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Your data matches 4 different statistics following compositions of up to 3 maps.
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Matching statistic: St000762
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St000762: Integer compositions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,1] => 3
[2] => 1
[1,1,1] => 6
[1,2] => 3
[2,1] => 1
[3] => 1
[1,1,1,1] => 10
[1,1,2] => 6
[1,2,1] => 3
[1,3] => 3
[2,1,1] => 1
[2,2] => 3
[3,1] => 1
[4] => 1
[1,1,1,1,1] => 15
[1,1,1,2] => 10
[1,1,2,1] => 6
[1,1,3] => 6
[1,2,1,1] => 3
[1,2,2] => 6
[1,3,1] => 3
[1,4] => 3
[2,1,1,1] => 1
[2,1,2] => 4
[2,2,1] => 3
[2,3] => 3
[3,1,1] => 1
[3,2] => 1
[4,1] => 1
[5] => 1
[1,1,1,1,1,1] => 21
[1,1,1,1,2] => 15
[1,1,1,2,1] => 10
[1,1,1,3] => 10
[1,1,2,1,1] => 6
[1,1,2,2] => 10
[1,1,3,1] => 6
[1,1,4] => 6
[1,2,1,1,1] => 3
[1,2,1,2] => 7
[1,2,2,1] => 6
[1,2,3] => 6
[1,3,1,1] => 3
[1,3,2] => 3
[1,4,1] => 3
[1,5] => 3
[2,1,1,1,1] => 1
[2,1,1,2] => 5
[2,1,2,1] => 4
[2,1,3] => 4
Description
The sum of the positions of the weak records of an integer composition.
A weak record is an element $a_i$ such that $a_i \geq a_j$ for all $j < i$. This statistic is the sum of their positions.
Matching statistic: St000456
Values
[1,1] => ([(0,1)],2)
=> ([],1)
=> ? ∊ {1,3}
[2] => ([],2)
=> ([],2)
=> ? ∊ {1,3}
[1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> ([],1)
=> ? ∊ {1,3,6}
[1,2] => ([(1,2)],3)
=> ([],2)
=> ? ∊ {1,3,6}
[2,1] => ([(0,2),(1,2)],3)
=> ([(0,1)],2)
=> 1
[3] => ([],3)
=> ([],3)
=> ? ∊ {1,3,6}
[1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([],1)
=> ? ∊ {1,3,3,6,10}
[1,1,2] => ([(1,2),(1,3),(2,3)],4)
=> ([],2)
=> ? ∊ {1,3,3,6,10}
[1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1)],2)
=> 1
[1,3] => ([(2,3)],4)
=> ([],3)
=> ? ∊ {1,3,3,6,10}
[2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1)],2)
=> 1
[2,2] => ([(1,3),(2,3)],4)
=> ([(1,2)],3)
=> ? ∊ {1,3,3,6,10}
[3,1] => ([(0,3),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,2)],3)
=> 3
[4] => ([],4)
=> ([],4)
=> ? ∊ {1,3,3,6,10}
[1,1,1,1,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([],1)
=> ? ∊ {1,1,3,3,4,6,6,10,15}
[1,1,1,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([],2)
=> ? ∊ {1,1,3,3,4,6,6,10,15}
[1,1,2,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1)],2)
=> 1
[1,1,3] => ([(2,3),(2,4),(3,4)],5)
=> ([],3)
=> ? ∊ {1,1,3,3,4,6,6,10,15}
[1,2,1,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1)],2)
=> 1
[1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,2)],3)
=> ? ∊ {1,1,3,3,4,6,6,10,15}
[1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 3
[1,4] => ([(3,4)],5)
=> ([],4)
=> ? ∊ {1,1,3,3,4,6,6,10,15}
[2,1,1,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1)],2)
=> 1
[2,1,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,2)],3)
=> ? ∊ {1,1,3,3,4,6,6,10,15}
[2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 3
[2,3] => ([(2,4),(3,4)],5)
=> ([(2,3)],4)
=> ? ∊ {1,1,3,3,4,6,6,10,15}
[3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 3
[3,2] => ([(1,4),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(2,3)],4)
=> ? ∊ {1,1,3,3,4,6,6,10,15}
[4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 6
[5] => ([],5)
=> ([],5)
=> ? ∊ {1,1,3,3,4,6,6,10,15}
[1,1,1,1,1,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([],1)
=> ? ∊ {1,1,1,1,3,3,3,4,4,5,6,6,7,10,10,15,21}
[1,1,1,1,2] => ([(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([],2)
=> ? ∊ {1,1,1,1,3,3,3,4,4,5,6,6,7,10,10,15,21}
[1,1,1,2,1] => ([(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1)],2)
=> 1
[1,1,1,3] => ([(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([],3)
=> ? ∊ {1,1,1,1,3,3,3,4,4,5,6,6,7,10,10,15,21}
[1,1,2,1,1] => ([(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1)],2)
=> 1
[1,1,2,2] => ([(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(1,2)],3)
=> ? ∊ {1,1,1,1,3,3,3,4,4,5,6,6,7,10,10,15,21}
[1,1,3,1] => ([(0,5),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> 3
[1,1,4] => ([(3,4),(3,5),(4,5)],6)
=> ([],4)
=> ? ∊ {1,1,1,1,3,3,3,4,4,5,6,6,7,10,10,15,21}
[1,2,1,1,1] => ([(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1)],2)
=> 1
[1,2,1,2] => ([(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(1,2)],3)
=> ? ∊ {1,1,1,1,3,3,3,4,4,5,6,6,7,10,10,15,21}
[1,2,2,1] => ([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> 3
[1,2,3] => ([(2,5),(3,4),(3,5),(4,5)],6)
=> ([(2,3)],4)
=> ? ∊ {1,1,1,1,3,3,3,4,4,5,6,6,7,10,10,15,21}
[1,3,1,1] => ([(0,4),(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> 3
[1,3,2] => ([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(1,2),(1,3),(2,3)],4)
=> ? ∊ {1,1,1,1,3,3,3,4,4,5,6,6,7,10,10,15,21}
[1,4,1] => ([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 6
[1,5] => ([(4,5)],6)
=> ([],5)
=> ? ∊ {1,1,1,1,3,3,3,4,4,5,6,6,7,10,10,15,21}
[2,1,1,1,1] => ([(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1)],2)
=> 1
[2,1,1,2] => ([(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(1,2)],3)
=> ? ∊ {1,1,1,1,3,3,3,4,4,5,6,6,7,10,10,15,21}
[2,1,2,1] => ([(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> 3
[2,1,3] => ([(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(2,3)],4)
=> ? ∊ {1,1,1,1,3,3,3,4,4,5,6,6,7,10,10,15,21}
[2,2,1,1] => ([(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> 3
[2,2,2] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(1,2),(1,3),(2,3)],4)
=> ? ∊ {1,1,1,1,3,3,3,4,4,5,6,6,7,10,10,15,21}
[2,3,1] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 6
[2,4] => ([(3,5),(4,5)],6)
=> ([(3,4)],5)
=> ? ∊ {1,1,1,1,3,3,3,4,4,5,6,6,7,10,10,15,21}
[3,1,1,1] => ([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> 3
[3,1,2] => ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(1,2),(1,3),(2,3)],4)
=> ? ∊ {1,1,1,1,3,3,3,4,4,5,6,6,7,10,10,15,21}
[3,2,1] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 6
[3,3] => ([(2,5),(3,5),(4,5)],6)
=> ([(2,3),(2,4),(3,4)],5)
=> ? ∊ {1,1,1,1,3,3,3,4,4,5,6,6,7,10,10,15,21}
[4,1,1] => ([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 6
[4,2] => ([(1,5),(2,5),(3,5),(4,5)],6)
=> ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {1,1,1,1,3,3,3,4,4,5,6,6,7,10,10,15,21}
[5,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 10
[6] => ([],6)
=> ([],6)
=> ? ∊ {1,1,1,1,3,3,3,4,4,5,6,6,7,10,10,15,21}
[1,1,1,1,1,1,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([],1)
=> ? ∊ {1,1,1,1,1,1,1,1,1,3,3,3,3,3,4,4,4,4,5,5,6,6,7,7,7,8,8,10,11,15,15,21,28}
[1,1,1,1,1,2] => ([(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([],2)
=> ? ∊ {1,1,1,1,1,1,1,1,1,3,3,3,3,3,4,4,4,4,5,5,6,6,7,7,7,8,8,10,11,15,15,21,28}
[1,1,1,1,2,1] => ([(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,1)],2)
=> 1
[1,1,1,1,3] => ([(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([],3)
=> ? ∊ {1,1,1,1,1,1,1,1,1,3,3,3,3,3,4,4,4,4,5,5,6,6,7,7,7,8,8,10,11,15,15,21,28}
[1,1,1,2,1,1] => ([(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,1)],2)
=> 1
[1,1,1,2,2] => ([(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(1,2)],3)
=> ? ∊ {1,1,1,1,1,1,1,1,1,3,3,3,3,3,4,4,4,4,5,5,6,6,7,7,7,8,8,10,11,15,15,21,28}
[1,1,1,3,1] => ([(0,6),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,1),(0,2),(1,2)],3)
=> 3
[1,1,1,4] => ([(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([],4)
=> ? ∊ {1,1,1,1,1,1,1,1,1,3,3,3,3,3,4,4,4,4,5,5,6,6,7,7,7,8,8,10,11,15,15,21,28}
[1,1,2,1,1,1] => ([(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,1)],2)
=> 1
[1,1,2,1,2] => ([(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(1,2)],3)
=> ? ∊ {1,1,1,1,1,1,1,1,1,3,3,3,3,3,4,4,4,4,5,5,6,6,7,7,7,8,8,10,11,15,15,21,28}
[1,1,2,2,1] => ([(0,6),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,1),(0,2),(1,2)],3)
=> 3
[1,1,2,3] => ([(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(2,3)],4)
=> ? ∊ {1,1,1,1,1,1,1,1,1,3,3,3,3,3,4,4,4,4,5,5,6,6,7,7,7,8,8,10,11,15,15,21,28}
[1,1,3,1,1] => ([(0,5),(0,6),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,1),(0,2),(1,2)],3)
=> 3
[1,1,3,2] => ([(1,6),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(1,2),(1,3),(2,3)],4)
=> ? ∊ {1,1,1,1,1,1,1,1,1,3,3,3,3,3,4,4,4,4,5,5,6,6,7,7,7,8,8,10,11,15,15,21,28}
[1,1,4,1] => ([(0,6),(1,6),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 6
[1,1,5] => ([(4,5),(4,6),(5,6)],7)
=> ([],5)
=> ? ∊ {1,1,1,1,1,1,1,1,1,3,3,3,3,3,4,4,4,4,5,5,6,6,7,7,7,8,8,10,11,15,15,21,28}
[1,2,1,1,1,1] => ([(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,1)],2)
=> 1
[1,2,1,1,2] => ([(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(1,2)],3)
=> ? ∊ {1,1,1,1,1,1,1,1,1,3,3,3,3,3,4,4,4,4,5,5,6,6,7,7,7,8,8,10,11,15,15,21,28}
[1,2,1,2,1] => ([(0,6),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,1),(0,2),(1,2)],3)
=> 3
[1,2,1,3] => ([(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(2,3)],4)
=> ? ∊ {1,1,1,1,1,1,1,1,1,3,3,3,3,3,4,4,4,4,5,5,6,6,7,7,7,8,8,10,11,15,15,21,28}
[1,2,2,1,1] => ([(0,5),(0,6),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,1),(0,2),(1,2)],3)
=> 3
[1,2,2,2] => ([(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(1,2),(1,3),(2,3)],4)
=> ? ∊ {1,1,1,1,1,1,1,1,1,3,3,3,3,3,4,4,4,4,5,5,6,6,7,7,7,8,8,10,11,15,15,21,28}
[1,2,3,1] => ([(0,6),(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 6
[1,2,4] => ([(3,6),(4,5),(4,6),(5,6)],7)
=> ([(3,4)],5)
=> ? ∊ {1,1,1,1,1,1,1,1,1,3,3,3,3,3,4,4,4,4,5,5,6,6,7,7,7,8,8,10,11,15,15,21,28}
[1,3,1,1,1] => ([(0,4),(0,5),(0,6),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,1),(0,2),(1,2)],3)
=> 3
[1,3,1,2] => ([(1,5),(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(1,2),(1,3),(2,3)],4)
=> ? ∊ {1,1,1,1,1,1,1,1,1,3,3,3,3,3,4,4,4,4,5,5,6,6,7,7,7,8,8,10,11,15,15,21,28}
[1,3,2,1] => ([(0,6),(1,5),(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 6
[1,4,1,1] => ([(0,5),(0,6),(1,5),(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 6
[1,5,1] => ([(0,6),(1,6),(2,6),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 10
[2,1,1,1,1,1] => ([(0,2),(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,1)],2)
=> 1
[2,1,1,2,1] => ([(0,6),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,1),(0,2),(1,2)],3)
=> 3
[2,1,2,1,1] => ([(0,5),(0,6),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,1),(0,2),(1,2)],3)
=> 3
[2,1,3,1] => ([(0,6),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 6
[2,2,1,1,1] => ([(0,4),(0,5),(0,6),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,1),(0,2),(1,2)],3)
=> 3
[2,2,2,1] => ([(0,6),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 6
[2,3,1,1] => ([(0,5),(0,6),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 6
[2,4,1] => ([(0,6),(1,6),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 10
[3,1,1,1,1] => ([(0,3),(0,4),(0,5),(0,6),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,1),(0,2),(1,2)],3)
=> 3
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: St001875
Mp00133: Integer compositions —delta morphism⟶ Integer compositions
Mp00180: Integer compositions —to ribbon⟶ Skew partitions
Mp00192: Skew partitions —dominating sublattice⟶ Lattices
St001875: Lattices ⟶ ℤResult quality: 9% ●values known / values provided: 9%●distinct values known / distinct values provided: 14%
Mp00180: Integer compositions —to ribbon⟶ Skew partitions
Mp00192: Skew partitions —dominating sublattice⟶ Lattices
St001875: Lattices ⟶ ℤResult quality: 9% ●values known / values provided: 9%●distinct values known / distinct values provided: 14%
Values
[1,1] => [2] => [[2],[]]
=> ([],1)
=> ? ∊ {1,3}
[2] => [1] => [[1],[]]
=> ([],1)
=> ? ∊ {1,3}
[1,1,1] => [3] => [[3],[]]
=> ([],1)
=> ? ∊ {1,1,3,6}
[1,2] => [1,1] => [[1,1],[]]
=> ([],1)
=> ? ∊ {1,1,3,6}
[2,1] => [1,1] => [[1,1],[]]
=> ([],1)
=> ? ∊ {1,1,3,6}
[3] => [1] => [[1],[]]
=> ([],1)
=> ? ∊ {1,1,3,6}
[1,1,1,1] => [4] => [[4],[]]
=> ([],1)
=> ? ∊ {1,1,1,3,3,3,6,10}
[1,1,2] => [2,1] => [[2,2],[1]]
=> ([],1)
=> ? ∊ {1,1,1,3,3,3,6,10}
[1,2,1] => [1,1,1] => [[1,1,1],[]]
=> ([],1)
=> ? ∊ {1,1,1,3,3,3,6,10}
[1,3] => [1,1] => [[1,1],[]]
=> ([],1)
=> ? ∊ {1,1,1,3,3,3,6,10}
[2,1,1] => [1,2] => [[2,1],[]]
=> ([],1)
=> ? ∊ {1,1,1,3,3,3,6,10}
[2,2] => [2] => [[2],[]]
=> ([],1)
=> ? ∊ {1,1,1,3,3,3,6,10}
[3,1] => [1,1] => [[1,1],[]]
=> ([],1)
=> ? ∊ {1,1,1,3,3,3,6,10}
[4] => [1] => [[1],[]]
=> ([],1)
=> ? ∊ {1,1,1,3,3,3,6,10}
[1,1,1,1,1] => [5] => [[5],[]]
=> ([],1)
=> ? ∊ {1,1,1,1,1,3,3,3,3,3,4,6,6,6,10,15}
[1,1,1,2] => [3,1] => [[3,3],[2]]
=> ([],1)
=> ? ∊ {1,1,1,1,1,3,3,3,3,3,4,6,6,6,10,15}
[1,1,2,1] => [2,1,1] => [[2,2,2],[1,1]]
=> ([],1)
=> ? ∊ {1,1,1,1,1,3,3,3,3,3,4,6,6,6,10,15}
[1,1,3] => [2,1] => [[2,2],[1]]
=> ([],1)
=> ? ∊ {1,1,1,1,1,3,3,3,3,3,4,6,6,6,10,15}
[1,2,1,1] => [1,1,2] => [[2,1,1],[]]
=> ([],1)
=> ? ∊ {1,1,1,1,1,3,3,3,3,3,4,6,6,6,10,15}
[1,2,2] => [1,2] => [[2,1],[]]
=> ([],1)
=> ? ∊ {1,1,1,1,1,3,3,3,3,3,4,6,6,6,10,15}
[1,3,1] => [1,1,1] => [[1,1,1],[]]
=> ([],1)
=> ? ∊ {1,1,1,1,1,3,3,3,3,3,4,6,6,6,10,15}
[1,4] => [1,1] => [[1,1],[]]
=> ([],1)
=> ? ∊ {1,1,1,1,1,3,3,3,3,3,4,6,6,6,10,15}
[2,1,1,1] => [1,3] => [[3,1],[]]
=> ([],1)
=> ? ∊ {1,1,1,1,1,3,3,3,3,3,4,6,6,6,10,15}
[2,1,2] => [1,1,1] => [[1,1,1],[]]
=> ([],1)
=> ? ∊ {1,1,1,1,1,3,3,3,3,3,4,6,6,6,10,15}
[2,2,1] => [2,1] => [[2,2],[1]]
=> ([],1)
=> ? ∊ {1,1,1,1,1,3,3,3,3,3,4,6,6,6,10,15}
[2,3] => [1,1] => [[1,1],[]]
=> ([],1)
=> ? ∊ {1,1,1,1,1,3,3,3,3,3,4,6,6,6,10,15}
[3,1,1] => [1,2] => [[2,1],[]]
=> ([],1)
=> ? ∊ {1,1,1,1,1,3,3,3,3,3,4,6,6,6,10,15}
[3,2] => [1,1] => [[1,1],[]]
=> ([],1)
=> ? ∊ {1,1,1,1,1,3,3,3,3,3,4,6,6,6,10,15}
[4,1] => [1,1] => [[1,1],[]]
=> ([],1)
=> ? ∊ {1,1,1,1,1,3,3,3,3,3,4,6,6,6,10,15}
[5] => [1] => [[1],[]]
=> ([],1)
=> ? ∊ {1,1,1,1,1,3,3,3,3,3,4,6,6,6,10,15}
[1,1,1,1,1,1] => [6] => [[6],[]]
=> ([],1)
=> ? ∊ {1,1,1,1,1,1,1,1,3,3,3,3,3,3,3,3,3,4,4,5,6,6,6,6,6,6,7,10,10,10,15,21}
[1,1,1,1,2] => [4,1] => [[4,4],[3]]
=> ([],1)
=> ? ∊ {1,1,1,1,1,1,1,1,3,3,3,3,3,3,3,3,3,4,4,5,6,6,6,6,6,6,7,10,10,10,15,21}
[1,1,1,2,1] => [3,1,1] => [[3,3,3],[2,2]]
=> ([],1)
=> ? ∊ {1,1,1,1,1,1,1,1,3,3,3,3,3,3,3,3,3,4,4,5,6,6,6,6,6,6,7,10,10,10,15,21}
[1,1,1,3] => [3,1] => [[3,3],[2]]
=> ([],1)
=> ? ∊ {1,1,1,1,1,1,1,1,3,3,3,3,3,3,3,3,3,4,4,5,6,6,6,6,6,6,7,10,10,10,15,21}
[1,1,2,1,1] => [2,1,2] => [[3,2,2],[1,1]]
=> ([(0,1)],2)
=> ? ∊ {1,1,1,1,1,1,1,1,3,3,3,3,3,3,3,3,3,4,4,5,6,6,6,6,6,6,7,10,10,10,15,21}
[1,1,2,2] => [2,2] => [[3,2],[1]]
=> ([(0,1)],2)
=> ? ∊ {1,1,1,1,1,1,1,1,3,3,3,3,3,3,3,3,3,4,4,5,6,6,6,6,6,6,7,10,10,10,15,21}
[1,1,3,1] => [2,1,1] => [[2,2,2],[1,1]]
=> ([],1)
=> ? ∊ {1,1,1,1,1,1,1,1,3,3,3,3,3,3,3,3,3,4,4,5,6,6,6,6,6,6,7,10,10,10,15,21}
[1,1,4] => [2,1] => [[2,2],[1]]
=> ([],1)
=> ? ∊ {1,1,1,1,1,1,1,1,3,3,3,3,3,3,3,3,3,4,4,5,6,6,6,6,6,6,7,10,10,10,15,21}
[1,2,1,1,1] => [1,1,3] => [[3,1,1],[]]
=> ([],1)
=> ? ∊ {1,1,1,1,1,1,1,1,3,3,3,3,3,3,3,3,3,4,4,5,6,6,6,6,6,6,7,10,10,10,15,21}
[1,2,1,2] => [1,1,1,1] => [[1,1,1,1],[]]
=> ([],1)
=> ? ∊ {1,1,1,1,1,1,1,1,3,3,3,3,3,3,3,3,3,4,4,5,6,6,6,6,6,6,7,10,10,10,15,21}
[1,2,2,1] => [1,2,1] => [[2,2,1],[1]]
=> ([(0,1)],2)
=> ? ∊ {1,1,1,1,1,1,1,1,3,3,3,3,3,3,3,3,3,4,4,5,6,6,6,6,6,6,7,10,10,10,15,21}
[1,2,3] => [1,1,1] => [[1,1,1],[]]
=> ([],1)
=> ? ∊ {1,1,1,1,1,1,1,1,3,3,3,3,3,3,3,3,3,4,4,5,6,6,6,6,6,6,7,10,10,10,15,21}
[1,3,1,1] => [1,1,2] => [[2,1,1],[]]
=> ([],1)
=> ? ∊ {1,1,1,1,1,1,1,1,3,3,3,3,3,3,3,3,3,4,4,5,6,6,6,6,6,6,7,10,10,10,15,21}
[1,3,2] => [1,1,1] => [[1,1,1],[]]
=> ([],1)
=> ? ∊ {1,1,1,1,1,1,1,1,3,3,3,3,3,3,3,3,3,4,4,5,6,6,6,6,6,6,7,10,10,10,15,21}
[1,4,1] => [1,1,1] => [[1,1,1],[]]
=> ([],1)
=> ? ∊ {1,1,1,1,1,1,1,1,3,3,3,3,3,3,3,3,3,4,4,5,6,6,6,6,6,6,7,10,10,10,15,21}
[1,5] => [1,1] => [[1,1],[]]
=> ([],1)
=> ? ∊ {1,1,1,1,1,1,1,1,3,3,3,3,3,3,3,3,3,4,4,5,6,6,6,6,6,6,7,10,10,10,15,21}
[2,1,1,1,1] => [1,4] => [[4,1],[]]
=> ([],1)
=> ? ∊ {1,1,1,1,1,1,1,1,3,3,3,3,3,3,3,3,3,4,4,5,6,6,6,6,6,6,7,10,10,10,15,21}
[2,1,1,2] => [1,2,1] => [[2,2,1],[1]]
=> ([(0,1)],2)
=> ? ∊ {1,1,1,1,1,1,1,1,3,3,3,3,3,3,3,3,3,4,4,5,6,6,6,6,6,6,7,10,10,10,15,21}
[2,1,2,1] => [1,1,1,1] => [[1,1,1,1],[]]
=> ([],1)
=> ? ∊ {1,1,1,1,1,1,1,1,3,3,3,3,3,3,3,3,3,4,4,5,6,6,6,6,6,6,7,10,10,10,15,21}
[2,1,3] => [1,1,1] => [[1,1,1],[]]
=> ([],1)
=> ? ∊ {1,1,1,1,1,1,1,1,3,3,3,3,3,3,3,3,3,4,4,5,6,6,6,6,6,6,7,10,10,10,15,21}
[1,1,2,2,1] => [2,2,1] => [[3,3,2],[2,1]]
=> ([(0,2),(2,1)],3)
=> 3
[1,2,2,1,1] => [1,2,2] => [[3,2,1],[1]]
=> ([(0,2),(2,1)],3)
=> 3
[1,1,1,2,1,1,1] => [3,1,3] => [[5,3,3],[2,2]]
=> ([(0,2),(2,1)],3)
=> 3
[1,1,1,2,2,1] => [3,2,1] => [[4,4,3],[3,2]]
=> ([(0,2),(2,1)],3)
=> 3
[1,1,2,1,1,2] => [2,1,2,1] => [[3,3,2,2],[2,1,1]]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 3
[1,1,2,2,1,1] => [2,2,2] => [[4,3,2],[2,1]]
=> ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> 4
[1,2,1,1,2,1] => [1,1,2,1,1] => [[2,2,2,1,1],[1,1]]
=> ([(0,2),(2,1)],3)
=> 3
[1,2,2,1,1,1] => [1,2,3] => [[4,2,1],[1]]
=> ([(0,2),(2,1)],3)
=> 3
[2,1,1,2,1,1] => [1,2,1,2] => [[3,2,2,1],[1,1]]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 3
[2,1,1,2,2] => [1,2,2] => [[3,2,1],[1]]
=> ([(0,2),(2,1)],3)
=> 3
[2,2,1,1,2] => [2,2,1] => [[3,3,2],[2,1]]
=> ([(0,2),(2,1)],3)
=> 3
[1,1,1,1,2,1,1,1] => [4,1,3] => [[6,4,4],[3,3]]
=> ([(0,2),(2,1)],3)
=> 3
[1,1,1,1,2,2,1] => [4,2,1] => [[5,5,4],[4,3]]
=> ([(0,2),(2,1)],3)
=> 3
[1,1,1,2,1,1,2] => [3,1,2,1] => [[4,4,3,3],[3,2,2]]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 3
[1,1,1,2,2,1,1] => [3,2,2] => [[5,4,3],[3,2]]
=> ([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6)
=> 5
[1,1,1,2,2,2] => [3,3] => [[5,3],[2]]
=> ([(0,2),(2,1)],3)
=> 3
[1,1,1,3,1,1,1] => [3,1,3] => [[5,3,3],[2,2]]
=> ([(0,2),(2,1)],3)
=> 3
[1,1,2,1,1,1,2] => [2,1,3,1] => [[4,4,2,2],[3,1,1]]
=> ([(0,2),(0,3),(1,4),(2,4),(3,1)],5)
=> 4
[1,1,2,1,1,2,1] => [2,1,2,1,1] => [[3,3,3,2,2],[2,2,1,1]]
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> 4
[1,1,2,1,1,3] => [2,1,2,1] => [[3,3,2,2],[2,1,1]]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 3
[1,1,2,2,1,1,1] => [2,2,3] => [[5,3,2],[2,1]]
=> ([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6)
=> 5
[1,1,2,2,1,2] => [2,2,1,1] => [[3,3,3,2],[2,2,1]]
=> ([(0,2),(2,1)],3)
=> 3
[1,1,2,2,2,1] => [2,3,1] => [[4,4,2],[3,1]]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 3
[1,1,2,2,3] => [2,2,1] => [[3,3,2],[2,1]]
=> ([(0,2),(2,1)],3)
=> 3
[1,1,3,1,1,2] => [2,1,2,1] => [[3,3,2,2],[2,1,1]]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 3
[1,1,3,3,1] => [2,2,1] => [[3,3,2],[2,1]]
=> ([(0,2),(2,1)],3)
=> 3
[1,2,1,1,1,2,1] => [1,1,3,1,1] => [[3,3,3,1,1],[2,2]]
=> ([(0,2),(2,1)],3)
=> 3
[1,2,1,1,2,1,1] => [1,1,2,1,2] => [[3,2,2,1,1],[1,1]]
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> 4
[1,2,1,1,2,2] => [1,1,2,2] => [[3,2,1,1],[1]]
=> ([(0,2),(2,1)],3)
=> 3
[1,2,1,1,3,1] => [1,1,2,1,1] => [[2,2,2,1,1],[1,1]]
=> ([(0,2),(2,1)],3)
=> 3
[1,2,2,1,1,1,1] => [1,2,4] => [[5,2,1],[1]]
=> ([(0,2),(2,1)],3)
=> 3
[1,2,2,1,1,2] => [1,2,2,1] => [[3,3,2,1],[2,1]]
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> 4
[1,2,2,2,1,1] => [1,3,2] => [[4,3,1],[2]]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 3
[1,3,1,1,2,1] => [1,1,2,1,1] => [[2,2,2,1,1],[1,1]]
=> ([(0,2),(2,1)],3)
=> 3
[1,3,3,1,1] => [1,2,2] => [[3,2,1],[1]]
=> ([(0,2),(2,1)],3)
=> 3
[2,1,1,1,2,1,1] => [1,3,1,2] => [[4,3,3,1],[2,2]]
=> ([(0,2),(0,3),(1,4),(2,4),(3,1)],5)
=> 4
[2,1,1,1,2,2] => [1,3,2] => [[4,3,1],[2]]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 3
[2,1,1,2,1,1,1] => [1,2,1,3] => [[4,2,2,1],[1,1]]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 3
[2,1,1,2,2,1] => [1,2,2,1] => [[3,3,2,1],[2,1]]
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> 4
[2,1,1,3,1,1] => [1,2,1,2] => [[3,2,2,1],[1,1]]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 3
[2,1,2,2,1,1] => [1,1,2,2] => [[3,2,1,1],[1]]
=> ([(0,2),(2,1)],3)
=> 3
[2,2,1,1,1,2] => [2,3,1] => [[4,4,2],[3,1]]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 3
[2,2,1,1,2,1] => [2,2,1,1] => [[3,3,3,2],[2,2,1]]
=> ([(0,2),(2,1)],3)
=> 3
[2,2,1,1,3] => [2,2,1] => [[3,3,2],[2,1]]
=> ([(0,2),(2,1)],3)
=> 3
[2,2,2,1,1,1] => [3,3] => [[5,3],[2]]
=> ([(0,2),(2,1)],3)
=> 3
[3,1,1,2,1,1] => [1,2,1,2] => [[3,2,2,1],[1,1]]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 3
[3,1,1,2,2] => [1,2,2] => [[3,2,1],[1]]
=> ([(0,2),(2,1)],3)
=> 3
[3,2,2,1,1] => [1,2,2] => [[3,2,1],[1]]
=> ([(0,2),(2,1)],3)
=> 3
Description
The number of simple modules with projective dimension at most 1.
Matching statistic: St000567
Mp00184: Integer compositions —to threshold graph⟶ Graphs
Mp00037: Graphs —to partition of connected components⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000567: Integer partitions ⟶ ℤResult quality: 9% ●values known / values provided: 9%●distinct values known / distinct values provided: 29%
Mp00037: Graphs —to partition of connected components⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000567: Integer partitions ⟶ ℤResult quality: 9% ●values known / values provided: 9%●distinct values known / distinct values provided: 29%
Values
[1,1] => ([(0,1)],2)
=> [2]
=> []
=> ? ∊ {1,3}
[2] => ([],2)
=> [1,1]
=> [1]
=> ? ∊ {1,3}
[1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> [3]
=> []
=> ? ∊ {1,3,6}
[1,2] => ([(1,2)],3)
=> [2,1]
=> [1]
=> ? ∊ {1,3,6}
[2,1] => ([(0,2),(1,2)],3)
=> [3]
=> []
=> ? ∊ {1,3,6}
[3] => ([],3)
=> [1,1,1]
=> [1,1]
=> 1
[1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [4]
=> []
=> ? ∊ {1,1,3,3,6,10}
[1,1,2] => ([(1,2),(1,3),(2,3)],4)
=> [3,1]
=> [1]
=> ? ∊ {1,1,3,3,6,10}
[1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> [4]
=> []
=> ? ∊ {1,1,3,3,6,10}
[1,3] => ([(2,3)],4)
=> [2,1,1]
=> [1,1]
=> 1
[2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [4]
=> []
=> ? ∊ {1,1,3,3,6,10}
[2,2] => ([(1,3),(2,3)],4)
=> [3,1]
=> [1]
=> ? ∊ {1,1,3,3,6,10}
[3,1] => ([(0,3),(1,3),(2,3)],4)
=> [4]
=> []
=> ? ∊ {1,1,3,3,6,10}
[4] => ([],4)
=> [1,1,1,1]
=> [1,1,1]
=> 3
[1,1,1,1,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> []
=> ? ∊ {1,1,1,3,3,3,3,4,6,6,10,15}
[1,1,1,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [4,1]
=> [1]
=> ? ∊ {1,1,1,3,3,3,3,4,6,6,10,15}
[1,1,2,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> []
=> ? ∊ {1,1,1,3,3,3,3,4,6,6,10,15}
[1,1,3] => ([(2,3),(2,4),(3,4)],5)
=> [3,1,1]
=> [1,1]
=> 1
[1,2,1,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> []
=> ? ∊ {1,1,1,3,3,3,3,4,6,6,10,15}
[1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> [4,1]
=> [1]
=> ? ∊ {1,1,1,3,3,3,3,4,6,6,10,15}
[1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> []
=> ? ∊ {1,1,1,3,3,3,3,4,6,6,10,15}
[1,4] => ([(3,4)],5)
=> [2,1,1,1]
=> [1,1,1]
=> 3
[2,1,1,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> []
=> ? ∊ {1,1,1,3,3,3,3,4,6,6,10,15}
[2,1,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [4,1]
=> [1]
=> ? ∊ {1,1,1,3,3,3,3,4,6,6,10,15}
[2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> []
=> ? ∊ {1,1,1,3,3,3,3,4,6,6,10,15}
[2,3] => ([(2,4),(3,4)],5)
=> [3,1,1]
=> [1,1]
=> 1
[3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> []
=> ? ∊ {1,1,1,3,3,3,3,4,6,6,10,15}
[3,2] => ([(1,4),(2,4),(3,4)],5)
=> [4,1]
=> [1]
=> ? ∊ {1,1,1,3,3,3,3,4,6,6,10,15}
[4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> [5]
=> []
=> ? ∊ {1,1,1,3,3,3,3,4,6,6,10,15}
[5] => ([],5)
=> [1,1,1,1,1]
=> [1,1,1,1]
=> 6
[1,1,1,1,1,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [6]
=> []
=> ? ∊ {1,1,1,1,3,3,3,3,3,3,3,4,4,5,6,6,6,6,6,7,10,10,15,21}
[1,1,1,1,2] => ([(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [5,1]
=> [1]
=> ? ∊ {1,1,1,1,3,3,3,3,3,3,3,4,4,5,6,6,6,6,6,7,10,10,15,21}
[1,1,1,2,1] => ([(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [6]
=> []
=> ? ∊ {1,1,1,1,3,3,3,3,3,3,3,4,4,5,6,6,6,6,6,7,10,10,15,21}
[1,1,1,3] => ([(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [4,1,1]
=> [1,1]
=> 1
[1,1,2,1,1] => ([(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [6]
=> []
=> ? ∊ {1,1,1,1,3,3,3,3,3,3,3,4,4,5,6,6,6,6,6,7,10,10,15,21}
[1,1,2,2] => ([(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [5,1]
=> [1]
=> ? ∊ {1,1,1,1,3,3,3,3,3,3,3,4,4,5,6,6,6,6,6,7,10,10,15,21}
[1,1,3,1] => ([(0,5),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [6]
=> []
=> ? ∊ {1,1,1,1,3,3,3,3,3,3,3,4,4,5,6,6,6,6,6,7,10,10,15,21}
[1,1,4] => ([(3,4),(3,5),(4,5)],6)
=> [3,1,1,1]
=> [1,1,1]
=> 3
[1,2,1,1,1] => ([(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [6]
=> []
=> ? ∊ {1,1,1,1,3,3,3,3,3,3,3,4,4,5,6,6,6,6,6,7,10,10,15,21}
[1,2,1,2] => ([(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [5,1]
=> [1]
=> ? ∊ {1,1,1,1,3,3,3,3,3,3,3,4,4,5,6,6,6,6,6,7,10,10,15,21}
[1,2,2,1] => ([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [6]
=> []
=> ? ∊ {1,1,1,1,3,3,3,3,3,3,3,4,4,5,6,6,6,6,6,7,10,10,15,21}
[1,2,3] => ([(2,5),(3,4),(3,5),(4,5)],6)
=> [4,1,1]
=> [1,1]
=> 1
[1,3,1,1] => ([(0,4),(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [6]
=> []
=> ? ∊ {1,1,1,1,3,3,3,3,3,3,3,4,4,5,6,6,6,6,6,7,10,10,15,21}
[1,3,2] => ([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> [5,1]
=> [1]
=> ? ∊ {1,1,1,1,3,3,3,3,3,3,3,4,4,5,6,6,6,6,6,7,10,10,15,21}
[1,4,1] => ([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> [6]
=> []
=> ? ∊ {1,1,1,1,3,3,3,3,3,3,3,4,4,5,6,6,6,6,6,7,10,10,15,21}
[1,5] => ([(4,5)],6)
=> [2,1,1,1,1]
=> [1,1,1,1]
=> 6
[2,1,1,1,1] => ([(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [6]
=> []
=> ? ∊ {1,1,1,1,3,3,3,3,3,3,3,4,4,5,6,6,6,6,6,7,10,10,15,21}
[2,1,1,2] => ([(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [5,1]
=> [1]
=> ? ∊ {1,1,1,1,3,3,3,3,3,3,3,4,4,5,6,6,6,6,6,7,10,10,15,21}
[2,1,2,1] => ([(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [6]
=> []
=> ? ∊ {1,1,1,1,3,3,3,3,3,3,3,4,4,5,6,6,6,6,6,7,10,10,15,21}
[2,1,3] => ([(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [4,1,1]
=> [1,1]
=> 1
[2,2,1,1] => ([(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [6]
=> []
=> ? ∊ {1,1,1,1,3,3,3,3,3,3,3,4,4,5,6,6,6,6,6,7,10,10,15,21}
[2,2,2] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [5,1]
=> [1]
=> ? ∊ {1,1,1,1,3,3,3,3,3,3,3,4,4,5,6,6,6,6,6,7,10,10,15,21}
[2,3,1] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [6]
=> []
=> ? ∊ {1,1,1,1,3,3,3,3,3,3,3,4,4,5,6,6,6,6,6,7,10,10,15,21}
[2,4] => ([(3,5),(4,5)],6)
=> [3,1,1,1]
=> [1,1,1]
=> 3
[3,1,1,1] => ([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [6]
=> []
=> ? ∊ {1,1,1,1,3,3,3,3,3,3,3,4,4,5,6,6,6,6,6,7,10,10,15,21}
[3,1,2] => ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [5,1]
=> [1]
=> ? ∊ {1,1,1,1,3,3,3,3,3,3,3,4,4,5,6,6,6,6,6,7,10,10,15,21}
[3,2,1] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [6]
=> []
=> ? ∊ {1,1,1,1,3,3,3,3,3,3,3,4,4,5,6,6,6,6,6,7,10,10,15,21}
[3,3] => ([(2,5),(3,5),(4,5)],6)
=> [4,1,1]
=> [1,1]
=> 1
[4,1,1] => ([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [6]
=> []
=> ? ∊ {1,1,1,1,3,3,3,3,3,3,3,4,4,5,6,6,6,6,6,7,10,10,15,21}
[4,2] => ([(1,5),(2,5),(3,5),(4,5)],6)
=> [5,1]
=> [1]
=> ? ∊ {1,1,1,1,3,3,3,3,3,3,3,4,4,5,6,6,6,6,6,7,10,10,15,21}
[5,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> [6]
=> []
=> ? ∊ {1,1,1,1,3,3,3,3,3,3,3,4,4,5,6,6,6,6,6,7,10,10,15,21}
[6] => ([],6)
=> [1,1,1,1,1,1]
=> [1,1,1,1,1]
=> 10
[1,1,1,1,1,1,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [7]
=> []
=> ? ∊ {1,1,1,1,1,1,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,5,5,6,6,6,6,6,6,6,6,6,6,7,7,7,8,8,10,10,10,10,10,11,15,15,21,28}
[1,1,1,1,1,2] => ([(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [6,1]
=> [1]
=> ? ∊ {1,1,1,1,1,1,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,5,5,6,6,6,6,6,6,6,6,6,6,7,7,7,8,8,10,10,10,10,10,11,15,15,21,28}
[1,1,1,1,2,1] => ([(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [7]
=> []
=> ? ∊ {1,1,1,1,1,1,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,5,5,6,6,6,6,6,6,6,6,6,6,7,7,7,8,8,10,10,10,10,10,11,15,15,21,28}
[1,1,1,1,3] => ([(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [5,1,1]
=> [1,1]
=> 1
[1,1,1,4] => ([(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [4,1,1,1]
=> [1,1,1]
=> 3
[1,1,2,3] => ([(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [5,1,1]
=> [1,1]
=> 1
[1,1,5] => ([(4,5),(4,6),(5,6)],7)
=> [3,1,1,1,1]
=> [1,1,1,1]
=> 6
[1,2,1,3] => ([(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [5,1,1]
=> [1,1]
=> 1
[1,2,4] => ([(3,6),(4,5),(4,6),(5,6)],7)
=> [4,1,1,1]
=> [1,1,1]
=> 3
[1,3,3] => ([(2,6),(3,6),(4,5),(4,6),(5,6)],7)
=> [5,1,1]
=> [1,1]
=> 1
[1,6] => ([(5,6)],7)
=> [2,1,1,1,1,1]
=> [1,1,1,1,1]
=> 10
[2,1,1,3] => ([(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [5,1,1]
=> [1,1]
=> 1
[2,1,4] => ([(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [4,1,1,1]
=> [1,1,1]
=> 3
[2,2,3] => ([(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [5,1,1]
=> [1,1]
=> 1
[2,5] => ([(4,6),(5,6)],7)
=> [3,1,1,1,1]
=> [1,1,1,1]
=> 6
[3,1,3] => ([(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [5,1,1]
=> [1,1]
=> 1
[3,4] => ([(3,6),(4,6),(5,6)],7)
=> [4,1,1,1]
=> [1,1,1]
=> 3
[4,3] => ([(2,6),(3,6),(4,6),(5,6)],7)
=> [5,1,1]
=> [1,1]
=> 1
[7] => ([],7)
=> [1,1,1,1,1,1,1]
=> [1,1,1,1,1,1]
=> 15
[1,1,1,5] => ([(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> [4,1,1,1,1]
=> [1,1,1,1]
=> 6
[1,1,3,3] => ([(2,7),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> [6,1,1]
=> [1,1]
=> 1
[1,2,2,3] => ([(2,7),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> [6,1,1]
=> [1,1]
=> 1
[1,3,1,3] => ([(2,6),(2,7),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> [6,1,1]
=> [1,1]
=> 1
[1,4,3] => ([(2,7),(3,7),(4,7),(5,6),(5,7),(6,7)],8)
=> [6,1,1]
=> [1,1]
=> 1
[2,1,2,3] => ([(2,7),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> [6,1,1]
=> [1,1]
=> 1
[2,2,1,3] => ([(2,6),(2,7),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> [6,1,1]
=> [1,1]
=> 1
[2,3,3] => ([(2,7),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> [6,1,1]
=> [1,1]
=> 1
[3,1,1,3] => ([(2,5),(2,6),(2,7),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> [6,1,1]
=> [1,1]
=> 1
[3,2,3] => ([(2,7),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> [6,1,1]
=> [1,1]
=> 1
[4,1,3] => ([(2,6),(2,7),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> [6,1,1]
=> [1,1]
=> 1
[4,4] => ([(3,7),(4,7),(5,7),(6,7)],8)
=> [5,1,1,1]
=> [1,1,1]
=> 3
[8] => ([],8)
=> [1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1]
=> 21
Description
The sum of the products of all pairs of parts.
This is the evaluation of the second elementary symmetric polynomial which is equal to
$$e_2(\lambda) = \binom{n+1}{2} - \sum_{i=1}^\ell\binom{\lambda_i+1}{2}$$
for a partition $\lambda = (\lambda_1,\dots,\lambda_\ell) \vdash n$, see [1].
This is the maximal number of inversions a permutation with the given shape can have, see [2, cor.2.4].
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