Your data matches 20 different statistics following compositions of up to 3 maps.
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Matching statistic: St000772
(load all 4 compositions to match this statistic)
Mapping
Mp00023: Dyck paths —to non-crossing permutation⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
St000772: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
 => [1] => ([],1)
 => 1
[1,1,0,0]
 => [2,1] => ([(0,1)],2)
 => 1
[1,1,0,1,0,0]
 => [2,3,1] => ([(0,2),(1,2)],3)
 => 1
[1,1,1,0,0,0]
 => [3,2,1] => ([(0,1),(0,2),(1,2)],3)
 => 2
[1,1,0,1,0,1,0,0]
 => [2,3,4,1] => ([(0,3),(1,3),(2,3)],4)
 => 2
[1,1,0,1,1,0,0,0]
 => [2,4,3,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => 1
[1,1,1,0,0,1,0,0]
 => [3,2,4,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => 1
[1,1,1,0,1,0,0,0]
 => [4,2,3,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 1
[1,1,1,1,0,0,0,0]
 => [4,3,2,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 3
[1,1,0,1,0,1,0,1,0,0]
 => [2,3,4,5,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 3
[1,1,0,1,0,1,1,0,0,0]
 => [2,3,5,4,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => 2
[1,1,0,1,1,0,0,1,0,0]
 => [2,4,3,5,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => 2
[1,1,0,1,1,0,1,0,0,0]
 => [2,5,3,4,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 1
[1,1,0,1,1,1,0,0,0,0]
 => [2,5,4,3,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 1
[1,1,1,0,0,1,0,1,0,0]
 => [3,2,4,5,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => 2
[1,1,1,0,0,1,1,0,0,0]
 => [3,2,5,4,1] => ([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
 => 1
[1,1,1,0,1,0,0,1,0,0]
 => [4,2,3,5,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 1
[1,1,1,0,1,0,1,0,0,0]
 => [5,2,3,4,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 2
[1,1,1,0,1,1,0,0,0,0]
 => [5,2,4,3,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 1
[1,1,1,1,0,0,0,1,0,0]
 => [4,3,2,5,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 1
[1,1,1,1,0,0,1,0,0,0]
 => [5,3,2,4,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 1
[1,1,1,1,0,1,0,0,0,0]
 => [5,3,4,2,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 1
[1,1,1,1,1,0,0,0,0,0]
 => [5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 4
[1,1,0,1,0,1,0,1,0,1,0,0]
 => [2,3,4,5,6,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
 => 4
[1,1,0,1,0,1,0,1,1,0,0,0]
 => [2,3,4,6,5,1] => ([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
 => 3
[1,1,0,1,0,1,1,0,0,1,0,0]
 => [2,3,5,4,6,1] => ([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
 => 3
[1,1,0,1,0,1,1,0,1,0,0,0]
 => [2,3,6,4,5,1] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 2
[1,1,0,1,0,1,1,1,0,0,0,0]
 => [2,3,6,5,4,1] => ([(0,5),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 2
[1,1,0,1,1,0,0,1,0,1,0,0]
 => [2,4,3,5,6,1] => ([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
 => 3
[1,1,0,1,1,0,0,1,1,0,0,0]
 => [2,4,3,6,5,1] => ([(0,5),(1,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)
 => 2
[1,1,0,1,1,0,1,0,0,1,0,0]
 => [2,5,3,4,6,1] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 2
[1,1,0,1,1,0,1,0,1,0,0,0]
 => [2,6,3,4,5,1] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 1
[1,1,0,1,1,0,1,1,0,0,0,0]
 => [2,6,3,5,4,1] => ([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 1
[1,1,0,1,1,1,0,0,0,1,0,0]
 => [2,5,4,3,6,1] => ([(0,5),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 2
[1,1,0,1,1,1,0,0,1,0,0,0]
 => [2,6,4,3,5,1] => ([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 1
[1,1,0,1,1,1,0,1,0,0,0,0]
 => [2,6,4,5,3,1] => ([(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 1
[1,1,0,1,1,1,1,0,0,0,0,0]
 => [2,6,5,4,3,1] => ([(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 1
[1,1,1,0,0,1,0,1,0,1,0,0]
 => [3,2,4,5,6,1] => ([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
 => 3
[1,1,1,0,0,1,0,1,1,0,0,0]
 => [3,2,4,6,5,1] => ([(0,5),(1,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)
 => 2
[1,1,1,0,0,1,1,0,0,1,0,0]
 => [3,2,5,4,6,1] => ([(0,5),(1,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)
 => 2
[1,1,1,0,0,1,1,0,1,0,0,0]
 => [3,2,6,4,5,1] => ([(0,1),(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 1
[1,1,1,0,0,1,1,1,0,0,0,0]
 => [3,2,6,5,4,1] => ([(0,1),(0,5),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 1
[1,1,1,0,1,0,0,1,0,1,0,0]
 => [4,2,3,5,6,1] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 2
[1,1,1,0,1,0,0,1,1,0,0,0]
 => [4,2,3,6,5,1] => ([(0,1),(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 1
[1,1,1,0,1,0,1,0,0,1,0,0]
 => [5,2,3,4,6,1] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 1
[1,1,1,0,1,0,1,0,1,0,0,0]
 => [6,2,3,4,5,1] => ([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 3
[1,1,1,0,1,0,1,1,0,0,0,0]
 => [6,2,3,5,4,1] => ([(0,4),(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 2
[1,1,1,0,1,1,0,0,0,1,0,0]
 => [5,2,4,3,6,1] => ([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 1
[1,1,1,0,1,1,0,0,1,0,0,0]
 => [6,2,4,3,5,1] => ([(0,4),(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 2
[1,1,1,0,1,1,0,1,0,0,0,0]
 => [6,2,4,5,3,1] => ([(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 1
Description
The multiplicity of the largest 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 $1$. The path on four vertices has eigenvalues $0, 4.7\dots, 6, 9.2\dots$ and therefore also statistic $1$. The graphs with statistic $n-1$, $n-2$ and $n-3$ have been characterised, see [1].
Matching statistic: St000773
Mapping
Mp00025: Dyck paths —to 132-avoiding permutation⟶ Permutations
Mp00252: Permutations —restriction⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
St000773: Graphs ⟶ ℤResult quality: 99% ●values known / values provided: 99%●distinct values known / distinct values provided: 100%
Values
[1,0]
 => [1] => [] => ([],0)
 => ? = 1
[1,1,0,0]
 => [1,2] => [1] => ([],1)
 => 1
[1,1,0,1,0,0]
 => [2,1,3] => [2,1] => ([(0,1)],2)
 => 1
[1,1,1,0,0,0]
 => [1,2,3] => [1,2] => ([],2)
 => 2
[1,1,0,1,0,1,0,0]
 => [3,2,1,4] => [3,2,1] => ([(0,1),(0,2),(1,2)],3)
 => 2
[1,1,0,1,1,0,0,0]
 => [2,3,1,4] => [2,3,1] => ([(0,2),(1,2)],3)
 => 1
[1,1,1,0,0,1,0,0]
 => [3,1,2,4] => [3,1,2] => ([(0,2),(1,2)],3)
 => 1
[1,1,1,0,1,0,0,0]
 => [2,1,3,4] => [2,1,3] => ([(1,2)],3)
 => 1
[1,1,1,1,0,0,0,0]
 => [1,2,3,4] => [1,2,3] => ([],3)
 => 3
[1,1,0,1,0,1,0,1,0,0]
 => [4,3,2,1,5] => [4,3,2,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 3
[1,1,0,1,0,1,1,0,0,0]
 => [3,4,2,1,5] => [3,4,2,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 2
[1,1,0,1,1,0,0,1,0,0]
 => [4,2,3,1,5] => [4,2,3,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 2
[1,1,0,1,1,0,1,0,0,0]
 => [3,2,4,1,5] => [3,2,4,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => 1
[1,1,0,1,1,1,0,0,0,0]
 => [2,3,4,1,5] => [2,3,4,1] => ([(0,3),(1,3),(2,3)],4)
 => 1
[1,1,1,0,0,1,0,1,0,0]
 => [4,3,1,2,5] => [4,3,1,2] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 2
[1,1,1,0,0,1,1,0,0,0]
 => [3,4,1,2,5] => [3,4,1,2] => ([(0,2),(0,3),(1,2),(1,3)],4)
 => 1
[1,1,1,0,1,0,0,1,0,0]
 => [4,2,1,3,5] => [4,2,1,3] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => 1
[1,1,1,0,1,0,1,0,0,0]
 => [3,2,1,4,5] => [3,2,1,4] => ([(1,2),(1,3),(2,3)],4)
 => 2
[1,1,1,0,1,1,0,0,0,0]
 => [2,3,1,4,5] => [2,3,1,4] => ([(1,3),(2,3)],4)
 => 1
[1,1,1,1,0,0,0,1,0,0]
 => [4,1,2,3,5] => [4,1,2,3] => ([(0,3),(1,3),(2,3)],4)
 => 1
[1,1,1,1,0,0,1,0,0,0]
 => [3,1,2,4,5] => [3,1,2,4] => ([(1,3),(2,3)],4)
 => 1
[1,1,1,1,0,1,0,0,0,0]
 => [2,1,3,4,5] => [2,1,3,4] => ([(2,3)],4)
 => 1
[1,1,1,1,1,0,0,0,0,0]
 => [1,2,3,4,5] => [1,2,3,4] => ([],4)
 => 4
[1,1,0,1,0,1,0,1,0,1,0,0]
 => [5,4,3,2,1,6] => [5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 4
[1,1,0,1,0,1,0,1,1,0,0,0]
 => [4,5,3,2,1,6] => [4,5,3,2,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 3
[1,1,0,1,0,1,1,0,0,1,0,0]
 => [5,3,4,2,1,6] => [5,3,4,2,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 3
[1,1,0,1,0,1,1,0,1,0,0,0]
 => [4,3,5,2,1,6] => [4,3,5,2,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 2
[1,1,0,1,0,1,1,1,0,0,0,0]
 => [3,4,5,2,1,6] => [3,4,5,2,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 2
[1,1,0,1,1,0,0,1,0,1,0,0]
 => [5,4,2,3,1,6] => [5,4,2,3,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 3
[1,1,0,1,1,0,0,1,1,0,0,0]
 => [4,5,2,3,1,6] => [4,5,2,3,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
 => 2
[1,1,0,1,1,0,1,0,0,1,0,0]
 => [5,3,2,4,1,6] => [5,3,2,4,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 2
[1,1,0,1,1,0,1,0,1,0,0,0]
 => [4,3,2,5,1,6] => [4,3,2,5,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 1
[1,1,0,1,1,0,1,1,0,0,0,0]
 => [3,4,2,5,1,6] => [3,4,2,5,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 1
[1,1,0,1,1,1,0,0,0,1,0,0]
 => [5,2,3,4,1,6] => [5,2,3,4,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 2
[1,1,0,1,1,1,0,0,1,0,0,0]
 => [4,2,3,5,1,6] => [4,2,3,5,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 1
[1,1,0,1,1,1,0,1,0,0,0,0]
 => [3,2,4,5,1,6] => [3,2,4,5,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => 1
[1,1,0,1,1,1,1,0,0,0,0,0]
 => [2,3,4,5,1,6] => [2,3,4,5,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 1
[1,1,1,0,0,1,0,1,0,1,0,0]
 => [5,4,3,1,2,6] => [5,4,3,1,2] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 3
[1,1,1,0,0,1,0,1,1,0,0,0]
 => [4,5,3,1,2,6] => [4,5,3,1,2] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
 => 2
[1,1,1,0,0,1,1,0,0,1,0,0]
 => [5,3,4,1,2,6] => [5,3,4,1,2] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
 => 2
[1,1,1,0,0,1,1,0,1,0,0,0]
 => [4,3,5,1,2,6] => [4,3,5,1,2] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
 => 1
[1,1,1,0,0,1,1,1,0,0,0,0]
 => [3,4,5,1,2,6] => [3,4,5,1,2] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
 => 1
[1,1,1,0,1,0,0,1,0,1,0,0]
 => [5,4,2,1,3,6] => [5,4,2,1,3] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 2
[1,1,1,0,1,0,0,1,1,0,0,0]
 => [4,5,2,1,3,6] => [4,5,2,1,3] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
 => 1
[1,1,1,0,1,0,1,0,0,1,0,0]
 => [5,3,2,1,4,6] => [5,3,2,1,4] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 1
[1,1,1,0,1,0,1,0,1,0,0,0]
 => [4,3,2,1,5,6] => [4,3,2,1,5] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 3
[1,1,1,0,1,0,1,1,0,0,0,0]
 => [3,4,2,1,5,6] => [3,4,2,1,5] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 2
[1,1,1,0,1,1,0,0,0,1,0,0]
 => [5,2,3,1,4,6] => [5,2,3,1,4] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 1
[1,1,1,0,1,1,0,0,1,0,0,0]
 => [4,2,3,1,5,6] => [4,2,3,1,5] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 2
[1,1,1,0,1,1,0,1,0,0,0,0]
 => [3,2,4,1,5,6] => [3,2,4,1,5] => ([(1,4),(2,3),(2,4),(3,4)],5)
 => 1
[1,1,1,0,1,1,1,0,0,0,0,0]
 => [2,3,4,1,5,6] => [2,3,4,1,5] => ([(1,4),(2,4),(3,4)],5)
 => 1
Description
The multiplicity of the largest Laplacian eigenvalue in a graph.
Matching statistic: St000234
(load all 3 compositions to match this statistic)
Mapping
Mp00027: Dyck paths —to partition⟶ Integer partitions
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00119: Dyck paths —to 321-avoiding permutation (Krattenthaler)⟶ Permutations
St000234: Permutations ⟶ ℤResult quality: 83% ●values known / values provided: 96%●distinct values known / distinct values provided: 83%
Values
[1,0]
 => []
 => []
 => [] => ? = 1
[1,1,0,0]
 => []
 => []
 => [] => ? = 1
[1,1,0,1,0,0]
 => [1]
 => [1,0,1,0]
 => [1,2] => 1
[1,1,1,0,0,0]
 => []
 => []
 => [] => ? = 2
[1,1,0,1,0,1,0,0]
 => [2,1]
 => [1,0,1,0,1,0]
 => [1,2,3] => 2
[1,1,0,1,1,0,0,0]
 => [1,1]
 => [1,0,1,1,0,0]
 => [1,3,2] => 1
[1,1,1,0,0,1,0,0]
 => [2]
 => [1,1,0,0,1,0]
 => [2,1,3] => 1
[1,1,1,0,1,0,0,0]
 => [1]
 => [1,0,1,0]
 => [1,2] => 1
[1,1,1,1,0,0,0,0]
 => []
 => []
 => [] => ? = 3
[1,1,0,1,0,1,0,1,0,0]
 => [3,2,1]
 => [1,0,1,0,1,0,1,0]
 => [1,2,3,4] => 3
[1,1,0,1,0,1,1,0,0,0]
 => [2,2,1]
 => [1,0,1,0,1,1,0,0]
 => [1,2,4,3] => 2
[1,1,0,1,1,0,0,1,0,0]
 => [3,1,1]
 => [1,0,1,1,0,0,1,0]
 => [1,3,2,4] => 2
[1,1,0,1,1,0,1,0,0,0]
 => [2,1,1]
 => [1,0,1,1,0,1,0,0]
 => [1,3,4,2] => 1
[1,1,0,1,1,1,0,0,0,0]
 => [1,1,1]
 => [1,0,1,1,1,0,0,0]
 => [1,4,2,3] => 1
[1,1,1,0,0,1,0,1,0,0]
 => [3,2]
 => [1,1,0,0,1,0,1,0]
 => [2,1,3,4] => 2
[1,1,1,0,0,1,1,0,0,0]
 => [2,2]
 => [1,1,0,0,1,1,0,0]
 => [2,1,4,3] => 1
[1,1,1,0,1,0,0,1,0,0]
 => [3,1]
 => [1,1,0,1,0,0,1,0]
 => [2,3,1,4] => 1
[1,1,1,0,1,0,1,0,0,0]
 => [2,1]
 => [1,0,1,0,1,0]
 => [1,2,3] => 2
[1,1,1,0,1,1,0,0,0,0]
 => [1,1]
 => [1,0,1,1,0,0]
 => [1,3,2] => 1
[1,1,1,1,0,0,0,1,0,0]
 => [3]
 => [1,1,1,0,0,0,1,0]
 => [3,1,2,4] => 1
[1,1,1,1,0,0,1,0,0,0]
 => [2]
 => [1,1,0,0,1,0]
 => [2,1,3] => 1
[1,1,1,1,0,1,0,0,0,0]
 => [1]
 => [1,0,1,0]
 => [1,2] => 1
[1,1,1,1,1,0,0,0,0,0]
 => []
 => []
 => [] => ? = 4
[1,1,0,1,0,1,0,1,0,1,0,0]
 => [4,3,2,1]
 => [1,0,1,0,1,0,1,0,1,0]
 => [1,2,3,4,5] => 4
[1,1,0,1,0,1,0,1,1,0,0,0]
 => [3,3,2,1]
 => [1,0,1,0,1,0,1,1,0,0]
 => [1,2,3,5,4] => 3
[1,1,0,1,0,1,1,0,0,1,0,0]
 => [4,2,2,1]
 => [1,0,1,0,1,1,0,0,1,0]
 => [1,2,4,3,5] => 3
[1,1,0,1,0,1,1,0,1,0,0,0]
 => [3,2,2,1]
 => [1,0,1,0,1,1,0,1,0,0]
 => [1,2,4,5,3] => 2
[1,1,0,1,0,1,1,1,0,0,0,0]
 => [2,2,2,1]
 => [1,0,1,0,1,1,1,0,0,0]
 => [1,2,5,3,4] => 2
[1,1,0,1,1,0,0,1,0,1,0,0]
 => [4,3,1,1]
 => [1,0,1,1,0,0,1,0,1,0]
 => [1,3,2,4,5] => 3
[1,1,0,1,1,0,0,1,1,0,0,0]
 => [3,3,1,1]
 => [1,0,1,1,0,0,1,1,0,0]
 => [1,3,2,5,4] => 2
[1,1,0,1,1,0,1,0,0,1,0,0]
 => [4,2,1,1]
 => [1,0,1,1,0,1,0,0,1,0]
 => [1,3,4,2,5] => 2
[1,1,0,1,1,0,1,0,1,0,0,0]
 => [3,2,1,1]
 => [1,0,1,1,0,1,0,1,0,0]
 => [1,3,4,5,2] => 1
[1,1,0,1,1,0,1,1,0,0,0,0]
 => [2,2,1,1]
 => [1,0,1,1,0,1,1,0,0,0]
 => [1,3,5,2,4] => 1
[1,1,0,1,1,1,0,0,0,1,0,0]
 => [4,1,1,1]
 => [1,0,1,1,1,0,0,0,1,0]
 => [1,4,2,3,5] => 2
[1,1,0,1,1,1,0,0,1,0,0,0]
 => [3,1,1,1]
 => [1,0,1,1,1,0,0,1,0,0]
 => [1,4,2,5,3] => 1
[1,1,0,1,1,1,0,1,0,0,0,0]
 => [2,1,1,1]
 => [1,0,1,1,1,0,1,0,0,0]
 => [1,4,5,2,3] => 1
[1,1,0,1,1,1,1,0,0,0,0,0]
 => [1,1,1,1]
 => [1,0,1,1,1,1,0,0,0,0]
 => [1,5,2,3,4] => 1
[1,1,1,0,0,1,0,1,0,1,0,0]
 => [4,3,2]
 => [1,1,0,0,1,0,1,0,1,0]
 => [2,1,3,4,5] => 3
[1,1,1,0,0,1,0,1,1,0,0,0]
 => [3,3,2]
 => [1,1,0,0,1,0,1,1,0,0]
 => [2,1,3,5,4] => 2
[1,1,1,0,0,1,1,0,0,1,0,0]
 => [4,2,2]
 => [1,1,0,0,1,1,0,0,1,0]
 => [2,1,4,3,5] => 2
[1,1,1,0,0,1,1,0,1,0,0,0]
 => [3,2,2]
 => [1,1,0,0,1,1,0,1,0,0]
 => [2,1,4,5,3] => 1
[1,1,1,0,0,1,1,1,0,0,0,0]
 => [2,2,2]
 => [1,1,0,0,1,1,1,0,0,0]
 => [2,1,5,3,4] => 1
[1,1,1,0,1,0,0,1,0,1,0,0]
 => [4,3,1]
 => [1,1,0,1,0,0,1,0,1,0]
 => [2,3,1,4,5] => 2
[1,1,1,0,1,0,0,1,1,0,0,0]
 => [3,3,1]
 => [1,1,0,1,0,0,1,1,0,0]
 => [2,3,1,5,4] => 1
[1,1,1,0,1,0,1,0,0,1,0,0]
 => [4,2,1]
 => [1,1,0,1,0,1,0,0,1,0]
 => [2,3,4,1,5] => 1
[1,1,1,0,1,0,1,0,1,0,0,0]
 => [3,2,1]
 => [1,0,1,0,1,0,1,0]
 => [1,2,3,4] => 3
[1,1,1,0,1,0,1,1,0,0,0,0]
 => [2,2,1]
 => [1,0,1,0,1,1,0,0]
 => [1,2,4,3] => 2
[1,1,1,0,1,1,0,0,0,1,0,0]
 => [4,1,1]
 => [1,1,0,1,1,0,0,0,1,0]
 => [2,4,1,3,5] => 1
[1,1,1,0,1,1,0,0,1,0,0,0]
 => [3,1,1]
 => [1,0,1,1,0,0,1,0]
 => [1,3,2,4] => 2
[1,1,1,0,1,1,0,1,0,0,0,0]
 => [2,1,1]
 => [1,0,1,1,0,1,0,0]
 => [1,3,4,2] => 1
[1,1,1,0,1,1,1,0,0,0,0,0]
 => [1,1,1]
 => [1,0,1,1,1,0,0,0]
 => [1,4,2,3] => 1
[1,1,1,1,0,0,0,1,0,1,0,0]
 => [4,3]
 => [1,1,1,0,0,0,1,0,1,0]
 => [3,1,2,4,5] => 2
[1,1,1,1,0,0,0,1,1,0,0,0]
 => [3,3]
 => [1,1,1,0,0,0,1,1,0,0]
 => [3,1,2,5,4] => 1
[1,1,1,1,0,0,1,0,0,1,0,0]
 => [4,2]
 => [1,1,1,0,0,1,0,0,1,0]
 => [3,1,4,2,5] => 1
[1,1,1,1,0,0,1,0,1,0,0,0]
 => [3,2]
 => [1,1,0,0,1,0,1,0]
 => [2,1,3,4] => 2
[1,1,1,1,1,1,0,0,0,0,0,0]
 => []
 => []
 => [] => ? = 5
[1,1,1,1,1,1,1,0,0,0,0,0,0,0]
 => []
 => []
 => [] => ? = 6
Description
The number of global ascents of a permutation. The global ascents are the integers $i$ such that $$C(\pi)=\{i\in [n-1] \mid \forall 1 \leq j \leq i < k \leq n: \pi(j) < \pi(k)\}.$$ Equivalently, by the pigeonhole principle, $$C(\pi)=\{i\in [n-1] \mid \forall 1 \leq j \leq i: \pi(j) \leq i \}.$$ For $n > 1$ it can also be described as an occurrence of the mesh pattern $$([1,2], \{(0,2),(1,0),(1,1),(2,0),(2,1) \})$$ or equivalently $$([1,2], \{(0,1),(0,2),(1,1),(1,2),(2,0) \}),$$ see [3]. According to [2], this is also the cardinality of the connectivity set of a permutation. The permutation is connected, when the connectivity set is empty. This gives [[oeis:A003319]].
Matching statistic: St000056
(load all 3 compositions to match this statistic)
Mapping
Mp00027: Dyck paths —to partition⟶ Integer partitions
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00119: Dyck paths —to 321-avoiding permutation (Krattenthaler)⟶ Permutations
St000056: Permutations ⟶ ℤResult quality: 83% ●values known / values provided: 96%●distinct values known / distinct values provided: 83%
Values
[1,0]
 => []
 => []
 => [] => ? = 1 + 1
[1,1,0,0]
 => []
 => []
 => [] => ? = 1 + 1
[1,1,0,1,0,0]
 => [1]
 => [1,0,1,0]
 => [1,2] => 2 = 1 + 1
[1,1,1,0,0,0]
 => []
 => []
 => [] => ? = 2 + 1
[1,1,0,1,0,1,0,0]
 => [2,1]
 => [1,0,1,0,1,0]
 => [1,2,3] => 3 = 2 + 1
[1,1,0,1,1,0,0,0]
 => [1,1]
 => [1,0,1,1,0,0]
 => [1,3,2] => 2 = 1 + 1
[1,1,1,0,0,1,0,0]
 => [2]
 => [1,1,0,0,1,0]
 => [2,1,3] => 2 = 1 + 1
[1,1,1,0,1,0,0,0]
 => [1]
 => [1,0,1,0]
 => [1,2] => 2 = 1 + 1
[1,1,1,1,0,0,0,0]
 => []
 => []
 => [] => ? = 3 + 1
[1,1,0,1,0,1,0,1,0,0]
 => [3,2,1]
 => [1,0,1,0,1,0,1,0]
 => [1,2,3,4] => 4 = 3 + 1
[1,1,0,1,0,1,1,0,0,0]
 => [2,2,1]
 => [1,0,1,0,1,1,0,0]
 => [1,2,4,3] => 3 = 2 + 1
[1,1,0,1,1,0,0,1,0,0]
 => [3,1,1]
 => [1,0,1,1,0,0,1,0]
 => [1,3,2,4] => 3 = 2 + 1
[1,1,0,1,1,0,1,0,0,0]
 => [2,1,1]
 => [1,0,1,1,0,1,0,0]
 => [1,3,4,2] => 2 = 1 + 1
[1,1,0,1,1,1,0,0,0,0]
 => [1,1,1]
 => [1,0,1,1,1,0,0,0]
 => [1,4,2,3] => 2 = 1 + 1
[1,1,1,0,0,1,0,1,0,0]
 => [3,2]
 => [1,1,0,0,1,0,1,0]
 => [2,1,3,4] => 3 = 2 + 1
[1,1,1,0,0,1,1,0,0,0]
 => [2,2]
 => [1,1,0,0,1,1,0,0]
 => [2,1,4,3] => 2 = 1 + 1
[1,1,1,0,1,0,0,1,0,0]
 => [3,1]
 => [1,1,0,1,0,0,1,0]
 => [2,3,1,4] => 2 = 1 + 1
[1,1,1,0,1,0,1,0,0,0]
 => [2,1]
 => [1,0,1,0,1,0]
 => [1,2,3] => 3 = 2 + 1
[1,1,1,0,1,1,0,0,0,0]
 => [1,1]
 => [1,0,1,1,0,0]
 => [1,3,2] => 2 = 1 + 1
[1,1,1,1,0,0,0,1,0,0]
 => [3]
 => [1,1,1,0,0,0,1,0]
 => [3,1,2,4] => 2 = 1 + 1
[1,1,1,1,0,0,1,0,0,0]
 => [2]
 => [1,1,0,0,1,0]
 => [2,1,3] => 2 = 1 + 1
[1,1,1,1,0,1,0,0,0,0]
 => [1]
 => [1,0,1,0]
 => [1,2] => 2 = 1 + 1
[1,1,1,1,1,0,0,0,0,0]
 => []
 => []
 => [] => ? = 4 + 1
[1,1,0,1,0,1,0,1,0,1,0,0]
 => [4,3,2,1]
 => [1,0,1,0,1,0,1,0,1,0]
 => [1,2,3,4,5] => 5 = 4 + 1
[1,1,0,1,0,1,0,1,1,0,0,0]
 => [3,3,2,1]
 => [1,0,1,0,1,0,1,1,0,0]
 => [1,2,3,5,4] => 4 = 3 + 1
[1,1,0,1,0,1,1,0,0,1,0,0]
 => [4,2,2,1]
 => [1,0,1,0,1,1,0,0,1,0]
 => [1,2,4,3,5] => 4 = 3 + 1
[1,1,0,1,0,1,1,0,1,0,0,0]
 => [3,2,2,1]
 => [1,0,1,0,1,1,0,1,0,0]
 => [1,2,4,5,3] => 3 = 2 + 1
[1,1,0,1,0,1,1,1,0,0,0,0]
 => [2,2,2,1]
 => [1,0,1,0,1,1,1,0,0,0]
 => [1,2,5,3,4] => 3 = 2 + 1
[1,1,0,1,1,0,0,1,0,1,0,0]
 => [4,3,1,1]
 => [1,0,1,1,0,0,1,0,1,0]
 => [1,3,2,4,5] => 4 = 3 + 1
[1,1,0,1,1,0,0,1,1,0,0,0]
 => [3,3,1,1]
 => [1,0,1,1,0,0,1,1,0,0]
 => [1,3,2,5,4] => 3 = 2 + 1
[1,1,0,1,1,0,1,0,0,1,0,0]
 => [4,2,1,1]
 => [1,0,1,1,0,1,0,0,1,0]
 => [1,3,4,2,5] => 3 = 2 + 1
[1,1,0,1,1,0,1,0,1,0,0,0]
 => [3,2,1,1]
 => [1,0,1,1,0,1,0,1,0,0]
 => [1,3,4,5,2] => 2 = 1 + 1
[1,1,0,1,1,0,1,1,0,0,0,0]
 => [2,2,1,1]
 => [1,0,1,1,0,1,1,0,0,0]
 => [1,3,5,2,4] => 2 = 1 + 1
[1,1,0,1,1,1,0,0,0,1,0,0]
 => [4,1,1,1]
 => [1,0,1,1,1,0,0,0,1,0]
 => [1,4,2,3,5] => 3 = 2 + 1
[1,1,0,1,1,1,0,0,1,0,0,0]
 => [3,1,1,1]
 => [1,0,1,1,1,0,0,1,0,0]
 => [1,4,2,5,3] => 2 = 1 + 1
[1,1,0,1,1,1,0,1,0,0,0,0]
 => [2,1,1,1]
 => [1,0,1,1,1,0,1,0,0,0]
 => [1,4,5,2,3] => 2 = 1 + 1
[1,1,0,1,1,1,1,0,0,0,0,0]
 => [1,1,1,1]
 => [1,0,1,1,1,1,0,0,0,0]
 => [1,5,2,3,4] => 2 = 1 + 1
[1,1,1,0,0,1,0,1,0,1,0,0]
 => [4,3,2]
 => [1,1,0,0,1,0,1,0,1,0]
 => [2,1,3,4,5] => 4 = 3 + 1
[1,1,1,0,0,1,0,1,1,0,0,0]
 => [3,3,2]
 => [1,1,0,0,1,0,1,1,0,0]
 => [2,1,3,5,4] => 3 = 2 + 1
[1,1,1,0,0,1,1,0,0,1,0,0]
 => [4,2,2]
 => [1,1,0,0,1,1,0,0,1,0]
 => [2,1,4,3,5] => 3 = 2 + 1
[1,1,1,0,0,1,1,0,1,0,0,0]
 => [3,2,2]
 => [1,1,0,0,1,1,0,1,0,0]
 => [2,1,4,5,3] => 2 = 1 + 1
[1,1,1,0,0,1,1,1,0,0,0,0]
 => [2,2,2]
 => [1,1,0,0,1,1,1,0,0,0]
 => [2,1,5,3,4] => 2 = 1 + 1
[1,1,1,0,1,0,0,1,0,1,0,0]
 => [4,3,1]
 => [1,1,0,1,0,0,1,0,1,0]
 => [2,3,1,4,5] => 3 = 2 + 1
[1,1,1,0,1,0,0,1,1,0,0,0]
 => [3,3,1]
 => [1,1,0,1,0,0,1,1,0,0]
 => [2,3,1,5,4] => 2 = 1 + 1
[1,1,1,0,1,0,1,0,0,1,0,0]
 => [4,2,1]
 => [1,1,0,1,0,1,0,0,1,0]
 => [2,3,4,1,5] => 2 = 1 + 1
[1,1,1,0,1,0,1,0,1,0,0,0]
 => [3,2,1]
 => [1,0,1,0,1,0,1,0]
 => [1,2,3,4] => 4 = 3 + 1
[1,1,1,0,1,0,1,1,0,0,0,0]
 => [2,2,1]
 => [1,0,1,0,1,1,0,0]
 => [1,2,4,3] => 3 = 2 + 1
[1,1,1,0,1,1,0,0,0,1,0,0]
 => [4,1,1]
 => [1,1,0,1,1,0,0,0,1,0]
 => [2,4,1,3,5] => 2 = 1 + 1
[1,1,1,0,1,1,0,0,1,0,0,0]
 => [3,1,1]
 => [1,0,1,1,0,0,1,0]
 => [1,3,2,4] => 3 = 2 + 1
[1,1,1,0,1,1,0,1,0,0,0,0]
 => [2,1,1]
 => [1,0,1,1,0,1,0,0]
 => [1,3,4,2] => 2 = 1 + 1
[1,1,1,0,1,1,1,0,0,0,0,0]
 => [1,1,1]
 => [1,0,1,1,1,0,0,0]
 => [1,4,2,3] => 2 = 1 + 1
[1,1,1,1,0,0,0,1,0,1,0,0]
 => [4,3]
 => [1,1,1,0,0,0,1,0,1,0]
 => [3,1,2,4,5] => 3 = 2 + 1
[1,1,1,1,0,0,0,1,1,0,0,0]
 => [3,3]
 => [1,1,1,0,0,0,1,1,0,0]
 => [3,1,2,5,4] => 2 = 1 + 1
[1,1,1,1,0,0,1,0,0,1,0,0]
 => [4,2]
 => [1,1,1,0,0,1,0,0,1,0]
 => [3,1,4,2,5] => 2 = 1 + 1
[1,1,1,1,0,0,1,0,1,0,0,0]
 => [3,2]
 => [1,1,0,0,1,0,1,0]
 => [2,1,3,4] => 3 = 2 + 1
[1,1,1,1,1,1,0,0,0,0,0,0]
 => []
 => []
 => [] => ? = 5 + 1
[1,1,1,1,1,1,1,0,0,0,0,0,0,0]
 => []
 => []
 => [] => ? = 6 + 1
Description
The decomposition (or block) number of a permutation. For $\pi \in \mathcal{S}_n$, this is given by $$\#\big\{ 1 \leq k \leq n : \{\pi_1,\ldots,\pi_k\} = \{1,\ldots,k\} \big\}.$$ This is also known as the number of connected components [1] or the number of blocks [2] of the permutation, considering it as a direct sum. This is one plus [[St000234]].
Matching statistic: St000061
Mapping
Mp00027: Dyck paths —to partition⟶ Integer partitions
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00029: Dyck paths —to binary tree: left tree, up step, right tree, down step⟶ Binary trees
St000061: Binary trees ⟶ ℤResult quality: 83% ●values known / values provided: 96%●distinct values known / distinct values provided: 83%
Values
[1,0]
 => []
 => []
 => .
 => ? = 1 + 1
[1,1,0,0]
 => []
 => []
 => .
 => ? = 1 + 1
[1,1,0,1,0,0]
 => [1]
 => [1,0,1,0]
 => [[.,.],.]
 => 2 = 1 + 1
[1,1,1,0,0,0]
 => []
 => []
 => .
 => ? = 2 + 1
[1,1,0,1,0,1,0,0]
 => [2,1]
 => [1,0,1,0,1,0]
 => [[[.,.],.],.]
 => 3 = 2 + 1
[1,1,0,1,1,0,0,0]
 => [1,1]
 => [1,0,1,1,0,0]
 => [[.,.],[.,.]]
 => 2 = 1 + 1
[1,1,1,0,0,1,0,0]
 => [2]
 => [1,1,0,0,1,0]
 => [[.,[.,.]],.]
 => 2 = 1 + 1
[1,1,1,0,1,0,0,0]
 => [1]
 => [1,0,1,0]
 => [[.,.],.]
 => 2 = 1 + 1
[1,1,1,1,0,0,0,0]
 => []
 => []
 => .
 => ? = 3 + 1
[1,1,0,1,0,1,0,1,0,0]
 => [3,2,1]
 => [1,0,1,0,1,0,1,0]
 => [[[[.,.],.],.],.]
 => 4 = 3 + 1
[1,1,0,1,0,1,1,0,0,0]
 => [2,2,1]
 => [1,0,1,0,1,1,0,0]
 => [[[.,.],.],[.,.]]
 => 3 = 2 + 1
[1,1,0,1,1,0,0,1,0,0]
 => [3,1,1]
 => [1,0,1,1,0,0,1,0]
 => [[[.,.],[.,.]],.]
 => 3 = 2 + 1
[1,1,0,1,1,0,1,0,0,0]
 => [2,1,1]
 => [1,0,1,1,0,1,0,0]
 => [[.,.],[[.,.],.]]
 => 2 = 1 + 1
[1,1,0,1,1,1,0,0,0,0]
 => [1,1,1]
 => [1,0,1,1,1,0,0,0]
 => [[.,.],[.,[.,.]]]
 => 2 = 1 + 1
[1,1,1,0,0,1,0,1,0,0]
 => [3,2]
 => [1,1,0,0,1,0,1,0]
 => [[[.,[.,.]],.],.]
 => 3 = 2 + 1
[1,1,1,0,0,1,1,0,0,0]
 => [2,2]
 => [1,1,0,0,1,1,0,0]
 => [[.,[.,.]],[.,.]]
 => 2 = 1 + 1
[1,1,1,0,1,0,0,1,0,0]
 => [3,1]
 => [1,1,0,1,0,0,1,0]
 => [[.,[[.,.],.]],.]
 => 2 = 1 + 1
[1,1,1,0,1,0,1,0,0,0]
 => [2,1]
 => [1,0,1,0,1,0]
 => [[[.,.],.],.]
 => 3 = 2 + 1
[1,1,1,0,1,1,0,0,0,0]
 => [1,1]
 => [1,0,1,1,0,0]
 => [[.,.],[.,.]]
 => 2 = 1 + 1
[1,1,1,1,0,0,0,1,0,0]
 => [3]
 => [1,1,1,0,0,0,1,0]
 => [[.,[.,[.,.]]],.]
 => 2 = 1 + 1
[1,1,1,1,0,0,1,0,0,0]
 => [2]
 => [1,1,0,0,1,0]
 => [[.,[.,.]],.]
 => 2 = 1 + 1
[1,1,1,1,0,1,0,0,0,0]
 => [1]
 => [1,0,1,0]
 => [[.,.],.]
 => 2 = 1 + 1
[1,1,1,1,1,0,0,0,0,0]
 => []
 => []
 => .
 => ? = 4 + 1
[1,1,0,1,0,1,0,1,0,1,0,0]
 => [4,3,2,1]
 => [1,0,1,0,1,0,1,0,1,0]
 => [[[[[.,.],.],.],.],.]
 => 5 = 4 + 1
[1,1,0,1,0,1,0,1,1,0,0,0]
 => [3,3,2,1]
 => [1,0,1,0,1,0,1,1,0,0]
 => [[[[.,.],.],.],[.,.]]
 => 4 = 3 + 1
[1,1,0,1,0,1,1,0,0,1,0,0]
 => [4,2,2,1]
 => [1,0,1,0,1,1,0,0,1,0]
 => [[[[.,.],.],[.,.]],.]
 => 4 = 3 + 1
[1,1,0,1,0,1,1,0,1,0,0,0]
 => [3,2,2,1]
 => [1,0,1,0,1,1,0,1,0,0]
 => [[[.,.],.],[[.,.],.]]
 => 3 = 2 + 1
[1,1,0,1,0,1,1,1,0,0,0,0]
 => [2,2,2,1]
 => [1,0,1,0,1,1,1,0,0,0]
 => [[[.,.],.],[.,[.,.]]]
 => 3 = 2 + 1
[1,1,0,1,1,0,0,1,0,1,0,0]
 => [4,3,1,1]
 => [1,0,1,1,0,0,1,0,1,0]
 => [[[[.,.],[.,.]],.],.]
 => 4 = 3 + 1
[1,1,0,1,1,0,0,1,1,0,0,0]
 => [3,3,1,1]
 => [1,0,1,1,0,0,1,1,0,0]
 => [[[.,.],[.,.]],[.,.]]
 => 3 = 2 + 1
[1,1,0,1,1,0,1,0,0,1,0,0]
 => [4,2,1,1]
 => [1,0,1,1,0,1,0,0,1,0]
 => [[[.,.],[[.,.],.]],.]
 => 3 = 2 + 1
[1,1,0,1,1,0,1,0,1,0,0,0]
 => [3,2,1,1]
 => [1,0,1,1,0,1,0,1,0,0]
 => [[.,.],[[[.,.],.],.]]
 => 2 = 1 + 1
[1,1,0,1,1,0,1,1,0,0,0,0]
 => [2,2,1,1]
 => [1,0,1,1,0,1,1,0,0,0]
 => [[.,.],[[.,.],[.,.]]]
 => 2 = 1 + 1
[1,1,0,1,1,1,0,0,0,1,0,0]
 => [4,1,1,1]
 => [1,0,1,1,1,0,0,0,1,0]
 => [[[.,.],[.,[.,.]]],.]
 => 3 = 2 + 1
[1,1,0,1,1,1,0,0,1,0,0,0]
 => [3,1,1,1]
 => [1,0,1,1,1,0,0,1,0,0]
 => [[.,.],[[.,[.,.]],.]]
 => 2 = 1 + 1
[1,1,0,1,1,1,0,1,0,0,0,0]
 => [2,1,1,1]
 => [1,0,1,1,1,0,1,0,0,0]
 => [[.,.],[.,[[.,.],.]]]
 => 2 = 1 + 1
[1,1,0,1,1,1,1,0,0,0,0,0]
 => [1,1,1,1]
 => [1,0,1,1,1,1,0,0,0,0]
 => [[.,.],[.,[.,[.,.]]]]
 => 2 = 1 + 1
[1,1,1,0,0,1,0,1,0,1,0,0]
 => [4,3,2]
 => [1,1,0,0,1,0,1,0,1,0]
 => [[[[.,[.,.]],.],.],.]
 => 4 = 3 + 1
[1,1,1,0,0,1,0,1,1,0,0,0]
 => [3,3,2]
 => [1,1,0,0,1,0,1,1,0,0]
 => [[[.,[.,.]],.],[.,.]]
 => 3 = 2 + 1
[1,1,1,0,0,1,1,0,0,1,0,0]
 => [4,2,2]
 => [1,1,0,0,1,1,0,0,1,0]
 => [[[.,[.,.]],[.,.]],.]
 => 3 = 2 + 1
[1,1,1,0,0,1,1,0,1,0,0,0]
 => [3,2,2]
 => [1,1,0,0,1,1,0,1,0,0]
 => [[.,[.,.]],[[.,.],.]]
 => 2 = 1 + 1
[1,1,1,0,0,1,1,1,0,0,0,0]
 => [2,2,2]
 => [1,1,0,0,1,1,1,0,0,0]
 => [[.,[.,.]],[.,[.,.]]]
 => 2 = 1 + 1
[1,1,1,0,1,0,0,1,0,1,0,0]
 => [4,3,1]
 => [1,1,0,1,0,0,1,0,1,0]
 => [[[.,[[.,.],.]],.],.]
 => 3 = 2 + 1
[1,1,1,0,1,0,0,1,1,0,0,0]
 => [3,3,1]
 => [1,1,0,1,0,0,1,1,0,0]
 => [[.,[[.,.],.]],[.,.]]
 => 2 = 1 + 1
[1,1,1,0,1,0,1,0,0,1,0,0]
 => [4,2,1]
 => [1,1,0,1,0,1,0,0,1,0]
 => [[.,[[[.,.],.],.]],.]
 => 2 = 1 + 1
[1,1,1,0,1,0,1,0,1,0,0,0]
 => [3,2,1]
 => [1,0,1,0,1,0,1,0]
 => [[[[.,.],.],.],.]
 => 4 = 3 + 1
[1,1,1,0,1,0,1,1,0,0,0,0]
 => [2,2,1]
 => [1,0,1,0,1,1,0,0]
 => [[[.,.],.],[.,.]]
 => 3 = 2 + 1
[1,1,1,0,1,1,0,0,0,1,0,0]
 => [4,1,1]
 => [1,1,0,1,1,0,0,0,1,0]
 => [[.,[[.,.],[.,.]]],.]
 => 2 = 1 + 1
[1,1,1,0,1,1,0,0,1,0,0,0]
 => [3,1,1]
 => [1,0,1,1,0,0,1,0]
 => [[[.,.],[.,.]],.]
 => 3 = 2 + 1
[1,1,1,0,1,1,0,1,0,0,0,0]
 => [2,1,1]
 => [1,0,1,1,0,1,0,0]
 => [[.,.],[[.,.],.]]
 => 2 = 1 + 1
[1,1,1,0,1,1,1,0,0,0,0,0]
 => [1,1,1]
 => [1,0,1,1,1,0,0,0]
 => [[.,.],[.,[.,.]]]
 => 2 = 1 + 1
[1,1,1,1,0,0,0,1,0,1,0,0]
 => [4,3]
 => [1,1,1,0,0,0,1,0,1,0]
 => [[[.,[.,[.,.]]],.],.]
 => 3 = 2 + 1
[1,1,1,1,0,0,0,1,1,0,0,0]
 => [3,3]
 => [1,1,1,0,0,0,1,1,0,0]
 => [[.,[.,[.,.]]],[.,.]]
 => 2 = 1 + 1
[1,1,1,1,0,0,1,0,0,1,0,0]
 => [4,2]
 => [1,1,1,0,0,1,0,0,1,0]
 => [[.,[[.,[.,.]],.]],.]
 => 2 = 1 + 1
[1,1,1,1,0,0,1,0,1,0,0,0]
 => [3,2]
 => [1,1,0,0,1,0,1,0]
 => [[[.,[.,.]],.],.]
 => 3 = 2 + 1
[1,1,1,1,1,1,0,0,0,0,0,0]
 => []
 => []
 => .
 => ? = 5 + 1
[1,1,1,1,1,1,1,0,0,0,0,0,0,0]
 => []
 => []
 => .
 => ? = 6 + 1
Description
The number of nodes on the left branch of a binary tree. Also corresponds to [[/StatisticsDatabase/St000011/|ST000011]] after applying the [[/BinaryTrees#Maps|Tamari bijection]] between binary trees and Dyck path.
Matching statistic: St000084
Mapping
Mp00027: Dyck paths —to partition⟶ Integer partitions
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00026: Dyck paths —to ordered tree⟶ Ordered trees
St000084: Ordered trees ⟶ ℤResult quality: 83% ●values known / values provided: 96%●distinct values known / distinct values provided: 83%
Values
[1,0]
 => []
 => []
 => []
 => ? = 1 + 1
[1,1,0,0]
 => []
 => []
 => []
 => ? = 1 + 1
[1,1,0,1,0,0]
 => [1]
 => [1,0,1,0]
 => [[],[]]
 => 2 = 1 + 1
[1,1,1,0,0,0]
 => []
 => []
 => []
 => ? = 2 + 1
[1,1,0,1,0,1,0,0]
 => [2,1]
 => [1,0,1,0,1,0]
 => [[],[],[]]
 => 3 = 2 + 1
[1,1,0,1,1,0,0,0]
 => [1,1]
 => [1,0,1,1,0,0]
 => [[],[[]]]
 => 2 = 1 + 1
[1,1,1,0,0,1,0,0]
 => [2]
 => [1,1,0,0,1,0]
 => [[[]],[]]
 => 2 = 1 + 1
[1,1,1,0,1,0,0,0]
 => [1]
 => [1,0,1,0]
 => [[],[]]
 => 2 = 1 + 1
[1,1,1,1,0,0,0,0]
 => []
 => []
 => []
 => ? = 3 + 1
[1,1,0,1,0,1,0,1,0,0]
 => [3,2,1]
 => [1,0,1,0,1,0,1,0]
 => [[],[],[],[]]
 => 4 = 3 + 1
[1,1,0,1,0,1,1,0,0,0]
 => [2,2,1]
 => [1,0,1,0,1,1,0,0]
 => [[],[],[[]]]
 => 3 = 2 + 1
[1,1,0,1,1,0,0,1,0,0]
 => [3,1,1]
 => [1,0,1,1,0,0,1,0]
 => [[],[[]],[]]
 => 3 = 2 + 1
[1,1,0,1,1,0,1,0,0,0]
 => [2,1,1]
 => [1,0,1,1,0,1,0,0]
 => [[],[[],[]]]
 => 2 = 1 + 1
[1,1,0,1,1,1,0,0,0,0]
 => [1,1,1]
 => [1,0,1,1,1,0,0,0]
 => [[],[[[]]]]
 => 2 = 1 + 1
[1,1,1,0,0,1,0,1,0,0]
 => [3,2]
 => [1,1,0,0,1,0,1,0]
 => [[[]],[],[]]
 => 3 = 2 + 1
[1,1,1,0,0,1,1,0,0,0]
 => [2,2]
 => [1,1,0,0,1,1,0,0]
 => [[[]],[[]]]
 => 2 = 1 + 1
[1,1,1,0,1,0,0,1,0,0]
 => [3,1]
 => [1,1,0,1,0,0,1,0]
 => [[[],[]],[]]
 => 2 = 1 + 1
[1,1,1,0,1,0,1,0,0,0]
 => [2,1]
 => [1,0,1,0,1,0]
 => [[],[],[]]
 => 3 = 2 + 1
[1,1,1,0,1,1,0,0,0,0]
 => [1,1]
 => [1,0,1,1,0,0]
 => [[],[[]]]
 => 2 = 1 + 1
[1,1,1,1,0,0,0,1,0,0]
 => [3]
 => [1,1,1,0,0,0,1,0]
 => [[[[]]],[]]
 => 2 = 1 + 1
[1,1,1,1,0,0,1,0,0,0]
 => [2]
 => [1,1,0,0,1,0]
 => [[[]],[]]
 => 2 = 1 + 1
[1,1,1,1,0,1,0,0,0,0]
 => [1]
 => [1,0,1,0]
 => [[],[]]
 => 2 = 1 + 1
[1,1,1,1,1,0,0,0,0,0]
 => []
 => []
 => []
 => ? = 4 + 1
[1,1,0,1,0,1,0,1,0,1,0,0]
 => [4,3,2,1]
 => [1,0,1,0,1,0,1,0,1,0]
 => [[],[],[],[],[]]
 => 5 = 4 + 1
[1,1,0,1,0,1,0,1,1,0,0,0]
 => [3,3,2,1]
 => [1,0,1,0,1,0,1,1,0,0]
 => [[],[],[],[[]]]
 => 4 = 3 + 1
[1,1,0,1,0,1,1,0,0,1,0,0]
 => [4,2,2,1]
 => [1,0,1,0,1,1,0,0,1,0]
 => [[],[],[[]],[]]
 => 4 = 3 + 1
[1,1,0,1,0,1,1,0,1,0,0,0]
 => [3,2,2,1]
 => [1,0,1,0,1,1,0,1,0,0]
 => [[],[],[[],[]]]
 => 3 = 2 + 1
[1,1,0,1,0,1,1,1,0,0,0,0]
 => [2,2,2,1]
 => [1,0,1,0,1,1,1,0,0,0]
 => [[],[],[[[]]]]
 => 3 = 2 + 1
[1,1,0,1,1,0,0,1,0,1,0,0]
 => [4,3,1,1]
 => [1,0,1,1,0,0,1,0,1,0]
 => [[],[[]],[],[]]
 => 4 = 3 + 1
[1,1,0,1,1,0,0,1,1,0,0,0]
 => [3,3,1,1]
 => [1,0,1,1,0,0,1,1,0,0]
 => [[],[[]],[[]]]
 => 3 = 2 + 1
[1,1,0,1,1,0,1,0,0,1,0,0]
 => [4,2,1,1]
 => [1,0,1,1,0,1,0,0,1,0]
 => [[],[[],[]],[]]
 => 3 = 2 + 1
[1,1,0,1,1,0,1,0,1,0,0,0]
 => [3,2,1,1]
 => [1,0,1,1,0,1,0,1,0,0]
 => [[],[[],[],[]]]
 => 2 = 1 + 1
[1,1,0,1,1,0,1,1,0,0,0,0]
 => [2,2,1,1]
 => [1,0,1,1,0,1,1,0,0,0]
 => [[],[[],[[]]]]
 => 2 = 1 + 1
[1,1,0,1,1,1,0,0,0,1,0,0]
 => [4,1,1,1]
 => [1,0,1,1,1,0,0,0,1,0]
 => [[],[[[]]],[]]
 => 3 = 2 + 1
[1,1,0,1,1,1,0,0,1,0,0,0]
 => [3,1,1,1]
 => [1,0,1,1,1,0,0,1,0,0]
 => [[],[[[]],[]]]
 => 2 = 1 + 1
[1,1,0,1,1,1,0,1,0,0,0,0]
 => [2,1,1,1]
 => [1,0,1,1,1,0,1,0,0,0]
 => [[],[[[],[]]]]
 => 2 = 1 + 1
[1,1,0,1,1,1,1,0,0,0,0,0]
 => [1,1,1,1]
 => [1,0,1,1,1,1,0,0,0,0]
 => [[],[[[[]]]]]
 => 2 = 1 + 1
[1,1,1,0,0,1,0,1,0,1,0,0]
 => [4,3,2]
 => [1,1,0,0,1,0,1,0,1,0]
 => [[[]],[],[],[]]
 => 4 = 3 + 1
[1,1,1,0,0,1,0,1,1,0,0,0]
 => [3,3,2]
 => [1,1,0,0,1,0,1,1,0,0]
 => [[[]],[],[[]]]
 => 3 = 2 + 1
[1,1,1,0,0,1,1,0,0,1,0,0]
 => [4,2,2]
 => [1,1,0,0,1,1,0,0,1,0]
 => [[[]],[[]],[]]
 => 3 = 2 + 1
[1,1,1,0,0,1,1,0,1,0,0,0]
 => [3,2,2]
 => [1,1,0,0,1,1,0,1,0,0]
 => [[[]],[[],[]]]
 => 2 = 1 + 1
[1,1,1,0,0,1,1,1,0,0,0,0]
 => [2,2,2]
 => [1,1,0,0,1,1,1,0,0,0]
 => [[[]],[[[]]]]
 => 2 = 1 + 1
[1,1,1,0,1,0,0,1,0,1,0,0]
 => [4,3,1]
 => [1,1,0,1,0,0,1,0,1,0]
 => [[[],[]],[],[]]
 => 3 = 2 + 1
[1,1,1,0,1,0,0,1,1,0,0,0]
 => [3,3,1]
 => [1,1,0,1,0,0,1,1,0,0]
 => [[[],[]],[[]]]
 => 2 = 1 + 1
[1,1,1,0,1,0,1,0,0,1,0,0]
 => [4,2,1]
 => [1,1,0,1,0,1,0,0,1,0]
 => [[[],[],[]],[]]
 => 2 = 1 + 1
[1,1,1,0,1,0,1,0,1,0,0,0]
 => [3,2,1]
 => [1,0,1,0,1,0,1,0]
 => [[],[],[],[]]
 => 4 = 3 + 1
[1,1,1,0,1,0,1,1,0,0,0,0]
 => [2,2,1]
 => [1,0,1,0,1,1,0,0]
 => [[],[],[[]]]
 => 3 = 2 + 1
[1,1,1,0,1,1,0,0,0,1,0,0]
 => [4,1,1]
 => [1,1,0,1,1,0,0,0,1,0]
 => [[[],[[]]],[]]
 => 2 = 1 + 1
[1,1,1,0,1,1,0,0,1,0,0,0]
 => [3,1,1]
 => [1,0,1,1,0,0,1,0]
 => [[],[[]],[]]
 => 3 = 2 + 1
[1,1,1,0,1,1,0,1,0,0,0,0]
 => [2,1,1]
 => [1,0,1,1,0,1,0,0]
 => [[],[[],[]]]
 => 2 = 1 + 1
[1,1,1,0,1,1,1,0,0,0,0,0]
 => [1,1,1]
 => [1,0,1,1,1,0,0,0]
 => [[],[[[]]]]
 => 2 = 1 + 1
[1,1,1,1,0,0,0,1,0,1,0,0]
 => [4,3]
 => [1,1,1,0,0,0,1,0,1,0]
 => [[[[]]],[],[]]
 => 3 = 2 + 1
[1,1,1,1,0,0,0,1,1,0,0,0]
 => [3,3]
 => [1,1,1,0,0,0,1,1,0,0]
 => [[[[]]],[[]]]
 => 2 = 1 + 1
[1,1,1,1,0,0,1,0,0,1,0,0]
 => [4,2]
 => [1,1,1,0,0,1,0,0,1,0]
 => [[[[]],[]],[]]
 => 2 = 1 + 1
[1,1,1,1,0,0,1,0,1,0,0,0]
 => [3,2]
 => [1,1,0,0,1,0,1,0]
 => [[[]],[],[]]
 => 3 = 2 + 1
[1,1,1,1,1,1,0,0,0,0,0,0]
 => []
 => []
 => []
 => ? = 5 + 1
[1,1,1,1,1,1,1,0,0,0,0,0,0,0]
 => []
 => []
 => []
 => ? = 6 + 1
Description
The number of subtrees.
Matching statistic: St000675
Mapping
Mp00027: Dyck paths —to partition⟶ Integer partitions
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00296: Dyck paths —Knuth-Krattenthaler⟶ Dyck paths
St000675: Dyck paths ⟶ ℤResult quality: 83% ●values known / values provided: 96%●distinct values known / distinct values provided: 83%
Values
[1,0]
 => []
 => []
 => []
 => ? = 1 + 1
[1,1,0,0]
 => []
 => []
 => []
 => ? = 1 + 1
[1,1,0,1,0,0]
 => [1]
 => [1,0,1,0]
 => [1,1,0,0]
 => 2 = 1 + 1
[1,1,1,0,0,0]
 => []
 => []
 => []
 => ? = 2 + 1
[1,1,0,1,0,1,0,0]
 => [2,1]
 => [1,0,1,0,1,0]
 => [1,1,1,0,0,0]
 => 3 = 2 + 1
[1,1,0,1,1,0,0,0]
 => [1,1]
 => [1,0,1,1,0,0]
 => [1,1,0,1,0,0]
 => 2 = 1 + 1
[1,1,1,0,0,1,0,0]
 => [2]
 => [1,1,0,0,1,0]
 => [1,0,1,0,1,0]
 => 2 = 1 + 1
[1,1,1,0,1,0,0,0]
 => [1]
 => [1,0,1,0]
 => [1,1,0,0]
 => 2 = 1 + 1
[1,1,1,1,0,0,0,0]
 => []
 => []
 => []
 => ? = 3 + 1
[1,1,0,1,0,1,0,1,0,0]
 => [3,2,1]
 => [1,0,1,0,1,0,1,0]
 => [1,1,1,1,0,0,0,0]
 => 4 = 3 + 1
[1,1,0,1,0,1,1,0,0,0]
 => [2,2,1]
 => [1,0,1,0,1,1,0,0]
 => [1,1,1,0,1,0,0,0]
 => 3 = 2 + 1
[1,1,0,1,1,0,0,1,0,0]
 => [3,1,1]
 => [1,0,1,1,0,0,1,0]
 => [1,1,0,1,0,1,0,0]
 => 3 = 2 + 1
[1,1,0,1,1,0,1,0,0,0]
 => [2,1,1]
 => [1,0,1,1,0,1,0,0]
 => [1,1,0,1,1,0,0,0]
 => 2 = 1 + 1
[1,1,0,1,1,1,0,0,0,0]
 => [1,1,1]
 => [1,0,1,1,1,0,0,0]
 => [1,1,1,0,0,1,0,0]
 => 2 = 1 + 1
[1,1,1,0,0,1,0,1,0,0]
 => [3,2]
 => [1,1,0,0,1,0,1,0]
 => [1,0,1,1,0,0,1,0]
 => 3 = 2 + 1
[1,1,1,0,0,1,1,0,0,0]
 => [2,2]
 => [1,1,0,0,1,1,0,0]
 => [1,0,1,0,1,0,1,0]
 => 2 = 1 + 1
[1,1,1,0,1,0,0,1,0,0]
 => [3,1]
 => [1,1,0,1,0,0,1,0]
 => [1,0,1,1,0,1,0,0]
 => 2 = 1 + 1
[1,1,1,0,1,0,1,0,0,0]
 => [2,1]
 => [1,0,1,0,1,0]
 => [1,1,1,0,0,0]
 => 3 = 2 + 1
[1,1,1,0,1,1,0,0,0,0]
 => [1,1]
 => [1,0,1,1,0,0]
 => [1,1,0,1,0,0]
 => 2 = 1 + 1
[1,1,1,1,0,0,0,1,0,0]
 => [3]
 => [1,1,1,0,0,0,1,0]
 => [1,1,0,1,0,0,1,0]
 => 2 = 1 + 1
[1,1,1,1,0,0,1,0,0,0]
 => [2]
 => [1,1,0,0,1,0]
 => [1,0,1,0,1,0]
 => 2 = 1 + 1
[1,1,1,1,0,1,0,0,0,0]
 => [1]
 => [1,0,1,0]
 => [1,1,0,0]
 => 2 = 1 + 1
[1,1,1,1,1,0,0,0,0,0]
 => []
 => []
 => []
 => ? = 4 + 1
[1,1,0,1,0,1,0,1,0,1,0,0]
 => [4,3,2,1]
 => [1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => 5 = 4 + 1
[1,1,0,1,0,1,0,1,1,0,0,0]
 => [3,3,2,1]
 => [1,0,1,0,1,0,1,1,0,0]
 => [1,1,1,1,0,1,0,0,0,0]
 => 4 = 3 + 1
[1,1,0,1,0,1,1,0,0,1,0,0]
 => [4,2,2,1]
 => [1,0,1,0,1,1,0,0,1,0]
 => [1,1,1,0,1,0,1,0,0,0]
 => 4 = 3 + 1
[1,1,0,1,0,1,1,0,1,0,0,0]
 => [3,2,2,1]
 => [1,0,1,0,1,1,0,1,0,0]
 => [1,1,1,0,1,1,0,0,0,0]
 => 3 = 2 + 1
[1,1,0,1,0,1,1,1,0,0,0,0]
 => [2,2,2,1]
 => [1,0,1,0,1,1,1,0,0,0]
 => [1,1,1,1,0,0,1,0,0,0]
 => 3 = 2 + 1
[1,1,0,1,1,0,0,1,0,1,0,0]
 => [4,3,1,1]
 => [1,0,1,1,0,0,1,0,1,0]
 => [1,1,0,1,1,0,0,1,0,0]
 => 4 = 3 + 1
[1,1,0,1,1,0,0,1,1,0,0,0]
 => [3,3,1,1]
 => [1,0,1,1,0,0,1,1,0,0]
 => [1,1,0,1,0,1,0,1,0,0]
 => 3 = 2 + 1
[1,1,0,1,1,0,1,0,0,1,0,0]
 => [4,2,1,1]
 => [1,0,1,1,0,1,0,0,1,0]
 => [1,1,0,1,1,0,1,0,0,0]
 => 3 = 2 + 1
[1,1,0,1,1,0,1,0,1,0,0,0]
 => [3,2,1,1]
 => [1,0,1,1,0,1,0,1,0,0]
 => [1,1,0,1,1,1,0,0,0,0]
 => 2 = 1 + 1
[1,1,0,1,1,0,1,1,0,0,0,0]
 => [2,2,1,1]
 => [1,0,1,1,0,1,1,0,0,0]
 => [1,1,0,1,0,1,1,0,0,0]
 => 2 = 1 + 1
[1,1,0,1,1,1,0,0,0,1,0,0]
 => [4,1,1,1]
 => [1,0,1,1,1,0,0,0,1,0]
 => [1,1,1,0,1,0,0,1,0,0]
 => 3 = 2 + 1
[1,1,0,1,1,1,0,0,1,0,0,0]
 => [3,1,1,1]
 => [1,0,1,1,1,0,0,1,0,0]
 => [1,1,1,0,0,1,0,1,0,0]
 => 2 = 1 + 1
[1,1,0,1,1,1,0,1,0,0,0,0]
 => [2,1,1,1]
 => [1,0,1,1,1,0,1,0,0,0]
 => [1,1,1,0,0,1,1,0,0,0]
 => 2 = 1 + 1
[1,1,0,1,1,1,1,0,0,0,0,0]
 => [1,1,1,1]
 => [1,0,1,1,1,1,0,0,0,0]
 => [1,1,1,1,0,0,0,1,0,0]
 => 2 = 1 + 1
[1,1,1,0,0,1,0,1,0,1,0,0]
 => [4,3,2]
 => [1,1,0,0,1,0,1,0,1,0]
 => [1,0,1,1,1,0,0,0,1,0]
 => 4 = 3 + 1
[1,1,1,0,0,1,0,1,1,0,0,0]
 => [3,3,2]
 => [1,1,0,0,1,0,1,1,0,0]
 => [1,0,1,1,0,1,0,0,1,0]
 => 3 = 2 + 1
[1,1,1,0,0,1,1,0,0,1,0,0]
 => [4,2,2]
 => [1,1,0,0,1,1,0,0,1,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => 3 = 2 + 1
[1,1,1,0,0,1,1,0,1,0,0,0]
 => [3,2,2]
 => [1,1,0,0,1,1,0,1,0,0]
 => [1,0,1,0,1,1,0,0,1,0]
 => 2 = 1 + 1
[1,1,1,0,0,1,1,1,0,0,0,0]
 => [2,2,2]
 => [1,1,0,0,1,1,1,0,0,0]
 => [1,0,1,1,0,0,1,0,1,0]
 => 2 = 1 + 1
[1,1,1,0,1,0,0,1,0,1,0,0]
 => [4,3,1]
 => [1,1,0,1,0,0,1,0,1,0]
 => [1,0,1,1,1,0,0,1,0,0]
 => 3 = 2 + 1
[1,1,1,0,1,0,0,1,1,0,0,0]
 => [3,3,1]
 => [1,1,0,1,0,0,1,1,0,0]
 => [1,0,1,1,0,1,0,1,0,0]
 => 2 = 1 + 1
[1,1,1,0,1,0,1,0,0,1,0,0]
 => [4,2,1]
 => [1,1,0,1,0,1,0,0,1,0]
 => [1,0,1,1,1,0,1,0,0,0]
 => 2 = 1 + 1
[1,1,1,0,1,0,1,0,1,0,0,0]
 => [3,2,1]
 => [1,0,1,0,1,0,1,0]
 => [1,1,1,1,0,0,0,0]
 => 4 = 3 + 1
[1,1,1,0,1,0,1,1,0,0,0,0]
 => [2,2,1]
 => [1,0,1,0,1,1,0,0]
 => [1,1,1,0,1,0,0,0]
 => 3 = 2 + 1
[1,1,1,0,1,1,0,0,0,1,0,0]
 => [4,1,1]
 => [1,1,0,1,1,0,0,0,1,0]
 => [1,0,1,0,1,0,1,1,0,0]
 => 2 = 1 + 1
[1,1,1,0,1,1,0,0,1,0,0,0]
 => [3,1,1]
 => [1,0,1,1,0,0,1,0]
 => [1,1,0,1,0,1,0,0]
 => 3 = 2 + 1
[1,1,1,0,1,1,0,1,0,0,0,0]
 => [2,1,1]
 => [1,0,1,1,0,1,0,0]
 => [1,1,0,1,1,0,0,0]
 => 2 = 1 + 1
[1,1,1,0,1,1,1,0,0,0,0,0]
 => [1,1,1]
 => [1,0,1,1,1,0,0,0]
 => [1,1,1,0,0,1,0,0]
 => 2 = 1 + 1
[1,1,1,1,0,0,0,1,0,1,0,0]
 => [4,3]
 => [1,1,1,0,0,0,1,0,1,0]
 => [1,1,0,1,1,0,0,0,1,0]
 => 3 = 2 + 1
[1,1,1,1,0,0,0,1,1,0,0,0]
 => [3,3]
 => [1,1,1,0,0,0,1,1,0,0]
 => [1,1,0,1,0,1,0,0,1,0]
 => 2 = 1 + 1
[1,1,1,1,0,0,1,0,0,1,0,0]
 => [4,2]
 => [1,1,1,0,0,1,0,0,1,0]
 => [1,1,0,0,1,0,1,0,1,0]
 => 2 = 1 + 1
[1,1,1,1,0,0,1,0,1,0,0,0]
 => [3,2]
 => [1,1,0,0,1,0,1,0]
 => [1,0,1,1,0,0,1,0]
 => 3 = 2 + 1
[1,1,1,1,1,1,0,0,0,0,0,0]
 => []
 => []
 => []
 => ? = 5 + 1
[1,1,1,1,1,1,1,0,0,0,0,0,0,0]
 => []
 => []
 => []
 => ? = 6 + 1
Description
The number of centered multitunnels of a Dyck path. This is the number of factorisations $D = A B C$ of a Dyck path, such that $B$ is a Dyck path and $A$ and $B$ have the same length.
Matching statistic: St000678
(load all 2 compositions to match this statistic)
Mapping
Mp00027: Dyck paths —to partition⟶ Integer partitions
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00132: Dyck paths —switch returns and last double rise⟶ Dyck paths
St000678: Dyck paths ⟶ ℤResult quality: 83% ●values known / values provided: 96%●distinct values known / distinct values provided: 83%
Values
[1,0]
 => []
 => []
 => []
 => ? = 1 + 1
[1,1,0,0]
 => []
 => []
 => []
 => ? = 1 + 1
[1,1,0,1,0,0]
 => [1]
 => [1,0,1,0]
 => [1,0,1,0]
 => 2 = 1 + 1
[1,1,1,0,0,0]
 => []
 => []
 => []
 => ? = 2 + 1
[1,1,0,1,0,1,0,0]
 => [2,1]
 => [1,0,1,0,1,0]
 => [1,0,1,0,1,0]
 => 3 = 2 + 1
[1,1,0,1,1,0,0,0]
 => [1,1]
 => [1,0,1,1,0,0]
 => [1,1,0,1,0,0]
 => 2 = 1 + 1
[1,1,1,0,0,1,0,0]
 => [2]
 => [1,1,0,0,1,0]
 => [1,1,0,0,1,0]
 => 2 = 1 + 1
[1,1,1,0,1,0,0,0]
 => [1]
 => [1,0,1,0]
 => [1,0,1,0]
 => 2 = 1 + 1
[1,1,1,1,0,0,0,0]
 => []
 => []
 => []
 => ? = 3 + 1
[1,1,0,1,0,1,0,1,0,0]
 => [3,2,1]
 => [1,0,1,0,1,0,1,0]
 => [1,0,1,0,1,0,1,0]
 => 4 = 3 + 1
[1,1,0,1,0,1,1,0,0,0]
 => [2,2,1]
 => [1,0,1,0,1,1,0,0]
 => [1,1,0,1,0,1,0,0]
 => 3 = 2 + 1
[1,1,0,1,1,0,0,1,0,0]
 => [3,1,1]
 => [1,0,1,1,0,0,1,0]
 => [1,1,0,1,0,0,1,0]
 => 3 = 2 + 1
[1,1,0,1,1,0,1,0,0,0]
 => [2,1,1]
 => [1,0,1,1,0,1,0,0]
 => [1,0,1,1,0,1,0,0]
 => 2 = 1 + 1
[1,1,0,1,1,1,0,0,0,0]
 => [1,1,1]
 => [1,0,1,1,1,0,0,0]
 => [1,1,1,0,1,0,0,0]
 => 2 = 1 + 1
[1,1,1,0,0,1,0,1,0,0]
 => [3,2]
 => [1,1,0,0,1,0,1,0]
 => [1,1,0,0,1,0,1,0]
 => 3 = 2 + 1
[1,1,1,0,0,1,1,0,0,0]
 => [2,2]
 => [1,1,0,0,1,1,0,0]
 => [1,1,1,0,0,1,0,0]
 => 2 = 1 + 1
[1,1,1,0,1,0,0,1,0,0]
 => [3,1]
 => [1,1,0,1,0,0,1,0]
 => [1,0,1,1,0,0,1,0]
 => 2 = 1 + 1
[1,1,1,0,1,0,1,0,0,0]
 => [2,1]
 => [1,0,1,0,1,0]
 => [1,0,1,0,1,0]
 => 3 = 2 + 1
[1,1,1,0,1,1,0,0,0,0]
 => [1,1]
 => [1,0,1,1,0,0]
 => [1,1,0,1,0,0]
 => 2 = 1 + 1
[1,1,1,1,0,0,0,1,0,0]
 => [3]
 => [1,1,1,0,0,0,1,0]
 => [1,1,1,0,0,0,1,0]
 => 2 = 1 + 1
[1,1,1,1,0,0,1,0,0,0]
 => [2]
 => [1,1,0,0,1,0]
 => [1,1,0,0,1,0]
 => 2 = 1 + 1
[1,1,1,1,0,1,0,0,0,0]
 => [1]
 => [1,0,1,0]
 => [1,0,1,0]
 => 2 = 1 + 1
[1,1,1,1,1,0,0,0,0,0]
 => []
 => []
 => []
 => ? = 4 + 1
[1,1,0,1,0,1,0,1,0,1,0,0]
 => [4,3,2,1]
 => [1,0,1,0,1,0,1,0,1,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => 5 = 4 + 1
[1,1,0,1,0,1,0,1,1,0,0,0]
 => [3,3,2,1]
 => [1,0,1,0,1,0,1,1,0,0]
 => [1,1,0,1,0,1,0,1,0,0]
 => 4 = 3 + 1
[1,1,0,1,0,1,1,0,0,1,0,0]
 => [4,2,2,1]
 => [1,0,1,0,1,1,0,0,1,0]
 => [1,1,0,1,0,1,0,0,1,0]
 => 4 = 3 + 1
[1,1,0,1,0,1,1,0,1,0,0,0]
 => [3,2,2,1]
 => [1,0,1,0,1,1,0,1,0,0]
 => [1,0,1,1,0,1,0,1,0,0]
 => 3 = 2 + 1
[1,1,0,1,0,1,1,1,0,0,0,0]
 => [2,2,2,1]
 => [1,0,1,0,1,1,1,0,0,0]
 => [1,1,1,0,1,0,1,0,0,0]
 => 3 = 2 + 1
[1,1,0,1,1,0,0,1,0,1,0,0]
 => [4,3,1,1]
 => [1,0,1,1,0,0,1,0,1,0]
 => [1,1,0,1,0,0,1,0,1,0]
 => 4 = 3 + 1
[1,1,0,1,1,0,0,1,1,0,0,0]
 => [3,3,1,1]
 => [1,0,1,1,0,0,1,1,0,0]
 => [1,1,1,0,1,0,0,1,0,0]
 => 3 = 2 + 1
[1,1,0,1,1,0,1,0,0,1,0,0]
 => [4,2,1,1]
 => [1,0,1,1,0,1,0,0,1,0]
 => [1,0,1,1,0,1,0,0,1,0]
 => 3 = 2 + 1
[1,1,0,1,1,0,1,0,1,0,0,0]
 => [3,2,1,1]
 => [1,0,1,1,0,1,0,1,0,0]
 => [1,0,1,0,1,1,0,1,0,0]
 => 2 = 1 + 1
[1,1,0,1,1,0,1,1,0,0,0,0]
 => [2,2,1,1]
 => [1,0,1,1,0,1,1,0,0,0]
 => [1,1,0,1,1,0,1,0,0,0]
 => 2 = 1 + 1
[1,1,0,1,1,1,0,0,0,1,0,0]
 => [4,1,1,1]
 => [1,0,1,1,1,0,0,0,1,0]
 => [1,1,1,0,1,0,0,0,1,0]
 => 3 = 2 + 1
[1,1,0,1,1,1,0,0,1,0,0,0]
 => [3,1,1,1]
 => [1,0,1,1,1,0,0,1,0,0]
 => [1,0,1,1,1,0,0,1,0,0]
 => 2 = 1 + 1
[1,1,0,1,1,1,0,1,0,0,0,0]
 => [2,1,1,1]
 => [1,0,1,1,1,0,1,0,0,0]
 => [1,0,1,1,1,0,1,0,0,0]
 => 2 = 1 + 1
[1,1,0,1,1,1,1,0,0,0,0,0]
 => [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 + 1
[1,1,1,0,0,1,0,1,0,1,0,0]
 => [4,3,2]
 => [1,1,0,0,1,0,1,0,1,0]
 => [1,1,0,0,1,0,1,0,1,0]
 => 4 = 3 + 1
[1,1,1,0,0,1,0,1,1,0,0,0]
 => [3,3,2]
 => [1,1,0,0,1,0,1,1,0,0]
 => [1,1,1,0,0,1,0,1,0,0]
 => 3 = 2 + 1
[1,1,1,0,0,1,1,0,0,1,0,0]
 => [4,2,2]
 => [1,1,0,0,1,1,0,0,1,0]
 => [1,1,1,0,0,1,0,0,1,0]
 => 3 = 2 + 1
[1,1,1,0,0,1,1,0,1,0,0,0]
 => [3,2,2]
 => [1,1,0,0,1,1,0,1,0,0]
 => [1,1,0,0,1,1,0,1,0,0]
 => 2 = 1 + 1
[1,1,1,0,0,1,1,1,0,0,0,0]
 => [2,2,2]
 => [1,1,0,0,1,1,1,0,0,0]
 => [1,1,1,1,0,0,1,0,0,0]
 => 2 = 1 + 1
[1,1,1,0,1,0,0,1,0,1,0,0]
 => [4,3,1]
 => [1,1,0,1,0,0,1,0,1,0]
 => [1,0,1,1,0,0,1,0,1,0]
 => 3 = 2 + 1
[1,1,1,0,1,0,0,1,1,0,0,0]
 => [3,3,1]
 => [1,1,0,1,0,0,1,1,0,0]
 => [1,1,0,1,1,0,0,1,0,0]
 => 2 = 1 + 1
[1,1,1,0,1,0,1,0,0,1,0,0]
 => [4,2,1]
 => [1,1,0,1,0,1,0,0,1,0]
 => [1,0,1,0,1,1,0,0,1,0]
 => 2 = 1 + 1
[1,1,1,0,1,0,1,0,1,0,0,0]
 => [3,2,1]
 => [1,0,1,0,1,0,1,0]
 => [1,0,1,0,1,0,1,0]
 => 4 = 3 + 1
[1,1,1,0,1,0,1,1,0,0,0,0]
 => [2,2,1]
 => [1,0,1,0,1,1,0,0]
 => [1,1,0,1,0,1,0,0]
 => 3 = 2 + 1
[1,1,1,0,1,1,0,0,0,1,0,0]
 => [4,1,1]
 => [1,1,0,1,1,0,0,0,1,0]
 => [1,1,0,1,1,0,0,0,1,0]
 => 2 = 1 + 1
[1,1,1,0,1,1,0,0,1,0,0,0]
 => [3,1,1]
 => [1,0,1,1,0,0,1,0]
 => [1,1,0,1,0,0,1,0]
 => 3 = 2 + 1
[1,1,1,0,1,1,0,1,0,0,0,0]
 => [2,1,1]
 => [1,0,1,1,0,1,0,0]
 => [1,0,1,1,0,1,0,0]
 => 2 = 1 + 1
[1,1,1,0,1,1,1,0,0,0,0,0]
 => [1,1,1]
 => [1,0,1,1,1,0,0,0]
 => [1,1,1,0,1,0,0,0]
 => 2 = 1 + 1
[1,1,1,1,0,0,0,1,0,1,0,0]
 => [4,3]
 => [1,1,1,0,0,0,1,0,1,0]
 => [1,1,1,0,0,0,1,0,1,0]
 => 3 = 2 + 1
[1,1,1,1,0,0,0,1,1,0,0,0]
 => [3,3]
 => [1,1,1,0,0,0,1,1,0,0]
 => [1,1,1,1,0,0,0,1,0,0]
 => 2 = 1 + 1
[1,1,1,1,0,0,1,0,0,1,0,0]
 => [4,2]
 => [1,1,1,0,0,1,0,0,1,0]
 => [1,1,0,0,1,1,0,0,1,0]
 => 2 = 1 + 1
[1,1,1,1,0,0,1,0,1,0,0,0]
 => [3,2]
 => [1,1,0,0,1,0,1,0]
 => [1,1,0,0,1,0,1,0]
 => 3 = 2 + 1
[1,1,1,1,1,1,0,0,0,0,0,0]
 => []
 => []
 => []
 => ? = 5 + 1
[1,1,1,1,1,1,1,0,0,0,0,0,0,0]
 => []
 => []
 => []
 => ? = 6 + 1
Description
The number of up steps after the last double rise of a Dyck path.
Matching statistic: St000843
Mapping
Mp00027: Dyck paths —to partition⟶ Integer partitions
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00146: Dyck paths —to tunnel matching⟶ Perfect matchings
St000843: Perfect matchings ⟶ ℤResult quality: 83% ●values known / values provided: 96%●distinct values known / distinct values provided: 83%
Values
[1,0]
 => []
 => []
 => []
 => ? = 1 + 1
[1,1,0,0]
 => []
 => []
 => []
 => ? = 1 + 1
[1,1,0,1,0,0]
 => [1]
 => [1,0,1,0]
 => [(1,2),(3,4)]
 => 2 = 1 + 1
[1,1,1,0,0,0]
 => []
 => []
 => []
 => ? = 2 + 1
[1,1,0,1,0,1,0,0]
 => [2,1]
 => [1,0,1,0,1,0]
 => [(1,2),(3,4),(5,6)]
 => 3 = 2 + 1
[1,1,0,1,1,0,0,0]
 => [1,1]
 => [1,0,1,1,0,0]
 => [(1,2),(3,6),(4,5)]
 => 2 = 1 + 1
[1,1,1,0,0,1,0,0]
 => [2]
 => [1,1,0,0,1,0]
 => [(1,4),(2,3),(5,6)]
 => 2 = 1 + 1
[1,1,1,0,1,0,0,0]
 => [1]
 => [1,0,1,0]
 => [(1,2),(3,4)]
 => 2 = 1 + 1
[1,1,1,1,0,0,0,0]
 => []
 => []
 => []
 => ? = 3 + 1
[1,1,0,1,0,1,0,1,0,0]
 => [3,2,1]
 => [1,0,1,0,1,0,1,0]
 => [(1,2),(3,4),(5,6),(7,8)]
 => 4 = 3 + 1
[1,1,0,1,0,1,1,0,0,0]
 => [2,2,1]
 => [1,0,1,0,1,1,0,0]
 => [(1,2),(3,4),(5,8),(6,7)]
 => 3 = 2 + 1
[1,1,0,1,1,0,0,1,0,0]
 => [3,1,1]
 => [1,0,1,1,0,0,1,0]
 => [(1,2),(3,6),(4,5),(7,8)]
 => 3 = 2 + 1
[1,1,0,1,1,0,1,0,0,0]
 => [2,1,1]
 => [1,0,1,1,0,1,0,0]
 => [(1,2),(3,8),(4,5),(6,7)]
 => 2 = 1 + 1
[1,1,0,1,1,1,0,0,0,0]
 => [1,1,1]
 => [1,0,1,1,1,0,0,0]
 => [(1,2),(3,8),(4,7),(5,6)]
 => 2 = 1 + 1
[1,1,1,0,0,1,0,1,0,0]
 => [3,2]
 => [1,1,0,0,1,0,1,0]
 => [(1,4),(2,3),(5,6),(7,8)]
 => 3 = 2 + 1
[1,1,1,0,0,1,1,0,0,0]
 => [2,2]
 => [1,1,0,0,1,1,0,0]
 => [(1,4),(2,3),(5,8),(6,7)]
 => 2 = 1 + 1
[1,1,1,0,1,0,0,1,0,0]
 => [3,1]
 => [1,1,0,1,0,0,1,0]
 => [(1,6),(2,3),(4,5),(7,8)]
 => 2 = 1 + 1
[1,1,1,0,1,0,1,0,0,0]
 => [2,1]
 => [1,0,1,0,1,0]
 => [(1,2),(3,4),(5,6)]
 => 3 = 2 + 1
[1,1,1,0,1,1,0,0,0,0]
 => [1,1]
 => [1,0,1,1,0,0]
 => [(1,2),(3,6),(4,5)]
 => 2 = 1 + 1
[1,1,1,1,0,0,0,1,0,0]
 => [3]
 => [1,1,1,0,0,0,1,0]
 => [(1,6),(2,5),(3,4),(7,8)]
 => 2 = 1 + 1
[1,1,1,1,0,0,1,0,0,0]
 => [2]
 => [1,1,0,0,1,0]
 => [(1,4),(2,3),(5,6)]
 => 2 = 1 + 1
[1,1,1,1,0,1,0,0,0,0]
 => [1]
 => [1,0,1,0]
 => [(1,2),(3,4)]
 => 2 = 1 + 1
[1,1,1,1,1,0,0,0,0,0]
 => []
 => []
 => []
 => ? = 4 + 1
[1,1,0,1,0,1,0,1,0,1,0,0]
 => [4,3,2,1]
 => [1,0,1,0,1,0,1,0,1,0]
 => [(1,2),(3,4),(5,6),(7,8),(9,10)]
 => 5 = 4 + 1
[1,1,0,1,0,1,0,1,1,0,0,0]
 => [3,3,2,1]
 => [1,0,1,0,1,0,1,1,0,0]
 => [(1,2),(3,4),(5,6),(7,10),(8,9)]
 => 4 = 3 + 1
[1,1,0,1,0,1,1,0,0,1,0,0]
 => [4,2,2,1]
 => [1,0,1,0,1,1,0,0,1,0]
 => [(1,2),(3,4),(5,8),(6,7),(9,10)]
 => 4 = 3 + 1
[1,1,0,1,0,1,1,0,1,0,0,0]
 => [3,2,2,1]
 => [1,0,1,0,1,1,0,1,0,0]
 => [(1,2),(3,4),(5,10),(6,7),(8,9)]
 => 3 = 2 + 1
[1,1,0,1,0,1,1,1,0,0,0,0]
 => [2,2,2,1]
 => [1,0,1,0,1,1,1,0,0,0]
 => [(1,2),(3,4),(5,10),(6,9),(7,8)]
 => 3 = 2 + 1
[1,1,0,1,1,0,0,1,0,1,0,0]
 => [4,3,1,1]
 => [1,0,1,1,0,0,1,0,1,0]
 => [(1,2),(3,6),(4,5),(7,8),(9,10)]
 => 4 = 3 + 1
[1,1,0,1,1,0,0,1,1,0,0,0]
 => [3,3,1,1]
 => [1,0,1,1,0,0,1,1,0,0]
 => [(1,2),(3,6),(4,5),(7,10),(8,9)]
 => 3 = 2 + 1
[1,1,0,1,1,0,1,0,0,1,0,0]
 => [4,2,1,1]
 => [1,0,1,1,0,1,0,0,1,0]
 => [(1,2),(3,8),(4,5),(6,7),(9,10)]
 => 3 = 2 + 1
[1,1,0,1,1,0,1,0,1,0,0,0]
 => [3,2,1,1]
 => [1,0,1,1,0,1,0,1,0,0]
 => [(1,2),(3,10),(4,5),(6,7),(8,9)]
 => 2 = 1 + 1
[1,1,0,1,1,0,1,1,0,0,0,0]
 => [2,2,1,1]
 => [1,0,1,1,0,1,1,0,0,0]
 => [(1,2),(3,10),(4,5),(6,9),(7,8)]
 => 2 = 1 + 1
[1,1,0,1,1,1,0,0,0,1,0,0]
 => [4,1,1,1]
 => [1,0,1,1,1,0,0,0,1,0]
 => [(1,2),(3,8),(4,7),(5,6),(9,10)]
 => 3 = 2 + 1
[1,1,0,1,1,1,0,0,1,0,0,0]
 => [3,1,1,1]
 => [1,0,1,1,1,0,0,1,0,0]
 => [(1,2),(3,10),(4,7),(5,6),(8,9)]
 => 2 = 1 + 1
[1,1,0,1,1,1,0,1,0,0,0,0]
 => [2,1,1,1]
 => [1,0,1,1,1,0,1,0,0,0]
 => [(1,2),(3,10),(4,9),(5,6),(7,8)]
 => 2 = 1 + 1
[1,1,0,1,1,1,1,0,0,0,0,0]
 => [1,1,1,1]
 => [1,0,1,1,1,1,0,0,0,0]
 => [(1,2),(3,10),(4,9),(5,8),(6,7)]
 => 2 = 1 + 1
[1,1,1,0,0,1,0,1,0,1,0,0]
 => [4,3,2]
 => [1,1,0,0,1,0,1,0,1,0]
 => [(1,4),(2,3),(5,6),(7,8),(9,10)]
 => 4 = 3 + 1
[1,1,1,0,0,1,0,1,1,0,0,0]
 => [3,3,2]
 => [1,1,0,0,1,0,1,1,0,0]
 => [(1,4),(2,3),(5,6),(7,10),(8,9)]
 => 3 = 2 + 1
[1,1,1,0,0,1,1,0,0,1,0,0]
 => [4,2,2]
 => [1,1,0,0,1,1,0,0,1,0]
 => [(1,4),(2,3),(5,8),(6,7),(9,10)]
 => 3 = 2 + 1
[1,1,1,0,0,1,1,0,1,0,0,0]
 => [3,2,2]
 => [1,1,0,0,1,1,0,1,0,0]
 => [(1,4),(2,3),(5,10),(6,7),(8,9)]
 => 2 = 1 + 1
[1,1,1,0,0,1,1,1,0,0,0,0]
 => [2,2,2]
 => [1,1,0,0,1,1,1,0,0,0]
 => [(1,4),(2,3),(5,10),(6,9),(7,8)]
 => 2 = 1 + 1
[1,1,1,0,1,0,0,1,0,1,0,0]
 => [4,3,1]
 => [1,1,0,1,0,0,1,0,1,0]
 => [(1,6),(2,3),(4,5),(7,8),(9,10)]
 => 3 = 2 + 1
[1,1,1,0,1,0,0,1,1,0,0,0]
 => [3,3,1]
 => [1,1,0,1,0,0,1,1,0,0]
 => [(1,6),(2,3),(4,5),(7,10),(8,9)]
 => 2 = 1 + 1
[1,1,1,0,1,0,1,0,0,1,0,0]
 => [4,2,1]
 => [1,1,0,1,0,1,0,0,1,0]
 => [(1,8),(2,3),(4,5),(6,7),(9,10)]
 => 2 = 1 + 1
[1,1,1,0,1,0,1,0,1,0,0,0]
 => [3,2,1]
 => [1,0,1,0,1,0,1,0]
 => [(1,2),(3,4),(5,6),(7,8)]
 => 4 = 3 + 1
[1,1,1,0,1,0,1,1,0,0,0,0]
 => [2,2,1]
 => [1,0,1,0,1,1,0,0]
 => [(1,2),(3,4),(5,8),(6,7)]
 => 3 = 2 + 1
[1,1,1,0,1,1,0,0,0,1,0,0]
 => [4,1,1]
 => [1,1,0,1,1,0,0,0,1,0]
 => [(1,8),(2,3),(4,7),(5,6),(9,10)]
 => 2 = 1 + 1
[1,1,1,0,1,1,0,0,1,0,0,0]
 => [3,1,1]
 => [1,0,1,1,0,0,1,0]
 => [(1,2),(3,6),(4,5),(7,8)]
 => 3 = 2 + 1
[1,1,1,0,1,1,0,1,0,0,0,0]
 => [2,1,1]
 => [1,0,1,1,0,1,0,0]
 => [(1,2),(3,8),(4,5),(6,7)]
 => 2 = 1 + 1
[1,1,1,0,1,1,1,0,0,0,0,0]
 => [1,1,1]
 => [1,0,1,1,1,0,0,0]
 => [(1,2),(3,8),(4,7),(5,6)]
 => 2 = 1 + 1
[1,1,1,1,0,0,0,1,0,1,0,0]
 => [4,3]
 => [1,1,1,0,0,0,1,0,1,0]
 => [(1,6),(2,5),(3,4),(7,8),(9,10)]
 => 3 = 2 + 1
[1,1,1,1,0,0,0,1,1,0,0,0]
 => [3,3]
 => [1,1,1,0,0,0,1,1,0,0]
 => [(1,6),(2,5),(3,4),(7,10),(8,9)]
 => 2 = 1 + 1
[1,1,1,1,0,0,1,0,0,1,0,0]
 => [4,2]
 => [1,1,1,0,0,1,0,0,1,0]
 => [(1,8),(2,5),(3,4),(6,7),(9,10)]
 => 2 = 1 + 1
[1,1,1,1,0,0,1,0,1,0,0,0]
 => [3,2]
 => [1,1,0,0,1,0,1,0]
 => [(1,4),(2,3),(5,6),(7,8)]
 => 3 = 2 + 1
[1,1,1,1,1,1,0,0,0,0,0,0]
 => []
 => []
 => []
 => ? = 5 + 1
[1,1,1,1,1,1,1,0,0,0,0,0,0,0]
 => []
 => []
 => []
 => ? = 6 + 1
Description
The decomposition number of a perfect matching. This is the number of integers $i$ such that all elements in $\{1,\dots,i\}$ are matched among themselves. Visually, it is the number of components of the arc diagram of the matching, where a component is a matching of a set of consecutive numbers $\{a,a+1,\dots,b\}$ such that there is no arc matching a number smaller than $a$ with a number larger than $b$. E.g., $\{(1,6),(2,4),(3,5)\}$ is a hairpin under a single edge - crossing nested by a single arc. Thus, this matching has one component. However, $\{(1,2),(3,6),(4,5)\}$ is a single edge to the left of a ladder (a pair of nested edges), so it has two components.
Matching statistic: St000991
(load all 2 compositions to match this statistic)
Mapping
Mp00027: Dyck paths —to partition⟶ Integer partitions
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00031: Dyck paths —to 312-avoiding permutation⟶ Permutations
St000991: Permutations ⟶ ℤResult quality: 83% ●values known / values provided: 96%●distinct values known / distinct values provided: 83%
Values
[1,0]
 => []
 => []
 => [] => ? = 1 + 1
[1,1,0,0]
 => []
 => []
 => [] => ? = 1 + 1
[1,1,0,1,0,0]
 => [1]
 => [1,0,1,0]
 => [1,2] => 2 = 1 + 1
[1,1,1,0,0,0]
 => []
 => []
 => [] => ? = 2 + 1
[1,1,0,1,0,1,0,0]
 => [2,1]
 => [1,0,1,0,1,0]
 => [1,2,3] => 3 = 2 + 1
[1,1,0,1,1,0,0,0]
 => [1,1]
 => [1,0,1,1,0,0]
 => [1,3,2] => 2 = 1 + 1
[1,1,1,0,0,1,0,0]
 => [2]
 => [1,1,0,0,1,0]
 => [2,1,3] => 2 = 1 + 1
[1,1,1,0,1,0,0,0]
 => [1]
 => [1,0,1,0]
 => [1,2] => 2 = 1 + 1
[1,1,1,1,0,0,0,0]
 => []
 => []
 => [] => ? = 3 + 1
[1,1,0,1,0,1,0,1,0,0]
 => [3,2,1]
 => [1,0,1,0,1,0,1,0]
 => [1,2,3,4] => 4 = 3 + 1
[1,1,0,1,0,1,1,0,0,0]
 => [2,2,1]
 => [1,0,1,0,1,1,0,0]
 => [1,2,4,3] => 3 = 2 + 1
[1,1,0,1,1,0,0,1,0,0]
 => [3,1,1]
 => [1,0,1,1,0,0,1,0]
 => [1,3,2,4] => 3 = 2 + 1
[1,1,0,1,1,0,1,0,0,0]
 => [2,1,1]
 => [1,0,1,1,0,1,0,0]
 => [1,3,4,2] => 2 = 1 + 1
[1,1,0,1,1,1,0,0,0,0]
 => [1,1,1]
 => [1,0,1,1,1,0,0,0]
 => [1,4,3,2] => 2 = 1 + 1
[1,1,1,0,0,1,0,1,0,0]
 => [3,2]
 => [1,1,0,0,1,0,1,0]
 => [2,1,3,4] => 3 = 2 + 1
[1,1,1,0,0,1,1,0,0,0]
 => [2,2]
 => [1,1,0,0,1,1,0,0]
 => [2,1,4,3] => 2 = 1 + 1
[1,1,1,0,1,0,0,1,0,0]
 => [3,1]
 => [1,1,0,1,0,0,1,0]
 => [2,3,1,4] => 2 = 1 + 1
[1,1,1,0,1,0,1,0,0,0]
 => [2,1]
 => [1,0,1,0,1,0]
 => [1,2,3] => 3 = 2 + 1
[1,1,1,0,1,1,0,0,0,0]
 => [1,1]
 => [1,0,1,1,0,0]
 => [1,3,2] => 2 = 1 + 1
[1,1,1,1,0,0,0,1,0,0]
 => [3]
 => [1,1,1,0,0,0,1,0]
 => [3,2,1,4] => 2 = 1 + 1
[1,1,1,1,0,0,1,0,0,0]
 => [2]
 => [1,1,0,0,1,0]
 => [2,1,3] => 2 = 1 + 1
[1,1,1,1,0,1,0,0,0,0]
 => [1]
 => [1,0,1,0]
 => [1,2] => 2 = 1 + 1
[1,1,1,1,1,0,0,0,0,0]
 => []
 => []
 => [] => ? = 4 + 1
[1,1,0,1,0,1,0,1,0,1,0,0]
 => [4,3,2,1]
 => [1,0,1,0,1,0,1,0,1,0]
 => [1,2,3,4,5] => 5 = 4 + 1
[1,1,0,1,0,1,0,1,1,0,0,0]
 => [3,3,2,1]
 => [1,0,1,0,1,0,1,1,0,0]
 => [1,2,3,5,4] => 4 = 3 + 1
[1,1,0,1,0,1,1,0,0,1,0,0]
 => [4,2,2,1]
 => [1,0,1,0,1,1,0,0,1,0]
 => [1,2,4,3,5] => 4 = 3 + 1
[1,1,0,1,0,1,1,0,1,0,0,0]
 => [3,2,2,1]
 => [1,0,1,0,1,1,0,1,0,0]
 => [1,2,4,5,3] => 3 = 2 + 1
[1,1,0,1,0,1,1,1,0,0,0,0]
 => [2,2,2,1]
 => [1,0,1,0,1,1,1,0,0,0]
 => [1,2,5,4,3] => 3 = 2 + 1
[1,1,0,1,1,0,0,1,0,1,0,0]
 => [4,3,1,1]
 => [1,0,1,1,0,0,1,0,1,0]
 => [1,3,2,4,5] => 4 = 3 + 1
[1,1,0,1,1,0,0,1,1,0,0,0]
 => [3,3,1,1]
 => [1,0,1,1,0,0,1,1,0,0]
 => [1,3,2,5,4] => 3 = 2 + 1
[1,1,0,1,1,0,1,0,0,1,0,0]
 => [4,2,1,1]
 => [1,0,1,1,0,1,0,0,1,0]
 => [1,3,4,2,5] => 3 = 2 + 1
[1,1,0,1,1,0,1,0,1,0,0,0]
 => [3,2,1,1]
 => [1,0,1,1,0,1,0,1,0,0]
 => [1,3,4,5,2] => 2 = 1 + 1
[1,1,0,1,1,0,1,1,0,0,0,0]
 => [2,2,1,1]
 => [1,0,1,1,0,1,1,0,0,0]
 => [1,3,5,4,2] => 2 = 1 + 1
[1,1,0,1,1,1,0,0,0,1,0,0]
 => [4,1,1,1]
 => [1,0,1,1,1,0,0,0,1,0]
 => [1,4,3,2,5] => 3 = 2 + 1
[1,1,0,1,1,1,0,0,1,0,0,0]
 => [3,1,1,1]
 => [1,0,1,1,1,0,0,1,0,0]
 => [1,4,3,5,2] => 2 = 1 + 1
[1,1,0,1,1,1,0,1,0,0,0,0]
 => [2,1,1,1]
 => [1,0,1,1,1,0,1,0,0,0]
 => [1,4,5,3,2] => 2 = 1 + 1
[1,1,0,1,1,1,1,0,0,0,0,0]
 => [1,1,1,1]
 => [1,0,1,1,1,1,0,0,0,0]
 => [1,5,4,3,2] => 2 = 1 + 1
[1,1,1,0,0,1,0,1,0,1,0,0]
 => [4,3,2]
 => [1,1,0,0,1,0,1,0,1,0]
 => [2,1,3,4,5] => 4 = 3 + 1
[1,1,1,0,0,1,0,1,1,0,0,0]
 => [3,3,2]
 => [1,1,0,0,1,0,1,1,0,0]
 => [2,1,3,5,4] => 3 = 2 + 1
[1,1,1,0,0,1,1,0,0,1,0,0]
 => [4,2,2]
 => [1,1,0,0,1,1,0,0,1,0]
 => [2,1,4,3,5] => 3 = 2 + 1
[1,1,1,0,0,1,1,0,1,0,0,0]
 => [3,2,2]
 => [1,1,0,0,1,1,0,1,0,0]
 => [2,1,4,5,3] => 2 = 1 + 1
[1,1,1,0,0,1,1,1,0,0,0,0]
 => [2,2,2]
 => [1,1,0,0,1,1,1,0,0,0]
 => [2,1,5,4,3] => 2 = 1 + 1
[1,1,1,0,1,0,0,1,0,1,0,0]
 => [4,3,1]
 => [1,1,0,1,0,0,1,0,1,0]
 => [2,3,1,4,5] => 3 = 2 + 1
[1,1,1,0,1,0,0,1,1,0,0,0]
 => [3,3,1]
 => [1,1,0,1,0,0,1,1,0,0]
 => [2,3,1,5,4] => 2 = 1 + 1
[1,1,1,0,1,0,1,0,0,1,0,0]
 => [4,2,1]
 => [1,1,0,1,0,1,0,0,1,0]
 => [2,3,4,1,5] => 2 = 1 + 1
[1,1,1,0,1,0,1,0,1,0,0,0]
 => [3,2,1]
 => [1,0,1,0,1,0,1,0]
 => [1,2,3,4] => 4 = 3 + 1
[1,1,1,0,1,0,1,1,0,0,0,0]
 => [2,2,1]
 => [1,0,1,0,1,1,0,0]
 => [1,2,4,3] => 3 = 2 + 1
[1,1,1,0,1,1,0,0,0,1,0,0]
 => [4,1,1]
 => [1,1,0,1,1,0,0,0,1,0]
 => [2,4,3,1,5] => 2 = 1 + 1
[1,1,1,0,1,1,0,0,1,0,0,0]
 => [3,1,1]
 => [1,0,1,1,0,0,1,0]
 => [1,3,2,4] => 3 = 2 + 1
[1,1,1,0,1,1,0,1,0,0,0,0]
 => [2,1,1]
 => [1,0,1,1,0,1,0,0]
 => [1,3,4,2] => 2 = 1 + 1
[1,1,1,0,1,1,1,0,0,0,0,0]
 => [1,1,1]
 => [1,0,1,1,1,0,0,0]
 => [1,4,3,2] => 2 = 1 + 1
[1,1,1,1,0,0,0,1,0,1,0,0]
 => [4,3]
 => [1,1,1,0,0,0,1,0,1,0]
 => [3,2,1,4,5] => 3 = 2 + 1
[1,1,1,1,0,0,0,1,1,0,0,0]
 => [3,3]
 => [1,1,1,0,0,0,1,1,0,0]
 => [3,2,1,5,4] => 2 = 1 + 1
[1,1,1,1,0,0,1,0,0,1,0,0]
 => [4,2]
 => [1,1,1,0,0,1,0,0,1,0]
 => [3,2,4,1,5] => 2 = 1 + 1
[1,1,1,1,0,0,1,0,1,0,0,0]
 => [3,2]
 => [1,1,0,0,1,0,1,0]
 => [2,1,3,4] => 3 = 2 + 1
[1,1,1,1,1,1,0,0,0,0,0,0]
 => []
 => []
 => [] => ? = 5 + 1
[1,1,1,1,1,1,1,0,0,0,0,0,0,0]
 => []
 => []
 => [] => ? = 6 + 1
Description
The number of right-to-left minima of a permutation. For the number of left-to-right maxima, see [[St000314]].
The following 10 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St001184Number of indecomposable injective modules with grade at least 1 in the corresponding Nakayama algebra. St001461The number of topologically connected components of the chord diagram of a permutation. St000439The position of the first down step of a Dyck path. 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. St001199The dominant dimension of $eAe$ for the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St001124The multiplicity of the standard representation in the Kronecker square corresponding to a partition. St001355Number of non-empty prefixes of a binary word that contain equally many 0's and 1's. St001462The number of factors of a standard tableaux under concatenation. St000455The second largest eigenvalue of a graph if it is integral. St001488The number of corners of a skew partition.