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Your data matches 27 different statistics following compositions of up to 3 maps.
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Matching statistic: St000550
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Values
([(0,1)],2)
=> ([(0,1)],2)
=> 2
([(1,2)],3)
=> ([(0,1)],2)
=> 2
([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 3
([(2,3)],4)
=> ([(0,1)],2)
=> 2
([(0,2),(0,3),(3,1)],4)
=> ([(0,1)],2)
=> 2
([(1,2),(2,3)],4)
=> ([(0,2),(2,1)],3)
=> 3
([(0,3),(1,2)],4)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 4
([(0,3),(1,2),(1,3)],4)
=> ([(0,2),(2,1)],3)
=> 3
([(0,2),(0,3),(1,2),(1,3)],4)
=> ([(0,1)],2)
=> 2
([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(2,1),(3,2)],4)
=> 4
([(0,3),(1,2),(2,3)],4)
=> ([(0,1)],2)
=> 2
([(3,4)],5)
=> ([(0,1)],2)
=> 2
([(0,2),(0,3),(0,4),(4,1)],5)
=> ([(0,1)],2)
=> 2
([(1,3),(1,4),(4,2)],5)
=> ([(0,1)],2)
=> 2
([(0,3),(0,4),(3,2),(4,1)],5)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 4
([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
=> ([(0,2),(2,1)],3)
=> 3
([(0,1),(0,2),(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,1)],2)
=> 2
([(2,3),(3,4)],5)
=> ([(0,2),(2,1)],3)
=> 3
([(0,4),(1,4),(4,2),(4,3)],5)
=> ([(0,1)],2)
=> 2
([(0,4),(1,4),(2,3)],5)
=> ([(0,1)],2)
=> 2
([(0,4),(1,4),(2,3),(2,4)],5)
=> ([(0,1)],2)
=> 2
([(0,4),(1,4),(2,3),(3,4)],5)
=> ([(0,1)],2)
=> 2
([(1,4),(2,3)],5)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 4
([(1,4),(2,3),(2,4)],5)
=> ([(0,2),(2,1)],3)
=> 3
([(0,4),(1,2),(1,4),(2,3)],5)
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> 5
([(0,3),(1,2),(1,3),(2,4),(3,4)],5)
=> ([(0,2),(2,1)],3)
=> 3
([(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,1)],2)
=> 2
([(0,3),(0,4),(1,3),(1,4),(4,2)],5)
=> ([(0,2),(2,1)],3)
=> 3
([(0,3),(0,4),(1,3),(1,4),(3,2),(4,2)],5)
=> ([(0,1)],2)
=> 2
([(0,4),(1,2),(1,4),(4,3)],5)
=> ([(0,3),(2,1),(3,2)],4)
=> 4
([(0,4),(1,2),(1,3)],5)
=> ([(0,1)],2)
=> 2
([(0,4),(1,2),(1,3),(1,4)],5)
=> ([(0,1)],2)
=> 2
([(0,2),(0,4),(3,1),(4,3)],5)
=> ([(0,2),(2,1)],3)
=> 3
([(0,2),(0,3),(1,4),(2,4),(3,1)],5)
=> ([(0,1)],2)
=> 2
([(0,3),(0,4),(1,2),(1,3),(2,4)],5)
=> ([(0,3),(2,1),(3,2)],4)
=> 4
([(0,3),(1,2),(1,4),(3,4)],5)
=> ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> 5
([(0,3),(0,4),(1,2),(2,3),(2,4)],5)
=> ([(0,2),(2,1)],3)
=> 3
([(1,4),(3,2),(4,3)],5)
=> ([(0,3),(2,1),(3,2)],4)
=> 4
([(1,4),(2,3),(3,4)],5)
=> ([(0,1)],2)
=> 2
([(0,4),(1,2),(2,4),(4,3)],5)
=> ([(0,1)],2)
=> 2
([(0,3),(1,4),(4,2)],5)
=> ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 6
([(0,4),(3,2),(4,1),(4,3)],5)
=> ([(0,1)],2)
=> 2
([(0,4),(1,2),(2,3),(2,4)],5)
=> ([(0,3),(2,1),(3,2)],4)
=> 4
([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 5
([(0,3),(1,2),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 4
([(0,4),(1,2),(2,3),(3,4)],5)
=> ([(0,2),(2,1)],3)
=> 3
([(4,5)],6)
=> ([(0,1)],2)
=> 2
([(0,2),(0,3),(0,4),(0,5),(5,1)],6)
=> ([(0,1)],2)
=> 2
([(1,3),(1,4),(1,5),(5,2)],6)
=> ([(0,1)],2)
=> 2
([(0,2),(0,3),(0,4),(2,5),(3,5),(4,1)],6)
=> ([(0,1)],2)
=> 2
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
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(load all 2 compositions to match this statistic)
Values
([(0,1)],2)
=> ([(0,1)],2)
=> 2
([(1,2)],3)
=> ([(0,1)],2)
=> 2
([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 3
([(2,3)],4)
=> ([(0,1)],2)
=> 2
([(0,2),(0,3),(3,1)],4)
=> ([(0,1)],2)
=> 2
([(1,2),(2,3)],4)
=> ([(0,2),(2,1)],3)
=> 3
([(0,3),(1,2)],4)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 4
([(0,3),(1,2),(1,3)],4)
=> ([(0,2),(2,1)],3)
=> 3
([(0,2),(0,3),(1,2),(1,3)],4)
=> ([(0,1)],2)
=> 2
([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(2,1),(3,2)],4)
=> 4
([(0,3),(1,2),(2,3)],4)
=> ([(0,1)],2)
=> 2
([(3,4)],5)
=> ([(0,1)],2)
=> 2
([(0,2),(0,3),(0,4),(4,1)],5)
=> ([(0,1)],2)
=> 2
([(1,3),(1,4),(4,2)],5)
=> ([(0,1)],2)
=> 2
([(0,3),(0,4),(3,2),(4,1)],5)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 4
([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
=> ([(0,2),(2,1)],3)
=> 3
([(0,1),(0,2),(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,1)],2)
=> 2
([(2,3),(3,4)],5)
=> ([(0,2),(2,1)],3)
=> 3
([(0,4),(1,4),(4,2),(4,3)],5)
=> ([(0,1)],2)
=> 2
([(0,4),(1,4),(2,3)],5)
=> ([(0,1)],2)
=> 2
([(0,4),(1,4),(2,3),(2,4)],5)
=> ([(0,1)],2)
=> 2
([(0,4),(1,4),(2,3),(3,4)],5)
=> ([(0,1)],2)
=> 2
([(1,4),(2,3)],5)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 4
([(1,4),(2,3),(2,4)],5)
=> ([(0,2),(2,1)],3)
=> 3
([(0,4),(1,2),(1,4),(2,3)],5)
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> 5
([(0,3),(1,2),(1,3),(2,4),(3,4)],5)
=> ([(0,2),(2,1)],3)
=> 3
([(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,1)],2)
=> 2
([(0,3),(0,4),(1,3),(1,4),(4,2)],5)
=> ([(0,2),(2,1)],3)
=> 3
([(0,3),(0,4),(1,3),(1,4),(3,2),(4,2)],5)
=> ([(0,1)],2)
=> 2
([(0,4),(1,2),(1,4),(4,3)],5)
=> ([(0,3),(2,1),(3,2)],4)
=> 4
([(0,4),(1,2),(1,3)],5)
=> ([(0,1)],2)
=> 2
([(0,4),(1,2),(1,3),(1,4)],5)
=> ([(0,1)],2)
=> 2
([(0,2),(0,4),(3,1),(4,3)],5)
=> ([(0,2),(2,1)],3)
=> 3
([(0,2),(0,3),(1,4),(2,4),(3,1)],5)
=> ([(0,1)],2)
=> 2
([(0,3),(0,4),(1,2),(1,3),(2,4)],5)
=> ([(0,3),(2,1),(3,2)],4)
=> 4
([(0,3),(1,2),(1,4),(3,4)],5)
=> ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> 5
([(0,3),(0,4),(1,2),(2,3),(2,4)],5)
=> ([(0,2),(2,1)],3)
=> 3
([(1,4),(3,2),(4,3)],5)
=> ([(0,3),(2,1),(3,2)],4)
=> 4
([(1,4),(2,3),(3,4)],5)
=> ([(0,1)],2)
=> 2
([(0,4),(1,2),(2,4),(4,3)],5)
=> ([(0,1)],2)
=> 2
([(0,3),(1,4),(4,2)],5)
=> ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 6
([(0,4),(3,2),(4,1),(4,3)],5)
=> ([(0,1)],2)
=> 2
([(0,4),(1,2),(2,3),(2,4)],5)
=> ([(0,3),(2,1),(3,2)],4)
=> 4
([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 5
([(0,3),(1,2),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 4
([(0,4),(1,2),(2,3),(3,4)],5)
=> ([(0,2),(2,1)],3)
=> 3
([(4,5)],6)
=> ([(0,1)],2)
=> 2
([(0,2),(0,3),(0,4),(0,5),(5,1)],6)
=> ([(0,1)],2)
=> 2
([(1,3),(1,4),(1,5),(5,2)],6)
=> ([(0,1)],2)
=> 2
([(0,2),(0,3),(0,4),(2,5),(3,5),(4,1)],6)
=> ([(0,1)],2)
=> 2
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
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(load all 2 compositions to match this statistic)
Values
([(0,1)],2)
=> ([(0,1)],2)
=> 2
([(1,2)],3)
=> ([(0,1)],2)
=> 2
([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 3
([(2,3)],4)
=> ([(0,1)],2)
=> 2
([(0,2),(0,3),(3,1)],4)
=> ([(0,1)],2)
=> 2
([(1,2),(2,3)],4)
=> ([(0,2),(2,1)],3)
=> 3
([(0,3),(1,2)],4)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 4
([(0,3),(1,2),(1,3)],4)
=> ([(0,2),(2,1)],3)
=> 3
([(0,2),(0,3),(1,2),(1,3)],4)
=> ([(0,1)],2)
=> 2
([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(2,1),(3,2)],4)
=> 4
([(0,3),(1,2),(2,3)],4)
=> ([(0,1)],2)
=> 2
([(3,4)],5)
=> ([(0,1)],2)
=> 2
([(0,2),(0,3),(0,4),(4,1)],5)
=> ([(0,1)],2)
=> 2
([(1,3),(1,4),(4,2)],5)
=> ([(0,1)],2)
=> 2
([(0,3),(0,4),(3,2),(4,1)],5)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 4
([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
=> ([(0,2),(2,1)],3)
=> 3
([(0,1),(0,2),(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,1)],2)
=> 2
([(2,3),(3,4)],5)
=> ([(0,2),(2,1)],3)
=> 3
([(0,4),(1,4),(4,2),(4,3)],5)
=> ([(0,1)],2)
=> 2
([(0,4),(1,4),(2,3)],5)
=> ([(0,1)],2)
=> 2
([(0,4),(1,4),(2,3),(2,4)],5)
=> ([(0,1)],2)
=> 2
([(0,4),(1,4),(2,3),(3,4)],5)
=> ([(0,1)],2)
=> 2
([(1,4),(2,3)],5)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 4
([(1,4),(2,3),(2,4)],5)
=> ([(0,2),(2,1)],3)
=> 3
([(0,4),(1,2),(1,4),(2,3)],5)
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> 5
([(0,3),(1,2),(1,3),(2,4),(3,4)],5)
=> ([(0,2),(2,1)],3)
=> 3
([(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,1)],2)
=> 2
([(0,3),(0,4),(1,3),(1,4),(4,2)],5)
=> ([(0,2),(2,1)],3)
=> 3
([(0,3),(0,4),(1,3),(1,4),(3,2),(4,2)],5)
=> ([(0,1)],2)
=> 2
([(0,4),(1,2),(1,4),(4,3)],5)
=> ([(0,3),(2,1),(3,2)],4)
=> 4
([(0,4),(1,2),(1,3)],5)
=> ([(0,1)],2)
=> 2
([(0,4),(1,2),(1,3),(1,4)],5)
=> ([(0,1)],2)
=> 2
([(0,2),(0,4),(3,1),(4,3)],5)
=> ([(0,2),(2,1)],3)
=> 3
([(0,2),(0,3),(1,4),(2,4),(3,1)],5)
=> ([(0,1)],2)
=> 2
([(0,3),(0,4),(1,2),(1,3),(2,4)],5)
=> ([(0,3),(2,1),(3,2)],4)
=> 4
([(0,3),(1,2),(1,4),(3,4)],5)
=> ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> 5
([(0,3),(0,4),(1,2),(2,3),(2,4)],5)
=> ([(0,2),(2,1)],3)
=> 3
([(1,4),(3,2),(4,3)],5)
=> ([(0,3),(2,1),(3,2)],4)
=> 4
([(1,4),(2,3),(3,4)],5)
=> ([(0,1)],2)
=> 2
([(0,4),(1,2),(2,4),(4,3)],5)
=> ([(0,1)],2)
=> 2
([(0,3),(1,4),(4,2)],5)
=> ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 6
([(0,4),(3,2),(4,1),(4,3)],5)
=> ([(0,1)],2)
=> 2
([(0,4),(1,2),(2,3),(2,4)],5)
=> ([(0,3),(2,1),(3,2)],4)
=> 4
([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 5
([(0,3),(1,2),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 4
([(0,4),(1,2),(2,3),(3,4)],5)
=> ([(0,2),(2,1)],3)
=> 3
([(4,5)],6)
=> ([(0,1)],2)
=> 2
([(0,2),(0,3),(0,4),(0,5),(5,1)],6)
=> ([(0,1)],2)
=> 2
([(1,3),(1,4),(1,5),(5,2)],6)
=> ([(0,1)],2)
=> 2
([(0,2),(0,3),(0,4),(2,5),(3,5),(4,1)],6)
=> ([(0,1)],2)
=> 2
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: St000070
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(load all 2 compositions to match this statistic)
Values
([(0,1)],2)
=> ([(0,1)],2)
=> ([],1)
=> 2
([(1,2)],3)
=> ([(0,1)],2)
=> ([],1)
=> 2
([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> ([(0,1)],2)
=> 3
([(2,3)],4)
=> ([(0,1)],2)
=> ([],1)
=> 2
([(0,2),(0,3),(3,1)],4)
=> ([(0,1)],2)
=> ([],1)
=> 2
([(1,2),(2,3)],4)
=> ([(0,2),(2,1)],3)
=> ([(0,1)],2)
=> 3
([(0,3),(1,2)],4)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([],2)
=> 4
([(0,3),(1,2),(1,3)],4)
=> ([(0,2),(2,1)],3)
=> ([(0,1)],2)
=> 3
([(0,2),(0,3),(1,2),(1,3)],4)
=> ([(0,1)],2)
=> ([],1)
=> 2
([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,2),(2,1)],3)
=> 4
([(0,3),(1,2),(2,3)],4)
=> ([(0,1)],2)
=> ([],1)
=> 2
([(3,4)],5)
=> ([(0,1)],2)
=> ([],1)
=> 2
([(0,2),(0,3),(0,4),(4,1)],5)
=> ([(0,1)],2)
=> ([],1)
=> 2
([(1,3),(1,4),(4,2)],5)
=> ([(0,1)],2)
=> ([],1)
=> 2
([(0,3),(0,4),(3,2),(4,1)],5)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([],2)
=> 4
([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
=> ([(0,2),(2,1)],3)
=> ([(0,1)],2)
=> 3
([(0,1),(0,2),(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,1)],2)
=> ([],1)
=> 2
([(2,3),(3,4)],5)
=> ([(0,2),(2,1)],3)
=> ([(0,1)],2)
=> 3
([(0,4),(1,4),(4,2),(4,3)],5)
=> ([(0,1)],2)
=> ([],1)
=> 2
([(0,4),(1,4),(2,3)],5)
=> ([(0,1)],2)
=> ([],1)
=> 2
([(0,4),(1,4),(2,3),(2,4)],5)
=> ([(0,1)],2)
=> ([],1)
=> 2
([(0,4),(1,4),(2,3),(3,4)],5)
=> ([(0,1)],2)
=> ([],1)
=> 2
([(1,4),(2,3)],5)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([],2)
=> 4
([(1,4),(2,3),(2,4)],5)
=> ([(0,2),(2,1)],3)
=> ([(0,1)],2)
=> 3
([(0,4),(1,2),(1,4),(2,3)],5)
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> ([(0,1),(0,2)],3)
=> 5
([(0,3),(1,2),(1,3),(2,4),(3,4)],5)
=> ([(0,2),(2,1)],3)
=> ([(0,1)],2)
=> 3
([(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,1)],2)
=> ([],1)
=> 2
([(0,3),(0,4),(1,3),(1,4),(4,2)],5)
=> ([(0,2),(2,1)],3)
=> ([(0,1)],2)
=> 3
([(0,3),(0,4),(1,3),(1,4),(3,2),(4,2)],5)
=> ([(0,1)],2)
=> ([],1)
=> 2
([(0,4),(1,2),(1,4),(4,3)],5)
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,2),(2,1)],3)
=> 4
([(0,4),(1,2),(1,3)],5)
=> ([(0,1)],2)
=> ([],1)
=> 2
([(0,4),(1,2),(1,3),(1,4)],5)
=> ([(0,1)],2)
=> ([],1)
=> 2
([(0,2),(0,4),(3,1),(4,3)],5)
=> ([(0,2),(2,1)],3)
=> ([(0,1)],2)
=> 3
([(0,2),(0,3),(1,4),(2,4),(3,1)],5)
=> ([(0,1)],2)
=> ([],1)
=> 2
([(0,3),(0,4),(1,2),(1,3),(2,4)],5)
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,2),(2,1)],3)
=> 4
([(0,3),(1,2),(1,4),(3,4)],5)
=> ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> ([(0,2),(1,2)],3)
=> 5
([(0,3),(0,4),(1,2),(2,3),(2,4)],5)
=> ([(0,2),(2,1)],3)
=> ([(0,1)],2)
=> 3
([(1,4),(3,2),(4,3)],5)
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,2),(2,1)],3)
=> 4
([(1,4),(2,3),(3,4)],5)
=> ([(0,1)],2)
=> ([],1)
=> 2
([(0,4),(1,2),(2,4),(4,3)],5)
=> ([(0,1)],2)
=> ([],1)
=> 2
([(0,3),(1,4),(4,2)],5)
=> ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ([(1,2)],3)
=> 6
([(0,4),(3,2),(4,1),(4,3)],5)
=> ([(0,1)],2)
=> ([],1)
=> 2
([(0,4),(1,2),(2,3),(2,4)],5)
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,2),(2,1)],3)
=> 4
([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,3),(2,1),(3,2)],4)
=> 5
([(0,3),(1,2),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([],2)
=> 4
([(0,4),(1,2),(2,3),(3,4)],5)
=> ([(0,2),(2,1)],3)
=> ([(0,1)],2)
=> 3
([(4,5)],6)
=> ([(0,1)],2)
=> ([],1)
=> 2
([(0,2),(0,3),(0,4),(0,5),(5,1)],6)
=> ([(0,1)],2)
=> ([],1)
=> 2
([(1,3),(1,4),(1,5),(5,2)],6)
=> ([(0,1)],2)
=> ([],1)
=> 2
([(0,2),(0,3),(0,4),(2,5),(3,5),(4,1)],6)
=> ([(0,1)],2)
=> ([],1)
=> 2
Description
The number of antichains in a poset.
An antichain in a poset $P$ is a subset of elements of $P$ which are pairwise incomparable.
An order ideal is a subset $I$ of $P$ such that $a\in I$ and $b \leq_P a$ implies $b \in I$. Since there is a one-to-one correspondence between antichains and order ideals, this statistic is also the number of order ideals in a poset.
Matching statistic: St000656
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Values
([(0,1)],2)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 2
([(1,2)],3)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 2
([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 3
([(2,3)],4)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 2
([(0,2),(0,3),(3,1)],4)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 2
([(1,2),(2,3)],4)
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 3
([(0,3),(1,2)],4)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 4
([(0,3),(1,2),(1,3)],4)
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 3
([(0,2),(0,3),(1,2),(1,3)],4)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 2
([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(2,1),(3,2)],4)
=> 4
([(0,3),(1,2),(2,3)],4)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 2
([(3,4)],5)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 2
([(0,2),(0,3),(0,4),(4,1)],5)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 2
([(1,3),(1,4),(4,2)],5)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 2
([(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)
=> 4
([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 3
([(0,1),(0,2),(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 2
([(2,3),(3,4)],5)
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 3
([(0,4),(1,4),(4,2),(4,3)],5)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 2
([(0,4),(1,4),(2,3)],5)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 2
([(0,4),(1,4),(2,3),(2,4)],5)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 2
([(0,4),(1,4),(2,3),(3,4)],5)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 2
([(1,4),(2,3)],5)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 4
([(1,4),(2,3),(2,4)],5)
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 3
([(0,4),(1,2),(1,4),(2,3)],5)
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> 5
([(0,3),(1,2),(1,3),(2,4),(3,4)],5)
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 3
([(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 2
([(0,3),(0,4),(1,3),(1,4),(4,2)],5)
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 3
([(0,3),(0,4),(1,3),(1,4),(3,2),(4,2)],5)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 2
([(0,4),(1,2),(1,4),(4,3)],5)
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(2,1),(3,2)],4)
=> 4
([(0,4),(1,2),(1,3)],5)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 2
([(0,4),(1,2),(1,3),(1,4)],5)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 2
([(0,2),(0,4),(3,1),(4,3)],5)
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 3
([(0,2),(0,3),(1,4),(2,4),(3,1)],5)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 2
([(0,3),(0,4),(1,2),(1,3),(2,4)],5)
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(2,1),(3,2)],4)
=> 4
([(0,3),(1,2),(1,4),(3,4)],5)
=> ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> 5
([(0,3),(0,4),(1,2),(2,3),(2,4)],5)
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 3
([(1,4),(3,2),(4,3)],5)
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(2,1),(3,2)],4)
=> 4
([(1,4),(2,3),(3,4)],5)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 2
([(0,4),(1,2),(2,4),(4,3)],5)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 2
([(0,3),(1,4),(4,2)],5)
=> ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 6
([(0,4),(3,2),(4,1),(4,3)],5)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 2
([(0,4),(1,2),(2,3),(2,4)],5)
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(2,1),(3,2)],4)
=> 4
([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 5
([(0,3),(1,2),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 4
([(0,4),(1,2),(2,3),(3,4)],5)
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 3
([(4,5)],6)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 2
([(0,2),(0,3),(0,4),(0,5),(5,1)],6)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 2
([(1,3),(1,4),(1,5),(5,2)],6)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 2
([(0,2),(0,3),(0,4),(2,5),(3,5),(4,1)],6)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 2
Description
The number of cuts of a poset.
A cut is a subset $A$ of the poset such that the set of lower bounds of the set of upper bounds of $A$ is exactly $A$.
Matching statistic: St001717
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Values
([(0,1)],2)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 2
([(1,2)],3)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 2
([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 3
([(2,3)],4)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 2
([(0,2),(0,3),(3,1)],4)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 2
([(1,2),(2,3)],4)
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 3
([(0,3),(1,2)],4)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 4
([(0,3),(1,2),(1,3)],4)
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 3
([(0,2),(0,3),(1,2),(1,3)],4)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 2
([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(2,1),(3,2)],4)
=> 4
([(0,3),(1,2),(2,3)],4)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 2
([(3,4)],5)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 2
([(0,2),(0,3),(0,4),(4,1)],5)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 2
([(1,3),(1,4),(4,2)],5)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 2
([(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)
=> 4
([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 3
([(0,1),(0,2),(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 2
([(2,3),(3,4)],5)
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 3
([(0,4),(1,4),(4,2),(4,3)],5)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 2
([(0,4),(1,4),(2,3)],5)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 2
([(0,4),(1,4),(2,3),(2,4)],5)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 2
([(0,4),(1,4),(2,3),(3,4)],5)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 2
([(1,4),(2,3)],5)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 4
([(1,4),(2,3),(2,4)],5)
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 3
([(0,4),(1,2),(1,4),(2,3)],5)
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> 5
([(0,3),(1,2),(1,3),(2,4),(3,4)],5)
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 3
([(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 2
([(0,3),(0,4),(1,3),(1,4),(4,2)],5)
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 3
([(0,3),(0,4),(1,3),(1,4),(3,2),(4,2)],5)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 2
([(0,4),(1,2),(1,4),(4,3)],5)
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(2,1),(3,2)],4)
=> 4
([(0,4),(1,2),(1,3)],5)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 2
([(0,4),(1,2),(1,3),(1,4)],5)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 2
([(0,2),(0,4),(3,1),(4,3)],5)
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 3
([(0,2),(0,3),(1,4),(2,4),(3,1)],5)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 2
([(0,3),(0,4),(1,2),(1,3),(2,4)],5)
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(2,1),(3,2)],4)
=> 4
([(0,3),(1,2),(1,4),(3,4)],5)
=> ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> 5
([(0,3),(0,4),(1,2),(2,3),(2,4)],5)
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 3
([(1,4),(3,2),(4,3)],5)
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(2,1),(3,2)],4)
=> 4
([(1,4),(2,3),(3,4)],5)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 2
([(0,4),(1,2),(2,4),(4,3)],5)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 2
([(0,3),(1,4),(4,2)],5)
=> ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 6
([(0,4),(3,2),(4,1),(4,3)],5)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 2
([(0,4),(1,2),(2,3),(2,4)],5)
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(2,1),(3,2)],4)
=> 4
([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 5
([(0,3),(1,2),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 4
([(0,4),(1,2),(2,3),(3,4)],5)
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 3
([(4,5)],6)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 2
([(0,2),(0,3),(0,4),(0,5),(5,1)],6)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 2
([(1,3),(1,4),(1,5),(5,2)],6)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 2
([(0,2),(0,3),(0,4),(2,5),(3,5),(4,1)],6)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 2
Description
The largest size of an interval in a poset.
Matching statistic: St001300
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Values
([(0,1)],2)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 1 = 2 - 1
([(1,2)],3)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 1 = 2 - 1
([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 2 = 3 - 1
([(2,3)],4)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 1 = 2 - 1
([(0,2),(0,3),(3,1)],4)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 1 = 2 - 1
([(1,2),(2,3)],4)
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 2 = 3 - 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 = 4 - 1
([(0,3),(1,2),(1,3)],4)
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 2 = 3 - 1
([(0,2),(0,3),(1,2),(1,3)],4)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 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 = 4 - 1
([(0,3),(1,2),(2,3)],4)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 1 = 2 - 1
([(3,4)],5)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 1 = 2 - 1
([(0,2),(0,3),(0,4),(4,1)],5)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 1 = 2 - 1
([(1,3),(1,4),(4,2)],5)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 1 = 2 - 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)
=> 3 = 4 - 1
([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 2 = 3 - 1
([(0,1),(0,2),(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 1 = 2 - 1
([(2,3),(3,4)],5)
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 2 = 3 - 1
([(0,4),(1,4),(4,2),(4,3)],5)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 1 = 2 - 1
([(0,4),(1,4),(2,3)],5)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 1 = 2 - 1
([(0,4),(1,4),(2,3),(2,4)],5)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 1 = 2 - 1
([(0,4),(1,4),(2,3),(3,4)],5)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 1 = 2 - 1
([(1,4),(2,3)],5)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 3 = 4 - 1
([(1,4),(2,3),(2,4)],5)
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 2 = 3 - 1
([(0,4),(1,2),(1,4),(2,3)],5)
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> 4 = 5 - 1
([(0,3),(1,2),(1,3),(2,4),(3,4)],5)
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 2 = 3 - 1
([(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 1 = 2 - 1
([(0,3),(0,4),(1,3),(1,4),(4,2)],5)
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 2 = 3 - 1
([(0,3),(0,4),(1,3),(1,4),(3,2),(4,2)],5)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 1 = 2 - 1
([(0,4),(1,2),(1,4),(4,3)],5)
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(2,1),(3,2)],4)
=> 3 = 4 - 1
([(0,4),(1,2),(1,3)],5)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 1 = 2 - 1
([(0,4),(1,2),(1,3),(1,4)],5)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 1 = 2 - 1
([(0,2),(0,4),(3,1),(4,3)],5)
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 2 = 3 - 1
([(0,2),(0,3),(1,4),(2,4),(3,1)],5)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 1 = 2 - 1
([(0,3),(0,4),(1,2),(1,3),(2,4)],5)
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(2,1),(3,2)],4)
=> 3 = 4 - 1
([(0,3),(1,2),(1,4),(3,4)],5)
=> ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> 4 = 5 - 1
([(0,3),(0,4),(1,2),(2,3),(2,4)],5)
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 2 = 3 - 1
([(1,4),(3,2),(4,3)],5)
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(2,1),(3,2)],4)
=> 3 = 4 - 1
([(1,4),(2,3),(3,4)],5)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 1 = 2 - 1
([(0,4),(1,2),(2,4),(4,3)],5)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 1 = 2 - 1
([(0,3),(1,4),(4,2)],5)
=> ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 5 = 6 - 1
([(0,4),(3,2),(4,1),(4,3)],5)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 1 = 2 - 1
([(0,4),(1,2),(2,3),(2,4)],5)
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(2,1),(3,2)],4)
=> 3 = 4 - 1
([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 4 = 5 - 1
([(0,3),(1,2),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 3 = 4 - 1
([(0,4),(1,2),(2,3),(3,4)],5)
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 2 = 3 - 1
([(4,5)],6)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 1 = 2 - 1
([(0,2),(0,3),(0,4),(0,5),(5,1)],6)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 1 = 2 - 1
([(1,3),(1,4),(1,5),(5,2)],6)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 1 = 2 - 1
([(0,2),(0,3),(0,4),(2,5),(3,5),(4,1)],6)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 1 = 2 - 1
Description
The rank of the boundary operator in degree 1 of the chain complex of the order complex of the poset.
Matching statistic: St000228
Values
([(0,1)],2)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> [2]
=> 2
([(1,2)],3)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> [2]
=> 2
([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> [3]
=> 3
([(2,3)],4)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> [2]
=> 2
([(0,2),(0,3),(3,1)],4)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> [2]
=> 2
([(1,2),(2,3)],4)
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> [3]
=> 3
([(0,3),(1,2)],4)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> [3,1]
=> 4
([(0,3),(1,2),(1,3)],4)
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> [3]
=> 3
([(0,2),(0,3),(1,2),(1,3)],4)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> [2]
=> 2
([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(2,1),(3,2)],4)
=> [4]
=> 4
([(0,3),(1,2),(2,3)],4)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> [2]
=> 2
([(3,4)],5)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> [2]
=> 2
([(0,2),(0,3),(0,4),(4,1)],5)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> [2]
=> 2
([(1,3),(1,4),(4,2)],5)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> [2]
=> 2
([(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)
=> [3,1]
=> 4
([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> [3]
=> 3
([(0,1),(0,2),(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> [2]
=> 2
([(2,3),(3,4)],5)
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> [3]
=> 3
([(0,4),(1,4),(4,2),(4,3)],5)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> [2]
=> 2
([(0,4),(1,4),(2,3)],5)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> [2]
=> 2
([(0,4),(1,4),(2,3),(2,4)],5)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> [2]
=> 2
([(0,4),(1,4),(2,3),(3,4)],5)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> [2]
=> 2
([(1,4),(2,3)],5)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> [3,1]
=> 4
([(1,4),(2,3),(2,4)],5)
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> [3]
=> 3
([(0,4),(1,2),(1,4),(2,3)],5)
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> [4,1]
=> 5
([(0,3),(1,2),(1,3),(2,4),(3,4)],5)
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> [3]
=> 3
([(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> [2]
=> 2
([(0,3),(0,4),(1,3),(1,4),(4,2)],5)
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> [3]
=> 3
([(0,3),(0,4),(1,3),(1,4),(3,2),(4,2)],5)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> [2]
=> 2
([(0,4),(1,2),(1,4),(4,3)],5)
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(2,1),(3,2)],4)
=> [4]
=> 4
([(0,4),(1,2),(1,3)],5)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> [2]
=> 2
([(0,4),(1,2),(1,3),(1,4)],5)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> [2]
=> 2
([(0,2),(0,4),(3,1),(4,3)],5)
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> [3]
=> 3
([(0,2),(0,3),(1,4),(2,4),(3,1)],5)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> [2]
=> 2
([(0,3),(0,4),(1,2),(1,3),(2,4)],5)
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(2,1),(3,2)],4)
=> [4]
=> 4
([(0,3),(1,2),(1,4),(3,4)],5)
=> ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> [4,1]
=> 5
([(0,3),(0,4),(1,2),(2,3),(2,4)],5)
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> [3]
=> 3
([(1,4),(3,2),(4,3)],5)
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(2,1),(3,2)],4)
=> [4]
=> 4
([(1,4),(2,3),(3,4)],5)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> [2]
=> 2
([(0,4),(1,2),(2,4),(4,3)],5)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> [2]
=> 2
([(0,3),(1,4),(4,2)],5)
=> ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> [4,2]
=> 6
([(0,4),(3,2),(4,1),(4,3)],5)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> [2]
=> 2
([(0,4),(1,2),(2,3),(2,4)],5)
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(2,1),(3,2)],4)
=> [4]
=> 4
([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> [5]
=> 5
([(0,3),(1,2),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> [3,1]
=> 4
([(0,4),(1,2),(2,3),(3,4)],5)
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> [3]
=> 3
([(4,5)],6)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> [2]
=> 2
([(0,2),(0,3),(0,4),(0,5),(5,1)],6)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> [2]
=> 2
([(1,3),(1,4),(1,5),(5,2)],6)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> [2]
=> 2
([(0,2),(0,3),(0,4),(2,5),(3,5),(4,1)],6)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> [2]
=> 2
Description
The size of a partition.
This statistic is the constant statistic of the level sets.
Matching statistic: St000479
Values
([(0,1)],2)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> ([],2)
=> 2
([(1,2)],3)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> ([],2)
=> 2
([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> ([],3)
=> 3
([(2,3)],4)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> ([],2)
=> 2
([(0,2),(0,3),(3,1)],4)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> ([],2)
=> 2
([(1,2),(2,3)],4)
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> ([],3)
=> 3
([(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)
=> 4
([(0,3),(1,2),(1,3)],4)
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> ([],3)
=> 3
([(0,2),(0,3),(1,2),(1,3)],4)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> ([],2)
=> 2
([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(2,1),(3,2)],4)
=> ([],4)
=> 4
([(0,3),(1,2),(2,3)],4)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> ([],2)
=> 2
([(3,4)],5)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> ([],2)
=> 2
([(0,2),(0,3),(0,4),(4,1)],5)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> ([],2)
=> 2
([(1,3),(1,4),(4,2)],5)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> ([],2)
=> 2
([(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)
=> 4
([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> ([],3)
=> 3
([(0,1),(0,2),(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> ([],2)
=> 2
([(2,3),(3,4)],5)
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> ([],3)
=> 3
([(0,4),(1,4),(4,2),(4,3)],5)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> ([],2)
=> 2
([(0,4),(1,4),(2,3)],5)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> ([],2)
=> 2
([(0,4),(1,4),(2,3),(2,4)],5)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> ([],2)
=> 2
([(0,4),(1,4),(2,3),(3,4)],5)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> ([],2)
=> 2
([(1,4),(2,3)],5)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(2,3)],4)
=> 4
([(1,4),(2,3),(2,4)],5)
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> ([],3)
=> 3
([(0,4),(1,2),(1,4),(2,3)],5)
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> ([(3,4)],5)
=> 5
([(0,3),(1,2),(1,3),(2,4),(3,4)],5)
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> ([],3)
=> 3
([(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> ([],2)
=> 2
([(0,3),(0,4),(1,3),(1,4),(4,2)],5)
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> ([],3)
=> 3
([(0,3),(0,4),(1,3),(1,4),(3,2),(4,2)],5)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> ([],2)
=> 2
([(0,4),(1,2),(1,4),(4,3)],5)
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(2,1),(3,2)],4)
=> ([],4)
=> 4
([(0,4),(1,2),(1,3)],5)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> ([],2)
=> 2
([(0,4),(1,2),(1,3),(1,4)],5)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> ([],2)
=> 2
([(0,2),(0,4),(3,1),(4,3)],5)
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> ([],3)
=> 3
([(0,2),(0,3),(1,4),(2,4),(3,1)],5)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> ([],2)
=> 2
([(0,3),(0,4),(1,2),(1,3),(2,4)],5)
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(2,1),(3,2)],4)
=> ([],4)
=> 4
([(0,3),(1,2),(1,4),(3,4)],5)
=> ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> ([(3,4)],5)
=> 5
([(0,3),(0,4),(1,2),(2,3),(2,4)],5)
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> ([],3)
=> 3
([(1,4),(3,2),(4,3)],5)
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(2,1),(3,2)],4)
=> ([],4)
=> 4
([(1,4),(2,3),(3,4)],5)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> ([],2)
=> 2
([(0,4),(1,2),(2,4),(4,3)],5)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> ([],2)
=> 2
([(0,3),(1,4),(4,2)],5)
=> ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ([(2,5),(3,4),(4,5)],6)
=> 6
([(0,4),(3,2),(4,1),(4,3)],5)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> ([],2)
=> 2
([(0,4),(1,2),(2,3),(2,4)],5)
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(2,1),(3,2)],4)
=> ([],4)
=> 4
([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([],5)
=> 5
([(0,3),(1,2),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(2,3)],4)
=> 4
([(0,4),(1,2),(2,3),(3,4)],5)
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> ([],3)
=> 3
([(4,5)],6)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> ([],2)
=> 2
([(0,2),(0,3),(0,4),(0,5),(5,1)],6)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> ([],2)
=> 2
([(1,3),(1,4),(1,5),(5,2)],6)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> ([],2)
=> 2
([(0,2),(0,3),(0,4),(2,5),(3,5),(4,1)],6)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> ([],2)
=> 2
Description
The Ramsey number of a graph.
This is the smallest integer $n$ such that every two-colouring of the edges of the complete graph $K_n$ contains a (not necessarily induced) monochromatic copy of the given graph. [1]
Thus, the Ramsey number of the complete graph $K_n$ is the ordinary Ramsey number $R(n,n)$. Very few of these numbers are known, in particular, it is only known that $43\leq R(5,5)\leq 48$. [2,3,4,5]
Matching statistic: St001318
Values
([(0,1)],2)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> ([],2)
=> 2
([(1,2)],3)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> ([],2)
=> 2
([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> ([],3)
=> 3
([(2,3)],4)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> ([],2)
=> 2
([(0,2),(0,3),(3,1)],4)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> ([],2)
=> 2
([(1,2),(2,3)],4)
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> ([],3)
=> 3
([(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)
=> 4
([(0,3),(1,2),(1,3)],4)
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> ([],3)
=> 3
([(0,2),(0,3),(1,2),(1,3)],4)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> ([],2)
=> 2
([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(2,1),(3,2)],4)
=> ([],4)
=> 4
([(0,3),(1,2),(2,3)],4)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> ([],2)
=> 2
([(3,4)],5)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> ([],2)
=> 2
([(0,2),(0,3),(0,4),(4,1)],5)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> ([],2)
=> 2
([(1,3),(1,4),(4,2)],5)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> ([],2)
=> 2
([(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)
=> 4
([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> ([],3)
=> 3
([(0,1),(0,2),(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> ([],2)
=> 2
([(2,3),(3,4)],5)
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> ([],3)
=> 3
([(0,4),(1,4),(4,2),(4,3)],5)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> ([],2)
=> 2
([(0,4),(1,4),(2,3)],5)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> ([],2)
=> 2
([(0,4),(1,4),(2,3),(2,4)],5)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> ([],2)
=> 2
([(0,4),(1,4),(2,3),(3,4)],5)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> ([],2)
=> 2
([(1,4),(2,3)],5)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(2,3)],4)
=> 4
([(1,4),(2,3),(2,4)],5)
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> ([],3)
=> 3
([(0,4),(1,2),(1,4),(2,3)],5)
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> ([(3,4)],5)
=> 5
([(0,3),(1,2),(1,3),(2,4),(3,4)],5)
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> ([],3)
=> 3
([(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> ([],2)
=> 2
([(0,3),(0,4),(1,3),(1,4),(4,2)],5)
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> ([],3)
=> 3
([(0,3),(0,4),(1,3),(1,4),(3,2),(4,2)],5)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> ([],2)
=> 2
([(0,4),(1,2),(1,4),(4,3)],5)
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(2,1),(3,2)],4)
=> ([],4)
=> 4
([(0,4),(1,2),(1,3)],5)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> ([],2)
=> 2
([(0,4),(1,2),(1,3),(1,4)],5)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> ([],2)
=> 2
([(0,2),(0,4),(3,1),(4,3)],5)
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> ([],3)
=> 3
([(0,2),(0,3),(1,4),(2,4),(3,1)],5)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> ([],2)
=> 2
([(0,3),(0,4),(1,2),(1,3),(2,4)],5)
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(2,1),(3,2)],4)
=> ([],4)
=> 4
([(0,3),(1,2),(1,4),(3,4)],5)
=> ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> ([(3,4)],5)
=> 5
([(0,3),(0,4),(1,2),(2,3),(2,4)],5)
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> ([],3)
=> 3
([(1,4),(3,2),(4,3)],5)
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(2,1),(3,2)],4)
=> ([],4)
=> 4
([(1,4),(2,3),(3,4)],5)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> ([],2)
=> 2
([(0,4),(1,2),(2,4),(4,3)],5)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> ([],2)
=> 2
([(0,3),(1,4),(4,2)],5)
=> ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ([(2,5),(3,4),(4,5)],6)
=> 6
([(0,4),(3,2),(4,1),(4,3)],5)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> ([],2)
=> 2
([(0,4),(1,2),(2,3),(2,4)],5)
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(2,1),(3,2)],4)
=> ([],4)
=> 4
([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([],5)
=> 5
([(0,3),(1,2),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(2,3)],4)
=> 4
([(0,4),(1,2),(2,3),(3,4)],5)
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> ([],3)
=> 3
([(4,5)],6)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> ([],2)
=> 2
([(0,2),(0,3),(0,4),(0,5),(5,1)],6)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> ([],2)
=> 2
([(1,3),(1,4),(1,5),(5,2)],6)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> ([],2)
=> 2
([(0,2),(0,3),(0,4),(2,5),(3,5),(4,1)],6)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> ([],2)
=> 2
Description
The number of vertices of the largest induced subforest with the same number of connected components of a graph.
The following 17 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
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. St000987The number of positive eigenvalues of the Laplacian matrix of the graph. St000189The number of elements in the poset. St001615The number of join prime elements of a lattice. St001617The dimension of the space of valuations of a lattice. St001875The number of simple modules with projective dimension at most 1. St001880The number of 2-Gorenstein indecomposable injective modules in the incidence algebra of the lattice. St000363The number of minimal vertex covers of a graph. St000422The energy of a graph, if it is integral. St000911The number of maximal antichains of maximal size in a poset. St000993The multiplicity of the largest part of an integer partition. 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. St001630The global dimension of the incidence algebra of the lattice over the rational numbers. St001878The projective dimension of the simple modules corresponding to the minimum of L in the incidence algebra of the lattice L. St001877Number of indecomposable injective modules with projective dimension 2.
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