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Your data matches 329 different statistics following compositions of up to 3 maps.
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Matching statistic: St001719
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Values
([],1)
=> ([],1)
=> 1
([],2)
=> ([],1)
=> 1
([(0,1)],2)
=> ([(0,1)],2)
=> 1
([],3)
=> ([],1)
=> 1
([(1,2)],3)
=> ([(0,1)],2)
=> 1
([(0,1),(0,2)],3)
=> ([],1)
=> 1
([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 1
([(0,2),(1,2)],3)
=> ([],1)
=> 1
([],4)
=> ([],1)
=> 1
([(2,3)],4)
=> ([(0,1)],2)
=> 1
([(1,2),(1,3)],4)
=> ([],1)
=> 1
([(0,1),(0,2),(0,3)],4)
=> ([],1)
=> 1
([(0,2),(0,3),(3,1)],4)
=> ([(0,1)],2)
=> 1
([(0,1),(0,2),(1,3),(2,3)],4)
=> ([],1)
=> 1
([(1,2),(2,3)],4)
=> ([(0,2),(2,1)],3)
=> 1
([(0,3),(3,1),(3,2)],4)
=> ([],1)
=> 1
([(1,3),(2,3)],4)
=> ([],1)
=> 1
([(0,3),(1,3),(3,2)],4)
=> ([],1)
=> 1
([(0,3),(1,3),(2,3)],4)
=> ([],1)
=> 1
([(0,3),(1,2)],4)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1
([(0,3),(1,2),(1,3)],4)
=> ([(0,2),(2,1)],3)
=> 1
([(0,2),(0,3),(1,2),(1,3)],4)
=> ([(0,1)],2)
=> 1
([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(2,1),(3,2)],4)
=> 1
([(0,3),(1,2),(2,3)],4)
=> ([(0,1)],2)
=> 1
([],5)
=> ([],1)
=> 1
([(3,4)],5)
=> ([(0,1)],2)
=> 1
([(2,3),(2,4)],5)
=> ([],1)
=> 1
([(1,2),(1,3),(1,4)],5)
=> ([],1)
=> 1
([(0,1),(0,2),(0,3),(0,4)],5)
=> ([],1)
=> 1
([(0,2),(0,3),(0,4),(4,1)],5)
=> ([(0,1)],2)
=> 1
([(0,1),(0,2),(0,3),(2,4),(3,4)],5)
=> ([],1)
=> 1
([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
=> ([],1)
=> 1
([(1,3),(1,4),(4,2)],5)
=> ([(0,1)],2)
=> 1
([(0,3),(0,4),(4,1),(4,2)],5)
=> ([],1)
=> 1
([(1,2),(1,3),(2,4),(3,4)],5)
=> ([],1)
=> 1
([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> ([],1)
=> 1
([(0,3),(0,4),(3,2),(4,1)],5)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1
([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
=> ([(0,2),(2,1)],3)
=> 1
([(0,1),(0,2),(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,1)],2)
=> 1
([(2,3),(3,4)],5)
=> ([(0,2),(2,1)],3)
=> 1
([(1,4),(4,2),(4,3)],5)
=> ([],1)
=> 1
([(0,4),(4,1),(4,2),(4,3)],5)
=> ([],1)
=> 1
([(2,4),(3,4)],5)
=> ([],1)
=> 1
([(1,4),(2,4),(4,3)],5)
=> ([],1)
=> 1
([(0,4),(1,4),(4,2),(4,3)],5)
=> ([(0,1)],2)
=> 1
([(1,4),(2,4),(3,4)],5)
=> ([],1)
=> 1
([(0,4),(1,4),(2,4),(4,3)],5)
=> ([],1)
=> 1
([(0,4),(1,4),(2,4),(3,4)],5)
=> ([],1)
=> 1
([(0,4),(1,4),(2,3)],5)
=> ([(0,1)],2)
=> 1
([(0,4),(1,3),(2,3),(2,4)],5)
=> ([],1)
=> 1
Description
The number of shortest chains of small intervals from the bottom to the top in a lattice.
An interval $[a, b]$ in a lattice is small if $b$ is a join of elements covering $a$.
Matching statistic: St001111
Values
([],1)
=> ([],1)
=> ([],1)
=> ([],1)
=> 1
([],2)
=> ([],1)
=> ([],1)
=> ([],1)
=> 1
([(0,1)],2)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> ([],2)
=> 1
([],3)
=> ([],1)
=> ([],1)
=> ([],1)
=> 1
([(1,2)],3)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> ([],2)
=> 1
([(0,1),(0,2)],3)
=> ([],1)
=> ([],1)
=> ([],1)
=> 1
([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> ([],3)
=> 1
([(0,2),(1,2)],3)
=> ([],1)
=> ([],1)
=> ([],1)
=> 1
([],4)
=> ([],1)
=> ([],1)
=> ([],1)
=> 1
([(2,3)],4)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> ([],2)
=> 1
([(1,2),(1,3)],4)
=> ([],1)
=> ([],1)
=> ([],1)
=> 1
([(0,1),(0,2),(0,3)],4)
=> ([],1)
=> ([],1)
=> ([],1)
=> 1
([(0,2),(0,3),(3,1)],4)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> ([],2)
=> 1
([(0,1),(0,2),(1,3),(2,3)],4)
=> ([],1)
=> ([],1)
=> ([],1)
=> 1
([(1,2),(2,3)],4)
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> ([],3)
=> 1
([(0,3),(3,1),(3,2)],4)
=> ([],1)
=> ([],1)
=> ([],1)
=> 1
([(1,3),(2,3)],4)
=> ([],1)
=> ([],1)
=> ([],1)
=> 1
([(0,3),(1,3),(3,2)],4)
=> ([],1)
=> ([],1)
=> ([],1)
=> 1
([(0,3),(1,3),(2,3)],4)
=> ([],1)
=> ([],1)
=> ([],1)
=> 1
([(0,3),(1,2)],4)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(2,3)],4)
=> 1
([(0,3),(1,2),(1,3)],4)
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> ([],3)
=> 1
([(0,2),(0,3),(1,2),(1,3)],4)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> ([],2)
=> 1
([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(2,1),(3,2)],4)
=> ([],4)
=> 1
([(0,3),(1,2),(2,3)],4)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> ([],2)
=> 1
([],5)
=> ([],1)
=> ([],1)
=> ([],1)
=> 1
([(3,4)],5)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> ([],2)
=> 1
([(2,3),(2,4)],5)
=> ([],1)
=> ([],1)
=> ([],1)
=> 1
([(1,2),(1,3),(1,4)],5)
=> ([],1)
=> ([],1)
=> ([],1)
=> 1
([(0,1),(0,2),(0,3),(0,4)],5)
=> ([],1)
=> ([],1)
=> ([],1)
=> 1
([(0,2),(0,3),(0,4),(4,1)],5)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> ([],2)
=> 1
([(0,1),(0,2),(0,3),(2,4),(3,4)],5)
=> ([],1)
=> ([],1)
=> ([],1)
=> 1
([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
=> ([],1)
=> ([],1)
=> ([],1)
=> 1
([(1,3),(1,4),(4,2)],5)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> ([],2)
=> 1
([(0,3),(0,4),(4,1),(4,2)],5)
=> ([],1)
=> ([],1)
=> ([],1)
=> 1
([(1,2),(1,3),(2,4),(3,4)],5)
=> ([],1)
=> ([],1)
=> ([],1)
=> 1
([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> ([],1)
=> ([],1)
=> ([],1)
=> 1
([(0,3),(0,4),(3,2),(4,1)],5)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(2,3)],4)
=> 1
([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> ([],3)
=> 1
([(0,1),(0,2),(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> ([],2)
=> 1
([(2,3),(3,4)],5)
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> ([],3)
=> 1
([(1,4),(4,2),(4,3)],5)
=> ([],1)
=> ([],1)
=> ([],1)
=> 1
([(0,4),(4,1),(4,2),(4,3)],5)
=> ([],1)
=> ([],1)
=> ([],1)
=> 1
([(2,4),(3,4)],5)
=> ([],1)
=> ([],1)
=> ([],1)
=> 1
([(1,4),(2,4),(4,3)],5)
=> ([],1)
=> ([],1)
=> ([],1)
=> 1
([(0,4),(1,4),(4,2),(4,3)],5)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> ([],2)
=> 1
([(1,4),(2,4),(3,4)],5)
=> ([],1)
=> ([],1)
=> ([],1)
=> 1
([(0,4),(1,4),(2,4),(4,3)],5)
=> ([],1)
=> ([],1)
=> ([],1)
=> 1
([(0,4),(1,4),(2,4),(3,4)],5)
=> ([],1)
=> ([],1)
=> ([],1)
=> 1
([(0,4),(1,4),(2,3)],5)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> ([],2)
=> 1
([(0,4),(1,3),(2,3),(2,4)],5)
=> ([],1)
=> ([],1)
=> ([],1)
=> 1
([(0,3),(1,4),(1,5),(3,5),(4,2)],6)
=> ([(0,3),(0,4),(1,6),(2,5),(3,7),(4,2),(4,7),(5,6),(7,1),(7,5)],8)
=> ([(0,3),(0,4),(1,6),(2,5),(3,7),(4,2),(4,7),(5,6),(7,1),(7,5)],8)
=> ([(2,7),(3,6),(4,5),(5,7),(6,7)],8)
=> ? ∊ {1,1,1,3}
([(0,5),(1,4),(4,2),(5,3)],6)
=> ([(0,3),(0,4),(1,7),(2,6),(3,2),(3,5),(4,1),(4,5),(5,6),(5,7),(6,8),(7,8)],9)
=> ([(0,3),(0,4),(1,7),(2,6),(3,2),(3,5),(4,1),(4,5),(5,6),(5,7),(6,8),(7,8)],9)
=> ([(2,5),(2,8),(3,4),(3,8),(4,7),(5,7),(6,7),(6,8),(7,8)],9)
=> ? ∊ {1,1,1,3}
([(0,5),(1,4),(2,3)],6)
=> ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)
=> ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)
=> ([(2,5),(2,6),(2,7),(3,4),(3,6),(3,7),(4,5),(4,7),(5,6)],8)
=> ? ∊ {1,1,1,3}
([(0,5),(1,3),(4,2),(5,4)],6)
=> ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ([(2,7),(3,6),(4,5),(4,6),(5,7),(6,7)],8)
=> ? ∊ {1,1,1,3}
Description
The weak 2-dynamic chromatic number of a graph.
A $k$-weak-dynamic coloring of a graph $G$ is a (non-proper) coloring of $G$ in such a way that each vertex $v$ sees at least $\min\{d(v), k\}$ colors in its neighborhood. The $k$-weak-dynamic number of a graph is the smallest number of colors needed to find an $k$-dynamic coloring.
This statistic records the $2$-weak-dynamic number of a graph.
Matching statistic: St001112
Values
([],1)
=> ([],1)
=> ([],1)
=> ([],1)
=> 1
([],2)
=> ([],1)
=> ([],1)
=> ([],1)
=> 1
([(0,1)],2)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> ([],2)
=> 1
([],3)
=> ([],1)
=> ([],1)
=> ([],1)
=> 1
([(1,2)],3)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> ([],2)
=> 1
([(0,1),(0,2)],3)
=> ([],1)
=> ([],1)
=> ([],1)
=> 1
([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> ([],3)
=> 1
([(0,2),(1,2)],3)
=> ([],1)
=> ([],1)
=> ([],1)
=> 1
([],4)
=> ([],1)
=> ([],1)
=> ([],1)
=> 1
([(2,3)],4)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> ([],2)
=> 1
([(1,2),(1,3)],4)
=> ([],1)
=> ([],1)
=> ([],1)
=> 1
([(0,1),(0,2),(0,3)],4)
=> ([],1)
=> ([],1)
=> ([],1)
=> 1
([(0,2),(0,3),(3,1)],4)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> ([],2)
=> 1
([(0,1),(0,2),(1,3),(2,3)],4)
=> ([],1)
=> ([],1)
=> ([],1)
=> 1
([(1,2),(2,3)],4)
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> ([],3)
=> 1
([(0,3),(3,1),(3,2)],4)
=> ([],1)
=> ([],1)
=> ([],1)
=> 1
([(1,3),(2,3)],4)
=> ([],1)
=> ([],1)
=> ([],1)
=> 1
([(0,3),(1,3),(3,2)],4)
=> ([],1)
=> ([],1)
=> ([],1)
=> 1
([(0,3),(1,3),(2,3)],4)
=> ([],1)
=> ([],1)
=> ([],1)
=> 1
([(0,3),(1,2)],4)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(2,3)],4)
=> 1
([(0,3),(1,2),(1,3)],4)
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> ([],3)
=> 1
([(0,2),(0,3),(1,2),(1,3)],4)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> ([],2)
=> 1
([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(2,1),(3,2)],4)
=> ([],4)
=> 1
([(0,3),(1,2),(2,3)],4)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> ([],2)
=> 1
([],5)
=> ([],1)
=> ([],1)
=> ([],1)
=> 1
([(3,4)],5)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> ([],2)
=> 1
([(2,3),(2,4)],5)
=> ([],1)
=> ([],1)
=> ([],1)
=> 1
([(1,2),(1,3),(1,4)],5)
=> ([],1)
=> ([],1)
=> ([],1)
=> 1
([(0,1),(0,2),(0,3),(0,4)],5)
=> ([],1)
=> ([],1)
=> ([],1)
=> 1
([(0,2),(0,3),(0,4),(4,1)],5)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> ([],2)
=> 1
([(0,1),(0,2),(0,3),(2,4),(3,4)],5)
=> ([],1)
=> ([],1)
=> ([],1)
=> 1
([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
=> ([],1)
=> ([],1)
=> ([],1)
=> 1
([(1,3),(1,4),(4,2)],5)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> ([],2)
=> 1
([(0,3),(0,4),(4,1),(4,2)],5)
=> ([],1)
=> ([],1)
=> ([],1)
=> 1
([(1,2),(1,3),(2,4),(3,4)],5)
=> ([],1)
=> ([],1)
=> ([],1)
=> 1
([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> ([],1)
=> ([],1)
=> ([],1)
=> 1
([(0,3),(0,4),(3,2),(4,1)],5)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(2,3)],4)
=> 1
([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> ([],3)
=> 1
([(0,1),(0,2),(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> ([],2)
=> 1
([(2,3),(3,4)],5)
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> ([],3)
=> 1
([(1,4),(4,2),(4,3)],5)
=> ([],1)
=> ([],1)
=> ([],1)
=> 1
([(0,4),(4,1),(4,2),(4,3)],5)
=> ([],1)
=> ([],1)
=> ([],1)
=> 1
([(2,4),(3,4)],5)
=> ([],1)
=> ([],1)
=> ([],1)
=> 1
([(1,4),(2,4),(4,3)],5)
=> ([],1)
=> ([],1)
=> ([],1)
=> 1
([(0,4),(1,4),(4,2),(4,3)],5)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> ([],2)
=> 1
([(1,4),(2,4),(3,4)],5)
=> ([],1)
=> ([],1)
=> ([],1)
=> 1
([(0,4),(1,4),(2,4),(4,3)],5)
=> ([],1)
=> ([],1)
=> ([],1)
=> 1
([(0,4),(1,4),(2,4),(3,4)],5)
=> ([],1)
=> ([],1)
=> ([],1)
=> 1
([(0,4),(1,4),(2,3)],5)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> ([],2)
=> 1
([(0,4),(1,3),(2,3),(2,4)],5)
=> ([],1)
=> ([],1)
=> ([],1)
=> 1
([(0,3),(1,4),(1,5),(3,5),(4,2)],6)
=> ([(0,3),(0,4),(1,6),(2,5),(3,7),(4,2),(4,7),(5,6),(7,1),(7,5)],8)
=> ([(0,3),(0,4),(1,6),(2,5),(3,7),(4,2),(4,7),(5,6),(7,1),(7,5)],8)
=> ([(2,7),(3,6),(4,5),(5,7),(6,7)],8)
=> ? ∊ {1,1,1,3}
([(0,5),(1,4),(4,2),(5,3)],6)
=> ([(0,3),(0,4),(1,7),(2,6),(3,2),(3,5),(4,1),(4,5),(5,6),(5,7),(6,8),(7,8)],9)
=> ([(0,3),(0,4),(1,7),(2,6),(3,2),(3,5),(4,1),(4,5),(5,6),(5,7),(6,8),(7,8)],9)
=> ([(2,5),(2,8),(3,4),(3,8),(4,7),(5,7),(6,7),(6,8),(7,8)],9)
=> ? ∊ {1,1,1,3}
([(0,5),(1,4),(2,3)],6)
=> ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)
=> ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)
=> ([(2,5),(2,6),(2,7),(3,4),(3,6),(3,7),(4,5),(4,7),(5,6)],8)
=> ? ∊ {1,1,1,3}
([(0,5),(1,3),(4,2),(5,4)],6)
=> ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ([(2,7),(3,6),(4,5),(4,6),(5,7),(6,7)],8)
=> ? ∊ {1,1,1,3}
Description
The 3-weak dynamic number of a graph.
A $k$-weak-dynamic coloring of a graph $G$ is a (non-proper) coloring of $G$ in such a way that each vertex $v$ sees at least $\min\{d(v), k\}$ colors in its neighborhood. The $k$-weak-dynamic number of a graph is the smallest number of colors needed to find an $k$-dynamic coloring.
This statistic records the $3$-weak-dynamic number of a graph.
Matching statistic: St001694
Values
([],1)
=> ([],1)
=> ([],1)
=> ([],1)
=> 1
([],2)
=> ([],1)
=> ([],1)
=> ([],1)
=> 1
([(0,1)],2)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> ([],2)
=> 1
([],3)
=> ([],1)
=> ([],1)
=> ([],1)
=> 1
([(1,2)],3)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> ([],2)
=> 1
([(0,1),(0,2)],3)
=> ([],1)
=> ([],1)
=> ([],1)
=> 1
([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> ([],3)
=> 1
([(0,2),(1,2)],3)
=> ([],1)
=> ([],1)
=> ([],1)
=> 1
([],4)
=> ([],1)
=> ([],1)
=> ([],1)
=> 1
([(2,3)],4)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> ([],2)
=> 1
([(1,2),(1,3)],4)
=> ([],1)
=> ([],1)
=> ([],1)
=> 1
([(0,1),(0,2),(0,3)],4)
=> ([],1)
=> ([],1)
=> ([],1)
=> 1
([(0,2),(0,3),(3,1)],4)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> ([],2)
=> 1
([(0,1),(0,2),(1,3),(2,3)],4)
=> ([],1)
=> ([],1)
=> ([],1)
=> 1
([(1,2),(2,3)],4)
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> ([],3)
=> 1
([(0,3),(3,1),(3,2)],4)
=> ([],1)
=> ([],1)
=> ([],1)
=> 1
([(1,3),(2,3)],4)
=> ([],1)
=> ([],1)
=> ([],1)
=> 1
([(0,3),(1,3),(3,2)],4)
=> ([],1)
=> ([],1)
=> ([],1)
=> 1
([(0,3),(1,3),(2,3)],4)
=> ([],1)
=> ([],1)
=> ([],1)
=> 1
([(0,3),(1,2)],4)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(2,3)],4)
=> 1
([(0,3),(1,2),(1,3)],4)
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> ([],3)
=> 1
([(0,2),(0,3),(1,2),(1,3)],4)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> ([],2)
=> 1
([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(2,1),(3,2)],4)
=> ([],4)
=> 1
([(0,3),(1,2),(2,3)],4)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> ([],2)
=> 1
([],5)
=> ([],1)
=> ([],1)
=> ([],1)
=> 1
([(3,4)],5)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> ([],2)
=> 1
([(2,3),(2,4)],5)
=> ([],1)
=> ([],1)
=> ([],1)
=> 1
([(1,2),(1,3),(1,4)],5)
=> ([],1)
=> ([],1)
=> ([],1)
=> 1
([(0,1),(0,2),(0,3),(0,4)],5)
=> ([],1)
=> ([],1)
=> ([],1)
=> 1
([(0,2),(0,3),(0,4),(4,1)],5)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> ([],2)
=> 1
([(0,1),(0,2),(0,3),(2,4),(3,4)],5)
=> ([],1)
=> ([],1)
=> ([],1)
=> 1
([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
=> ([],1)
=> ([],1)
=> ([],1)
=> 1
([(1,3),(1,4),(4,2)],5)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> ([],2)
=> 1
([(0,3),(0,4),(4,1),(4,2)],5)
=> ([],1)
=> ([],1)
=> ([],1)
=> 1
([(1,2),(1,3),(2,4),(3,4)],5)
=> ([],1)
=> ([],1)
=> ([],1)
=> 1
([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> ([],1)
=> ([],1)
=> ([],1)
=> 1
([(0,3),(0,4),(3,2),(4,1)],5)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(2,3)],4)
=> 1
([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> ([],3)
=> 1
([(0,1),(0,2),(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> ([],2)
=> 1
([(2,3),(3,4)],5)
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> ([],3)
=> 1
([(1,4),(4,2),(4,3)],5)
=> ([],1)
=> ([],1)
=> ([],1)
=> 1
([(0,4),(4,1),(4,2),(4,3)],5)
=> ([],1)
=> ([],1)
=> ([],1)
=> 1
([(2,4),(3,4)],5)
=> ([],1)
=> ([],1)
=> ([],1)
=> 1
([(1,4),(2,4),(4,3)],5)
=> ([],1)
=> ([],1)
=> ([],1)
=> 1
([(0,4),(1,4),(4,2),(4,3)],5)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> ([],2)
=> 1
([(1,4),(2,4),(3,4)],5)
=> ([],1)
=> ([],1)
=> ([],1)
=> 1
([(0,4),(1,4),(2,4),(4,3)],5)
=> ([],1)
=> ([],1)
=> ([],1)
=> 1
([(0,4),(1,4),(2,4),(3,4)],5)
=> ([],1)
=> ([],1)
=> ([],1)
=> 1
([(0,4),(1,4),(2,3)],5)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> ([],2)
=> 1
([(0,4),(1,3),(2,3),(2,4)],5)
=> ([],1)
=> ([],1)
=> ([],1)
=> 1
([(0,3),(1,4),(1,5),(3,5),(4,2)],6)
=> ([(0,3),(0,4),(1,6),(2,5),(3,7),(4,2),(4,7),(5,6),(7,1),(7,5)],8)
=> ([(0,3),(0,4),(1,6),(2,5),(3,7),(4,2),(4,7),(5,6),(7,1),(7,5)],8)
=> ([(2,7),(3,6),(4,5),(5,7),(6,7)],8)
=> ? ∊ {1,1,1,3}
([(0,5),(1,4),(4,2),(5,3)],6)
=> ([(0,3),(0,4),(1,7),(2,6),(3,2),(3,5),(4,1),(4,5),(5,6),(5,7),(6,8),(7,8)],9)
=> ([(0,3),(0,4),(1,7),(2,6),(3,2),(3,5),(4,1),(4,5),(5,6),(5,7),(6,8),(7,8)],9)
=> ([(2,5),(2,8),(3,4),(3,8),(4,7),(5,7),(6,7),(6,8),(7,8)],9)
=> ? ∊ {1,1,1,3}
([(0,5),(1,4),(2,3)],6)
=> ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)
=> ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)
=> ([(2,5),(2,6),(2,7),(3,4),(3,6),(3,7),(4,5),(4,7),(5,6)],8)
=> ? ∊ {1,1,1,3}
([(0,5),(1,3),(4,2),(5,4)],6)
=> ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ([(2,7),(3,6),(4,5),(4,6),(5,7),(6,7)],8)
=> ? ∊ {1,1,1,3}
Description
The number of maximal dissociation sets in a graph.
Matching statistic: St001716
Values
([],1)
=> ([],1)
=> ([],1)
=> ([],1)
=> 1
([],2)
=> ([],1)
=> ([],1)
=> ([],1)
=> 1
([(0,1)],2)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> ([],2)
=> 1
([],3)
=> ([],1)
=> ([],1)
=> ([],1)
=> 1
([(1,2)],3)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> ([],2)
=> 1
([(0,1),(0,2)],3)
=> ([],1)
=> ([],1)
=> ([],1)
=> 1
([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> ([],3)
=> 1
([(0,2),(1,2)],3)
=> ([],1)
=> ([],1)
=> ([],1)
=> 1
([],4)
=> ([],1)
=> ([],1)
=> ([],1)
=> 1
([(2,3)],4)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> ([],2)
=> 1
([(1,2),(1,3)],4)
=> ([],1)
=> ([],1)
=> ([],1)
=> 1
([(0,1),(0,2),(0,3)],4)
=> ([],1)
=> ([],1)
=> ([],1)
=> 1
([(0,2),(0,3),(3,1)],4)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> ([],2)
=> 1
([(0,1),(0,2),(1,3),(2,3)],4)
=> ([],1)
=> ([],1)
=> ([],1)
=> 1
([(1,2),(2,3)],4)
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> ([],3)
=> 1
([(0,3),(3,1),(3,2)],4)
=> ([],1)
=> ([],1)
=> ([],1)
=> 1
([(1,3),(2,3)],4)
=> ([],1)
=> ([],1)
=> ([],1)
=> 1
([(0,3),(1,3),(3,2)],4)
=> ([],1)
=> ([],1)
=> ([],1)
=> 1
([(0,3),(1,3),(2,3)],4)
=> ([],1)
=> ([],1)
=> ([],1)
=> 1
([(0,3),(1,2)],4)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(2,3)],4)
=> 1
([(0,3),(1,2),(1,3)],4)
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> ([],3)
=> 1
([(0,2),(0,3),(1,2),(1,3)],4)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> ([],2)
=> 1
([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(2,1),(3,2)],4)
=> ([],4)
=> 1
([(0,3),(1,2),(2,3)],4)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> ([],2)
=> 1
([],5)
=> ([],1)
=> ([],1)
=> ([],1)
=> 1
([(3,4)],5)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> ([],2)
=> 1
([(2,3),(2,4)],5)
=> ([],1)
=> ([],1)
=> ([],1)
=> 1
([(1,2),(1,3),(1,4)],5)
=> ([],1)
=> ([],1)
=> ([],1)
=> 1
([(0,1),(0,2),(0,3),(0,4)],5)
=> ([],1)
=> ([],1)
=> ([],1)
=> 1
([(0,2),(0,3),(0,4),(4,1)],5)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> ([],2)
=> 1
([(0,1),(0,2),(0,3),(2,4),(3,4)],5)
=> ([],1)
=> ([],1)
=> ([],1)
=> 1
([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
=> ([],1)
=> ([],1)
=> ([],1)
=> 1
([(1,3),(1,4),(4,2)],5)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> ([],2)
=> 1
([(0,3),(0,4),(4,1),(4,2)],5)
=> ([],1)
=> ([],1)
=> ([],1)
=> 1
([(1,2),(1,3),(2,4),(3,4)],5)
=> ([],1)
=> ([],1)
=> ([],1)
=> 1
([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> ([],1)
=> ([],1)
=> ([],1)
=> 1
([(0,3),(0,4),(3,2),(4,1)],5)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(2,3)],4)
=> 1
([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> ([],3)
=> 1
([(0,1),(0,2),(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> ([],2)
=> 1
([(2,3),(3,4)],5)
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> ([],3)
=> 1
([(1,4),(4,2),(4,3)],5)
=> ([],1)
=> ([],1)
=> ([],1)
=> 1
([(0,4),(4,1),(4,2),(4,3)],5)
=> ([],1)
=> ([],1)
=> ([],1)
=> 1
([(2,4),(3,4)],5)
=> ([],1)
=> ([],1)
=> ([],1)
=> 1
([(1,4),(2,4),(4,3)],5)
=> ([],1)
=> ([],1)
=> ([],1)
=> 1
([(0,4),(1,4),(4,2),(4,3)],5)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> ([],2)
=> 1
([(1,4),(2,4),(3,4)],5)
=> ([],1)
=> ([],1)
=> ([],1)
=> 1
([(0,4),(1,4),(2,4),(4,3)],5)
=> ([],1)
=> ([],1)
=> ([],1)
=> 1
([(0,4),(1,4),(2,4),(3,4)],5)
=> ([],1)
=> ([],1)
=> ([],1)
=> 1
([(0,4),(1,4),(2,3)],5)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> ([],2)
=> 1
([(0,4),(1,3),(2,3),(2,4)],5)
=> ([],1)
=> ([],1)
=> ([],1)
=> 1
([(0,3),(1,4),(1,5),(3,5),(4,2)],6)
=> ([(0,3),(0,4),(1,6),(2,5),(3,7),(4,2),(4,7),(5,6),(7,1),(7,5)],8)
=> ([(0,3),(0,4),(1,6),(2,5),(3,7),(4,2),(4,7),(5,6),(7,1),(7,5)],8)
=> ([(2,7),(3,6),(4,5),(5,7),(6,7)],8)
=> ? ∊ {1,1,1,3}
([(0,5),(1,4),(4,2),(5,3)],6)
=> ([(0,3),(0,4),(1,7),(2,6),(3,2),(3,5),(4,1),(4,5),(5,6),(5,7),(6,8),(7,8)],9)
=> ([(0,3),(0,4),(1,7),(2,6),(3,2),(3,5),(4,1),(4,5),(5,6),(5,7),(6,8),(7,8)],9)
=> ([(2,5),(2,8),(3,4),(3,8),(4,7),(5,7),(6,7),(6,8),(7,8)],9)
=> ? ∊ {1,1,1,3}
([(0,5),(1,4),(2,3)],6)
=> ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)
=> ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)
=> ([(2,5),(2,6),(2,7),(3,4),(3,6),(3,7),(4,5),(4,7),(5,6)],8)
=> ? ∊ {1,1,1,3}
([(0,5),(1,3),(4,2),(5,4)],6)
=> ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ([(2,7),(3,6),(4,5),(4,6),(5,7),(6,7)],8)
=> ? ∊ {1,1,1,3}
Description
The 1-improper chromatic number of a graph.
This is the least number of colours in a vertex-colouring, such that each vertex has at most one neighbour with the same colour.
Matching statistic: St001774
Values
([],1)
=> ([],1)
=> ([],1)
=> ([],1)
=> 1
([],2)
=> ([],1)
=> ([],1)
=> ([],1)
=> 1
([(0,1)],2)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> ([],2)
=> 1
([],3)
=> ([],1)
=> ([],1)
=> ([],1)
=> 1
([(1,2)],3)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> ([],2)
=> 1
([(0,1),(0,2)],3)
=> ([],1)
=> ([],1)
=> ([],1)
=> 1
([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> ([],3)
=> 1
([(0,2),(1,2)],3)
=> ([],1)
=> ([],1)
=> ([],1)
=> 1
([],4)
=> ([],1)
=> ([],1)
=> ([],1)
=> 1
([(2,3)],4)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> ([],2)
=> 1
([(1,2),(1,3)],4)
=> ([],1)
=> ([],1)
=> ([],1)
=> 1
([(0,1),(0,2),(0,3)],4)
=> ([],1)
=> ([],1)
=> ([],1)
=> 1
([(0,2),(0,3),(3,1)],4)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> ([],2)
=> 1
([(0,1),(0,2),(1,3),(2,3)],4)
=> ([],1)
=> ([],1)
=> ([],1)
=> 1
([(1,2),(2,3)],4)
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> ([],3)
=> 1
([(0,3),(3,1),(3,2)],4)
=> ([],1)
=> ([],1)
=> ([],1)
=> 1
([(1,3),(2,3)],4)
=> ([],1)
=> ([],1)
=> ([],1)
=> 1
([(0,3),(1,3),(3,2)],4)
=> ([],1)
=> ([],1)
=> ([],1)
=> 1
([(0,3),(1,3),(2,3)],4)
=> ([],1)
=> ([],1)
=> ([],1)
=> 1
([(0,3),(1,2)],4)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(2,3)],4)
=> 1
([(0,3),(1,2),(1,3)],4)
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> ([],3)
=> 1
([(0,2),(0,3),(1,2),(1,3)],4)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> ([],2)
=> 1
([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(2,1),(3,2)],4)
=> ([],4)
=> 1
([(0,3),(1,2),(2,3)],4)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> ([],2)
=> 1
([],5)
=> ([],1)
=> ([],1)
=> ([],1)
=> 1
([(3,4)],5)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> ([],2)
=> 1
([(2,3),(2,4)],5)
=> ([],1)
=> ([],1)
=> ([],1)
=> 1
([(1,2),(1,3),(1,4)],5)
=> ([],1)
=> ([],1)
=> ([],1)
=> 1
([(0,1),(0,2),(0,3),(0,4)],5)
=> ([],1)
=> ([],1)
=> ([],1)
=> 1
([(0,2),(0,3),(0,4),(4,1)],5)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> ([],2)
=> 1
([(0,1),(0,2),(0,3),(2,4),(3,4)],5)
=> ([],1)
=> ([],1)
=> ([],1)
=> 1
([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
=> ([],1)
=> ([],1)
=> ([],1)
=> 1
([(1,3),(1,4),(4,2)],5)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> ([],2)
=> 1
([(0,3),(0,4),(4,1),(4,2)],5)
=> ([],1)
=> ([],1)
=> ([],1)
=> 1
([(1,2),(1,3),(2,4),(3,4)],5)
=> ([],1)
=> ([],1)
=> ([],1)
=> 1
([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> ([],1)
=> ([],1)
=> ([],1)
=> 1
([(0,3),(0,4),(3,2),(4,1)],5)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(2,3)],4)
=> 1
([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> ([],3)
=> 1
([(0,1),(0,2),(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> ([],2)
=> 1
([(2,3),(3,4)],5)
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> ([],3)
=> 1
([(1,4),(4,2),(4,3)],5)
=> ([],1)
=> ([],1)
=> ([],1)
=> 1
([(0,4),(4,1),(4,2),(4,3)],5)
=> ([],1)
=> ([],1)
=> ([],1)
=> 1
([(2,4),(3,4)],5)
=> ([],1)
=> ([],1)
=> ([],1)
=> 1
([(1,4),(2,4),(4,3)],5)
=> ([],1)
=> ([],1)
=> ([],1)
=> 1
([(0,4),(1,4),(4,2),(4,3)],5)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> ([],2)
=> 1
([(1,4),(2,4),(3,4)],5)
=> ([],1)
=> ([],1)
=> ([],1)
=> 1
([(0,4),(1,4),(2,4),(4,3)],5)
=> ([],1)
=> ([],1)
=> ([],1)
=> 1
([(0,4),(1,4),(2,4),(3,4)],5)
=> ([],1)
=> ([],1)
=> ([],1)
=> 1
([(0,4),(1,4),(2,3)],5)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> ([],2)
=> 1
([(0,4),(1,3),(2,3),(2,4)],5)
=> ([],1)
=> ([],1)
=> ([],1)
=> 1
([(0,3),(1,4),(1,5),(3,5),(4,2)],6)
=> ([(0,3),(0,4),(1,6),(2,5),(3,7),(4,2),(4,7),(5,6),(7,1),(7,5)],8)
=> ([(0,3),(0,4),(1,6),(2,5),(3,7),(4,2),(4,7),(5,6),(7,1),(7,5)],8)
=> ([(2,7),(3,6),(4,5),(5,7),(6,7)],8)
=> ? ∊ {1,1,1,3}
([(0,5),(1,4),(4,2),(5,3)],6)
=> ([(0,3),(0,4),(1,7),(2,6),(3,2),(3,5),(4,1),(4,5),(5,6),(5,7),(6,8),(7,8)],9)
=> ([(0,3),(0,4),(1,7),(2,6),(3,2),(3,5),(4,1),(4,5),(5,6),(5,7),(6,8),(7,8)],9)
=> ([(2,5),(2,8),(3,4),(3,8),(4,7),(5,7),(6,7),(6,8),(7,8)],9)
=> ? ∊ {1,1,1,3}
([(0,5),(1,4),(2,3)],6)
=> ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)
=> ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)
=> ([(2,5),(2,6),(2,7),(3,4),(3,6),(3,7),(4,5),(4,7),(5,6)],8)
=> ? ∊ {1,1,1,3}
([(0,5),(1,3),(4,2),(5,4)],6)
=> ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ([(2,7),(3,6),(4,5),(4,6),(5,7),(6,7)],8)
=> ? ∊ {1,1,1,3}
Description
The degree of the minimal polynomial of the smallest eigenvalue of a graph.
Matching statistic: St001775
Values
([],1)
=> ([],1)
=> ([],1)
=> ([],1)
=> 1
([],2)
=> ([],1)
=> ([],1)
=> ([],1)
=> 1
([(0,1)],2)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> ([],2)
=> 1
([],3)
=> ([],1)
=> ([],1)
=> ([],1)
=> 1
([(1,2)],3)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> ([],2)
=> 1
([(0,1),(0,2)],3)
=> ([],1)
=> ([],1)
=> ([],1)
=> 1
([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> ([],3)
=> 1
([(0,2),(1,2)],3)
=> ([],1)
=> ([],1)
=> ([],1)
=> 1
([],4)
=> ([],1)
=> ([],1)
=> ([],1)
=> 1
([(2,3)],4)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> ([],2)
=> 1
([(1,2),(1,3)],4)
=> ([],1)
=> ([],1)
=> ([],1)
=> 1
([(0,1),(0,2),(0,3)],4)
=> ([],1)
=> ([],1)
=> ([],1)
=> 1
([(0,2),(0,3),(3,1)],4)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> ([],2)
=> 1
([(0,1),(0,2),(1,3),(2,3)],4)
=> ([],1)
=> ([],1)
=> ([],1)
=> 1
([(1,2),(2,3)],4)
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> ([],3)
=> 1
([(0,3),(3,1),(3,2)],4)
=> ([],1)
=> ([],1)
=> ([],1)
=> 1
([(1,3),(2,3)],4)
=> ([],1)
=> ([],1)
=> ([],1)
=> 1
([(0,3),(1,3),(3,2)],4)
=> ([],1)
=> ([],1)
=> ([],1)
=> 1
([(0,3),(1,3),(2,3)],4)
=> ([],1)
=> ([],1)
=> ([],1)
=> 1
([(0,3),(1,2)],4)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(2,3)],4)
=> 1
([(0,3),(1,2),(1,3)],4)
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> ([],3)
=> 1
([(0,2),(0,3),(1,2),(1,3)],4)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> ([],2)
=> 1
([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(2,1),(3,2)],4)
=> ([],4)
=> 1
([(0,3),(1,2),(2,3)],4)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> ([],2)
=> 1
([],5)
=> ([],1)
=> ([],1)
=> ([],1)
=> 1
([(3,4)],5)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> ([],2)
=> 1
([(2,3),(2,4)],5)
=> ([],1)
=> ([],1)
=> ([],1)
=> 1
([(1,2),(1,3),(1,4)],5)
=> ([],1)
=> ([],1)
=> ([],1)
=> 1
([(0,1),(0,2),(0,3),(0,4)],5)
=> ([],1)
=> ([],1)
=> ([],1)
=> 1
([(0,2),(0,3),(0,4),(4,1)],5)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> ([],2)
=> 1
([(0,1),(0,2),(0,3),(2,4),(3,4)],5)
=> ([],1)
=> ([],1)
=> ([],1)
=> 1
([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
=> ([],1)
=> ([],1)
=> ([],1)
=> 1
([(1,3),(1,4),(4,2)],5)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> ([],2)
=> 1
([(0,3),(0,4),(4,1),(4,2)],5)
=> ([],1)
=> ([],1)
=> ([],1)
=> 1
([(1,2),(1,3),(2,4),(3,4)],5)
=> ([],1)
=> ([],1)
=> ([],1)
=> 1
([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> ([],1)
=> ([],1)
=> ([],1)
=> 1
([(0,3),(0,4),(3,2),(4,1)],5)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(2,3)],4)
=> 1
([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> ([],3)
=> 1
([(0,1),(0,2),(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> ([],2)
=> 1
([(2,3),(3,4)],5)
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> ([],3)
=> 1
([(1,4),(4,2),(4,3)],5)
=> ([],1)
=> ([],1)
=> ([],1)
=> 1
([(0,4),(4,1),(4,2),(4,3)],5)
=> ([],1)
=> ([],1)
=> ([],1)
=> 1
([(2,4),(3,4)],5)
=> ([],1)
=> ([],1)
=> ([],1)
=> 1
([(1,4),(2,4),(4,3)],5)
=> ([],1)
=> ([],1)
=> ([],1)
=> 1
([(0,4),(1,4),(4,2),(4,3)],5)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> ([],2)
=> 1
([(1,4),(2,4),(3,4)],5)
=> ([],1)
=> ([],1)
=> ([],1)
=> 1
([(0,4),(1,4),(2,4),(4,3)],5)
=> ([],1)
=> ([],1)
=> ([],1)
=> 1
([(0,4),(1,4),(2,4),(3,4)],5)
=> ([],1)
=> ([],1)
=> ([],1)
=> 1
([(0,4),(1,4),(2,3)],5)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> ([],2)
=> 1
([(0,4),(1,3),(2,3),(2,4)],5)
=> ([],1)
=> ([],1)
=> ([],1)
=> 1
([(0,3),(1,4),(1,5),(3,5),(4,2)],6)
=> ([(0,3),(0,4),(1,6),(2,5),(3,7),(4,2),(4,7),(5,6),(7,1),(7,5)],8)
=> ([(0,3),(0,4),(1,6),(2,5),(3,7),(4,2),(4,7),(5,6),(7,1),(7,5)],8)
=> ([(2,7),(3,6),(4,5),(5,7),(6,7)],8)
=> ? ∊ {1,1,1,3}
([(0,5),(1,4),(4,2),(5,3)],6)
=> ([(0,3),(0,4),(1,7),(2,6),(3,2),(3,5),(4,1),(4,5),(5,6),(5,7),(6,8),(7,8)],9)
=> ([(0,3),(0,4),(1,7),(2,6),(3,2),(3,5),(4,1),(4,5),(5,6),(5,7),(6,8),(7,8)],9)
=> ([(2,5),(2,8),(3,4),(3,8),(4,7),(5,7),(6,7),(6,8),(7,8)],9)
=> ? ∊ {1,1,1,3}
([(0,5),(1,4),(2,3)],6)
=> ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)
=> ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)
=> ([(2,5),(2,6),(2,7),(3,4),(3,6),(3,7),(4,5),(4,7),(5,6)],8)
=> ? ∊ {1,1,1,3}
([(0,5),(1,3),(4,2),(5,4)],6)
=> ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ([(2,7),(3,6),(4,5),(4,6),(5,7),(6,7)],8)
=> ? ∊ {1,1,1,3}
Description
The degree of the minimal polynomial of the largest eigenvalue of a graph.
Matching statistic: St001776
Values
([],1)
=> ([],1)
=> ([],1)
=> ([],1)
=> 1
([],2)
=> ([],1)
=> ([],1)
=> ([],1)
=> 1
([(0,1)],2)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> ([],2)
=> 1
([],3)
=> ([],1)
=> ([],1)
=> ([],1)
=> 1
([(1,2)],3)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> ([],2)
=> 1
([(0,1),(0,2)],3)
=> ([],1)
=> ([],1)
=> ([],1)
=> 1
([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> ([],3)
=> 1
([(0,2),(1,2)],3)
=> ([],1)
=> ([],1)
=> ([],1)
=> 1
([],4)
=> ([],1)
=> ([],1)
=> ([],1)
=> 1
([(2,3)],4)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> ([],2)
=> 1
([(1,2),(1,3)],4)
=> ([],1)
=> ([],1)
=> ([],1)
=> 1
([(0,1),(0,2),(0,3)],4)
=> ([],1)
=> ([],1)
=> ([],1)
=> 1
([(0,2),(0,3),(3,1)],4)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> ([],2)
=> 1
([(0,1),(0,2),(1,3),(2,3)],4)
=> ([],1)
=> ([],1)
=> ([],1)
=> 1
([(1,2),(2,3)],4)
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> ([],3)
=> 1
([(0,3),(3,1),(3,2)],4)
=> ([],1)
=> ([],1)
=> ([],1)
=> 1
([(1,3),(2,3)],4)
=> ([],1)
=> ([],1)
=> ([],1)
=> 1
([(0,3),(1,3),(3,2)],4)
=> ([],1)
=> ([],1)
=> ([],1)
=> 1
([(0,3),(1,3),(2,3)],4)
=> ([],1)
=> ([],1)
=> ([],1)
=> 1
([(0,3),(1,2)],4)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(2,3)],4)
=> 1
([(0,3),(1,2),(1,3)],4)
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> ([],3)
=> 1
([(0,2),(0,3),(1,2),(1,3)],4)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> ([],2)
=> 1
([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(2,1),(3,2)],4)
=> ([],4)
=> 1
([(0,3),(1,2),(2,3)],4)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> ([],2)
=> 1
([],5)
=> ([],1)
=> ([],1)
=> ([],1)
=> 1
([(3,4)],5)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> ([],2)
=> 1
([(2,3),(2,4)],5)
=> ([],1)
=> ([],1)
=> ([],1)
=> 1
([(1,2),(1,3),(1,4)],5)
=> ([],1)
=> ([],1)
=> ([],1)
=> 1
([(0,1),(0,2),(0,3),(0,4)],5)
=> ([],1)
=> ([],1)
=> ([],1)
=> 1
([(0,2),(0,3),(0,4),(4,1)],5)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> ([],2)
=> 1
([(0,1),(0,2),(0,3),(2,4),(3,4)],5)
=> ([],1)
=> ([],1)
=> ([],1)
=> 1
([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
=> ([],1)
=> ([],1)
=> ([],1)
=> 1
([(1,3),(1,4),(4,2)],5)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> ([],2)
=> 1
([(0,3),(0,4),(4,1),(4,2)],5)
=> ([],1)
=> ([],1)
=> ([],1)
=> 1
([(1,2),(1,3),(2,4),(3,4)],5)
=> ([],1)
=> ([],1)
=> ([],1)
=> 1
([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> ([],1)
=> ([],1)
=> ([],1)
=> 1
([(0,3),(0,4),(3,2),(4,1)],5)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(2,3)],4)
=> 1
([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> ([],3)
=> 1
([(0,1),(0,2),(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> ([],2)
=> 1
([(2,3),(3,4)],5)
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> ([],3)
=> 1
([(1,4),(4,2),(4,3)],5)
=> ([],1)
=> ([],1)
=> ([],1)
=> 1
([(0,4),(4,1),(4,2),(4,3)],5)
=> ([],1)
=> ([],1)
=> ([],1)
=> 1
([(2,4),(3,4)],5)
=> ([],1)
=> ([],1)
=> ([],1)
=> 1
([(1,4),(2,4),(4,3)],5)
=> ([],1)
=> ([],1)
=> ([],1)
=> 1
([(0,4),(1,4),(4,2),(4,3)],5)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> ([],2)
=> 1
([(1,4),(2,4),(3,4)],5)
=> ([],1)
=> ([],1)
=> ([],1)
=> 1
([(0,4),(1,4),(2,4),(4,3)],5)
=> ([],1)
=> ([],1)
=> ([],1)
=> 1
([(0,4),(1,4),(2,4),(3,4)],5)
=> ([],1)
=> ([],1)
=> ([],1)
=> 1
([(0,4),(1,4),(2,3)],5)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> ([],2)
=> 1
([(0,4),(1,3),(2,3),(2,4)],5)
=> ([],1)
=> ([],1)
=> ([],1)
=> 1
([(0,3),(1,4),(1,5),(3,5),(4,2)],6)
=> ([(0,3),(0,4),(1,6),(2,5),(3,7),(4,2),(4,7),(5,6),(7,1),(7,5)],8)
=> ([(0,3),(0,4),(1,6),(2,5),(3,7),(4,2),(4,7),(5,6),(7,1),(7,5)],8)
=> ([(2,7),(3,6),(4,5),(5,7),(6,7)],8)
=> ? ∊ {1,1,1,3}
([(0,5),(1,4),(4,2),(5,3)],6)
=> ([(0,3),(0,4),(1,7),(2,6),(3,2),(3,5),(4,1),(4,5),(5,6),(5,7),(6,8),(7,8)],9)
=> ([(0,3),(0,4),(1,7),(2,6),(3,2),(3,5),(4,1),(4,5),(5,6),(5,7),(6,8),(7,8)],9)
=> ([(2,5),(2,8),(3,4),(3,8),(4,7),(5,7),(6,7),(6,8),(7,8)],9)
=> ? ∊ {1,1,1,3}
([(0,5),(1,4),(2,3)],6)
=> ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)
=> ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)
=> ([(2,5),(2,6),(2,7),(3,4),(3,6),(3,7),(4,5),(4,7),(5,6)],8)
=> ? ∊ {1,1,1,3}
([(0,5),(1,3),(4,2),(5,4)],6)
=> ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ([(2,7),(3,6),(4,5),(4,6),(5,7),(6,7)],8)
=> ? ∊ {1,1,1,3}
Description
The degree of the minimal polynomial of the largest Laplacian eigenvalue of a graph.
Matching statistic: St000370
(load all 5 compositions to match this statistic)
(load all 5 compositions to match this statistic)
Values
([],1)
=> ([],1)
=> ([],1)
=> ([],1)
=> 0 = 1 - 1
([],2)
=> ([(0,1)],2)
=> ([],1)
=> ([],1)
=> 0 = 1 - 1
([(0,1)],2)
=> ([],2)
=> ([],2)
=> ([(0,1)],2)
=> 0 = 1 - 1
([],3)
=> ([(0,1),(0,2),(1,2)],3)
=> ([],1)
=> ([],1)
=> 0 = 1 - 1
([(1,2)],3)
=> ([(0,2),(1,2)],3)
=> ([(0,1)],2)
=> ([],2)
=> 0 = 1 - 1
([(0,1),(0,2)],3)
=> ([(1,2)],3)
=> ([],2)
=> ([(0,1)],2)
=> 0 = 1 - 1
([(0,2),(2,1)],3)
=> ([],3)
=> ([],3)
=> ([(0,1),(0,2),(1,2)],3)
=> 0 = 1 - 1
([(0,2),(1,2)],3)
=> ([(1,2)],3)
=> ([],2)
=> ([(0,1)],2)
=> 0 = 1 - 1
([],4)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([],1)
=> ([],1)
=> 0 = 1 - 1
([(2,3)],4)
=> ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1)],2)
=> ([],2)
=> 0 = 1 - 1
([(1,2),(1,3)],4)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1)],2)
=> ([],2)
=> 0 = 1 - 1
([(0,1),(0,2),(0,3)],4)
=> ([(1,2),(1,3),(2,3)],4)
=> ([],2)
=> ([(0,1)],2)
=> 0 = 1 - 1
([(0,2),(0,3),(3,1)],4)
=> ([(1,3),(2,3)],4)
=> ([(1,2)],3)
=> ([(0,2),(1,2)],3)
=> 0 = 1 - 1
([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(2,3)],4)
=> ([],3)
=> ([(0,1),(0,2),(1,2)],3)
=> 0 = 1 - 1
([(1,2),(2,3)],4)
=> ([(0,3),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,2)],3)
=> ([],3)
=> 0 = 1 - 1
([(0,3),(3,1),(3,2)],4)
=> ([(2,3)],4)
=> ([],3)
=> ([(0,1),(0,2),(1,2)],3)
=> 0 = 1 - 1
([(1,3),(2,3)],4)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1)],2)
=> ([],2)
=> 0 = 1 - 1
([(0,3),(1,3),(3,2)],4)
=> ([(2,3)],4)
=> ([],3)
=> ([(0,1),(0,2),(1,2)],3)
=> 0 = 1 - 1
([(0,3),(1,3),(2,3)],4)
=> ([(1,2),(1,3),(2,3)],4)
=> ([],2)
=> ([(0,1)],2)
=> 0 = 1 - 1
([(0,3),(1,2)],4)
=> ([(0,2),(0,3),(1,2),(1,3)],4)
=> ([(0,2),(0,3),(1,2),(1,3)],4)
=> ([(0,3),(1,2)],4)
=> 0 = 1 - 1
([(0,3),(1,2),(1,3)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> ([(0,2),(1,2)],3)
=> ([(1,2)],3)
=> 0 = 1 - 1
([(0,2),(0,3),(1,2),(1,3)],4)
=> ([(0,3),(1,2)],4)
=> ([],2)
=> ([(0,1)],2)
=> 0 = 1 - 1
([(0,3),(2,1),(3,2)],4)
=> ([],4)
=> ([],4)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 0 = 1 - 1
([(0,3),(1,2),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> ([(1,2)],3)
=> ([(0,2),(1,2)],3)
=> 0 = 1 - 1
([],5)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([],1)
=> ([],1)
=> 0 = 1 - 1
([(3,4)],5)
=> ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1)],2)
=> ([],2)
=> 0 = 1 - 1
([(2,3),(2,4)],5)
=> ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1)],2)
=> ([],2)
=> 0 = 1 - 1
([(1,2),(1,3),(1,4)],5)
=> ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1)],2)
=> ([],2)
=> 0 = 1 - 1
([(0,1),(0,2),(0,3),(0,4)],5)
=> ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([],2)
=> ([(0,1)],2)
=> 0 = 1 - 1
([(0,2),(0,3),(0,4),(4,1)],5)
=> ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,2)],3)
=> ([(0,2),(1,2)],3)
=> 0 = 1 - 1
([(0,1),(0,2),(0,3),(2,4),(3,4)],5)
=> ([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,2)],3)
=> ([(0,2),(1,2)],3)
=> 0 = 1 - 1
([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
=> ([(2,3),(2,4),(3,4)],5)
=> ([],3)
=> ([(0,1),(0,2),(1,2)],3)
=> 0 = 1 - 1
([(1,3),(1,4),(4,2)],5)
=> ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> ([],3)
=> 0 = 1 - 1
([(0,3),(0,4),(4,1),(4,2)],5)
=> ([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,2)],3)
=> ([(0,2),(1,2)],3)
=> 0 = 1 - 1
([(1,2),(1,3),(2,4),(3,4)],5)
=> ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> ([],3)
=> 0 = 1 - 1
([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> ([(3,4)],5)
=> ([],4)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 0 = 1 - 1
([(0,3),(0,4),(3,2),(4,1)],5)
=> ([(1,3),(1,4),(2,3),(2,4)],5)
=> ([(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
=> 0 = 1 - 1
([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
=> ([(1,4),(2,3),(3,4)],5)
=> ([(1,3),(2,3)],4)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> 0 = 1 - 1
([(0,1),(0,2),(1,3),(1,4),(2,3),(2,4)],5)
=> ([(1,4),(2,3)],5)
=> ([],3)
=> ([(0,1),(0,2),(1,2)],3)
=> 0 = 1 - 1
([(2,3),(3,4)],5)
=> ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> ([],3)
=> 0 = 1 - 1
([(1,4),(4,2),(4,3)],5)
=> ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> ([],3)
=> 0 = 1 - 1
([(0,4),(4,1),(4,2),(4,3)],5)
=> ([(2,3),(2,4),(3,4)],5)
=> ([],3)
=> ([(0,1),(0,2),(1,2)],3)
=> 0 = 1 - 1
([(2,4),(3,4)],5)
=> ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1)],2)
=> ([],2)
=> 0 = 1 - 1
([(1,4),(2,4),(4,3)],5)
=> ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> ([],3)
=> 0 = 1 - 1
([(0,4),(1,4),(4,2),(4,3)],5)
=> ([(1,4),(2,3)],5)
=> ([],3)
=> ([(0,1),(0,2),(1,2)],3)
=> 0 = 1 - 1
([(1,4),(2,4),(3,4)],5)
=> ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1)],2)
=> ([],2)
=> 0 = 1 - 1
([(0,4),(1,4),(2,4),(4,3)],5)
=> ([(2,3),(2,4),(3,4)],5)
=> ([],3)
=> ([(0,1),(0,2),(1,2)],3)
=> 0 = 1 - 1
([(0,4),(1,4),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([],2)
=> ([(0,1)],2)
=> 0 = 1 - 1
([(0,4),(1,4),(2,3)],5)
=> ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,2),(0,3),(1,2),(1,3)],4)
=> ([(0,3),(1,2)],4)
=> 0 = 1 - 1
([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(0,2),(0,3),(1,2),(1,3)],4)
=> ([(0,3),(1,2)],4)
=> 0 = 1 - 1
([(0,3),(1,4),(1,5),(3,5),(4,2)],6)
=> ([(0,4),(0,5),(1,2),(1,3),(2,4),(2,5),(3,4),(3,5)],6)
=> ([(0,3),(0,6),(0,7),(1,2),(1,5),(1,7),(2,4),(2,6),(3,4),(3,5),(4,5),(4,6),(5,7),(6,7)],8)
=> ([(0,3),(0,6),(0,7),(1,2),(1,5),(1,7),(2,4),(2,6),(3,4),(3,5),(4,5),(4,6),(5,7),(6,7)],8)
=> ? ∊ {1,2,2,3} - 1
([(0,5),(1,4),(4,2),(5,3)],6)
=> ([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5)],6)
=> ([(0,5),(0,6),(0,7),(0,8),(1,3),(1,4),(1,7),(1,8),(2,3),(2,4),(2,5),(2,6),(3,6),(3,8),(4,5),(4,7),(5,8),(6,7)],9)
=> ([(0,5),(0,6),(0,7),(0,8),(1,3),(1,4),(1,7),(1,8),(2,3),(2,4),(2,5),(2,6),(3,6),(3,8),(4,5),(4,7),(5,8),(6,7)],9)
=> ? ∊ {1,2,2,3} - 1
([(0,5),(1,4),(2,3)],6)
=> ([(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> ([(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(1,2),(1,3),(1,4),(1,5),(1,6),(1,7),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7)],8)
=> ([(0,7),(1,6),(2,5),(3,4)],8)
=> ? ∊ {1,2,2,3} - 1
([(0,5),(1,3),(4,2),(5,4)],6)
=> ([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> ([(0,4),(0,5),(0,6),(0,7),(1,2),(1,3),(1,6),(1,7),(2,3),(2,5),(2,7),(3,4),(3,7),(4,5),(4,6),(5,6)],8)
=> ([(0,5),(0,6),(0,7),(1,2),(1,3),(1,4),(2,6),(2,7),(3,5),(3,7),(4,5),(4,6)],8)
=> ? ∊ {1,2,2,3} - 1
Description
The genus of a graph.
This is the smallest genus of an oriented surface on which the graph can be embedded without crossings. One can indeed compute the genus as the sum of the genuses for the connected components.
Matching statistic: St001306
Values
([],1)
=> ([],1)
=> ([],1)
=> ([],1)
=> 0 = 1 - 1
([],2)
=> ([],1)
=> ([],1)
=> ([],1)
=> 0 = 1 - 1
([(0,1)],2)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> ([],2)
=> 0 = 1 - 1
([],3)
=> ([],1)
=> ([],1)
=> ([],1)
=> 0 = 1 - 1
([(1,2)],3)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> ([],2)
=> 0 = 1 - 1
([(0,1),(0,2)],3)
=> ([],1)
=> ([],1)
=> ([],1)
=> 0 = 1 - 1
([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> ([],3)
=> 0 = 1 - 1
([(0,2),(1,2)],3)
=> ([],1)
=> ([],1)
=> ([],1)
=> 0 = 1 - 1
([],4)
=> ([],1)
=> ([],1)
=> ([],1)
=> 0 = 1 - 1
([(2,3)],4)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> ([],2)
=> 0 = 1 - 1
([(1,2),(1,3)],4)
=> ([],1)
=> ([],1)
=> ([],1)
=> 0 = 1 - 1
([(0,1),(0,2),(0,3)],4)
=> ([],1)
=> ([],1)
=> ([],1)
=> 0 = 1 - 1
([(0,2),(0,3),(3,1)],4)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> ([],2)
=> 0 = 1 - 1
([(0,1),(0,2),(1,3),(2,3)],4)
=> ([],1)
=> ([],1)
=> ([],1)
=> 0 = 1 - 1
([(1,2),(2,3)],4)
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> ([],3)
=> 0 = 1 - 1
([(0,3),(3,1),(3,2)],4)
=> ([],1)
=> ([],1)
=> ([],1)
=> 0 = 1 - 1
([(1,3),(2,3)],4)
=> ([],1)
=> ([],1)
=> ([],1)
=> 0 = 1 - 1
([(0,3),(1,3),(3,2)],4)
=> ([],1)
=> ([],1)
=> ([],1)
=> 0 = 1 - 1
([(0,3),(1,3),(2,3)],4)
=> ([],1)
=> ([],1)
=> ([],1)
=> 0 = 1 - 1
([(0,3),(1,2)],4)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(2,3)],4)
=> 0 = 1 - 1
([(0,3),(1,2),(1,3)],4)
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> ([],3)
=> 0 = 1 - 1
([(0,2),(0,3),(1,2),(1,3)],4)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> ([],2)
=> 0 = 1 - 1
([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(2,1),(3,2)],4)
=> ([],4)
=> 0 = 1 - 1
([(0,3),(1,2),(2,3)],4)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> ([],2)
=> 0 = 1 - 1
([],5)
=> ([],1)
=> ([],1)
=> ([],1)
=> 0 = 1 - 1
([(3,4)],5)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> ([],2)
=> 0 = 1 - 1
([(2,3),(2,4)],5)
=> ([],1)
=> ([],1)
=> ([],1)
=> 0 = 1 - 1
([(1,2),(1,3),(1,4)],5)
=> ([],1)
=> ([],1)
=> ([],1)
=> 0 = 1 - 1
([(0,1),(0,2),(0,3),(0,4)],5)
=> ([],1)
=> ([],1)
=> ([],1)
=> 0 = 1 - 1
([(0,2),(0,3),(0,4),(4,1)],5)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> ([],2)
=> 0 = 1 - 1
([(0,1),(0,2),(0,3),(2,4),(3,4)],5)
=> ([],1)
=> ([],1)
=> ([],1)
=> 0 = 1 - 1
([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
=> ([],1)
=> ([],1)
=> ([],1)
=> 0 = 1 - 1
([(1,3),(1,4),(4,2)],5)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> ([],2)
=> 0 = 1 - 1
([(0,3),(0,4),(4,1),(4,2)],5)
=> ([],1)
=> ([],1)
=> ([],1)
=> 0 = 1 - 1
([(1,2),(1,3),(2,4),(3,4)],5)
=> ([],1)
=> ([],1)
=> ([],1)
=> 0 = 1 - 1
([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> ([],1)
=> ([],1)
=> ([],1)
=> 0 = 1 - 1
([(0,3),(0,4),(3,2),(4,1)],5)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(2,3)],4)
=> 0 = 1 - 1
([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> ([],3)
=> 0 = 1 - 1
([(0,1),(0,2),(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> ([],2)
=> 0 = 1 - 1
([(2,3),(3,4)],5)
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> ([],3)
=> 0 = 1 - 1
([(1,4),(4,2),(4,3)],5)
=> ([],1)
=> ([],1)
=> ([],1)
=> 0 = 1 - 1
([(0,4),(4,1),(4,2),(4,3)],5)
=> ([],1)
=> ([],1)
=> ([],1)
=> 0 = 1 - 1
([(2,4),(3,4)],5)
=> ([],1)
=> ([],1)
=> ([],1)
=> 0 = 1 - 1
([(1,4),(2,4),(4,3)],5)
=> ([],1)
=> ([],1)
=> ([],1)
=> 0 = 1 - 1
([(0,4),(1,4),(4,2),(4,3)],5)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> ([],2)
=> 0 = 1 - 1
([(1,4),(2,4),(3,4)],5)
=> ([],1)
=> ([],1)
=> ([],1)
=> 0 = 1 - 1
([(0,4),(1,4),(2,4),(4,3)],5)
=> ([],1)
=> ([],1)
=> ([],1)
=> 0 = 1 - 1
([(0,4),(1,4),(2,4),(3,4)],5)
=> ([],1)
=> ([],1)
=> ([],1)
=> 0 = 1 - 1
([(0,4),(1,4),(2,3)],5)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> ([],2)
=> 0 = 1 - 1
([(0,4),(1,3),(2,3),(2,4)],5)
=> ([],1)
=> ([],1)
=> ([],1)
=> 0 = 1 - 1
([(0,3),(1,4),(1,5),(3,5),(4,2)],6)
=> ([(0,3),(0,4),(1,6),(2,5),(3,7),(4,2),(4,7),(5,6),(7,1),(7,5)],8)
=> ([(0,3),(0,4),(1,6),(2,5),(3,7),(4,2),(4,7),(5,6),(7,1),(7,5)],8)
=> ([(2,7),(3,6),(4,5),(5,7),(6,7)],8)
=> ? ∊ {1,1,1,3} - 1
([(0,5),(1,4),(4,2),(5,3)],6)
=> ([(0,3),(0,4),(1,7),(2,6),(3,2),(3,5),(4,1),(4,5),(5,6),(5,7),(6,8),(7,8)],9)
=> ([(0,3),(0,4),(1,7),(2,6),(3,2),(3,5),(4,1),(4,5),(5,6),(5,7),(6,8),(7,8)],9)
=> ([(2,5),(2,8),(3,4),(3,8),(4,7),(5,7),(6,7),(6,8),(7,8)],9)
=> ? ∊ {1,1,1,3} - 1
([(0,5),(1,4),(2,3)],6)
=> ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)
=> ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)
=> ([(2,5),(2,6),(2,7),(3,4),(3,6),(3,7),(4,5),(4,7),(5,6)],8)
=> ? ∊ {1,1,1,3} - 1
([(0,5),(1,3),(4,2),(5,4)],6)
=> ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ([(2,7),(3,6),(4,5),(4,6),(5,7),(6,7)],8)
=> ? ∊ {1,1,1,3} - 1
Description
The number of induced paths on four vertices in a graph.
The following 319 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St001350Half of the Albertson index of a graph. St001723The differential of a graph. St001724The 2-packing differential of a graph. St001798The difference of the number of edges in a graph and the number of edges in the complement of the Turán graph. St000447The number of pairs of vertices of a graph with distance 3. St001575The minimal number of edges to add or remove to make a graph edge transitive. St001577The minimal number of edges to add or remove to make a graph a cograph. St001703The villainy of a graph. St001738The minimal order of a graph which is not an induced subgraph of the given graph. St001518The number of graphs with the same ordinary spectrum as the given graph. St001496The number of graphs with the same Laplacian spectrum as the given graph. St001613The binary logarithm of the size of the center of a lattice. St001681The number of inclusion-wise minimal subsets of a lattice, whose meet is the bottom element. St001881The number of factors of a lattice as a Cartesian product of lattices. St001325The minimal number of occurrences of the comparability-pattern in a linear ordering of the vertices of the graph. St001677The number of non-degenerate subsets of a lattice whose meet is the bottom element. St001845The number of join irreducibles minus the rank of a lattice. St000914The sum of the values of the Möbius function of a poset. St000882The number of connected components of short braid edges in the graph of braid moves of a permutation. St000879The number of long braid edges in the graph of braid moves of a permutation. St000095The number of triangles of a graph. St000274The number of perfect matchings of a graph. St000303The determinant of the product of the incidence matrix and its transpose of a graph divided by $4$. St001572The minimal number of edges to remove to make a graph bipartite. St001573The minimal number of edges to remove to make a graph triangle-free. St001690The length of a longest path in a graph such that after removing the paths edges, every vertex of the path has distance two from some other vertex of the path. St001783The number of odd automorphisms of a graph. St001871The number of triconnected components of a graph. St000781The number of proper colouring schemes of a Ferrers diagram. St001621The number of atoms of a lattice. St001901The largest multiplicity of an irreducible representation contained in the higher Lie character for an integer partition. St001934The number of monotone factorisations of genus zero of a permutation of given cycle type. St000181The number of connected components of the Hasse diagram for the poset. St001307The number of induced stars on four vertices in a graph. St001347The number of pairs of vertices of a graph having the same neighbourhood. St001603The number of colourings of a polygon such that the multiplicities of a colour are given by a partition. St001820The size of the image of the pop stack sorting operator. St001846The number of elements which do not have a complement in the lattice. St001605The number of colourings of a cycle such that the multiplicities of colours are given by a partition. St000031The number of cycles in the cycle decomposition of a permutation. St000022The number of fixed points of a permutation. St000153The number of adjacent cycles of a permutation. St001465The number of adjacent transpositions in the cycle decomposition of a permutation. St000056The decomposition (or block) number of a permutation. St000162The number of nontrivial cycles in the cycle decomposition of a permutation. St000486The number of cycles of length at least 3 of a permutation. St000694The number of affine bounded permutations that project to a given permutation. St000788The number of nesting-similar perfect matchings of a perfect matching. St001081The number of minimal length factorizations of a permutation into star transpositions. St001174The Gorenstein dimension of the algebra $A/I$ when $I$ is the tilting module corresponding to the permutation in the Auslander algebra of $K[x]/(x^n)$. St001208The number of connected components of the quiver of $A/T$ when $T$ is the 1-tilting module corresponding to the permutation in the Auslander algebra $A$ of $K[x]/(x^n)$. St001256Number of simple reflexive modules that are 2-stable reflexive. St001461The number of topologically connected components of the chord diagram of a permutation. St001493The number of simple modules with maximal even projective dimension in the corresponding Nakayama algebra. St001507The sum of projective dimension of simple modules with even projective dimension divided by 2 in the LNakayama algebra corresponding to Dyck paths. St001590The crossing number of a perfect matching. St001661Half the permanent of the Identity matrix plus the permutation matrix associated to the permutation. St001830The chord expansion number of a perfect matching. St001832The number of non-crossing perfect matchings in the chord expansion of a perfect matching. St001859The number of factors of the Stanley symmetric function associated with a permutation. St000221The number of strong fixed points of a permutation. St000279The size of the preimage of the map 'cycle-as-one-line notation' from Permutations to Permutations. St000375The number of non weak exceedences of a permutation that are mid-points of a decreasing subsequence of length $3$. St000488The number of cycles of a permutation of length at most 2. St000623The number of occurrences of the pattern 52341 in a permutation. St000666The number of right tethers of a permutation. St000689The maximal n such that the minimal generator-cogenerator module in the LNakayama algebra of a Dyck path is n-rigid. St000787The number of flips required to make a perfect matching noncrossing. St001165Number of simple modules with even projective dimension in the corresponding Nakayama algebra. St001204Call a CNakayama algebra (a Nakayama algebra with a cyclic quiver) with Kupisch series $L=[c_0,c_1,...,c_{n−1}]$ such that $n=c_0 < c_i$ for all $i > 0$ a special CNakayama algebra. St001292The injective dimension of the tensor product of two copies of the dual of the Nakayama algebra associated to a Dyck path. St001314The number of tilting modules of arbitrary projective dimension that have no simple modules as a direct summand in the corresponding Nakayama algebra. St001359The number of permutations in the equivalence class of a permutation obtained by taking inverses of cycles. St001381The fertility of a permutation. St001444The rank of the skew-symmetric form which is non-zero on crossing arcs of a perfect matching. St001466The number of transpositions swapping cyclically adjacent numbers in a permutation. St001549The number of restricted non-inversions between exceedances. St001551The number of restricted non-inversions between exceedances where the rightmost exceedance is linked. St001552The number of inversions between excedances and fixed points of a permutation. St001663The number of occurrences of the Hertzsprung pattern 132 in a permutation. St001810The number of fixed points of a permutation smaller than its largest moved point. St001811The Castelnuovo-Mumford regularity of a permutation. St001837The number of occurrences of a 312 pattern in the restricted growth word of a perfect matching. St001850The number of Hecke atoms of a permutation. St001906Half of the difference between the total displacement and the number of inversions and the reflection length of a permutation. St001195The global dimension of the algebra $A/AfA$ of the corresponding Nakayama algebra $A$ with minimal left faithful projective-injective module $Af$. St000771The largest multiplicity of a distance Laplacian eigenvalue in a connected graph. St000772The multiplicity of the largest distance Laplacian eigenvalue in a connected graph. St001271The competition number of a graph. St000422The energy of a graph, if it is integral. St000718The largest Laplacian eigenvalue of a graph if it is integral. St000455The second largest eigenvalue of a graph if it is integral. St001890The maximum magnitude of the Möbius function of a poset. St001878The projective dimension of the simple modules corresponding to the minimum of L in the incidence algebra of the lattice L. St000908The length of the shortest maximal antichain in a poset. St001532The leading coefficient of the Poincare polynomial of the poset cone. St001533The largest coefficient of the Poincare polynomial of the poset cone. St001301The first Betti number of the order complex associated with the poset. St001396Number of triples of incomparable elements in a finite poset. St001964The interval resolution global dimension of a poset. St001634The trace of the Coxeter matrix of the incidence algebra of a poset. St000454The largest eigenvalue of a graph if it is integral. St000759The smallest missing part in an integer partition. St000475The number of parts equal to 1 in a partition. St000618The number of self-evacuating tableaux of given shape. St001432The order dimension of the partition. St001599The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on rooted trees. St001604The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on polygons. St001780The order of promotion on the set of standard tableaux of given shape. St001899The total number of irreducible representations contained in the higher Lie character for an integer partition. St001900The number of distinct irreducible representations contained in the higher Lie character for an integer partition. St001908The number of semistandard tableaux of distinct weight whose maximal entry is the length of the partition. St001924The number of cells in an integer partition whose arm and leg length coincide. St001785The number of ways to obtain a partition as the multiset of antidiagonal lengths of the Ferrers diagram of a partition. St000629The defect of a binary word. St001657The number of twos in an integer partition. St001722The number of minimal chains with small intervals between a binary word and the top element. St001593This is the number of standard Young tableaux of the given shifted shape. 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. St000706The product of the factorials of the multiplicities of an integer partition. St000993The multiplicity of the largest part of an integer partition. St001568The smallest positive integer that does not appear twice in the partition. St001385The number of conjugacy classes of subgroups with connected subgroups of sizes prescribed by an integer partition. St001484The number of singletons of an integer partition. St001588The number of distinct odd parts smaller than the largest even part in an integer partition. St000913The number of ways to refine the partition into singletons. St000752The Grundy value for the game 'Couples are forever' on an integer partition. St001714The number of subpartitions of an integer partition that do not dominate the conjugate subpartition. St000847The number of standard Young tableaux whose descent set is the binary word. St000921The number of internal inversions of a binary word. St001730The number of times the path corresponding to a binary word crosses the base line. St000456The monochromatic index of a connected graph. St000297The number of leading ones in a binary word. St000011The number of touch points (or returns) of a Dyck path. St000326The position of the first one in a binary word after appending a 1 at the end. St000386The number of factors DDU in a Dyck path. St001490The number of connected components of a skew partition. St001001The number of indecomposable modules with projective and injective dimension equal to the global dimension of the Nakayama algebra corresponding to the Dyck path. St001371The length of the longest Yamanouchi prefix of a binary word. St001803The maximal overlap of the cylindrical tableau associated with a tableau. St001804The minimal height of the rectangular inner shape in a cylindrical tableau associated to a tableau. St000655The length of the minimal rise of a Dyck path. St001786The number of total orderings of the north steps of a Dyck path such that steps after the k-th east step are not among the first k positions in the order. St000439The position of the first down step of a Dyck path. St000630The length of the shortest palindromic decomposition of a binary word. St000785The number of distinct colouring schemes of a 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. St000052The number of valleys of a Dyck path not on the x-axis. St000296The length of the symmetric border of a binary word. St000897The number of different multiplicities of parts of an integer partition. St001031The height of the bicoloured Motzkin path associated with the Dyck path. St000980The number of boxes weakly below the path and above the diagonal that lie below at least two peaks. St001107The number of times one can erase the first up and the last down step in a Dyck path and still remain a Dyck path. St001172The number of 1-rises at odd height of a Dyck path. St001584The area statistic between a Dyck path and its bounce path. St000678The number of up steps after the last double rise of a Dyck path. St000745The index of the last row whose first entry is the row number in a standard Young tableau. St001696The natural major index of a standard Young tableau. St001732The number of peaks visible from the left. St001037The number of inner corners of the upper path of the parallelogram polyomino associated with the Dyck path. St001141The number of occurrences of hills of size 3 in a Dyck path. St000659The number of rises of length at least 2 of a Dyck path. St000790The number of pairs of centered tunnels, one strictly containing the other, of a Dyck path. St001137Number of simple modules that are 3-regular in the corresponding Nakayama algebra. St000929The constant term of the character polynomial of an integer partition. St000057The Shynar inversion number of a standard tableau. St000687The dimension of $Hom(I,P)$ for the LNakayama algebra of a Dyck path. St000685The dominant dimension of the LNakayama algebra associated to a Dyck path. St001063Numbers of 3-torsionfree simple modules in the corresponding Nakayama algebra. St001064Number of simple modules in the corresponding Nakayama algebra that are 3-syzygy modules. St001483The number of simple module modules that appear in the socle of the regular module but have no nontrivial selfextensions with the regular module. St000966Number of peaks minus the global dimension of the corresponding LNakayama algebra. St001025Number of simple modules with projective dimension 4 in the Nakayama algebra corresponding to the Dyck path. St001125The number of simple modules that satisfy the 2-regular condition in the corresponding Nakayama algebra. St001276The number of 2-regular indecomposable modules in the corresponding Nakayama algebra. St001877Number of indecomposable injective modules with projective dimension 2. St001011Number of simple modules of projective dimension 2 in the Nakayama algebra corresponding to the Dyck path. St001471The magnitude of a Dyck path. St001066The number of simple reflexive modules in the corresponding Nakayama algebra. St001501The dominant dimension of magnitude 1 Nakayama algebras. St001932The number of pairs of singleton blocks in the noncrossing set partition corresponding to a Dyck path, that can be merged to create another noncrossing set partition. St000791The number of pairs of left tunnels, one strictly containing the other, of a Dyck path. St001139The number of occurrences of hills of size 2 in a Dyck path. St000930The k-Gorenstein degree of the corresponding Nakayama algebra with linear quiver. St001202Call a CNakayama algebra (a Nakayama algebra with a cyclic quiver) with Kupisch series $L=[c_0,c_1,...,c_{n−1}]$ such that $n=c_0 < c_i$ for all $i > 0$ a special CNakayama algebra. St001289The vector space dimension of the n-fold tensor product of D(A), where n is maximal such that this n-fold tensor product is nonzero. St001021Sum of the differences between projective and codominant dimension of the non-projective indecomposable injective modules in the Nakayama algebra corresponding to the Dyck path. St001089Number of indecomposable projective non-injective modules minus the number of indecomposable projective non-injective modules with dominant dimension equal to the injective dimension in the corresponding Nakayama algebra. St001140Number of indecomposable modules with projective and injective dimension at least two in the corresponding Nakayama algebra. St001163The number of simple modules with dominant dimension at least three in the corresponding Nakayama algebra. St001193The dimension of $Ext_A^1(A/AeA,A)$ in the corresponding Nakayama algebra $A$ such that $eA$ is a minimal faithful projective-injective module. St001216The number of indecomposable injective modules in the corresponding Nakayama algebra that have non-vanishing second Ext-group with the regular module. St001221The number of simple modules in the corresponding LNakayama algebra that have 2 dimensional second Extension group with the regular module. St001222Number of simple modules in the corresponding LNakayama algebra that have a unique 2-extension with the regular module. St001229The vector space dimension of the first extension group between the Jacobson radical J and J^2. St001230The number of simple modules with injective dimension equal to the dominant dimension equal to one and the dual property. St001264The smallest index i such that the i-th simple module has projective dimension equal to the global dimension of the corresponding Nakayama algebra. St001274The number of indecomposable injective modules with projective dimension equal to two. St001275The projective dimension of the second term in a minimal injective coresolution of the regular module. St001290The first natural number n such that the tensor product of n copies of D(A) is zero for the corresponding Nakayama algebra A. St000069The number of maximal elements of a poset. St000025The number of initial rises of a Dyck path. St000026The position of the first return of a Dyck path. St001135The projective dimension of the first simple module in the Nakayama algebra corresponding to the Dyck path. St001570The minimal number of edges to add to make a graph Hamiltonian. St001695The natural comajor index of a standard Young tableau. St001698The comajor index of a standard tableau minus the weighted size of its shape. St001699The major index of a standard tableau minus the weighted size of its shape. St001712The number of natural descents of a standard Young tableau. St001462The number of factors of a standard tableaux under concatenation. St000284The Plancherel distribution on integer partitions. St000698The number of 2-rim hooks removed from an integer partition to obtain its associated 2-core. St000704The number of semistandard tableaux on a given integer partition with minimal maximal entry. St000901The cube of the number of standard Young tableaux with shape given by the partition. St001128The exponens consonantiae of a partition. St001006Number of simple modules with projective dimension equal to the global dimension of the Nakayama algebra corresponding to the Dyck path. St001013Number of indecomposable injective modules with codominant dimension equal to the global dimension in the Nakayama algebra corresponding to the Dyck path. St001188The number of simple modules $S$ with grade $\inf \{ i \geq 0 | Ext^i(S,A) \neq 0 \}$ at least two in the Nakayama algebra $A$ corresponding to the Dyck path. St001212The number of simple modules in the corresponding Nakayama algebra that have non-zero second Ext-group with the regular module. St001244The number of simple modules of projective dimension one that are not 1-regular for the Nakayama algebra associated to a Dyck path. St001113Number of indecomposable projective non-injective modules with reflexive Auslander-Reiten sequences in the corresponding Nakayama algebra. St001198The number of simple modules in the algebra $eAe$ with projective dimension at most 1 in the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St001206The maximal dimension of an indecomposable projective $eAe$-module (that is the height of the corresponding Dyck path) of the corresponding Nakayama algebra with minimal faithful projective-injective module $eA$. St001219Number of simple modules S in the corresponding Nakayama algebra such that the Auslander-Reiten sequence ending at S has the property that all modules in the exact sequence are reflexive. St001239The largest vector space dimension of the double dual of a simple module in the corresponding Nakayama algebra. St000999Number of indecomposable projective module with injective dimension equal to the global dimension in the Nakayama algebra corresponding to the Dyck path. St001192The maximal dimension of $Ext_A^2(S,A)$ for a simple module $S$ over the corresponding Nakayama algebra $A$. St000954Number of times the corresponding LNakayama algebra has $Ext^i(D(A),A)=0$ for $i>0$. St001181Number of indecomposable injective modules with grade at least 3 in the corresponding Nakayama algebra. St001231The 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. St001234The number of indecomposable three dimensional modules with projective dimension one. St000068The number of minimal elements in a poset. St000449The number of pairs of vertices of a graph with distance 4. St001236The dominant dimension of the corresponding Comp-Nakayama algebra. St000658The number of rises of length 2 of a Dyck path. St001199The dominant dimension of $eAe$ for the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St000674The number of hills of a Dyck path. St001217The projective dimension of the indecomposable injective module I[n-2] in the corresponding Nakayama algebra with simples enumerated from 0 to n-1. St001257The dominant dimension of the double dual of A/J when A is the corresponding Nakayama algebra with Jacobson radical J. St001159Number of simple modules with dominant dimension equal to the global dimension in the corresponding Nakayama algebra. St000335The difference of lower and upper interactions. St000968We make a CNakayama algebra out of the LNakayama algebra (corresponding to the Dyck path) $[c_0,c_1,...,c_{n−1}]$ by adding $c_0$ to $c_{n−1}$. St001184Number of indecomposable injective modules with grade at least 1 in the corresponding Nakayama algebra. St001185The number of indecomposable injective modules of grade at least 2 in the corresponding Nakayama algebra. St001201The grade of the simple module $S_0$ in the special CNakayama algebra corresponding to the Dyck path. St001481The minimal height of a peak of a Dyck path. St000951The dimension of $Ext^{1}(D(A),A)$ of the corresponding LNakayama algebra. St001008Number of indecomposable injective modules with projective dimension 1 in the Nakayama algebra corresponding to the Dyck path. St001122The multiplicity of the sign representation in the Kronecker square corresponding to a partition. St001226The number of integers i such that the radical of the i-th indecomposable projective module has vanishing first extension group with the Jacobson radical J in the corresponding Nakayama algebra. St001487The number of inner corners of a skew partition. St001435The number of missing boxes in the first row. St001438The number of missing boxes of a skew partition. St001232The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2. St000097The order of the largest clique of the graph. St001118The acyclic chromatic index of a graph. St001491The number of indecomposable projective-injective modules in the algebra corresponding to a subset. St000741The Colin de Verdière graph invariant. St000098The chromatic number of a graph. St000266The number of spanning subgraphs of a graph with the same connected components. St000267The number of maximal spanning forests contained in a graph. St000773The multiplicity of the largest Laplacian eigenvalue in a graph. St000775The multiplicity of the largest eigenvalue in a graph. St001272The number of graphs with the same degree sequence. St001316The domatic number of a graph. St001395The number of strictly unfriendly partitions of a graph. St001475The evaluation of the Tutte polynomial of the graph at (x,y) equal to (1,0). St001476The evaluation of the Tutte polynomial of the graph at (x,y) equal to (1,-1). St001546The number of monomials in the Tutte polynomial of a graph. St000283The size of the preimage of the map 'to graph' from Binary trees to Graphs. St000323The minimal crossing number of a graph. St000351The determinant of the adjacency matrix of a graph. St000368The Altshuler-Steinberg determinant of a graph. St000379The number of Hamiltonian cycles in a graph. St000403The Szeged index minus the Wiener index of a graph. St000636The hull number of a graph. St000637The length of the longest cycle in a graph. St000671The maximin edge-connectivity for choosing a subgraph. St000699The toughness times the least common multiple of 1,. St000948The chromatic discriminant of a graph. St001029The size of the core of a graph. St001069The coefficient of the monomial xy of the Tutte polynomial of the graph. St001109The number of proper colourings of a graph with as few colours as possible. St001119The length of a shortest maximal path in a graph. St001281The normalized isoperimetric number of a graph. St001305The number of induced cycles on four vertices in a graph. St001309The number of four-cliques in a graph. St001310The number of induced diamond graphs in a graph. St001311The cyclomatic number of a graph. St001317The minimal number of occurrences of the forest-pattern in a linear ordering of the vertices of the graph. St001320The minimal number of occurrences of the path-pattern in a linear ordering of the vertices of the graph. St001323The independence gap of a graph. St001324The minimal number of occurrences of the chordal-pattern in a linear ordering of the vertices of the graph. St001326The minimal number of occurrences of the interval-pattern in a linear ordering of the vertices of the graph. St001328The minimal number of occurrences of the bipartite-pattern in a linear ordering of the vertices of the graph. St001329The minimal number of occurrences of the outerplanar pattern in a linear ordering of the vertices of the graph. St001331The size of the minimal feedback vertex set. St001334The minimal number of occurrences of the 3-colorable pattern in a linear ordering of the vertices of the graph. St001335The cardinality of a minimal cycle-isolating set of a graph. St001336The minimal number of vertices in a graph whose complement is triangle-free. St001357The maximal degree of a regular spanning subgraph of a graph. St001367The smallest number which does not occur as degree of a vertex in a graph. St001638The book thickness of a graph. St001654The monophonic hull number of a graph. St001689The number of celebrities in a graph. St001702The absolute value of the determinant of the adjacency matrix of a graph. St001736The total number of cycles in a graph. St001793The difference between the clique number and the chromatic number of a graph. St001794Half the number of sets of vertices in a graph which are dominating and non-blocking. St001795The binary logarithm of the evaluation of the Tutte polynomial of the graph at (x,y) equal to (-1,-1). St001796The absolute value of the quotient of the Tutte polynomial of the graph at (1,1) and (-1,-1). St001797The number of overfull subgraphs of a graph.
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