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Your data matches 20 different statistics following compositions of up to 3 maps.
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Matching statistic: St000089
St000089: Integer compositions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => 0
[1,1] => 0
[2] => 0
[1,1,1] => 0
[1,2] => 1
[2,1] => 1
[3] => 0
[1,1,1,1] => 0
[1,1,2] => 1
[1,2,1] => 2
[1,3] => 2
[2,1,1] => 1
[2,2] => 0
[3,1] => 2
[4] => 0
[1,1,1,1,1] => 0
[1,1,1,2] => 1
[1,1,2,1] => 2
[1,1,3] => 2
[1,2,1,1] => 2
[1,2,2] => 1
[1,3,1] => 4
[1,4] => 3
[2,1,1,1] => 1
[2,1,2] => 2
[2,2,1] => 1
[2,3] => 1
[3,1,1] => 2
[3,2] => 1
[4,1] => 3
[5] => 0
[1,1,1,1,1,1] => 0
[1,1,1,1,2] => 1
[1,1,1,2,1] => 2
[1,1,1,3] => 2
[1,1,2,1,1] => 2
[1,1,2,2] => 1
[1,1,3,1] => 4
[1,1,4] => 3
[1,2,1,1,1] => 2
[1,2,1,2] => 3
[1,2,2,1] => 2
[1,2,3] => 2
[1,3,1,1] => 4
[1,3,2] => 3
[1,4,1] => 6
[1,5] => 4
[2,1,1,1,1] => 1
[2,1,1,2] => 2
[2,1,2,1] => 3
Description
The absolute variation of a composition.
Matching statistic: St000777
Values
[1] => ([],1)
=> ([],1)
=> 1 = 0 + 1
[1,1] => ([(0,1)],2)
=> ([],1)
=> 1 = 0 + 1
[2] => ([],2)
=> ([],2)
=> ? = 0 + 1
[1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> ([(0,1),(0,2),(1,2)],3)
=> 2 = 1 + 1
[1,2] => ([(1,2)],3)
=> ([],2)
=> ? ∊ {0,1} + 1
[2,1] => ([(0,2),(1,2)],3)
=> ([],1)
=> 1 = 0 + 1
[3] => ([],3)
=> ([],3)
=> ? ∊ {0,1} + 1
[1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 2 = 1 + 1
[1,1,2] => ([(1,2),(1,3),(2,3)],4)
=> ([(1,2),(1,3),(2,3)],4)
=> ? ∊ {0,0,2,2} + 1
[1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,2)],3)
=> 2 = 1 + 1
[1,3] => ([(2,3)],4)
=> ([],3)
=> ? ∊ {0,0,2,2} + 1
[2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3 = 2 + 1
[2,2] => ([(1,3),(2,3)],4)
=> ([],2)
=> ? ∊ {0,0,2,2} + 1
[3,1] => ([(0,3),(1,3),(2,3)],4)
=> ([],1)
=> 1 = 0 + 1
[4] => ([],4)
=> ([],4)
=> ? ∊ {0,0,2,2} + 1
[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)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2 = 1 + 1
[1,1,1,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {0,1,1,1,2,2,3,4} + 1
[1,1,2,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 2 = 1 + 1
[1,1,3] => ([(2,3),(2,4),(3,4)],5)
=> ([(2,3),(2,4),(3,4)],5)
=> ? ∊ {0,1,1,1,2,2,3,4} + 1
[1,2,1,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 4 = 3 + 1
[1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(2,3)],4)
=> ? ∊ {0,1,1,1,2,2,3,4} + 1
[1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 2 = 1 + 1
[1,4] => ([(3,4)],5)
=> ([],4)
=> ? ∊ {0,1,1,1,2,2,3,4} + 1
[2,1,1,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = 2 + 1
[2,1,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {0,1,1,1,2,2,3,4} + 1
[2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3 = 2 + 1
[2,3] => ([(2,4),(3,4)],5)
=> ([],3)
=> ? ∊ {0,1,1,1,2,2,3,4} + 1
[3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = 2 + 1
[3,2] => ([(1,4),(2,4),(3,4)],5)
=> ([],2)
=> ? ∊ {0,1,1,1,2,2,3,4} + 1
[4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> ([],1)
=> 1 = 0 + 1
[5] => ([],5)
=> ([],5)
=> ? ∊ {0,1,1,1,2,2,3,4} + 1
[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)
=> ([(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)
=> 2 = 1 + 1
[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)
=> ([(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? ∊ {0,0,0,2,2,2,2,2,3,3,3,3,4,4,4,6} + 1
[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),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2 = 1 + 1
[1,1,1,3] => ([(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? ∊ {0,0,0,2,2,2,2,2,3,3,3,3,4,4,4,6} + 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)
=> ([(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)
=> 4 = 3 + 1
[1,1,2,2] => ([(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {0,0,0,2,2,2,2,2,3,3,3,3,4,4,4,6} + 1
[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),(0,3),(1,2),(1,3),(2,3)],4)
=> 2 = 1 + 1
[1,1,4] => ([(3,4),(3,5),(4,5)],6)
=> ([(3,4),(3,5),(4,5)],6)
=> ? ∊ {0,0,0,2,2,2,2,2,3,3,3,3,4,4,4,6} + 1
[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)
=> ([(0,3),(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)
=> 4 = 3 + 1
[1,2,1,2] => ([(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? ∊ {0,0,0,2,2,2,2,2,3,3,3,3,4,4,4,6} + 1
[1,2,2,1] => ([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 4 = 3 + 1
[1,2,3] => ([(2,5),(3,4),(3,5),(4,5)],6)
=> ([(2,3),(2,4),(3,4)],5)
=> ? ∊ {0,0,0,2,2,2,2,2,3,3,3,3,4,4,4,6} + 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)
=> ([(0,4),(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 4 = 3 + 1
[1,3,2] => ([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(1,2),(1,3),(2,3)],4)
=> ? ∊ {0,0,0,2,2,2,2,2,3,3,3,3,4,4,4,6} + 1
[1,4,1] => ([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> 2 = 1 + 1
[1,5] => ([(4,5)],6)
=> ([],5)
=> ? ∊ {0,0,0,2,2,2,2,2,3,3,3,3,4,4,4,6} + 1
[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)
=> ([(0,2),(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)
=> 3 = 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)
=> ([(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? ∊ {0,0,0,2,2,2,2,2,3,3,3,3,4,4,4,6} + 1
[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)
=> ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = 2 + 1
[2,1,3] => ([(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? ∊ {0,0,0,2,2,2,2,2,3,3,3,3,4,4,4,6} + 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)
=> ([(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 5 = 4 + 1
[2,2,2] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {0,0,0,2,2,2,2,2,3,3,3,3,4,4,4,6} + 1
[2,3,1] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3 = 2 + 1
[2,4] => ([(3,5),(4,5)],6)
=> ([],4)
=> ? ∊ {0,0,0,2,2,2,2,2,3,3,3,3,4,4,4,6} + 1
[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)
=> ([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 3 = 2 + 1
[3,1,2] => ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? ∊ {0,0,0,2,2,2,2,2,3,3,3,3,4,4,4,6} + 1
[3,2,1] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = 2 + 1
[3,3] => ([(2,5),(3,5),(4,5)],6)
=> ([],3)
=> ? ∊ {0,0,0,2,2,2,2,2,3,3,3,3,4,4,4,6} + 1
[4,1,1] => ([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 3 = 2 + 1
[4,2] => ([(1,5),(2,5),(3,5),(4,5)],6)
=> ([],2)
=> ? ∊ {0,0,0,2,2,2,2,2,3,3,3,3,4,4,4,6} + 1
[5,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> ([],1)
=> 1 = 0 + 1
[6] => ([],6)
=> ([],6)
=> ? ∊ {0,0,0,2,2,2,2,2,3,3,3,3,4,4,4,6} + 1
[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)
=> ([(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)
=> 2 = 1 + 1
[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)
=> ([(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)
=> ? ∊ {0,1,1,1,1,1,2,2,2,2,2,2,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,6,6,8} + 1
[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),(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)
=> 2 = 1 + 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)
=> ([(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? ∊ {0,1,1,1,1,1,2,2,2,2,2,2,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,6,6,8} + 1
[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)
=> ([(0,5),(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)
=> 4 = 3 + 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),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? ∊ {0,1,1,1,1,1,2,2,2,2,2,2,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,6,6,8} + 1
[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),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2 = 1 + 1
[1,1,1,4] => ([(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? ∊ {0,1,1,1,1,1,2,2,2,2,2,2,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,6,6,8} + 1
[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)
=> ([(0,4),(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)
=> 4 = 3 + 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)
=> ([(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? ∊ {0,1,1,1,1,1,2,2,2,2,2,2,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,6,6,8} + 1
[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)
=> ([(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 4 = 3 + 1
[1,1,2,3] => ([(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? ∊ {0,1,1,1,1,1,2,2,2,2,2,2,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,6,6,8} + 1
[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)
=> ([(0,5),(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)
=> 4 = 3 + 1
[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),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {0,1,1,1,1,1,2,2,2,2,2,2,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,6,6,8} + 1
[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)
=> 2 = 1 + 1
[1,1,5] => ([(4,5),(4,6),(5,6)],7)
=> ([(4,5),(4,6),(5,6)],7)
=> ? ∊ {0,1,1,1,1,1,2,2,2,2,2,2,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,6,6,8} + 1
[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)
=> ([(0,3),(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)
=> 4 = 3 + 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)
=> ([(1,4),(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)
=> ? ∊ {0,1,1,1,1,1,2,2,2,2,2,2,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,6,6,8} + 1
[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)
=> ([(0,3),(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)
=> 4 = 3 + 1
[1,2,1,3] => ([(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? ∊ {0,1,1,1,1,1,2,2,2,2,2,2,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,6,6,8} + 1
[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)
=> ([(0,5),(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)
=> 6 = 5 + 1
[1,2,2,2] => ([(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? ∊ {0,1,1,1,1,1,2,2,2,2,2,2,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,6,6,8} + 1
[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)
=> ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 4 = 3 + 1
[1,2,4] => ([(3,6),(4,5),(4,6),(5,6)],7)
=> ([(3,4),(3,5),(4,5)],6)
=> ? ∊ {0,1,1,1,1,1,2,2,2,2,2,2,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,6,6,8} + 1
[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)
=> ([(0,4),(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)
=> 4 = 3 + 1
[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)
=> ([(1,5),(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? ∊ {0,1,1,1,1,1,2,2,2,2,2,2,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,6,6,8} + 1
[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)
=> ([(0,4),(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 4 = 3 + 1
[1,3,3] => ([(2,6),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(2,3),(2,4),(3,4)],5)
=> ? ∊ {0,1,1,1,1,1,2,2,2,2,2,2,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,6,6,8} + 1
[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)
=> ([(0,5),(0,6),(1,5),(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 4 = 3 + 1
[1,4,2] => ([(1,6),(2,6),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(1,2),(1,3),(2,3)],4)
=> ? ∊ {0,1,1,1,1,1,2,2,2,2,2,2,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,6,6,8} + 1
[1,5,1] => ([(0,6),(1,6),(2,6),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,1),(0,2),(1,2)],3)
=> 2 = 1 + 1
[1,6] => ([(5,6)],7)
=> ([],6)
=> ? ∊ {0,1,1,1,1,1,2,2,2,2,2,2,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,6,6,8} + 1
[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)
=> ([(0,2),(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)
=> 3 = 2 + 1
[2,1,1,1,2] => ([(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)
=> ([(1,3),(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)
=> ? ∊ {0,1,1,1,1,1,2,2,2,2,2,2,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,6,6,8} + 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)
=> ([(0,2),(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)
=> 3 = 2 + 1
[2,1,1,3] => ([(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? ∊ {0,1,1,1,1,1,2,2,2,2,2,2,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,6,6,8} + 1
[2,1,2,2] => ([(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? ∊ {0,1,1,1,1,1,2,2,2,2,2,2,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,6,6,8} + 1
Description
The number of distinct eigenvalues of the distance Laplacian of a connected graph.
Matching statistic: St001645
Values
[1] => ([],1)
=> ([],1)
=> 1 = 0 + 1
[1,1] => ([(0,1)],2)
=> ([],1)
=> 1 = 0 + 1
[2] => ([],2)
=> ([],2)
=> ? = 0 + 1
[1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> ([],1)
=> 1 = 0 + 1
[1,2] => ([(1,2)],3)
=> ([],2)
=> ? ∊ {0,1} + 1
[2,1] => ([(0,2),(1,2)],3)
=> ([(0,1)],2)
=> 2 = 1 + 1
[3] => ([],3)
=> ([],3)
=> ? ∊ {0,1} + 1
[1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([],1)
=> 1 = 0 + 1
[1,1,2] => ([(1,2),(1,3),(2,3)],4)
=> ([],2)
=> ? ∊ {0,0,2,2} + 1
[1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1)],2)
=> 2 = 1 + 1
[1,3] => ([(2,3)],4)
=> ([],3)
=> ? ∊ {0,0,2,2} + 1
[2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1)],2)
=> 2 = 1 + 1
[2,2] => ([(1,3),(2,3)],4)
=> ([(1,2)],3)
=> ? ∊ {0,0,2,2} + 1
[3,1] => ([(0,3),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,2)],3)
=> 3 = 2 + 1
[4] => ([],4)
=> ([],4)
=> ? ∊ {0,0,2,2} + 1
[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 = 0 + 1
[1,1,1,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([],2)
=> ? ∊ {0,1,1,1,2,2,3,4} + 1
[1,1,2,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1)],2)
=> 2 = 1 + 1
[1,1,3] => ([(2,3),(2,4),(3,4)],5)
=> ([],3)
=> ? ∊ {0,1,1,1,2,2,3,4} + 1
[1,2,1,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1)],2)
=> 2 = 1 + 1
[1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,2)],3)
=> ? ∊ {0,1,1,1,2,2,3,4} + 1
[1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 3 = 2 + 1
[1,4] => ([(3,4)],5)
=> ([],4)
=> ? ∊ {0,1,1,1,2,2,3,4} + 1
[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)
=> 2 = 1 + 1
[2,1,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,2)],3)
=> ? ∊ {0,1,1,1,2,2,3,4} + 1
[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 + 1
[2,3] => ([(2,4),(3,4)],5)
=> ([(2,3)],4)
=> ? ∊ {0,1,1,1,2,2,3,4} + 1
[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 = 2 + 1
[3,2] => ([(1,4),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(2,3)],4)
=> ? ∊ {0,1,1,1,2,2,3,4} + 1
[4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 4 = 3 + 1
[5] => ([],5)
=> ([],5)
=> ? ∊ {0,1,1,1,2,2,3,4} + 1
[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 = 0 + 1
[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)
=> ? ∊ {0,0,0,2,2,2,2,2,3,3,3,3,4,4,4,6} + 1
[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)
=> 2 = 1 + 1
[1,1,1,3] => ([(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([],3)
=> ? ∊ {0,0,0,2,2,2,2,2,3,3,3,3,4,4,4,6} + 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)
=> ([(0,1)],2)
=> 2 = 1 + 1
[1,1,2,2] => ([(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(1,2)],3)
=> ? ∊ {0,0,0,2,2,2,2,2,3,3,3,3,4,4,4,6} + 1
[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 = 2 + 1
[1,1,4] => ([(3,4),(3,5),(4,5)],6)
=> ([],4)
=> ? ∊ {0,0,0,2,2,2,2,2,3,3,3,3,4,4,4,6} + 1
[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)
=> 2 = 1 + 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)
=> ? ∊ {0,0,0,2,2,2,2,2,3,3,3,3,4,4,4,6} + 1
[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 = 2 + 1
[1,2,3] => ([(2,5),(3,4),(3,5),(4,5)],6)
=> ([(2,3)],4)
=> ? ∊ {0,0,0,2,2,2,2,2,3,3,3,3,4,4,4,6} + 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)
=> ([(0,1),(0,2),(1,2)],3)
=> 3 = 2 + 1
[1,3,2] => ([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(1,2),(1,3),(2,3)],4)
=> ? ∊ {0,0,0,2,2,2,2,2,3,3,3,3,4,4,4,6} + 1
[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)
=> 4 = 3 + 1
[1,5] => ([(4,5)],6)
=> ([],5)
=> ? ∊ {0,0,0,2,2,2,2,2,3,3,3,3,4,4,4,6} + 1
[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)
=> 2 = 1 + 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)
=> ? ∊ {0,0,0,2,2,2,2,2,3,3,3,3,4,4,4,6} + 1
[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
[2,1,3] => ([(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(2,3)],4)
=> ? ∊ {0,0,0,2,2,2,2,2,3,3,3,3,4,4,4,6} + 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)
=> ([(0,1),(0,2),(1,2)],3)
=> 3 = 2 + 1
[2,2,2] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(1,2),(1,3),(2,3)],4)
=> ? ∊ {0,0,0,2,2,2,2,2,3,3,3,3,4,4,4,6} + 1
[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)
=> 4 = 3 + 1
[2,4] => ([(3,5),(4,5)],6)
=> ([(3,4)],5)
=> ? ∊ {0,0,0,2,2,2,2,2,3,3,3,3,4,4,4,6} + 1
[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 = 2 + 1
[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)
=> ? ∊ {0,0,0,2,2,2,2,2,3,3,3,3,4,4,4,6} + 1
[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)
=> 4 = 3 + 1
[3,3] => ([(2,5),(3,5),(4,5)],6)
=> ([(2,3),(2,4),(3,4)],5)
=> ? ∊ {0,0,0,2,2,2,2,2,3,3,3,3,4,4,4,6} + 1
[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)
=> 4 = 3 + 1
[4,2] => ([(1,5),(2,5),(3,5),(4,5)],6)
=> ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {0,0,0,2,2,2,2,2,3,3,3,3,4,4,4,6} + 1
[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)
=> 5 = 4 + 1
[6] => ([],6)
=> ([],6)
=> ? ∊ {0,0,0,2,2,2,2,2,3,3,3,3,4,4,4,6} + 1
[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 = 0 + 1
[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)
=> ? ∊ {0,1,1,1,1,1,2,2,2,2,2,2,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,6,6,8} + 1
[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)
=> 2 = 1 + 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)
=> ? ∊ {0,1,1,1,1,1,2,2,2,2,2,2,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,6,6,8} + 1
[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)
=> 2 = 1 + 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)
=> ? ∊ {0,1,1,1,1,1,2,2,2,2,2,2,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,6,6,8} + 1
[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 = 2 + 1
[1,1,1,4] => ([(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([],4)
=> ? ∊ {0,1,1,1,1,1,2,2,2,2,2,2,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,6,6,8} + 1
[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)
=> 2 = 1 + 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)
=> ? ∊ {0,1,1,1,1,1,2,2,2,2,2,2,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,6,6,8} + 1
[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 = 2 + 1
[1,1,2,3] => ([(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(2,3)],4)
=> ? ∊ {0,1,1,1,1,1,2,2,2,2,2,2,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,6,6,8} + 1
[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 = 2 + 1
[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)
=> ? ∊ {0,1,1,1,1,1,2,2,2,2,2,2,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,6,6,8} + 1
[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)
=> 4 = 3 + 1
[1,1,5] => ([(4,5),(4,6),(5,6)],7)
=> ([],5)
=> ? ∊ {0,1,1,1,1,1,2,2,2,2,2,2,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,6,6,8} + 1
[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)
=> 2 = 1 + 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)
=> ? ∊ {0,1,1,1,1,1,2,2,2,2,2,2,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,6,6,8} + 1
[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 = 2 + 1
[1,2,1,3] => ([(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(2,3)],4)
=> ? ∊ {0,1,1,1,1,1,2,2,2,2,2,2,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,6,6,8} + 1
[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 = 2 + 1
[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)
=> ? ∊ {0,1,1,1,1,1,2,2,2,2,2,2,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,6,6,8} + 1
[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)
=> 4 = 3 + 1
[1,2,4] => ([(3,6),(4,5),(4,6),(5,6)],7)
=> ([(3,4)],5)
=> ? ∊ {0,1,1,1,1,1,2,2,2,2,2,2,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,6,6,8} + 1
[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 = 2 + 1
[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)
=> ? ∊ {0,1,1,1,1,1,2,2,2,2,2,2,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,6,6,8} + 1
[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)
=> 4 = 3 + 1
[1,3,3] => ([(2,6),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(2,3),(2,4),(3,4)],5)
=> ? ∊ {0,1,1,1,1,1,2,2,2,2,2,2,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,6,6,8} + 1
[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)
=> 4 = 3 + 1
[1,4,2] => ([(1,6),(2,6),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {0,1,1,1,1,1,2,2,2,2,2,2,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,6,6,8} + 1
[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)
=> 5 = 4 + 1
[1,6] => ([(5,6)],7)
=> ([],6)
=> ? ∊ {0,1,1,1,1,1,2,2,2,2,2,2,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,6,6,8} + 1
[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)
=> 2 = 1 + 1
[2,1,1,1,2] => ([(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)
=> ([(1,2)],3)
=> ? ∊ {0,1,1,1,1,1,2,2,2,2,2,2,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,6,6,8} + 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
[2,1,1,3] => ([(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(2,3)],4)
=> ? ∊ {0,1,1,1,1,1,2,2,2,2,2,2,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,6,6,8} + 1
[2,1,2,2] => ([(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(1,2),(1,3),(2,3)],4)
=> ? ∊ {0,1,1,1,1,1,2,2,2,2,2,2,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,6,6,8} + 1
Description
The pebbling number of a connected graph.
Matching statistic: St001330
Mp00133: Integer compositions —delta morphism⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
Mp00203: Graphs —cone⟶ Graphs
St001330: Graphs ⟶ ℤResult quality: 49% ●values known / values provided: 49%●distinct values known / distinct values provided: 62%
Mp00184: Integer compositions —to threshold graph⟶ Graphs
Mp00203: Graphs —cone⟶ Graphs
St001330: Graphs ⟶ ℤResult quality: 49% ●values known / values provided: 49%●distinct values known / distinct values provided: 62%
Values
[1] => [1] => ([],1)
=> ([(0,1)],2)
=> 2 = 0 + 2
[1,1] => [2] => ([],2)
=> ([(0,2),(1,2)],3)
=> 2 = 0 + 2
[2] => [1] => ([],1)
=> ([(0,1)],2)
=> 2 = 0 + 2
[1,1,1] => [3] => ([],3)
=> ([(0,3),(1,3),(2,3)],4)
=> 2 = 0 + 2
[1,2] => [1,1] => ([(0,1)],2)
=> ([(0,1),(0,2),(1,2)],3)
=> 3 = 1 + 2
[2,1] => [1,1] => ([(0,1)],2)
=> ([(0,1),(0,2),(1,2)],3)
=> 3 = 1 + 2
[3] => [1] => ([],1)
=> ([(0,1)],2)
=> 2 = 0 + 2
[1,1,1,1] => [4] => ([],4)
=> ([(0,4),(1,4),(2,4),(3,4)],5)
=> 2 = 0 + 2
[1,1,2] => [2,1] => ([(0,2),(1,2)],3)
=> ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ? ∊ {2,2} + 2
[1,2,1] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 4 = 2 + 2
[1,3] => [1,1] => ([(0,1)],2)
=> ([(0,1),(0,2),(1,2)],3)
=> 3 = 1 + 2
[2,1,1] => [1,2] => ([(1,2)],3)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> ? ∊ {2,2} + 2
[2,2] => [2] => ([],2)
=> ([(0,2),(1,2)],3)
=> 2 = 0 + 2
[3,1] => [1,1] => ([(0,1)],2)
=> ([(0,1),(0,2),(1,2)],3)
=> 3 = 1 + 2
[4] => [1] => ([],1)
=> ([(0,1)],2)
=> 2 = 0 + 2
[1,1,1,1,1] => [5] => ([],5)
=> ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> 2 = 0 + 2
[1,1,1,2] => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {1,1,2,2,2,3,3,4} + 2
[1,1,2,1] => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {1,1,2,2,2,3,3,4} + 2
[1,1,3] => [2,1] => ([(0,2),(1,2)],3)
=> ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ? ∊ {1,1,2,2,2,3,3,4} + 2
[1,2,1,1] => [1,1,2] => ([(1,2),(1,3),(2,3)],4)
=> ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {1,1,2,2,2,3,3,4} + 2
[1,2,2] => [1,2] => ([(1,2)],3)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> ? ∊ {1,1,2,2,2,3,3,4} + 2
[1,3,1] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 4 = 2 + 2
[1,4] => [1,1] => ([(0,1)],2)
=> ([(0,1),(0,2),(1,2)],3)
=> 3 = 1 + 2
[2,1,1,1] => [1,3] => ([(2,3)],4)
=> ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {1,1,2,2,2,3,3,4} + 2
[2,1,2] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 4 = 2 + 2
[2,2,1] => [2,1] => ([(0,2),(1,2)],3)
=> ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ? ∊ {1,1,2,2,2,3,3,4} + 2
[2,3] => [1,1] => ([(0,1)],2)
=> ([(0,1),(0,2),(1,2)],3)
=> 3 = 1 + 2
[3,1,1] => [1,2] => ([(1,2)],3)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> ? ∊ {1,1,2,2,2,3,3,4} + 2
[3,2] => [1,1] => ([(0,1)],2)
=> ([(0,1),(0,2),(1,2)],3)
=> 3 = 1 + 2
[4,1] => [1,1] => ([(0,1)],2)
=> ([(0,1),(0,2),(1,2)],3)
=> 3 = 1 + 2
[5] => [1] => ([],1)
=> ([(0,1)],2)
=> 2 = 0 + 2
[1,1,1,1,1,1] => [6] => ([],6)
=> ([(0,6),(1,6),(2,6),(3,6),(4,6),(5,6)],7)
=> 2 = 0 + 2
[1,1,1,1,2] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> ([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? ∊ {2,2,2,2,3,3,3,3,3,3,4,4,4,4,6} + 2
[1,1,1,2,1] => [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(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)
=> ? ∊ {2,2,2,2,3,3,3,3,3,3,4,4,4,4,6} + 2
[1,1,1,3] => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {2,2,2,2,3,3,3,3,3,3,4,4,4,4,6} + 2
[1,1,2,1,1] => [2,1,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? ∊ {2,2,2,2,3,3,3,3,3,3,4,4,4,4,6} + 2
[1,1,2,2] => [2,2] => ([(1,3),(2,3)],4)
=> ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {2,2,2,2,3,3,3,3,3,3,4,4,4,4,6} + 2
[1,1,3,1] => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {2,2,2,2,3,3,3,3,3,3,4,4,4,4,6} + 2
[1,1,4] => [2,1] => ([(0,2),(1,2)],3)
=> ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ? ∊ {2,2,2,2,3,3,3,3,3,3,4,4,4,4,6} + 2
[1,2,1,1,1] => [1,1,3] => ([(2,3),(2,4),(3,4)],5)
=> ([(0,5),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? ∊ {2,2,2,2,3,3,3,3,3,3,4,4,4,4,6} + 2
[1,2,1,2] => [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 5 = 3 + 2
[1,2,2,1] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {2,2,2,2,3,3,3,3,3,3,4,4,4,4,6} + 2
[1,2,3] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 4 = 2 + 2
[1,3,1,1] => [1,1,2] => ([(1,2),(1,3),(2,3)],4)
=> ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {2,2,2,2,3,3,3,3,3,3,4,4,4,4,6} + 2
[1,3,2] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 4 = 2 + 2
[1,4,1] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 4 = 2 + 2
[1,5] => [1,1] => ([(0,1)],2)
=> ([(0,1),(0,2),(1,2)],3)
=> 3 = 1 + 2
[2,1,1,1,1] => [1,4] => ([(3,4)],5)
=> ([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> ? ∊ {2,2,2,2,3,3,3,3,3,3,4,4,4,4,6} + 2
[2,1,1,2] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {2,2,2,2,3,3,3,3,3,3,4,4,4,4,6} + 2
[2,1,2,1] => [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 5 = 3 + 2
[2,1,3] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 4 = 2 + 2
[2,2,1,1] => [2,2] => ([(1,3),(2,3)],4)
=> ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {2,2,2,2,3,3,3,3,3,3,4,4,4,4,6} + 2
[2,2,2] => [3] => ([],3)
=> ([(0,3),(1,3),(2,3)],4)
=> 2 = 0 + 2
[2,3,1] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 4 = 2 + 2
[2,4] => [1,1] => ([(0,1)],2)
=> ([(0,1),(0,2),(1,2)],3)
=> 3 = 1 + 2
[3,1,1,1] => [1,3] => ([(2,3)],4)
=> ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {2,2,2,2,3,3,3,3,3,3,4,4,4,4,6} + 2
[3,1,2] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 4 = 2 + 2
[3,2,1] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 4 = 2 + 2
[3,3] => [2] => ([],2)
=> ([(0,2),(1,2)],3)
=> 2 = 0 + 2
[4,1,1] => [1,2] => ([(1,2)],3)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> ? ∊ {2,2,2,2,3,3,3,3,3,3,4,4,4,4,6} + 2
[4,2] => [1,1] => ([(0,1)],2)
=> ([(0,1),(0,2),(1,2)],3)
=> 3 = 1 + 2
[5,1] => [1,1] => ([(0,1)],2)
=> ([(0,1),(0,2),(1,2)],3)
=> 3 = 1 + 2
[6] => [1] => ([],1)
=> ([(0,1)],2)
=> 2 = 0 + 2
[1,1,1,1,1,1,1] => [7] => ([],7)
=> ([(0,7),(1,7),(2,7),(3,7),(4,7),(5,7),(6,7)],8)
=> 2 = 0 + 2
[1,1,1,1,1,2] => [5,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> ([(0,5),(0,6),(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? ∊ {1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,6,6,8} + 2
[1,1,1,1,2,1] => [4,1,1] => ([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,4),(0,5),(0,6),(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? ∊ {1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,6,6,8} + 2
[1,1,1,1,3] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> ([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? ∊ {1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,6,6,8} + 2
[1,1,1,2,1,1] => [3,1,2] => ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,6),(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? ∊ {1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,6,6,8} + 2
[1,1,1,2,2] => [3,2] => ([(1,4),(2,4),(3,4)],5)
=> ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? ∊ {1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,6,6,8} + 2
[1,1,1,3,1] => [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(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)
=> ? ∊ {1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,6,6,8} + 2
[1,1,1,4] => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,6,6,8} + 2
[1,1,2,1,1,1] => [2,1,3] => ([(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,6),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? ∊ {1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,6,6,8} + 2
[1,1,2,1,2] => [2,1,1,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(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)
=> ? ∊ {1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,6,6,8} + 2
[1,1,2,2,1] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? ∊ {1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,6,6,8} + 2
[1,1,2,3] => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,6,6,8} + 2
[1,1,3,1,1] => [2,1,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? ∊ {1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,6,6,8} + 2
[1,1,3,2] => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,6,6,8} + 2
[1,1,4,1] => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,6,6,8} + 2
[1,1,5] => [2,1] => ([(0,2),(1,2)],3)
=> ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ? ∊ {1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,6,6,8} + 2
[1,2,1,1,1,1] => [1,1,4] => ([(3,4),(3,5),(4,5)],6)
=> ([(0,6),(1,6),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? ∊ {1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,6,6,8} + 2
[1,2,1,1,2] => [1,1,2,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(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)
=> ? ∊ {1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,6,6,8} + 2
[1,2,1,2,1] => [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)
=> ([(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 = 4 + 2
[1,2,1,3] => [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 5 = 3 + 2
[1,2,2,1,1] => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? ∊ {1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,6,6,8} + 2
[1,2,2,2] => [1,3] => ([(2,3)],4)
=> ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,6,6,8} + 2
[1,2,3,1] => [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 5 = 3 + 2
[1,2,4] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 4 = 2 + 2
[1,3,1,1,1] => [1,1,3] => ([(2,3),(2,4),(3,4)],5)
=> ([(0,5),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? ∊ {1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,6,6,8} + 2
[1,3,1,2] => [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 5 = 3 + 2
[1,3,2,1] => [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 5 = 3 + 2
[1,3,3] => [1,2] => ([(1,2)],3)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> ? ∊ {1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,6,6,8} + 2
[1,4,1,1] => [1,1,2] => ([(1,2),(1,3),(2,3)],4)
=> ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,6,6,8} + 2
[1,4,2] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 4 = 2 + 2
[1,5,1] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 4 = 2 + 2
[1,6] => [1,1] => ([(0,1)],2)
=> ([(0,1),(0,2),(1,2)],3)
=> 3 = 1 + 2
[2,1,1,1,1,1] => [1,5] => ([(4,5)],6)
=> ([(0,6),(1,6),(2,6),(3,6),(4,5),(4,6),(5,6)],7)
=> ? ∊ {1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,6,6,8} + 2
[2,1,1,1,2] => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? ∊ {1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,6,6,8} + 2
[2,1,1,2,1] => [1,2,1,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(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)
=> ? ∊ {1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,6,6,8} + 2
[2,1,3,1] => [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 5 = 3 + 2
[2,1,4] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 4 = 2 + 2
Description
The hat guessing number of a graph.
Suppose that each vertex of a graph corresponds to a player, wearing a hat whose color is arbitrarily chosen from a set of $q$ possible colors. Each player can see the hat colors of his neighbors, but not his own hat color. All of the players are asked to guess their own hat colors simultaneously, according to a predetermined guessing strategy and the hat colors they see, where no communication between them is allowed. The hat guessing number $HG(G)$ of a graph $G$ is the largest integer $q$ such that there exists a guessing strategy guaranteeing at least one correct guess for any hat assignment of $q$ possible colors.
Because it suffices that a single player guesses correctly, the hat guessing number of a graph is the maximum of the hat guessing numbers of its connected components.
Matching statistic: St000668
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00180: Integer compositions —to ribbon⟶ Skew partitions
Mp00183: Skew partitions —inner shape⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000668: Integer partitions ⟶ ℤResult quality: 35% ●values known / values provided: 35%●distinct values known / distinct values provided: 62%
Mp00183: Skew partitions —inner shape⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000668: Integer partitions ⟶ ℤResult quality: 35% ●values known / values provided: 35%●distinct values known / distinct values provided: 62%
Values
[1] => [[1],[]]
=> []
=> ?
=> ? = 0
[1,1] => [[1,1],[]]
=> []
=> ?
=> ? ∊ {0,0}
[2] => [[2],[]]
=> []
=> ?
=> ? ∊ {0,0}
[1,1,1] => [[1,1,1],[]]
=> []
=> ?
=> ? ∊ {0,0,1,1}
[1,2] => [[2,1],[]]
=> []
=> ?
=> ? ∊ {0,0,1,1}
[2,1] => [[2,2],[1]]
=> [1]
=> []
=> ? ∊ {0,0,1,1}
[3] => [[3],[]]
=> []
=> ?
=> ? ∊ {0,0,1,1}
[1,1,1,1] => [[1,1,1,1],[]]
=> []
=> ?
=> ? ∊ {0,0,0,1,1,2,2,2}
[1,1,2] => [[2,1,1],[]]
=> []
=> ?
=> ? ∊ {0,0,0,1,1,2,2,2}
[1,2,1] => [[2,2,1],[1]]
=> [1]
=> []
=> ? ∊ {0,0,0,1,1,2,2,2}
[1,3] => [[3,1],[]]
=> []
=> ?
=> ? ∊ {0,0,0,1,1,2,2,2}
[2,1,1] => [[2,2,2],[1,1]]
=> [1,1]
=> [1]
=> ? ∊ {0,0,0,1,1,2,2,2}
[2,2] => [[3,2],[1]]
=> [1]
=> []
=> ? ∊ {0,0,0,1,1,2,2,2}
[3,1] => [[3,3],[2]]
=> [2]
=> []
=> ? ∊ {0,0,0,1,1,2,2,2}
[4] => [[4],[]]
=> []
=> ?
=> ? ∊ {0,0,0,1,1,2,2,2}
[1,1,1,1,1] => [[1,1,1,1,1],[]]
=> []
=> ?
=> ? ∊ {0,0,1,1,1,1,1,2,2,2,2,3,3,4}
[1,1,1,2] => [[2,1,1,1],[]]
=> []
=> ?
=> ? ∊ {0,0,1,1,1,1,1,2,2,2,2,3,3,4}
[1,1,2,1] => [[2,2,1,1],[1]]
=> [1]
=> []
=> ? ∊ {0,0,1,1,1,1,1,2,2,2,2,3,3,4}
[1,1,3] => [[3,1,1],[]]
=> []
=> ?
=> ? ∊ {0,0,1,1,1,1,1,2,2,2,2,3,3,4}
[1,2,1,1] => [[2,2,2,1],[1,1]]
=> [1,1]
=> [1]
=> ? ∊ {0,0,1,1,1,1,1,2,2,2,2,3,3,4}
[1,2,2] => [[3,2,1],[1]]
=> [1]
=> []
=> ? ∊ {0,0,1,1,1,1,1,2,2,2,2,3,3,4}
[1,3,1] => [[3,3,1],[2]]
=> [2]
=> []
=> ? ∊ {0,0,1,1,1,1,1,2,2,2,2,3,3,4}
[1,4] => [[4,1],[]]
=> []
=> ?
=> ? ∊ {0,0,1,1,1,1,1,2,2,2,2,3,3,4}
[2,1,1,1] => [[2,2,2,2],[1,1,1]]
=> [1,1,1]
=> [1,1]
=> 1
[2,1,2] => [[3,2,2],[1,1]]
=> [1,1]
=> [1]
=> ? ∊ {0,0,1,1,1,1,1,2,2,2,2,3,3,4}
[2,2,1] => [[3,3,2],[2,1]]
=> [2,1]
=> [1]
=> ? ∊ {0,0,1,1,1,1,1,2,2,2,2,3,3,4}
[2,3] => [[4,2],[1]]
=> [1]
=> []
=> ? ∊ {0,0,1,1,1,1,1,2,2,2,2,3,3,4}
[3,1,1] => [[3,3,3],[2,2]]
=> [2,2]
=> [2]
=> 2
[3,2] => [[4,3],[2]]
=> [2]
=> []
=> ? ∊ {0,0,1,1,1,1,1,2,2,2,2,3,3,4}
[4,1] => [[4,4],[3]]
=> [3]
=> []
=> ? ∊ {0,0,1,1,1,1,1,2,2,2,2,3,3,4}
[5] => [[5],[]]
=> []
=> ?
=> ? ∊ {0,0,1,1,1,1,1,2,2,2,2,3,3,4}
[1,1,1,1,1,1] => [[1,1,1,1,1,1],[]]
=> []
=> ?
=> ? ∊ {0,0,0,0,2,2,2,2,2,2,3,3,3,3,3,3,3,4,4,4,4,6}
[1,1,1,1,2] => [[2,1,1,1,1],[]]
=> []
=> ?
=> ? ∊ {0,0,0,0,2,2,2,2,2,2,3,3,3,3,3,3,3,4,4,4,4,6}
[1,1,1,2,1] => [[2,2,1,1,1],[1]]
=> [1]
=> []
=> ? ∊ {0,0,0,0,2,2,2,2,2,2,3,3,3,3,3,3,3,4,4,4,4,6}
[1,1,1,3] => [[3,1,1,1],[]]
=> []
=> ?
=> ? ∊ {0,0,0,0,2,2,2,2,2,2,3,3,3,3,3,3,3,4,4,4,4,6}
[1,1,2,1,1] => [[2,2,2,1,1],[1,1]]
=> [1,1]
=> [1]
=> ? ∊ {0,0,0,0,2,2,2,2,2,2,3,3,3,3,3,3,3,4,4,4,4,6}
[1,1,2,2] => [[3,2,1,1],[1]]
=> [1]
=> []
=> ? ∊ {0,0,0,0,2,2,2,2,2,2,3,3,3,3,3,3,3,4,4,4,4,6}
[1,1,3,1] => [[3,3,1,1],[2]]
=> [2]
=> []
=> ? ∊ {0,0,0,0,2,2,2,2,2,2,3,3,3,3,3,3,3,4,4,4,4,6}
[1,1,4] => [[4,1,1],[]]
=> []
=> ?
=> ? ∊ {0,0,0,0,2,2,2,2,2,2,3,3,3,3,3,3,3,4,4,4,4,6}
[1,2,1,1,1] => [[2,2,2,2,1],[1,1,1]]
=> [1,1,1]
=> [1,1]
=> 1
[1,2,1,2] => [[3,2,2,1],[1,1]]
=> [1,1]
=> [1]
=> ? ∊ {0,0,0,0,2,2,2,2,2,2,3,3,3,3,3,3,3,4,4,4,4,6}
[1,2,2,1] => [[3,3,2,1],[2,1]]
=> [2,1]
=> [1]
=> ? ∊ {0,0,0,0,2,2,2,2,2,2,3,3,3,3,3,3,3,4,4,4,4,6}
[1,2,3] => [[4,2,1],[1]]
=> [1]
=> []
=> ? ∊ {0,0,0,0,2,2,2,2,2,2,3,3,3,3,3,3,3,4,4,4,4,6}
[1,3,1,1] => [[3,3,3,1],[2,2]]
=> [2,2]
=> [2]
=> 2
[1,3,2] => [[4,3,1],[2]]
=> [2]
=> []
=> ? ∊ {0,0,0,0,2,2,2,2,2,2,3,3,3,3,3,3,3,4,4,4,4,6}
[1,4,1] => [[4,4,1],[3]]
=> [3]
=> []
=> ? ∊ {0,0,0,0,2,2,2,2,2,2,3,3,3,3,3,3,3,4,4,4,4,6}
[1,5] => [[5,1],[]]
=> []
=> ?
=> ? ∊ {0,0,0,0,2,2,2,2,2,2,3,3,3,3,3,3,3,4,4,4,4,6}
[2,1,1,1,1] => [[2,2,2,2,2],[1,1,1,1]]
=> [1,1,1,1]
=> [1,1,1]
=> 1
[2,1,1,2] => [[3,2,2,2],[1,1,1]]
=> [1,1,1]
=> [1,1]
=> 1
[2,1,2,1] => [[3,3,2,2],[2,1,1]]
=> [2,1,1]
=> [1,1]
=> 1
[2,1,3] => [[4,2,2],[1,1]]
=> [1,1]
=> [1]
=> ? ∊ {0,0,0,0,2,2,2,2,2,2,3,3,3,3,3,3,3,4,4,4,4,6}
[2,2,1,1] => [[3,3,3,2],[2,2,1]]
=> [2,2,1]
=> [2,1]
=> 2
[2,2,2] => [[4,3,2],[2,1]]
=> [2,1]
=> [1]
=> ? ∊ {0,0,0,0,2,2,2,2,2,2,3,3,3,3,3,3,3,4,4,4,4,6}
[2,3,1] => [[4,4,2],[3,1]]
=> [3,1]
=> [1]
=> ? ∊ {0,0,0,0,2,2,2,2,2,2,3,3,3,3,3,3,3,4,4,4,4,6}
[2,4] => [[5,2],[1]]
=> [1]
=> []
=> ? ∊ {0,0,0,0,2,2,2,2,2,2,3,3,3,3,3,3,3,4,4,4,4,6}
[3,1,1,1] => [[3,3,3,3],[2,2,2]]
=> [2,2,2]
=> [2,2]
=> 2
[3,1,2] => [[4,3,3],[2,2]]
=> [2,2]
=> [2]
=> 2
[3,2,1] => [[4,4,3],[3,2]]
=> [3,2]
=> [2]
=> 2
[3,3] => [[5,3],[2]]
=> [2]
=> []
=> ? ∊ {0,0,0,0,2,2,2,2,2,2,3,3,3,3,3,3,3,4,4,4,4,6}
[4,1,1] => [[4,4,4],[3,3]]
=> [3,3]
=> [3]
=> 3
[4,2] => [[5,4],[3]]
=> [3]
=> []
=> ? ∊ {0,0,0,0,2,2,2,2,2,2,3,3,3,3,3,3,3,4,4,4,4,6}
[5,1] => [[5,5],[4]]
=> [4]
=> []
=> ? ∊ {0,0,0,0,2,2,2,2,2,2,3,3,3,3,3,3,3,4,4,4,4,6}
[1,1,2,1,1,1] => [[2,2,2,2,1,1],[1,1,1]]
=> [1,1,1]
=> [1,1]
=> 1
[1,1,3,1,1] => [[3,3,3,1,1],[2,2]]
=> [2,2]
=> [2]
=> 2
[1,2,1,1,1,1] => [[2,2,2,2,2,1],[1,1,1,1]]
=> [1,1,1,1]
=> [1,1,1]
=> 1
[1,2,1,1,2] => [[3,2,2,2,1],[1,1,1]]
=> [1,1,1]
=> [1,1]
=> 1
[1,2,1,2,1] => [[3,3,2,2,1],[2,1,1]]
=> [2,1,1]
=> [1,1]
=> 1
[1,2,2,1,1] => [[3,3,3,2,1],[2,2,1]]
=> [2,2,1]
=> [2,1]
=> 2
[1,3,1,1,1] => [[3,3,3,3,1],[2,2,2]]
=> [2,2,2]
=> [2,2]
=> 2
[1,3,1,2] => [[4,3,3,1],[2,2]]
=> [2,2]
=> [2]
=> 2
[1,3,2,1] => [[4,4,3,1],[3,2]]
=> [3,2]
=> [2]
=> 2
[1,4,1,1] => [[4,4,4,1],[3,3]]
=> [3,3]
=> [3]
=> 3
[2,1,1,1,1,1] => [[2,2,2,2,2,2],[1,1,1,1,1]]
=> [1,1,1,1,1]
=> [1,1,1,1]
=> 1
[2,1,1,1,2] => [[3,2,2,2,2],[1,1,1,1]]
=> [1,1,1,1]
=> [1,1,1]
=> 1
[2,1,1,2,1] => [[3,3,2,2,2],[2,1,1,1]]
=> [2,1,1,1]
=> [1,1,1]
=> 1
[2,1,1,3] => [[4,2,2,2],[1,1,1]]
=> [1,1,1]
=> [1,1]
=> 1
[2,1,2,1,1] => [[3,3,3,2,2],[2,2,1,1]]
=> [2,2,1,1]
=> [2,1,1]
=> 2
[2,1,2,2] => [[4,3,2,2],[2,1,1]]
=> [2,1,1]
=> [1,1]
=> 1
[2,1,3,1] => [[4,4,2,2],[3,1,1]]
=> [3,1,1]
=> [1,1]
=> 1
[2,2,1,1,1] => [[3,3,3,3,2],[2,2,2,1]]
=> [2,2,2,1]
=> [2,2,1]
=> 2
[2,2,1,2] => [[4,3,3,2],[2,2,1]]
=> [2,2,1]
=> [2,1]
=> 2
[2,2,2,1] => [[4,4,3,2],[3,2,1]]
=> [3,2,1]
=> [2,1]
=> 2
[2,3,1,1] => [[4,4,4,2],[3,3,1]]
=> [3,3,1]
=> [3,1]
=> 3
[3,1,1,1,1] => [[3,3,3,3,3],[2,2,2,2]]
=> [2,2,2,2]
=> [2,2,2]
=> 2
[3,1,1,2] => [[4,3,3,3],[2,2,2]]
=> [2,2,2]
=> [2,2]
=> 2
[3,1,2,1] => [[4,4,3,3],[3,2,2]]
=> [3,2,2]
=> [2,2]
=> 2
[3,1,3] => [[5,3,3],[2,2]]
=> [2,2]
=> [2]
=> 2
[3,2,1,1] => [[4,4,4,3],[3,3,2]]
=> [3,3,2]
=> [3,2]
=> 6
[3,2,2] => [[5,4,3],[3,2]]
=> [3,2]
=> [2]
=> 2
[3,3,1] => [[5,5,3],[4,2]]
=> [4,2]
=> [2]
=> 2
[4,1,1,1] => [[4,4,4,4],[3,3,3]]
=> [3,3,3]
=> [3,3]
=> 3
[4,1,2] => [[5,4,4],[3,3]]
=> [3,3]
=> [3]
=> 3
[4,2,1] => [[5,5,4],[4,3]]
=> [4,3]
=> [3]
=> 3
[5,1,1] => [[5,5,5],[4,4]]
=> [4,4]
=> [4]
=> 4
Description
The least common multiple of the parts of the partition.
Matching statistic: St000770
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00180: Integer compositions —to ribbon⟶ Skew partitions
Mp00183: Skew partitions —inner shape⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000770: Integer partitions ⟶ ℤResult quality: 35% ●values known / values provided: 35%●distinct values known / distinct values provided: 75%
Mp00183: Skew partitions —inner shape⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000770: Integer partitions ⟶ ℤResult quality: 35% ●values known / values provided: 35%●distinct values known / distinct values provided: 75%
Values
[1] => [[1],[]]
=> []
=> ?
=> ? = 0
[1,1] => [[1,1],[]]
=> []
=> ?
=> ? ∊ {0,0}
[2] => [[2],[]]
=> []
=> ?
=> ? ∊ {0,0}
[1,1,1] => [[1,1,1],[]]
=> []
=> ?
=> ? ∊ {0,0,1,1}
[1,2] => [[2,1],[]]
=> []
=> ?
=> ? ∊ {0,0,1,1}
[2,1] => [[2,2],[1]]
=> [1]
=> []
=> ? ∊ {0,0,1,1}
[3] => [[3],[]]
=> []
=> ?
=> ? ∊ {0,0,1,1}
[1,1,1,1] => [[1,1,1,1],[]]
=> []
=> ?
=> ? ∊ {0,0,0,1,1,2,2,2}
[1,1,2] => [[2,1,1],[]]
=> []
=> ?
=> ? ∊ {0,0,0,1,1,2,2,2}
[1,2,1] => [[2,2,1],[1]]
=> [1]
=> []
=> ? ∊ {0,0,0,1,1,2,2,2}
[1,3] => [[3,1],[]]
=> []
=> ?
=> ? ∊ {0,0,0,1,1,2,2,2}
[2,1,1] => [[2,2,2],[1,1]]
=> [1,1]
=> [1]
=> ? ∊ {0,0,0,1,1,2,2,2}
[2,2] => [[3,2],[1]]
=> [1]
=> []
=> ? ∊ {0,0,0,1,1,2,2,2}
[3,1] => [[3,3],[2]]
=> [2]
=> []
=> ? ∊ {0,0,0,1,1,2,2,2}
[4] => [[4],[]]
=> []
=> ?
=> ? ∊ {0,0,0,1,1,2,2,2}
[1,1,1,1,1] => [[1,1,1,1,1],[]]
=> []
=> ?
=> ? ∊ {0,0,1,1,1,1,1,2,2,2,2,3,3,4}
[1,1,1,2] => [[2,1,1,1],[]]
=> []
=> ?
=> ? ∊ {0,0,1,1,1,1,1,2,2,2,2,3,3,4}
[1,1,2,1] => [[2,2,1,1],[1]]
=> [1]
=> []
=> ? ∊ {0,0,1,1,1,1,1,2,2,2,2,3,3,4}
[1,1,3] => [[3,1,1],[]]
=> []
=> ?
=> ? ∊ {0,0,1,1,1,1,1,2,2,2,2,3,3,4}
[1,2,1,1] => [[2,2,2,1],[1,1]]
=> [1,1]
=> [1]
=> ? ∊ {0,0,1,1,1,1,1,2,2,2,2,3,3,4}
[1,2,2] => [[3,2,1],[1]]
=> [1]
=> []
=> ? ∊ {0,0,1,1,1,1,1,2,2,2,2,3,3,4}
[1,3,1] => [[3,3,1],[2]]
=> [2]
=> []
=> ? ∊ {0,0,1,1,1,1,1,2,2,2,2,3,3,4}
[1,4] => [[4,1],[]]
=> []
=> ?
=> ? ∊ {0,0,1,1,1,1,1,2,2,2,2,3,3,4}
[2,1,1,1] => [[2,2,2,2],[1,1,1]]
=> [1,1,1]
=> [1,1]
=> 1
[2,1,2] => [[3,2,2],[1,1]]
=> [1,1]
=> [1]
=> ? ∊ {0,0,1,1,1,1,1,2,2,2,2,3,3,4}
[2,2,1] => [[3,3,2],[2,1]]
=> [2,1]
=> [1]
=> ? ∊ {0,0,1,1,1,1,1,2,2,2,2,3,3,4}
[2,3] => [[4,2],[1]]
=> [1]
=> []
=> ? ∊ {0,0,1,1,1,1,1,2,2,2,2,3,3,4}
[3,1,1] => [[3,3,3],[2,2]]
=> [2,2]
=> [2]
=> 2
[3,2] => [[4,3],[2]]
=> [2]
=> []
=> ? ∊ {0,0,1,1,1,1,1,2,2,2,2,3,3,4}
[4,1] => [[4,4],[3]]
=> [3]
=> []
=> ? ∊ {0,0,1,1,1,1,1,2,2,2,2,3,3,4}
[5] => [[5],[]]
=> []
=> ?
=> ? ∊ {0,0,1,1,1,1,1,2,2,2,2,3,3,4}
[1,1,1,1,1,1] => [[1,1,1,1,1,1],[]]
=> []
=> ?
=> ? ∊ {0,0,0,0,2,2,2,2,2,2,2,3,3,3,3,3,3,3,4,4,4,6}
[1,1,1,1,2] => [[2,1,1,1,1],[]]
=> []
=> ?
=> ? ∊ {0,0,0,0,2,2,2,2,2,2,2,3,3,3,3,3,3,3,4,4,4,6}
[1,1,1,2,1] => [[2,2,1,1,1],[1]]
=> [1]
=> []
=> ? ∊ {0,0,0,0,2,2,2,2,2,2,2,3,3,3,3,3,3,3,4,4,4,6}
[1,1,1,3] => [[3,1,1,1],[]]
=> []
=> ?
=> ? ∊ {0,0,0,0,2,2,2,2,2,2,2,3,3,3,3,3,3,3,4,4,4,6}
[1,1,2,1,1] => [[2,2,2,1,1],[1,1]]
=> [1,1]
=> [1]
=> ? ∊ {0,0,0,0,2,2,2,2,2,2,2,3,3,3,3,3,3,3,4,4,4,6}
[1,1,2,2] => [[3,2,1,1],[1]]
=> [1]
=> []
=> ? ∊ {0,0,0,0,2,2,2,2,2,2,2,3,3,3,3,3,3,3,4,4,4,6}
[1,1,3,1] => [[3,3,1,1],[2]]
=> [2]
=> []
=> ? ∊ {0,0,0,0,2,2,2,2,2,2,2,3,3,3,3,3,3,3,4,4,4,6}
[1,1,4] => [[4,1,1],[]]
=> []
=> ?
=> ? ∊ {0,0,0,0,2,2,2,2,2,2,2,3,3,3,3,3,3,3,4,4,4,6}
[1,2,1,1,1] => [[2,2,2,2,1],[1,1,1]]
=> [1,1,1]
=> [1,1]
=> 1
[1,2,1,2] => [[3,2,2,1],[1,1]]
=> [1,1]
=> [1]
=> ? ∊ {0,0,0,0,2,2,2,2,2,2,2,3,3,3,3,3,3,3,4,4,4,6}
[1,2,2,1] => [[3,3,2,1],[2,1]]
=> [2,1]
=> [1]
=> ? ∊ {0,0,0,0,2,2,2,2,2,2,2,3,3,3,3,3,3,3,4,4,4,6}
[1,2,3] => [[4,2,1],[1]]
=> [1]
=> []
=> ? ∊ {0,0,0,0,2,2,2,2,2,2,2,3,3,3,3,3,3,3,4,4,4,6}
[1,3,1,1] => [[3,3,3,1],[2,2]]
=> [2,2]
=> [2]
=> 2
[1,3,2] => [[4,3,1],[2]]
=> [2]
=> []
=> ? ∊ {0,0,0,0,2,2,2,2,2,2,2,3,3,3,3,3,3,3,4,4,4,6}
[1,4,1] => [[4,4,1],[3]]
=> [3]
=> []
=> ? ∊ {0,0,0,0,2,2,2,2,2,2,2,3,3,3,3,3,3,3,4,4,4,6}
[1,5] => [[5,1],[]]
=> []
=> ?
=> ? ∊ {0,0,0,0,2,2,2,2,2,2,2,3,3,3,3,3,3,3,4,4,4,6}
[2,1,1,1,1] => [[2,2,2,2,2],[1,1,1,1]]
=> [1,1,1,1]
=> [1,1,1]
=> 1
[2,1,1,2] => [[3,2,2,2],[1,1,1]]
=> [1,1,1]
=> [1,1]
=> 1
[2,1,2,1] => [[3,3,2,2],[2,1,1]]
=> [2,1,1]
=> [1,1]
=> 1
[2,1,3] => [[4,2,2],[1,1]]
=> [1,1]
=> [1]
=> ? ∊ {0,0,0,0,2,2,2,2,2,2,2,3,3,3,3,3,3,3,4,4,4,6}
[2,2,1,1] => [[3,3,3,2],[2,2,1]]
=> [2,2,1]
=> [2,1]
=> 4
[2,2,2] => [[4,3,2],[2,1]]
=> [2,1]
=> [1]
=> ? ∊ {0,0,0,0,2,2,2,2,2,2,2,3,3,3,3,3,3,3,4,4,4,6}
[2,3,1] => [[4,4,2],[3,1]]
=> [3,1]
=> [1]
=> ? ∊ {0,0,0,0,2,2,2,2,2,2,2,3,3,3,3,3,3,3,4,4,4,6}
[2,4] => [[5,2],[1]]
=> [1]
=> []
=> ? ∊ {0,0,0,0,2,2,2,2,2,2,2,3,3,3,3,3,3,3,4,4,4,6}
[3,1,1,1] => [[3,3,3,3],[2,2,2]]
=> [2,2,2]
=> [2,2]
=> 2
[3,1,2] => [[4,3,3],[2,2]]
=> [2,2]
=> [2]
=> 2
[3,2,1] => [[4,4,3],[3,2]]
=> [3,2]
=> [2]
=> 2
[3,3] => [[5,3],[2]]
=> [2]
=> []
=> ? ∊ {0,0,0,0,2,2,2,2,2,2,2,3,3,3,3,3,3,3,4,4,4,6}
[4,1,1] => [[4,4,4],[3,3]]
=> [3,3]
=> [3]
=> 3
[4,2] => [[5,4],[3]]
=> [3]
=> []
=> ? ∊ {0,0,0,0,2,2,2,2,2,2,2,3,3,3,3,3,3,3,4,4,4,6}
[5,1] => [[5,5],[4]]
=> [4]
=> []
=> ? ∊ {0,0,0,0,2,2,2,2,2,2,2,3,3,3,3,3,3,3,4,4,4,6}
[1,1,2,1,1,1] => [[2,2,2,2,1,1],[1,1,1]]
=> [1,1,1]
=> [1,1]
=> 1
[1,1,3,1,1] => [[3,3,3,1,1],[2,2]]
=> [2,2]
=> [2]
=> 2
[1,2,1,1,1,1] => [[2,2,2,2,2,1],[1,1,1,1]]
=> [1,1,1,1]
=> [1,1,1]
=> 1
[1,2,1,1,2] => [[3,2,2,2,1],[1,1,1]]
=> [1,1,1]
=> [1,1]
=> 1
[1,2,1,2,1] => [[3,3,2,2,1],[2,1,1]]
=> [2,1,1]
=> [1,1]
=> 1
[1,2,2,1,1] => [[3,3,3,2,1],[2,2,1]]
=> [2,2,1]
=> [2,1]
=> 4
[1,3,1,1,1] => [[3,3,3,3,1],[2,2,2]]
=> [2,2,2]
=> [2,2]
=> 2
[1,3,1,2] => [[4,3,3,1],[2,2]]
=> [2,2]
=> [2]
=> 2
[1,3,2,1] => [[4,4,3,1],[3,2]]
=> [3,2]
=> [2]
=> 2
[1,4,1,1] => [[4,4,4,1],[3,3]]
=> [3,3]
=> [3]
=> 3
[2,1,1,1,1,1] => [[2,2,2,2,2,2],[1,1,1,1,1]]
=> [1,1,1,1,1]
=> [1,1,1,1]
=> 1
[2,1,1,1,2] => [[3,2,2,2,2],[1,1,1,1]]
=> [1,1,1,1]
=> [1,1,1]
=> 1
[2,1,1,2,1] => [[3,3,2,2,2],[2,1,1,1]]
=> [2,1,1,1]
=> [1,1,1]
=> 1
[2,1,1,3] => [[4,2,2,2],[1,1,1]]
=> [1,1,1]
=> [1,1]
=> 1
[2,1,2,1,1] => [[3,3,3,2,2],[2,2,1,1]]
=> [2,2,1,1]
=> [2,1,1]
=> 5
[2,1,2,2] => [[4,3,2,2],[2,1,1]]
=> [2,1,1]
=> [1,1]
=> 1
[2,1,3,1] => [[4,4,2,2],[3,1,1]]
=> [3,1,1]
=> [1,1]
=> 1
[2,2,1,1,1] => [[3,3,3,3,2],[2,2,2,1]]
=> [2,2,2,1]
=> [2,2,1]
=> 4
[2,2,1,2] => [[4,3,3,2],[2,2,1]]
=> [2,2,1]
=> [2,1]
=> 4
[2,2,2,1] => [[4,4,3,2],[3,2,1]]
=> [3,2,1]
=> [2,1]
=> 4
[2,3,1,1] => [[4,4,4,2],[3,3,1]]
=> [3,3,1]
=> [3,1]
=> 5
[3,1,1,1,1] => [[3,3,3,3,3],[2,2,2,2]]
=> [2,2,2,2]
=> [2,2,2]
=> 2
[3,1,1,2] => [[4,3,3,3],[2,2,2]]
=> [2,2,2]
=> [2,2]
=> 2
[3,1,2,1] => [[4,4,3,3],[3,2,2]]
=> [3,2,2]
=> [2,2]
=> 2
[3,1,3] => [[5,3,3],[2,2]]
=> [2,2]
=> [2]
=> 2
[3,2,1,1] => [[4,4,4,3],[3,3,2]]
=> [3,3,2]
=> [3,2]
=> 6
[3,2,2] => [[5,4,3],[3,2]]
=> [3,2]
=> [2]
=> 2
[3,3,1] => [[5,5,3],[4,2]]
=> [4,2]
=> [2]
=> 2
[4,1,1,1] => [[4,4,4,4],[3,3,3]]
=> [3,3,3]
=> [3,3]
=> 3
[4,1,2] => [[5,4,4],[3,3]]
=> [3,3]
=> [3]
=> 3
[4,2,1] => [[5,5,4],[4,3]]
=> [4,3]
=> [3]
=> 3
[5,1,1] => [[5,5,5],[4,4]]
=> [4,4]
=> [4]
=> 4
Description
The major index of an integer partition when read from bottom to top.
This is the sum of the positions of the corners of the shape of an integer partition when reading from bottom to top.
For example, the partition $\lambda = (8,6,6,4,3,3)$ has corners at positions 3,6,9, and 13, giving a major index of 31.
Matching statistic: St000937
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Mp00180: Integer compositions —to ribbon⟶ Skew partitions
Mp00183: Skew partitions —inner shape⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000937: Integer partitions ⟶ ℤResult quality: 35% ●values known / values provided: 35%●distinct values known / distinct values provided: 62%
Mp00183: Skew partitions —inner shape⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000937: Integer partitions ⟶ ℤResult quality: 35% ●values known / values provided: 35%●distinct values known / distinct values provided: 62%
Values
[1] => [[1],[]]
=> []
=> ?
=> ? = 0
[1,1] => [[1,1],[]]
=> []
=> ?
=> ? ∊ {0,0}
[2] => [[2],[]]
=> []
=> ?
=> ? ∊ {0,0}
[1,1,1] => [[1,1,1],[]]
=> []
=> ?
=> ? ∊ {0,0,1,1}
[1,2] => [[2,1],[]]
=> []
=> ?
=> ? ∊ {0,0,1,1}
[2,1] => [[2,2],[1]]
=> [1]
=> []
=> ? ∊ {0,0,1,1}
[3] => [[3],[]]
=> []
=> ?
=> ? ∊ {0,0,1,1}
[1,1,1,1] => [[1,1,1,1],[]]
=> []
=> ?
=> ? ∊ {0,0,0,1,1,2,2,2}
[1,1,2] => [[2,1,1],[]]
=> []
=> ?
=> ? ∊ {0,0,0,1,1,2,2,2}
[1,2,1] => [[2,2,1],[1]]
=> [1]
=> []
=> ? ∊ {0,0,0,1,1,2,2,2}
[1,3] => [[3,1],[]]
=> []
=> ?
=> ? ∊ {0,0,0,1,1,2,2,2}
[2,1,1] => [[2,2,2],[1,1]]
=> [1,1]
=> [1]
=> ? ∊ {0,0,0,1,1,2,2,2}
[2,2] => [[3,2],[1]]
=> [1]
=> []
=> ? ∊ {0,0,0,1,1,2,2,2}
[3,1] => [[3,3],[2]]
=> [2]
=> []
=> ? ∊ {0,0,0,1,1,2,2,2}
[4] => [[4],[]]
=> []
=> ?
=> ? ∊ {0,0,0,1,1,2,2,2}
[1,1,1,1,1] => [[1,1,1,1,1],[]]
=> []
=> ?
=> ? ∊ {0,0,1,1,1,1,1,2,2,2,2,3,3,4}
[1,1,1,2] => [[2,1,1,1],[]]
=> []
=> ?
=> ? ∊ {0,0,1,1,1,1,1,2,2,2,2,3,3,4}
[1,1,2,1] => [[2,2,1,1],[1]]
=> [1]
=> []
=> ? ∊ {0,0,1,1,1,1,1,2,2,2,2,3,3,4}
[1,1,3] => [[3,1,1],[]]
=> []
=> ?
=> ? ∊ {0,0,1,1,1,1,1,2,2,2,2,3,3,4}
[1,2,1,1] => [[2,2,2,1],[1,1]]
=> [1,1]
=> [1]
=> ? ∊ {0,0,1,1,1,1,1,2,2,2,2,3,3,4}
[1,2,2] => [[3,2,1],[1]]
=> [1]
=> []
=> ? ∊ {0,0,1,1,1,1,1,2,2,2,2,3,3,4}
[1,3,1] => [[3,3,1],[2]]
=> [2]
=> []
=> ? ∊ {0,0,1,1,1,1,1,2,2,2,2,3,3,4}
[1,4] => [[4,1],[]]
=> []
=> ?
=> ? ∊ {0,0,1,1,1,1,1,2,2,2,2,3,3,4}
[2,1,1,1] => [[2,2,2,2],[1,1,1]]
=> [1,1,1]
=> [1,1]
=> 1
[2,1,2] => [[3,2,2],[1,1]]
=> [1,1]
=> [1]
=> ? ∊ {0,0,1,1,1,1,1,2,2,2,2,3,3,4}
[2,2,1] => [[3,3,2],[2,1]]
=> [2,1]
=> [1]
=> ? ∊ {0,0,1,1,1,1,1,2,2,2,2,3,3,4}
[2,3] => [[4,2],[1]]
=> [1]
=> []
=> ? ∊ {0,0,1,1,1,1,1,2,2,2,2,3,3,4}
[3,1,1] => [[3,3,3],[2,2]]
=> [2,2]
=> [2]
=> 2
[3,2] => [[4,3],[2]]
=> [2]
=> []
=> ? ∊ {0,0,1,1,1,1,1,2,2,2,2,3,3,4}
[4,1] => [[4,4],[3]]
=> [3]
=> []
=> ? ∊ {0,0,1,1,1,1,1,2,2,2,2,3,3,4}
[5] => [[5],[]]
=> []
=> ?
=> ? ∊ {0,0,1,1,1,1,1,2,2,2,2,3,3,4}
[1,1,1,1,1,1] => [[1,1,1,1,1,1],[]]
=> []
=> ?
=> ? ∊ {0,0,0,0,2,2,2,2,2,2,3,3,3,3,3,3,3,4,4,4,4,6}
[1,1,1,1,2] => [[2,1,1,1,1],[]]
=> []
=> ?
=> ? ∊ {0,0,0,0,2,2,2,2,2,2,3,3,3,3,3,3,3,4,4,4,4,6}
[1,1,1,2,1] => [[2,2,1,1,1],[1]]
=> [1]
=> []
=> ? ∊ {0,0,0,0,2,2,2,2,2,2,3,3,3,3,3,3,3,4,4,4,4,6}
[1,1,1,3] => [[3,1,1,1],[]]
=> []
=> ?
=> ? ∊ {0,0,0,0,2,2,2,2,2,2,3,3,3,3,3,3,3,4,4,4,4,6}
[1,1,2,1,1] => [[2,2,2,1,1],[1,1]]
=> [1,1]
=> [1]
=> ? ∊ {0,0,0,0,2,2,2,2,2,2,3,3,3,3,3,3,3,4,4,4,4,6}
[1,1,2,2] => [[3,2,1,1],[1]]
=> [1]
=> []
=> ? ∊ {0,0,0,0,2,2,2,2,2,2,3,3,3,3,3,3,3,4,4,4,4,6}
[1,1,3,1] => [[3,3,1,1],[2]]
=> [2]
=> []
=> ? ∊ {0,0,0,0,2,2,2,2,2,2,3,3,3,3,3,3,3,4,4,4,4,6}
[1,1,4] => [[4,1,1],[]]
=> []
=> ?
=> ? ∊ {0,0,0,0,2,2,2,2,2,2,3,3,3,3,3,3,3,4,4,4,4,6}
[1,2,1,1,1] => [[2,2,2,2,1],[1,1,1]]
=> [1,1,1]
=> [1,1]
=> 1
[1,2,1,2] => [[3,2,2,1],[1,1]]
=> [1,1]
=> [1]
=> ? ∊ {0,0,0,0,2,2,2,2,2,2,3,3,3,3,3,3,3,4,4,4,4,6}
[1,2,2,1] => [[3,3,2,1],[2,1]]
=> [2,1]
=> [1]
=> ? ∊ {0,0,0,0,2,2,2,2,2,2,3,3,3,3,3,3,3,4,4,4,4,6}
[1,2,3] => [[4,2,1],[1]]
=> [1]
=> []
=> ? ∊ {0,0,0,0,2,2,2,2,2,2,3,3,3,3,3,3,3,4,4,4,4,6}
[1,3,1,1] => [[3,3,3,1],[2,2]]
=> [2,2]
=> [2]
=> 2
[1,3,2] => [[4,3,1],[2]]
=> [2]
=> []
=> ? ∊ {0,0,0,0,2,2,2,2,2,2,3,3,3,3,3,3,3,4,4,4,4,6}
[1,4,1] => [[4,4,1],[3]]
=> [3]
=> []
=> ? ∊ {0,0,0,0,2,2,2,2,2,2,3,3,3,3,3,3,3,4,4,4,4,6}
[1,5] => [[5,1],[]]
=> []
=> ?
=> ? ∊ {0,0,0,0,2,2,2,2,2,2,3,3,3,3,3,3,3,4,4,4,4,6}
[2,1,1,1,1] => [[2,2,2,2,2],[1,1,1,1]]
=> [1,1,1,1]
=> [1,1,1]
=> 2
[2,1,1,2] => [[3,2,2,2],[1,1,1]]
=> [1,1,1]
=> [1,1]
=> 1
[2,1,2,1] => [[3,3,2,2],[2,1,1]]
=> [2,1,1]
=> [1,1]
=> 1
[2,1,3] => [[4,2,2],[1,1]]
=> [1,1]
=> [1]
=> ? ∊ {0,0,0,0,2,2,2,2,2,2,3,3,3,3,3,3,3,4,4,4,4,6}
[2,2,1,1] => [[3,3,3,2],[2,2,1]]
=> [2,2,1]
=> [2,1]
=> 1
[2,2,2] => [[4,3,2],[2,1]]
=> [2,1]
=> [1]
=> ? ∊ {0,0,0,0,2,2,2,2,2,2,3,3,3,3,3,3,3,4,4,4,4,6}
[2,3,1] => [[4,4,2],[3,1]]
=> [3,1]
=> [1]
=> ? ∊ {0,0,0,0,2,2,2,2,2,2,3,3,3,3,3,3,3,4,4,4,4,6}
[2,4] => [[5,2],[1]]
=> [1]
=> []
=> ? ∊ {0,0,0,0,2,2,2,2,2,2,3,3,3,3,3,3,3,4,4,4,4,6}
[3,1,1,1] => [[3,3,3,3],[2,2,2]]
=> [2,2,2]
=> [2,2]
=> 2
[3,1,2] => [[4,3,3],[2,2]]
=> [2,2]
=> [2]
=> 2
[3,2,1] => [[4,4,3],[3,2]]
=> [3,2]
=> [2]
=> 2
[3,3] => [[5,3],[2]]
=> [2]
=> []
=> ? ∊ {0,0,0,0,2,2,2,2,2,2,3,3,3,3,3,3,3,4,4,4,4,6}
[4,1,1] => [[4,4,4],[3,3]]
=> [3,3]
=> [3]
=> 3
[4,2] => [[5,4],[3]]
=> [3]
=> []
=> ? ∊ {0,0,0,0,2,2,2,2,2,2,3,3,3,3,3,3,3,4,4,4,4,6}
[5,1] => [[5,5],[4]]
=> [4]
=> []
=> ? ∊ {0,0,0,0,2,2,2,2,2,2,3,3,3,3,3,3,3,4,4,4,4,6}
[1,1,2,1,1,1] => [[2,2,2,2,1,1],[1,1,1]]
=> [1,1,1]
=> [1,1]
=> 1
[1,1,3,1,1] => [[3,3,3,1,1],[2,2]]
=> [2,2]
=> [2]
=> 2
[1,2,1,1,1,1] => [[2,2,2,2,2,1],[1,1,1,1]]
=> [1,1,1,1]
=> [1,1,1]
=> 2
[1,2,1,1,2] => [[3,2,2,2,1],[1,1,1]]
=> [1,1,1]
=> [1,1]
=> 1
[1,2,1,2,1] => [[3,3,2,2,1],[2,1,1]]
=> [2,1,1]
=> [1,1]
=> 1
[1,2,2,1,1] => [[3,3,3,2,1],[2,2,1]]
=> [2,2,1]
=> [2,1]
=> 1
[1,3,1,1,1] => [[3,3,3,3,1],[2,2,2]]
=> [2,2,2]
=> [2,2]
=> 2
[1,3,1,2] => [[4,3,3,1],[2,2]]
=> [2,2]
=> [2]
=> 2
[1,3,2,1] => [[4,4,3,1],[3,2]]
=> [3,2]
=> [2]
=> 2
[1,4,1,1] => [[4,4,4,1],[3,3]]
=> [3,3]
=> [3]
=> 3
[2,1,1,1,1,1] => [[2,2,2,2,2,2],[1,1,1,1,1]]
=> [1,1,1,1,1]
=> [1,1,1,1]
=> 3
[2,1,1,1,2] => [[3,2,2,2,2],[1,1,1,1]]
=> [1,1,1,1]
=> [1,1,1]
=> 2
[2,1,1,2,1] => [[3,3,2,2,2],[2,1,1,1]]
=> [2,1,1,1]
=> [1,1,1]
=> 2
[2,1,1,3] => [[4,2,2,2],[1,1,1]]
=> [1,1,1]
=> [1,1]
=> 1
[2,1,2,1,1] => [[3,3,3,2,2],[2,2,1,1]]
=> [2,2,1,1]
=> [2,1,1]
=> 2
[2,1,2,2] => [[4,3,2,2],[2,1,1]]
=> [2,1,1]
=> [1,1]
=> 1
[2,1,3,1] => [[4,4,2,2],[3,1,1]]
=> [3,1,1]
=> [1,1]
=> 1
[2,2,1,1,1] => [[3,3,3,3,2],[2,2,2,1]]
=> [2,2,2,1]
=> [2,2,1]
=> 3
[2,2,1,2] => [[4,3,3,2],[2,2,1]]
=> [2,2,1]
=> [2,1]
=> 1
[2,2,2,1] => [[4,4,3,2],[3,2,1]]
=> [3,2,1]
=> [2,1]
=> 1
[2,3,1,1] => [[4,4,4,2],[3,3,1]]
=> [3,3,1]
=> [3,1]
=> 2
[3,1,1,1,1] => [[3,3,3,3,3],[2,2,2,2]]
=> [2,2,2,2]
=> [2,2,2]
=> 5
[3,1,1,2] => [[4,3,3,3],[2,2,2]]
=> [2,2,2]
=> [2,2]
=> 2
[3,1,2,1] => [[4,4,3,3],[3,2,2]]
=> [3,2,2]
=> [2,2]
=> 2
[3,1,3] => [[5,3,3],[2,2]]
=> [2,2]
=> [2]
=> 2
[3,2,1,1] => [[4,4,4,3],[3,3,2]]
=> [3,3,2]
=> [3,2]
=> 4
[3,2,2] => [[5,4,3],[3,2]]
=> [3,2]
=> [2]
=> 2
[3,3,1] => [[5,5,3],[4,2]]
=> [4,2]
=> [2]
=> 2
[4,1,1,1] => [[4,4,4,4],[3,3,3]]
=> [3,3,3]
=> [3,3]
=> 5
[4,1,2] => [[5,4,4],[3,3]]
=> [3,3]
=> [3]
=> 3
[4,2,1] => [[5,5,4],[4,3]]
=> [4,3]
=> [3]
=> 3
[5,1,1] => [[5,5,5],[4,4]]
=> [4,4]
=> [4]
=> 5
Description
The number of positive values of the symmetric group character corresponding to the partition.
For example, the character values of the irreducible representation $S^{(2,2)}$ are $2$ on the conjugacy classes $(4)$ and $(2,2)$, $0$ on the conjugacy classes $(3,1)$ and $(1,1,1,1)$, and $-1$ on the conjugacy class $(2,1,1)$. Therefore, the statistic on the partition $(2,2)$ is $2$.
Matching statistic: St000939
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00180: Integer compositions —to ribbon⟶ Skew partitions
Mp00183: Skew partitions —inner shape⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000939: Integer partitions ⟶ ℤResult quality: 35% ●values known / values provided: 35%●distinct values known / distinct values provided: 75%
Mp00183: Skew partitions —inner shape⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000939: Integer partitions ⟶ ℤResult quality: 35% ●values known / values provided: 35%●distinct values known / distinct values provided: 75%
Values
[1] => [[1],[]]
=> []
=> ?
=> ? = 0
[1,1] => [[1,1],[]]
=> []
=> ?
=> ? ∊ {0,0}
[2] => [[2],[]]
=> []
=> ?
=> ? ∊ {0,0}
[1,1,1] => [[1,1,1],[]]
=> []
=> ?
=> ? ∊ {0,0,1,1}
[1,2] => [[2,1],[]]
=> []
=> ?
=> ? ∊ {0,0,1,1}
[2,1] => [[2,2],[1]]
=> [1]
=> []
=> ? ∊ {0,0,1,1}
[3] => [[3],[]]
=> []
=> ?
=> ? ∊ {0,0,1,1}
[1,1,1,1] => [[1,1,1,1],[]]
=> []
=> ?
=> ? ∊ {0,0,0,1,1,2,2,2}
[1,1,2] => [[2,1,1],[]]
=> []
=> ?
=> ? ∊ {0,0,0,1,1,2,2,2}
[1,2,1] => [[2,2,1],[1]]
=> [1]
=> []
=> ? ∊ {0,0,0,1,1,2,2,2}
[1,3] => [[3,1],[]]
=> []
=> ?
=> ? ∊ {0,0,0,1,1,2,2,2}
[2,1,1] => [[2,2,2],[1,1]]
=> [1,1]
=> [1]
=> ? ∊ {0,0,0,1,1,2,2,2}
[2,2] => [[3,2],[1]]
=> [1]
=> []
=> ? ∊ {0,0,0,1,1,2,2,2}
[3,1] => [[3,3],[2]]
=> [2]
=> []
=> ? ∊ {0,0,0,1,1,2,2,2}
[4] => [[4],[]]
=> []
=> ?
=> ? ∊ {0,0,0,1,1,2,2,2}
[1,1,1,1,1] => [[1,1,1,1,1],[]]
=> []
=> ?
=> ? ∊ {0,0,1,1,1,1,1,2,2,2,2,3,3,4}
[1,1,1,2] => [[2,1,1,1],[]]
=> []
=> ?
=> ? ∊ {0,0,1,1,1,1,1,2,2,2,2,3,3,4}
[1,1,2,1] => [[2,2,1,1],[1]]
=> [1]
=> []
=> ? ∊ {0,0,1,1,1,1,1,2,2,2,2,3,3,4}
[1,1,3] => [[3,1,1],[]]
=> []
=> ?
=> ? ∊ {0,0,1,1,1,1,1,2,2,2,2,3,3,4}
[1,2,1,1] => [[2,2,2,1],[1,1]]
=> [1,1]
=> [1]
=> ? ∊ {0,0,1,1,1,1,1,2,2,2,2,3,3,4}
[1,2,2] => [[3,2,1],[1]]
=> [1]
=> []
=> ? ∊ {0,0,1,1,1,1,1,2,2,2,2,3,3,4}
[1,3,1] => [[3,3,1],[2]]
=> [2]
=> []
=> ? ∊ {0,0,1,1,1,1,1,2,2,2,2,3,3,4}
[1,4] => [[4,1],[]]
=> []
=> ?
=> ? ∊ {0,0,1,1,1,1,1,2,2,2,2,3,3,4}
[2,1,1,1] => [[2,2,2,2],[1,1,1]]
=> [1,1,1]
=> [1,1]
=> 2
[2,1,2] => [[3,2,2],[1,1]]
=> [1,1]
=> [1]
=> ? ∊ {0,0,1,1,1,1,1,2,2,2,2,3,3,4}
[2,2,1] => [[3,3,2],[2,1]]
=> [2,1]
=> [1]
=> ? ∊ {0,0,1,1,1,1,1,2,2,2,2,3,3,4}
[2,3] => [[4,2],[1]]
=> [1]
=> []
=> ? ∊ {0,0,1,1,1,1,1,2,2,2,2,3,3,4}
[3,1,1] => [[3,3,3],[2,2]]
=> [2,2]
=> [2]
=> 1
[3,2] => [[4,3],[2]]
=> [2]
=> []
=> ? ∊ {0,0,1,1,1,1,1,2,2,2,2,3,3,4}
[4,1] => [[4,4],[3]]
=> [3]
=> []
=> ? ∊ {0,0,1,1,1,1,1,2,2,2,2,3,3,4}
[5] => [[5],[]]
=> []
=> ?
=> ? ∊ {0,0,1,1,1,1,1,2,2,2,2,3,3,4}
[1,1,1,1,1,1] => [[1,1,1,1,1,1],[]]
=> []
=> ?
=> ? ∊ {0,0,0,0,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,6}
[1,1,1,1,2] => [[2,1,1,1,1],[]]
=> []
=> ?
=> ? ∊ {0,0,0,0,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,6}
[1,1,1,2,1] => [[2,2,1,1,1],[1]]
=> [1]
=> []
=> ? ∊ {0,0,0,0,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,6}
[1,1,1,3] => [[3,1,1,1],[]]
=> []
=> ?
=> ? ∊ {0,0,0,0,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,6}
[1,1,2,1,1] => [[2,2,2,1,1],[1,1]]
=> [1,1]
=> [1]
=> ? ∊ {0,0,0,0,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,6}
[1,1,2,2] => [[3,2,1,1],[1]]
=> [1]
=> []
=> ? ∊ {0,0,0,0,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,6}
[1,1,3,1] => [[3,3,1,1],[2]]
=> [2]
=> []
=> ? ∊ {0,0,0,0,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,6}
[1,1,4] => [[4,1,1],[]]
=> []
=> ?
=> ? ∊ {0,0,0,0,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,6}
[1,2,1,1,1] => [[2,2,2,2,1],[1,1,1]]
=> [1,1,1]
=> [1,1]
=> 2
[1,2,1,2] => [[3,2,2,1],[1,1]]
=> [1,1]
=> [1]
=> ? ∊ {0,0,0,0,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,6}
[1,2,2,1] => [[3,3,2,1],[2,1]]
=> [2,1]
=> [1]
=> ? ∊ {0,0,0,0,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,6}
[1,2,3] => [[4,2,1],[1]]
=> [1]
=> []
=> ? ∊ {0,0,0,0,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,6}
[1,3,1,1] => [[3,3,3,1],[2,2]]
=> [2,2]
=> [2]
=> 1
[1,3,2] => [[4,3,1],[2]]
=> [2]
=> []
=> ? ∊ {0,0,0,0,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,6}
[1,4,1] => [[4,4,1],[3]]
=> [3]
=> []
=> ? ∊ {0,0,0,0,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,6}
[1,5] => [[5,1],[]]
=> []
=> ?
=> ? ∊ {0,0,0,0,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,6}
[2,1,1,1,1] => [[2,2,2,2,2],[1,1,1,1]]
=> [1,1,1,1]
=> [1,1,1]
=> 3
[2,1,1,2] => [[3,2,2,2],[1,1,1]]
=> [1,1,1]
=> [1,1]
=> 2
[2,1,2,1] => [[3,3,2,2],[2,1,1]]
=> [2,1,1]
=> [1,1]
=> 2
[2,1,3] => [[4,2,2],[1,1]]
=> [1,1]
=> [1]
=> ? ∊ {0,0,0,0,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,6}
[2,2,1,1] => [[3,3,3,2],[2,2,1]]
=> [2,2,1]
=> [2,1]
=> 1
[2,2,2] => [[4,3,2],[2,1]]
=> [2,1]
=> [1]
=> ? ∊ {0,0,0,0,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,6}
[2,3,1] => [[4,4,2],[3,1]]
=> [3,1]
=> [1]
=> ? ∊ {0,0,0,0,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,6}
[2,4] => [[5,2],[1]]
=> [1]
=> []
=> ? ∊ {0,0,0,0,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,6}
[3,1,1,1] => [[3,3,3,3],[2,2,2]]
=> [2,2,2]
=> [2,2]
=> 3
[3,1,2] => [[4,3,3],[2,2]]
=> [2,2]
=> [2]
=> 1
[3,2,1] => [[4,4,3],[3,2]]
=> [3,2]
=> [2]
=> 1
[3,3] => [[5,3],[2]]
=> [2]
=> []
=> ? ∊ {0,0,0,0,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,6}
[4,1,1] => [[4,4,4],[3,3]]
=> [3,3]
=> [3]
=> 2
[4,2] => [[5,4],[3]]
=> [3]
=> []
=> ? ∊ {0,0,0,0,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,6}
[5,1] => [[5,5],[4]]
=> [4]
=> []
=> ? ∊ {0,0,0,0,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,6}
[1,1,2,1,1,1] => [[2,2,2,2,1,1],[1,1,1]]
=> [1,1,1]
=> [1,1]
=> 2
[1,1,3,1,1] => [[3,3,3,1,1],[2,2]]
=> [2,2]
=> [2]
=> 1
[1,2,1,1,1,1] => [[2,2,2,2,2,1],[1,1,1,1]]
=> [1,1,1,1]
=> [1,1,1]
=> 3
[1,2,1,1,2] => [[3,2,2,2,1],[1,1,1]]
=> [1,1,1]
=> [1,1]
=> 2
[1,2,1,2,1] => [[3,3,2,2,1],[2,1,1]]
=> [2,1,1]
=> [1,1]
=> 2
[1,2,2,1,1] => [[3,3,3,2,1],[2,2,1]]
=> [2,2,1]
=> [2,1]
=> 1
[1,3,1,1,1] => [[3,3,3,3,1],[2,2,2]]
=> [2,2,2]
=> [2,2]
=> 3
[1,3,1,2] => [[4,3,3,1],[2,2]]
=> [2,2]
=> [2]
=> 1
[1,3,2,1] => [[4,4,3,1],[3,2]]
=> [3,2]
=> [2]
=> 1
[1,4,1,1] => [[4,4,4,1],[3,3]]
=> [3,3]
=> [3]
=> 2
[2,1,1,1,1,1] => [[2,2,2,2,2,2],[1,1,1,1,1]]
=> [1,1,1,1,1]
=> [1,1,1,1]
=> 5
[2,1,1,1,2] => [[3,2,2,2,2],[1,1,1,1]]
=> [1,1,1,1]
=> [1,1,1]
=> 3
[2,1,1,2,1] => [[3,3,2,2,2],[2,1,1,1]]
=> [2,1,1,1]
=> [1,1,1]
=> 3
[2,1,1,3] => [[4,2,2,2],[1,1,1]]
=> [1,1,1]
=> [1,1]
=> 2
[2,1,2,1,1] => [[3,3,3,2,2],[2,2,1,1]]
=> [2,2,1,1]
=> [2,1,1]
=> 2
[2,1,2,2] => [[4,3,2,2],[2,1,1]]
=> [2,1,1]
=> [1,1]
=> 2
[2,1,3,1] => [[4,4,2,2],[3,1,1]]
=> [3,1,1]
=> [1,1]
=> 2
[2,2,1,1,1] => [[3,3,3,3,2],[2,2,2,1]]
=> [2,2,2,1]
=> [2,2,1]
=> 4
[2,2,1,2] => [[4,3,3,2],[2,2,1]]
=> [2,2,1]
=> [2,1]
=> 1
[2,2,2,1] => [[4,4,3,2],[3,2,1]]
=> [3,2,1]
=> [2,1]
=> 1
[2,3,1,1] => [[4,4,4,2],[3,3,1]]
=> [3,3,1]
=> [3,1]
=> 2
[3,1,1,1,1] => [[3,3,3,3,3],[2,2,2,2]]
=> [2,2,2,2]
=> [2,2,2]
=> 5
[3,1,1,2] => [[4,3,3,3],[2,2,2]]
=> [2,2,2]
=> [2,2]
=> 3
[3,1,2,1] => [[4,4,3,3],[3,2,2]]
=> [3,2,2]
=> [2,2]
=> 3
[3,1,3] => [[5,3,3],[2,2]]
=> [2,2]
=> [2]
=> 1
[3,2,1,1] => [[4,4,4,3],[3,3,2]]
=> [3,3,2]
=> [3,2]
=> 3
[3,2,2] => [[5,4,3],[3,2]]
=> [3,2]
=> [2]
=> 1
[3,3,1] => [[5,5,3],[4,2]]
=> [4,2]
=> [2]
=> 1
[4,1,1,1] => [[4,4,4,4],[3,3,3]]
=> [3,3,3]
=> [3,3]
=> 6
[4,1,2] => [[5,4,4],[3,3]]
=> [3,3]
=> [3]
=> 2
[4,2,1] => [[5,5,4],[4,3]]
=> [4,3]
=> [3]
=> 2
[5,1,1] => [[5,5,5],[4,4]]
=> [4,4]
=> [4]
=> 2
Description
The number of characters of the symmetric group whose value on the partition is positive.
Matching statistic: St000681
Mp00184: Integer compositions —to threshold graph⟶ Graphs
Mp00037: Graphs —to partition of connected components⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000681: Integer partitions ⟶ ℤResult quality: 24% ●values known / values provided: 24%●distinct values known / distinct values provided: 62%
Mp00037: Graphs —to partition of connected components⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000681: Integer partitions ⟶ ℤResult quality: 24% ●values known / values provided: 24%●distinct values known / distinct values provided: 62%
Values
[1] => ([],1)
=> [1]
=> []
=> ? = 0
[1,1] => ([(0,1)],2)
=> [2]
=> []
=> ? ∊ {0,0}
[2] => ([],2)
=> [1,1]
=> [1]
=> ? ∊ {0,0}
[1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> [3]
=> []
=> ? ∊ {0,0,1}
[1,2] => ([(1,2)],3)
=> [2,1]
=> [1]
=> ? ∊ {0,0,1}
[2,1] => ([(0,2),(1,2)],3)
=> [3]
=> []
=> ? ∊ {0,0,1}
[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]
=> []
=> ? ∊ {0,0,0,1,2,2}
[1,1,2] => ([(1,2),(1,3),(2,3)],4)
=> [3,1]
=> [1]
=> ? ∊ {0,0,0,1,2,2}
[1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> [4]
=> []
=> ? ∊ {0,0,0,1,2,2}
[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]
=> []
=> ? ∊ {0,0,0,1,2,2}
[2,2] => ([(1,3),(2,3)],4)
=> [3,1]
=> [1]
=> ? ∊ {0,0,0,1,2,2}
[3,1] => ([(0,3),(1,3),(2,3)],4)
=> [4]
=> []
=> ? ∊ {0,0,0,1,2,2}
[4] => ([],4)
=> [1,1,1,1]
=> [1,1,1]
=> 2
[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]
=> []
=> ? ∊ {0,0,1,1,1,1,2,2,2,2,3,4}
[1,1,1,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [4,1]
=> [1]
=> ? ∊ {0,0,1,1,1,1,2,2,2,2,3,4}
[1,1,2,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> []
=> ? ∊ {0,0,1,1,1,1,2,2,2,2,3,4}
[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]
=> []
=> ? ∊ {0,0,1,1,1,1,2,2,2,2,3,4}
[1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> [4,1]
=> [1]
=> ? ∊ {0,0,1,1,1,1,2,2,2,2,3,4}
[1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> []
=> ? ∊ {0,0,1,1,1,1,2,2,2,2,3,4}
[1,4] => ([(3,4)],5)
=> [2,1,1,1]
=> [1,1,1]
=> 2
[2,1,1,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> []
=> ? ∊ {0,0,1,1,1,1,2,2,2,2,3,4}
[2,1,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [4,1]
=> [1]
=> ? ∊ {0,0,1,1,1,1,2,2,2,2,3,4}
[2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> []
=> ? ∊ {0,0,1,1,1,1,2,2,2,2,3,4}
[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]
=> []
=> ? ∊ {0,0,1,1,1,1,2,2,2,2,3,4}
[3,2] => ([(1,4),(2,4),(3,4)],5)
=> [4,1]
=> [1]
=> ? ∊ {0,0,1,1,1,1,2,2,2,2,3,4}
[4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> [5]
=> []
=> ? ∊ {0,0,1,1,1,1,2,2,2,2,3,4}
[5] => ([],5)
=> [1,1,1,1,1]
=> [1,1,1,1]
=> 3
[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]
=> []
=> ? ∊ {0,0,0,0,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,4,4,4,6}
[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]
=> ? ∊ {0,0,0,0,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,4,4,4,6}
[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]
=> []
=> ? ∊ {0,0,0,0,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,4,4,4,6}
[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]
=> []
=> ? ∊ {0,0,0,0,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,4,4,4,6}
[1,1,2,2] => ([(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [5,1]
=> [1]
=> ? ∊ {0,0,0,0,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,4,4,4,6}
[1,1,3,1] => ([(0,5),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [6]
=> []
=> ? ∊ {0,0,0,0,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,4,4,4,6}
[1,1,4] => ([(3,4),(3,5),(4,5)],6)
=> [3,1,1,1]
=> [1,1,1]
=> 2
[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]
=> []
=> ? ∊ {0,0,0,0,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,4,4,4,6}
[1,2,1,2] => ([(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [5,1]
=> [1]
=> ? ∊ {0,0,0,0,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,4,4,4,6}
[1,2,2,1] => ([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [6]
=> []
=> ? ∊ {0,0,0,0,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,4,4,4,6}
[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]
=> []
=> ? ∊ {0,0,0,0,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,4,4,4,6}
[1,3,2] => ([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> [5,1]
=> [1]
=> ? ∊ {0,0,0,0,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,4,4,4,6}
[1,4,1] => ([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> [6]
=> []
=> ? ∊ {0,0,0,0,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,4,4,4,6}
[1,5] => ([(4,5)],6)
=> [2,1,1,1,1]
=> [1,1,1,1]
=> 3
[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]
=> []
=> ? ∊ {0,0,0,0,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,4,4,4,6}
[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]
=> ? ∊ {0,0,0,0,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,4,4,4,6}
[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]
=> []
=> ? ∊ {0,0,0,0,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,4,4,4,6}
[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]
=> []
=> ? ∊ {0,0,0,0,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,4,4,4,6}
[2,2,2] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [5,1]
=> [1]
=> ? ∊ {0,0,0,0,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,4,4,4,6}
[2,3,1] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [6]
=> []
=> ? ∊ {0,0,0,0,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,4,4,4,6}
[2,4] => ([(3,5),(4,5)],6)
=> [3,1,1,1]
=> [1,1,1]
=> 2
[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]
=> []
=> ? ∊ {0,0,0,0,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,4,4,4,6}
[3,1,2] => ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [5,1]
=> [1]
=> ? ∊ {0,0,0,0,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,4,4,4,6}
[3,2,1] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [6]
=> []
=> ? ∊ {0,0,0,0,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,4,4,4,6}
[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]
=> []
=> ? ∊ {0,0,0,0,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,4,4,4,6}
[4,2] => ([(1,5),(2,5),(3,5),(4,5)],6)
=> [5,1]
=> [1]
=> ? ∊ {0,0,0,0,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,4,4,4,6}
[5,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> [6]
=> []
=> ? ∊ {0,0,0,0,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,4,4,4,6}
[6] => ([],6)
=> [1,1,1,1,1,1]
=> [1,1,1,1,1]
=> 4
[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]
=> []
=> ? ∊ {0,0,1,1,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,6,6,8}
[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]
=> ? ∊ {0,0,1,1,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,6,6,8}
[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]
=> 2
[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]
=> 3
[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]
=> 2
[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]
=> 4
[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]
=> 2
[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]
=> 3
[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]
=> 2
[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]
=> 5
Description
The Grundy value of Chomp on Ferrers diagrams.
Players take turns and choose a cell of the diagram, cutting off all cells below and to the right of this cell in English notation. The player who is left with the single cell partition looses. The traditional version is played on chocolate bars, see [1].
This statistic is the Grundy value of the partition, that is, the smallest non-negative integer which does not occur as value of a partition obtained by a single move.
Matching statistic: St000706
Mp00133: Integer compositions —delta morphism⟶ Integer compositions
Mp00180: Integer compositions —to ribbon⟶ Skew partitions
Mp00183: Skew partitions —inner shape⟶ Integer partitions
St000706: Integer partitions ⟶ ℤResult quality: 20% ●values known / values provided: 20%●distinct values known / distinct values provided: 38%
Mp00180: Integer compositions —to ribbon⟶ Skew partitions
Mp00183: Skew partitions —inner shape⟶ Integer partitions
St000706: Integer partitions ⟶ ℤResult quality: 20% ●values known / values provided: 20%●distinct values known / distinct values provided: 38%
Values
[1] => [1] => [[1],[]]
=> []
=> ? = 0
[1,1] => [2] => [[2],[]]
=> []
=> ? ∊ {0,0}
[2] => [1] => [[1],[]]
=> []
=> ? ∊ {0,0}
[1,1,1] => [3] => [[3],[]]
=> []
=> ? ∊ {0,0,1,1}
[1,2] => [1,1] => [[1,1],[]]
=> []
=> ? ∊ {0,0,1,1}
[2,1] => [1,1] => [[1,1],[]]
=> []
=> ? ∊ {0,0,1,1}
[3] => [1] => [[1],[]]
=> []
=> ? ∊ {0,0,1,1}
[1,1,1,1] => [4] => [[4],[]]
=> []
=> ? ∊ {0,0,0,1,1,2,2,2}
[1,1,2] => [2,1] => [[2,2],[1]]
=> [1]
=> ? ∊ {0,0,0,1,1,2,2,2}
[1,2,1] => [1,1,1] => [[1,1,1],[]]
=> []
=> ? ∊ {0,0,0,1,1,2,2,2}
[1,3] => [1,1] => [[1,1],[]]
=> []
=> ? ∊ {0,0,0,1,1,2,2,2}
[2,1,1] => [1,2] => [[2,1],[]]
=> []
=> ? ∊ {0,0,0,1,1,2,2,2}
[2,2] => [2] => [[2],[]]
=> []
=> ? ∊ {0,0,0,1,1,2,2,2}
[3,1] => [1,1] => [[1,1],[]]
=> []
=> ? ∊ {0,0,0,1,1,2,2,2}
[4] => [1] => [[1],[]]
=> []
=> ? ∊ {0,0,0,1,1,2,2,2}
[1,1,1,1,1] => [5] => [[5],[]]
=> []
=> ? ∊ {0,0,1,1,1,1,1,2,2,2,2,3,3,4}
[1,1,1,2] => [3,1] => [[3,3],[2]]
=> [2]
=> 1
[1,1,2,1] => [2,1,1] => [[2,2,2],[1,1]]
=> [1,1]
=> 2
[1,1,3] => [2,1] => [[2,2],[1]]
=> [1]
=> ? ∊ {0,0,1,1,1,1,1,2,2,2,2,3,3,4}
[1,2,1,1] => [1,1,2] => [[2,1,1],[]]
=> []
=> ? ∊ {0,0,1,1,1,1,1,2,2,2,2,3,3,4}
[1,2,2] => [1,2] => [[2,1],[]]
=> []
=> ? ∊ {0,0,1,1,1,1,1,2,2,2,2,3,3,4}
[1,3,1] => [1,1,1] => [[1,1,1],[]]
=> []
=> ? ∊ {0,0,1,1,1,1,1,2,2,2,2,3,3,4}
[1,4] => [1,1] => [[1,1],[]]
=> []
=> ? ∊ {0,0,1,1,1,1,1,2,2,2,2,3,3,4}
[2,1,1,1] => [1,3] => [[3,1],[]]
=> []
=> ? ∊ {0,0,1,1,1,1,1,2,2,2,2,3,3,4}
[2,1,2] => [1,1,1] => [[1,1,1],[]]
=> []
=> ? ∊ {0,0,1,1,1,1,1,2,2,2,2,3,3,4}
[2,2,1] => [2,1] => [[2,2],[1]]
=> [1]
=> ? ∊ {0,0,1,1,1,1,1,2,2,2,2,3,3,4}
[2,3] => [1,1] => [[1,1],[]]
=> []
=> ? ∊ {0,0,1,1,1,1,1,2,2,2,2,3,3,4}
[3,1,1] => [1,2] => [[2,1],[]]
=> []
=> ? ∊ {0,0,1,1,1,1,1,2,2,2,2,3,3,4}
[3,2] => [1,1] => [[1,1],[]]
=> []
=> ? ∊ {0,0,1,1,1,1,1,2,2,2,2,3,3,4}
[4,1] => [1,1] => [[1,1],[]]
=> []
=> ? ∊ {0,0,1,1,1,1,1,2,2,2,2,3,3,4}
[5] => [1] => [[1],[]]
=> []
=> ? ∊ {0,0,1,1,1,1,1,2,2,2,2,3,3,4}
[1,1,1,1,1,1] => [6] => [[6],[]]
=> []
=> ? ∊ {0,0,0,0,1,1,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,4,4,6}
[1,1,1,1,2] => [4,1] => [[4,4],[3]]
=> [3]
=> 1
[1,1,1,2,1] => [3,1,1] => [[3,3,3],[2,2]]
=> [2,2]
=> 2
[1,1,1,3] => [3,1] => [[3,3],[2]]
=> [2]
=> 1
[1,1,2,1,1] => [2,1,2] => [[3,2,2],[1,1]]
=> [1,1]
=> 2
[1,1,2,2] => [2,2] => [[3,2],[1]]
=> [1]
=> ? ∊ {0,0,0,0,1,1,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,4,4,6}
[1,1,3,1] => [2,1,1] => [[2,2,2],[1,1]]
=> [1,1]
=> 2
[1,1,4] => [2,1] => [[2,2],[1]]
=> [1]
=> ? ∊ {0,0,0,0,1,1,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,4,4,6}
[1,2,1,1,1] => [1,1,3] => [[3,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,1,1,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,4,4,6}
[1,2,1,2] => [1,1,1,1] => [[1,1,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,1,1,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,4,4,6}
[1,2,2,1] => [1,2,1] => [[2,2,1],[1]]
=> [1]
=> ? ∊ {0,0,0,0,1,1,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,4,4,6}
[1,2,3] => [1,1,1] => [[1,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,1,1,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,4,4,6}
[1,3,1,1] => [1,1,2] => [[2,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,1,1,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,4,4,6}
[1,3,2] => [1,1,1] => [[1,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,1,1,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,4,4,6}
[1,4,1] => [1,1,1] => [[1,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,1,1,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,4,4,6}
[1,5] => [1,1] => [[1,1],[]]
=> []
=> ? ∊ {0,0,0,0,1,1,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,4,4,6}
[2,1,1,1,1] => [1,4] => [[4,1],[]]
=> []
=> ? ∊ {0,0,0,0,1,1,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,4,4,6}
[2,1,1,2] => [1,2,1] => [[2,2,1],[1]]
=> [1]
=> ? ∊ {0,0,0,0,1,1,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,4,4,6}
[2,1,2,1] => [1,1,1,1] => [[1,1,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,1,1,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,4,4,6}
[2,1,3] => [1,1,1] => [[1,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,1,1,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,4,4,6}
[2,2,1,1] => [2,2] => [[3,2],[1]]
=> [1]
=> ? ∊ {0,0,0,0,1,1,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,4,4,6}
[2,2,2] => [3] => [[3],[]]
=> []
=> ? ∊ {0,0,0,0,1,1,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,4,4,6}
[2,3,1] => [1,1,1] => [[1,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,1,1,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,4,4,6}
[2,4] => [1,1] => [[1,1],[]]
=> []
=> ? ∊ {0,0,0,0,1,1,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,4,4,6}
[3,1,1,1] => [1,3] => [[3,1],[]]
=> []
=> ? ∊ {0,0,0,0,1,1,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,4,4,6}
[3,1,2] => [1,1,1] => [[1,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,1,1,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,4,4,6}
[1,1,1,1,1,2] => [5,1] => [[5,5],[4]]
=> [4]
=> 1
[1,1,1,1,2,1] => [4,1,1] => [[4,4,4],[3,3]]
=> [3,3]
=> 2
[1,1,1,1,3] => [4,1] => [[4,4],[3]]
=> [3]
=> 1
[1,1,1,2,1,1] => [3,1,2] => [[4,3,3],[2,2]]
=> [2,2]
=> 2
[1,1,1,2,2] => [3,2] => [[4,3],[2]]
=> [2]
=> 1
[1,1,1,3,1] => [3,1,1] => [[3,3,3],[2,2]]
=> [2,2]
=> 2
[1,1,1,4] => [3,1] => [[3,3],[2]]
=> [2]
=> 1
[1,1,2,1,1,1] => [2,1,3] => [[4,2,2],[1,1]]
=> [1,1]
=> 2
[1,1,2,1,2] => [2,1,1,1] => [[2,2,2,2],[1,1,1]]
=> [1,1,1]
=> 6
[1,1,2,2,1] => [2,2,1] => [[3,3,2],[2,1]]
=> [2,1]
=> 1
[1,1,2,3] => [2,1,1] => [[2,2,2],[1,1]]
=> [1,1]
=> 2
[1,1,3,1,1] => [2,1,2] => [[3,2,2],[1,1]]
=> [1,1]
=> 2
[1,1,3,2] => [2,1,1] => [[2,2,2],[1,1]]
=> [1,1]
=> 2
[1,1,4,1] => [2,1,1] => [[2,2,2],[1,1]]
=> [1,1]
=> 2
[2,1,1,1,2] => [1,3,1] => [[3,3,1],[2]]
=> [2]
=> 1
[2,1,1,2,1] => [1,2,1,1] => [[2,2,2,1],[1,1]]
=> [1,1]
=> 2
[2,2,1,2] => [2,1,1] => [[2,2,2],[1,1]]
=> [1,1]
=> 2
[2,2,2,1] => [3,1] => [[3,3],[2]]
=> [2]
=> 1
Description
The product of the factorials of the multiplicities of an integer partition.
The following 10 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000714The number of semistandard Young tableau of given shape, with entries at most 2. St000993The multiplicity of the largest part of an integer partition. St001568The smallest positive integer that does not appear twice in the partition. St001624The breadth of a lattice. St000454The largest eigenvalue of a graph if it is integral. St001060The distinguishing index of a graph. St001605The number of colourings of a cycle such that the multiplicities of colours are given by a partition. St001423The number of distinct cubes in a binary word. St001556The number of inversions of the third entry of a permutation. St001811The Castelnuovo-Mumford regularity of a permutation.
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