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Your data matches 6 different statistics following compositions of up to 3 maps.
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Matching statistic: St001850
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St001850: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => 1
[1,2] => 1
[2,1] => 1
[1,2,3] => 1
[1,3,2] => 1
[2,1,3] => 1
[2,3,1] => 0
[3,1,2] => 0
[3,2,1] => 3
[1,2,3,4] => 1
[1,2,4,3] => 1
[1,3,2,4] => 1
[1,3,4,2] => 0
[1,4,2,3] => 0
[1,4,3,2] => 3
[2,1,3,4] => 1
[2,1,4,3] => 1
[2,3,1,4] => 0
[2,3,4,1] => 0
[2,4,1,3] => 0
[2,4,3,1] => 0
[3,1,2,4] => 0
[3,1,4,2] => 0
[3,2,1,4] => 3
[3,2,4,1] => 0
[3,4,1,2] => 1
[3,4,2,1] => 0
[4,1,2,3] => 0
[4,1,3,2] => 0
[4,2,1,3] => 0
[4,2,3,1] => 5
[4,3,1,2] => 0
[4,3,2,1] => 7
[1,2,3,4,5] => 1
[1,2,3,5,4] => 1
[1,2,4,3,5] => 1
[1,2,4,5,3] => 0
[1,2,5,3,4] => 0
[1,2,5,4,3] => 3
[1,3,2,4,5] => 1
[1,3,2,5,4] => 1
[1,3,4,2,5] => 0
[1,3,4,5,2] => 0
[1,3,5,2,4] => 0
[1,3,5,4,2] => 0
[1,4,2,3,5] => 0
[1,4,2,5,3] => 0
[1,4,3,2,5] => 3
[1,4,3,5,2] => 0
[1,4,5,2,3] => 1
Description
The number of Hecke atoms of a permutation.
For a permutation $z\in\mathfrak S_n$, this is the cardinality of the set
$$
\{ w\in\mathfrak S_n | w^{-1} \star w = z\},
$$
where $\star$ denotes the Demazure product. Note that $w\mapsto w^{-1}\star w$ is a surjection onto the set of involutions.
Matching statistic: St000455
Values
[1] => ([],1)
=> ([],1)
=> ([],1)
=> ? = 1
[1,2] => ([],2)
=> ([],2)
=> ([],2)
=> ? = 1
[2,1] => ([(0,1)],2)
=> ([],1)
=> ([],1)
=> ? = 1
[1,2,3] => ([],3)
=> ([],3)
=> ([],3)
=> ? = 1
[1,3,2] => ([(1,2)],3)
=> ([],2)
=> ([],2)
=> ? = 1
[2,1,3] => ([(1,2)],3)
=> ([],2)
=> ([],2)
=> ? = 1
[2,3,1] => ([(0,2),(1,2)],3)
=> ([(0,2),(1,2)],3)
=> ([(0,2),(1,2)],3)
=> 0
[3,1,2] => ([(0,2),(1,2)],3)
=> ([(0,2),(1,2)],3)
=> ([(0,2),(1,2)],3)
=> 0
[3,2,1] => ([(0,1),(0,2),(1,2)],3)
=> ([],1)
=> ([],1)
=> ? = 3
[1,2,3,4] => ([],4)
=> ([],4)
=> ([],4)
=> ? = 1
[1,2,4,3] => ([(2,3)],4)
=> ([],3)
=> ([],3)
=> ? = 1
[1,3,2,4] => ([(2,3)],4)
=> ([],3)
=> ([],3)
=> ? = 1
[1,3,4,2] => ([(1,3),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> 0
[1,4,2,3] => ([(1,3),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> 0
[1,4,3,2] => ([(1,2),(1,3),(2,3)],4)
=> ([],2)
=> ([],2)
=> ? = 3
[2,1,3,4] => ([(2,3)],4)
=> ([],3)
=> ([],3)
=> ? = 1
[2,1,4,3] => ([(0,3),(1,2)],4)
=> ([],2)
=> ([],2)
=> ? = 1
[2,3,1,4] => ([(1,3),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> 0
[2,3,4,1] => ([(0,3),(1,3),(2,3)],4)
=> ([(0,3),(1,3),(2,3)],4)
=> ([(0,3),(1,3),(2,3)],4)
=> 0
[2,4,1,3] => ([(0,3),(1,2),(2,3)],4)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> ? = 0
[2,4,3,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,2),(1,2)],3)
=> ([(0,2),(1,2)],3)
=> 0
[3,1,2,4] => ([(1,3),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> 0
[3,1,4,2] => ([(0,3),(1,2),(2,3)],4)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> ? = 0
[3,2,1,4] => ([(1,2),(1,3),(2,3)],4)
=> ([],2)
=> ([],2)
=> ? = 3
[3,2,4,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,2),(1,2)],3)
=> ([(0,2),(1,2)],3)
=> 0
[3,4,1,2] => ([(0,2),(0,3),(1,2),(1,3)],4)
=> ([],1)
=> ([],1)
=> ? = 1
[3,4,2,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([],1)
=> ([],1)
=> ? = 0
[4,1,2,3] => ([(0,3),(1,3),(2,3)],4)
=> ([(0,3),(1,3),(2,3)],4)
=> ([(0,3),(1,3),(2,3)],4)
=> 0
[4,1,3,2] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,2),(1,2)],3)
=> ([(0,2),(1,2)],3)
=> 0
[4,2,1,3] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,2),(1,2)],3)
=> ([(0,2),(1,2)],3)
=> 0
[4,2,3,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([],1)
=> ([],1)
=> ? = 5
[4,3,1,2] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([],1)
=> ([],1)
=> ? = 0
[4,3,2,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([],1)
=> ([],1)
=> ? = 7
[1,2,3,4,5] => ([],5)
=> ([],5)
=> ([],5)
=> ? = 1
[1,2,3,5,4] => ([(3,4)],5)
=> ([],4)
=> ([],4)
=> ? = 1
[1,2,4,3,5] => ([(3,4)],5)
=> ([],4)
=> ([],4)
=> ? = 1
[1,2,4,5,3] => ([(2,4),(3,4)],5)
=> ([(2,4),(3,4)],5)
=> ([(2,4),(3,4)],5)
=> 0
[1,2,5,3,4] => ([(2,4),(3,4)],5)
=> ([(2,4),(3,4)],5)
=> ([(2,4),(3,4)],5)
=> 0
[1,2,5,4,3] => ([(2,3),(2,4),(3,4)],5)
=> ([],3)
=> ([],3)
=> ? = 3
[1,3,2,4,5] => ([(3,4)],5)
=> ([],4)
=> ([],4)
=> ? = 1
[1,3,2,5,4] => ([(1,4),(2,3)],5)
=> ([],3)
=> ([],3)
=> ? = 1
[1,3,4,2,5] => ([(2,4),(3,4)],5)
=> ([(2,4),(3,4)],5)
=> ([(2,4),(3,4)],5)
=> 0
[1,3,4,5,2] => ([(1,4),(2,4),(3,4)],5)
=> ([(1,4),(2,4),(3,4)],5)
=> ([(1,4),(2,4),(3,4)],5)
=> 0
[1,3,5,2,4] => ([(1,4),(2,3),(3,4)],5)
=> ([(1,5),(2,4),(3,4),(3,5)],6)
=> ([(1,2),(1,5),(2,4),(3,4),(3,5),(4,5)],6)
=> ? = 0
[1,3,5,4,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,3),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> 0
[1,4,2,3,5] => ([(2,4),(3,4)],5)
=> ([(2,4),(3,4)],5)
=> ([(2,4),(3,4)],5)
=> 0
[1,4,2,5,3] => ([(1,4),(2,3),(3,4)],5)
=> ([(1,5),(2,4),(3,4),(3,5)],6)
=> ([(1,2),(1,5),(2,4),(3,4),(3,5),(4,5)],6)
=> ? = 0
[1,4,3,2,5] => ([(2,3),(2,4),(3,4)],5)
=> ([],3)
=> ([],3)
=> ? = 3
[1,4,3,5,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,3),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> 0
[1,4,5,2,3] => ([(1,3),(1,4),(2,3),(2,4)],5)
=> ([],2)
=> ([],2)
=> ? = 1
[1,4,5,3,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([],2)
=> ([],2)
=> ? = 0
[1,5,2,3,4] => ([(1,4),(2,4),(3,4)],5)
=> ([(1,4),(2,4),(3,4)],5)
=> ([(1,4),(2,4),(3,4)],5)
=> 0
[1,5,2,4,3] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,3),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> 0
[1,5,3,2,4] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,3),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> 0
[1,5,3,4,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([],2)
=> ([],2)
=> ? = 5
[1,5,4,2,3] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([],2)
=> ([],2)
=> ? = 0
[1,5,4,3,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([],2)
=> ([],2)
=> ? = 7
[2,1,3,4,5] => ([(3,4)],5)
=> ([],4)
=> ([],4)
=> ? = 1
[2,1,3,5,4] => ([(1,4),(2,3)],5)
=> ([],3)
=> ([],3)
=> ? = 1
[2,1,4,3,5] => ([(1,4),(2,3)],5)
=> ([],3)
=> ([],3)
=> ? = 1
[2,1,4,5,3] => ([(0,1),(2,4),(3,4)],5)
=> ([(1,3),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> 0
[2,1,5,3,4] => ([(0,1),(2,4),(3,4)],5)
=> ([(1,3),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> 0
[2,1,5,4,3] => ([(0,1),(2,3),(2,4),(3,4)],5)
=> ([],2)
=> ([],2)
=> ? = 3
[2,3,1,4,5] => ([(2,4),(3,4)],5)
=> ([(2,4),(3,4)],5)
=> ([(2,4),(3,4)],5)
=> 0
[2,3,1,5,4] => ([(0,1),(2,4),(3,4)],5)
=> ([(1,3),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> 0
[2,3,4,1,5] => ([(1,4),(2,4),(3,4)],5)
=> ([(1,4),(2,4),(3,4)],5)
=> ([(1,4),(2,4),(3,4)],5)
=> 0
[2,3,4,5,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> ([(0,4),(1,4),(2,4),(3,4)],5)
=> ([(0,4),(1,4),(2,4),(3,4)],5)
=> 0
[2,3,5,1,4] => ([(0,4),(1,4),(2,3),(3,4)],5)
=> ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
=> ([(0,2),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6)
=> ? = 0
[2,3,5,4,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(1,3),(2,3)],4)
=> ([(0,3),(1,3),(2,3)],4)
=> 0
[2,4,1,3,5] => ([(1,4),(2,3),(3,4)],5)
=> ([(1,5),(2,4),(3,4),(3,5)],6)
=> ([(1,2),(1,5),(2,4),(3,4),(3,5),(4,5)],6)
=> ? = 0
[2,4,1,5,3] => ([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,6),(1,5),(2,3),(2,4),(3,5),(4,6)],7)
=> ([(0,1),(0,3),(0,4),(0,6),(1,2),(1,4),(1,5),(2,3),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 0
[2,4,3,1,5] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,3),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> 0
[2,4,3,5,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(1,3),(2,3)],4)
=> ([(0,3),(1,3),(2,3)],4)
=> 0
[2,4,5,1,3] => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> ([(0,2),(1,2)],3)
=> ([(0,2),(1,2)],3)
=> 0
[2,4,5,3,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,2),(1,2)],3)
=> ([(0,2),(1,2)],3)
=> 0
[2,5,1,3,4] => ([(0,4),(1,4),(2,3),(3,4)],5)
=> ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
=> ([(0,2),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6)
=> ? = 0
[2,5,1,4,3] => ([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> ? = 0
[2,5,3,1,4] => ([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> ? = 0
[2,5,3,4,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,2),(1,2)],3)
=> ([(0,2),(1,2)],3)
=> 0
[2,5,4,1,3] => ([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(0,2),(1,2)],3)
=> ([(0,2),(1,2)],3)
=> 0
[2,5,4,3,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,2),(1,2)],3)
=> ([(0,2),(1,2)],3)
=> 0
[3,1,2,4,5] => ([(2,4),(3,4)],5)
=> ([(2,4),(3,4)],5)
=> ([(2,4),(3,4)],5)
=> 0
[3,1,2,5,4] => ([(0,1),(2,4),(3,4)],5)
=> ([(1,3),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> 0
[3,1,4,2,5] => ([(1,4),(2,3),(3,4)],5)
=> ([(1,5),(2,4),(3,4),(3,5)],6)
=> ([(1,2),(1,5),(2,4),(3,4),(3,5),(4,5)],6)
=> ? = 0
[3,1,4,5,2] => ([(0,4),(1,4),(2,3),(3,4)],5)
=> ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
=> ([(0,2),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6)
=> ? = 0
[3,1,5,2,4] => ([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,6),(1,5),(2,3),(2,4),(3,5),(4,6)],7)
=> ([(0,1),(0,3),(0,4),(0,6),(1,2),(1,4),(1,5),(2,3),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 0
[3,1,5,4,2] => ([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> ? = 0
[3,2,1,4,5] => ([(2,3),(2,4),(3,4)],5)
=> ([],3)
=> ([],3)
=> ? = 3
[3,2,4,1,5] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,3),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> 0
[3,2,4,5,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(1,3),(2,3)],4)
=> ([(0,3),(1,3),(2,3)],4)
=> 0
[3,2,5,4,1] => ([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
=> ([(0,2),(1,2)],3)
=> ([(0,2),(1,2)],3)
=> 0
[3,4,1,5,2] => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> ([(0,2),(1,2)],3)
=> ([(0,2),(1,2)],3)
=> 0
[3,4,2,5,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,2),(1,2)],3)
=> ([(0,2),(1,2)],3)
=> 0
[3,5,1,2,4] => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> ([(0,2),(1,2)],3)
=> ([(0,2),(1,2)],3)
=> 0
[3,5,2,1,4] => ([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(0,2),(1,2)],3)
=> ([(0,2),(1,2)],3)
=> 0
[4,1,2,3,5] => ([(1,4),(2,4),(3,4)],5)
=> ([(1,4),(2,4),(3,4)],5)
=> ([(1,4),(2,4),(3,4)],5)
=> 0
[4,1,3,2,5] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,3),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> 0
[4,1,5,2,3] => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> ([(0,2),(1,2)],3)
=> ([(0,2),(1,2)],3)
=> 0
[4,1,5,3,2] => ([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(0,2),(1,2)],3)
=> ([(0,2),(1,2)],3)
=> 0
[4,2,1,3,5] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,3),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> 0
Description
The second largest eigenvalue of a graph if it is integral.
This statistic is undefined if the second largest eigenvalue of the graph is not integral.
Chapter 4 of [1] provides lots of context.
Matching statistic: St000456
Values
[1] => ([],1)
=> ([],1)
=> ? = 1 + 1
[1,2] => ([],2)
=> ([],2)
=> ? = 1 + 1
[2,1] => ([(0,1)],2)
=> ([],1)
=> ? = 1 + 1
[1,2,3] => ([],3)
=> ([],3)
=> ? = 1 + 1
[1,3,2] => ([(1,2)],3)
=> ([],2)
=> ? = 1 + 1
[2,1,3] => ([(1,2)],3)
=> ([],2)
=> ? = 1 + 1
[2,3,1] => ([(0,2),(1,2)],3)
=> ([(0,2),(1,2)],3)
=> 1 = 0 + 1
[3,1,2] => ([(0,2),(1,2)],3)
=> ([(0,2),(1,2)],3)
=> 1 = 0 + 1
[3,2,1] => ([(0,1),(0,2),(1,2)],3)
=> ([],1)
=> ? = 3 + 1
[1,2,3,4] => ([],4)
=> ([],4)
=> ? = 1 + 1
[1,2,4,3] => ([(2,3)],4)
=> ([],3)
=> ? = 1 + 1
[1,3,2,4] => ([(2,3)],4)
=> ([],3)
=> ? = 1 + 1
[1,3,4,2] => ([(1,3),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> ? = 0 + 1
[1,4,2,3] => ([(1,3),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> ? = 0 + 1
[1,4,3,2] => ([(1,2),(1,3),(2,3)],4)
=> ([],2)
=> ? = 3 + 1
[2,1,3,4] => ([(2,3)],4)
=> ([],3)
=> ? = 1 + 1
[2,1,4,3] => ([(0,3),(1,2)],4)
=> ([],2)
=> ? = 1 + 1
[2,3,1,4] => ([(1,3),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> ? = 0 + 1
[2,3,4,1] => ([(0,3),(1,3),(2,3)],4)
=> ([(0,3),(1,3),(2,3)],4)
=> 1 = 0 + 1
[2,4,1,3] => ([(0,3),(1,2),(2,3)],4)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 1 = 0 + 1
[2,4,3,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,2),(1,2)],3)
=> 1 = 0 + 1
[3,1,2,4] => ([(1,3),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> ? = 0 + 1
[3,1,4,2] => ([(0,3),(1,2),(2,3)],4)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 1 = 0 + 1
[3,2,1,4] => ([(1,2),(1,3),(2,3)],4)
=> ([],2)
=> ? = 3 + 1
[3,2,4,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,2),(1,2)],3)
=> 1 = 0 + 1
[3,4,1,2] => ([(0,2),(0,3),(1,2),(1,3)],4)
=> ([],1)
=> ? = 1 + 1
[3,4,2,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([],1)
=> ? = 0 + 1
[4,1,2,3] => ([(0,3),(1,3),(2,3)],4)
=> ([(0,3),(1,3),(2,3)],4)
=> 1 = 0 + 1
[4,1,3,2] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,2),(1,2)],3)
=> 1 = 0 + 1
[4,2,1,3] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,2),(1,2)],3)
=> 1 = 0 + 1
[4,2,3,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([],1)
=> ? = 5 + 1
[4,3,1,2] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([],1)
=> ? = 0 + 1
[4,3,2,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([],1)
=> ? = 7 + 1
[1,2,3,4,5] => ([],5)
=> ([],5)
=> ? = 1 + 1
[1,2,3,5,4] => ([(3,4)],5)
=> ([],4)
=> ? = 1 + 1
[1,2,4,3,5] => ([(3,4)],5)
=> ([],4)
=> ? = 1 + 1
[1,2,4,5,3] => ([(2,4),(3,4)],5)
=> ([(2,4),(3,4)],5)
=> ? = 0 + 1
[1,2,5,3,4] => ([(2,4),(3,4)],5)
=> ([(2,4),(3,4)],5)
=> ? = 0 + 1
[1,2,5,4,3] => ([(2,3),(2,4),(3,4)],5)
=> ([],3)
=> ? = 3 + 1
[1,3,2,4,5] => ([(3,4)],5)
=> ([],4)
=> ? = 1 + 1
[1,3,2,5,4] => ([(1,4),(2,3)],5)
=> ([],3)
=> ? = 1 + 1
[1,3,4,2,5] => ([(2,4),(3,4)],5)
=> ([(2,4),(3,4)],5)
=> ? = 0 + 1
[1,3,4,5,2] => ([(1,4),(2,4),(3,4)],5)
=> ([(1,4),(2,4),(3,4)],5)
=> ? = 0 + 1
[1,3,5,2,4] => ([(1,4),(2,3),(3,4)],5)
=> ([(1,5),(2,4),(3,4),(3,5)],6)
=> ? = 0 + 1
[1,3,5,4,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,3),(2,3)],4)
=> ? = 0 + 1
[1,4,2,3,5] => ([(2,4),(3,4)],5)
=> ([(2,4),(3,4)],5)
=> ? = 0 + 1
[1,4,2,5,3] => ([(1,4),(2,3),(3,4)],5)
=> ([(1,5),(2,4),(3,4),(3,5)],6)
=> ? = 0 + 1
[1,4,3,2,5] => ([(2,3),(2,4),(3,4)],5)
=> ([],3)
=> ? = 3 + 1
[1,4,3,5,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,3),(2,3)],4)
=> ? = 0 + 1
[1,4,5,2,3] => ([(1,3),(1,4),(2,3),(2,4)],5)
=> ([],2)
=> ? = 1 + 1
[1,4,5,3,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([],2)
=> ? = 0 + 1
[1,5,2,3,4] => ([(1,4),(2,4),(3,4)],5)
=> ([(1,4),(2,4),(3,4)],5)
=> ? = 0 + 1
[1,5,2,4,3] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,3),(2,3)],4)
=> ? = 0 + 1
[1,5,3,2,4] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,3),(2,3)],4)
=> ? = 0 + 1
[1,5,3,4,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([],2)
=> ? = 5 + 1
[1,5,4,2,3] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([],2)
=> ? = 0 + 1
[1,5,4,3,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([],2)
=> ? = 7 + 1
[2,1,3,4,5] => ([(3,4)],5)
=> ([],4)
=> ? = 1 + 1
[2,1,3,5,4] => ([(1,4),(2,3)],5)
=> ([],3)
=> ? = 1 + 1
[2,1,4,3,5] => ([(1,4),(2,3)],5)
=> ([],3)
=> ? = 1 + 1
[2,3,4,5,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> ([(0,4),(1,4),(2,4),(3,4)],5)
=> 1 = 0 + 1
[2,3,5,1,4] => ([(0,4),(1,4),(2,3),(3,4)],5)
=> ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
=> 1 = 0 + 1
[2,3,5,4,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(1,3),(2,3)],4)
=> 1 = 0 + 1
[2,4,1,5,3] => ([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,6),(1,5),(2,3),(2,4),(3,5),(4,6)],7)
=> 1 = 0 + 1
[2,4,3,5,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(1,3),(2,3)],4)
=> 1 = 0 + 1
[2,4,5,1,3] => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> ([(0,2),(1,2)],3)
=> 1 = 0 + 1
[2,4,5,3,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,2),(1,2)],3)
=> 1 = 0 + 1
[2,5,1,3,4] => ([(0,4),(1,4),(2,3),(3,4)],5)
=> ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
=> 1 = 0 + 1
[2,5,1,4,3] => ([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 1 = 0 + 1
[2,5,3,1,4] => ([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 1 = 0 + 1
[2,5,3,4,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,2),(1,2)],3)
=> 1 = 0 + 1
[2,5,4,1,3] => ([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(0,2),(1,2)],3)
=> 1 = 0 + 1
[2,5,4,3,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,2),(1,2)],3)
=> 1 = 0 + 1
[3,1,4,5,2] => ([(0,4),(1,4),(2,3),(3,4)],5)
=> ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
=> 1 = 0 + 1
[3,1,5,2,4] => ([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,6),(1,5),(2,3),(2,4),(3,5),(4,6)],7)
=> 1 = 0 + 1
[3,1,5,4,2] => ([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 1 = 0 + 1
[3,2,4,5,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(1,3),(2,3)],4)
=> 1 = 0 + 1
[3,2,5,1,4] => ([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 1 = 0 + 1
[3,2,5,4,1] => ([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
=> ([(0,2),(1,2)],3)
=> 1 = 0 + 1
[3,4,1,5,2] => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> ([(0,2),(1,2)],3)
=> 1 = 0 + 1
[3,4,2,5,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,2),(1,2)],3)
=> 1 = 0 + 1
[3,5,1,2,4] => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> ([(0,2),(1,2)],3)
=> 1 = 0 + 1
[3,5,2,1,4] => ([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(0,2),(1,2)],3)
=> 1 = 0 + 1
[4,1,2,5,3] => ([(0,4),(1,4),(2,3),(3,4)],5)
=> ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
=> 1 = 0 + 1
[4,1,3,5,2] => ([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 1 = 0 + 1
[4,1,5,2,3] => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> ([(0,2),(1,2)],3)
=> 1 = 0 + 1
[4,1,5,3,2] => ([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(0,2),(1,2)],3)
=> 1 = 0 + 1
[4,2,1,5,3] => ([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 1 = 0 + 1
[4,2,3,5,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,2),(1,2)],3)
=> 1 = 0 + 1
[4,3,1,5,2] => ([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(0,2),(1,2)],3)
=> 1 = 0 + 1
[4,3,2,5,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,2),(1,2)],3)
=> 1 = 0 + 1
[5,1,2,3,4] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> ([(0,4),(1,4),(2,4),(3,4)],5)
=> 1 = 0 + 1
[5,1,2,4,3] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(1,3),(2,3)],4)
=> 1 = 0 + 1
[5,1,3,2,4] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(1,3),(2,3)],4)
=> 1 = 0 + 1
[5,1,3,4,2] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,2),(1,2)],3)
=> 1 = 0 + 1
[5,1,4,2,3] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,2),(1,2)],3)
=> 1 = 0 + 1
[5,1,4,3,2] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,2),(1,2)],3)
=> 1 = 0 + 1
[5,2,1,3,4] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(1,3),(2,3)],4)
=> 1 = 0 + 1
[5,2,1,4,3] => ([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
=> ([(0,2),(1,2)],3)
=> 1 = 0 + 1
[5,2,3,1,4] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,2),(1,2)],3)
=> 1 = 0 + 1
Description
The monochromatic index of a connected graph.
This is the maximal number of colours such that there is a colouring of the edges where any two vertices can be joined by a monochromatic path.
For example, a circle graph other than the triangle can be coloured with at most two colours: one edge blue, all the others red.
Matching statistic: St000264
Values
[1] => ([],1)
=> ([],1)
=> ([],1)
=> ? = 1 + 3
[1,2] => ([],2)
=> ([],2)
=> ([],2)
=> ? = 1 + 3
[2,1] => ([(0,1)],2)
=> ([],1)
=> ([],1)
=> ? = 1 + 3
[1,2,3] => ([],3)
=> ([],3)
=> ([],3)
=> ? = 1 + 3
[1,3,2] => ([(1,2)],3)
=> ([],2)
=> ([],2)
=> ? = 1 + 3
[2,1,3] => ([(1,2)],3)
=> ([],2)
=> ([],2)
=> ? = 1 + 3
[2,3,1] => ([(0,2),(1,2)],3)
=> ([(0,2),(1,2)],3)
=> ([(0,2),(1,2)],3)
=> ? = 0 + 3
[3,1,2] => ([(0,2),(1,2)],3)
=> ([(0,2),(1,2)],3)
=> ([(0,2),(1,2)],3)
=> ? = 0 + 3
[3,2,1] => ([(0,1),(0,2),(1,2)],3)
=> ([],1)
=> ([],1)
=> ? = 3 + 3
[1,2,3,4] => ([],4)
=> ([],4)
=> ([],4)
=> ? = 1 + 3
[1,2,4,3] => ([(2,3)],4)
=> ([],3)
=> ([],3)
=> ? = 1 + 3
[1,3,2,4] => ([(2,3)],4)
=> ([],3)
=> ([],3)
=> ? = 1 + 3
[1,3,4,2] => ([(1,3),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> ? = 0 + 3
[1,4,2,3] => ([(1,3),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> ? = 0 + 3
[1,4,3,2] => ([(1,2),(1,3),(2,3)],4)
=> ([],2)
=> ([],2)
=> ? = 3 + 3
[2,1,3,4] => ([(2,3)],4)
=> ([],3)
=> ([],3)
=> ? = 1 + 3
[2,1,4,3] => ([(0,3),(1,2)],4)
=> ([],2)
=> ([],2)
=> ? = 1 + 3
[2,3,1,4] => ([(1,3),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> ? = 0 + 3
[2,3,4,1] => ([(0,3),(1,3),(2,3)],4)
=> ([(0,3),(1,3),(2,3)],4)
=> ([(0,3),(1,3),(2,3)],4)
=> ? = 0 + 3
[2,4,1,3] => ([(0,3),(1,2),(2,3)],4)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> 3 = 0 + 3
[2,4,3,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,2),(1,2)],3)
=> ([(0,2),(1,2)],3)
=> ? = 0 + 3
[3,1,2,4] => ([(1,3),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> ? = 0 + 3
[3,1,4,2] => ([(0,3),(1,2),(2,3)],4)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> 3 = 0 + 3
[3,2,1,4] => ([(1,2),(1,3),(2,3)],4)
=> ([],2)
=> ([],2)
=> ? = 3 + 3
[3,2,4,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,2),(1,2)],3)
=> ([(0,2),(1,2)],3)
=> ? = 0 + 3
[3,4,1,2] => ([(0,2),(0,3),(1,2),(1,3)],4)
=> ([],1)
=> ([],1)
=> ? = 1 + 3
[3,4,2,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([],1)
=> ([],1)
=> ? = 0 + 3
[4,1,2,3] => ([(0,3),(1,3),(2,3)],4)
=> ([(0,3),(1,3),(2,3)],4)
=> ([(0,3),(1,3),(2,3)],4)
=> ? = 0 + 3
[4,1,3,2] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,2),(1,2)],3)
=> ([(0,2),(1,2)],3)
=> ? = 0 + 3
[4,2,1,3] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,2),(1,2)],3)
=> ([(0,2),(1,2)],3)
=> ? = 0 + 3
[4,2,3,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([],1)
=> ([],1)
=> ? = 5 + 3
[4,3,1,2] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([],1)
=> ([],1)
=> ? = 0 + 3
[4,3,2,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([],1)
=> ([],1)
=> ? = 7 + 3
[1,2,3,4,5] => ([],5)
=> ([],5)
=> ([],5)
=> ? = 1 + 3
[1,2,3,5,4] => ([(3,4)],5)
=> ([],4)
=> ([],4)
=> ? = 1 + 3
[1,2,4,3,5] => ([(3,4)],5)
=> ([],4)
=> ([],4)
=> ? = 1 + 3
[1,2,4,5,3] => ([(2,4),(3,4)],5)
=> ([(2,4),(3,4)],5)
=> ([(2,4),(3,4)],5)
=> ? = 0 + 3
[1,2,5,3,4] => ([(2,4),(3,4)],5)
=> ([(2,4),(3,4)],5)
=> ([(2,4),(3,4)],5)
=> ? = 0 + 3
[1,2,5,4,3] => ([(2,3),(2,4),(3,4)],5)
=> ([],3)
=> ([],3)
=> ? = 3 + 3
[1,3,2,4,5] => ([(3,4)],5)
=> ([],4)
=> ([],4)
=> ? = 1 + 3
[1,3,2,5,4] => ([(1,4),(2,3)],5)
=> ([],3)
=> ([],3)
=> ? = 1 + 3
[1,3,4,2,5] => ([(2,4),(3,4)],5)
=> ([(2,4),(3,4)],5)
=> ([(2,4),(3,4)],5)
=> ? = 0 + 3
[1,3,4,5,2] => ([(1,4),(2,4),(3,4)],5)
=> ([(1,4),(2,4),(3,4)],5)
=> ([(1,4),(2,4),(3,4)],5)
=> ? = 0 + 3
[1,3,5,2,4] => ([(1,4),(2,3),(3,4)],5)
=> ([(1,5),(2,4),(3,4),(3,5)],6)
=> ([(1,2),(1,5),(2,4),(3,4),(3,5),(4,5)],6)
=> 3 = 0 + 3
[1,3,5,4,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,3),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> ? = 0 + 3
[1,4,2,3,5] => ([(2,4),(3,4)],5)
=> ([(2,4),(3,4)],5)
=> ([(2,4),(3,4)],5)
=> ? = 0 + 3
[1,4,2,5,3] => ([(1,4),(2,3),(3,4)],5)
=> ([(1,5),(2,4),(3,4),(3,5)],6)
=> ([(1,2),(1,5),(2,4),(3,4),(3,5),(4,5)],6)
=> 3 = 0 + 3
[1,4,3,2,5] => ([(2,3),(2,4),(3,4)],5)
=> ([],3)
=> ([],3)
=> ? = 3 + 3
[1,4,3,5,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,3),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> ? = 0 + 3
[1,4,5,2,3] => ([(1,3),(1,4),(2,3),(2,4)],5)
=> ([],2)
=> ([],2)
=> ? = 1 + 3
[1,4,5,3,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([],2)
=> ([],2)
=> ? = 0 + 3
[1,5,2,3,4] => ([(1,4),(2,4),(3,4)],5)
=> ([(1,4),(2,4),(3,4)],5)
=> ([(1,4),(2,4),(3,4)],5)
=> ? = 0 + 3
[1,5,2,4,3] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,3),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> ? = 0 + 3
[1,5,3,2,4] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,3),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> ? = 0 + 3
[2,3,5,1,4] => ([(0,4),(1,4),(2,3),(3,4)],5)
=> ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
=> ([(0,2),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6)
=> 3 = 0 + 3
[2,4,1,3,5] => ([(1,4),(2,3),(3,4)],5)
=> ([(1,5),(2,4),(3,4),(3,5)],6)
=> ([(1,2),(1,5),(2,4),(3,4),(3,5),(4,5)],6)
=> 3 = 0 + 3
[2,4,1,5,3] => ([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,6),(1,5),(2,3),(2,4),(3,5),(4,6)],7)
=> ([(0,1),(0,3),(0,4),(0,6),(1,2),(1,4),(1,5),(2,3),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 3 = 0 + 3
[2,5,1,3,4] => ([(0,4),(1,4),(2,3),(3,4)],5)
=> ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
=> ([(0,2),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6)
=> 3 = 0 + 3
[2,5,1,4,3] => ([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> 3 = 0 + 3
[2,5,3,1,4] => ([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> 3 = 0 + 3
[3,1,4,2,5] => ([(1,4),(2,3),(3,4)],5)
=> ([(1,5),(2,4),(3,4),(3,5)],6)
=> ([(1,2),(1,5),(2,4),(3,4),(3,5),(4,5)],6)
=> 3 = 0 + 3
[3,1,4,5,2] => ([(0,4),(1,4),(2,3),(3,4)],5)
=> ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
=> ([(0,2),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6)
=> 3 = 0 + 3
[3,1,5,2,4] => ([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,6),(1,5),(2,3),(2,4),(3,5),(4,6)],7)
=> ([(0,1),(0,3),(0,4),(0,6),(1,2),(1,4),(1,5),(2,3),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 3 = 0 + 3
[3,1,5,4,2] => ([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> 3 = 0 + 3
[3,2,5,1,4] => ([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> 3 = 0 + 3
[4,1,2,5,3] => ([(0,4),(1,4),(2,3),(3,4)],5)
=> ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
=> ([(0,2),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6)
=> 3 = 0 + 3
[4,1,3,5,2] => ([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> 3 = 0 + 3
[4,2,1,5,3] => ([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> 3 = 0 + 3
[1,2,4,6,3,5] => ([(2,5),(3,4),(4,5)],6)
=> ([(2,6),(3,5),(4,5),(4,6)],7)
=> ([(2,3),(2,6),(3,5),(4,5),(4,6),(5,6)],7)
=> 3 = 0 + 3
[1,2,5,3,6,4] => ([(2,5),(3,4),(4,5)],6)
=> ([(2,6),(3,5),(4,5),(4,6)],7)
=> ([(2,3),(2,6),(3,5),(4,5),(4,6),(5,6)],7)
=> 3 = 0 + 3
[1,3,4,6,2,5] => ([(1,5),(2,5),(3,4),(4,5)],6)
=> ([(1,6),(2,6),(3,4),(4,5),(5,6)],7)
=> ([(1,3),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(4,5),(4,6),(5,6)],7)
=> 3 = 0 + 3
[1,3,5,2,4,6] => ([(2,5),(3,4),(4,5)],6)
=> ([(2,6),(3,5),(4,5),(4,6)],7)
=> ([(2,3),(2,6),(3,5),(4,5),(4,6),(5,6)],7)
=> 3 = 0 + 3
[1,3,6,2,4,5] => ([(1,5),(2,5),(3,4),(4,5)],6)
=> ([(1,6),(2,6),(3,4),(4,5),(5,6)],7)
=> ([(1,3),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(4,5),(4,6),(5,6)],7)
=> 3 = 0 + 3
[1,3,6,2,5,4] => ([(1,4),(2,3),(2,5),(3,5),(4,5)],6)
=> ([(1,5),(2,4),(3,4),(3,5)],6)
=> ([(1,2),(1,5),(2,4),(3,4),(3,5),(4,5)],6)
=> 3 = 0 + 3
[1,3,6,4,2,5] => ([(1,5),(2,4),(3,4),(3,5),(4,5)],6)
=> ([(1,5),(2,4),(3,4),(3,5)],6)
=> ([(1,2),(1,5),(2,4),(3,4),(3,5),(4,5)],6)
=> 3 = 0 + 3
[1,4,2,5,3,6] => ([(2,5),(3,4),(4,5)],6)
=> ([(2,6),(3,5),(4,5),(4,6)],7)
=> ([(2,3),(2,6),(3,5),(4,5),(4,6),(5,6)],7)
=> 3 = 0 + 3
[1,4,2,5,6,3] => ([(1,5),(2,5),(3,4),(4,5)],6)
=> ([(1,6),(2,6),(3,4),(4,5),(5,6)],7)
=> ([(1,3),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(4,5),(4,6),(5,6)],7)
=> 3 = 0 + 3
[1,4,2,6,5,3] => ([(1,4),(2,3),(2,5),(3,5),(4,5)],6)
=> ([(1,5),(2,4),(3,4),(3,5)],6)
=> ([(1,2),(1,5),(2,4),(3,4),(3,5),(4,5)],6)
=> 3 = 0 + 3
[1,4,3,6,2,5] => ([(1,4),(2,3),(2,5),(3,5),(4,5)],6)
=> ([(1,5),(2,4),(3,4),(3,5)],6)
=> ([(1,2),(1,5),(2,4),(3,4),(3,5),(4,5)],6)
=> 3 = 0 + 3
[1,5,2,3,6,4] => ([(1,5),(2,5),(3,4),(4,5)],6)
=> ([(1,6),(2,6),(3,4),(4,5),(5,6)],7)
=> ([(1,3),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(4,5),(4,6),(5,6)],7)
=> 3 = 0 + 3
[1,5,2,4,6,3] => ([(1,5),(2,4),(3,4),(3,5),(4,5)],6)
=> ([(1,5),(2,4),(3,4),(3,5)],6)
=> ([(1,2),(1,5),(2,4),(3,4),(3,5),(4,5)],6)
=> 3 = 0 + 3
[1,5,3,2,6,4] => ([(1,4),(2,3),(2,5),(3,5),(4,5)],6)
=> ([(1,5),(2,4),(3,4),(3,5)],6)
=> ([(1,2),(1,5),(2,4),(3,4),(3,5),(4,5)],6)
=> 3 = 0 + 3
[2,1,4,6,3,5] => ([(0,1),(2,5),(3,4),(4,5)],6)
=> ([(1,5),(2,4),(3,4),(3,5)],6)
=> ([(1,2),(1,5),(2,4),(3,4),(3,5),(4,5)],6)
=> 3 = 0 + 3
[2,1,5,3,6,4] => ([(0,1),(2,5),(3,4),(4,5)],6)
=> ([(1,5),(2,4),(3,4),(3,5)],6)
=> ([(1,2),(1,5),(2,4),(3,4),(3,5),(4,5)],6)
=> 3 = 0 + 3
[2,3,4,6,1,5] => ([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
=> ([(0,6),(1,6),(2,6),(3,4),(4,5),(5,6)],7)
=> ([(0,2),(0,6),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 3 = 0 + 3
[2,3,5,1,4,6] => ([(1,5),(2,5),(3,4),(4,5)],6)
=> ([(1,6),(2,6),(3,4),(4,5),(5,6)],7)
=> ([(1,3),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(4,5),(4,6),(5,6)],7)
=> 3 = 0 + 3
[2,3,6,1,4,5] => ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> ([(0,6),(1,6),(2,5),(3,5),(4,5),(4,6)],7)
=> ([(0,1),(0,5),(0,6),(1,3),(1,4),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 3 = 0 + 3
[2,3,6,1,5,4] => ([(0,4),(1,4),(2,3),(2,5),(3,5),(4,5)],6)
=> ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
=> ([(0,2),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6)
=> 3 = 0 + 3
[2,3,6,4,1,5] => ([(0,5),(1,5),(2,4),(3,4),(3,5),(4,5)],6)
=> ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
=> ([(0,2),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6)
=> 3 = 0 + 3
[2,4,1,3,5,6] => ([(2,5),(3,4),(4,5)],6)
=> ([(2,6),(3,5),(4,5),(4,6)],7)
=> ([(2,3),(2,6),(3,5),(4,5),(4,6),(5,6)],7)
=> 3 = 0 + 3
[2,4,1,3,6,5] => ([(0,1),(2,5),(3,4),(4,5)],6)
=> ([(1,5),(2,4),(3,4),(3,5)],6)
=> ([(1,2),(1,5),(2,4),(3,4),(3,5),(4,5)],6)
=> 3 = 0 + 3
[2,4,1,6,5,3] => ([(0,4),(1,2),(1,5),(2,5),(3,4),(3,5)],6)
=> ([(0,6),(1,5),(2,3),(2,4),(3,5),(4,6)],7)
=> ([(0,1),(0,3),(0,4),(0,6),(1,2),(1,4),(1,5),(2,3),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 3 = 0 + 3
[2,4,3,6,1,5] => ([(0,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)
=> ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
=> ([(0,2),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6)
=> 3 = 0 + 3
[2,4,5,1,6,3] => ([(0,5),(1,4),(2,4),(2,5),(3,4),(3,5)],6)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> 3 = 0 + 3
[2,4,6,1,3,5] => ([(0,5),(1,4),(2,3),(2,4),(3,5),(4,5)],6)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> 3 = 0 + 3
[2,4,6,3,1,5] => ([(0,5),(1,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> 3 = 0 + 3
[2,5,1,3,4,6] => ([(1,5),(2,5),(3,4),(4,5)],6)
=> ([(1,6),(2,6),(3,4),(4,5),(5,6)],7)
=> ([(1,3),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(4,5),(4,6),(5,6)],7)
=> 3 = 0 + 3
[2,5,1,4,3,6] => ([(1,4),(2,3),(2,5),(3,5),(4,5)],6)
=> ([(1,5),(2,4),(3,4),(3,5)],6)
=> ([(1,2),(1,5),(2,4),(3,4),(3,5),(4,5)],6)
=> 3 = 0 + 3
[2,5,1,4,6,3] => ([(0,4),(1,2),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,6),(1,5),(2,3),(2,4),(3,5),(4,6)],7)
=> ([(0,1),(0,3),(0,4),(0,6),(1,2),(1,4),(1,5),(2,3),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 3 = 0 + 3
[2,5,1,6,3,4] => ([(0,4),(1,2),(1,3),(2,5),(3,5),(4,5)],6)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> 3 = 0 + 3
Description
The girth of a graph, which is not a tree.
This is the length of the shortest cycle in the graph.
Matching statistic: St001604
Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
Mp00060: Permutations —Robinson-Schensted tableau shape⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St001604: Integer partitions ⟶ ℤResult quality: 8% ●values known / values provided: 13%●distinct values known / distinct values provided: 8%
Mp00060: Permutations —Robinson-Schensted tableau shape⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St001604: Integer partitions ⟶ ℤResult quality: 8% ●values known / values provided: 13%●distinct values known / distinct values provided: 8%
Values
[1] => [1] => [1]
=> []
=> ? = 1
[1,2] => [1,2] => [2]
=> []
=> ? = 1
[2,1] => [1,2] => [2]
=> []
=> ? = 1
[1,2,3] => [1,2,3] => [3]
=> []
=> ? = 1
[1,3,2] => [1,2,3] => [3]
=> []
=> ? = 1
[2,1,3] => [1,2,3] => [3]
=> []
=> ? = 1
[2,3,1] => [1,2,3] => [3]
=> []
=> ? = 0
[3,1,2] => [1,3,2] => [2,1]
=> [1]
=> ? = 0
[3,2,1] => [1,3,2] => [2,1]
=> [1]
=> ? = 3
[1,2,3,4] => [1,2,3,4] => [4]
=> []
=> ? = 1
[1,2,4,3] => [1,2,3,4] => [4]
=> []
=> ? = 1
[1,3,2,4] => [1,2,3,4] => [4]
=> []
=> ? = 1
[1,3,4,2] => [1,2,3,4] => [4]
=> []
=> ? = 0
[1,4,2,3] => [1,2,4,3] => [3,1]
=> [1]
=> ? = 0
[1,4,3,2] => [1,2,4,3] => [3,1]
=> [1]
=> ? = 3
[2,1,3,4] => [1,2,3,4] => [4]
=> []
=> ? = 1
[2,1,4,3] => [1,2,3,4] => [4]
=> []
=> ? = 1
[2,3,1,4] => [1,2,3,4] => [4]
=> []
=> ? = 0
[2,3,4,1] => [1,2,3,4] => [4]
=> []
=> ? = 0
[2,4,1,3] => [1,2,4,3] => [3,1]
=> [1]
=> ? = 0
[2,4,3,1] => [1,2,4,3] => [3,1]
=> [1]
=> ? = 0
[3,1,2,4] => [1,3,2,4] => [3,1]
=> [1]
=> ? = 0
[3,1,4,2] => [1,3,4,2] => [3,1]
=> [1]
=> ? = 0
[3,2,1,4] => [1,3,2,4] => [3,1]
=> [1]
=> ? = 3
[3,2,4,1] => [1,3,4,2] => [3,1]
=> [1]
=> ? = 0
[3,4,1,2] => [1,3,2,4] => [3,1]
=> [1]
=> ? = 1
[3,4,2,1] => [1,3,2,4] => [3,1]
=> [1]
=> ? = 0
[4,1,2,3] => [1,4,3,2] => [2,1,1]
=> [1,1]
=> ? = 0
[4,1,3,2] => [1,4,2,3] => [3,1]
=> [1]
=> ? = 0
[4,2,1,3] => [1,4,3,2] => [2,1,1]
=> [1,1]
=> ? = 0
[4,2,3,1] => [1,4,2,3] => [3,1]
=> [1]
=> ? = 5
[4,3,1,2] => [1,4,2,3] => [3,1]
=> [1]
=> ? = 0
[4,3,2,1] => [1,4,2,3] => [3,1]
=> [1]
=> ? = 7
[1,2,3,4,5] => [1,2,3,4,5] => [5]
=> []
=> ? = 1
[1,2,3,5,4] => [1,2,3,4,5] => [5]
=> []
=> ? = 1
[1,2,4,3,5] => [1,2,3,4,5] => [5]
=> []
=> ? = 1
[1,2,4,5,3] => [1,2,3,4,5] => [5]
=> []
=> ? = 0
[1,2,5,3,4] => [1,2,3,5,4] => [4,1]
=> [1]
=> ? = 0
[1,2,5,4,3] => [1,2,3,5,4] => [4,1]
=> [1]
=> ? = 3
[1,3,2,4,5] => [1,2,3,4,5] => [5]
=> []
=> ? = 1
[1,3,2,5,4] => [1,2,3,4,5] => [5]
=> []
=> ? = 1
[1,3,4,2,5] => [1,2,3,4,5] => [5]
=> []
=> ? = 0
[1,3,4,5,2] => [1,2,3,4,5] => [5]
=> []
=> ? = 0
[1,3,5,2,4] => [1,2,3,5,4] => [4,1]
=> [1]
=> ? = 0
[1,3,5,4,2] => [1,2,3,5,4] => [4,1]
=> [1]
=> ? = 0
[1,4,2,3,5] => [1,2,4,3,5] => [4,1]
=> [1]
=> ? = 0
[1,4,2,5,3] => [1,2,4,5,3] => [4,1]
=> [1]
=> ? = 0
[1,4,3,2,5] => [1,2,4,3,5] => [4,1]
=> [1]
=> ? = 3
[1,4,3,5,2] => [1,2,4,5,3] => [4,1]
=> [1]
=> ? = 0
[1,4,5,2,3] => [1,2,4,3,5] => [4,1]
=> [1]
=> ? = 1
[5,1,2,3,4] => [1,5,4,3,2] => [2,1,1,1]
=> [1,1,1]
=> 0
[5,2,1,3,4] => [1,5,4,3,2] => [2,1,1,1]
=> [1,1,1]
=> 0
[1,6,2,3,4,5] => [1,2,6,5,4,3] => [3,1,1,1]
=> [1,1,1]
=> 0
[1,6,3,2,4,5] => [1,2,6,5,4,3] => [3,1,1,1]
=> [1,1,1]
=> 0
[2,6,1,3,4,5] => [1,2,6,5,4,3] => [3,1,1,1]
=> [1,1,1]
=> 0
[2,6,3,1,4,5] => [1,2,6,5,4,3] => [3,1,1,1]
=> [1,1,1]
=> 0
[3,1,6,2,4,5] => [1,3,6,5,4,2] => [3,1,1,1]
=> [1,1,1]
=> 0
[3,1,6,4,2,5] => [1,3,6,5,2,4] => [3,2,1]
=> [2,1]
=> 0
[3,2,6,1,4,5] => [1,3,6,5,4,2] => [3,1,1,1]
=> [1,1,1]
=> 0
[3,2,6,4,1,5] => [1,3,6,5,2,4] => [3,2,1]
=> [2,1]
=> 0
[3,4,6,1,2,5] => [1,3,6,5,2,4] => [3,2,1]
=> [2,1]
=> 0
[3,4,6,2,1,5] => [1,3,6,5,2,4] => [3,2,1]
=> [2,1]
=> 0
[3,5,6,1,4,2] => [1,3,6,2,5,4] => [3,2,1]
=> [2,1]
=> 0
[3,5,6,2,4,1] => [1,3,6,2,5,4] => [3,2,1]
=> [2,1]
=> 0
[3,5,6,4,1,2] => [1,3,6,2,5,4] => [3,2,1]
=> [2,1]
=> 0
[3,5,6,4,2,1] => [1,3,6,2,5,4] => [3,2,1]
=> [2,1]
=> 0
[3,6,1,2,4,5] => [1,3,2,6,5,4] => [3,2,1]
=> [2,1]
=> 0
[3,6,1,4,2,5] => [1,3,2,6,5,4] => [3,2,1]
=> [2,1]
=> 0
[3,6,2,1,4,5] => [1,3,2,6,5,4] => [3,2,1]
=> [2,1]
=> 0
[3,6,2,4,1,5] => [1,3,2,6,5,4] => [3,2,1]
=> [2,1]
=> 0
[4,1,2,6,3,5] => [1,4,6,5,3,2] => [3,1,1,1]
=> [1,1,1]
=> 0
[4,1,2,6,5,3] => [1,4,6,3,2,5] => [3,2,1]
=> [2,1]
=> 0
[4,1,3,6,2,5] => [1,4,6,5,2,3] => [3,2,1]
=> [2,1]
=> 0
[4,1,5,6,2,3] => [1,4,6,3,5,2] => [3,2,1]
=> [2,1]
=> 0
[4,1,6,3,2,5] => [1,4,3,6,5,2] => [3,2,1]
=> [2,1]
=> 0
[4,1,6,3,5,2] => [1,4,3,6,2,5] => [3,2,1]
=> [2,1]
=> 0
[4,2,1,6,3,5] => [1,4,6,5,3,2] => [3,1,1,1]
=> [1,1,1]
=> 0
[4,2,1,6,5,3] => [1,4,6,3,2,5] => [3,2,1]
=> [2,1]
=> 0
[4,2,3,6,1,5] => [1,4,6,5,2,3] => [3,2,1]
=> [2,1]
=> 0
[4,2,5,6,1,3] => [1,4,6,3,5,2] => [3,2,1]
=> [2,1]
=> 0
[4,2,6,3,1,5] => [1,4,3,6,5,2] => [3,2,1]
=> [2,1]
=> 0
[4,2,6,3,5,1] => [1,4,3,6,2,5] => [3,2,1]
=> [2,1]
=> 0
[4,3,1,6,2,5] => [1,4,6,5,2,3] => [3,2,1]
=> [2,1]
=> 0
[4,3,2,6,1,5] => [1,4,6,5,2,3] => [3,2,1]
=> [2,1]
=> 0
[4,5,1,6,2,3] => [1,4,6,3,2,5] => [3,2,1]
=> [2,1]
=> 0
[4,5,1,6,3,2] => [1,4,6,2,5,3] => [3,2,1]
=> [2,1]
=> 0
[4,5,2,6,1,3] => [1,4,6,3,2,5] => [3,2,1]
=> [2,1]
=> 0
[4,5,2,6,3,1] => [1,4,6,2,5,3] => [3,2,1]
=> [2,1]
=> 0
[4,5,3,6,1,2] => [1,4,6,2,5,3] => [3,2,1]
=> [2,1]
=> 0
[4,5,3,6,2,1] => [1,4,6,2,5,3] => [3,2,1]
=> [2,1]
=> 0
[4,5,6,3,1,2] => [1,4,3,6,2,5] => [3,2,1]
=> [2,1]
=> 0
[4,5,6,3,2,1] => [1,4,3,6,2,5] => [3,2,1]
=> [2,1]
=> 0
[4,6,1,2,3,5] => [1,4,2,6,5,3] => [3,2,1]
=> [2,1]
=> 0
[4,6,1,3,2,5] => [1,4,3,2,6,5] => [3,2,1]
=> [2,1]
=> 0
[4,6,1,3,5,2] => [1,4,3,2,6,5] => [3,2,1]
=> [2,1]
=> 0
[4,6,2,1,3,5] => [1,4,2,6,5,3] => [3,2,1]
=> [2,1]
=> 0
[4,6,2,3,1,5] => [1,4,3,2,6,5] => [3,2,1]
=> [2,1]
=> 0
[4,6,2,3,5,1] => [1,4,3,2,6,5] => [3,2,1]
=> [2,1]
=> 0
[4,6,3,1,2,5] => [1,4,2,6,5,3] => [3,2,1]
=> [2,1]
=> 0
[4,6,3,2,1,5] => [1,4,2,6,5,3] => [3,2,1]
=> [2,1]
=> 0
Description
The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on polygons.
Equivalently, this is the multiplicity of the irreducible representation corresponding to a partition in the cycle index of the dihedral group.
This statistic is only defined for partitions of size at least 3, to avoid ambiguity.
Matching statistic: St001851
(load all 8 compositions to match this statistic)
(load all 8 compositions to match this statistic)
Mp00170: Permutations —to signed permutation⟶ Signed permutations
St001851: Signed permutations ⟶ ℤResult quality: 5% ●values known / values provided: 5%●distinct values known / distinct values provided: 38%
St001851: Signed permutations ⟶ ℤResult quality: 5% ●values known / values provided: 5%●distinct values known / distinct values provided: 38%
Values
[1] => [1] => 1
[1,2] => [1,2] => 1
[2,1] => [2,1] => 1
[1,2,3] => [1,2,3] => 1
[1,3,2] => [1,3,2] => 1
[2,1,3] => [2,1,3] => 1
[2,3,1] => [2,3,1] => 0
[3,1,2] => [3,1,2] => 0
[3,2,1] => [3,2,1] => 3
[1,2,3,4] => [1,2,3,4] => 1
[1,2,4,3] => [1,2,4,3] => 1
[1,3,2,4] => [1,3,2,4] => 1
[1,3,4,2] => [1,3,4,2] => 0
[1,4,2,3] => [1,4,2,3] => 0
[1,4,3,2] => [1,4,3,2] => 3
[2,1,3,4] => [2,1,3,4] => 1
[2,1,4,3] => [2,1,4,3] => 1
[2,3,1,4] => [2,3,1,4] => 0
[2,3,4,1] => [2,3,4,1] => 0
[2,4,1,3] => [2,4,1,3] => 0
[2,4,3,1] => [2,4,3,1] => 0
[3,1,2,4] => [3,1,2,4] => 0
[3,1,4,2] => [3,1,4,2] => 0
[3,2,1,4] => [3,2,1,4] => 3
[3,2,4,1] => [3,2,4,1] => 0
[3,4,1,2] => [3,4,1,2] => 1
[3,4,2,1] => [3,4,2,1] => 0
[4,1,2,3] => [4,1,2,3] => 0
[4,1,3,2] => [4,1,3,2] => 0
[4,2,1,3] => [4,2,1,3] => 0
[4,2,3,1] => [4,2,3,1] => 5
[4,3,1,2] => [4,3,1,2] => 0
[4,3,2,1] => [4,3,2,1] => 7
[1,2,3,4,5] => [1,2,3,4,5] => 1
[1,2,3,5,4] => [1,2,3,5,4] => 1
[1,2,4,3,5] => [1,2,4,3,5] => 1
[1,2,4,5,3] => [1,2,4,5,3] => 0
[1,2,5,3,4] => [1,2,5,3,4] => 0
[1,2,5,4,3] => [1,2,5,4,3] => 3
[1,3,2,4,5] => [1,3,2,4,5] => 1
[1,3,2,5,4] => [1,3,2,5,4] => 1
[1,3,4,2,5] => [1,3,4,2,5] => 0
[1,3,4,5,2] => [1,3,4,5,2] => 0
[1,3,5,2,4] => [1,3,5,2,4] => 0
[1,3,5,4,2] => [1,3,5,4,2] => 0
[1,4,2,3,5] => [1,4,2,3,5] => 0
[1,4,2,5,3] => [1,4,2,5,3] => 0
[1,4,3,2,5] => [1,4,3,2,5] => 3
[1,4,3,5,2] => [1,4,3,5,2] => 0
[1,4,5,2,3] => [1,4,5,2,3] => 1
[2,1,3,4,5] => [2,1,3,4,5] => ? = 1
[2,1,3,5,4] => [2,1,3,5,4] => ? = 1
[2,1,4,3,5] => [2,1,4,3,5] => ? = 1
[2,1,4,5,3] => [2,1,4,5,3] => ? = 0
[2,1,5,3,4] => [2,1,5,3,4] => ? = 0
[2,1,5,4,3] => [2,1,5,4,3] => ? = 3
[2,3,1,4,5] => [2,3,1,4,5] => ? = 0
[2,3,1,5,4] => [2,3,1,5,4] => ? = 0
[2,3,4,1,5] => [2,3,4,1,5] => ? = 0
[2,3,4,5,1] => [2,3,4,5,1] => ? = 0
[2,3,5,1,4] => [2,3,5,1,4] => ? = 0
[2,3,5,4,1] => [2,3,5,4,1] => ? = 0
[2,4,1,3,5] => [2,4,1,3,5] => ? = 0
[2,4,1,5,3] => [2,4,1,5,3] => ? = 0
[2,4,3,1,5] => [2,4,3,1,5] => ? = 0
[2,4,3,5,1] => [2,4,3,5,1] => ? = 0
[2,4,5,1,3] => [2,4,5,1,3] => ? = 0
[2,4,5,3,1] => [2,4,5,3,1] => ? = 0
[2,5,1,3,4] => [2,5,1,3,4] => ? = 0
[2,5,1,4,3] => [2,5,1,4,3] => ? = 0
[2,5,3,1,4] => [2,5,3,1,4] => ? = 0
[2,5,3,4,1] => [2,5,3,4,1] => ? = 0
[2,5,4,1,3] => [2,5,4,1,3] => ? = 0
[2,5,4,3,1] => [2,5,4,3,1] => ? = 0
[3,1,2,4,5] => [3,1,2,4,5] => ? = 0
[3,1,2,5,4] => [3,1,2,5,4] => ? = 0
[3,1,4,2,5] => [3,1,4,2,5] => ? = 0
[3,1,4,5,2] => [3,1,4,5,2] => ? = 0
[3,1,5,2,4] => [3,1,5,2,4] => ? = 0
[3,1,5,4,2] => [3,1,5,4,2] => ? = 0
[3,2,1,4,5] => [3,2,1,4,5] => ? = 3
[3,2,1,5,4] => [3,2,1,5,4] => ? = 3
[3,2,4,1,5] => [3,2,4,1,5] => ? = 0
[3,2,4,5,1] => [3,2,4,5,1] => ? = 0
[3,2,5,1,4] => [3,2,5,1,4] => ? = 0
[3,2,5,4,1] => [3,2,5,4,1] => ? = 0
[3,4,1,2,5] => [3,4,1,2,5] => ? = 1
[3,4,1,5,2] => [3,4,1,5,2] => ? = 0
[3,4,2,1,5] => [3,4,2,1,5] => ? = 0
[3,4,2,5,1] => [3,4,2,5,1] => ? = 0
[3,4,5,1,2] => [3,4,5,1,2] => ? = 0
[3,4,5,2,1] => [3,4,5,2,1] => ? = 0
[3,5,1,2,4] => [3,5,1,2,4] => ? = 0
[3,5,1,4,2] => [3,5,1,4,2] => ? = 3
[3,5,2,1,4] => [3,5,2,1,4] => ? = 0
[3,5,2,4,1] => [3,5,2,4,1] => ? = 0
[3,5,4,1,2] => [3,5,4,1,2] => ? = 0
[3,5,4,2,1] => [3,5,4,2,1] => ? = 0
[4,1,2,3,5] => [4,1,2,3,5] => ? = 0
[4,1,2,5,3] => [4,1,2,5,3] => ? = 0
Description
The number of Hecke atoms of a signed permutation.
For a signed permutation $z\in\mathfrak H_n$, this is the cardinality of the set
$$
\{ w\in\mathfrak H_n | w^{-1} \star w = z\},
$$
where $\star$ denotes the Demazure product. Note that $w\mapsto w^{-1}\star w$ is a surjection onto the set of involutions.
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