
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
 => [1,1,1,1,1] => [5] => 1
([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [2,1,2] => [1,3,1] => 2
([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => [1,1,1,1,1] => [5] => 1
([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [1,1,1,1,1] => [5] => 1
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
 => [1,1,2,1] => [3,2] => 2
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [2,2,1] => [1,2,2] => 2
([(0,4),(1,3),(2,3),(2,4)],5)
 => [1,1,1,1,1] => [5] => 1
([(0,1),(2,3),(2,4),(3,4)],5)
 => [2,1,2] => [1,3,1] => 2
([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
 => [1,1,1,1,1] => [5] => 1
([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
 => [1,2,1,1] => [2,3] => 2
([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
 => [2,2,1] => [1,2,2] => 2
([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => [1,1,1,1,1] => [5] => 1
([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
 => [1,1,1,1,1] => [5] => 1
([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
 => [1,1,1,1,1] => [5] => 1
([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [3,2] => [1,1,2,1] => 3
([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [1,2,1,1] => [2,3] => 2
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [2,1,1,1] => [1,4] => 2
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
 => [1,1,1,1,1] => [5] => 1
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
 => [2,2,1] => [1,2,2] => 2
Description
The global dimension of the corresponding Comp-Nakayama algebra. We identify the composition [n1-1,n2-1,...,nr-1] with the Nakayama algebra with Kupisch series [n1,n1-1,...,2,n2,n2-1,...,2,...,nr,nr-1,...,3,2,1]. We call such Nakayama algebras with Kupisch series corresponding to a integer composition "Comp-Nakayama algebra".
Matching statistic: St000010
Mapping
Mp00152: Graphs —Laplacian multiplicities⟶ Integer compositions
Mp00040: Integer compositions —to partition⟶ Integer partitions
Mp00044: Integer partitions —conjugate⟶ Integer partitions
St000010: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
([],1)
 => [1] => [1]
 => [1]
 => 1
([],2)
 => [2] => [2]
 => [1,1]
 => 2
([(0,1)],2)
 => [1,1] => [1,1]
 => [2]
 => 1
([],3)
 => [3] => [3]
 => [1,1,1]
 => 3
([(1,2)],3)
 => [1,2] => [2,1]
 => [2,1]
 => 2
([(0,2),(1,2)],3)
 => [1,1,1] => [1,1,1]
 => [3]
 => 1
([(0,1),(0,2),(1,2)],3)
 => [2,1] => [2,1]
 => [2,1]
 => 2
([],4)
 => [4] => [4]
 => [1,1,1,1]
 => 4
([(2,3)],4)
 => [1,3] => [3,1]
 => [2,1,1]
 => 3
([(1,3),(2,3)],4)
 => [1,1,2] => [2,1,1]
 => [3,1]
 => 2
([(0,3),(1,3),(2,3)],4)
 => [1,2,1] => [2,1,1]
 => [3,1]
 => 2
([(0,3),(1,2)],4)
 => [2,2] => [2,2]
 => [2,2]
 => 2
([(0,3),(1,2),(2,3)],4)
 => [1,1,1,1] => [1,1,1,1]
 => [4]
 => 1
([(1,2),(1,3),(2,3)],4)
 => [2,2] => [2,2]
 => [2,2]
 => 2
([(0,3),(1,2),(1,3),(2,3)],4)
 => [1,1,1,1] => [1,1,1,1]
 => [4]
 => 1
([(0,2),(0,3),(1,2),(1,3)],4)
 => [1,2,1] => [2,1,1]
 => [3,1]
 => 2
([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => [2,1,1] => [2,1,1]
 => [3,1]
 => 2
([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => [3,1] => [3,1]
 => [2,1,1]
 => 3
([],5)
 => [5] => [5]
 => [1,1,1,1,1]
 => 5
([(3,4)],5)
 => [1,4] => [4,1]
 => [2,1,1,1]
 => 4
([(2,4),(3,4)],5)
 => [1,1,3] => [3,1,1]
 => [3,1,1]
 => 3
([(1,4),(2,4),(3,4)],5)
 => [1,2,2] => [2,2,1]
 => [3,2]
 => 2
([(0,4),(1,4),(2,4),(3,4)],5)
 => [1,3,1] => [3,1,1]
 => [3,1,1]
 => 3
([(1,4),(2,3)],5)
 => [2,3] => [3,2]
 => [2,2,1]
 => 3
([(1,4),(2,3),(3,4)],5)
 => [1,1,1,2] => [2,1,1,1]
 => [4,1]
 => 2
([(0,1),(2,4),(3,4)],5)
 => [1,1,1,2] => [2,1,1,1]
 => [4,1]
 => 2
([(2,3),(2,4),(3,4)],5)
 => [2,3] => [3,2]
 => [2,2,1]
 => 3
([(0,4),(1,4),(2,3),(3,4)],5)
 => [1,1,1,1,1] => [1,1,1,1,1]
 => [5]
 => 1
([(1,4),(2,3),(2,4),(3,4)],5)
 => [1,1,1,2] => [2,1,1,1]
 => [4,1]
 => 2
([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => [1,1,2,1] => [2,1,1,1]
 => [4,1]
 => 2
([(1,3),(1,4),(2,3),(2,4)],5)
 => [1,2,2] => [2,2,1]
 => [3,2]
 => 2
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
 => [1,1,1,1,1] => [1,1,1,1,1]
 => [5]
 => 1
([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [2,1,2] => [2,2,1]
 => [3,2]
 => 2
([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => [1,1,1,1,1] => [1,1,1,1,1]
 => [5]
 => 1
([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [1,1,1,1,1] => [1,1,1,1,1]
 => [5]
 => 1
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
 => [1,1,2,1] => [2,1,1,1]
 => [4,1]
 => 2
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [2,2,1] => [2,2,1]
 => [3,2]
 => 2
([(0,4),(1,3),(2,3),(2,4)],5)
 => [1,1,1,1,1] => [1,1,1,1,1]
 => [5]
 => 1
([(0,1),(2,3),(2,4),(3,4)],5)
 => [2,1,2] => [2,2,1]
 => [3,2]
 => 2
([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
 => [1,1,1,1,1] => [1,1,1,1,1]
 => [5]
 => 1
([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
 => [1,2,1,1] => [2,1,1,1]
 => [4,1]
 => 2
([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
 => [2,2,1] => [2,2,1]
 => [3,2]
 => 2
([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => [1,1,1,1,1] => [1,1,1,1,1]
 => [5]
 => 1
([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
 => [1,1,1,1,1] => [1,1,1,1,1]
 => [5]
 => 1
([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
 => [1,1,1,1,1] => [1,1,1,1,1]
 => [5]
 => 1
([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [3,2] => [3,2]
 => [2,2,1]
 => 3
([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [1,2,1,1] => [2,1,1,1]
 => [4,1]
 => 2
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [2,1,1,1] => [2,1,1,1]
 => [4,1]
 => 2
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
 => [1,1,1,1,1] => [1,1,1,1,1]
 => [5]
 => 1
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
 => [2,2,1] => [2,2,1]
 => [3,2]
 => 2
Description
The length of the partition.
Matching statistic: St000025
Mapping
Mp00152: Graphs —Laplacian multiplicities⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
Mp00121: Dyck paths —Cori-Le Borgne involution⟶ Dyck paths
St000025: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
([],1)
 => [1] => [1,0]
 => [1,0]
 => 1
([],2)
 => [2] => [1,1,0,0]
 => [1,1,0,0]
 => 2
([(0,1)],2)
 => [1,1] => [1,0,1,0]
 => [1,0,1,0]
 => 1
([],3)
 => [3] => [1,1,1,0,0,0]
 => [1,1,1,0,0,0]
 => 3
([(1,2)],3)
 => [1,2] => [1,0,1,1,0,0]
 => [1,1,0,1,0,0]
 => 2
([(0,2),(1,2)],3)
 => [1,1,1] => [1,0,1,0,1,0]
 => [1,0,1,0,1,0]
 => 1
([(0,1),(0,2),(1,2)],3)
 => [2,1] => [1,1,0,0,1,0]
 => [1,1,0,0,1,0]
 => 2
([],4)
 => [4] => [1,1,1,1,0,0,0,0]
 => [1,1,1,1,0,0,0,0]
 => 4
([(2,3)],4)
 => [1,3] => [1,0,1,1,1,0,0,0]
 => [1,1,1,0,1,0,0,0]
 => 3
([(1,3),(2,3)],4)
 => [1,1,2] => [1,0,1,0,1,1,0,0]
 => [1,1,0,1,0,1,0,0]
 => 2
([(0,3),(1,3),(2,3)],4)
 => [1,2,1] => [1,0,1,1,0,0,1,0]
 => [1,1,0,1,0,0,1,0]
 => 2
([(0,3),(1,2)],4)
 => [2,2] => [1,1,0,0,1,1,0,0]
 => [1,1,0,0,1,1,0,0]
 => 2
([(0,3),(1,2),(2,3)],4)
 => [1,1,1,1] => [1,0,1,0,1,0,1,0]
 => [1,0,1,0,1,0,1,0]
 => 1
([(1,2),(1,3),(2,3)],4)
 => [2,2] => [1,1,0,0,1,1,0,0]
 => [1,1,0,0,1,1,0,0]
 => 2
([(0,3),(1,2),(1,3),(2,3)],4)
 => [1,1,1,1] => [1,0,1,0,1,0,1,0]
 => [1,0,1,0,1,0,1,0]
 => 1
([(0,2),(0,3),(1,2),(1,3)],4)
 => [1,2,1] => [1,0,1,1,0,0,1,0]
 => [1,1,0,1,0,0,1,0]
 => 2
([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => [2,1,1] => [1,1,0,0,1,0,1,0]
 => [1,1,0,0,1,0,1,0]
 => 2
([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => [3,1] => [1,1,1,0,0,0,1,0]
 => [1,1,1,0,0,1,0,0]
 => 3
([],5)
 => [5] => [1,1,1,1,1,0,0,0,0,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => 5
([(3,4)],5)
 => [1,4] => [1,0,1,1,1,1,0,0,0,0]
 => [1,1,1,1,0,1,0,0,0,0]
 => 4
([(2,4),(3,4)],5)
 => [1,1,3] => [1,0,1,0,1,1,1,0,0,0]
 => [1,1,1,0,1,0,1,0,0,0]
 => 3
([(1,4),(2,4),(3,4)],5)
 => [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
 => [1,1,0,0,1,1,0,1,0,0]
 => 2
([(0,4),(1,4),(2,4),(3,4)],5)
 => [1,3,1] => [1,0,1,1,1,0,0,0,1,0]
 => [1,1,1,0,1,0,0,1,0,0]
 => 3
([(1,4),(2,3)],5)
 => [2,3] => [1,1,0,0,1,1,1,0,0,0]
 => [1,1,1,0,0,1,1,0,0,0]
 => 3
([(1,4),(2,3),(3,4)],5)
 => [1,1,1,2] => [1,0,1,0,1,0,1,1,0,0]
 => [1,1,0,1,0,1,0,1,0,0]
 => 2
([(0,1),(2,4),(3,4)],5)
 => [1,1,1,2] => [1,0,1,0,1,0,1,1,0,0]
 => [1,1,0,1,0,1,0,1,0,0]
 => 2
([(2,3),(2,4),(3,4)],5)
 => [2,3] => [1,1,0,0,1,1,1,0,0,0]
 => [1,1,1,0,0,1,1,0,0,0]
 => 3
([(0,4),(1,4),(2,3),(3,4)],5)
 => [1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => 1
([(1,4),(2,3),(2,4),(3,4)],5)
 => [1,1,1,2] => [1,0,1,0,1,0,1,1,0,0]
 => [1,1,0,1,0,1,0,1,0,0]
 => 2
([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => [1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
 => [1,1,0,1,0,1,0,0,1,0]
 => 2
([(1,3),(1,4),(2,3),(2,4)],5)
 => [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
 => [1,1,0,0,1,1,0,1,0,0]
 => 2
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
 => [1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => 1
([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
 => [1,1,0,1,0,0,1,1,0,0]
 => 2
([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => [1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => 1
([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => 1
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
 => [1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
 => [1,1,0,1,0,1,0,0,1,0]
 => 2
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
 => [1,1,0,0,1,1,0,0,1,0]
 => 2
([(0,4),(1,3),(2,3),(2,4)],5)
 => [1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => 1
([(0,1),(2,3),(2,4),(3,4)],5)
 => [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
 => [1,1,0,1,0,0,1,1,0,0]
 => 2
([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
 => [1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => 1
([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
 => [1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
 => [1,1,0,1,0,0,1,0,1,0]
 => 2
([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
 => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
 => [1,1,0,0,1,1,0,0,1,0]
 => 2
([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => [1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => 1
([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
 => [1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => 1
([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
 => [1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => 1
([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [3,2] => [1,1,1,0,0,0,1,1,0,0]
 => [1,1,1,0,0,0,1,1,0,0]
 => 3
([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
 => [1,1,0,1,0,0,1,0,1,0]
 => 2
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
 => [1,1,0,0,1,0,1,0,1,0]
 => 2
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
 => [1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => 1
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
 => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
 => [1,1,0,0,1,1,0,0,1,0]
 => 2
Description
The number of initial rises of a Dyck path. In other words, this is the height of the first peak of $D$.
Matching statistic: St000062
Mapping
Mp00152: Graphs —Laplacian multiplicities⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
Mp00025: Dyck paths —to 132-avoiding permutation⟶ Permutations
St000062: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
([],1)
 => [1] => [1,0]
 => [1] => 1
([],2)
 => [2] => [1,1,0,0]
 => [1,2] => 2
([(0,1)],2)
 => [1,1] => [1,0,1,0]
 => [2,1] => 1
([],3)
 => [3] => [1,1,1,0,0,0]
 => [1,2,3] => 3
([(1,2)],3)
 => [1,2] => [1,0,1,1,0,0]
 => [2,3,1] => 2
([(0,2),(1,2)],3)
 => [1,1,1] => [1,0,1,0,1,0]
 => [3,2,1] => 1
([(0,1),(0,2),(1,2)],3)
 => [2,1] => [1,1,0,0,1,0]
 => [3,1,2] => 2
([],4)
 => [4] => [1,1,1,1,0,0,0,0]
 => [1,2,3,4] => 4
([(2,3)],4)
 => [1,3] => [1,0,1,1,1,0,0,0]
 => [2,3,4,1] => 3
([(1,3),(2,3)],4)
 => [1,1,2] => [1,0,1,0,1,1,0,0]
 => [3,4,2,1] => 2
([(0,3),(1,3),(2,3)],4)
 => [1,2,1] => [1,0,1,1,0,0,1,0]
 => [4,2,3,1] => 2
([(0,3),(1,2)],4)
 => [2,2] => [1,1,0,0,1,1,0,0]
 => [3,4,1,2] => 2
([(0,3),(1,2),(2,3)],4)
 => [1,1,1,1] => [1,0,1,0,1,0,1,0]
 => [4,3,2,1] => 1
([(1,2),(1,3),(2,3)],4)
 => [2,2] => [1,1,0,0,1,1,0,0]
 => [3,4,1,2] => 2
([(0,3),(1,2),(1,3),(2,3)],4)
 => [1,1,1,1] => [1,0,1,0,1,0,1,0]
 => [4,3,2,1] => 1
([(0,2),(0,3),(1,2),(1,3)],4)
 => [1,2,1] => [1,0,1,1,0,0,1,0]
 => [4,2,3,1] => 2
([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => [2,1,1] => [1,1,0,0,1,0,1,0]
 => [4,3,1,2] => 2
([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => [3,1] => [1,1,1,0,0,0,1,0]
 => [4,1,2,3] => 3
([],5)
 => [5] => [1,1,1,1,1,0,0,0,0,0]
 => [1,2,3,4,5] => 5
([(3,4)],5)
 => [1,4] => [1,0,1,1,1,1,0,0,0,0]
 => [2,3,4,5,1] => 4
([(2,4),(3,4)],5)
 => [1,1,3] => [1,0,1,0,1,1,1,0,0,0]
 => [3,4,5,2,1] => 3
([(1,4),(2,4),(3,4)],5)
 => [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
 => [4,5,2,3,1] => 2
([(0,4),(1,4),(2,4),(3,4)],5)
 => [1,3,1] => [1,0,1,1,1,0,0,0,1,0]
 => [5,2,3,4,1] => 3
([(1,4),(2,3)],5)
 => [2,3] => [1,1,0,0,1,1,1,0,0,0]
 => [3,4,5,1,2] => 3
([(1,4),(2,3),(3,4)],5)
 => [1,1,1,2] => [1,0,1,0,1,0,1,1,0,0]
 => [4,5,3,2,1] => 2
([(0,1),(2,4),(3,4)],5)
 => [1,1,1,2] => [1,0,1,0,1,0,1,1,0,0]
 => [4,5,3,2,1] => 2
([(2,3),(2,4),(3,4)],5)
 => [2,3] => [1,1,0,0,1,1,1,0,0,0]
 => [3,4,5,1,2] => 3
([(0,4),(1,4),(2,3),(3,4)],5)
 => [1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
 => [5,4,3,2,1] => 1
([(1,4),(2,3),(2,4),(3,4)],5)
 => [1,1,1,2] => [1,0,1,0,1,0,1,1,0,0]
 => [4,5,3,2,1] => 2
([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => [1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
 => [5,3,4,2,1] => 2
([(1,3),(1,4),(2,3),(2,4)],5)
 => [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
 => [4,5,2,3,1] => 2
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
 => [1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
 => [5,4,3,2,1] => 1
([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
 => [4,5,3,1,2] => 2
([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => [1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
 => [5,4,3,2,1] => 1
([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
 => [5,4,3,2,1] => 1
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
 => [1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
 => [5,3,4,2,1] => 2
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
 => [5,3,4,1,2] => 2
([(0,4),(1,3),(2,3),(2,4)],5)
 => [1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
 => [5,4,3,2,1] => 1
([(0,1),(2,3),(2,4),(3,4)],5)
 => [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
 => [4,5,3,1,2] => 2
([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
 => [1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
 => [5,4,3,2,1] => 1
([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
 => [1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
 => [5,4,2,3,1] => 2
([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
 => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
 => [5,3,4,1,2] => 2
([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => [1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
 => [5,4,3,2,1] => 1
([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
 => [1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
 => [5,4,3,2,1] => 1
([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
 => [1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
 => [5,4,3,2,1] => 1
([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [3,2] => [1,1,1,0,0,0,1,1,0,0]
 => [4,5,1,2,3] => 3
([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
 => [5,4,2,3,1] => 2
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
 => [5,4,3,1,2] => 2
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
 => [1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
 => [5,4,3,2,1] => 1
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
 => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
 => [5,3,4,1,2] => 2
Description
The length of the longest increasing subsequence of the permutation.
Matching statistic: St000166
Mapping
Mp00152: Graphs —Laplacian multiplicities⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
Mp00026: Dyck paths —to ordered tree⟶ Ordered trees
St000166: Ordered trees ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
([],1)
 => [1] => [1,0]
 => [[]]
 => 1
([],2)
 => [2] => [1,1,0,0]
 => [[[]]]
 => 2
([(0,1)],2)
 => [1,1] => [1,0,1,0]
 => [[],[]]
 => 1
([],3)
 => [3] => [1,1,1,0,0,0]
 => [[[[]]]]
 => 3
([(1,2)],3)
 => [1,2] => [1,0,1,1,0,0]
 => [[],[[]]]
 => 2
([(0,2),(1,2)],3)
 => [1,1,1] => [1,0,1,0,1,0]
 => [[],[],[]]
 => 1
([(0,1),(0,2),(1,2)],3)
 => [2,1] => [1,1,0,0,1,0]
 => [[[]],[]]
 => 2
([],4)
 => [4] => [1,1,1,1,0,0,0,0]
 => [[[[[]]]]]
 => 4
([(2,3)],4)
 => [1,3] => [1,0,1,1,1,0,0,0]
 => [[],[[[]]]]
 => 3
([(1,3),(2,3)],4)
 => [1,1,2] => [1,0,1,0,1,1,0,0]
 => [[],[],[[]]]
 => 2
([(0,3),(1,3),(2,3)],4)
 => [1,2,1] => [1,0,1,1,0,0,1,0]
 => [[],[[]],[]]
 => 2
([(0,3),(1,2)],4)
 => [2,2] => [1,1,0,0,1,1,0,0]
 => [[[]],[[]]]
 => 2
([(0,3),(1,2),(2,3)],4)
 => [1,1,1,1] => [1,0,1,0,1,0,1,0]
 => [[],[],[],[]]
 => 1
([(1,2),(1,3),(2,3)],4)
 => [2,2] => [1,1,0,0,1,1,0,0]
 => [[[]],[[]]]
 => 2
([(0,3),(1,2),(1,3),(2,3)],4)
 => [1,1,1,1] => [1,0,1,0,1,0,1,0]
 => [[],[],[],[]]
 => 1
([(0,2),(0,3),(1,2),(1,3)],4)
 => [1,2,1] => [1,0,1,1,0,0,1,0]
 => [[],[[]],[]]
 => 2
([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => [2,1,1] => [1,1,0,0,1,0,1,0]
 => [[[]],[],[]]
 => 2
([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => [3,1] => [1,1,1,0,0,0,1,0]
 => [[[[]]],[]]
 => 3
([],5)
 => [5] => [1,1,1,1,1,0,0,0,0,0]
 => [[[[[[]]]]]]
 => 5
([(3,4)],5)
 => [1,4] => [1,0,1,1,1,1,0,0,0,0]
 => [[],[[[[]]]]]
 => 4
([(2,4),(3,4)],5)
 => [1,1,3] => [1,0,1,0,1,1,1,0,0,0]
 => [[],[],[[[]]]]
 => 3
([(1,4),(2,4),(3,4)],5)
 => [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
 => [[],[[]],[[]]]
 => 2
([(0,4),(1,4),(2,4),(3,4)],5)
 => [1,3,1] => [1,0,1,1,1,0,0,0,1,0]
 => [[],[[[]]],[]]
 => 3
([(1,4),(2,3)],5)
 => [2,3] => [1,1,0,0,1,1,1,0,0,0]
 => [[[]],[[[]]]]
 => 3
([(1,4),(2,3),(3,4)],5)
 => [1,1,1,2] => [1,0,1,0,1,0,1,1,0,0]
 => [[],[],[],[[]]]
 => 2
([(0,1),(2,4),(3,4)],5)
 => [1,1,1,2] => [1,0,1,0,1,0,1,1,0,0]
 => [[],[],[],[[]]]
 => 2
([(2,3),(2,4),(3,4)],5)
 => [2,3] => [1,1,0,0,1,1,1,0,0,0]
 => [[[]],[[[]]]]
 => 3
([(0,4),(1,4),(2,3),(3,4)],5)
 => [1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
 => [[],[],[],[],[]]
 => 1
([(1,4),(2,3),(2,4),(3,4)],5)
 => [1,1,1,2] => [1,0,1,0,1,0,1,1,0,0]
 => [[],[],[],[[]]]
 => 2
([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => [1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
 => [[],[],[[]],[]]
 => 2
([(1,3),(1,4),(2,3),(2,4)],5)
 => [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
 => [[],[[]],[[]]]
 => 2
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
 => [1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
 => [[],[],[],[],[]]
 => 1
([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
 => [[[]],[],[[]]]
 => 2
([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => [1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
 => [[],[],[],[],[]]
 => 1
([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
 => [[],[],[],[],[]]
 => 1
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
 => [1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
 => [[],[],[[]],[]]
 => 2
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
 => [[[]],[[]],[]]
 => 2
([(0,4),(1,3),(2,3),(2,4)],5)
 => [1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
 => [[],[],[],[],[]]
 => 1
([(0,1),(2,3),(2,4),(3,4)],5)
 => [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
 => [[[]],[],[[]]]
 => 2
([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
 => [1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
 => [[],[],[],[],[]]
 => 1
([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
 => [1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
 => [[],[[]],[],[]]
 => 2
([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
 => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
 => [[[]],[[]],[]]
 => 2
([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => [1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
 => [[],[],[],[],[]]
 => 1
([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
 => [1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
 => [[],[],[],[],[]]
 => 1
([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
 => [1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
 => [[],[],[],[],[]]
 => 1
([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [3,2] => [1,1,1,0,0,0,1,1,0,0]
 => [[[[]]],[[]]]
 => 3
([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
 => [[],[[]],[],[]]
 => 2
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
 => [[[]],[],[],[]]
 => 2
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
 => [1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
 => [[],[],[],[],[]]
 => 1
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
 => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
 => [[[]],[[]],[]]
 => 2
Description
The depth minus 1 of an ordered tree. The ordered trees of size $n$ are bijection with the Dyck paths of size $n-1$, and this statistic then corresponds to [[St000013]].
The following 189 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000308The height of the tree associated to a permutation. St000392The length of the longest run of ones in a binary word. St000442The maximal area to the right of an up step of a Dyck path. St000451The length of the longest pattern of the form k 1 2. St000628The balance of a binary word. St000676The number of odd rises of a Dyck path. St000684The global dimension of the LNakayama algebra associated to a Dyck path. St000686The finitistic dominant dimension of a Dyck path. St000720The size of the largest partition in the oscillating tableau corresponding to the perfect matching. St000734The last entry in the first row of a standard tableau. St000930The k-Gorenstein degree of the corresponding Nakayama algebra with linear quiver. St000982The length of the longest constant subword. St000983The length of the longest alternating subword. St001203We associate to a CNakayama algebra (a Nakayama algebra with a cyclic quiver) with Kupisch series $L=[c_0,c_1,...,c_{n-1}]$ such that $n=c_0 < c_i$ for all $i > 0$ a Dyck path as follows: St001210Gives the maximal vector space dimension of the first Ext-group between an indecomposable module X and the regular module A, when A is the Nakayama algebra corresponding to the Dyck path. St001239The largest vector space dimension of the double dual of a simple module in the corresponding Nakayama algebra. St001372The length of a longest cyclic run of ones of a binary word. St001530The depth of a Dyck path. St001652The length of a longest interval of consecutive numbers. St001809The index of the step at the first peak of maximal height in a Dyck path. St000094The depth of an ordered tree. St000141The maximum drop size of a permutation. St000209Maximum difference of elements in cycles. St000225Difference between largest and smallest parts in a partition. St000306The bounce count of a Dyck path. St000439The position of the first down step of a Dyck path. St000521The number of distinct subtrees of an ordered tree. St000662The staircase size of the code of a permutation. St001046The maximal number of arcs nesting a given arc of a perfect matching. St001090The number of pop-stack-sorts needed to sort a permutation. St001192The maximal dimension of $Ext_A^2(S,A)$ for a simple module $S$ over the corresponding Nakayama algebra $A$. St001294The maximal torsionfree index of a simple non-projective module in the corresponding Nakayama algebra. St001296The maximal torsionfree index of an indecomposable non-projective module in the corresponding Nakayama algebra. St000444The length of the maximal rise of a Dyck path. St000485The length of the longest cycle of a permutation. St000844The size of the largest block in the direct sum decomposition of a permutation. St000887The maximal number of nonzero entries on a diagonal of a permutation matrix. St001039The maximal height of a column in the parallelogram polyomino associated with a Dyck path. St001062The maximal size of a block of a set partition. St000503The maximal difference between two elements in a common block. St000956The maximal displacement of a permutation. St001291The number of indecomposable summands of the tensor product of two copies of the dual of the Nakayama algebra associated to a Dyck path. St001330The hat guessing number of a graph. St000028The number of stack-sorts needed to sort a permutation. St000454The largest eigenvalue of a graph if it is integral. St000260The radius of a connected graph. St001570The minimal number of edges to add to make a graph Hamiltonian. St001060The distinguishing index of a graph. St000777The number of distinct eigenvalues of the distance Laplacian of a connected graph. St000259The diameter of a connected graph. St000741The Colin de Verdière graph invariant. St001165Number of simple modules with even projective dimension in the corresponding Nakayama algebra. St000455The second largest eigenvalue of a graph if it is integral. St001118The acyclic chromatic index of a graph. St000993The multiplicity of the largest part of an integer partition. St001198The number of simple modules in the algebra $eAe$ with projective dimension at most 1 in the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St001206The maximal dimension of an indecomposable projective $eAe$-module (that is the height of the corresponding Dyck path) of the corresponding Nakayama algebra with minimal faithful projective-injective module $eA$. St001568The smallest positive integer that does not appear twice in the partition. St000464The Schultz index of a connected graph. St001545The second Elser number of a connected graph. St001704The size of the largest multi-subset-intersection of the deck of a graph with the deck of another graph. St001630The global dimension of the incidence algebra of the lattice over the rational numbers. St001878The projective dimension of the simple modules corresponding to the minimum of L in the incidence algebra of the lattice L. St001644The dimension of a graph. St000952Gives the number of irreducible factors of the Coxeter polynomial of the Dyck path over the rational numbers. St001200The number of simple modules in $eAe$ with projective dimension at most 2 in the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St001289The vector space dimension of the n-fold tensor product of D(A), where n is maximal such that this n-fold tensor product is nonzero. St000460The hook length of the last cell along the main diagonal of an integer partition. St000870The product of the hook lengths of the diagonal cells in an integer partition. St001914The size of the orbit of an integer partition in Bulgarian solitaire. St000015The number of peaks of a Dyck path. St000144The pyramid weight of the Dyck path. St000228The size of a partition. St000293The number of inversions of a binary word. St000318The number of addable cells of the Ferrers diagram of an integer partition. St000335The difference of lower and upper interactions. St000380Half of the maximal perimeter of a rectangle fitting into the diagram of an integer partition. St000384The maximal part of the shifted composition of an integer partition. St000393The number of strictly increasing runs in a binary word. St000395The sum of the heights of the peaks of a Dyck path. St000419The number of Dyck paths that are weakly above the Dyck path, except for the path itself. St000443The number of long tunnels of a Dyck path. St000459The hook length of the base cell of a partition. St000519The largest length of a factor maximising the subword complexity. St000531The leading coefficient of the rook polynomial of an integer partition. St000543The size of the conjugacy class of a binary word. St000626The minimal period of a binary word. St000630The length of the shortest palindromic decomposition of a binary word. St000678The number of up steps after the last double rise of a Dyck path. St000759The smallest missing part in an integer partition. St000784The maximum of the length and the largest part of the integer partition. St000922The minimal number such that all substrings of this length are unique. St000955Number of times one has $Ext^i(D(A),A)>0$ for $i>0$ for the corresponding LNakayama algebra. St000968We make a CNakayama algebra out of the LNakayama algebra (corresponding to the Dyck path) $[c_0,c_1,...,c_{n−1}]$ by adding $c_0$ to $c_{n−1}$. St000969We make a CNakayama algebra out of the LNakayama algebra (corresponding to the Dyck path) $[c_0,c_1,...,c_{n-1}]$ by adding $c_0$ to $c_{n-1}$. St001007Number of simple modules with projective dimension 1 in the Nakayama algebra corresponding to the Dyck path. St001014Number of indecomposable injective modules with codominant dimension equal to the dominant dimension of the Nakayama algebra corresponding to the Dyck path. St001020Sum of the codominant dimensions of the non-projective indecomposable injective modules of the Nakayama algebra corresponding to the Dyck path. St001023Number of simple modules with projective dimension at most 3 in the Nakayama algebra corresponding to the Dyck path. St001024Maximum of dominant dimensions of the simple modules in the Nakayama algebra corresponding to the Dyck path. St001028Number of simple modules with injective dimension equal to the dominant dimension in the Nakayama algebra corresponding to the Dyck path. St001038The minimal height of a column in the parallelogram polyomino associated with the Dyck path. St001068Number of torsionless simple modules in the corresponding Nakayama algebra. St001088Number of indecomposable projective non-injective modules with dominant dimension equal to the injective dimension in the corresponding Nakayama algebra. St001135The projective dimension of the first simple module in the Nakayama algebra corresponding to the Dyck path. St001170Number of indecomposable injective modules whose socle has projective dimension at most g-1 when g denotes the global dimension in the corresponding Nakayama algebra. St001180Number of indecomposable injective modules with projective dimension at most 1. St001183The maximum of $projdim(S)+injdim(S)$ over all simple modules in the Nakayama algebra corresponding to the Dyck path. St001187The number of simple modules with grade at least one in the corresponding Nakayama algebra. St001190Number of simple modules with projective dimension at most 4 in the corresponding Nakayama algebra. St001201The grade of the simple module $S_0$ in the special CNakayama algebra corresponding to the Dyck path. St001202Call a CNakayama algebra (a Nakayama algebra with a cyclic quiver) with Kupisch series $L=[c_0,c_1,...,c_{n−1}]$ such that $n=c_0 < c_i$ for all $i > 0$ a special CNakayama algebra. St001211The number of simple modules in the corresponding Nakayama algebra that have vanishing second Ext-group with the regular module. St001224Let X be the direct sum of all simple modules of the corresponding Nakayama algebra. St001226The number of integers i such that the radical of the i-th indecomposable projective module has vanishing first extension group with the Jacobson radical J in the corresponding Nakayama algebra. St001237The number of simple modules with injective dimension at most one or dominant dimension at least one. St001258Gives the maximum of injective plus projective dimension of an indecomposable module over the corresponding Nakayama algebra. St001267The length of the Lyndon factorization of the binary word. St001290The first natural number n such that the tensor product of n copies of D(A) is zero for the corresponding Nakayama algebra A. St001348The bounce of the parallelogram polyomino associated with the Dyck path. St001416The length of a longest palindromic factor of a binary word. St001417The length of a longest palindromic subword of a binary word. St001436The index of a given binary word in the lex-order among all its cyclic shifts. St001471The magnitude of a Dyck path. St001488The number of corners of a skew partition. St001492The number of simple modules that do not appear in the socle of the regular module or have no nontrivial selfextensions with the regular module in the corresponding Nakayama algebra. St001500The global dimension of magnitude 1 Nakayama algebras. St001504The sum of all indegrees of vertices with indegree at least two in the resolution quiver of a Nakayama algebra corresponding to the Dyck path. St001505The number of elements generated by the Dyck path as a map in the full transformation monoid. St001523The degree of symmetry of a Dyck path. St001526The Loewy length of the Auslander-Reiten translate of the regular module as a bimodule of the Nakayama algebra corresponding to the Dyck path. St001659The number of ways to place as many non-attacking rooks as possible on a Ferrers board. St001660The number of ways to place as many non-attacking rooks as possible on a skew Ferrers board. St001733The number of weak left to right maxima of a Dyck path. St001814The number of partitions interlacing the given partition. St001885The number of binary words with the same proper border set. St001959The product of the heights of the peaks of a Dyck path. St001514The dimension of the top of the Auslander-Reiten translate of the regular modules as a bimodule. St001515The vector space dimension of the socle of the first syzygy module of the regular module (as a bimodule). St000620The number of standard tableaux of shape equal to the given partition such that the minimal cyclic descent is odd. St000668The least common multiple of the parts of the partition. St000681The Grundy value of Chomp on Ferrers diagrams. St000698The number of 2-rim hooks removed from an integer partition to obtain its associated 2-core. St000704The number of semistandard tableaux on a given integer partition with minimal maximal entry. St000708The product of the parts of an integer partition. St000933The number of multipartitions of sizes given by an integer partition. St000937The number of positive values of the symmetric group character corresponding to the partition. St001128The exponens consonantiae of a partition. St000939The number of characters of the symmetric group whose value on the partition is positive. St001232The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2. St000772The multiplicity of the largest distance Laplacian eigenvalue in a connected graph. St001491The number of indecomposable projective-injective modules in the algebra corresponding to a subset. St001589The nesting number of a perfect matching. St001431Half of the Loewy length minus one of a modified stable Auslander algebra of the Nakayama algebra corresponding to the Dyck path. St001553The number of indecomposable summands of the square of the Jacobson radical as a bimodule in the Nakayama algebra corresponding to the Dyck path. St000770The major index of an integer partition when read from bottom to top. St000815The number of semistandard Young tableaux of partition weight of given shape. St000675The number of centered multitunnels of a Dyck path. St000744The length of the path to the largest entry in a standard Young tableau. St001499The number of indecomposable projective-injective modules of a magnitude 1 Nakayama algebra. St001603The number of colourings of a polygon such that the multiplicities of a colour are given by a partition. St000264The girth of a graph, which is not a tree. St001605The number of colourings of a cycle such that the multiplicities of colours are given by a partition. St001621The number of atoms of a lattice. St001624The breadth of a lattice. St001199The dominant dimension of $eAe$ for the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St000207Number of integral Gelfand-Tsetlin polytopes with prescribed top row and integer composition weight. St000208Number of integral Gelfand-Tsetlin polytopes with prescribed top row and integer partition weight. St000618The number of self-evacuating tableaux of given shape. St000667The greatest common divisor of the parts of the partition. St000755The number of real roots of the characteristic polynomial of a linear recurrence associated with an integer partition. St000781The number of proper colouring schemes of a Ferrers diagram. St001250The number of parts of a partition that are not congruent 0 modulo 3. St001360The number of covering relations in Young's lattice below a partition. St001380The number of monomer-dimer tilings of a Ferrers diagram. St001389The number of partitions of the same length below the given integer partition. St001432The order dimension of the partition. St001527The cyclic permutation representation number of an integer partition. St001571The Cartan determinant of the integer partition. St001599The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on rooted trees. St001780The order of promotion on the set of standard tableaux of given shape. St001899The total number of irreducible representations contained in the higher Lie character for an integer partition. St001900The number of distinct irreducible representations contained in the higher Lie character for an integer partition. St001901The largest multiplicity of an irreducible representation contained in the higher Lie character for an integer partition. St001908The number of semistandard tableaux of distinct weight whose maximal entry is the length of the partition. St001924The number of cells in an integer partition whose arm and leg length coincide. St001933The largest multiplicity of a part in an integer partition. St001934The number of monotone factorisations of genus zero of a permutation of given cycle type. St000776The maximal multiplicity of an eigenvalue in a graph.