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Your data matches 142 different statistics following compositions of up to 3 maps.
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Matching statistic: St000929
Mp00274: Graphs —block-cut tree⟶ Graphs
Mp00037: Graphs —to partition of connected components⟶ Integer partitions
St000929: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00037: Graphs —to partition of connected components⟶ Integer partitions
St000929: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
([],2)
=> ([],2)
=> [1,1]
=> 1
([],3)
=> ([],3)
=> [1,1,1]
=> 1
([(1,2)],3)
=> ([],2)
=> [1,1]
=> 1
([(0,2),(1,2)],3)
=> ([(0,2),(1,2)],3)
=> [3]
=> 0
([],4)
=> ([],4)
=> [1,1,1,1]
=> 1
([(2,3)],4)
=> ([],3)
=> [1,1,1]
=> 1
([(1,3),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> [3,1]
=> 0
([(0,3),(1,3),(2,3)],4)
=> ([(0,3),(1,3),(2,3)],4)
=> [4]
=> 0
([(0,3),(1,2)],4)
=> ([],2)
=> [1,1]
=> 1
([(0,3),(1,2),(2,3)],4)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> [5]
=> 0
([(1,2),(1,3),(2,3)],4)
=> ([],2)
=> [1,1]
=> 1
([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,2),(1,2)],3)
=> [3]
=> 0
([],5)
=> ([],5)
=> [1,1,1,1,1]
=> 1
([(3,4)],5)
=> ([],4)
=> [1,1,1,1]
=> 1
([(2,4),(3,4)],5)
=> ([(2,4),(3,4)],5)
=> [3,1,1]
=> 0
([(1,4),(2,4),(3,4)],5)
=> ([(1,4),(2,4),(3,4)],5)
=> [4,1]
=> 0
([(0,4),(1,4),(2,4),(3,4)],5)
=> ([(0,4),(1,4),(2,4),(3,4)],5)
=> [5]
=> 0
([(1,4),(2,3)],5)
=> ([],3)
=> [1,1,1]
=> 1
([(1,4),(2,3),(3,4)],5)
=> ([(1,5),(2,4),(3,4),(3,5)],6)
=> [5,1]
=> 0
([(0,1),(2,4),(3,4)],5)
=> ([(1,3),(2,3)],4)
=> [3,1]
=> 0
([(2,3),(2,4),(3,4)],5)
=> ([],3)
=> [1,1,1]
=> 1
([(0,4),(1,4),(2,3),(3,4)],5)
=> ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
=> [6]
=> 0
([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,3),(2,3)],4)
=> [3,1]
=> 0
([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(1,3),(2,3)],4)
=> [4]
=> 0
([(1,3),(1,4),(2,3),(2,4)],5)
=> ([],2)
=> [1,1]
=> 1
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> ([(0,2),(1,2)],3)
=> [3]
=> 0
([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([],2)
=> [1,1]
=> 1
([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> [5]
=> 0
([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,2),(1,2)],3)
=> [3]
=> 0
([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,6),(1,5),(2,3),(2,4),(3,5),(4,6)],7)
=> [7]
=> 0
([(0,1),(2,3),(2,4),(3,4)],5)
=> ([],2)
=> [1,1]
=> 1
([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> [5]
=> 0
([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
=> ([(0,2),(1,2)],3)
=> [3]
=> 0
([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(0,2),(1,2)],3)
=> [3]
=> 0
([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([],2)
=> [1,1]
=> 1
([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,2),(1,2)],3)
=> [3]
=> 0
([],6)
=> ([],6)
=> [1,1,1,1,1,1]
=> 1
([(4,5)],6)
=> ([],5)
=> [1,1,1,1,1]
=> 1
([(3,5),(4,5)],6)
=> ([(3,5),(4,5)],6)
=> [3,1,1,1]
=> 0
([(2,5),(3,5),(4,5)],6)
=> ([(2,5),(3,5),(4,5)],6)
=> [4,1,1]
=> 0
([(1,5),(2,5),(3,5),(4,5)],6)
=> ([(1,5),(2,5),(3,5),(4,5)],6)
=> [5,1]
=> 0
([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> [6]
=> 0
([(2,5),(3,4)],6)
=> ([],4)
=> [1,1,1,1]
=> 1
([(2,5),(3,4),(4,5)],6)
=> ([(2,6),(3,5),(4,5),(4,6)],7)
=> [5,1,1]
=> 0
([(1,2),(3,5),(4,5)],6)
=> ([(2,4),(3,4)],5)
=> [3,1,1]
=> 0
([(3,4),(3,5),(4,5)],6)
=> ([],4)
=> [1,1,1,1]
=> 1
([(1,5),(2,5),(3,4),(4,5)],6)
=> ([(1,6),(2,6),(3,4),(4,5),(5,6)],7)
=> [6,1]
=> 0
([(0,1),(2,5),(3,5),(4,5)],6)
=> ([(1,4),(2,4),(3,4)],5)
=> [4,1]
=> 0
([(2,5),(3,4),(3,5),(4,5)],6)
=> ([(2,4),(3,4)],5)
=> [3,1,1]
=> 0
([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
=> ([(0,6),(1,6),(2,6),(3,4),(4,5),(5,6)],7)
=> [7]
=> 0
Description
The constant term of the character polynomial of an integer partition.
The definition of the character polynomial can be found in [1]. Indeed, this constant term is $0$ for partitions $\lambda \neq 1^n$ and $1$ for $\lambda = 1^n$.
Matching statistic: St001593
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00274: Graphs —block-cut tree⟶ Graphs
Mp00037: Graphs —to partition of connected components⟶ Integer partitions
Mp00044: Integer partitions —conjugate⟶ Integer partitions
St001593: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00037: Graphs —to partition of connected components⟶ Integer partitions
Mp00044: Integer partitions —conjugate⟶ Integer partitions
St001593: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
([],2)
=> ([],2)
=> [1,1]
=> [2]
=> 1
([],3)
=> ([],3)
=> [1,1,1]
=> [3]
=> 1
([(1,2)],3)
=> ([],2)
=> [1,1]
=> [2]
=> 1
([(0,2),(1,2)],3)
=> ([(0,2),(1,2)],3)
=> [3]
=> [1,1,1]
=> 0
([],4)
=> ([],4)
=> [1,1,1,1]
=> [4]
=> 1
([(2,3)],4)
=> ([],3)
=> [1,1,1]
=> [3]
=> 1
([(1,3),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> [3,1]
=> [2,1,1]
=> 0
([(0,3),(1,3),(2,3)],4)
=> ([(0,3),(1,3),(2,3)],4)
=> [4]
=> [1,1,1,1]
=> 0
([(0,3),(1,2)],4)
=> ([],2)
=> [1,1]
=> [2]
=> 1
([(0,3),(1,2),(2,3)],4)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> [5]
=> [1,1,1,1,1]
=> 0
([(1,2),(1,3),(2,3)],4)
=> ([],2)
=> [1,1]
=> [2]
=> 1
([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,2),(1,2)],3)
=> [3]
=> [1,1,1]
=> 0
([],5)
=> ([],5)
=> [1,1,1,1,1]
=> [5]
=> 1
([(3,4)],5)
=> ([],4)
=> [1,1,1,1]
=> [4]
=> 1
([(2,4),(3,4)],5)
=> ([(2,4),(3,4)],5)
=> [3,1,1]
=> [3,1,1]
=> 0
([(1,4),(2,4),(3,4)],5)
=> ([(1,4),(2,4),(3,4)],5)
=> [4,1]
=> [2,1,1,1]
=> 0
([(0,4),(1,4),(2,4),(3,4)],5)
=> ([(0,4),(1,4),(2,4),(3,4)],5)
=> [5]
=> [1,1,1,1,1]
=> 0
([(1,4),(2,3)],5)
=> ([],3)
=> [1,1,1]
=> [3]
=> 1
([(1,4),(2,3),(3,4)],5)
=> ([(1,5),(2,4),(3,4),(3,5)],6)
=> [5,1]
=> [2,1,1,1,1]
=> 0
([(0,1),(2,4),(3,4)],5)
=> ([(1,3),(2,3)],4)
=> [3,1]
=> [2,1,1]
=> 0
([(2,3),(2,4),(3,4)],5)
=> ([],3)
=> [1,1,1]
=> [3]
=> 1
([(0,4),(1,4),(2,3),(3,4)],5)
=> ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
=> [6]
=> [1,1,1,1,1,1]
=> 0
([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,3),(2,3)],4)
=> [3,1]
=> [2,1,1]
=> 0
([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(1,3),(2,3)],4)
=> [4]
=> [1,1,1,1]
=> 0
([(1,3),(1,4),(2,3),(2,4)],5)
=> ([],2)
=> [1,1]
=> [2]
=> 1
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> ([(0,2),(1,2)],3)
=> [3]
=> [1,1,1]
=> 0
([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([],2)
=> [1,1]
=> [2]
=> 1
([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> [5]
=> [1,1,1,1,1]
=> 0
([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,2),(1,2)],3)
=> [3]
=> [1,1,1]
=> 0
([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,6),(1,5),(2,3),(2,4),(3,5),(4,6)],7)
=> [7]
=> [1,1,1,1,1,1,1]
=> 0
([(0,1),(2,3),(2,4),(3,4)],5)
=> ([],2)
=> [1,1]
=> [2]
=> 1
([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> [5]
=> [1,1,1,1,1]
=> 0
([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
=> ([(0,2),(1,2)],3)
=> [3]
=> [1,1,1]
=> 0
([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(0,2),(1,2)],3)
=> [3]
=> [1,1,1]
=> 0
([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([],2)
=> [1,1]
=> [2]
=> 1
([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,2),(1,2)],3)
=> [3]
=> [1,1,1]
=> 0
([],6)
=> ([],6)
=> [1,1,1,1,1,1]
=> [6]
=> 1
([(4,5)],6)
=> ([],5)
=> [1,1,1,1,1]
=> [5]
=> 1
([(3,5),(4,5)],6)
=> ([(3,5),(4,5)],6)
=> [3,1,1,1]
=> [4,1,1]
=> 0
([(2,5),(3,5),(4,5)],6)
=> ([(2,5),(3,5),(4,5)],6)
=> [4,1,1]
=> [3,1,1,1]
=> 0
([(1,5),(2,5),(3,5),(4,5)],6)
=> ([(1,5),(2,5),(3,5),(4,5)],6)
=> [5,1]
=> [2,1,1,1,1]
=> 0
([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> [6]
=> [1,1,1,1,1,1]
=> 0
([(2,5),(3,4)],6)
=> ([],4)
=> [1,1,1,1]
=> [4]
=> 1
([(2,5),(3,4),(4,5)],6)
=> ([(2,6),(3,5),(4,5),(4,6)],7)
=> [5,1,1]
=> [3,1,1,1,1]
=> 0
([(1,2),(3,5),(4,5)],6)
=> ([(2,4),(3,4)],5)
=> [3,1,1]
=> [3,1,1]
=> 0
([(3,4),(3,5),(4,5)],6)
=> ([],4)
=> [1,1,1,1]
=> [4]
=> 1
([(1,5),(2,5),(3,4),(4,5)],6)
=> ([(1,6),(2,6),(3,4),(4,5),(5,6)],7)
=> [6,1]
=> [2,1,1,1,1,1]
=> 0
([(0,1),(2,5),(3,5),(4,5)],6)
=> ([(1,4),(2,4),(3,4)],5)
=> [4,1]
=> [2,1,1,1]
=> 0
([(2,5),(3,4),(3,5),(4,5)],6)
=> ([(2,4),(3,4)],5)
=> [3,1,1]
=> [3,1,1]
=> 0
([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
=> ([(0,6),(1,6),(2,6),(3,4),(4,5),(5,6)],7)
=> [7]
=> [1,1,1,1,1,1,1]
=> 0
Description
This is the number of standard Young tableaux of the given shifted shape.
For an integer partition $\lambda = (\lambda_1,\dots,\lambda_k)$, the shifted diagram is obtained by moving the $i$-th row in the diagram $i-1$ boxes to the right, i.e.,
$$\lambda^∗ = \{(i, j) | 1 \leq i \leq k, i \leq j \leq \lambda_i + i − 1 \}.$$
In particular, this statistic is zero if and only if $\lambda_{i+1} = \lambda_i$ for some $i$.
Matching statistic: St000268
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Values
([],2)
=> ([],2)
=> ([],1)
=> 1
([],3)
=> ([],3)
=> ([],1)
=> 1
([(1,2)],3)
=> ([],2)
=> ([],1)
=> 1
([(0,2),(1,2)],3)
=> ([(0,2),(1,2)],3)
=> ([(0,1)],2)
=> 0
([],4)
=> ([],4)
=> ([],1)
=> 1
([(2,3)],4)
=> ([],3)
=> ([],1)
=> 1
([(1,3),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> ([(1,2)],3)
=> 0
([(0,3),(1,3),(2,3)],4)
=> ([(0,3),(1,3),(2,3)],4)
=> ([(0,1)],2)
=> 0
([(0,3),(1,2)],4)
=> ([],2)
=> ([],1)
=> 1
([(0,3),(1,2),(2,3)],4)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 0
([(1,2),(1,3),(2,3)],4)
=> ([],2)
=> ([],1)
=> 1
([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,2),(1,2)],3)
=> ([(0,1)],2)
=> 0
([],5)
=> ([],5)
=> ([],1)
=> 1
([(3,4)],5)
=> ([],4)
=> ([],1)
=> 1
([(2,4),(3,4)],5)
=> ([(2,4),(3,4)],5)
=> ([(1,2)],3)
=> 0
([(1,4),(2,4),(3,4)],5)
=> ([(1,4),(2,4),(3,4)],5)
=> ([(1,2)],3)
=> 0
([(0,4),(1,4),(2,4),(3,4)],5)
=> ([(0,4),(1,4),(2,4),(3,4)],5)
=> ([(0,1)],2)
=> 0
([(1,4),(2,3)],5)
=> ([],3)
=> ([],1)
=> 1
([(1,4),(2,3),(3,4)],5)
=> ([(1,5),(2,4),(3,4),(3,5)],6)
=> ([(1,5),(2,4),(3,4),(3,5)],6)
=> 0
([(0,1),(2,4),(3,4)],5)
=> ([(1,3),(2,3)],4)
=> ([(1,2)],3)
=> 0
([(2,3),(2,4),(3,4)],5)
=> ([],3)
=> ([],1)
=> 1
([(0,4),(1,4),(2,3),(3,4)],5)
=> ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 0
([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,3),(2,3)],4)
=> ([(1,2)],3)
=> 0
([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(1,3),(2,3)],4)
=> ([(0,1)],2)
=> 0
([(1,3),(1,4),(2,3),(2,4)],5)
=> ([],2)
=> ([],1)
=> 1
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> ([(0,2),(1,2)],3)
=> ([(0,1)],2)
=> 0
([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([],2)
=> ([],1)
=> 1
([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 0
([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,2),(1,2)],3)
=> ([(0,1)],2)
=> 0
([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,6),(1,5),(2,3),(2,4),(3,5),(4,6)],7)
=> ([(0,6),(1,5),(2,3),(2,4),(3,5),(4,6)],7)
=> 0
([(0,1),(2,3),(2,4),(3,4)],5)
=> ([],2)
=> ([],1)
=> 1
([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 0
([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
=> ([(0,2),(1,2)],3)
=> ([(0,1)],2)
=> 0
([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(0,2),(1,2)],3)
=> ([(0,1)],2)
=> 0
([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([],2)
=> ([],1)
=> 1
([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,2),(1,2)],3)
=> ([(0,1)],2)
=> 0
([],6)
=> ([],6)
=> ([],1)
=> 1
([(4,5)],6)
=> ([],5)
=> ([],1)
=> 1
([(3,5),(4,5)],6)
=> ([(3,5),(4,5)],6)
=> ([(1,2)],3)
=> 0
([(2,5),(3,5),(4,5)],6)
=> ([(2,5),(3,5),(4,5)],6)
=> ([(1,2)],3)
=> 0
([(1,5),(2,5),(3,5),(4,5)],6)
=> ([(1,5),(2,5),(3,5),(4,5)],6)
=> ([(1,2)],3)
=> 0
([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> ([(0,1)],2)
=> 0
([(2,5),(3,4)],6)
=> ([],4)
=> ([],1)
=> 1
([(2,5),(3,4),(4,5)],6)
=> ([(2,6),(3,5),(4,5),(4,6)],7)
=> ([(1,5),(2,4),(3,4),(3,5)],6)
=> 0
([(1,2),(3,5),(4,5)],6)
=> ([(2,4),(3,4)],5)
=> ([(1,2)],3)
=> 0
([(3,4),(3,5),(4,5)],6)
=> ([],4)
=> ([],1)
=> 1
([(1,5),(2,5),(3,4),(4,5)],6)
=> ([(1,6),(2,6),(3,4),(4,5),(5,6)],7)
=> ([(1,5),(2,4),(3,4),(3,5)],6)
=> 0
([(0,1),(2,5),(3,5),(4,5)],6)
=> ([(1,4),(2,4),(3,4)],5)
=> ([(1,2)],3)
=> 0
([(2,5),(3,4),(3,5),(4,5)],6)
=> ([(2,4),(3,4)],5)
=> ([(1,2)],3)
=> 0
([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
=> ([(0,6),(1,6),(2,6),(3,4),(4,5),(5,6)],7)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 0
([(1,5),(2,4),(3,4),(3,5)],6)
=> ([(1,7),(2,6),(3,4),(3,5),(4,6),(5,7)],8)
=> ([(1,7),(2,6),(3,4),(3,5),(4,6),(5,7)],8)
=> ? = 0
([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
=> ([(0,6),(1,5),(2,7),(3,5),(3,7),(4,6),(4,7)],8)
=> ([(0,6),(1,5),(2,7),(3,5),(3,7),(4,6),(4,7)],8)
=> ? = 0
([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
=> ([(0,8),(1,7),(2,3),(2,4),(3,5),(4,6),(5,7),(6,8)],9)
=> ([(0,8),(1,7),(2,3),(2,4),(3,5),(4,6),(5,7),(6,8)],9)
=> ? = 0
([(1,5),(2,3),(2,6),(3,6),(4,5),(4,6)],7)
=> ([(1,7),(2,6),(3,4),(3,5),(4,6),(5,7)],8)
=> ([(1,7),(2,6),(3,4),(3,5),(4,6),(5,7)],8)
=> ? = 0
([(1,5),(2,3),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(1,7),(2,6),(3,4),(3,5),(4,6),(5,7)],8)
=> ([(1,7),(2,6),(3,4),(3,5),(4,6),(5,7)],8)
=> ? = 0
([(0,6),(1,5),(2,3),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,6),(1,5),(2,7),(3,5),(3,7),(4,6),(4,7)],8)
=> ([(0,6),(1,5),(2,7),(3,5),(3,7),(4,6),(4,7)],8)
=> ? = 0
([(0,6),(1,4),(2,5),(2,6),(3,4),(3,5),(5,6)],7)
=> ([(0,8),(1,7),(2,3),(2,4),(3,5),(4,6),(5,7),(6,8)],9)
=> ([(0,8),(1,7),(2,3),(2,4),(3,5),(4,6),(5,7),(6,8)],9)
=> ? = 0
([(0,1),(2,5),(3,4),(4,6),(5,6)],7)
=> ([(1,7),(2,6),(3,4),(3,5),(4,6),(5,7)],8)
=> ([(1,7),(2,6),(3,4),(3,5),(4,6),(5,7)],8)
=> ? = 0
([(0,5),(1,4),(2,3),(2,6),(3,6),(4,6),(5,6)],7)
=> ([(0,6),(1,5),(2,7),(3,5),(3,7),(4,6),(4,7)],8)
=> ([(0,6),(1,5),(2,7),(3,5),(3,7),(4,6),(4,7)],8)
=> ? = 0
([(0,4),(1,3),(2,5),(2,6),(3,5),(4,6),(5,6)],7)
=> ([(0,8),(1,7),(2,3),(2,4),(3,5),(4,6),(5,7),(6,8)],9)
=> ([(0,8),(1,7),(2,3),(2,4),(3,5),(4,6),(5,7),(6,8)],9)
=> ? = 0
([(0,6),(1,4),(2,3),(2,5),(3,5),(4,6),(5,6)],7)
=> ([(0,6),(1,5),(2,7),(3,5),(3,7),(4,6),(4,7)],8)
=> ([(0,6),(1,5),(2,7),(3,5),(3,7),(4,6),(4,7)],8)
=> ? = 0
([(0,5),(1,4),(1,5),(2,3),(2,6),(3,6),(4,6)],7)
=> ([(0,8),(1,7),(2,3),(2,4),(3,5),(4,6),(5,7),(6,8)],9)
=> ([(0,8),(1,7),(2,3),(2,4),(3,5),(4,6),(5,7),(6,8)],9)
=> ? = 0
Description
The number of strongly connected orientations of a graph.
Matching statistic: St000344
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Values
([],2)
=> ([],2)
=> ([],1)
=> 1
([],3)
=> ([],3)
=> ([],1)
=> 1
([(1,2)],3)
=> ([],2)
=> ([],1)
=> 1
([(0,2),(1,2)],3)
=> ([(0,2),(1,2)],3)
=> ([(0,1)],2)
=> 0
([],4)
=> ([],4)
=> ([],1)
=> 1
([(2,3)],4)
=> ([],3)
=> ([],1)
=> 1
([(1,3),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> ([(1,2)],3)
=> 0
([(0,3),(1,3),(2,3)],4)
=> ([(0,3),(1,3),(2,3)],4)
=> ([(0,1)],2)
=> 0
([(0,3),(1,2)],4)
=> ([],2)
=> ([],1)
=> 1
([(0,3),(1,2),(2,3)],4)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 0
([(1,2),(1,3),(2,3)],4)
=> ([],2)
=> ([],1)
=> 1
([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,2),(1,2)],3)
=> ([(0,1)],2)
=> 0
([],5)
=> ([],5)
=> ([],1)
=> 1
([(3,4)],5)
=> ([],4)
=> ([],1)
=> 1
([(2,4),(3,4)],5)
=> ([(2,4),(3,4)],5)
=> ([(1,2)],3)
=> 0
([(1,4),(2,4),(3,4)],5)
=> ([(1,4),(2,4),(3,4)],5)
=> ([(1,2)],3)
=> 0
([(0,4),(1,4),(2,4),(3,4)],5)
=> ([(0,4),(1,4),(2,4),(3,4)],5)
=> ([(0,1)],2)
=> 0
([(1,4),(2,3)],5)
=> ([],3)
=> ([],1)
=> 1
([(1,4),(2,3),(3,4)],5)
=> ([(1,5),(2,4),(3,4),(3,5)],6)
=> ([(1,5),(2,4),(3,4),(3,5)],6)
=> 0
([(0,1),(2,4),(3,4)],5)
=> ([(1,3),(2,3)],4)
=> ([(1,2)],3)
=> 0
([(2,3),(2,4),(3,4)],5)
=> ([],3)
=> ([],1)
=> 1
([(0,4),(1,4),(2,3),(3,4)],5)
=> ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 0
([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,3),(2,3)],4)
=> ([(1,2)],3)
=> 0
([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(1,3),(2,3)],4)
=> ([(0,1)],2)
=> 0
([(1,3),(1,4),(2,3),(2,4)],5)
=> ([],2)
=> ([],1)
=> 1
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> ([(0,2),(1,2)],3)
=> ([(0,1)],2)
=> 0
([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([],2)
=> ([],1)
=> 1
([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 0
([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,2),(1,2)],3)
=> ([(0,1)],2)
=> 0
([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,6),(1,5),(2,3),(2,4),(3,5),(4,6)],7)
=> ([(0,6),(1,5),(2,3),(2,4),(3,5),(4,6)],7)
=> 0
([(0,1),(2,3),(2,4),(3,4)],5)
=> ([],2)
=> ([],1)
=> 1
([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 0
([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
=> ([(0,2),(1,2)],3)
=> ([(0,1)],2)
=> 0
([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(0,2),(1,2)],3)
=> ([(0,1)],2)
=> 0
([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([],2)
=> ([],1)
=> 1
([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,2),(1,2)],3)
=> ([(0,1)],2)
=> 0
([],6)
=> ([],6)
=> ([],1)
=> 1
([(4,5)],6)
=> ([],5)
=> ([],1)
=> 1
([(3,5),(4,5)],6)
=> ([(3,5),(4,5)],6)
=> ([(1,2)],3)
=> 0
([(2,5),(3,5),(4,5)],6)
=> ([(2,5),(3,5),(4,5)],6)
=> ([(1,2)],3)
=> 0
([(1,5),(2,5),(3,5),(4,5)],6)
=> ([(1,5),(2,5),(3,5),(4,5)],6)
=> ([(1,2)],3)
=> 0
([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> ([(0,1)],2)
=> 0
([(2,5),(3,4)],6)
=> ([],4)
=> ([],1)
=> 1
([(2,5),(3,4),(4,5)],6)
=> ([(2,6),(3,5),(4,5),(4,6)],7)
=> ([(1,5),(2,4),(3,4),(3,5)],6)
=> 0
([(1,2),(3,5),(4,5)],6)
=> ([(2,4),(3,4)],5)
=> ([(1,2)],3)
=> 0
([(3,4),(3,5),(4,5)],6)
=> ([],4)
=> ([],1)
=> 1
([(1,5),(2,5),(3,4),(4,5)],6)
=> ([(1,6),(2,6),(3,4),(4,5),(5,6)],7)
=> ([(1,5),(2,4),(3,4),(3,5)],6)
=> 0
([(0,1),(2,5),(3,5),(4,5)],6)
=> ([(1,4),(2,4),(3,4)],5)
=> ([(1,2)],3)
=> 0
([(2,5),(3,4),(3,5),(4,5)],6)
=> ([(2,4),(3,4)],5)
=> ([(1,2)],3)
=> 0
([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
=> ([(0,6),(1,6),(2,6),(3,4),(4,5),(5,6)],7)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 0
([(1,5),(2,4),(3,4),(3,5)],6)
=> ([(1,7),(2,6),(3,4),(3,5),(4,6),(5,7)],8)
=> ([(1,7),(2,6),(3,4),(3,5),(4,6),(5,7)],8)
=> ? = 0
([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
=> ([(0,6),(1,5),(2,7),(3,5),(3,7),(4,6),(4,7)],8)
=> ([(0,6),(1,5),(2,7),(3,5),(3,7),(4,6),(4,7)],8)
=> ? = 0
([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
=> ([(0,8),(1,7),(2,3),(2,4),(3,5),(4,6),(5,7),(6,8)],9)
=> ([(0,8),(1,7),(2,3),(2,4),(3,5),(4,6),(5,7),(6,8)],9)
=> ? = 0
([(1,5),(2,3),(2,6),(3,6),(4,5),(4,6)],7)
=> ([(1,7),(2,6),(3,4),(3,5),(4,6),(5,7)],8)
=> ([(1,7),(2,6),(3,4),(3,5),(4,6),(5,7)],8)
=> ? = 0
([(1,5),(2,3),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(1,7),(2,6),(3,4),(3,5),(4,6),(5,7)],8)
=> ([(1,7),(2,6),(3,4),(3,5),(4,6),(5,7)],8)
=> ? = 0
([(0,6),(1,5),(2,3),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,6),(1,5),(2,7),(3,5),(3,7),(4,6),(4,7)],8)
=> ([(0,6),(1,5),(2,7),(3,5),(3,7),(4,6),(4,7)],8)
=> ? = 0
([(0,6),(1,4),(2,5),(2,6),(3,4),(3,5),(5,6)],7)
=> ([(0,8),(1,7),(2,3),(2,4),(3,5),(4,6),(5,7),(6,8)],9)
=> ([(0,8),(1,7),(2,3),(2,4),(3,5),(4,6),(5,7),(6,8)],9)
=> ? = 0
([(0,1),(2,5),(3,4),(4,6),(5,6)],7)
=> ([(1,7),(2,6),(3,4),(3,5),(4,6),(5,7)],8)
=> ([(1,7),(2,6),(3,4),(3,5),(4,6),(5,7)],8)
=> ? = 0
([(0,5),(1,4),(2,3),(2,6),(3,6),(4,6),(5,6)],7)
=> ([(0,6),(1,5),(2,7),(3,5),(3,7),(4,6),(4,7)],8)
=> ([(0,6),(1,5),(2,7),(3,5),(3,7),(4,6),(4,7)],8)
=> ? = 0
([(0,4),(1,3),(2,5),(2,6),(3,5),(4,6),(5,6)],7)
=> ([(0,8),(1,7),(2,3),(2,4),(3,5),(4,6),(5,7),(6,8)],9)
=> ([(0,8),(1,7),(2,3),(2,4),(3,5),(4,6),(5,7),(6,8)],9)
=> ? = 0
([(0,6),(1,4),(2,3),(2,5),(3,5),(4,6),(5,6)],7)
=> ([(0,6),(1,5),(2,7),(3,5),(3,7),(4,6),(4,7)],8)
=> ([(0,6),(1,5),(2,7),(3,5),(3,7),(4,6),(4,7)],8)
=> ? = 0
([(0,5),(1,4),(1,5),(2,3),(2,6),(3,6),(4,6)],7)
=> ([(0,8),(1,7),(2,3),(2,4),(3,5),(4,6),(5,7),(6,8)],9)
=> ([(0,8),(1,7),(2,3),(2,4),(3,5),(4,6),(5,7),(6,8)],9)
=> ? = 0
Description
The number of strongly connected outdegree sequences of a graph.
This is the evaluation of the Tutte polynomial at $x=0$ and $y=1$. According to [1,2], the set of strongly connected outdegree sequences is in bijection with strongly connected minimal orientations and also with external spanning trees.
Matching statistic: St001073
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Values
([],2)
=> ([],2)
=> ([],1)
=> 1
([],3)
=> ([],3)
=> ([],1)
=> 1
([(1,2)],3)
=> ([],2)
=> ([],1)
=> 1
([(0,2),(1,2)],3)
=> ([(0,2),(1,2)],3)
=> ([(0,1)],2)
=> 0
([],4)
=> ([],4)
=> ([],1)
=> 1
([(2,3)],4)
=> ([],3)
=> ([],1)
=> 1
([(1,3),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> ([(1,2)],3)
=> 0
([(0,3),(1,3),(2,3)],4)
=> ([(0,3),(1,3),(2,3)],4)
=> ([(0,1)],2)
=> 0
([(0,3),(1,2)],4)
=> ([],2)
=> ([],1)
=> 1
([(0,3),(1,2),(2,3)],4)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 0
([(1,2),(1,3),(2,3)],4)
=> ([],2)
=> ([],1)
=> 1
([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,2),(1,2)],3)
=> ([(0,1)],2)
=> 0
([],5)
=> ([],5)
=> ([],1)
=> 1
([(3,4)],5)
=> ([],4)
=> ([],1)
=> 1
([(2,4),(3,4)],5)
=> ([(2,4),(3,4)],5)
=> ([(1,2)],3)
=> 0
([(1,4),(2,4),(3,4)],5)
=> ([(1,4),(2,4),(3,4)],5)
=> ([(1,2)],3)
=> 0
([(0,4),(1,4),(2,4),(3,4)],5)
=> ([(0,4),(1,4),(2,4),(3,4)],5)
=> ([(0,1)],2)
=> 0
([(1,4),(2,3)],5)
=> ([],3)
=> ([],1)
=> 1
([(1,4),(2,3),(3,4)],5)
=> ([(1,5),(2,4),(3,4),(3,5)],6)
=> ([(1,5),(2,4),(3,4),(3,5)],6)
=> 0
([(0,1),(2,4),(3,4)],5)
=> ([(1,3),(2,3)],4)
=> ([(1,2)],3)
=> 0
([(2,3),(2,4),(3,4)],5)
=> ([],3)
=> ([],1)
=> 1
([(0,4),(1,4),(2,3),(3,4)],5)
=> ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 0
([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,3),(2,3)],4)
=> ([(1,2)],3)
=> 0
([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(1,3),(2,3)],4)
=> ([(0,1)],2)
=> 0
([(1,3),(1,4),(2,3),(2,4)],5)
=> ([],2)
=> ([],1)
=> 1
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> ([(0,2),(1,2)],3)
=> ([(0,1)],2)
=> 0
([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([],2)
=> ([],1)
=> 1
([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 0
([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,2),(1,2)],3)
=> ([(0,1)],2)
=> 0
([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,6),(1,5),(2,3),(2,4),(3,5),(4,6)],7)
=> ([(0,6),(1,5),(2,3),(2,4),(3,5),(4,6)],7)
=> 0
([(0,1),(2,3),(2,4),(3,4)],5)
=> ([],2)
=> ([],1)
=> 1
([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 0
([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
=> ([(0,2),(1,2)],3)
=> ([(0,1)],2)
=> 0
([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(0,2),(1,2)],3)
=> ([(0,1)],2)
=> 0
([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([],2)
=> ([],1)
=> 1
([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,2),(1,2)],3)
=> ([(0,1)],2)
=> 0
([],6)
=> ([],6)
=> ([],1)
=> 1
([(4,5)],6)
=> ([],5)
=> ([],1)
=> 1
([(3,5),(4,5)],6)
=> ([(3,5),(4,5)],6)
=> ([(1,2)],3)
=> 0
([(2,5),(3,5),(4,5)],6)
=> ([(2,5),(3,5),(4,5)],6)
=> ([(1,2)],3)
=> 0
([(1,5),(2,5),(3,5),(4,5)],6)
=> ([(1,5),(2,5),(3,5),(4,5)],6)
=> ([(1,2)],3)
=> 0
([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> ([(0,1)],2)
=> 0
([(2,5),(3,4)],6)
=> ([],4)
=> ([],1)
=> 1
([(2,5),(3,4),(4,5)],6)
=> ([(2,6),(3,5),(4,5),(4,6)],7)
=> ([(1,5),(2,4),(3,4),(3,5)],6)
=> 0
([(1,2),(3,5),(4,5)],6)
=> ([(2,4),(3,4)],5)
=> ([(1,2)],3)
=> 0
([(3,4),(3,5),(4,5)],6)
=> ([],4)
=> ([],1)
=> 1
([(1,5),(2,5),(3,4),(4,5)],6)
=> ([(1,6),(2,6),(3,4),(4,5),(5,6)],7)
=> ([(1,5),(2,4),(3,4),(3,5)],6)
=> 0
([(0,1),(2,5),(3,5),(4,5)],6)
=> ([(1,4),(2,4),(3,4)],5)
=> ([(1,2)],3)
=> 0
([(2,5),(3,4),(3,5),(4,5)],6)
=> ([(2,4),(3,4)],5)
=> ([(1,2)],3)
=> 0
([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
=> ([(0,6),(1,6),(2,6),(3,4),(4,5),(5,6)],7)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 0
([(1,5),(2,4),(3,4),(3,5)],6)
=> ([(1,7),(2,6),(3,4),(3,5),(4,6),(5,7)],8)
=> ([(1,7),(2,6),(3,4),(3,5),(4,6),(5,7)],8)
=> ? = 0
([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
=> ([(0,6),(1,5),(2,7),(3,5),(3,7),(4,6),(4,7)],8)
=> ([(0,6),(1,5),(2,7),(3,5),(3,7),(4,6),(4,7)],8)
=> ? = 0
([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
=> ([(0,8),(1,7),(2,3),(2,4),(3,5),(4,6),(5,7),(6,8)],9)
=> ([(0,8),(1,7),(2,3),(2,4),(3,5),(4,6),(5,7),(6,8)],9)
=> ? = 0
([(1,5),(2,3),(2,6),(3,6),(4,5),(4,6)],7)
=> ([(1,7),(2,6),(3,4),(3,5),(4,6),(5,7)],8)
=> ([(1,7),(2,6),(3,4),(3,5),(4,6),(5,7)],8)
=> ? = 0
([(1,5),(2,3),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(1,7),(2,6),(3,4),(3,5),(4,6),(5,7)],8)
=> ([(1,7),(2,6),(3,4),(3,5),(4,6),(5,7)],8)
=> ? = 0
([(0,6),(1,5),(2,3),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,6),(1,5),(2,7),(3,5),(3,7),(4,6),(4,7)],8)
=> ([(0,6),(1,5),(2,7),(3,5),(3,7),(4,6),(4,7)],8)
=> ? = 0
([(0,6),(1,4),(2,5),(2,6),(3,4),(3,5),(5,6)],7)
=> ([(0,8),(1,7),(2,3),(2,4),(3,5),(4,6),(5,7),(6,8)],9)
=> ([(0,8),(1,7),(2,3),(2,4),(3,5),(4,6),(5,7),(6,8)],9)
=> ? = 0
([(0,1),(2,5),(3,4),(4,6),(5,6)],7)
=> ([(1,7),(2,6),(3,4),(3,5),(4,6),(5,7)],8)
=> ([(1,7),(2,6),(3,4),(3,5),(4,6),(5,7)],8)
=> ? = 0
([(0,5),(1,4),(2,3),(2,6),(3,6),(4,6),(5,6)],7)
=> ([(0,6),(1,5),(2,7),(3,5),(3,7),(4,6),(4,7)],8)
=> ([(0,6),(1,5),(2,7),(3,5),(3,7),(4,6),(4,7)],8)
=> ? = 0
([(0,4),(1,3),(2,5),(2,6),(3,5),(4,6),(5,6)],7)
=> ([(0,8),(1,7),(2,3),(2,4),(3,5),(4,6),(5,7),(6,8)],9)
=> ([(0,8),(1,7),(2,3),(2,4),(3,5),(4,6),(5,7),(6,8)],9)
=> ? = 0
([(0,6),(1,4),(2,3),(2,5),(3,5),(4,6),(5,6)],7)
=> ([(0,6),(1,5),(2,7),(3,5),(3,7),(4,6),(4,7)],8)
=> ([(0,6),(1,5),(2,7),(3,5),(3,7),(4,6),(4,7)],8)
=> ? = 0
([(0,5),(1,4),(1,5),(2,3),(2,6),(3,6),(4,6)],7)
=> ([(0,8),(1,7),(2,3),(2,4),(3,5),(4,6),(5,7),(6,8)],9)
=> ([(0,8),(1,7),(2,3),(2,4),(3,5),(4,6),(5,7),(6,8)],9)
=> ? = 0
Description
The number of nowhere zero 3-flows of a graph.
This is the absolute value of the evaluation of the Tutte polynomial of the graph at $(x,y)=(0,-2)$.
For $4$-regular graphs, this coincides with the number of Eulerian orientations.
Matching statistic: St001477
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Values
([],2)
=> ([],2)
=> ([],1)
=> 1
([],3)
=> ([],3)
=> ([],1)
=> 1
([(1,2)],3)
=> ([],2)
=> ([],1)
=> 1
([(0,2),(1,2)],3)
=> ([(0,2),(1,2)],3)
=> ([(0,1)],2)
=> 0
([],4)
=> ([],4)
=> ([],1)
=> 1
([(2,3)],4)
=> ([],3)
=> ([],1)
=> 1
([(1,3),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> ([(1,2)],3)
=> 0
([(0,3),(1,3),(2,3)],4)
=> ([(0,3),(1,3),(2,3)],4)
=> ([(0,1)],2)
=> 0
([(0,3),(1,2)],4)
=> ([],2)
=> ([],1)
=> 1
([(0,3),(1,2),(2,3)],4)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 0
([(1,2),(1,3),(2,3)],4)
=> ([],2)
=> ([],1)
=> 1
([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,2),(1,2)],3)
=> ([(0,1)],2)
=> 0
([],5)
=> ([],5)
=> ([],1)
=> 1
([(3,4)],5)
=> ([],4)
=> ([],1)
=> 1
([(2,4),(3,4)],5)
=> ([(2,4),(3,4)],5)
=> ([(1,2)],3)
=> 0
([(1,4),(2,4),(3,4)],5)
=> ([(1,4),(2,4),(3,4)],5)
=> ([(1,2)],3)
=> 0
([(0,4),(1,4),(2,4),(3,4)],5)
=> ([(0,4),(1,4),(2,4),(3,4)],5)
=> ([(0,1)],2)
=> 0
([(1,4),(2,3)],5)
=> ([],3)
=> ([],1)
=> 1
([(1,4),(2,3),(3,4)],5)
=> ([(1,5),(2,4),(3,4),(3,5)],6)
=> ([(1,5),(2,4),(3,4),(3,5)],6)
=> 0
([(0,1),(2,4),(3,4)],5)
=> ([(1,3),(2,3)],4)
=> ([(1,2)],3)
=> 0
([(2,3),(2,4),(3,4)],5)
=> ([],3)
=> ([],1)
=> 1
([(0,4),(1,4),(2,3),(3,4)],5)
=> ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 0
([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,3),(2,3)],4)
=> ([(1,2)],3)
=> 0
([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(1,3),(2,3)],4)
=> ([(0,1)],2)
=> 0
([(1,3),(1,4),(2,3),(2,4)],5)
=> ([],2)
=> ([],1)
=> 1
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> ([(0,2),(1,2)],3)
=> ([(0,1)],2)
=> 0
([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([],2)
=> ([],1)
=> 1
([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 0
([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,2),(1,2)],3)
=> ([(0,1)],2)
=> 0
([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,6),(1,5),(2,3),(2,4),(3,5),(4,6)],7)
=> ([(0,6),(1,5),(2,3),(2,4),(3,5),(4,6)],7)
=> 0
([(0,1),(2,3),(2,4),(3,4)],5)
=> ([],2)
=> ([],1)
=> 1
([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 0
([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
=> ([(0,2),(1,2)],3)
=> ([(0,1)],2)
=> 0
([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(0,2),(1,2)],3)
=> ([(0,1)],2)
=> 0
([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([],2)
=> ([],1)
=> 1
([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,2),(1,2)],3)
=> ([(0,1)],2)
=> 0
([],6)
=> ([],6)
=> ([],1)
=> 1
([(4,5)],6)
=> ([],5)
=> ([],1)
=> 1
([(3,5),(4,5)],6)
=> ([(3,5),(4,5)],6)
=> ([(1,2)],3)
=> 0
([(2,5),(3,5),(4,5)],6)
=> ([(2,5),(3,5),(4,5)],6)
=> ([(1,2)],3)
=> 0
([(1,5),(2,5),(3,5),(4,5)],6)
=> ([(1,5),(2,5),(3,5),(4,5)],6)
=> ([(1,2)],3)
=> 0
([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> ([(0,1)],2)
=> 0
([(2,5),(3,4)],6)
=> ([],4)
=> ([],1)
=> 1
([(2,5),(3,4),(4,5)],6)
=> ([(2,6),(3,5),(4,5),(4,6)],7)
=> ([(1,5),(2,4),(3,4),(3,5)],6)
=> 0
([(1,2),(3,5),(4,5)],6)
=> ([(2,4),(3,4)],5)
=> ([(1,2)],3)
=> 0
([(3,4),(3,5),(4,5)],6)
=> ([],4)
=> ([],1)
=> 1
([(1,5),(2,5),(3,4),(4,5)],6)
=> ([(1,6),(2,6),(3,4),(4,5),(5,6)],7)
=> ([(1,5),(2,4),(3,4),(3,5)],6)
=> 0
([(0,1),(2,5),(3,5),(4,5)],6)
=> ([(1,4),(2,4),(3,4)],5)
=> ([(1,2)],3)
=> 0
([(2,5),(3,4),(3,5),(4,5)],6)
=> ([(2,4),(3,4)],5)
=> ([(1,2)],3)
=> 0
([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
=> ([(0,6),(1,6),(2,6),(3,4),(4,5),(5,6)],7)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 0
([(1,5),(2,4),(3,4),(3,5)],6)
=> ([(1,7),(2,6),(3,4),(3,5),(4,6),(5,7)],8)
=> ([(1,7),(2,6),(3,4),(3,5),(4,6),(5,7)],8)
=> ? = 0
([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
=> ([(0,6),(1,5),(2,7),(3,5),(3,7),(4,6),(4,7)],8)
=> ([(0,6),(1,5),(2,7),(3,5),(3,7),(4,6),(4,7)],8)
=> ? = 0
([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
=> ([(0,8),(1,7),(2,3),(2,4),(3,5),(4,6),(5,7),(6,8)],9)
=> ([(0,8),(1,7),(2,3),(2,4),(3,5),(4,6),(5,7),(6,8)],9)
=> ? = 0
([(1,5),(2,3),(2,6),(3,6),(4,5),(4,6)],7)
=> ([(1,7),(2,6),(3,4),(3,5),(4,6),(5,7)],8)
=> ([(1,7),(2,6),(3,4),(3,5),(4,6),(5,7)],8)
=> ? = 0
([(1,5),(2,3),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(1,7),(2,6),(3,4),(3,5),(4,6),(5,7)],8)
=> ([(1,7),(2,6),(3,4),(3,5),(4,6),(5,7)],8)
=> ? = 0
([(0,6),(1,5),(2,3),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,6),(1,5),(2,7),(3,5),(3,7),(4,6),(4,7)],8)
=> ([(0,6),(1,5),(2,7),(3,5),(3,7),(4,6),(4,7)],8)
=> ? = 0
([(0,6),(1,4),(2,5),(2,6),(3,4),(3,5),(5,6)],7)
=> ([(0,8),(1,7),(2,3),(2,4),(3,5),(4,6),(5,7),(6,8)],9)
=> ([(0,8),(1,7),(2,3),(2,4),(3,5),(4,6),(5,7),(6,8)],9)
=> ? = 0
([(0,1),(2,5),(3,4),(4,6),(5,6)],7)
=> ([(1,7),(2,6),(3,4),(3,5),(4,6),(5,7)],8)
=> ([(1,7),(2,6),(3,4),(3,5),(4,6),(5,7)],8)
=> ? = 0
([(0,5),(1,4),(2,3),(2,6),(3,6),(4,6),(5,6)],7)
=> ([(0,6),(1,5),(2,7),(3,5),(3,7),(4,6),(4,7)],8)
=> ([(0,6),(1,5),(2,7),(3,5),(3,7),(4,6),(4,7)],8)
=> ? = 0
([(0,4),(1,3),(2,5),(2,6),(3,5),(4,6),(5,6)],7)
=> ([(0,8),(1,7),(2,3),(2,4),(3,5),(4,6),(5,7),(6,8)],9)
=> ([(0,8),(1,7),(2,3),(2,4),(3,5),(4,6),(5,7),(6,8)],9)
=> ? = 0
([(0,6),(1,4),(2,3),(2,5),(3,5),(4,6),(5,6)],7)
=> ([(0,6),(1,5),(2,7),(3,5),(3,7),(4,6),(4,7)],8)
=> ([(0,6),(1,5),(2,7),(3,5),(3,7),(4,6),(4,7)],8)
=> ? = 0
([(0,5),(1,4),(1,5),(2,3),(2,6),(3,6),(4,6)],7)
=> ([(0,8),(1,7),(2,3),(2,4),(3,5),(4,6),(5,7),(6,8)],9)
=> ([(0,8),(1,7),(2,3),(2,4),(3,5),(4,6),(5,7),(6,8)],9)
=> ? = 0
Description
The number of nowhere zero 5-flows of a graph.
Matching statistic: St001478
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Values
([],2)
=> ([],2)
=> ([],1)
=> 1
([],3)
=> ([],3)
=> ([],1)
=> 1
([(1,2)],3)
=> ([],2)
=> ([],1)
=> 1
([(0,2),(1,2)],3)
=> ([(0,2),(1,2)],3)
=> ([(0,1)],2)
=> 0
([],4)
=> ([],4)
=> ([],1)
=> 1
([(2,3)],4)
=> ([],3)
=> ([],1)
=> 1
([(1,3),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> ([(1,2)],3)
=> 0
([(0,3),(1,3),(2,3)],4)
=> ([(0,3),(1,3),(2,3)],4)
=> ([(0,1)],2)
=> 0
([(0,3),(1,2)],4)
=> ([],2)
=> ([],1)
=> 1
([(0,3),(1,2),(2,3)],4)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 0
([(1,2),(1,3),(2,3)],4)
=> ([],2)
=> ([],1)
=> 1
([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,2),(1,2)],3)
=> ([(0,1)],2)
=> 0
([],5)
=> ([],5)
=> ([],1)
=> 1
([(3,4)],5)
=> ([],4)
=> ([],1)
=> 1
([(2,4),(3,4)],5)
=> ([(2,4),(3,4)],5)
=> ([(1,2)],3)
=> 0
([(1,4),(2,4),(3,4)],5)
=> ([(1,4),(2,4),(3,4)],5)
=> ([(1,2)],3)
=> 0
([(0,4),(1,4),(2,4),(3,4)],5)
=> ([(0,4),(1,4),(2,4),(3,4)],5)
=> ([(0,1)],2)
=> 0
([(1,4),(2,3)],5)
=> ([],3)
=> ([],1)
=> 1
([(1,4),(2,3),(3,4)],5)
=> ([(1,5),(2,4),(3,4),(3,5)],6)
=> ([(1,5),(2,4),(3,4),(3,5)],6)
=> 0
([(0,1),(2,4),(3,4)],5)
=> ([(1,3),(2,3)],4)
=> ([(1,2)],3)
=> 0
([(2,3),(2,4),(3,4)],5)
=> ([],3)
=> ([],1)
=> 1
([(0,4),(1,4),(2,3),(3,4)],5)
=> ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 0
([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,3),(2,3)],4)
=> ([(1,2)],3)
=> 0
([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(1,3),(2,3)],4)
=> ([(0,1)],2)
=> 0
([(1,3),(1,4),(2,3),(2,4)],5)
=> ([],2)
=> ([],1)
=> 1
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> ([(0,2),(1,2)],3)
=> ([(0,1)],2)
=> 0
([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([],2)
=> ([],1)
=> 1
([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 0
([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,2),(1,2)],3)
=> ([(0,1)],2)
=> 0
([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,6),(1,5),(2,3),(2,4),(3,5),(4,6)],7)
=> ([(0,6),(1,5),(2,3),(2,4),(3,5),(4,6)],7)
=> 0
([(0,1),(2,3),(2,4),(3,4)],5)
=> ([],2)
=> ([],1)
=> 1
([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 0
([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
=> ([(0,2),(1,2)],3)
=> ([(0,1)],2)
=> 0
([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(0,2),(1,2)],3)
=> ([(0,1)],2)
=> 0
([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([],2)
=> ([],1)
=> 1
([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,2),(1,2)],3)
=> ([(0,1)],2)
=> 0
([],6)
=> ([],6)
=> ([],1)
=> 1
([(4,5)],6)
=> ([],5)
=> ([],1)
=> 1
([(3,5),(4,5)],6)
=> ([(3,5),(4,5)],6)
=> ([(1,2)],3)
=> 0
([(2,5),(3,5),(4,5)],6)
=> ([(2,5),(3,5),(4,5)],6)
=> ([(1,2)],3)
=> 0
([(1,5),(2,5),(3,5),(4,5)],6)
=> ([(1,5),(2,5),(3,5),(4,5)],6)
=> ([(1,2)],3)
=> 0
([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> ([(0,1)],2)
=> 0
([(2,5),(3,4)],6)
=> ([],4)
=> ([],1)
=> 1
([(2,5),(3,4),(4,5)],6)
=> ([(2,6),(3,5),(4,5),(4,6)],7)
=> ([(1,5),(2,4),(3,4),(3,5)],6)
=> 0
([(1,2),(3,5),(4,5)],6)
=> ([(2,4),(3,4)],5)
=> ([(1,2)],3)
=> 0
([(3,4),(3,5),(4,5)],6)
=> ([],4)
=> ([],1)
=> 1
([(1,5),(2,5),(3,4),(4,5)],6)
=> ([(1,6),(2,6),(3,4),(4,5),(5,6)],7)
=> ([(1,5),(2,4),(3,4),(3,5)],6)
=> 0
([(0,1),(2,5),(3,5),(4,5)],6)
=> ([(1,4),(2,4),(3,4)],5)
=> ([(1,2)],3)
=> 0
([(2,5),(3,4),(3,5),(4,5)],6)
=> ([(2,4),(3,4)],5)
=> ([(1,2)],3)
=> 0
([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
=> ([(0,6),(1,6),(2,6),(3,4),(4,5),(5,6)],7)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 0
([(1,5),(2,4),(3,4),(3,5)],6)
=> ([(1,7),(2,6),(3,4),(3,5),(4,6),(5,7)],8)
=> ([(1,7),(2,6),(3,4),(3,5),(4,6),(5,7)],8)
=> ? = 0
([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
=> ([(0,6),(1,5),(2,7),(3,5),(3,7),(4,6),(4,7)],8)
=> ([(0,6),(1,5),(2,7),(3,5),(3,7),(4,6),(4,7)],8)
=> ? = 0
([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
=> ([(0,8),(1,7),(2,3),(2,4),(3,5),(4,6),(5,7),(6,8)],9)
=> ([(0,8),(1,7),(2,3),(2,4),(3,5),(4,6),(5,7),(6,8)],9)
=> ? = 0
([(1,5),(2,3),(2,6),(3,6),(4,5),(4,6)],7)
=> ([(1,7),(2,6),(3,4),(3,5),(4,6),(5,7)],8)
=> ([(1,7),(2,6),(3,4),(3,5),(4,6),(5,7)],8)
=> ? = 0
([(1,5),(2,3),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(1,7),(2,6),(3,4),(3,5),(4,6),(5,7)],8)
=> ([(1,7),(2,6),(3,4),(3,5),(4,6),(5,7)],8)
=> ? = 0
([(0,6),(1,5),(2,3),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,6),(1,5),(2,7),(3,5),(3,7),(4,6),(4,7)],8)
=> ([(0,6),(1,5),(2,7),(3,5),(3,7),(4,6),(4,7)],8)
=> ? = 0
([(0,6),(1,4),(2,5),(2,6),(3,4),(3,5),(5,6)],7)
=> ([(0,8),(1,7),(2,3),(2,4),(3,5),(4,6),(5,7),(6,8)],9)
=> ([(0,8),(1,7),(2,3),(2,4),(3,5),(4,6),(5,7),(6,8)],9)
=> ? = 0
([(0,1),(2,5),(3,4),(4,6),(5,6)],7)
=> ([(1,7),(2,6),(3,4),(3,5),(4,6),(5,7)],8)
=> ([(1,7),(2,6),(3,4),(3,5),(4,6),(5,7)],8)
=> ? = 0
([(0,5),(1,4),(2,3),(2,6),(3,6),(4,6),(5,6)],7)
=> ([(0,6),(1,5),(2,7),(3,5),(3,7),(4,6),(4,7)],8)
=> ([(0,6),(1,5),(2,7),(3,5),(3,7),(4,6),(4,7)],8)
=> ? = 0
([(0,4),(1,3),(2,5),(2,6),(3,5),(4,6),(5,6)],7)
=> ([(0,8),(1,7),(2,3),(2,4),(3,5),(4,6),(5,7),(6,8)],9)
=> ([(0,8),(1,7),(2,3),(2,4),(3,5),(4,6),(5,7),(6,8)],9)
=> ? = 0
([(0,6),(1,4),(2,3),(2,5),(3,5),(4,6),(5,6)],7)
=> ([(0,6),(1,5),(2,7),(3,5),(3,7),(4,6),(4,7)],8)
=> ([(0,6),(1,5),(2,7),(3,5),(3,7),(4,6),(4,7)],8)
=> ? = 0
([(0,5),(1,4),(1,5),(2,3),(2,6),(3,6),(4,6)],7)
=> ([(0,8),(1,7),(2,3),(2,4),(3,5),(4,6),(5,7),(6,8)],9)
=> ([(0,8),(1,7),(2,3),(2,4),(3,5),(4,6),(5,7),(6,8)],9)
=> ? = 0
Description
The number of nowhere zero 4-flows of a graph.
Matching statistic: St001795
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Values
([],2)
=> ([],2)
=> ([],1)
=> 1
([],3)
=> ([],3)
=> ([],1)
=> 1
([(1,2)],3)
=> ([],2)
=> ([],1)
=> 1
([(0,2),(1,2)],3)
=> ([(0,2),(1,2)],3)
=> ([(0,1)],2)
=> 0
([],4)
=> ([],4)
=> ([],1)
=> 1
([(2,3)],4)
=> ([],3)
=> ([],1)
=> 1
([(1,3),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> ([(1,2)],3)
=> 0
([(0,3),(1,3),(2,3)],4)
=> ([(0,3),(1,3),(2,3)],4)
=> ([(0,1)],2)
=> 0
([(0,3),(1,2)],4)
=> ([],2)
=> ([],1)
=> 1
([(0,3),(1,2),(2,3)],4)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 0
([(1,2),(1,3),(2,3)],4)
=> ([],2)
=> ([],1)
=> 1
([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,2),(1,2)],3)
=> ([(0,1)],2)
=> 0
([],5)
=> ([],5)
=> ([],1)
=> 1
([(3,4)],5)
=> ([],4)
=> ([],1)
=> 1
([(2,4),(3,4)],5)
=> ([(2,4),(3,4)],5)
=> ([(1,2)],3)
=> 0
([(1,4),(2,4),(3,4)],5)
=> ([(1,4),(2,4),(3,4)],5)
=> ([(1,2)],3)
=> 0
([(0,4),(1,4),(2,4),(3,4)],5)
=> ([(0,4),(1,4),(2,4),(3,4)],5)
=> ([(0,1)],2)
=> 0
([(1,4),(2,3)],5)
=> ([],3)
=> ([],1)
=> 1
([(1,4),(2,3),(3,4)],5)
=> ([(1,5),(2,4),(3,4),(3,5)],6)
=> ([(1,5),(2,4),(3,4),(3,5)],6)
=> 0
([(0,1),(2,4),(3,4)],5)
=> ([(1,3),(2,3)],4)
=> ([(1,2)],3)
=> 0
([(2,3),(2,4),(3,4)],5)
=> ([],3)
=> ([],1)
=> 1
([(0,4),(1,4),(2,3),(3,4)],5)
=> ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 0
([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,3),(2,3)],4)
=> ([(1,2)],3)
=> 0
([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(1,3),(2,3)],4)
=> ([(0,1)],2)
=> 0
([(1,3),(1,4),(2,3),(2,4)],5)
=> ([],2)
=> ([],1)
=> 1
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> ([(0,2),(1,2)],3)
=> ([(0,1)],2)
=> 0
([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([],2)
=> ([],1)
=> 1
([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 0
([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,2),(1,2)],3)
=> ([(0,1)],2)
=> 0
([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,6),(1,5),(2,3),(2,4),(3,5),(4,6)],7)
=> ([(0,6),(1,5),(2,3),(2,4),(3,5),(4,6)],7)
=> 0
([(0,1),(2,3),(2,4),(3,4)],5)
=> ([],2)
=> ([],1)
=> 1
([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 0
([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
=> ([(0,2),(1,2)],3)
=> ([(0,1)],2)
=> 0
([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(0,2),(1,2)],3)
=> ([(0,1)],2)
=> 0
([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([],2)
=> ([],1)
=> 1
([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,2),(1,2)],3)
=> ([(0,1)],2)
=> 0
([],6)
=> ([],6)
=> ([],1)
=> 1
([(4,5)],6)
=> ([],5)
=> ([],1)
=> 1
([(3,5),(4,5)],6)
=> ([(3,5),(4,5)],6)
=> ([(1,2)],3)
=> 0
([(2,5),(3,5),(4,5)],6)
=> ([(2,5),(3,5),(4,5)],6)
=> ([(1,2)],3)
=> 0
([(1,5),(2,5),(3,5),(4,5)],6)
=> ([(1,5),(2,5),(3,5),(4,5)],6)
=> ([(1,2)],3)
=> 0
([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> ([(0,1)],2)
=> 0
([(2,5),(3,4)],6)
=> ([],4)
=> ([],1)
=> 1
([(2,5),(3,4),(4,5)],6)
=> ([(2,6),(3,5),(4,5),(4,6)],7)
=> ([(1,5),(2,4),(3,4),(3,5)],6)
=> 0
([(1,2),(3,5),(4,5)],6)
=> ([(2,4),(3,4)],5)
=> ([(1,2)],3)
=> 0
([(3,4),(3,5),(4,5)],6)
=> ([],4)
=> ([],1)
=> 1
([(1,5),(2,5),(3,4),(4,5)],6)
=> ([(1,6),(2,6),(3,4),(4,5),(5,6)],7)
=> ([(1,5),(2,4),(3,4),(3,5)],6)
=> 0
([(0,1),(2,5),(3,5),(4,5)],6)
=> ([(1,4),(2,4),(3,4)],5)
=> ([(1,2)],3)
=> 0
([(2,5),(3,4),(3,5),(4,5)],6)
=> ([(2,4),(3,4)],5)
=> ([(1,2)],3)
=> 0
([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
=> ([(0,6),(1,6),(2,6),(3,4),(4,5),(5,6)],7)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 0
([(1,5),(2,4),(3,4),(3,5)],6)
=> ([(1,7),(2,6),(3,4),(3,5),(4,6),(5,7)],8)
=> ([(1,7),(2,6),(3,4),(3,5),(4,6),(5,7)],8)
=> ? = 0
([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
=> ([(0,6),(1,5),(2,7),(3,5),(3,7),(4,6),(4,7)],8)
=> ([(0,6),(1,5),(2,7),(3,5),(3,7),(4,6),(4,7)],8)
=> ? = 0
([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
=> ([(0,8),(1,7),(2,3),(2,4),(3,5),(4,6),(5,7),(6,8)],9)
=> ([(0,8),(1,7),(2,3),(2,4),(3,5),(4,6),(5,7),(6,8)],9)
=> ? = 0
([(1,5),(2,3),(2,6),(3,6),(4,5),(4,6)],7)
=> ([(1,7),(2,6),(3,4),(3,5),(4,6),(5,7)],8)
=> ([(1,7),(2,6),(3,4),(3,5),(4,6),(5,7)],8)
=> ? = 0
([(1,5),(2,3),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(1,7),(2,6),(3,4),(3,5),(4,6),(5,7)],8)
=> ([(1,7),(2,6),(3,4),(3,5),(4,6),(5,7)],8)
=> ? = 0
([(0,6),(1,5),(2,3),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,6),(1,5),(2,7),(3,5),(3,7),(4,6),(4,7)],8)
=> ([(0,6),(1,5),(2,7),(3,5),(3,7),(4,6),(4,7)],8)
=> ? = 0
([(0,6),(1,4),(2,5),(2,6),(3,4),(3,5),(5,6)],7)
=> ([(0,8),(1,7),(2,3),(2,4),(3,5),(4,6),(5,7),(6,8)],9)
=> ([(0,8),(1,7),(2,3),(2,4),(3,5),(4,6),(5,7),(6,8)],9)
=> ? = 0
([(0,1),(2,5),(3,4),(4,6),(5,6)],7)
=> ([(1,7),(2,6),(3,4),(3,5),(4,6),(5,7)],8)
=> ([(1,7),(2,6),(3,4),(3,5),(4,6),(5,7)],8)
=> ? = 0
([(0,5),(1,4),(2,3),(2,6),(3,6),(4,6),(5,6)],7)
=> ([(0,6),(1,5),(2,7),(3,5),(3,7),(4,6),(4,7)],8)
=> ([(0,6),(1,5),(2,7),(3,5),(3,7),(4,6),(4,7)],8)
=> ? = 0
([(0,4),(1,3),(2,5),(2,6),(3,5),(4,6),(5,6)],7)
=> ([(0,8),(1,7),(2,3),(2,4),(3,5),(4,6),(5,7),(6,8)],9)
=> ([(0,8),(1,7),(2,3),(2,4),(3,5),(4,6),(5,7),(6,8)],9)
=> ? = 0
([(0,6),(1,4),(2,3),(2,5),(3,5),(4,6),(5,6)],7)
=> ([(0,6),(1,5),(2,7),(3,5),(3,7),(4,6),(4,7)],8)
=> ([(0,6),(1,5),(2,7),(3,5),(3,7),(4,6),(4,7)],8)
=> ? = 0
([(0,5),(1,4),(1,5),(2,3),(2,6),(3,6),(4,6)],7)
=> ([(0,8),(1,7),(2,3),(2,4),(3,5),(4,6),(5,7),(6,8)],9)
=> ([(0,8),(1,7),(2,3),(2,4),(3,5),(4,6),(5,7),(6,8)],9)
=> ? = 0
Description
The binary logarithm of the evaluation of the Tutte polynomial of the graph at (x,y) equal to (-1,-1).
Matching statistic: St001675
Mp00274: Graphs —block-cut tree⟶ Graphs
Mp00324: Graphs —chromatic difference sequence⟶ Integer compositions
St001675: Integer compositions ⟶ ℤResult quality: 98% ●values known / values provided: 98%●distinct values known / distinct values provided: 100%
Mp00324: Graphs —chromatic difference sequence⟶ Integer compositions
St001675: Integer compositions ⟶ ℤResult quality: 98% ●values known / values provided: 98%●distinct values known / distinct values provided: 100%
Values
([],2)
=> ([],2)
=> [2] => 1
([],3)
=> ([],3)
=> [3] => 1
([(1,2)],3)
=> ([],2)
=> [2] => 1
([(0,2),(1,2)],3)
=> ([(0,2),(1,2)],3)
=> [2,1] => 0
([],4)
=> ([],4)
=> [4] => 1
([(2,3)],4)
=> ([],3)
=> [3] => 1
([(1,3),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> [3,1] => 0
([(0,3),(1,3),(2,3)],4)
=> ([(0,3),(1,3),(2,3)],4)
=> [3,1] => 0
([(0,3),(1,2)],4)
=> ([],2)
=> [2] => 1
([(0,3),(1,2),(2,3)],4)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> [3,2] => 0
([(1,2),(1,3),(2,3)],4)
=> ([],2)
=> [2] => 1
([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,2),(1,2)],3)
=> [2,1] => 0
([],5)
=> ([],5)
=> [5] => 1
([(3,4)],5)
=> ([],4)
=> [4] => 1
([(2,4),(3,4)],5)
=> ([(2,4),(3,4)],5)
=> [4,1] => 0
([(1,4),(2,4),(3,4)],5)
=> ([(1,4),(2,4),(3,4)],5)
=> [4,1] => 0
([(0,4),(1,4),(2,4),(3,4)],5)
=> ([(0,4),(1,4),(2,4),(3,4)],5)
=> [4,1] => 0
([(1,4),(2,3)],5)
=> ([],3)
=> [3] => 1
([(1,4),(2,3),(3,4)],5)
=> ([(1,5),(2,4),(3,4),(3,5)],6)
=> [4,2] => 0
([(0,1),(2,4),(3,4)],5)
=> ([(1,3),(2,3)],4)
=> [3,1] => 0
([(2,3),(2,4),(3,4)],5)
=> ([],3)
=> [3] => 1
([(0,4),(1,4),(2,3),(3,4)],5)
=> ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
=> [4,2] => 0
([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,3),(2,3)],4)
=> [3,1] => 0
([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(1,3),(2,3)],4)
=> [3,1] => 0
([(1,3),(1,4),(2,3),(2,4)],5)
=> ([],2)
=> [2] => 1
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> ([(0,2),(1,2)],3)
=> [2,1] => 0
([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([],2)
=> [2] => 1
([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> [3,2] => 0
([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,2),(1,2)],3)
=> [2,1] => 0
([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,6),(1,5),(2,3),(2,4),(3,5),(4,6)],7)
=> [4,3] => 0
([(0,1),(2,3),(2,4),(3,4)],5)
=> ([],2)
=> [2] => 1
([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> [3,2] => 0
([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
=> ([(0,2),(1,2)],3)
=> [2,1] => 0
([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(0,2),(1,2)],3)
=> [2,1] => 0
([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([],2)
=> [2] => 1
([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,2),(1,2)],3)
=> [2,1] => 0
([],6)
=> ([],6)
=> [6] => 1
([(4,5)],6)
=> ([],5)
=> [5] => 1
([(3,5),(4,5)],6)
=> ([(3,5),(4,5)],6)
=> [5,1] => 0
([(2,5),(3,5),(4,5)],6)
=> ([(2,5),(3,5),(4,5)],6)
=> [5,1] => 0
([(1,5),(2,5),(3,5),(4,5)],6)
=> ([(1,5),(2,5),(3,5),(4,5)],6)
=> [5,1] => 0
([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> [5,1] => 0
([(2,5),(3,4)],6)
=> ([],4)
=> [4] => 1
([(2,5),(3,4),(4,5)],6)
=> ([(2,6),(3,5),(4,5),(4,6)],7)
=> [5,2] => 0
([(1,2),(3,5),(4,5)],6)
=> ([(2,4),(3,4)],5)
=> [4,1] => 0
([(3,4),(3,5),(4,5)],6)
=> ([],4)
=> [4] => 1
([(1,5),(2,5),(3,4),(4,5)],6)
=> ([(1,6),(2,6),(3,4),(4,5),(5,6)],7)
=> [5,2] => 0
([(0,1),(2,5),(3,5),(4,5)],6)
=> ([(1,4),(2,4),(3,4)],5)
=> [4,1] => 0
([(2,5),(3,4),(3,5),(4,5)],6)
=> ([(2,4),(3,4)],5)
=> [4,1] => 0
([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
=> ([(0,6),(1,6),(2,6),(3,4),(4,5),(5,6)],7)
=> [5,2] => 0
([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
=> ([(0,6),(1,7),(2,7),(3,4),(3,5),(4,6),(5,7)],8)
=> ? => ? = 0
([(1,5),(2,4),(3,4),(3,5)],6)
=> ([(1,7),(2,6),(3,4),(3,5),(4,6),(5,7)],8)
=> ? => ? = 0
([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
=> ([(0,6),(1,5),(2,7),(3,5),(3,7),(4,6),(4,7)],8)
=> ? => ? = 0
([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
=> ([(0,8),(1,7),(2,3),(2,4),(3,5),(4,6),(5,7),(6,8)],9)
=> ? => ? = 0
([(0,6),(1,6),(2,3),(3,5),(4,5),(4,6),(5,6)],7)
=> ([(0,6),(1,7),(2,7),(3,4),(3,5),(4,6),(5,7)],8)
=> ? => ? = 0
([(1,5),(2,3),(2,6),(3,6),(4,5),(4,6)],7)
=> ([(1,7),(2,6),(3,4),(3,5),(4,6),(5,7)],8)
=> ? => ? = 0
([(1,5),(2,3),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(1,7),(2,6),(3,4),(3,5),(4,6),(5,7)],8)
=> ? => ? = 0
([(0,5),(1,6),(2,3),(2,6),(3,6),(4,5),(4,6)],7)
=> ([(0,6),(1,7),(2,7),(3,4),(3,5),(4,6),(5,7)],8)
=> ? => ? = 0
([(0,6),(1,5),(2,3),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,6),(1,5),(2,7),(3,5),(3,7),(4,6),(4,7)],8)
=> ? => ? = 0
([(0,6),(1,6),(2,3),(2,5),(3,5),(4,5),(4,6)],7)
=> ([(0,6),(1,7),(2,7),(3,4),(3,5),(4,6),(5,7)],8)
=> ? => ? = 0
([(0,6),(1,4),(2,5),(2,6),(3,4),(3,5),(5,6)],7)
=> ([(0,8),(1,7),(2,3),(2,4),(3,5),(4,6),(5,7),(6,8)],9)
=> ? => ? = 0
([(0,4),(1,4),(2,5),(3,5),(3,6),(4,6),(5,6)],7)
=> ([(0,6),(1,7),(2,7),(3,4),(3,5),(4,6),(5,7)],8)
=> ? => ? = 0
([(0,1),(2,5),(3,4),(4,6),(5,6)],7)
=> ([(1,7),(2,6),(3,4),(3,5),(4,6),(5,7)],8)
=> ? => ? = 0
([(0,5),(1,4),(2,3),(2,6),(3,6),(4,6),(5,6)],7)
=> ([(0,6),(1,5),(2,7),(3,5),(3,7),(4,6),(4,7)],8)
=> ? => ? = 0
([(0,4),(1,3),(2,5),(2,6),(3,5),(4,6),(5,6)],7)
=> ([(0,8),(1,7),(2,3),(2,4),(3,5),(4,6),(5,7),(6,8)],9)
=> ? => ? = 0
([(0,6),(1,4),(2,3),(2,5),(3,5),(4,6),(5,6)],7)
=> ([(0,6),(1,5),(2,7),(3,5),(3,7),(4,6),(4,7)],8)
=> ? => ? = 0
([(0,5),(1,4),(1,5),(2,3),(2,6),(3,6),(4,6)],7)
=> ([(0,8),(1,7),(2,3),(2,4),(3,5),(4,6),(5,7),(6,8)],9)
=> ? => ? = 0
Description
The number of parts equal to the part in the reversed composition.
Matching statistic: St000309
Values
([],2)
=> ([],2)
=> ([],1)
=> 1
([],3)
=> ([],3)
=> ([],1)
=> 1
([(1,2)],3)
=> ([],2)
=> ([],1)
=> 1
([(0,2),(1,2)],3)
=> ([(0,2),(1,2)],3)
=> ([(0,1)],2)
=> 0
([],4)
=> ([],4)
=> ([],1)
=> 1
([(2,3)],4)
=> ([],3)
=> ([],1)
=> 1
([(1,3),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> ([(0,1)],2)
=> 0
([(0,3),(1,3),(2,3)],4)
=> ([(0,3),(1,3),(2,3)],4)
=> ([(0,1)],2)
=> 0
([(0,3),(1,2)],4)
=> ([],2)
=> ([],1)
=> 1
([(0,3),(1,2),(2,3)],4)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,1)],2)
=> 0
([(1,2),(1,3),(2,3)],4)
=> ([],2)
=> ([],1)
=> 1
([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,2),(1,2)],3)
=> ([(0,1)],2)
=> 0
([],5)
=> ([],5)
=> ([],1)
=> 1
([(3,4)],5)
=> ([],4)
=> ([],1)
=> 1
([(2,4),(3,4)],5)
=> ([(2,4),(3,4)],5)
=> ([(0,1)],2)
=> 0
([(1,4),(2,4),(3,4)],5)
=> ([(1,4),(2,4),(3,4)],5)
=> ([(0,1)],2)
=> 0
([(0,4),(1,4),(2,4),(3,4)],5)
=> ([(0,4),(1,4),(2,4),(3,4)],5)
=> ([(0,1)],2)
=> 0
([(1,4),(2,3)],5)
=> ([],3)
=> ([],1)
=> 1
([(1,4),(2,3),(3,4)],5)
=> ([(1,5),(2,4),(3,4),(3,5)],6)
=> ([(0,1)],2)
=> 0
([(0,1),(2,4),(3,4)],5)
=> ([(1,3),(2,3)],4)
=> ([(0,1)],2)
=> 0
([(2,3),(2,4),(3,4)],5)
=> ([],3)
=> ([],1)
=> 1
([(0,4),(1,4),(2,3),(3,4)],5)
=> ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
=> ([(0,1)],2)
=> 0
([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,3),(2,3)],4)
=> ([(0,1)],2)
=> 0
([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(1,3),(2,3)],4)
=> ([(0,1)],2)
=> 0
([(1,3),(1,4),(2,3),(2,4)],5)
=> ([],2)
=> ([],1)
=> 1
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> ([(0,2),(1,2)],3)
=> ([(0,1)],2)
=> 0
([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([],2)
=> ([],1)
=> 1
([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,1)],2)
=> 0
([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,2),(1,2)],3)
=> ([(0,1)],2)
=> 0
([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,6),(1,5),(2,3),(2,4),(3,5),(4,6)],7)
=> ?
=> ? = 0
([(0,1),(2,3),(2,4),(3,4)],5)
=> ([],2)
=> ([],1)
=> 1
([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,1)],2)
=> 0
([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
=> ([(0,2),(1,2)],3)
=> ([(0,1)],2)
=> 0
([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(0,2),(1,2)],3)
=> ([(0,1)],2)
=> 0
([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([],2)
=> ([],1)
=> 1
([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,2),(1,2)],3)
=> ([(0,1)],2)
=> 0
([],6)
=> ([],6)
=> ([],1)
=> 1
([(4,5)],6)
=> ([],5)
=> ([],1)
=> 1
([(3,5),(4,5)],6)
=> ([(3,5),(4,5)],6)
=> ([(0,1)],2)
=> 0
([(2,5),(3,5),(4,5)],6)
=> ([(2,5),(3,5),(4,5)],6)
=> ([(0,1)],2)
=> 0
([(1,5),(2,5),(3,5),(4,5)],6)
=> ([(1,5),(2,5),(3,5),(4,5)],6)
=> ([(0,1)],2)
=> 0
([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> ([(0,1)],2)
=> 0
([(2,5),(3,4)],6)
=> ([],4)
=> ([],1)
=> 1
([(2,5),(3,4),(4,5)],6)
=> ([(2,6),(3,5),(4,5),(4,6)],7)
=> ?
=> ? = 0
([(1,2),(3,5),(4,5)],6)
=> ([(2,4),(3,4)],5)
=> ([(0,1)],2)
=> 0
([(3,4),(3,5),(4,5)],6)
=> ([],4)
=> ([],1)
=> 1
([(1,5),(2,5),(3,4),(4,5)],6)
=> ([(1,6),(2,6),(3,4),(4,5),(5,6)],7)
=> ?
=> ? = 0
([(0,1),(2,5),(3,5),(4,5)],6)
=> ([(1,4),(2,4),(3,4)],5)
=> ([(0,1)],2)
=> 0
([(2,5),(3,4),(3,5),(4,5)],6)
=> ([(2,4),(3,4)],5)
=> ([(0,1)],2)
=> 0
([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
=> ([(0,6),(1,6),(2,6),(3,4),(4,5),(5,6)],7)
=> ?
=> ? = 0
([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(1,4),(2,4),(3,4)],5)
=> ([(0,1)],2)
=> 0
([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,4),(1,4),(2,4),(3,4)],5)
=> ([(0,1)],2)
=> 0
([(2,4),(2,5),(3,4),(3,5)],6)
=> ([],3)
=> ([],1)
=> 1
([(0,5),(1,5),(2,4),(3,4)],6)
=> ([(0,5),(1,5),(2,4),(3,4)],6)
=> ([(0,1)],2)
=> 0
([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
=> ([(0,6),(1,7),(2,7),(3,4),(3,5),(4,6),(5,7)],8)
=> ?
=> ? = 0
([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> ([(0,6),(1,6),(2,5),(3,5),(4,5),(4,6)],7)
=> ?
=> ? = 0
([(1,5),(2,4),(3,4),(3,5)],6)
=> ([(1,7),(2,6),(3,4),(3,5),(4,6),(5,7)],8)
=> ?
=> ? = 0
([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
=> ([(0,6),(1,5),(2,7),(3,5),(3,7),(4,6),(4,7)],8)
=> ?
=> ? = 0
([(0,5),(1,4),(2,3),(3,4),(3,5),(4,5)],6)
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> ?
=> ? = 0
([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
=> ([(0,8),(1,7),(2,3),(2,4),(3,5),(4,6),(5,7),(6,8)],9)
=> ?
=> ? = 0
([(0,4),(1,2),(1,5),(2,5),(3,4),(3,5)],6)
=> ([(0,6),(1,5),(2,3),(2,4),(3,5),(4,6)],7)
=> ?
=> ? = 0
([(0,4),(1,2),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,6),(1,5),(2,3),(2,4),(3,5),(4,6)],7)
=> ?
=> ? = 0
([],7)
=> ([],7)
=> ?
=> ? = 1
([(4,6),(5,6)],7)
=> ([(4,6),(5,6)],7)
=> ?
=> ? = 0
([(3,6),(4,6),(5,6)],7)
=> ([(3,6),(4,6),(5,6)],7)
=> ?
=> ? = 0
([(2,6),(3,6),(4,6),(5,6)],7)
=> ([(2,6),(3,6),(4,6),(5,6)],7)
=> ?
=> ? = 0
([(1,6),(2,6),(3,6),(4,6),(5,6)],7)
=> ([(1,6),(2,6),(3,6),(4,6),(5,6)],7)
=> ?
=> ? = 0
([(0,6),(1,6),(2,6),(3,6),(4,6),(5,6)],7)
=> ([(0,6),(1,6),(2,6),(3,6),(4,6),(5,6)],7)
=> ?
=> ? = 0
([(1,6),(2,6),(3,5),(4,5)],7)
=> ([(1,6),(2,6),(3,5),(4,5)],7)
=> ?
=> ? = 0
([(0,6),(1,6),(2,6),(3,5),(4,5)],7)
=> ([(0,6),(1,6),(2,6),(3,5),(4,5)],7)
=> ?
=> ? = 0
([(2,6),(3,5),(4,5),(4,6),(5,6)],7)
=> ([(2,6),(3,5),(4,5),(4,6)],7)
=> ?
=> ? = 0
([(1,6),(2,6),(3,5),(4,5),(4,6),(5,6)],7)
=> ([(1,6),(2,6),(3,4),(4,5),(5,6)],7)
=> ?
=> ? = 0
([(0,6),(1,6),(2,6),(3,5),(4,5),(4,6),(5,6)],7)
=> ([(0,6),(1,6),(2,6),(3,4),(4,5),(5,6)],7)
=> ?
=> ? = 0
([(0,6),(1,6),(2,5),(3,5),(4,5),(4,6),(5,6)],7)
=> ([(0,6),(1,6),(2,5),(3,5),(4,5),(4,6)],7)
=> ?
=> ? = 0
([(1,2),(3,6),(4,5),(5,6)],7)
=> ([(2,6),(3,5),(4,5),(4,6)],7)
=> ?
=> ? = 0
([(0,1),(2,6),(3,6),(4,5),(5,6)],7)
=> ([(1,6),(2,6),(3,4),(4,5),(5,6)],7)
=> ?
=> ? = 0
([(2,5),(3,4),(3,6),(4,6),(5,6)],7)
=> ([(2,6),(3,5),(4,5),(4,6)],7)
=> ?
=> ? = 0
([(1,5),(2,6),(3,4),(3,6),(4,6),(5,6)],7)
=> ([(1,6),(2,6),(3,4),(4,5),(5,6)],7)
=> ?
=> ? = 0
([(0,6),(1,6),(2,3),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,6),(1,6),(2,6),(3,4),(4,5),(5,6)],7)
=> ?
=> ? = 0
([(0,6),(1,6),(2,3),(3,5),(4,5),(4,6),(5,6)],7)
=> ([(0,6),(1,7),(2,7),(3,4),(3,5),(4,6),(5,7)],8)
=> ?
=> ? = 0
([(1,5),(2,3),(2,6),(3,6),(4,5),(4,6)],7)
=> ([(1,7),(2,6),(3,4),(3,5),(4,6),(5,7)],8)
=> ?
=> ? = 0
([(1,5),(2,3),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(1,7),(2,6),(3,4),(3,5),(4,6),(5,7)],8)
=> ?
=> ? = 0
([(1,5),(2,5),(3,4),(3,6),(4,6),(5,6)],7)
=> ([(1,6),(2,6),(3,4),(4,5),(5,6)],7)
=> ?
=> ? = 0
([(0,5),(1,6),(2,3),(2,6),(3,6),(4,5),(4,6)],7)
=> ([(0,6),(1,7),(2,7),(3,4),(3,5),(4,6),(5,7)],8)
=> ?
=> ? = 0
([(0,6),(1,5),(2,3),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,6),(1,5),(2,7),(3,5),(3,7),(4,6),(4,7)],8)
=> ?
=> ? = 0
([(0,6),(1,5),(2,5),(3,4),(3,6),(4,6),(5,6)],7)
=> ([(0,6),(1,6),(2,5),(3,5),(4,5),(4,6)],7)
=> ?
=> ? = 0
([(0,6),(1,6),(2,6),(3,4),(3,5),(4,5),(5,6)],7)
=> ([(0,6),(1,6),(2,6),(3,4),(4,5),(5,6)],7)
=> ?
=> ? = 0
([(0,6),(1,5),(2,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> ?
=> ? = 0
([(0,6),(1,4),(2,5),(2,6),(3,5),(3,6),(4,5)],7)
=> ([(0,6),(1,5),(2,3),(2,4),(3,5),(4,6)],7)
=> ?
=> ? = 0
([(0,6),(1,6),(2,3),(2,5),(3,5),(4,5),(4,6)],7)
=> ([(0,6),(1,7),(2,7),(3,4),(3,5),(4,6),(5,7)],8)
=> ?
=> ? = 0
([(0,6),(1,4),(2,5),(2,6),(3,5),(3,6),(4,5),(5,6)],7)
=> ([(0,6),(1,5),(2,3),(2,4),(3,5),(4,6)],7)
=> ?
=> ? = 0
([(0,4),(1,6),(2,5),(3,5),(3,6),(4,5),(4,6)],7)
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> ?
=> ? = 0
([(0,6),(1,2),(2,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,6),(1,5),(2,3),(2,4),(3,5),(4,6)],7)
=> ?
=> ? = 0
([(0,6),(1,4),(2,3),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> ?
=> ? = 0
([(0,5),(1,2),(1,3),(2,6),(3,6),(4,5),(4,6)],7)
=> ([(0,6),(1,5),(2,3),(2,4),(3,5),(4,6)],7)
=> ?
=> ? = 0
([(0,6),(1,4),(2,5),(2,6),(3,4),(3,5),(5,6)],7)
=> ([(0,8),(1,7),(2,3),(2,4),(3,5),(4,6),(5,7),(6,8)],9)
=> ?
=> ? = 0
([(0,4),(1,4),(2,5),(3,5),(3,6),(4,6),(5,6)],7)
=> ([(0,6),(1,7),(2,7),(3,4),(3,5),(4,6),(5,7)],8)
=> ?
=> ? = 0
([(0,4),(1,4),(1,6),(2,5),(2,6),(3,5),(3,6),(5,6)],7)
=> ([(0,6),(1,5),(2,3),(2,4),(3,5),(4,6)],7)
=> ?
=> ? = 0
([(0,5),(1,4),(2,4),(2,6),(3,5),(3,6),(4,6),(5,6)],7)
=> ([(0,6),(1,5),(2,3),(2,4),(3,5),(4,6)],7)
=> ?
=> ? = 0
([(0,6),(1,5),(2,4),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> ?
=> ? = 0
Description
The number of vertices with even degree.
The following 132 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000315The number of isolated vertices of a graph. St000986The multiplicity of the eigenvalue zero of the adjacency matrix of the graph. St001353The number of prime nodes in the modular decomposition of a graph. St001356The number of vertices in prime modules of a graph. St001367The smallest number which does not occur as degree of a vertex in a graph. St001691The number of kings in a graph. St000302The determinant of the distance matrix of a connected graph. St000351The determinant of the adjacency matrix of a graph. St000096The number of spanning trees of a graph. St000271The chromatic index of a graph. St000276The size of the preimage of the map 'to graph' from Ordered trees to Graphs. St000283The size of the preimage of the map 'to graph' from Binary trees to Graphs. St000450The number of edges minus the number of vertices plus 2 of a graph. St000948The chromatic discriminant of a graph. St001057The Grundy value of the game of creating an independent set in a graph. St001707The length of a longest path in a graph such that the remaining vertices can be partitioned into two sets of the same size without edges between them. St001796The absolute value of the quotient of the Tutte polynomial of the graph at (1,1) and (-1,-1). St000379The number of Hamiltonian cycles in a graph. St001592The maximal number of simple paths between any two different vertices of a graph. St000455The second largest eigenvalue of a graph if it is integral. St000456The monochromatic index of a connected graph. St001118The acyclic chromatic index of a graph. St001281The normalized isoperimetric number of a graph. St000464The Schultz index of a connected graph. St001545The second Elser number of a connected graph. St001704The size of the largest multi-subset-intersection of the deck of a graph with the deck of another graph. St000264The girth of a graph, which is not a tree. St000699The toughness times the least common multiple of 1,. St000259The diameter of a connected graph. St000260The radius of a connected graph. St000777The number of distinct eigenvalues of the distance Laplacian of a connected graph. St001060The distinguishing index of a graph. St001570The minimal number of edges to add to make a graph Hamiltonian. St001330The hat guessing number of a graph. St000771The largest multiplicity of a distance Laplacian eigenvalue in a connected graph. St000772The multiplicity of the largest distance Laplacian eigenvalue in a connected graph. St001645The pebbling number of a connected graph. St000466The Gutman (or modified Schultz) index of a connected graph. St000467The hyper-Wiener index of a connected graph. St000145The Dyson rank of a partition. St000175Degree of the polynomial counting the number of semistandard Young tableaux when stretching the shape. St000205Number of non-integral Gelfand-Tsetlin polytopes with prescribed top row and partition weight. St000206Number of non-integral Gelfand-Tsetlin polytopes with prescribed top row and integer composition weight. St000225Difference between largest and smallest parts in a partition. St000319The spin of an integer partition. St000320The dinv adjustment of an integer partition. St000506The number of standard desarrangement tableaux of shape equal to the given partition. St000749The smallest integer d such that the restriction of the representation corresponding to a partition of n to the symmetric group on n-d letters has a constituent of odd degree. St000944The 3-degree of an integer partition. St001175The size of a partition minus the hook length of the base cell. St001176The size of a partition minus its first part. St001177Twice the mean value of the major index among all standard Young tableaux of a partition. St001178Twelve times the variance of the major index among all standard Young tableaux of a partition. St001248Sum of the even parts of a partition. St001279The sum of the parts of an integer partition that are at least two. St001280The number of parts of an integer partition that are at least two. St001384The number of boxes in the diagram of a partition that do not lie in the largest triangle it contains. St001392The largest nonnegative integer which is not a part and is smaller than the largest part of the partition. St001440The number of standard Young tableaux whose major index is congruent one modulo the size of a given integer partition. St001541The Gini index of an integer partition. St001586The number of odd parts smaller than the largest even part in an integer partition. St001587Half of the largest even part of an integer partition. St001657The number of twos in an integer partition. St001714The number of subpartitions of an integer partition that do not dominate the conjugate subpartition. St001767The largest minimal number of arrows pointing to a cell in the Ferrers diagram in any assignment. St001912The length of the preperiod in Bulgarian solitaire corresponding to an integer partition. St001918The degree of the cyclic sieving polynomial corresponding to an integer partition. St001961The sum of the greatest common divisors of all pairs of parts. St000046The largest eigenvalue of the random to random operator acting on the simple module corresponding to the given partition. St000137The Grundy value of an integer partition. St000207Number of integral Gelfand-Tsetlin polytopes with prescribed top row and integer composition weight. St000208Number of integral Gelfand-Tsetlin polytopes with prescribed top row and integer partition weight. St000460The hook length of the last cell along the main diagonal of an integer partition. St000474Dyson's crank of a partition. St000618The number of self-evacuating tableaux of given shape. St000667The greatest common divisor of the parts of the partition. St000755The number of real roots of the characteristic polynomial of a linear recurrence associated with an integer partition. St000781The number of proper colouring schemes of a Ferrers diagram. St000870The product of the hook lengths of the diagonal cells in an integer partition. St001122The multiplicity of the sign representation in the Kronecker square corresponding to a partition. St001247The number of parts of a partition that are not congruent 2 modulo 3. St001249Sum of the odd parts of a partition. St001250The number of parts of a partition that are not congruent 0 modulo 3. St001262The dimension of the maximal parabolic seaweed algebra corresponding to the partition. St001283The number of finite solvable groups that are realised by the given partition over the complex numbers. St001284The number of finite groups that are realised by the given partition over the complex numbers. St001360The number of covering relations in Young's lattice below a partition. St001364The number of permutations whose cube equals a fixed permutation of given cycle type. St001378The product of the cohook lengths of the integer partition. St001380The number of monomer-dimer tilings of a Ferrers diagram. St001383The BG-rank of an integer partition. St001389The number of partitions of the same length below the given integer partition. St001432The order dimension of the partition. St001442The number of standard Young tableaux whose major index is divisible by the size of a given integer partition. St001525The number of symmetric hooks on the diagonal of a partition. St001527The cyclic permutation representation number of an integer partition. St001529The number of monomials in the expansion of the nabla operator applied to the power-sum symmetric function indexed by the partition. St001561The value of the elementary symmetric function evaluated at 1. St001562The value of the complete homogeneous symmetric function evaluated at 1. St001563The value of the power-sum symmetric function evaluated at 1. St001564The value of the forgotten symmetric functions when all variables set to 1. St001571The Cartan determinant of the integer partition. St001599The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on rooted trees. St001600The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on simple graphs. St001601The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on trees. St001602The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on endofunctions. St001606The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on set partitions. St001607The number of coloured graphs such that the multiplicities of colours are given by a partition. St001608The number of coloured rooted trees such that the multiplicities of colours are given by a partition. St001609The number of coloured trees such that the multiplicities of colours are given by a partition. St001610The number of coloured endofunctions such that the multiplicities of colours are given by a partition. St001611The number of multiset partitions such that the multiplicities of elements are given by a partition. St001627The number of coloured connected graphs such that the multiplicities of colours are given by a partition. St001628The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on simple connected graphs. St001763The Hurwitz number of an integer partition. St001780The order of promotion on the set of standard tableaux of given shape. St001785The number of ways to obtain a partition as the multiset of antidiagonal lengths of the Ferrers diagram of a partition. St001899The total number of irreducible representations contained in the higher Lie character for an integer partition. St001900The number of distinct irreducible representations contained in the higher Lie character for an integer partition. St001901The largest multiplicity of an irreducible representation contained in the higher Lie character for an integer partition. St001908The number of semistandard tableaux of distinct weight whose maximal entry is the length of the partition. St001913The number of preimages of an integer partition in Bulgarian solitaire. St001914The size of the orbit of an integer partition in Bulgarian solitaire. St001924The number of cells in an integer partition whose arm and leg length coincide. St001933The largest multiplicity of a part in an integer partition. St001934The number of monotone factorisations of genus zero of a permutation of given cycle type. St001936The number of transitive factorisations of a permutation of given cycle type into star transpositions. St001938The number of transitive monotone factorizations of genus zero of a permutation of given cycle type. St001939The number of parts that are equal to their multiplicity in the integer partition. St001940The number of distinct parts that are equal to their multiplicity in the integer partition. St001943The sum of the squares of the hook lengths of an integer partition. St001625The Möbius invariant of a lattice.
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