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Your data matches 41 different statistics following compositions of up to 3 maps.
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Matching statistic: St000771
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(load all 5 compositions to match this statistic)
Values
[1] => ([],1)
=> ([],1)
=> 1
[1,1] => ([(0,1)],2)
=> ([(0,1)],2)
=> 1
[2] => ([],2)
=> ([],1)
=> 1
[1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> ([(0,1),(0,2),(1,2)],3)
=> 2
[2,1] => ([(0,2),(1,2)],3)
=> ([(0,1)],2)
=> 1
[3] => ([],3)
=> ([],1)
=> 1
[1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3
[1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> 1
[2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,2)],3)
=> 2
[3,1] => ([(0,3),(1,3),(2,3)],4)
=> ([(0,1)],2)
=> 1
[4] => ([],4)
=> ([],1)
=> 1
[1,1,1,1,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 4
[1,1,2,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
[1,2,1,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
[1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> 1
[2,1,1,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3
[2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> 1
[3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 2
[4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> ([(0,1)],2)
=> 1
[5] => ([],5)
=> ([],1)
=> 1
[1,1,1,1,1,1] => ([(0,1),(0,2),(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)
=> ([(0,1),(0,2),(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)
=> 5
[1,1,1,2,1] => ([(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 3
[1,1,2,1,1] => ([(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)
=> ([(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
[1,1,3,1] => ([(0,5),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
[1,2,1,1,1] => ([(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)
=> ([(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
[1,2,2,1] => ([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 1
[1,3,1,1] => ([(0,4),(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
[1,4,1] => ([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> 1
[2,1,1,1,1] => ([(0,2),(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)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 4
[2,1,2,1] => ([(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
[2,2,1,1] => ([(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
[2,3,1] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> 1
[3,1,1,1] => ([(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)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3
[3,2,1] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> 1
[4,1,1] => ([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> 2
[5,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> ([(0,1)],2)
=> 1
[6] => ([],6)
=> ([],1)
=> 1
[1,1,1,1,1,1,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 6
[1,1,1,1,2,1] => ([(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 4
[1,1,1,2,1,1] => ([(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 3
[1,1,1,3,1] => ([(0,6),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 3
[1,1,2,1,1,1] => ([(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 3
[1,1,2,2,1] => ([(0,6),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,6),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 2
[1,1,3,1,1] => ([(0,5),(0,6),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(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
[1,1,4,1] => ([(0,6),(1,6),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
[1,2,1,1,1,1] => ([(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 4
[1,2,1,2,1] => ([(0,6),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,6),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 2
[1,2,2,1,1] => ([(0,5),(0,6),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,5),(0,6),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 2
[1,2,3,1] => ([(0,6),(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 1
[1,3,1,1,1] => ([(0,4),(0,5),(0,6),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(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
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: St000774
Values
[1] => ([],1)
=> ([],1)
=> 1
[1,1] => ([(0,1)],2)
=> ([(0,1)],2)
=> 1
[2] => ([],2)
=> ([],1)
=> 1
[1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> ([(0,1),(0,2),(1,2)],3)
=> 2
[2,1] => ([(0,2),(1,2)],3)
=> ([(0,1)],2)
=> 1
[3] => ([],3)
=> ([],1)
=> 1
[1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3
[1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> 1
[2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,2)],3)
=> 2
[3,1] => ([(0,3),(1,3),(2,3)],4)
=> ([(0,1)],2)
=> 1
[4] => ([],4)
=> ([],1)
=> 1
[1,1,1,1,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 4
[1,1,2,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
[1,2,1,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
[1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> 1
[2,1,1,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3
[2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> 1
[3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 2
[4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> ([(0,1)],2)
=> 1
[5] => ([],5)
=> ([],1)
=> 1
[1,1,1,1,1,1] => ([(0,1),(0,2),(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)
=> ([(0,1),(0,2),(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)
=> 5
[1,1,1,2,1] => ([(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 3
[1,1,2,1,1] => ([(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)
=> ([(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
[1,1,3,1] => ([(0,5),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
[1,2,1,1,1] => ([(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)
=> ([(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
[1,2,2,1] => ([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 1
[1,3,1,1] => ([(0,4),(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
[1,4,1] => ([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> 1
[2,1,1,1,1] => ([(0,2),(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)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 4
[2,1,2,1] => ([(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
[2,2,1,1] => ([(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
[2,3,1] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> 1
[3,1,1,1] => ([(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)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3
[3,2,1] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> 1
[4,1,1] => ([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> 2
[5,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> ([(0,1)],2)
=> 1
[6] => ([],6)
=> ([],1)
=> 1
[1,1,1,1,1,1,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 6
[1,1,1,1,2,1] => ([(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 4
[1,1,1,2,1,1] => ([(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 3
[1,1,1,3,1] => ([(0,6),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 3
[1,1,2,1,1,1] => ([(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 3
[1,1,2,2,1] => ([(0,6),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,6),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 2
[1,1,3,1,1] => ([(0,5),(0,6),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(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
[1,1,4,1] => ([(0,6),(1,6),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
[1,2,1,1,1,1] => ([(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 4
[1,2,1,2,1] => ([(0,6),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,6),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 2
[1,2,2,1,1] => ([(0,5),(0,6),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,5),(0,6),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 2
[1,2,3,1] => ([(0,6),(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 1
[1,3,1,1,1] => ([(0,4),(0,5),(0,6),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(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
Description
The maximal multiplicity of a Laplacian eigenvalue in a graph.
Matching statistic: St000381
Mp00184: Integer compositions —to threshold graph⟶ Graphs
Mp00247: Graphs —de-duplicate⟶ Graphs
Mp00152: Graphs —Laplacian multiplicities⟶ Integer compositions
St000381: Integer compositions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00247: Graphs —de-duplicate⟶ Graphs
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)
=> [1] => 1
[1,1] => ([(0,1)],2)
=> ([(0,1)],2)
=> [1,1] => 1
[2] => ([],2)
=> ([],1)
=> [1] => 1
[1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> ([(0,1),(0,2),(1,2)],3)
=> [2,1] => 2
[2,1] => ([(0,2),(1,2)],3)
=> ([(0,1)],2)
=> [1,1] => 1
[3] => ([],3)
=> ([],1)
=> [1] => 1
[1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [3,1] => 3
[1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> [1,1,1,1] => 1
[2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,2)],3)
=> [2,1] => 2
[3,1] => ([(0,3),(1,3),(2,3)],4)
=> ([(0,1)],2)
=> [1,1] => 1
[4] => ([],4)
=> ([],1)
=> [1] => 1
[1,1,1,1,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [4,1] => 4
[1,1,2,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [1,2,1,1] => 2
[1,2,1,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [2,1,1,1] => 2
[1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> [1,1,1,1] => 1
[2,1,1,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [3,1] => 3
[2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> [1,1,1,1] => 1
[3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> [2,1] => 2
[4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> ([(0,1)],2)
=> [1,1] => 1
[5] => ([],5)
=> ([],1)
=> [1] => 1
[1,1,1,1,1,1] => ([(0,1),(0,2),(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)
=> ([(0,1),(0,2),(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)
=> [5,1] => 5
[1,1,1,2,1] => ([(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [1,3,1,1] => 3
[1,1,2,1,1] => ([(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)
=> ([(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)
=> [2,2,1,1] => 2
[1,1,3,1] => ([(0,5),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [1,2,1,1] => 2
[1,2,1,1,1] => ([(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)
=> ([(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)
=> [3,1,1,1] => 3
[1,2,2,1] => ([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [1,1,1,1,1,1] => 1
[1,3,1,1] => ([(0,4),(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [2,1,1,1] => 2
[1,4,1] => ([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> [1,1,1,1] => 1
[2,1,1,1,1] => ([(0,2),(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)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [4,1] => 4
[2,1,2,1] => ([(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [1,2,1,1] => 2
[2,2,1,1] => ([(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [2,1,1,1] => 2
[2,3,1] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> [1,1,1,1] => 1
[3,1,1,1] => ([(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)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [3,1] => 3
[3,2,1] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> [1,1,1,1] => 1
[4,1,1] => ([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> [2,1] => 2
[5,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> ([(0,1)],2)
=> [1,1] => 1
[6] => ([],6)
=> ([],1)
=> [1] => 1
[1,1,1,1,1,1,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [6,1] => 6
[1,1,1,1,2,1] => ([(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [1,4,1,1] => 4
[1,1,1,2,1,1] => ([(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [2,3,1,1] => 3
[1,1,1,3,1] => ([(0,6),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [1,3,1,1] => 3
[1,1,2,1,1,1] => ([(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [3,2,1,1] => 3
[1,1,2,2,1] => ([(0,6),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,6),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [1,1,2,1,1,1] => 2
[1,1,3,1,1] => ([(0,5),(0,6),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(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)
=> [2,2,1,1] => 2
[1,1,4,1] => ([(0,6),(1,6),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [1,2,1,1] => 2
[1,2,1,1,1,1] => ([(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [4,1,1,1] => 4
[1,2,1,2,1] => ([(0,6),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,6),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [1,2,1,1,1,1] => 2
[1,2,2,1,1] => ([(0,5),(0,6),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,5),(0,6),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [2,1,1,1,1,1] => 2
[1,2,3,1] => ([(0,6),(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [1,1,1,1,1,1] => 1
[1,3,1,1,1] => ([(0,4),(0,5),(0,6),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(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)
=> [3,1,1,1] => 3
Description
The largest part of an integer composition.
Matching statistic: St000808
Mp00184: Integer compositions —to threshold graph⟶ Graphs
Mp00247: Graphs —de-duplicate⟶ Graphs
Mp00152: Graphs —Laplacian multiplicities⟶ Integer compositions
St000808: Integer compositions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00247: Graphs —de-duplicate⟶ Graphs
Mp00152: Graphs —Laplacian multiplicities⟶ Integer compositions
St000808: Integer compositions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => ([],1)
=> ([],1)
=> [1] => 1
[1,1] => ([(0,1)],2)
=> ([(0,1)],2)
=> [1,1] => 1
[2] => ([],2)
=> ([],1)
=> [1] => 1
[1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> ([(0,1),(0,2),(1,2)],3)
=> [2,1] => 2
[2,1] => ([(0,2),(1,2)],3)
=> ([(0,1)],2)
=> [1,1] => 1
[3] => ([],3)
=> ([],1)
=> [1] => 1
[1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [3,1] => 3
[1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> [1,1,1,1] => 1
[2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,2)],3)
=> [2,1] => 2
[3,1] => ([(0,3),(1,3),(2,3)],4)
=> ([(0,1)],2)
=> [1,1] => 1
[4] => ([],4)
=> ([],1)
=> [1] => 1
[1,1,1,1,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [4,1] => 4
[1,1,2,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [1,2,1,1] => 2
[1,2,1,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [2,1,1,1] => 2
[1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> [1,1,1,1] => 1
[2,1,1,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [3,1] => 3
[2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> [1,1,1,1] => 1
[3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> [2,1] => 2
[4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> ([(0,1)],2)
=> [1,1] => 1
[5] => ([],5)
=> ([],1)
=> [1] => 1
[1,1,1,1,1,1] => ([(0,1),(0,2),(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)
=> ([(0,1),(0,2),(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)
=> [5,1] => 5
[1,1,1,2,1] => ([(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [1,3,1,1] => 3
[1,1,2,1,1] => ([(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)
=> ([(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)
=> [2,2,1,1] => 2
[1,1,3,1] => ([(0,5),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [1,2,1,1] => 2
[1,2,1,1,1] => ([(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)
=> ([(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)
=> [3,1,1,1] => 3
[1,2,2,1] => ([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [1,1,1,1,1,1] => 1
[1,3,1,1] => ([(0,4),(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [2,1,1,1] => 2
[1,4,1] => ([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> [1,1,1,1] => 1
[2,1,1,1,1] => ([(0,2),(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)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [4,1] => 4
[2,1,2,1] => ([(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [1,2,1,1] => 2
[2,2,1,1] => ([(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [2,1,1,1] => 2
[2,3,1] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> [1,1,1,1] => 1
[3,1,1,1] => ([(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)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [3,1] => 3
[3,2,1] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> [1,1,1,1] => 1
[4,1,1] => ([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> [2,1] => 2
[5,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> ([(0,1)],2)
=> [1,1] => 1
[6] => ([],6)
=> ([],1)
=> [1] => 1
[1,1,1,1,1,1,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [6,1] => 6
[1,1,1,1,2,1] => ([(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [1,4,1,1] => 4
[1,1,1,2,1,1] => ([(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [2,3,1,1] => 3
[1,1,1,3,1] => ([(0,6),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [1,3,1,1] => 3
[1,1,2,1,1,1] => ([(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [3,2,1,1] => 3
[1,1,2,2,1] => ([(0,6),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,6),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [1,1,2,1,1,1] => 2
[1,1,3,1,1] => ([(0,5),(0,6),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(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)
=> [2,2,1,1] => 2
[1,1,4,1] => ([(0,6),(1,6),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [1,2,1,1] => 2
[1,2,1,1,1,1] => ([(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [4,1,1,1] => 4
[1,2,1,2,1] => ([(0,6),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,6),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [1,2,1,1,1,1] => 2
[1,2,2,1,1] => ([(0,5),(0,6),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,5),(0,6),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [2,1,1,1,1,1] => 2
[1,2,3,1] => ([(0,6),(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [1,1,1,1,1,1] => 1
[1,3,1,1,1] => ([(0,4),(0,5),(0,6),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(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)
=> [3,1,1,1] => 3
Description
The number of up steps of the associated bargraph.
Interpret the composition as the sequence of heights of the bars of a bargraph. This statistic is the number of up steps.
Matching statistic: St001933
Mp00180: Integer compositions —to ribbon⟶ Skew partitions
Mp00182: Skew partitions —outer shape⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St001933: Integer partitions ⟶ ℤResult quality: 90% ●values known / values provided: 90%●distinct values known / distinct values provided: 100%
Mp00182: Skew partitions —outer shape⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St001933: Integer partitions ⟶ ℤResult quality: 90% ●values known / values provided: 90%●distinct values known / distinct values provided: 100%
Values
[1] => [[1],[]]
=> [1]
=> []
=> ? = 1
[1,1] => [[1,1],[]]
=> [1,1]
=> [1]
=> 1
[2] => [[2],[]]
=> [2]
=> []
=> ? = 1
[1,1,1] => [[1,1,1],[]]
=> [1,1,1]
=> [1,1]
=> 2
[2,1] => [[2,2],[1]]
=> [2,2]
=> [2]
=> 1
[3] => [[3],[]]
=> [3]
=> []
=> ? = 1
[1,1,1,1] => [[1,1,1,1],[]]
=> [1,1,1,1]
=> [1,1,1]
=> 3
[1,2,1] => [[2,2,1],[1]]
=> [2,2,1]
=> [2,1]
=> 1
[2,1,1] => [[2,2,2],[1,1]]
=> [2,2,2]
=> [2,2]
=> 2
[3,1] => [[3,3],[2]]
=> [3,3]
=> [3]
=> 1
[4] => [[4],[]]
=> [4]
=> []
=> ? = 1
[1,1,1,1,1] => [[1,1,1,1,1],[]]
=> [1,1,1,1,1]
=> [1,1,1,1]
=> 4
[1,1,2,1] => [[2,2,1,1],[1]]
=> [2,2,1,1]
=> [2,1,1]
=> 2
[1,2,1,1] => [[2,2,2,1],[1,1]]
=> [2,2,2,1]
=> [2,2,1]
=> 2
[1,3,1] => [[3,3,1],[2]]
=> [3,3,1]
=> [3,1]
=> 1
[2,1,1,1] => [[2,2,2,2],[1,1,1]]
=> [2,2,2,2]
=> [2,2,2]
=> 3
[2,2,1] => [[3,3,2],[2,1]]
=> [3,3,2]
=> [3,2]
=> 1
[3,1,1] => [[3,3,3],[2,2]]
=> [3,3,3]
=> [3,3]
=> 2
[4,1] => [[4,4],[3]]
=> [4,4]
=> [4]
=> 1
[5] => [[5],[]]
=> [5]
=> []
=> ? = 1
[1,1,1,1,1,1] => [[1,1,1,1,1,1],[]]
=> [1,1,1,1,1,1]
=> [1,1,1,1,1]
=> 5
[1,1,1,2,1] => [[2,2,1,1,1],[1]]
=> [2,2,1,1,1]
=> [2,1,1,1]
=> 3
[1,1,2,1,1] => [[2,2,2,1,1],[1,1]]
=> [2,2,2,1,1]
=> [2,2,1,1]
=> 2
[1,1,3,1] => [[3,3,1,1],[2]]
=> [3,3,1,1]
=> [3,1,1]
=> 2
[1,2,1,1,1] => [[2,2,2,2,1],[1,1,1]]
=> [2,2,2,2,1]
=> [2,2,2,1]
=> 3
[1,2,2,1] => [[3,3,2,1],[2,1]]
=> [3,3,2,1]
=> [3,2,1]
=> 1
[1,3,1,1] => [[3,3,3,1],[2,2]]
=> [3,3,3,1]
=> [3,3,1]
=> 2
[1,4,1] => [[4,4,1],[3]]
=> [4,4,1]
=> [4,1]
=> 1
[2,1,1,1,1] => [[2,2,2,2,2],[1,1,1,1]]
=> [2,2,2,2,2]
=> [2,2,2,2]
=> 4
[2,1,2,1] => [[3,3,2,2],[2,1,1]]
=> [3,3,2,2]
=> [3,2,2]
=> 2
[2,2,1,1] => [[3,3,3,2],[2,2,1]]
=> [3,3,3,2]
=> [3,3,2]
=> 2
[2,3,1] => [[4,4,2],[3,1]]
=> [4,4,2]
=> [4,2]
=> 1
[3,1,1,1] => [[3,3,3,3],[2,2,2]]
=> [3,3,3,3]
=> [3,3,3]
=> 3
[3,2,1] => [[4,4,3],[3,2]]
=> [4,4,3]
=> [4,3]
=> 1
[4,1,1] => [[4,4,4],[3,3]]
=> [4,4,4]
=> [4,4]
=> 2
[5,1] => [[5,5],[4]]
=> [5,5]
=> [5]
=> 1
[6] => [[6],[]]
=> [6]
=> []
=> ? = 1
[1,1,1,1,1,1,1] => [[1,1,1,1,1,1,1],[]]
=> [1,1,1,1,1,1,1]
=> [1,1,1,1,1,1]
=> 6
[1,1,1,1,2,1] => [[2,2,1,1,1,1],[1]]
=> [2,2,1,1,1,1]
=> [2,1,1,1,1]
=> 4
[1,1,1,2,1,1] => [[2,2,2,1,1,1],[1,1]]
=> [2,2,2,1,1,1]
=> [2,2,1,1,1]
=> 3
[1,1,1,3,1] => [[3,3,1,1,1],[2]]
=> [3,3,1,1,1]
=> [3,1,1,1]
=> 3
[1,1,2,1,1,1] => [[2,2,2,2,1,1],[1,1,1]]
=> [2,2,2,2,1,1]
=> [2,2,2,1,1]
=> 3
[1,1,2,2,1] => [[3,3,2,1,1],[2,1]]
=> [3,3,2,1,1]
=> [3,2,1,1]
=> 2
[1,1,3,1,1] => [[3,3,3,1,1],[2,2]]
=> [3,3,3,1,1]
=> [3,3,1,1]
=> 2
[1,1,4,1] => [[4,4,1,1],[3]]
=> [4,4,1,1]
=> [4,1,1]
=> 2
[1,2,1,1,1,1] => [[2,2,2,2,2,1],[1,1,1,1]]
=> [2,2,2,2,2,1]
=> [2,2,2,2,1]
=> 4
[1,2,1,2,1] => [[3,3,2,2,1],[2,1,1]]
=> [3,3,2,2,1]
=> [3,2,2,1]
=> 2
[1,2,2,1,1] => [[3,3,3,2,1],[2,2,1]]
=> [3,3,3,2,1]
=> [3,3,2,1]
=> 2
[1,2,3,1] => [[4,4,2,1],[3,1]]
=> [4,4,2,1]
=> [4,2,1]
=> 1
[1,3,1,1,1] => [[3,3,3,3,1],[2,2,2]]
=> [3,3,3,3,1]
=> [3,3,3,1]
=> 3
[1,3,2,1] => [[4,4,3,1],[3,2]]
=> [4,4,3,1]
=> [4,3,1]
=> 1
[1,4,1,1] => [[4,4,4,1],[3,3]]
=> [4,4,4,1]
=> [4,4,1]
=> 2
[1,5,1] => [[5,5,1],[4]]
=> [5,5,1]
=> [5,1]
=> 1
[2,1,1,1,1,1] => [[2,2,2,2,2,2],[1,1,1,1,1]]
=> [2,2,2,2,2,2]
=> [2,2,2,2,2]
=> 5
[2,1,1,2,1] => [[3,3,2,2,2],[2,1,1,1]]
=> [3,3,2,2,2]
=> [3,2,2,2]
=> 3
[2,1,2,1,1] => [[3,3,3,2,2],[2,2,1,1]]
=> [3,3,3,2,2]
=> [3,3,2,2]
=> 2
[7] => [[7],[]]
=> [7]
=> []
=> ? = 1
Description
The largest multiplicity of a part in an integer partition.
Matching statistic: St000684
Mp00231: Integer compositions —bounce path⟶ Dyck paths
Mp00027: Dyck paths —to partition⟶ Integer partitions
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
St000684: Dyck paths ⟶ ℤResult quality: 90% ●values known / values provided: 90%●distinct values known / distinct values provided: 100%
Mp00027: Dyck paths —to partition⟶ Integer partitions
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
St000684: Dyck paths ⟶ ℤResult quality: 90% ●values known / values provided: 90%●distinct values known / distinct values provided: 100%
Values
[1] => [1,0]
=> []
=> []
=> ? = 1 + 1
[1,1] => [1,0,1,0]
=> [1]
=> [1,0,1,0]
=> 2 = 1 + 1
[2] => [1,1,0,0]
=> []
=> []
=> ? = 1 + 1
[1,1,1] => [1,0,1,0,1,0]
=> [2,1]
=> [1,0,1,0,1,0]
=> 3 = 2 + 1
[2,1] => [1,1,0,0,1,0]
=> [2]
=> [1,1,0,0,1,0]
=> 2 = 1 + 1
[3] => [1,1,1,0,0,0]
=> []
=> []
=> ? = 1 + 1
[1,1,1,1] => [1,0,1,0,1,0,1,0]
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> 4 = 3 + 1
[1,2,1] => [1,0,1,1,0,0,1,0]
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> 2 = 1 + 1
[2,1,1] => [1,1,0,0,1,0,1,0]
=> [3,2]
=> [1,1,0,0,1,0,1,0]
=> 3 = 2 + 1
[3,1] => [1,1,1,0,0,0,1,0]
=> [3]
=> [1,1,1,0,0,0,1,0]
=> 2 = 1 + 1
[4] => [1,1,1,1,0,0,0,0]
=> []
=> []
=> ? = 1 + 1
[1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
=> [4,3,2,1]
=> [1,0,1,0,1,0,1,0,1,0]
=> 5 = 4 + 1
[1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
=> [4,2,2,1]
=> [1,0,1,0,1,1,0,0,1,0]
=> 3 = 2 + 1
[1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
=> [4,3,1,1]
=> [1,0,1,1,0,0,1,0,1,0]
=> 3 = 2 + 1
[1,3,1] => [1,0,1,1,1,0,0,0,1,0]
=> [4,1,1,1]
=> [1,0,1,1,1,0,0,0,1,0]
=> 2 = 1 + 1
[2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
=> [4,3,2]
=> [1,1,0,0,1,0,1,0,1,0]
=> 4 = 3 + 1
[2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> [4,2,2]
=> [1,1,0,0,1,1,0,0,1,0]
=> 2 = 1 + 1
[3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> [4,3]
=> [1,1,1,0,0,0,1,0,1,0]
=> 3 = 2 + 1
[4,1] => [1,1,1,1,0,0,0,0,1,0]
=> [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> 2 = 1 + 1
[5] => [1,1,1,1,1,0,0,0,0,0]
=> []
=> []
=> ? = 1 + 1
[1,1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0,1,0]
=> [5,4,3,2,1]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> 6 = 5 + 1
[1,1,1,2,1] => [1,0,1,0,1,0,1,1,0,0,1,0]
=> [5,3,3,2,1]
=> [1,0,1,0,1,0,1,1,0,0,1,0]
=> 4 = 3 + 1
[1,1,2,1,1] => [1,0,1,0,1,1,0,0,1,0,1,0]
=> [5,4,2,2,1]
=> [1,0,1,0,1,1,0,0,1,0,1,0]
=> 3 = 2 + 1
[1,1,3,1] => [1,0,1,0,1,1,1,0,0,0,1,0]
=> [5,2,2,2,1]
=> [1,0,1,0,1,1,1,0,0,0,1,0]
=> 3 = 2 + 1
[1,2,1,1,1] => [1,0,1,1,0,0,1,0,1,0,1,0]
=> [5,4,3,1,1]
=> [1,0,1,1,0,0,1,0,1,0,1,0]
=> 4 = 3 + 1
[1,2,2,1] => [1,0,1,1,0,0,1,1,0,0,1,0]
=> [5,3,3,1,1]
=> [1,0,1,1,0,0,1,1,0,0,1,0]
=> 2 = 1 + 1
[1,3,1,1] => [1,0,1,1,1,0,0,0,1,0,1,0]
=> [5,4,1,1,1]
=> [1,0,1,1,1,0,0,0,1,0,1,0]
=> 3 = 2 + 1
[1,4,1] => [1,0,1,1,1,1,0,0,0,0,1,0]
=> [5,1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0,1,0]
=> 2 = 1 + 1
[2,1,1,1,1] => [1,1,0,0,1,0,1,0,1,0,1,0]
=> [5,4,3,2]
=> [1,1,0,0,1,0,1,0,1,0,1,0]
=> 5 = 4 + 1
[2,1,2,1] => [1,1,0,0,1,0,1,1,0,0,1,0]
=> [5,3,3,2]
=> [1,1,0,0,1,0,1,1,0,0,1,0]
=> 3 = 2 + 1
[2,2,1,1] => [1,1,0,0,1,1,0,0,1,0,1,0]
=> [5,4,2,2]
=> [1,1,0,0,1,1,0,0,1,0,1,0]
=> 3 = 2 + 1
[2,3,1] => [1,1,0,0,1,1,1,0,0,0,1,0]
=> [5,2,2,2]
=> [1,1,0,0,1,1,1,0,0,0,1,0]
=> 2 = 1 + 1
[3,1,1,1] => [1,1,1,0,0,0,1,0,1,0,1,0]
=> [5,4,3]
=> [1,1,1,0,0,0,1,0,1,0,1,0]
=> 4 = 3 + 1
[3,2,1] => [1,1,1,0,0,0,1,1,0,0,1,0]
=> [5,3,3]
=> [1,1,1,0,0,0,1,1,0,0,1,0]
=> 2 = 1 + 1
[4,1,1] => [1,1,1,1,0,0,0,0,1,0,1,0]
=> [5,4]
=> [1,1,1,1,0,0,0,0,1,0,1,0]
=> 3 = 2 + 1
[5,1] => [1,1,1,1,1,0,0,0,0,0,1,0]
=> [5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> 2 = 1 + 1
[6] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> []
=> []
=> ? = 1 + 1
[1,1,1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [6,5,4,3,2,1]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> 7 = 6 + 1
[1,1,1,1,2,1] => [1,0,1,0,1,0,1,0,1,1,0,0,1,0]
=> [6,4,4,3,2,1]
=> [1,0,1,0,1,0,1,0,1,1,0,0,1,0]
=> 5 = 4 + 1
[1,1,1,2,1,1] => [1,0,1,0,1,0,1,1,0,0,1,0,1,0]
=> [6,5,3,3,2,1]
=> [1,0,1,0,1,0,1,1,0,0,1,0,1,0]
=> 4 = 3 + 1
[1,1,1,3,1] => [1,0,1,0,1,0,1,1,1,0,0,0,1,0]
=> [6,3,3,3,2,1]
=> [1,0,1,0,1,0,1,1,1,0,0,0,1,0]
=> 4 = 3 + 1
[1,1,2,1,1,1] => [1,0,1,0,1,1,0,0,1,0,1,0,1,0]
=> [6,5,4,2,2,1]
=> [1,0,1,0,1,1,0,0,1,0,1,0,1,0]
=> 4 = 3 + 1
[1,1,2,2,1] => [1,0,1,0,1,1,0,0,1,1,0,0,1,0]
=> [6,4,4,2,2,1]
=> [1,0,1,0,1,1,0,0,1,1,0,0,1,0]
=> 3 = 2 + 1
[1,1,3,1,1] => [1,0,1,0,1,1,1,0,0,0,1,0,1,0]
=> [6,5,2,2,2,1]
=> [1,0,1,0,1,1,1,0,0,0,1,0,1,0]
=> 3 = 2 + 1
[1,1,4,1] => [1,0,1,0,1,1,1,1,0,0,0,0,1,0]
=> [6,2,2,2,2,1]
=> [1,0,1,0,1,1,1,1,0,0,0,0,1,0]
=> 3 = 2 + 1
[1,2,1,1,1,1] => [1,0,1,1,0,0,1,0,1,0,1,0,1,0]
=> [6,5,4,3,1,1]
=> [1,0,1,1,0,0,1,0,1,0,1,0,1,0]
=> 5 = 4 + 1
[1,2,1,2,1] => [1,0,1,1,0,0,1,0,1,1,0,0,1,0]
=> [6,4,4,3,1,1]
=> [1,0,1,1,0,0,1,0,1,1,0,0,1,0]
=> 3 = 2 + 1
[1,2,2,1,1] => [1,0,1,1,0,0,1,1,0,0,1,0,1,0]
=> [6,5,3,3,1,1]
=> [1,0,1,1,0,0,1,1,0,0,1,0,1,0]
=> 3 = 2 + 1
[1,2,3,1] => [1,0,1,1,0,0,1,1,1,0,0,0,1,0]
=> [6,3,3,3,1,1]
=> [1,0,1,1,0,0,1,1,1,0,0,0,1,0]
=> 2 = 1 + 1
[1,3,1,1,1] => [1,0,1,1,1,0,0,0,1,0,1,0,1,0]
=> [6,5,4,1,1,1]
=> [1,0,1,1,1,0,0,0,1,0,1,0,1,0]
=> 4 = 3 + 1
[1,3,2,1] => [1,0,1,1,1,0,0,0,1,1,0,0,1,0]
=> [6,4,4,1,1,1]
=> [1,0,1,1,1,0,0,0,1,1,0,0,1,0]
=> 2 = 1 + 1
[1,4,1,1] => [1,0,1,1,1,1,0,0,0,0,1,0,1,0]
=> [6,5,1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0,1,0,1,0]
=> 3 = 2 + 1
[1,5,1] => [1,0,1,1,1,1,1,0,0,0,0,0,1,0]
=> [6,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,0,0,1,0]
=> 2 = 1 + 1
[2,1,1,1,1,1] => [1,1,0,0,1,0,1,0,1,0,1,0,1,0]
=> [6,5,4,3,2]
=> [1,1,0,0,1,0,1,0,1,0,1,0,1,0]
=> 6 = 5 + 1
[2,1,1,2,1] => [1,1,0,0,1,0,1,0,1,1,0,0,1,0]
=> [6,4,4,3,2]
=> [1,1,0,0,1,0,1,0,1,1,0,0,1,0]
=> 4 = 3 + 1
[2,1,2,1,1] => [1,1,0,0,1,0,1,1,0,0,1,0,1,0]
=> [6,5,3,3,2]
=> [1,1,0,0,1,0,1,1,0,0,1,0,1,0]
=> 3 = 2 + 1
[7] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> []
=> []
=> ? = 1 + 1
Description
The global dimension of the LNakayama algebra associated to a Dyck path.
An n-LNakayama algebra is a quiver algebra with a directed line as a connected quiver with $n$ points for $n \geq 2$. Number those points from the left to the right by $0,1,\ldots,n-1$.
The algebra is then uniquely determined by the dimension $c_i$ of the projective indecomposable modules at point $i$. Such algebras are then uniquely determined by lists of the form $[c_0,c_1,...,c_{n-1}]$ with the conditions: $c_{n-1}=1$ and $c_i -1 \leq c_{i+1}$ for all $i$. The number of such algebras is then the $n-1$-st Catalan number $C_{n-1}$.
One can get also an interpretation with Dyck paths by associating the top boundary of the Auslander-Reiten quiver (which is a Dyck path) to those algebras. Example: [3,4,3,3,2,1] corresponds to the Dyck path [1,1,0,1,1,0,0,1,0,0].
Conjecture: that there is an explicit bijection between $n$-LNakayama algebras with global dimension bounded by $m$ and Dyck paths with height at most $m$.
Examples:
* For $m=2$, the number of Dyck paths with global dimension at most $m$ starts for $n \geq 2$ with 1,2, 4, 8, 16, 32, 64, 128, 256, 512, 1024, 2048, 4096, 8192.
* For $m=3$, the number of Dyck paths with global dimension at most $m$ starts for $n \geq 2$ with 1, 2, 5, 13, 34, 89, 233, 610, 1597, 4181, 10946, 28657, 75025, 196418.
Matching statistic: St001118
Mp00231: Integer compositions —bounce path⟶ Dyck paths
Mp00129: Dyck paths —to 321-avoiding permutation (Billey-Jockusch-Stanley)⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
St001118: Graphs ⟶ ℤResult quality: 89% ●values known / values provided: 89%●distinct values known / distinct values provided: 100%
Mp00129: Dyck paths —to 321-avoiding permutation (Billey-Jockusch-Stanley)⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
St001118: Graphs ⟶ ℤResult quality: 89% ●values known / values provided: 89%●distinct values known / distinct values provided: 100%
Values
[1] => [1,0]
=> [1] => ([],1)
=> ? = 1
[1,1] => [1,0,1,0]
=> [2,1] => ([(0,1)],2)
=> 1
[2] => [1,1,0,0]
=> [1,2] => ([],2)
=> ? = 1
[1,1,1] => [1,0,1,0,1,0]
=> [2,3,1] => ([(0,2),(1,2)],3)
=> 2
[2,1] => [1,1,0,0,1,0]
=> [1,3,2] => ([(1,2)],3)
=> 1
[3] => [1,1,1,0,0,0]
=> [1,2,3] => ([],3)
=> ? = 1
[1,1,1,1] => [1,0,1,0,1,0,1,0]
=> [2,3,4,1] => ([(0,3),(1,3),(2,3)],4)
=> 3
[1,2,1] => [1,0,1,1,0,0,1,0]
=> [2,1,4,3] => ([(0,3),(1,2)],4)
=> 1
[2,1,1] => [1,1,0,0,1,0,1,0]
=> [1,3,4,2] => ([(1,3),(2,3)],4)
=> 2
[3,1] => [1,1,1,0,0,0,1,0]
=> [1,2,4,3] => ([(2,3)],4)
=> 1
[4] => [1,1,1,1,0,0,0,0]
=> [1,2,3,4] => ([],4)
=> ? = 1
[1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
=> [2,3,4,5,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 4
[1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
=> [2,3,1,5,4] => ([(0,1),(2,4),(3,4)],5)
=> 2
[1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
=> [2,1,4,5,3] => ([(0,1),(2,4),(3,4)],5)
=> 2
[1,3,1] => [1,0,1,1,1,0,0,0,1,0]
=> [2,1,3,5,4] => ([(1,4),(2,3)],5)
=> 1
[2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
=> [1,3,4,5,2] => ([(1,4),(2,4),(3,4)],5)
=> 3
[2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> [1,3,2,5,4] => ([(1,4),(2,3)],5)
=> 1
[3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> [1,2,4,5,3] => ([(2,4),(3,4)],5)
=> 2
[4,1] => [1,1,1,1,0,0,0,0,1,0]
=> [1,2,3,5,4] => ([(3,4)],5)
=> 1
[5] => [1,1,1,1,1,0,0,0,0,0]
=> [1,2,3,4,5] => ([],5)
=> ? = 1
[1,1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0,1,0]
=> [2,3,4,5,6,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> 5
[1,1,1,2,1] => [1,0,1,0,1,0,1,1,0,0,1,0]
=> [2,3,4,1,6,5] => ([(0,1),(2,5),(3,5),(4,5)],6)
=> 3
[1,1,2,1,1] => [1,0,1,0,1,1,0,0,1,0,1,0]
=> [2,3,1,5,6,4] => ([(0,5),(1,5),(2,4),(3,4)],6)
=> 2
[1,1,3,1] => [1,0,1,0,1,1,1,0,0,0,1,0]
=> [2,3,1,4,6,5] => ([(1,2),(3,5),(4,5)],6)
=> 2
[1,2,1,1,1] => [1,0,1,1,0,0,1,0,1,0,1,0]
=> [2,1,4,5,6,3] => ([(0,1),(2,5),(3,5),(4,5)],6)
=> 3
[1,2,2,1] => [1,0,1,1,0,0,1,1,0,0,1,0]
=> [2,1,4,3,6,5] => ([(0,5),(1,4),(2,3)],6)
=> 1
[1,3,1,1] => [1,0,1,1,1,0,0,0,1,0,1,0]
=> [2,1,3,5,6,4] => ([(1,2),(3,5),(4,5)],6)
=> 2
[1,4,1] => [1,0,1,1,1,1,0,0,0,0,1,0]
=> [2,1,3,4,6,5] => ([(2,5),(3,4)],6)
=> 1
[2,1,1,1,1] => [1,1,0,0,1,0,1,0,1,0,1,0]
=> [1,3,4,5,6,2] => ([(1,5),(2,5),(3,5),(4,5)],6)
=> 4
[2,1,2,1] => [1,1,0,0,1,0,1,1,0,0,1,0]
=> [1,3,4,2,6,5] => ([(1,2),(3,5),(4,5)],6)
=> 2
[2,2,1,1] => [1,1,0,0,1,1,0,0,1,0,1,0]
=> [1,3,2,5,6,4] => ([(1,2),(3,5),(4,5)],6)
=> 2
[2,3,1] => [1,1,0,0,1,1,1,0,0,0,1,0]
=> [1,3,2,4,6,5] => ([(2,5),(3,4)],6)
=> 1
[3,1,1,1] => [1,1,1,0,0,0,1,0,1,0,1,0]
=> [1,2,4,5,6,3] => ([(2,5),(3,5),(4,5)],6)
=> 3
[3,2,1] => [1,1,1,0,0,0,1,1,0,0,1,0]
=> [1,2,4,3,6,5] => ([(2,5),(3,4)],6)
=> 1
[4,1,1] => [1,1,1,1,0,0,0,0,1,0,1,0]
=> [1,2,3,5,6,4] => ([(3,5),(4,5)],6)
=> 2
[5,1] => [1,1,1,1,1,0,0,0,0,0,1,0]
=> [1,2,3,4,6,5] => ([(4,5)],6)
=> 1
[6] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,2,3,4,5,6] => ([],6)
=> ? = 1
[1,1,1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [2,3,4,5,6,7,1] => ([(0,6),(1,6),(2,6),(3,6),(4,6),(5,6)],7)
=> 6
[1,1,1,1,2,1] => [1,0,1,0,1,0,1,0,1,1,0,0,1,0]
=> [2,3,4,5,1,7,6] => ([(0,1),(2,6),(3,6),(4,6),(5,6)],7)
=> 4
[1,1,1,2,1,1] => [1,0,1,0,1,0,1,1,0,0,1,0,1,0]
=> [2,3,4,1,6,7,5] => ([(0,6),(1,6),(2,6),(3,5),(4,5)],7)
=> 3
[1,1,1,3,1] => [1,0,1,0,1,0,1,1,1,0,0,0,1,0]
=> [2,3,4,1,5,7,6] => ([(1,2),(3,6),(4,6),(5,6)],7)
=> 3
[1,1,2,1,1,1] => [1,0,1,0,1,1,0,0,1,0,1,0,1,0]
=> [2,3,1,5,6,7,4] => ([(0,6),(1,6),(2,6),(3,5),(4,5)],7)
=> 3
[1,1,2,2,1] => [1,0,1,0,1,1,0,0,1,1,0,0,1,0]
=> [2,3,1,5,4,7,6] => ([(0,3),(1,2),(4,6),(5,6)],7)
=> 2
[1,1,3,1,1] => [1,0,1,0,1,1,1,0,0,0,1,0,1,0]
=> [2,3,1,4,6,7,5] => ([(1,6),(2,6),(3,5),(4,5)],7)
=> 2
[1,1,4,1] => [1,0,1,0,1,1,1,1,0,0,0,0,1,0]
=> [2,3,1,4,5,7,6] => ([(2,3),(4,6),(5,6)],7)
=> 2
[1,2,1,1,1,1] => [1,0,1,1,0,0,1,0,1,0,1,0,1,0]
=> [2,1,4,5,6,7,3] => ([(0,1),(2,6),(3,6),(4,6),(5,6)],7)
=> 4
[1,2,1,2,1] => [1,0,1,1,0,0,1,0,1,1,0,0,1,0]
=> [2,1,4,5,3,7,6] => ([(0,3),(1,2),(4,6),(5,6)],7)
=> 2
[1,2,2,1,1] => [1,0,1,1,0,0,1,1,0,0,1,0,1,0]
=> [2,1,4,3,6,7,5] => ([(0,3),(1,2),(4,6),(5,6)],7)
=> 2
[1,2,3,1] => [1,0,1,1,0,0,1,1,1,0,0,0,1,0]
=> [2,1,4,3,5,7,6] => ([(1,6),(2,5),(3,4)],7)
=> 1
[1,3,1,1,1] => [1,0,1,1,1,0,0,0,1,0,1,0,1,0]
=> [2,1,3,5,6,7,4] => ([(1,2),(3,6),(4,6),(5,6)],7)
=> 3
[1,3,2,1] => [1,0,1,1,1,0,0,0,1,1,0,0,1,0]
=> [2,1,3,5,4,7,6] => ([(1,6),(2,5),(3,4)],7)
=> 1
[1,4,1,1] => [1,0,1,1,1,1,0,0,0,0,1,0,1,0]
=> [2,1,3,4,6,7,5] => ([(2,3),(4,6),(5,6)],7)
=> 2
[1,5,1] => [1,0,1,1,1,1,1,0,0,0,0,0,1,0]
=> [2,1,3,4,5,7,6] => ([(3,6),(4,5)],7)
=> 1
[2,1,1,1,1,1] => [1,1,0,0,1,0,1,0,1,0,1,0,1,0]
=> [1,3,4,5,6,7,2] => ([(1,6),(2,6),(3,6),(4,6),(5,6)],7)
=> 5
[2,1,1,2,1] => [1,1,0,0,1,0,1,0,1,1,0,0,1,0]
=> [1,3,4,5,2,7,6] => ([(1,2),(3,6),(4,6),(5,6)],7)
=> 3
[2,1,2,1,1] => [1,1,0,0,1,0,1,1,0,0,1,0,1,0]
=> [1,3,4,2,6,7,5] => ([(1,6),(2,6),(3,5),(4,5)],7)
=> 2
[7] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [1,2,3,4,5,6,7] => ([],7)
=> ? = 1
[4,1,1,1,1] => [1,1,1,1,0,0,0,0,1,0,1,0,1,0,1,0]
=> [1,2,3,5,6,7,8,4] => ([(3,7),(4,7),(5,7),(6,7)],8)
=> ? = 4
Description
The acyclic chromatic index of a graph.
An acyclic edge coloring of a graph is a proper colouring of the edges of a graph such that the union of the edges colored with any two given colours is a forest.
The smallest number of colours such that such a colouring exists is the acyclic chromatic index.
Matching statistic: St000686
Mp00231: Integer compositions —bounce path⟶ Dyck paths
Mp00027: Dyck paths —to partition⟶ Integer partitions
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
St000686: Dyck paths ⟶ ℤResult quality: 89% ●values known / values provided: 89%●distinct values known / distinct values provided: 100%
Mp00027: Dyck paths —to partition⟶ Integer partitions
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
St000686: Dyck paths ⟶ ℤResult quality: 89% ●values known / values provided: 89%●distinct values known / distinct values provided: 100%
Values
[1] => [1,0]
=> []
=> []
=> ? = 1 + 1
[1,1] => [1,0,1,0]
=> [1]
=> [1,0,1,0]
=> 2 = 1 + 1
[2] => [1,1,0,0]
=> []
=> []
=> ? = 1 + 1
[1,1,1] => [1,0,1,0,1,0]
=> [2,1]
=> [1,0,1,0,1,0]
=> 3 = 2 + 1
[2,1] => [1,1,0,0,1,0]
=> [2]
=> [1,1,0,0,1,0]
=> 2 = 1 + 1
[3] => [1,1,1,0,0,0]
=> []
=> []
=> ? = 1 + 1
[1,1,1,1] => [1,0,1,0,1,0,1,0]
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> 4 = 3 + 1
[1,2,1] => [1,0,1,1,0,0,1,0]
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> 2 = 1 + 1
[2,1,1] => [1,1,0,0,1,0,1,0]
=> [3,2]
=> [1,1,0,0,1,0,1,0]
=> 3 = 2 + 1
[3,1] => [1,1,1,0,0,0,1,0]
=> [3]
=> [1,1,1,0,0,0,1,0]
=> 2 = 1 + 1
[4] => [1,1,1,1,0,0,0,0]
=> []
=> []
=> ? = 1 + 1
[1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
=> [4,3,2,1]
=> [1,0,1,0,1,0,1,0,1,0]
=> 5 = 4 + 1
[1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
=> [4,2,2,1]
=> [1,0,1,0,1,1,0,0,1,0]
=> 3 = 2 + 1
[1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
=> [4,3,1,1]
=> [1,0,1,1,0,0,1,0,1,0]
=> 3 = 2 + 1
[1,3,1] => [1,0,1,1,1,0,0,0,1,0]
=> [4,1,1,1]
=> [1,0,1,1,1,0,0,0,1,0]
=> 2 = 1 + 1
[2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
=> [4,3,2]
=> [1,1,0,0,1,0,1,0,1,0]
=> 4 = 3 + 1
[2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> [4,2,2]
=> [1,1,0,0,1,1,0,0,1,0]
=> 2 = 1 + 1
[3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> [4,3]
=> [1,1,1,0,0,0,1,0,1,0]
=> 3 = 2 + 1
[4,1] => [1,1,1,1,0,0,0,0,1,0]
=> [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> 2 = 1 + 1
[5] => [1,1,1,1,1,0,0,0,0,0]
=> []
=> []
=> ? = 1 + 1
[1,1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0,1,0]
=> [5,4,3,2,1]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> 6 = 5 + 1
[1,1,1,2,1] => [1,0,1,0,1,0,1,1,0,0,1,0]
=> [5,3,3,2,1]
=> [1,0,1,0,1,0,1,1,0,0,1,0]
=> 4 = 3 + 1
[1,1,2,1,1] => [1,0,1,0,1,1,0,0,1,0,1,0]
=> [5,4,2,2,1]
=> [1,0,1,0,1,1,0,0,1,0,1,0]
=> 3 = 2 + 1
[1,1,3,1] => [1,0,1,0,1,1,1,0,0,0,1,0]
=> [5,2,2,2,1]
=> [1,0,1,0,1,1,1,0,0,0,1,0]
=> 3 = 2 + 1
[1,2,1,1,1] => [1,0,1,1,0,0,1,0,1,0,1,0]
=> [5,4,3,1,1]
=> [1,0,1,1,0,0,1,0,1,0,1,0]
=> 4 = 3 + 1
[1,2,2,1] => [1,0,1,1,0,0,1,1,0,0,1,0]
=> [5,3,3,1,1]
=> [1,0,1,1,0,0,1,1,0,0,1,0]
=> 2 = 1 + 1
[1,3,1,1] => [1,0,1,1,1,0,0,0,1,0,1,0]
=> [5,4,1,1,1]
=> [1,0,1,1,1,0,0,0,1,0,1,0]
=> 3 = 2 + 1
[1,4,1] => [1,0,1,1,1,1,0,0,0,0,1,0]
=> [5,1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0,1,0]
=> 2 = 1 + 1
[2,1,1,1,1] => [1,1,0,0,1,0,1,0,1,0,1,0]
=> [5,4,3,2]
=> [1,1,0,0,1,0,1,0,1,0,1,0]
=> 5 = 4 + 1
[2,1,2,1] => [1,1,0,0,1,0,1,1,0,0,1,0]
=> [5,3,3,2]
=> [1,1,0,0,1,0,1,1,0,0,1,0]
=> 3 = 2 + 1
[2,2,1,1] => [1,1,0,0,1,1,0,0,1,0,1,0]
=> [5,4,2,2]
=> [1,1,0,0,1,1,0,0,1,0,1,0]
=> 3 = 2 + 1
[2,3,1] => [1,1,0,0,1,1,1,0,0,0,1,0]
=> [5,2,2,2]
=> [1,1,0,0,1,1,1,0,0,0,1,0]
=> 2 = 1 + 1
[3,1,1,1] => [1,1,1,0,0,0,1,0,1,0,1,0]
=> [5,4,3]
=> [1,1,1,0,0,0,1,0,1,0,1,0]
=> 4 = 3 + 1
[3,2,1] => [1,1,1,0,0,0,1,1,0,0,1,0]
=> [5,3,3]
=> [1,1,1,0,0,0,1,1,0,0,1,0]
=> 2 = 1 + 1
[4,1,1] => [1,1,1,1,0,0,0,0,1,0,1,0]
=> [5,4]
=> [1,1,1,1,0,0,0,0,1,0,1,0]
=> 3 = 2 + 1
[5,1] => [1,1,1,1,1,0,0,0,0,0,1,0]
=> [5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> 2 = 1 + 1
[6] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> []
=> []
=> ? = 1 + 1
[1,1,1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [6,5,4,3,2,1]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> 7 = 6 + 1
[1,1,1,1,2,1] => [1,0,1,0,1,0,1,0,1,1,0,0,1,0]
=> [6,4,4,3,2,1]
=> [1,0,1,0,1,0,1,0,1,1,0,0,1,0]
=> 5 = 4 + 1
[1,1,1,2,1,1] => [1,0,1,0,1,0,1,1,0,0,1,0,1,0]
=> [6,5,3,3,2,1]
=> [1,0,1,0,1,0,1,1,0,0,1,0,1,0]
=> 4 = 3 + 1
[1,1,1,3,1] => [1,0,1,0,1,0,1,1,1,0,0,0,1,0]
=> [6,3,3,3,2,1]
=> [1,0,1,0,1,0,1,1,1,0,0,0,1,0]
=> 4 = 3 + 1
[1,1,2,1,1,1] => [1,0,1,0,1,1,0,0,1,0,1,0,1,0]
=> [6,5,4,2,2,1]
=> [1,0,1,0,1,1,0,0,1,0,1,0,1,0]
=> 4 = 3 + 1
[1,1,2,2,1] => [1,0,1,0,1,1,0,0,1,1,0,0,1,0]
=> [6,4,4,2,2,1]
=> [1,0,1,0,1,1,0,0,1,1,0,0,1,0]
=> 3 = 2 + 1
[1,1,3,1,1] => [1,0,1,0,1,1,1,0,0,0,1,0,1,0]
=> [6,5,2,2,2,1]
=> [1,0,1,0,1,1,1,0,0,0,1,0,1,0]
=> 3 = 2 + 1
[1,1,4,1] => [1,0,1,0,1,1,1,1,0,0,0,0,1,0]
=> [6,2,2,2,2,1]
=> [1,0,1,0,1,1,1,1,0,0,0,0,1,0]
=> 3 = 2 + 1
[1,2,1,1,1,1] => [1,0,1,1,0,0,1,0,1,0,1,0,1,0]
=> [6,5,4,3,1,1]
=> [1,0,1,1,0,0,1,0,1,0,1,0,1,0]
=> 5 = 4 + 1
[1,2,1,2,1] => [1,0,1,1,0,0,1,0,1,1,0,0,1,0]
=> [6,4,4,3,1,1]
=> [1,0,1,1,0,0,1,0,1,1,0,0,1,0]
=> 3 = 2 + 1
[1,2,2,1,1] => [1,0,1,1,0,0,1,1,0,0,1,0,1,0]
=> [6,5,3,3,1,1]
=> [1,0,1,1,0,0,1,1,0,0,1,0,1,0]
=> 3 = 2 + 1
[1,2,3,1] => [1,0,1,1,0,0,1,1,1,0,0,0,1,0]
=> [6,3,3,3,1,1]
=> [1,0,1,1,0,0,1,1,1,0,0,0,1,0]
=> 2 = 1 + 1
[1,3,1,1,1] => [1,0,1,1,1,0,0,0,1,0,1,0,1,0]
=> [6,5,4,1,1,1]
=> [1,0,1,1,1,0,0,0,1,0,1,0,1,0]
=> 4 = 3 + 1
[1,3,2,1] => [1,0,1,1,1,0,0,0,1,1,0,0,1,0]
=> [6,4,4,1,1,1]
=> [1,0,1,1,1,0,0,0,1,1,0,0,1,0]
=> 2 = 1 + 1
[1,4,1,1] => [1,0,1,1,1,1,0,0,0,0,1,0,1,0]
=> [6,5,1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0,1,0,1,0]
=> 3 = 2 + 1
[1,5,1] => [1,0,1,1,1,1,1,0,0,0,0,0,1,0]
=> [6,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,0,0,1,0]
=> 2 = 1 + 1
[2,1,1,1,1,1] => [1,1,0,0,1,0,1,0,1,0,1,0,1,0]
=> [6,5,4,3,2]
=> [1,1,0,0,1,0,1,0,1,0,1,0,1,0]
=> 6 = 5 + 1
[2,1,1,2,1] => [1,1,0,0,1,0,1,0,1,1,0,0,1,0]
=> [6,4,4,3,2]
=> [1,1,0,0,1,0,1,0,1,1,0,0,1,0]
=> 4 = 3 + 1
[2,1,2,1,1] => [1,1,0,0,1,0,1,1,0,0,1,0,1,0]
=> [6,5,3,3,2]
=> [1,1,0,0,1,0,1,1,0,0,1,0,1,0]
=> 3 = 2 + 1
[7] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> []
=> []
=> ? = 1 + 1
[4,1,1,1,1] => [1,1,1,1,0,0,0,0,1,0,1,0,1,0,1,0]
=> [7,6,5,4]
=> [1,1,1,1,0,0,0,0,1,0,1,0,1,0,1,0]
=> ? = 4 + 1
Description
The finitistic dominant dimension of a Dyck path.
To every LNakayama algebra there is a corresponding Dyck path, see also [[St000684]]. We associate the finitistic dominant dimension of the algebra to the corresponding Dyck path.
Matching statistic: St001596
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00041: Integer compositions —conjugate⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
Mp00233: Dyck paths —skew partition⟶ Skew partitions
St001596: Skew partitions ⟶ ℤResult quality: 50% ●values known / values provided: 55%●distinct values known / distinct values provided: 50%
Mp00231: Integer compositions —bounce path⟶ Dyck paths
Mp00233: Dyck paths —skew partition⟶ Skew partitions
St001596: Skew partitions ⟶ ℤResult quality: 50% ●values known / values provided: 55%●distinct values known / distinct values provided: 50%
Values
[1] => [1] => [1,0]
=> [[1],[]]
=> 0 = 1 - 1
[1,1] => [2] => [1,1,0,0]
=> [[2],[]]
=> 0 = 1 - 1
[2] => [1,1] => [1,0,1,0]
=> [[1,1],[]]
=> 0 = 1 - 1
[1,1,1] => [3] => [1,1,1,0,0,0]
=> [[2,2],[]]
=> 1 = 2 - 1
[2,1] => [2,1] => [1,1,0,0,1,0]
=> [[2,2],[1]]
=> 0 = 1 - 1
[3] => [1,1,1] => [1,0,1,0,1,0]
=> [[1,1,1],[]]
=> 0 = 1 - 1
[1,1,1,1] => [4] => [1,1,1,1,0,0,0,0]
=> [[3,3],[]]
=> 2 = 3 - 1
[1,2,1] => [2,2] => [1,1,0,0,1,1,0,0]
=> [[3,2],[1]]
=> 0 = 1 - 1
[2,1,1] => [3,1] => [1,1,1,0,0,0,1,0]
=> [[2,2,2],[1]]
=> 1 = 2 - 1
[3,1] => [2,1,1] => [1,1,0,0,1,0,1,0]
=> [[2,2,2],[1,1]]
=> 0 = 1 - 1
[4] => [1,1,1,1] => [1,0,1,0,1,0,1,0]
=> [[1,1,1,1],[]]
=> 0 = 1 - 1
[1,1,1,1,1] => [5] => [1,1,1,1,1,0,0,0,0,0]
=> [[3,3,3],[]]
=> ? = 4 - 1
[1,1,2,1] => [2,3] => [1,1,0,0,1,1,1,0,0,0]
=> [[3,3,2],[1,1]]
=> 1 = 2 - 1
[1,2,1,1] => [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> [[3,2,2],[1]]
=> 1 = 2 - 1
[1,3,1] => [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
=> [[3,2,2],[1,1]]
=> 0 = 1 - 1
[2,1,1,1] => [4,1] => [1,1,1,1,0,0,0,0,1,0]
=> [[3,3,3],[2]]
=> 2 = 3 - 1
[2,2,1] => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> [[3,3,2],[2,1]]
=> 0 = 1 - 1
[3,1,1] => [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> [[2,2,2,2],[1,1]]
=> 1 = 2 - 1
[4,1] => [2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
=> [[2,2,2,2],[1,1,1]]
=> 0 = 1 - 1
[5] => [1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
=> [[1,1,1,1,1],[]]
=> 0 = 1 - 1
[1,1,1,1,1,1] => [6] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> [[4,4,4],[]]
=> ? = 5 - 1
[1,1,1,2,1] => [2,4] => [1,1,0,0,1,1,1,1,0,0,0,0]
=> [[4,4,2],[1,1]]
=> ? = 3 - 1
[1,1,2,1,1] => [3,3] => [1,1,1,0,0,0,1,1,1,0,0,0]
=> [[3,3,2,2],[1,1]]
=> ? = 2 - 1
[1,1,3,1] => [2,1,3] => [1,1,0,0,1,0,1,1,1,0,0,0]
=> [[3,3,2,2],[1,1,1]]
=> 1 = 2 - 1
[1,2,1,1,1] => [4,2] => [1,1,1,1,0,0,0,0,1,1,0,0]
=> [[4,3,3],[2]]
=> ? = 3 - 1
[1,2,2,1] => [2,2,2] => [1,1,0,0,1,1,0,0,1,1,0,0]
=> [[4,3,2],[2,1]]
=> 0 = 1 - 1
[1,3,1,1] => [3,1,2] => [1,1,1,0,0,0,1,0,1,1,0,0]
=> [[3,2,2,2],[1,1]]
=> 1 = 2 - 1
[1,4,1] => [2,1,1,2] => [1,1,0,0,1,0,1,0,1,1,0,0]
=> [[3,2,2,2],[1,1,1]]
=> 0 = 1 - 1
[2,1,1,1,1] => [5,1] => [1,1,1,1,1,0,0,0,0,0,1,0]
=> [[3,3,3,3],[2]]
=> ? = 4 - 1
[2,1,2,1] => [2,3,1] => [1,1,0,0,1,1,1,0,0,0,1,0]
=> [[3,3,3,2],[2,1,1]]
=> 1 = 2 - 1
[2,2,1,1] => [3,2,1] => [1,1,1,0,0,0,1,1,0,0,1,0]
=> [[3,3,2,2],[2,1]]
=> 1 = 2 - 1
[2,3,1] => [2,1,2,1] => [1,1,0,0,1,0,1,1,0,0,1,0]
=> [[3,3,2,2],[2,1,1]]
=> 0 = 1 - 1
[3,1,1,1] => [4,1,1] => [1,1,1,1,0,0,0,0,1,0,1,0]
=> [[3,3,3,3],[2,2]]
=> ? = 3 - 1
[3,2,1] => [2,2,1,1] => [1,1,0,0,1,1,0,0,1,0,1,0]
=> [[3,3,3,2],[2,2,1]]
=> 0 = 1 - 1
[4,1,1] => [3,1,1,1] => [1,1,1,0,0,0,1,0,1,0,1,0]
=> [[2,2,2,2,2],[1,1,1]]
=> 1 = 2 - 1
[5,1] => [2,1,1,1,1] => [1,1,0,0,1,0,1,0,1,0,1,0]
=> [[2,2,2,2,2],[1,1,1,1]]
=> 0 = 1 - 1
[6] => [1,1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0,1,0]
=> [[1,1,1,1,1,1],[]]
=> 0 = 1 - 1
[1,1,1,1,1,1,1] => [7] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [[4,4,4,4],[]]
=> ? = 6 - 1
[1,1,1,1,2,1] => [2,5] => [1,1,0,0,1,1,1,1,1,0,0,0,0,0]
=> [[4,4,4,2],[1,1,1]]
=> ? = 4 - 1
[1,1,1,2,1,1] => [3,4] => [1,1,1,0,0,0,1,1,1,1,0,0,0,0]
=> [[4,4,2,2],[1,1]]
=> ? = 3 - 1
[1,1,1,3,1] => [2,1,4] => [1,1,0,0,1,0,1,1,1,1,0,0,0,0]
=> [[4,4,2,2],[1,1,1]]
=> ? = 3 - 1
[1,1,2,1,1,1] => [4,3] => [1,1,1,1,0,0,0,0,1,1,1,0,0,0]
=> [[4,4,3,3],[2,2]]
=> ? = 3 - 1
[1,1,2,2,1] => [2,2,3] => [1,1,0,0,1,1,0,0,1,1,1,0,0,0]
=> [[4,4,3,2],[2,2,1]]
=> ? = 2 - 1
[1,1,3,1,1] => [3,1,3] => [1,1,1,0,0,0,1,0,1,1,1,0,0,0]
=> [[3,3,2,2,2],[1,1,1]]
=> ? = 2 - 1
[1,1,4,1] => [2,1,1,3] => [1,1,0,0,1,0,1,0,1,1,1,0,0,0]
=> [[3,3,2,2,2],[1,1,1,1]]
=> ? = 2 - 1
[1,2,1,1,1,1] => [5,2] => [1,1,1,1,1,0,0,0,0,0,1,1,0,0]
=> [[4,3,3,3],[2]]
=> ? = 4 - 1
[1,2,1,2,1] => [2,3,2] => [1,1,0,0,1,1,1,0,0,0,1,1,0,0]
=> [[4,3,3,2],[2,1,1]]
=> ? = 2 - 1
[1,2,2,1,1] => [3,2,2] => [1,1,1,0,0,0,1,1,0,0,1,1,0,0]
=> [[4,3,2,2],[2,1]]
=> ? = 2 - 1
[1,2,3,1] => [2,1,2,2] => [1,1,0,0,1,0,1,1,0,0,1,1,0,0]
=> [[4,3,2,2],[2,1,1]]
=> 0 = 1 - 1
[1,3,1,1,1] => [4,1,2] => [1,1,1,1,0,0,0,0,1,0,1,1,0,0]
=> [[4,3,3,3],[2,2]]
=> ? = 3 - 1
[1,3,2,1] => [2,2,1,2] => [1,1,0,0,1,1,0,0,1,0,1,1,0,0]
=> [[4,3,3,2],[2,2,1]]
=> 0 = 1 - 1
[1,4,1,1] => [3,1,1,2] => [1,1,1,0,0,0,1,0,1,0,1,1,0,0]
=> [[3,2,2,2,2],[1,1,1]]
=> ? = 2 - 1
[1,5,1] => [2,1,1,1,2] => [1,1,0,0,1,0,1,0,1,0,1,1,0,0]
=> [[3,2,2,2,2],[1,1,1,1]]
=> 0 = 1 - 1
[2,1,1,1,1,1] => [6,1] => [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> [[4,4,4,4],[3]]
=> ? = 5 - 1
[2,1,1,2,1] => [2,4,1] => [1,1,0,0,1,1,1,1,0,0,0,0,1,0]
=> [[4,4,4,2],[3,1,1]]
=> ? = 3 - 1
[2,1,2,1,1] => [3,3,1] => [1,1,1,0,0,0,1,1,1,0,0,0,1,0]
=> [[3,3,3,2,2],[2,1,1]]
=> ? = 2 - 1
[2,1,3,1] => [2,1,3,1] => [1,1,0,0,1,0,1,1,1,0,0,0,1,0]
=> [[3,3,3,2,2],[2,1,1,1]]
=> ? = 2 - 1
[2,2,1,1,1] => [4,2,1] => [1,1,1,1,0,0,0,0,1,1,0,0,1,0]
=> [[4,4,3,3],[3,2]]
=> ? = 3 - 1
[2,2,2,1] => [2,2,2,1] => [1,1,0,0,1,1,0,0,1,1,0,0,1,0]
=> [[4,4,3,2],[3,2,1]]
=> 0 = 1 - 1
[2,3,1,1] => [3,1,2,1] => [1,1,1,0,0,0,1,0,1,1,0,0,1,0]
=> [[3,3,2,2,2],[2,1,1]]
=> ? = 2 - 1
[2,4,1] => [2,1,1,2,1] => [1,1,0,0,1,0,1,0,1,1,0,0,1,0]
=> [[3,3,2,2,2],[2,1,1,1]]
=> 0 = 1 - 1
[3,1,1,1,1] => [5,1,1] => [1,1,1,1,1,0,0,0,0,0,1,0,1,0]
=> [[3,3,3,3,3],[2,2]]
=> ? = 4 - 1
[3,1,2,1] => [2,3,1,1] => [1,1,0,0,1,1,1,0,0,0,1,0,1,0]
=> [[3,3,3,3,2],[2,2,1,1]]
=> ? = 2 - 1
[3,2,1,1] => [3,2,1,1] => [1,1,1,0,0,0,1,1,0,0,1,0,1,0]
=> [[3,3,3,2,2],[2,2,1]]
=> ? = 2 - 1
[3,3,1] => [2,1,2,1,1] => [1,1,0,0,1,0,1,1,0,0,1,0,1,0]
=> [[3,3,3,2,2],[2,2,1,1]]
=> 0 = 1 - 1
[4,1,1,1] => [4,1,1,1] => [1,1,1,1,0,0,0,0,1,0,1,0,1,0]
=> [[3,3,3,3,3],[2,2,2]]
=> ? = 3 - 1
[4,2,1] => [2,2,1,1,1] => [1,1,0,0,1,1,0,0,1,0,1,0,1,0]
=> [[3,3,3,3,2],[2,2,2,1]]
=> 0 = 1 - 1
[5,1,1] => [3,1,1,1,1] => [1,1,1,0,0,0,1,0,1,0,1,0,1,0]
=> [[2,2,2,2,2,2],[1,1,1,1]]
=> ? = 2 - 1
[6,1] => [2,1,1,1,1,1] => [1,1,0,0,1,0,1,0,1,0,1,0,1,0]
=> [[2,2,2,2,2,2],[1,1,1,1,1]]
=> 0 = 1 - 1
[7] => [1,1,1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [[1,1,1,1,1,1,1],[]]
=> 0 = 1 - 1
[4,1,1,1,1] => [5,1,1,1] => [1,1,1,1,1,0,0,0,0,0,1,0,1,0,1,0]
=> [[3,3,3,3,3,3],[2,2,2]]
=> ? = 4 - 1
Description
The number of two-by-two squares inside a skew partition.
This is, the number of cells $(i,j)$ in a skew partition for which the box $(i+1,j+1)$ is also a cell inside the skew partition.
Matching statistic: St001231
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00041: Integer compositions —conjugate⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
Mp00121: Dyck paths —Cori-Le Borgne involution⟶ Dyck paths
St001231: Dyck paths ⟶ ℤResult quality: 52% ●values known / values provided: 52%●distinct values known / distinct values provided: 83%
Mp00231: Integer compositions —bounce path⟶ Dyck paths
Mp00121: Dyck paths —Cori-Le Borgne involution⟶ Dyck paths
St001231: Dyck paths ⟶ ℤResult quality: 52% ●values known / values provided: 52%●distinct values known / distinct values provided: 83%
Values
[1] => [1] => [1,0]
=> [1,0]
=> 0 = 1 - 1
[1,1] => [2] => [1,1,0,0]
=> [1,1,0,0]
=> 0 = 1 - 1
[2] => [1,1] => [1,0,1,0]
=> [1,0,1,0]
=> 0 = 1 - 1
[1,1,1] => [3] => [1,1,1,0,0,0]
=> [1,1,1,0,0,0]
=> 1 = 2 - 1
[2,1] => [2,1] => [1,1,0,0,1,0]
=> [1,1,0,0,1,0]
=> 0 = 1 - 1
[3] => [1,1,1] => [1,0,1,0,1,0]
=> [1,0,1,0,1,0]
=> 0 = 1 - 1
[1,1,1,1] => [4] => [1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> 2 = 3 - 1
[1,2,1] => [2,2] => [1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> 0 = 1 - 1
[2,1,1] => [3,1] => [1,1,1,0,0,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> 1 = 2 - 1
[3,1] => [2,1,1] => [1,1,0,0,1,0,1,0]
=> [1,1,0,0,1,0,1,0]
=> 0 = 1 - 1
[4] => [1,1,1,1] => [1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0]
=> 0 = 1 - 1
[1,1,1,1,1] => [5] => [1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> 3 = 4 - 1
[1,1,2,1] => [2,3] => [1,1,0,0,1,1,1,0,0,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> 1 = 2 - 1
[1,2,1,1] => [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> 1 = 2 - 1
[1,3,1] => [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
=> [1,1,0,1,0,0,1,1,0,0]
=> 0 = 1 - 1
[2,1,1,1] => [4,1] => [1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> 2 = 3 - 1
[2,2,1] => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> 0 = 1 - 1
[3,1,1] => [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> 1 = 2 - 1
[4,1] => [2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> 0 = 1 - 1
[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]
=> 0 = 1 - 1
[1,1,1,1,1,1] => [6] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0]
=> 4 = 5 - 1
[1,1,1,2,1] => [2,4] => [1,1,0,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,1,1,0,0,0,0]
=> 2 = 3 - 1
[1,1,2,1,1] => [3,3] => [1,1,1,0,0,0,1,1,1,0,0,0]
=> [1,1,1,0,0,0,1,1,1,0,0,0]
=> 1 = 2 - 1
[1,1,3,1] => [2,1,3] => [1,1,0,0,1,0,1,1,1,0,0,0]
=> [1,1,1,0,1,0,0,1,1,0,0,0]
=> 1 = 2 - 1
[1,2,1,1,1] => [4,2] => [1,1,1,1,0,0,0,0,1,1,0,0]
=> [1,1,1,1,0,0,0,1,1,0,0,0]
=> 2 = 3 - 1
[1,2,2,1] => [2,2,2] => [1,1,0,0,1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,0,1,1,0,0]
=> 0 = 1 - 1
[1,3,1,1] => [3,1,2] => [1,1,1,0,0,0,1,0,1,1,0,0]
=> [1,1,1,0,0,0,1,1,0,1,0,0]
=> 1 = 2 - 1
[1,4,1] => [2,1,1,2] => [1,1,0,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,0,1,1,0,0]
=> 0 = 1 - 1
[2,1,1,1,1] => [5,1] => [1,1,1,1,1,0,0,0,0,0,1,0]
=> [1,1,1,1,1,0,0,1,0,0,0,0]
=> 3 = 4 - 1
[2,1,2,1] => [2,3,1] => [1,1,0,0,1,1,1,0,0,0,1,0]
=> [1,1,1,0,0,1,1,0,0,1,0,0]
=> 1 = 2 - 1
[2,2,1,1] => [3,2,1] => [1,1,1,0,0,0,1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0,1,1,0,0]
=> 1 = 2 - 1
[2,3,1] => [2,1,2,1] => [1,1,0,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,0,0,1,1,0,0,1,0]
=> 0 = 1 - 1
[3,1,1,1] => [4,1,1] => [1,1,1,1,0,0,0,0,1,0,1,0]
=> [1,1,1,1,0,0,1,0,1,0,0,0]
=> 2 = 3 - 1
[3,2,1] => [2,2,1,1] => [1,1,0,0,1,1,0,0,1,0,1,0]
=> [1,1,0,0,1,1,0,0,1,0,1,0]
=> 0 = 1 - 1
[4,1,1] => [3,1,1,1] => [1,1,1,0,0,0,1,0,1,0,1,0]
=> [1,1,1,0,0,1,0,1,0,1,0,0]
=> 1 = 2 - 1
[5,1] => [2,1,1,1,1] => [1,1,0,0,1,0,1,0,1,0,1,0]
=> [1,1,0,0,1,0,1,0,1,0,1,0]
=> 0 = 1 - 1
[6] => [1,1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> 0 = 1 - 1
[1,1,1,1,1,1,1] => [7] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> ? = 6 - 1
[1,1,1,1,2,1] => [2,5] => [1,1,0,0,1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1,0,0,1,1,0,0,0,0,0]
=> ? = 4 - 1
[1,1,1,2,1,1] => [3,4] => [1,1,1,0,0,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,1,1,1,0,0,0,0]
=> ? = 3 - 1
[1,1,1,3,1] => [2,1,4] => [1,1,0,0,1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,1,0,0,1,1,0,0,0,0]
=> ? = 3 - 1
[1,1,2,1,1,1] => [4,3] => [1,1,1,1,0,0,0,0,1,1,1,0,0,0]
=> [1,1,1,1,0,0,0,0,1,1,1,0,0,0]
=> ? = 3 - 1
[1,1,2,2,1] => [2,2,3] => [1,1,0,0,1,1,0,0,1,1,1,0,0,0]
=> [1,1,1,0,0,1,1,0,0,1,1,0,0,0]
=> ? = 2 - 1
[1,1,3,1,1] => [3,1,3] => [1,1,1,0,0,0,1,0,1,1,1,0,0,0]
=> [1,1,1,0,1,0,0,0,1,1,1,0,0,0]
=> ? = 2 - 1
[1,1,4,1] => [2,1,1,3] => [1,1,0,0,1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,0,1,0,1,0,0,1,1,0,0,0]
=> ? = 2 - 1
[1,2,1,1,1,1] => [5,2] => [1,1,1,1,1,0,0,0,0,0,1,1,0,0]
=> [1,1,1,1,1,0,0,0,1,1,0,0,0,0]
=> ? = 4 - 1
[1,2,1,2,1] => [2,3,2] => [1,1,0,0,1,1,1,0,0,0,1,1,0,0]
=> [1,1,1,0,0,1,1,0,0,0,1,1,0,0]
=> ? = 2 - 1
[1,2,2,1,1] => [3,2,2] => [1,1,1,0,0,0,1,1,0,0,1,1,0,0]
=> [1,1,1,0,0,0,1,1,0,0,1,1,0,0]
=> ? = 2 - 1
[1,2,3,1] => [2,1,2,2] => [1,1,0,0,1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,1,0,0,1,1,0,0]
=> ? = 1 - 1
[1,3,1,1,1] => [4,1,2] => [1,1,1,1,0,0,0,0,1,0,1,1,0,0]
=> [1,1,1,1,0,0,0,1,1,0,1,0,0,0]
=> ? = 3 - 1
[1,3,2,1] => [2,2,1,2] => [1,1,0,0,1,1,0,0,1,0,1,1,0,0]
=> [1,1,0,1,0,0,1,1,0,0,1,1,0,0]
=> ? = 1 - 1
[1,4,1,1] => [3,1,1,2] => [1,1,1,0,0,0,1,0,1,0,1,1,0,0]
=> [1,1,1,0,0,0,1,1,0,1,0,1,0,0]
=> ? = 2 - 1
[1,5,1] => [2,1,1,1,2] => [1,1,0,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,1,0,0,1,1,0,0]
=> ? = 1 - 1
[2,1,1,1,1,1] => [6,1] => [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> [1,1,1,1,1,1,0,0,1,0,0,0,0,0]
=> ? = 5 - 1
[2,1,1,2,1] => [2,4,1] => [1,1,0,0,1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,1,0,0,1,1,0,0,1,0,0,0]
=> ? = 3 - 1
[2,1,2,1,1] => [3,3,1] => [1,1,1,0,0,0,1,1,1,0,0,0,1,0]
=> [1,1,1,0,0,0,1,1,1,0,0,1,0,0]
=> ? = 2 - 1
[2,1,3,1] => [2,1,3,1] => [1,1,0,0,1,0,1,1,1,0,0,0,1,0]
=> [1,1,1,0,1,0,0,1,1,0,0,1,0,0]
=> ? = 2 - 1
[2,2,1,1,1] => [4,2,1] => [1,1,1,1,0,0,0,0,1,1,0,0,1,0]
=> [1,1,1,1,0,0,1,0,0,1,1,0,0,0]
=> ? = 3 - 1
[2,2,2,1] => [2,2,2,1] => [1,1,0,0,1,1,0,0,1,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0,1,1,0,0,1,0]
=> ? = 1 - 1
[2,3,1,1] => [3,1,2,1] => [1,1,1,0,0,0,1,0,1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0,1,1,0,1,0,0]
=> ? = 2 - 1
[2,4,1] => [2,1,1,2,1] => [1,1,0,0,1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,0,1,0,0,1,1,0,0,1,0]
=> ? = 1 - 1
[3,1,1,1,1] => [5,1,1] => [1,1,1,1,1,0,0,0,0,0,1,0,1,0]
=> [1,1,1,1,1,0,0,1,0,1,0,0,0,0]
=> ? = 4 - 1
[3,1,2,1] => [2,3,1,1] => [1,1,0,0,1,1,1,0,0,0,1,0,1,0]
=> [1,1,1,0,0,1,1,0,0,1,0,1,0,0]
=> ? = 2 - 1
[3,2,1,1] => [3,2,1,1] => [1,1,1,0,0,0,1,1,0,0,1,0,1,0]
=> [1,1,1,0,0,1,0,1,0,0,1,1,0,0]
=> ? = 2 - 1
[3,3,1] => [2,1,2,1,1] => [1,1,0,0,1,0,1,1,0,0,1,0,1,0]
=> [1,1,0,1,0,0,1,1,0,0,1,0,1,0]
=> ? = 1 - 1
[4,1,1,1] => [4,1,1,1] => [1,1,1,1,0,0,0,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,1,0,1,0,1,0,0,0]
=> ? = 3 - 1
[4,2,1] => [2,2,1,1,1] => [1,1,0,0,1,1,0,0,1,0,1,0,1,0]
=> [1,1,0,0,1,1,0,0,1,0,1,0,1,0]
=> ? = 1 - 1
[5,1,1] => [3,1,1,1,1] => [1,1,1,0,0,0,1,0,1,0,1,0,1,0]
=> [1,1,1,0,0,1,0,1,0,1,0,1,0,0]
=> ? = 2 - 1
[6,1] => [2,1,1,1,1,1] => [1,1,0,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,0,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 1 - 1
[7] => [1,1,1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 1 - 1
[4,1,1,1,1] => [5,1,1,1] => [1,1,1,1,1,0,0,0,0,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,1,0,1,0,1,0,0,0,0]
=> ? = 4 - 1
Description
The number of simple modules that are non-projective and non-injective with the property that they have projective dimension equal to one and that also the Auslander-Reiten translates of the module and the inverse Auslander-Reiten translate of the module have the same projective dimension.
Actually the same statistics results for algebras with at most 7 simple modules when dropping the assumption that the module has projective dimension one. The author is not sure whether this holds in general.
The following 31 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St001234The number of indecomposable three dimensional modules with projective dimension one. 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. St000930The k-Gorenstein degree of the corresponding Nakayama algebra with linear quiver. St001530The depth of a Dyck path. St001549The number of restricted non-inversions between exceedances. St001060The distinguishing index of a graph. St001238The number of simple modules S such that the Auslander-Reiten translate of S is isomorphic to the Nakayama functor applied to the second syzygy of S. St001314The number of tilting modules of arbitrary projective dimension that have no simple modules as a direct summand in the corresponding Nakayama algebra. St001066The number of simple reflexive modules in the corresponding Nakayama algebra. St000365The number of double ascents of a permutation. St000954Number of times the corresponding LNakayama algebra has $Ext^i(D(A),A)=0$ for $i>0$. St001811The Castelnuovo-Mumford regularity of a permutation. St001624The breadth of a lattice. St001845The number of join irreducibles minus the rank of a lattice. St001862The number of crossings of a signed permutation. St001882The number of occurrences of a type-B 231 pattern in a signed permutation. St000732The number of double deficiencies of a permutation. St001186Number of simple modules with grade at least 3 in the corresponding Nakayama algebra. St001651The Frankl number of a lattice. St001906Half of the difference between the total displacement and the number of inversions and the reflection length of a permutation. St001942The number of loops of the quiver corresponding to the reduced incidence algebra of a poset. St001095The number of non-isomorphic posets with precisely one further covering relation. St000772The multiplicity of the largest distance Laplacian eigenvalue in a connected graph. St000777The number of distinct eigenvalues of the distance Laplacian of a connected graph. St001645The pebbling number of a connected graph. St000259The diameter of a connected graph. St000260The radius of a connected graph. St000302The determinant of the distance matrix of a connected graph. St000466The Gutman (or modified Schultz) index of a connected graph. St000467The hyper-Wiener index of a connected graph.
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