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Your data matches 213 different statistics following compositions of up to 3 maps.
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Matching statistic: St000912
Values
([],1)
=> 1 = 0 + 1
([],2)
=> 1 = 0 + 1
([(0,1)],2)
=> 2 = 1 + 1
([],3)
=> 1 = 0 + 1
([(1,2)],3)
=> 2 = 1 + 1
([(0,1),(0,2)],3)
=> 2 = 1 + 1
([(0,2),(2,1)],3)
=> 3 = 2 + 1
([(0,2),(1,2)],3)
=> 2 = 1 + 1
([],4)
=> 1 = 0 + 1
([(2,3)],4)
=> 2 = 1 + 1
([(1,2),(1,3)],4)
=> 2 = 1 + 1
([(0,1),(0,2),(0,3)],4)
=> 2 = 1 + 1
([(0,2),(0,3),(3,1)],4)
=> 3 = 2 + 1
([(0,1),(0,2),(1,3),(2,3)],4)
=> 3 = 2 + 1
([(1,2),(2,3)],4)
=> 3 = 2 + 1
([(0,3),(3,1),(3,2)],4)
=> 3 = 2 + 1
([(1,3),(2,3)],4)
=> 2 = 1 + 1
([(0,3),(1,3),(3,2)],4)
=> 3 = 2 + 1
([(0,3),(1,3),(2,3)],4)
=> 2 = 1 + 1
([(0,3),(1,2)],4)
=> 4 = 3 + 1
([(0,3),(1,2),(1,3)],4)
=> 3 = 2 + 1
([(0,2),(0,3),(1,2),(1,3)],4)
=> 2 = 1 + 1
([(0,3),(2,1),(3,2)],4)
=> 4 = 3 + 1
([(0,3),(1,2),(2,3)],4)
=> 3 = 2 + 1
([],5)
=> 1 = 0 + 1
([(3,4)],5)
=> 2 = 1 + 1
([(2,3),(2,4)],5)
=> 2 = 1 + 1
([(1,2),(1,3),(1,4)],5)
=> 2 = 1 + 1
([(0,1),(0,2),(0,3),(0,4)],5)
=> 2 = 1 + 1
([(0,2),(0,3),(0,4),(4,1)],5)
=> 3 = 2 + 1
([(0,1),(0,2),(0,3),(2,4),(3,4)],5)
=> 3 = 2 + 1
([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
=> 3 = 2 + 1
([(1,3),(1,4),(4,2)],5)
=> 3 = 2 + 1
([(0,3),(0,4),(4,1),(4,2)],5)
=> 3 = 2 + 1
([(1,2),(1,3),(2,4),(3,4)],5)
=> 3 = 2 + 1
([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> 4 = 3 + 1
([(0,3),(0,4),(3,2),(4,1)],5)
=> 5 = 4 + 1
([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
=> 4 = 3 + 1
([(0,1),(0,2),(1,3),(1,4),(2,3),(2,4)],5)
=> 3 = 2 + 1
([(2,3),(3,4)],5)
=> 3 = 2 + 1
([(1,4),(4,2),(4,3)],5)
=> 3 = 2 + 1
([(0,4),(4,1),(4,2),(4,3)],5)
=> 3 = 2 + 1
([(2,4),(3,4)],5)
=> 2 = 1 + 1
([(1,4),(2,4),(4,3)],5)
=> 3 = 2 + 1
([(0,4),(1,4),(4,2),(4,3)],5)
=> 3 = 2 + 1
([(1,4),(2,4),(3,4)],5)
=> 2 = 1 + 1
([(0,4),(1,4),(2,4),(4,3)],5)
=> 3 = 2 + 1
([(0,4),(1,4),(2,4),(3,4)],5)
=> 2 = 1 + 1
([(0,4),(1,4),(2,3)],5)
=> 4 = 3 + 1
([(0,4),(1,3),(2,3),(2,4)],5)
=> 4 = 3 + 1
Description
The number of maximal antichains in a poset.
Matching statistic: St000550
Values
([],1)
=> ([],1)
=> 1 = 0 + 1
([],2)
=> ([],1)
=> 1 = 0 + 1
([(0,1)],2)
=> ([(0,1)],2)
=> 2 = 1 + 1
([],3)
=> ([],1)
=> 1 = 0 + 1
([(1,2)],3)
=> ([(0,1)],2)
=> 2 = 1 + 1
([(0,1),(0,2)],3)
=> ([(0,1)],2)
=> 2 = 1 + 1
([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 3 = 2 + 1
([(0,2),(1,2)],3)
=> ([(0,1)],2)
=> 2 = 1 + 1
([],4)
=> ([],1)
=> 1 = 0 + 1
([(2,3)],4)
=> ([(0,1)],2)
=> 2 = 1 + 1
([(1,2),(1,3)],4)
=> ([(0,1)],2)
=> 2 = 1 + 1
([(0,1),(0,2),(0,3)],4)
=> ([(0,1)],2)
=> 2 = 1 + 1
([(0,2),(0,3),(3,1)],4)
=> ([(0,2),(2,1)],3)
=> 3 = 2 + 1
([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(0,2),(2,1)],3)
=> 3 = 2 + 1
([(1,2),(2,3)],4)
=> ([(0,2),(2,1)],3)
=> 3 = 2 + 1
([(0,3),(3,1),(3,2)],4)
=> ([(0,2),(2,1)],3)
=> 3 = 2 + 1
([(1,3),(2,3)],4)
=> ([(0,1)],2)
=> 2 = 1 + 1
([(0,3),(1,3),(3,2)],4)
=> ([(0,2),(2,1)],3)
=> 3 = 2 + 1
([(0,3),(1,3),(2,3)],4)
=> ([(0,1)],2)
=> 2 = 1 + 1
([(0,3),(1,2)],4)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 4 = 3 + 1
([(0,3),(1,2),(1,3)],4)
=> ([(0,2),(2,1)],3)
=> 3 = 2 + 1
([(0,2),(0,3),(1,2),(1,3)],4)
=> ([(0,1)],2)
=> 2 = 1 + 1
([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(2,1),(3,2)],4)
=> 4 = 3 + 1
([(0,3),(1,2),(2,3)],4)
=> ([(0,2),(2,1)],3)
=> 3 = 2 + 1
([],5)
=> ([],1)
=> 1 = 0 + 1
([(3,4)],5)
=> ([(0,1)],2)
=> 2 = 1 + 1
([(2,3),(2,4)],5)
=> ([(0,1)],2)
=> 2 = 1 + 1
([(1,2),(1,3),(1,4)],5)
=> ([(0,1)],2)
=> 2 = 1 + 1
([(0,1),(0,2),(0,3),(0,4)],5)
=> ([(0,1)],2)
=> 2 = 1 + 1
([(0,2),(0,3),(0,4),(4,1)],5)
=> ([(0,2),(2,1)],3)
=> 3 = 2 + 1
([(0,1),(0,2),(0,3),(2,4),(3,4)],5)
=> ([(0,2),(2,1)],3)
=> 3 = 2 + 1
([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
=> ([(0,2),(2,1)],3)
=> 3 = 2 + 1
([(1,3),(1,4),(4,2)],5)
=> ([(0,2),(2,1)],3)
=> 3 = 2 + 1
([(0,3),(0,4),(4,1),(4,2)],5)
=> ([(0,2),(2,1)],3)
=> 3 = 2 + 1
([(1,2),(1,3),(2,4),(3,4)],5)
=> ([(0,2),(2,1)],3)
=> 3 = 2 + 1
([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> ([(0,3),(2,1),(3,2)],4)
=> 4 = 3 + 1
([(0,3),(0,4),(3,2),(4,1)],5)
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> 5 = 4 + 1
([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
=> ([(0,3),(2,1),(3,2)],4)
=> 4 = 3 + 1
([(0,1),(0,2),(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,2),(2,1)],3)
=> 3 = 2 + 1
([(2,3),(3,4)],5)
=> ([(0,2),(2,1)],3)
=> 3 = 2 + 1
([(1,4),(4,2),(4,3)],5)
=> ([(0,2),(2,1)],3)
=> 3 = 2 + 1
([(0,4),(4,1),(4,2),(4,3)],5)
=> ([(0,2),(2,1)],3)
=> 3 = 2 + 1
([(2,4),(3,4)],5)
=> ([(0,1)],2)
=> 2 = 1 + 1
([(1,4),(2,4),(4,3)],5)
=> ([(0,2),(2,1)],3)
=> 3 = 2 + 1
([(0,4),(1,4),(4,2),(4,3)],5)
=> ([(0,2),(2,1)],3)
=> 3 = 2 + 1
([(1,4),(2,4),(3,4)],5)
=> ([(0,1)],2)
=> 2 = 1 + 1
([(0,4),(1,4),(2,4),(4,3)],5)
=> ([(0,2),(2,1)],3)
=> 3 = 2 + 1
([(0,4),(1,4),(2,4),(3,4)],5)
=> ([(0,1)],2)
=> 2 = 1 + 1
([(0,4),(1,4),(2,3)],5)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 4 = 3 + 1
([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 4 = 3 + 1
Description
The number of modular elements of a lattice.
A pair $(x, y)$ of elements of a lattice $L$ is a modular pair if for every $z\geq y$ we have that $(y\vee x) \wedge z = y \vee (x \wedge z)$. An element $x$ is left-modular if $(x, y)$ is a modular pair for every $y\in L$, and is modular if both $(x, y)$ and $(y, x)$ are modular pairs for every $y\in L$.
Matching statistic: St000551
Values
([],1)
=> ([],1)
=> 1 = 0 + 1
([],2)
=> ([],1)
=> 1 = 0 + 1
([(0,1)],2)
=> ([(0,1)],2)
=> 2 = 1 + 1
([],3)
=> ([],1)
=> 1 = 0 + 1
([(1,2)],3)
=> ([(0,1)],2)
=> 2 = 1 + 1
([(0,1),(0,2)],3)
=> ([(0,1)],2)
=> 2 = 1 + 1
([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 3 = 2 + 1
([(0,2),(1,2)],3)
=> ([(0,1)],2)
=> 2 = 1 + 1
([],4)
=> ([],1)
=> 1 = 0 + 1
([(2,3)],4)
=> ([(0,1)],2)
=> 2 = 1 + 1
([(1,2),(1,3)],4)
=> ([(0,1)],2)
=> 2 = 1 + 1
([(0,1),(0,2),(0,3)],4)
=> ([(0,1)],2)
=> 2 = 1 + 1
([(0,2),(0,3),(3,1)],4)
=> ([(0,2),(2,1)],3)
=> 3 = 2 + 1
([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(0,2),(2,1)],3)
=> 3 = 2 + 1
([(1,2),(2,3)],4)
=> ([(0,2),(2,1)],3)
=> 3 = 2 + 1
([(0,3),(3,1),(3,2)],4)
=> ([(0,2),(2,1)],3)
=> 3 = 2 + 1
([(1,3),(2,3)],4)
=> ([(0,1)],2)
=> 2 = 1 + 1
([(0,3),(1,3),(3,2)],4)
=> ([(0,2),(2,1)],3)
=> 3 = 2 + 1
([(0,3),(1,3),(2,3)],4)
=> ([(0,1)],2)
=> 2 = 1 + 1
([(0,3),(1,2)],4)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 4 = 3 + 1
([(0,3),(1,2),(1,3)],4)
=> ([(0,2),(2,1)],3)
=> 3 = 2 + 1
([(0,2),(0,3),(1,2),(1,3)],4)
=> ([(0,1)],2)
=> 2 = 1 + 1
([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(2,1),(3,2)],4)
=> 4 = 3 + 1
([(0,3),(1,2),(2,3)],4)
=> ([(0,2),(2,1)],3)
=> 3 = 2 + 1
([],5)
=> ([],1)
=> 1 = 0 + 1
([(3,4)],5)
=> ([(0,1)],2)
=> 2 = 1 + 1
([(2,3),(2,4)],5)
=> ([(0,1)],2)
=> 2 = 1 + 1
([(1,2),(1,3),(1,4)],5)
=> ([(0,1)],2)
=> 2 = 1 + 1
([(0,1),(0,2),(0,3),(0,4)],5)
=> ([(0,1)],2)
=> 2 = 1 + 1
([(0,2),(0,3),(0,4),(4,1)],5)
=> ([(0,2),(2,1)],3)
=> 3 = 2 + 1
([(0,1),(0,2),(0,3),(2,4),(3,4)],5)
=> ([(0,2),(2,1)],3)
=> 3 = 2 + 1
([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
=> ([(0,2),(2,1)],3)
=> 3 = 2 + 1
([(1,3),(1,4),(4,2)],5)
=> ([(0,2),(2,1)],3)
=> 3 = 2 + 1
([(0,3),(0,4),(4,1),(4,2)],5)
=> ([(0,2),(2,1)],3)
=> 3 = 2 + 1
([(1,2),(1,3),(2,4),(3,4)],5)
=> ([(0,2),(2,1)],3)
=> 3 = 2 + 1
([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> ([(0,3),(2,1),(3,2)],4)
=> 4 = 3 + 1
([(0,3),(0,4),(3,2),(4,1)],5)
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> 5 = 4 + 1
([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
=> ([(0,3),(2,1),(3,2)],4)
=> 4 = 3 + 1
([(0,1),(0,2),(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,2),(2,1)],3)
=> 3 = 2 + 1
([(2,3),(3,4)],5)
=> ([(0,2),(2,1)],3)
=> 3 = 2 + 1
([(1,4),(4,2),(4,3)],5)
=> ([(0,2),(2,1)],3)
=> 3 = 2 + 1
([(0,4),(4,1),(4,2),(4,3)],5)
=> ([(0,2),(2,1)],3)
=> 3 = 2 + 1
([(2,4),(3,4)],5)
=> ([(0,1)],2)
=> 2 = 1 + 1
([(1,4),(2,4),(4,3)],5)
=> ([(0,2),(2,1)],3)
=> 3 = 2 + 1
([(0,4),(1,4),(4,2),(4,3)],5)
=> ([(0,2),(2,1)],3)
=> 3 = 2 + 1
([(1,4),(2,4),(3,4)],5)
=> ([(0,1)],2)
=> 2 = 1 + 1
([(0,4),(1,4),(2,4),(4,3)],5)
=> ([(0,2),(2,1)],3)
=> 3 = 2 + 1
([(0,4),(1,4),(2,4),(3,4)],5)
=> ([(0,1)],2)
=> 2 = 1 + 1
([(0,4),(1,4),(2,3)],5)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 4 = 3 + 1
([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 4 = 3 + 1
Description
The number of left modular elements of a lattice.
A pair $(x, y)$ of elements of a lattice $L$ is a modular pair if for every $z\geq y$ we have that $(y\vee x) \wedge z = y \vee (x \wedge z)$. An element $x$ is left-modular if $(x, y)$ is a modular pair for every $y\in L$.
Matching statistic: St001616
Values
([],1)
=> ([],1)
=> 1 = 0 + 1
([],2)
=> ([],1)
=> 1 = 0 + 1
([(0,1)],2)
=> ([(0,1)],2)
=> 2 = 1 + 1
([],3)
=> ([],1)
=> 1 = 0 + 1
([(1,2)],3)
=> ([(0,1)],2)
=> 2 = 1 + 1
([(0,1),(0,2)],3)
=> ([(0,1)],2)
=> 2 = 1 + 1
([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 3 = 2 + 1
([(0,2),(1,2)],3)
=> ([(0,1)],2)
=> 2 = 1 + 1
([],4)
=> ([],1)
=> 1 = 0 + 1
([(2,3)],4)
=> ([(0,1)],2)
=> 2 = 1 + 1
([(1,2),(1,3)],4)
=> ([(0,1)],2)
=> 2 = 1 + 1
([(0,1),(0,2),(0,3)],4)
=> ([(0,1)],2)
=> 2 = 1 + 1
([(0,2),(0,3),(3,1)],4)
=> ([(0,2),(2,1)],3)
=> 3 = 2 + 1
([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(0,2),(2,1)],3)
=> 3 = 2 + 1
([(1,2),(2,3)],4)
=> ([(0,2),(2,1)],3)
=> 3 = 2 + 1
([(0,3),(3,1),(3,2)],4)
=> ([(0,2),(2,1)],3)
=> 3 = 2 + 1
([(1,3),(2,3)],4)
=> ([(0,1)],2)
=> 2 = 1 + 1
([(0,3),(1,3),(3,2)],4)
=> ([(0,2),(2,1)],3)
=> 3 = 2 + 1
([(0,3),(1,3),(2,3)],4)
=> ([(0,1)],2)
=> 2 = 1 + 1
([(0,3),(1,2)],4)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 4 = 3 + 1
([(0,3),(1,2),(1,3)],4)
=> ([(0,2),(2,1)],3)
=> 3 = 2 + 1
([(0,2),(0,3),(1,2),(1,3)],4)
=> ([(0,1)],2)
=> 2 = 1 + 1
([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(2,1),(3,2)],4)
=> 4 = 3 + 1
([(0,3),(1,2),(2,3)],4)
=> ([(0,2),(2,1)],3)
=> 3 = 2 + 1
([],5)
=> ([],1)
=> 1 = 0 + 1
([(3,4)],5)
=> ([(0,1)],2)
=> 2 = 1 + 1
([(2,3),(2,4)],5)
=> ([(0,1)],2)
=> 2 = 1 + 1
([(1,2),(1,3),(1,4)],5)
=> ([(0,1)],2)
=> 2 = 1 + 1
([(0,1),(0,2),(0,3),(0,4)],5)
=> ([(0,1)],2)
=> 2 = 1 + 1
([(0,2),(0,3),(0,4),(4,1)],5)
=> ([(0,2),(2,1)],3)
=> 3 = 2 + 1
([(0,1),(0,2),(0,3),(2,4),(3,4)],5)
=> ([(0,2),(2,1)],3)
=> 3 = 2 + 1
([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
=> ([(0,2),(2,1)],3)
=> 3 = 2 + 1
([(1,3),(1,4),(4,2)],5)
=> ([(0,2),(2,1)],3)
=> 3 = 2 + 1
([(0,3),(0,4),(4,1),(4,2)],5)
=> ([(0,2),(2,1)],3)
=> 3 = 2 + 1
([(1,2),(1,3),(2,4),(3,4)],5)
=> ([(0,2),(2,1)],3)
=> 3 = 2 + 1
([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> ([(0,3),(2,1),(3,2)],4)
=> 4 = 3 + 1
([(0,3),(0,4),(3,2),(4,1)],5)
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> 5 = 4 + 1
([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
=> ([(0,3),(2,1),(3,2)],4)
=> 4 = 3 + 1
([(0,1),(0,2),(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,2),(2,1)],3)
=> 3 = 2 + 1
([(2,3),(3,4)],5)
=> ([(0,2),(2,1)],3)
=> 3 = 2 + 1
([(1,4),(4,2),(4,3)],5)
=> ([(0,2),(2,1)],3)
=> 3 = 2 + 1
([(0,4),(4,1),(4,2),(4,3)],5)
=> ([(0,2),(2,1)],3)
=> 3 = 2 + 1
([(2,4),(3,4)],5)
=> ([(0,1)],2)
=> 2 = 1 + 1
([(1,4),(2,4),(4,3)],5)
=> ([(0,2),(2,1)],3)
=> 3 = 2 + 1
([(0,4),(1,4),(4,2),(4,3)],5)
=> ([(0,2),(2,1)],3)
=> 3 = 2 + 1
([(1,4),(2,4),(3,4)],5)
=> ([(0,1)],2)
=> 2 = 1 + 1
([(0,4),(1,4),(2,4),(4,3)],5)
=> ([(0,2),(2,1)],3)
=> 3 = 2 + 1
([(0,4),(1,4),(2,4),(3,4)],5)
=> ([(0,1)],2)
=> 2 = 1 + 1
([(0,4),(1,4),(2,3)],5)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 4 = 3 + 1
([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 4 = 3 + 1
Description
The number of neutral elements in a lattice.
An element $e$ of the lattice $L$ is neutral if the sublattice generated by $e$, $x$ and $y$ is distributive for all $x, y \in L$.
Matching statistic: St001300
Values
([],1)
=> ([],1)
=> ([],1)
=> 0
([],2)
=> ([],1)
=> ([],1)
=> 0
([(0,1)],2)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 1
([],3)
=> ([],1)
=> ([],1)
=> 0
([(1,2)],3)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 1
([(0,1),(0,2)],3)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 1
([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 2
([(0,2),(1,2)],3)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 1
([],4)
=> ([],1)
=> ([],1)
=> 0
([(2,3)],4)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 1
([(1,2),(1,3)],4)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 1
([(0,1),(0,2),(0,3)],4)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 1
([(0,2),(0,3),(3,1)],4)
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 2
([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 2
([(1,2),(2,3)],4)
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 2
([(0,3),(3,1),(3,2)],4)
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 2
([(1,3),(2,3)],4)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 1
([(0,3),(1,3),(3,2)],4)
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 2
([(0,3),(1,3),(2,3)],4)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 1
([(0,3),(1,2)],4)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 3
([(0,3),(1,2),(1,3)],4)
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 2
([(0,2),(0,3),(1,2),(1,3)],4)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 1
([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(2,1),(3,2)],4)
=> 3
([(0,3),(1,2),(2,3)],4)
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 2
([],5)
=> ([],1)
=> ([],1)
=> 0
([(3,4)],5)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 1
([(2,3),(2,4)],5)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 1
([(1,2),(1,3),(1,4)],5)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 1
([(0,1),(0,2),(0,3),(0,4)],5)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 1
([(0,2),(0,3),(0,4),(4,1)],5)
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 2
([(0,1),(0,2),(0,3),(2,4),(3,4)],5)
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 2
([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 2
([(1,3),(1,4),(4,2)],5)
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 2
([(0,3),(0,4),(4,1),(4,2)],5)
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 2
([(1,2),(1,3),(2,4),(3,4)],5)
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 2
([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(2,1),(3,2)],4)
=> 3
([(0,3),(0,4),(3,2),(4,1)],5)
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> 4
([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(2,1),(3,2)],4)
=> 3
([(0,1),(0,2),(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 2
([(2,3),(3,4)],5)
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 2
([(1,4),(4,2),(4,3)],5)
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 2
([(0,4),(4,1),(4,2),(4,3)],5)
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 2
([(2,4),(3,4)],5)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 1
([(1,4),(2,4),(4,3)],5)
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 2
([(0,4),(1,4),(4,2),(4,3)],5)
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 2
([(1,4),(2,4),(3,4)],5)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 1
([(0,4),(1,4),(2,4),(4,3)],5)
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 2
([(0,4),(1,4),(2,4),(3,4)],5)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 1
([(0,4),(1,4),(2,3)],5)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 3
([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 3
Description
The rank of the boundary operator in degree 1 of the chain complex of the order complex of the poset.
Matching statistic: St000010
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00198: Posets —incomparability graph⟶ Graphs
Mp00251: Graphs —clique sizes⟶ Integer partitions
St000010: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00251: Graphs —clique sizes⟶ Integer partitions
St000010: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
([],1)
=> ([],1)
=> [1]
=> 1 = 0 + 1
([],2)
=> ([(0,1)],2)
=> [2]
=> 1 = 0 + 1
([(0,1)],2)
=> ([],2)
=> [1,1]
=> 2 = 1 + 1
([],3)
=> ([(0,1),(0,2),(1,2)],3)
=> [3]
=> 1 = 0 + 1
([(1,2)],3)
=> ([(0,2),(1,2)],3)
=> [2,2]
=> 2 = 1 + 1
([(0,1),(0,2)],3)
=> ([(1,2)],3)
=> [2,1]
=> 2 = 1 + 1
([(0,2),(2,1)],3)
=> ([],3)
=> [1,1,1]
=> 3 = 2 + 1
([(0,2),(1,2)],3)
=> ([(1,2)],3)
=> [2,1]
=> 2 = 1 + 1
([],4)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [4]
=> 1 = 0 + 1
([(2,3)],4)
=> ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [3,3]
=> 2 = 1 + 1
([(1,2),(1,3)],4)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> [3,2]
=> 2 = 1 + 1
([(0,1),(0,2),(0,3)],4)
=> ([(1,2),(1,3),(2,3)],4)
=> [3,1]
=> 2 = 1 + 1
([(0,2),(0,3),(3,1)],4)
=> ([(1,3),(2,3)],4)
=> [2,2,1]
=> 3 = 2 + 1
([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(2,3)],4)
=> [2,1,1]
=> 3 = 2 + 1
([(1,2),(2,3)],4)
=> ([(0,3),(1,3),(2,3)],4)
=> [2,2,2]
=> 3 = 2 + 1
([(0,3),(3,1),(3,2)],4)
=> ([(2,3)],4)
=> [2,1,1]
=> 3 = 2 + 1
([(1,3),(2,3)],4)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> [3,2]
=> 2 = 1 + 1
([(0,3),(1,3),(3,2)],4)
=> ([(2,3)],4)
=> [2,1,1]
=> 3 = 2 + 1
([(0,3),(1,3),(2,3)],4)
=> ([(1,2),(1,3),(2,3)],4)
=> [3,1]
=> 2 = 1 + 1
([(0,3),(1,2)],4)
=> ([(0,2),(0,3),(1,2),(1,3)],4)
=> [2,2,2,2]
=> 4 = 3 + 1
([(0,3),(1,2),(1,3)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> [2,2,2]
=> 3 = 2 + 1
([(0,2),(0,3),(1,2),(1,3)],4)
=> ([(0,3),(1,2)],4)
=> [2,2]
=> 2 = 1 + 1
([(0,3),(2,1),(3,2)],4)
=> ([],4)
=> [1,1,1,1]
=> 4 = 3 + 1
([(0,3),(1,2),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> [2,2,1]
=> 3 = 2 + 1
([],5)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> 1 = 0 + 1
([(3,4)],5)
=> ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [4,4]
=> 2 = 1 + 1
([(2,3),(2,4)],5)
=> ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [4,3]
=> 2 = 1 + 1
([(1,2),(1,3),(1,4)],5)
=> ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [4,2]
=> 2 = 1 + 1
([(0,1),(0,2),(0,3),(0,4)],5)
=> ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [4,1]
=> 2 = 1 + 1
([(0,2),(0,3),(0,4),(4,1)],5)
=> ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [3,3,1]
=> 3 = 2 + 1
([(0,1),(0,2),(0,3),(2,4),(3,4)],5)
=> ([(1,4),(2,3),(2,4),(3,4)],5)
=> [3,2,1]
=> 3 = 2 + 1
([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
=> ([(2,3),(2,4),(3,4)],5)
=> [3,1,1]
=> 3 = 2 + 1
([(1,3),(1,4),(4,2)],5)
=> ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [3,3,2]
=> 3 = 2 + 1
([(0,3),(0,4),(4,1),(4,2)],5)
=> ([(1,4),(2,3),(2,4),(3,4)],5)
=> [3,2,1]
=> 3 = 2 + 1
([(1,2),(1,3),(2,4),(3,4)],5)
=> ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> [3,2,2]
=> 3 = 2 + 1
([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> ([(3,4)],5)
=> [2,1,1,1]
=> 4 = 3 + 1
([(0,3),(0,4),(3,2),(4,1)],5)
=> ([(1,3),(1,4),(2,3),(2,4)],5)
=> [2,2,2,2,1]
=> 5 = 4 + 1
([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
=> ([(1,4),(2,3),(3,4)],5)
=> [2,2,2,1]
=> 4 = 3 + 1
([(0,1),(0,2),(1,3),(1,4),(2,3),(2,4)],5)
=> ([(1,4),(2,3)],5)
=> [2,2,1]
=> 3 = 2 + 1
([(2,3),(3,4)],5)
=> ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [3,3,3]
=> 3 = 2 + 1
([(1,4),(4,2),(4,3)],5)
=> ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> [3,2,2]
=> 3 = 2 + 1
([(0,4),(4,1),(4,2),(4,3)],5)
=> ([(2,3),(2,4),(3,4)],5)
=> [3,1,1]
=> 3 = 2 + 1
([(2,4),(3,4)],5)
=> ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [4,3]
=> 2 = 1 + 1
([(1,4),(2,4),(4,3)],5)
=> ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> [3,2,2]
=> 3 = 2 + 1
([(0,4),(1,4),(4,2),(4,3)],5)
=> ([(1,4),(2,3)],5)
=> [2,2,1]
=> 3 = 2 + 1
([(1,4),(2,4),(3,4)],5)
=> ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [4,2]
=> 2 = 1 + 1
([(0,4),(1,4),(2,4),(4,3)],5)
=> ([(2,3),(2,4),(3,4)],5)
=> [3,1,1]
=> 3 = 2 + 1
([(0,4),(1,4),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [4,1]
=> 2 = 1 + 1
([(0,4),(1,4),(2,3)],5)
=> ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
=> [3,3,2,2]
=> 4 = 3 + 1
([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> [3,2,2,2]
=> 4 = 3 + 1
Description
The length of the partition.
Matching statistic: St000189
Values
([],1)
=> ([],1)
=> ([],1)
=> 1 = 0 + 1
([],2)
=> ([],1)
=> ([],1)
=> 1 = 0 + 1
([(0,1)],2)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 2 = 1 + 1
([],3)
=> ([],1)
=> ([],1)
=> 1 = 0 + 1
([(1,2)],3)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 2 = 1 + 1
([(0,1),(0,2)],3)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 2 = 1 + 1
([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 3 = 2 + 1
([(0,2),(1,2)],3)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 2 = 1 + 1
([],4)
=> ([],1)
=> ([],1)
=> 1 = 0 + 1
([(2,3)],4)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 2 = 1 + 1
([(1,2),(1,3)],4)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 2 = 1 + 1
([(0,1),(0,2),(0,3)],4)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 2 = 1 + 1
([(0,2),(0,3),(3,1)],4)
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 3 = 2 + 1
([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 3 = 2 + 1
([(1,2),(2,3)],4)
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 3 = 2 + 1
([(0,3),(3,1),(3,2)],4)
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 3 = 2 + 1
([(1,3),(2,3)],4)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 2 = 1 + 1
([(0,3),(1,3),(3,2)],4)
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 3 = 2 + 1
([(0,3),(1,3),(2,3)],4)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 2 = 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)
=> 4 = 3 + 1
([(0,3),(1,2),(1,3)],4)
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 3 = 2 + 1
([(0,2),(0,3),(1,2),(1,3)],4)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 2 = 1 + 1
([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(2,1),(3,2)],4)
=> 4 = 3 + 1
([(0,3),(1,2),(2,3)],4)
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 3 = 2 + 1
([],5)
=> ([],1)
=> ([],1)
=> 1 = 0 + 1
([(3,4)],5)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 2 = 1 + 1
([(2,3),(2,4)],5)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 2 = 1 + 1
([(1,2),(1,3),(1,4)],5)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 2 = 1 + 1
([(0,1),(0,2),(0,3),(0,4)],5)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 2 = 1 + 1
([(0,2),(0,3),(0,4),(4,1)],5)
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 3 = 2 + 1
([(0,1),(0,2),(0,3),(2,4),(3,4)],5)
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 3 = 2 + 1
([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 3 = 2 + 1
([(1,3),(1,4),(4,2)],5)
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 3 = 2 + 1
([(0,3),(0,4),(4,1),(4,2)],5)
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 3 = 2 + 1
([(1,2),(1,3),(2,4),(3,4)],5)
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 3 = 2 + 1
([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(2,1),(3,2)],4)
=> 4 = 3 + 1
([(0,3),(0,4),(3,2),(4,1)],5)
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> 5 = 4 + 1
([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(2,1),(3,2)],4)
=> 4 = 3 + 1
([(0,1),(0,2),(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 3 = 2 + 1
([(2,3),(3,4)],5)
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 3 = 2 + 1
([(1,4),(4,2),(4,3)],5)
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 3 = 2 + 1
([(0,4),(4,1),(4,2),(4,3)],5)
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 3 = 2 + 1
([(2,4),(3,4)],5)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 2 = 1 + 1
([(1,4),(2,4),(4,3)],5)
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 3 = 2 + 1
([(0,4),(1,4),(4,2),(4,3)],5)
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 3 = 2 + 1
([(1,4),(2,4),(3,4)],5)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 2 = 1 + 1
([(0,4),(1,4),(2,4),(4,3)],5)
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 3 = 2 + 1
([(0,4),(1,4),(2,4),(3,4)],5)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 2 = 1 + 1
([(0,4),(1,4),(2,3)],5)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 4 = 3 + 1
([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 4 = 3 + 1
Description
The number of elements in the poset.
Matching statistic: St001304
(load all 6 compositions to match this statistic)
(load all 6 compositions to match this statistic)
Values
([],1)
=> ([],1)
=> ([],1)
=> 1 = 0 + 1
([],2)
=> ([(0,1)],2)
=> ([],2)
=> 1 = 0 + 1
([(0,1)],2)
=> ([],2)
=> ([(0,1)],2)
=> 2 = 1 + 1
([],3)
=> ([(0,1),(0,2),(1,2)],3)
=> ([],3)
=> 1 = 0 + 1
([(1,2)],3)
=> ([(0,2),(1,2)],3)
=> ([(1,2)],3)
=> 2 = 1 + 1
([(0,1),(0,2)],3)
=> ([(1,2)],3)
=> ([(0,2),(1,2)],3)
=> 2 = 1 + 1
([(0,2),(2,1)],3)
=> ([],3)
=> ([(0,1),(0,2),(1,2)],3)
=> 3 = 2 + 1
([(0,2),(1,2)],3)
=> ([(1,2)],3)
=> ([(0,2),(1,2)],3)
=> 2 = 1 + 1
([],4)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([],4)
=> 1 = 0 + 1
([(2,3)],4)
=> ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(2,3)],4)
=> 2 = 1 + 1
([(1,2),(1,3)],4)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> 2 = 1 + 1
([(0,1),(0,2),(0,3)],4)
=> ([(1,2),(1,3),(2,3)],4)
=> ([(0,3),(1,3),(2,3)],4)
=> 2 = 1 + 1
([(0,2),(0,3),(3,1)],4)
=> ([(1,3),(2,3)],4)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> 3 = 2 + 1
([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(2,3)],4)
=> ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3 = 2 + 1
([(1,2),(2,3)],4)
=> ([(0,3),(1,3),(2,3)],4)
=> ([(1,2),(1,3),(2,3)],4)
=> 3 = 2 + 1
([(0,3),(3,1),(3,2)],4)
=> ([(2,3)],4)
=> ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3 = 2 + 1
([(1,3),(2,3)],4)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> 2 = 1 + 1
([(0,3),(1,3),(3,2)],4)
=> ([(2,3)],4)
=> ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3 = 2 + 1
([(0,3),(1,3),(2,3)],4)
=> ([(1,2),(1,3),(2,3)],4)
=> ([(0,3),(1,3),(2,3)],4)
=> 2 = 1 + 1
([(0,3),(1,2)],4)
=> ([(0,2),(0,3),(1,2),(1,3)],4)
=> ([(0,3),(1,2)],4)
=> 4 = 3 + 1
([(0,3),(1,2),(1,3)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> 3 = 2 + 1
([(0,2),(0,3),(1,2),(1,3)],4)
=> ([(0,3),(1,2)],4)
=> ([(0,2),(0,3),(1,2),(1,3)],4)
=> 2 = 1 + 1
([(0,3),(2,1),(3,2)],4)
=> ([],4)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 4 = 3 + 1
([(0,3),(1,2),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> 3 = 2 + 1
([],5)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([],5)
=> 1 = 0 + 1
([(3,4)],5)
=> ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(3,4)],5)
=> 2 = 1 + 1
([(2,3),(2,4)],5)
=> ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(2,4),(3,4)],5)
=> 2 = 1 + 1
([(1,2),(1,3),(1,4)],5)
=> ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,4),(2,4),(3,4)],5)
=> 2 = 1 + 1
([(0,1),(0,2),(0,3),(0,4)],5)
=> ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(1,4),(2,4),(3,4)],5)
=> 2 = 1 + 1
([(0,2),(0,3),(0,4),(4,1)],5)
=> ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = 2 + 1
([(0,1),(0,2),(0,3),(2,4),(3,4)],5)
=> ([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = 2 + 1
([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
=> ([(2,3),(2,4),(3,4)],5)
=> ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = 2 + 1
([(1,3),(1,4),(4,2)],5)
=> ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = 2 + 1
([(0,3),(0,4),(4,1),(4,2)],5)
=> ([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = 2 + 1
([(1,2),(1,3),(2,4),(3,4)],5)
=> ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = 2 + 1
([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> ([(3,4)],5)
=> ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 4 = 3 + 1
([(0,3),(0,4),(3,2),(4,1)],5)
=> ([(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
=> 5 = 4 + 1
([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
=> ([(1,4),(2,3),(3,4)],5)
=> ([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
=> 4 = 3 + 1
([(0,1),(0,2),(1,3),(1,4),(2,3),(2,4)],5)
=> ([(1,4),(2,3)],5)
=> ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> 3 = 2 + 1
([(2,3),(3,4)],5)
=> ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(2,3),(2,4),(3,4)],5)
=> 3 = 2 + 1
([(1,4),(4,2),(4,3)],5)
=> ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = 2 + 1
([(0,4),(4,1),(4,2),(4,3)],5)
=> ([(2,3),(2,4),(3,4)],5)
=> ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = 2 + 1
([(2,4),(3,4)],5)
=> ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(2,4),(3,4)],5)
=> 2 = 1 + 1
([(1,4),(2,4),(4,3)],5)
=> ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = 2 + 1
([(0,4),(1,4),(4,2),(4,3)],5)
=> ([(1,4),(2,3)],5)
=> ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> 3 = 2 + 1
([(1,4),(2,4),(3,4)],5)
=> ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,4),(2,4),(3,4)],5)
=> 2 = 1 + 1
([(0,4),(1,4),(2,4),(4,3)],5)
=> ([(2,3),(2,4),(3,4)],5)
=> ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = 2 + 1
([(0,4),(1,4),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(1,4),(2,4),(3,4)],5)
=> 2 = 1 + 1
([(0,4),(1,4),(2,3)],5)
=> ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,1),(2,4),(3,4)],5)
=> 4 = 3 + 1
([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 4 = 3 + 1
Description
The number of maximally independent sets of vertices of a graph.
An '''independent set''' of vertices of a graph is a set of vertices no two of which are adjacent. If a set of vertices is independent then so is every subset. This statistic counts the number of maximally independent sets of vertices.
Matching statistic: St001717
Values
([],1)
=> ([],1)
=> ([],1)
=> 1 = 0 + 1
([],2)
=> ([],1)
=> ([],1)
=> 1 = 0 + 1
([(0,1)],2)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 2 = 1 + 1
([],3)
=> ([],1)
=> ([],1)
=> 1 = 0 + 1
([(1,2)],3)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 2 = 1 + 1
([(0,1),(0,2)],3)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 2 = 1 + 1
([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 3 = 2 + 1
([(0,2),(1,2)],3)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 2 = 1 + 1
([],4)
=> ([],1)
=> ([],1)
=> 1 = 0 + 1
([(2,3)],4)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 2 = 1 + 1
([(1,2),(1,3)],4)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 2 = 1 + 1
([(0,1),(0,2),(0,3)],4)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 2 = 1 + 1
([(0,2),(0,3),(3,1)],4)
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 3 = 2 + 1
([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 3 = 2 + 1
([(1,2),(2,3)],4)
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 3 = 2 + 1
([(0,3),(3,1),(3,2)],4)
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 3 = 2 + 1
([(1,3),(2,3)],4)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 2 = 1 + 1
([(0,3),(1,3),(3,2)],4)
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 3 = 2 + 1
([(0,3),(1,3),(2,3)],4)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 2 = 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)
=> 4 = 3 + 1
([(0,3),(1,2),(1,3)],4)
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 3 = 2 + 1
([(0,2),(0,3),(1,2),(1,3)],4)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 2 = 1 + 1
([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(2,1),(3,2)],4)
=> 4 = 3 + 1
([(0,3),(1,2),(2,3)],4)
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 3 = 2 + 1
([],5)
=> ([],1)
=> ([],1)
=> 1 = 0 + 1
([(3,4)],5)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 2 = 1 + 1
([(2,3),(2,4)],5)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 2 = 1 + 1
([(1,2),(1,3),(1,4)],5)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 2 = 1 + 1
([(0,1),(0,2),(0,3),(0,4)],5)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 2 = 1 + 1
([(0,2),(0,3),(0,4),(4,1)],5)
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 3 = 2 + 1
([(0,1),(0,2),(0,3),(2,4),(3,4)],5)
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 3 = 2 + 1
([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 3 = 2 + 1
([(1,3),(1,4),(4,2)],5)
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 3 = 2 + 1
([(0,3),(0,4),(4,1),(4,2)],5)
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 3 = 2 + 1
([(1,2),(1,3),(2,4),(3,4)],5)
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 3 = 2 + 1
([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(2,1),(3,2)],4)
=> 4 = 3 + 1
([(0,3),(0,4),(3,2),(4,1)],5)
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> 5 = 4 + 1
([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(2,1),(3,2)],4)
=> 4 = 3 + 1
([(0,1),(0,2),(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 3 = 2 + 1
([(2,3),(3,4)],5)
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 3 = 2 + 1
([(1,4),(4,2),(4,3)],5)
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 3 = 2 + 1
([(0,4),(4,1),(4,2),(4,3)],5)
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 3 = 2 + 1
([(2,4),(3,4)],5)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 2 = 1 + 1
([(1,4),(2,4),(4,3)],5)
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 3 = 2 + 1
([(0,4),(1,4),(4,2),(4,3)],5)
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 3 = 2 + 1
([(1,4),(2,4),(3,4)],5)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 2 = 1 + 1
([(0,4),(1,4),(2,4),(4,3)],5)
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 3 = 2 + 1
([(0,4),(1,4),(2,4),(3,4)],5)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 2 = 1 + 1
([(0,4),(1,4),(2,3)],5)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 4 = 3 + 1
([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 4 = 3 + 1
Description
The largest size of an interval in a poset.
Matching statistic: St000987
Values
([],1)
=> ([],1)
=> ([],1)
=> ([],1)
=> 0
([],2)
=> ([],1)
=> ([],1)
=> ([],1)
=> 0
([(0,1)],2)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 1
([],3)
=> ([],1)
=> ([],1)
=> ([],1)
=> 0
([(1,2)],3)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 1
([(0,1),(0,2)],3)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 1
([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> ([(0,2),(1,2)],3)
=> 2
([(0,2),(1,2)],3)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 1
([],4)
=> ([],1)
=> ([],1)
=> ([],1)
=> 0
([(2,3)],4)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 1
([(1,2),(1,3)],4)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 1
([(0,1),(0,2),(0,3)],4)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 1
([(0,2),(0,3),(3,1)],4)
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> ([(0,2),(1,2)],3)
=> 2
([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> ([(0,2),(1,2)],3)
=> 2
([(1,2),(2,3)],4)
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> ([(0,2),(1,2)],3)
=> 2
([(0,3),(3,1),(3,2)],4)
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> ([(0,2),(1,2)],3)
=> 2
([(1,3),(2,3)],4)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 1
([(0,3),(1,3),(3,2)],4)
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> ([(0,2),(1,2)],3)
=> 2
([(0,3),(1,3),(2,3)],4)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 1
([(0,3),(1,2)],4)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(0,2),(0,3),(1,2),(1,3)],4)
=> 3
([(0,3),(1,2),(1,3)],4)
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> ([(0,2),(1,2)],3)
=> 2
([(0,2),(0,3),(1,2),(1,3)],4)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 1
([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> 3
([(0,3),(1,2),(2,3)],4)
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> ([(0,2),(1,2)],3)
=> 2
([],5)
=> ([],1)
=> ([],1)
=> ([],1)
=> 0
([(3,4)],5)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 1
([(2,3),(2,4)],5)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 1
([(1,2),(1,3),(1,4)],5)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 1
([(0,1),(0,2),(0,3),(0,4)],5)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 1
([(0,2),(0,3),(0,4),(4,1)],5)
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> ([(0,2),(1,2)],3)
=> 2
([(0,1),(0,2),(0,3),(2,4),(3,4)],5)
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> ([(0,2),(1,2)],3)
=> 2
([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> ([(0,2),(1,2)],3)
=> 2
([(1,3),(1,4),(4,2)],5)
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> ([(0,2),(1,2)],3)
=> 2
([(0,3),(0,4),(4,1),(4,2)],5)
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> ([(0,2),(1,2)],3)
=> 2
([(1,2),(1,3),(2,4),(3,4)],5)
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> ([(0,2),(1,2)],3)
=> 2
([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> 3
([(0,3),(0,4),(3,2),(4,1)],5)
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> 4
([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> 3
([(0,1),(0,2),(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> ([(0,2),(1,2)],3)
=> 2
([(2,3),(3,4)],5)
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> ([(0,2),(1,2)],3)
=> 2
([(1,4),(4,2),(4,3)],5)
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> ([(0,2),(1,2)],3)
=> 2
([(0,4),(4,1),(4,2),(4,3)],5)
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> ([(0,2),(1,2)],3)
=> 2
([(2,4),(3,4)],5)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 1
([(1,4),(2,4),(4,3)],5)
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> ([(0,2),(1,2)],3)
=> 2
([(0,4),(1,4),(4,2),(4,3)],5)
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> ([(0,2),(1,2)],3)
=> 2
([(1,4),(2,4),(3,4)],5)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 1
([(0,4),(1,4),(2,4),(4,3)],5)
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> ([(0,2),(1,2)],3)
=> 2
([(0,4),(1,4),(2,4),(3,4)],5)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 1
([(0,4),(1,4),(2,3)],5)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(0,2),(0,3),(1,2),(1,3)],4)
=> 3
([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(0,2),(0,3),(1,2),(1,3)],4)
=> 3
Description
The number of positive eigenvalues of the Laplacian matrix of the graph.
This is the number of vertices minus the number of connected components of the graph.
The following 203 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St001120The length of a longest path in a graph. St000147The largest part of an integer partition. St000228The size of a partition. St000479The Ramsey number of a graph. St001318The number of vertices of the largest induced subforest with the same number of connected components of a graph. St001321The number of vertices of the largest induced subforest of a graph. St001342The number of vertices in the center of a graph. St001622The number of join-irreducible elements of a lattice. 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. St001615The number of join prime elements of a lattice. St001617The dimension of the space of valuations of a lattice. St001118The acyclic chromatic index of a graph. St000741The Colin de Verdière graph invariant. St001232The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2. St000053The number of valleys of the Dyck path. St001506Half the projective dimension of the unique simple module with even projective dimension in a magnitude 1 Nakayama algebra. St001873For a Nakayama algebra corresponding to a Dyck path, we define the matrix C with entries the Hom-spaces between $e_i J$ and $e_j J$ (the radical of the indecomposable projective modules). St000676The number of odd rises of a Dyck path. St001068Number of torsionless simple modules in the corresponding Nakayama algebra. St001471The magnitude of a Dyck path. St001505The number of elements generated by the Dyck path as a map in the full transformation monoid. St000998Number of indecomposable projective modules with injective dimension smaller than or equal to the dominant dimension in the Nakayama algebra corresponding to the Dyck path. St001060The distinguishing index of a graph. St000259The diameter of a connected graph. St000329The number of evenly positioned ascents of the Dyck path, with the initial position equal to 1. St001508The degree of the standard monomial associated to a Dyck path relative to the diagonal boundary. St000331The number of upper interactions of a Dyck path. St001142The projective dimension of the socle of the regular module as a bimodule in the Nakayama algebra corresponding to the Dyck path. St001169Number of simple modules with projective dimension at least two in the corresponding Nakayama algebra. St001194The injective dimension of $A/AfA$ in the corresponding Nakayama algebra $A$ when $Af$ is the minimal faithful projective-injective left $A$-module St001212The number of simple modules in the corresponding Nakayama algebra that have non-zero second Ext-group with the regular module. St001215Let X be the direct sum of all simple modules of the corresponding Nakayama algebra. St001227The vector space dimension of the first extension group between the socle of the regular module and the Jacobson radical of 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. St001509The degree of the standard monomial associated to a Dyck path relative to the trivial lower boundary. St000015The number of peaks of a Dyck path. St001165Number of simple modules with even projective dimension in the corresponding Nakayama algebra. St000144The pyramid weight of the Dyck path. St001183The maximum of $projdim(S)+injdim(S)$ over all simple modules in the Nakayama algebra corresponding to the Dyck path. St001258Gives the maximum of injective plus projective dimension of an indecomposable module over the corresponding Nakayama algebra. St001065Number of indecomposable reflexive modules in the corresponding Nakayama algebra. St001179Number of indecomposable injective modules with projective dimension at most 2 in the corresponding Nakayama algebra. St001237The number of simple modules with injective dimension at most one or dominant dimension at least one. St001240The number of indecomposable modules e_i J^2 that have injective dimension at most one in the corresponding Nakayama algebra St000937The number of positive values of the symmetric group character corresponding to the partition. St001553The number of indecomposable summands of the square of the Jacobson radical as a bimodule in the Nakayama algebra corresponding to the Dyck path. St001515The vector space dimension of the socle of the first syzygy module of the regular module (as a bimodule). St001526The Loewy length of the Auslander-Reiten translate of the regular module as a bimodule of the Nakayama algebra corresponding to the Dyck path. St001514The dimension of the top of the Auslander-Reiten translate of the regular modules as a bimodule. St001613The binary logarithm of the size of the center of a lattice. St001291The number of indecomposable summands of the tensor product of two copies of the dual of the Nakayama algebra associated to a Dyck path. St001229The vector space dimension of the first extension group between the Jacobson radical J and J^2. St001480The number of simple summands of the module J^2/J^3. St001820The size of the image of the pop stack sorting operator. St001720The minimal length of a chain of small intervals in a lattice. St000454The largest eigenvalue of a graph if it is integral. St000755The number of real roots of the characteristic polynomial of a linear recurrence associated with an integer partition. St000260The radius of a connected graph. St001431Half of the Loewy length minus one of a modified stable Auslander algebra of the Nakayama algebra corresponding to the Dyck path. St001568The smallest positive integer that does not appear twice in the partition. St001498The normalised height of a Nakayama algebra with magnitude 1. St000777The number of distinct eigenvalues of the distance Laplacian of a connected graph. St000271The chromatic index of a graph. St000460The hook length of the last cell along the main diagonal of an integer partition. St000870The product of the hook lengths of the diagonal cells in an integer partition. St001360The number of covering relations in Young's lattice below a partition. St001380The number of monomer-dimer tilings of a Ferrers diagram. 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 $eA$. St000455The second largest eigenvalue of a graph if it is integral. St001630The global dimension of the incidence algebra of the lattice over the rational numbers. St000717The number of ordinal summands of a poset. St000906The length of the shortest maximal chain in a poset. St001637The number of (upper) dissectors of a poset. St000939The number of characters of the symmetric group whose value on the partition is positive. St000993The multiplicity of the largest part of an integer partition. St001605The number of colourings of a cycle such that the multiplicities of colours are given by a partition. St001876The number of 2-regular simple modules in the incidence algebra of the lattice. St001632The number of indecomposable injective modules $I$ with $dim Ext^1(I,A)=1$ for the incidence algebra A of a poset. St001624The breadth of a lattice. St001626The number of maximal proper sublattices of a lattice. St000456The monochromatic index of a connected graph. St001633The number of simple modules with projective dimension two in the incidence algebra of the poset. St000171The degree of the graph. St000272The treewidth of a graph. St000362The size of a minimal vertex cover of a graph. St000387The matching number of a graph. St000535The rank-width of a graph. St000536The pathwidth of a graph. St000537The cutwidth of a graph. St000552The number of cut vertices of a graph. St000985The number of positive eigenvalues of the adjacency matrix of the graph. St001270The bandwidth of a graph. St001277The degeneracy of a graph. St001305The number of induced cycles on four vertices 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. 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. St001331The size of the minimal feedback vertex set. St001333The cardinality of a minimal edge-isolating set of a graph. St001349The number of different graphs obtained from the given graph by removing an edge. St001358The largest degree of a regular subgraph of a graph. St001393The induced matching number of a graph. St001592The maximal number of simple paths between any two different vertices of a graph. St001603The number of colourings of a polygon such that the multiplicities of a colour are given by a partition. St001638The book thickness of a graph. St001644The dimension of a graph. St001689The number of celebrities in a graph. St001743The discrepancy of a graph. St001792The arboricity of a graph. St001826The maximal number of leaves on a vertex of a graph. St001962The proper pathwidth of a graph. St000172The Grundy number of a graph. St000363The number of minimal vertex covers of a graph. St000388The number of orbits of vertices of a graph under automorphisms. St000482The (zero)-forcing number of a graph. St000544The cop number of a graph. St000553The number of blocks of a graph. St000778The metric dimension of a graph. St000785The number of distinct colouring schemes of a graph. St001029The size of the core of a graph. St001108The 2-dynamic chromatic number of a graph. St001110The 3-dynamic chromatic number of a graph. St001116The game chromatic number of a graph. St001261The Castelnuovo-Mumford regularity of a graph. St001271The competition number of a graph. St001352The number of internal nodes in the modular decomposition of a graph. St001367The smallest number which does not occur as degree of a vertex in a graph. St001494The Alon-Tarsi number of a graph. St001580The acyclic chromatic number of a graph. St001670The connected partition number of a graph. St001674The number of vertices of the largest induced star graph in the graph. St001725The harmonious chromatic number of a graph. St001829The common independence number of a graph. St001883The mutual visibility number of a graph. St001951The number of factors in the disjoint direct product decomposition of the automorphism group of a graph. St001963The tree-depth of a graph. St000469The distinguishing number of a graph. St000723The maximal cardinality of a set of vertices with the same neighbourhood in a graph. St000776The maximal multiplicity of an eigenvalue in a graph. St000986The multiplicity of the eigenvalue zero of the adjacency matrix of the graph. St001366The maximal multiplicity of a degree of a vertex of a graph. St001691The number of kings in a graph. St001704The size of the largest multi-subset-intersection of the deck of a graph with the deck of another graph. St001746The coalition number of a graph. St001844The maximal degree of a generator of the invariant ring of the automorphism group of a graph. St001877Number of indecomposable injective modules with projective dimension 2. St001879The number of indecomposable summands of the top of the first syzygy of the dual of the regular module in the incidence algebra of the lattice. St001880The number of 2-Gorenstein indecomposable injective modules in the incidence algebra of the lattice. St001200The number of simple modules in $eAe$ with projective dimension at most 2 in the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St001878The projective dimension of the simple modules corresponding to the minimum of L in the incidence algebra of the lattice L. St000632The jump number of the poset. St000848The balance constant multiplied with the number of linear extensions of a poset. St000849The number of 1/3-balanced pairs in a poset. St000298The order dimension or Dushnik-Miller dimension of a poset. St000640The rank of the largest boolean interval in a poset. St000845The maximal number of elements covered by an element in a poset. St000846The maximal number of elements covering an element of a poset. St000910The number of maximal chains of minimal length in a poset. St001105The number of greedy linear extensions of a poset. St001106The number of supergreedy linear extensions of a poset. St000907The number of maximal antichains of minimal length in a poset. St001195The global dimension of the algebra $A/AfA$ of the corresponding Nakayama algebra $A$ with minimal left faithful projective-injective module $Af$. St000284The Plancherel distribution on integer partitions. St000668The least common multiple of the parts of the partition. St000681The Grundy value of Chomp on Ferrers diagrams. St000698The number of 2-rim hooks removed from an integer partition to obtain its associated 2-core. St000704The number of semistandard tableaux on a given integer partition with minimal maximal entry. St000707The product of the factorials of the parts. St000708The product of the parts of an integer partition. St000770The major index of an integer partition when read from bottom to top. St000815The number of semistandard Young tableaux of partition weight of given shape. St000933The number of multipartitions of sizes given by an integer partition. St001128The exponens consonantiae of a partition. St001000Number of indecomposable modules with projective dimension equal to the global dimension in the Nakayama algebra corresponding to the Dyck path. St000307The number of rowmotion orbits of a poset. St001117The game chromatic index of a graph. St001574The minimal number of edges to add or remove to make a graph regular. St001576The minimal number of edges to add or remove to make a graph vertex transitive. St001742The difference of the maximal and the minimal degree in a graph. St001812The biclique partition number of a graph. St000450The number of edges minus the number of vertices plus 2 of a graph. St000822The Hadwiger number of the graph. St001330The hat guessing number of a graph. St001734The lettericity of a graph. St000315The number of isolated vertices of a graph. St001642The Prague dimension of a graph. St001570The minimal number of edges to add to make a graph Hamiltonian. St000264The girth of a graph, which is not a tree. St000510The number of invariant oriented cycles when acting with a permutation of given cycle type. St000515The number of invariant set partitions when acting with a permutation of given cycle type. St000567The sum of the products of all pairs of parts. St000620The number of standard tableaux of shape equal to the given partition such that the minimal cyclic descent is odd. St000714The number of semistandard Young tableau of given shape, with entries at most 2. St000901The cube of the number of standard Young tableaux with shape given by the partition. St000929The constant term of the character polynomial of an integer partition. St001099The coefficient times the product of the factorials of the parts of the monomial symmetric function indexed by the partition in the formal group law for leaf labelled binary trees. St001100The coefficient times the product of the factorials of the parts of the monomial symmetric function indexed by the partition in the formal group law for leaf labelled trees. St001101The coefficient times the product of the factorials of the parts of the monomial symmetric function indexed by the partition in the formal group law for increasing trees. St001123The multiplicity of the dual of the standard representation in the Kronecker square corresponding to a partition. St001491The number of indecomposable projective-injective modules in the algebra corresponding to a subset.
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