- FindStat

Your data matches 171 different statistics following compositions of up to 3 maps.
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Matching statistic: St001810
(load all 33 compositions to match this statistic)

Mapping

St001810: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

[1] => 0
[1,2] => 0
[2,1] => 0
[1,2,3] => 0
[1,3,2] => 1
[2,1,3] => 0
[2,3,1] => 0
[3,1,2] => 0
[3,2,1] => 1
[1,2,3,4] => 0
[1,2,4,3] => 2
[1,3,2,4] => 1
[1,3,4,2] => 1
[1,4,2,3] => 1
[1,4,3,2] => 2
[2,1,3,4] => 0
[2,1,4,3] => 0
[2,3,1,4] => 0
[2,3,4,1] => 0
[2,4,1,3] => 0
[2,4,3,1] => 1
[3,1,2,4] => 0
[3,1,4,2] => 0
[3,2,1,4] => 1
[3,2,4,1] => 1
[3,4,1,2] => 0
[3,4,2,1] => 0
[4,1,2,3] => 0
[4,1,3,2] => 1
[4,2,1,3] => 1
[4,2,3,1] => 2
[4,3,1,2] => 0
[4,3,2,1] => 0
[1,2,3,4,5] => 0
[1,2,3,5,4] => 3
[1,2,4,3,5] => 2
[1,2,4,5,3] => 2
[1,2,5,3,4] => 2
[1,2,5,4,3] => 3
[1,3,2,4,5] => 1
[1,3,2,5,4] => 1
[1,3,4,2,5] => 1
[1,3,4,5,2] => 1
[1,3,5,2,4] => 1
[1,3,5,4,2] => 2
[1,4,2,3,5] => 1
[1,4,2,5,3] => 1
[1,4,3,2,5] => 2
[1,4,3,5,2] => 2
[1,4,5,2,3] => 1

Description

The number of fixed points of a permutation smaller than its largest moved point.

Matching statistic: St001644

Mapping

Mp00160: Permutations —graph of inversions⟶ Graphs
Mp00274: Graphs —block-cut tree⟶ Graphs
Mp00157: Graphs —connected complement⟶ Graphs
St001644: Graphs ⟶ ℤResult quality: 60% ●values known / values provided: 85%●distinct values known / distinct values provided: 60%

Values

[1] => ([],1)
 => ([],1)
 => ([],1)
 => 0
[1,2] => ([],2)
 => ([],2)
 => ([],2)
 => 0
[2,1] => ([(0,1)],2)
 => ([],1)
 => ([],1)
 => 0
[1,2,3] => ([],3)
 => ([],3)
 => ([],3)
 => 0
[1,3,2] => ([(1,2)],3)
 => ([],2)
 => ([],2)
 => 0
[2,1,3] => ([(1,2)],3)
 => ([],2)
 => ([],2)
 => 0
[2,3,1] => ([(0,2),(1,2)],3)
 => ([(0,2),(1,2)],3)
 => ([(0,2),(1,2)],3)
 => 1
[3,1,2] => ([(0,2),(1,2)],3)
 => ([(0,2),(1,2)],3)
 => ([(0,2),(1,2)],3)
 => 1
[3,2,1] => ([(0,1),(0,2),(1,2)],3)
 => ([],1)
 => ([],1)
 => 0
[1,2,3,4] => ([],4)
 => ([],4)
 => ([],4)
 => 0
[1,2,4,3] => ([(2,3)],4)
 => ([],3)
 => ([],3)
 => 0
[1,3,2,4] => ([(2,3)],4)
 => ([],3)
 => ([],3)
 => 0
[1,3,4,2] => ([(1,3),(2,3)],4)
 => ([(1,3),(2,3)],4)
 => ([(1,3),(2,3)],4)
 => 1
[1,4,2,3] => ([(1,3),(2,3)],4)
 => ([(1,3),(2,3)],4)
 => ([(1,3),(2,3)],4)
 => 1
[1,4,3,2] => ([(1,2),(1,3),(2,3)],4)
 => ([],2)
 => ([],2)
 => 0
[2,1,3,4] => ([(2,3)],4)
 => ([],3)
 => ([],3)
 => 0
[2,1,4,3] => ([(0,3),(1,2)],4)
 => ([],2)
 => ([],2)
 => 0
[2,3,1,4] => ([(1,3),(2,3)],4)
 => ([(1,3),(2,3)],4)
 => ([(1,3),(2,3)],4)
 => 1
[2,3,4,1] => ([(0,3),(1,3),(2,3)],4)
 => ([(0,3),(1,3),(2,3)],4)
 => ([(0,3),(1,3),(2,3)],4)
 => 2
[2,4,1,3] => ([(0,3),(1,2),(2,3)],4)
 => ([(0,4),(1,3),(2,3),(2,4)],5)
 => ([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => ? ∊ {0,2}
[2,4,3,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => ([(0,2),(1,2)],3)
 => ([(0,2),(1,2)],3)
 => 1
[3,1,2,4] => ([(1,3),(2,3)],4)
 => ([(1,3),(2,3)],4)
 => ([(1,3),(2,3)],4)
 => 1
[3,1,4,2] => ([(0,3),(1,2),(2,3)],4)
 => ([(0,4),(1,3),(2,3),(2,4)],5)
 => ([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => ? ∊ {0,2}
[3,2,1,4] => ([(1,2),(1,3),(2,3)],4)
 => ([],2)
 => ([],2)
 => 0
[3,2,4,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => ([(0,2),(1,2)],3)
 => ([(0,2),(1,2)],3)
 => 1
[3,4,1,2] => ([(0,2),(0,3),(1,2),(1,3)],4)
 => ([],1)
 => ([],1)
 => 0
[3,4,2,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => ([],1)
 => ([],1)
 => 0
[4,1,2,3] => ([(0,3),(1,3),(2,3)],4)
 => ([(0,3),(1,3),(2,3)],4)
 => ([(0,3),(1,3),(2,3)],4)
 => 2
[4,1,3,2] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => ([(0,2),(1,2)],3)
 => ([(0,2),(1,2)],3)
 => 1
[4,2,1,3] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => ([(0,2),(1,2)],3)
 => ([(0,2),(1,2)],3)
 => 1
[4,2,3,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => ([],1)
 => ([],1)
 => 0
[4,3,1,2] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => ([],1)
 => ([],1)
 => 0
[4,3,2,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => ([],1)
 => ([],1)
 => 0
[1,2,3,4,5] => ([],5)
 => ([],5)
 => ([],5)
 => 0
[1,2,3,5,4] => ([(3,4)],5)
 => ([],4)
 => ([],4)
 => 0
[1,2,4,3,5] => ([(3,4)],5)
 => ([],4)
 => ([],4)
 => 0
[1,2,4,5,3] => ([(2,4),(3,4)],5)
 => ([(2,4),(3,4)],5)
 => ([(2,4),(3,4)],5)
 => 1
[1,2,5,3,4] => ([(2,4),(3,4)],5)
 => ([(2,4),(3,4)],5)
 => ([(2,4),(3,4)],5)
 => 1
[1,2,5,4,3] => ([(2,3),(2,4),(3,4)],5)
 => ([],3)
 => ([],3)
 => 0
[1,3,2,4,5] => ([(3,4)],5)
 => ([],4)
 => ([],4)
 => 0
[1,3,2,5,4] => ([(1,4),(2,3)],5)
 => ([],3)
 => ([],3)
 => 0
[1,3,4,2,5] => ([(2,4),(3,4)],5)
 => ([(2,4),(3,4)],5)
 => ([(2,4),(3,4)],5)
 => 1
[1,3,4,5,2] => ([(1,4),(2,4),(3,4)],5)
 => ([(1,4),(2,4),(3,4)],5)
 => ([(1,4),(2,4),(3,4)],5)
 => 2
[1,3,5,2,4] => ([(1,4),(2,3),(3,4)],5)
 => ([(1,5),(2,4),(3,4),(3,5)],6)
 => ([(1,2),(1,5),(2,4),(3,4),(3,5),(4,5)],6)
 => ? ∊ {0,0,0,0,0,1,1,1,1,2,2,2,3,3,3,3}
[1,3,5,4,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
 => ([(1,3),(2,3)],4)
 => ([(1,3),(2,3)],4)
 => 1
[1,4,2,3,5] => ([(2,4),(3,4)],5)
 => ([(2,4),(3,4)],5)
 => ([(2,4),(3,4)],5)
 => 1
[1,4,2,5,3] => ([(1,4),(2,3),(3,4)],5)
 => ([(1,5),(2,4),(3,4),(3,5)],6)
 => ([(1,2),(1,5),(2,4),(3,4),(3,5),(4,5)],6)
 => ? ∊ {0,0,0,0,0,1,1,1,1,2,2,2,3,3,3,3}
[1,4,3,2,5] => ([(2,3),(2,4),(3,4)],5)
 => ([],3)
 => ([],3)
 => 0
[1,4,3,5,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
 => ([(1,3),(2,3)],4)
 => ([(1,3),(2,3)],4)
 => 1
[1,4,5,2,3] => ([(1,3),(1,4),(2,3),(2,4)],5)
 => ([],2)
 => ([],2)
 => 0
[1,4,5,3,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([],2)
 => ([],2)
 => 0
[1,5,2,3,4] => ([(1,4),(2,4),(3,4)],5)
 => ([(1,4),(2,4),(3,4)],5)
 => ([(1,4),(2,4),(3,4)],5)
 => 2
[1,5,2,4,3] => ([(1,4),(2,3),(2,4),(3,4)],5)
 => ([(1,3),(2,3)],4)
 => ([(1,3),(2,3)],4)
 => 1
[1,5,3,2,4] => ([(1,4),(2,3),(2,4),(3,4)],5)
 => ([(1,3),(2,3)],4)
 => ([(1,3),(2,3)],4)
 => 1
[2,3,5,1,4] => ([(0,4),(1,4),(2,3),(3,4)],5)
 => ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
 => ([(0,2),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6)
 => ? ∊ {0,0,0,0,0,1,1,1,1,2,2,2,3,3,3,3}
[2,4,1,3,5] => ([(1,4),(2,3),(3,4)],5)
 => ([(1,5),(2,4),(3,4),(3,5)],6)
 => ([(1,2),(1,5),(2,4),(3,4),(3,5),(4,5)],6)
 => ? ∊ {0,0,0,0,0,1,1,1,1,2,2,2,3,3,3,3}
[2,4,1,5,3] => ([(0,4),(1,3),(2,3),(2,4)],5)
 => ([(0,6),(1,5),(2,3),(2,4),(3,5),(4,6)],7)
 => ([(0,1),(0,3),(0,4),(0,6),(1,2),(1,4),(1,5),(2,3),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? ∊ {0,0,0,0,0,1,1,1,1,2,2,2,3,3,3,3}
[2,5,1,3,4] => ([(0,4),(1,4),(2,3),(3,4)],5)
 => ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
 => ([(0,2),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6)
 => ? ∊ {0,0,0,0,0,1,1,1,1,2,2,2,3,3,3,3}
[2,5,1,4,3] => ([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
 => ([(0,4),(1,3),(2,3),(2,4)],5)
 => ([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => ? ∊ {0,0,0,0,0,1,1,1,1,2,2,2,3,3,3,3}
[2,5,3,1,4] => ([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => ([(0,4),(1,3),(2,3),(2,4)],5)
 => ([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => ? ∊ {0,0,0,0,0,1,1,1,1,2,2,2,3,3,3,3}
[3,1,4,2,5] => ([(1,4),(2,3),(3,4)],5)
 => ([(1,5),(2,4),(3,4),(3,5)],6)
 => ([(1,2),(1,5),(2,4),(3,4),(3,5),(4,5)],6)
 => ? ∊ {0,0,0,0,0,1,1,1,1,2,2,2,3,3,3,3}
[3,1,4,5,2] => ([(0,4),(1,4),(2,3),(3,4)],5)
 => ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
 => ([(0,2),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6)
 => ? ∊ {0,0,0,0,0,1,1,1,1,2,2,2,3,3,3,3}
[3,1,5,2,4] => ([(0,4),(1,3),(2,3),(2,4)],5)
 => ([(0,6),(1,5),(2,3),(2,4),(3,5),(4,6)],7)
 => ([(0,1),(0,3),(0,4),(0,6),(1,2),(1,4),(1,5),(2,3),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? ∊ {0,0,0,0,0,1,1,1,1,2,2,2,3,3,3,3}
[3,1,5,4,2] => ([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
 => ([(0,4),(1,3),(2,3),(2,4)],5)
 => ([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => ? ∊ {0,0,0,0,0,1,1,1,1,2,2,2,3,3,3,3}
[3,2,5,1,4] => ([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
 => ([(0,4),(1,3),(2,3),(2,4)],5)
 => ([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => ? ∊ {0,0,0,0,0,1,1,1,1,2,2,2,3,3,3,3}
[4,1,2,5,3] => ([(0,4),(1,4),(2,3),(3,4)],5)
 => ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
 => ([(0,2),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6)
 => ? ∊ {0,0,0,0,0,1,1,1,1,2,2,2,3,3,3,3}
[4,1,3,5,2] => ([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => ([(0,4),(1,3),(2,3),(2,4)],5)
 => ([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => ? ∊ {0,0,0,0,0,1,1,1,1,2,2,2,3,3,3,3}
[4,2,1,5,3] => ([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
 => ([(0,4),(1,3),(2,3),(2,4)],5)
 => ([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => ? ∊ {0,0,0,0,0,1,1,1,1,2,2,2,3,3,3,3}
[1,2,4,6,3,5] => ([(2,5),(3,4),(4,5)],6)
 => ([(2,6),(3,5),(4,5),(4,6)],7)
 => ([(2,3),(2,6),(3,5),(4,5),(4,6),(5,6)],7)
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4}
[1,2,5,3,6,4] => ([(2,5),(3,4),(4,5)],6)
 => ([(2,6),(3,5),(4,5),(4,6)],7)
 => ([(2,3),(2,6),(3,5),(4,5),(4,6),(5,6)],7)
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4}
[1,3,4,6,2,5] => ([(1,5),(2,5),(3,4),(4,5)],6)
 => ([(1,6),(2,6),(3,4),(4,5),(5,6)],7)
 => ([(1,3),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(4,5),(4,6),(5,6)],7)
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4}
[1,3,5,2,4,6] => ([(2,5),(3,4),(4,5)],6)
 => ([(2,6),(3,5),(4,5),(4,6)],7)
 => ([(2,3),(2,6),(3,5),(4,5),(4,6),(5,6)],7)
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4}
[1,3,5,2,6,4] => ([(1,5),(2,4),(3,4),(3,5)],6)
 => ([(1,7),(2,6),(3,4),(3,5),(4,6),(5,7)],8)
 => ([(1,2),(1,4),(1,5),(1,7),(2,3),(2,5),(2,6),(3,4),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4}
[1,3,6,2,4,5] => ([(1,5),(2,5),(3,4),(4,5)],6)
 => ([(1,6),(2,6),(3,4),(4,5),(5,6)],7)
 => ([(1,3),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(4,5),(4,6),(5,6)],7)
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4}
[1,3,6,2,5,4] => ([(1,4),(2,3),(2,5),(3,5),(4,5)],6)
 => ([(1,5),(2,4),(3,4),(3,5)],6)
 => ([(1,2),(1,5),(2,4),(3,4),(3,5),(4,5)],6)
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4}
[1,3,6,4,2,5] => ([(1,5),(2,4),(3,4),(3,5),(4,5)],6)
 => ([(1,5),(2,4),(3,4),(3,5)],6)
 => ([(1,2),(1,5),(2,4),(3,4),(3,5),(4,5)],6)
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4}
[1,4,2,5,3,6] => ([(2,5),(3,4),(4,5)],6)
 => ([(2,6),(3,5),(4,5),(4,6)],7)
 => ([(2,3),(2,6),(3,5),(4,5),(4,6),(5,6)],7)
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4}
[1,4,2,5,6,3] => ([(1,5),(2,5),(3,4),(4,5)],6)
 => ([(1,6),(2,6),(3,4),(4,5),(5,6)],7)
 => ([(1,3),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(4,5),(4,6),(5,6)],7)
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4}
[1,4,2,6,3,5] => ([(1,5),(2,4),(3,4),(3,5)],6)
 => ([(1,7),(2,6),(3,4),(3,5),(4,6),(5,7)],8)
 => ([(1,2),(1,4),(1,5),(1,7),(2,3),(2,5),(2,6),(3,4),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4}
[1,4,2,6,5,3] => ([(1,4),(2,3),(2,5),(3,5),(4,5)],6)
 => ([(1,5),(2,4),(3,4),(3,5)],6)
 => ([(1,2),(1,5),(2,4),(3,4),(3,5),(4,5)],6)
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4}
[1,4,3,6,2,5] => ([(1,4),(2,3),(2,5),(3,5),(4,5)],6)
 => ([(1,5),(2,4),(3,4),(3,5)],6)
 => ([(1,2),(1,5),(2,4),(3,4),(3,5),(4,5)],6)
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4}
[1,5,2,3,6,4] => ([(1,5),(2,5),(3,4),(4,5)],6)
 => ([(1,6),(2,6),(3,4),(4,5),(5,6)],7)
 => ([(1,3),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(4,5),(4,6),(5,6)],7)
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4}
[1,5,2,4,6,3] => ([(1,5),(2,4),(3,4),(3,5),(4,5)],6)
 => ([(1,5),(2,4),(3,4),(3,5)],6)
 => ([(1,2),(1,5),(2,4),(3,4),(3,5),(4,5)],6)
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4}
[1,5,3,2,6,4] => ([(1,4),(2,3),(2,5),(3,5),(4,5)],6)
 => ([(1,5),(2,4),(3,4),(3,5)],6)
 => ([(1,2),(1,5),(2,4),(3,4),(3,5),(4,5)],6)
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4}
[2,1,4,6,3,5] => ([(0,1),(2,5),(3,4),(4,5)],6)
 => ([(1,5),(2,4),(3,4),(3,5)],6)
 => ([(1,2),(1,5),(2,4),(3,4),(3,5),(4,5)],6)
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4}
[2,1,5,3,6,4] => ([(0,1),(2,5),(3,4),(4,5)],6)
 => ([(1,5),(2,4),(3,4),(3,5)],6)
 => ([(1,2),(1,5),(2,4),(3,4),(3,5),(4,5)],6)
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4}
[2,3,4,6,1,5] => ([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
 => ([(0,6),(1,6),(2,6),(3,4),(4,5),(5,6)],7)
 => ([(0,2),(0,6),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4}
[2,3,5,1,4,6] => ([(1,5),(2,5),(3,4),(4,5)],6)
 => ([(1,6),(2,6),(3,4),(4,5),(5,6)],7)
 => ([(1,3),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(4,5),(4,6),(5,6)],7)
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4}
[2,3,5,1,6,4] => ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
 => ([(0,6),(1,7),(2,7),(3,4),(3,5),(4,6),(5,7)],8)
 => ([(0,2),(0,3),(0,4),(0,7),(1,3),(1,4),(1,5),(1,6),(1,7),(2,3),(2,5),(2,6),(2,7),(3,5),(3,6),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4}
[2,3,6,1,4,5] => ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
 => ([(0,6),(1,6),(2,5),(3,5),(4,5),(4,6)],7)
 => ([(0,1),(0,5),(0,6),(1,3),(1,4),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4}
[2,3,6,1,5,4] => ([(0,4),(1,4),(2,3),(2,5),(3,5),(4,5)],6)
 => ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
 => ([(0,2),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6)
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4}
[2,3,6,4,1,5] => ([(0,5),(1,5),(2,4),(3,4),(3,5),(4,5)],6)
 => ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
 => ([(0,2),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6)
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4}
[2,4,1,3,5,6] => ([(2,5),(3,4),(4,5)],6)
 => ([(2,6),(3,5),(4,5),(4,6)],7)
 => ([(2,3),(2,6),(3,5),(4,5),(4,6),(5,6)],7)
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4}
[2,4,1,3,6,5] => ([(0,1),(2,5),(3,4),(4,5)],6)
 => ([(1,5),(2,4),(3,4),(3,5)],6)
 => ([(1,2),(1,5),(2,4),(3,4),(3,5),(4,5)],6)
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4}
[2,4,1,5,3,6] => ([(1,5),(2,4),(3,4),(3,5)],6)
 => ([(1,7),(2,6),(3,4),(3,5),(4,6),(5,7)],8)
 => ([(1,2),(1,4),(1,5),(1,7),(2,3),(2,5),(2,6),(3,4),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4}
[2,4,1,5,6,3] => ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
 => ([(0,6),(1,7),(2,7),(3,4),(3,5),(4,6),(5,7)],8)
 => ([(0,2),(0,3),(0,4),(0,7),(1,3),(1,4),(1,5),(1,6),(1,7),(2,3),(2,5),(2,6),(2,7),(3,5),(3,6),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4}
[2,4,1,6,3,5] => ([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
 => ([(0,8),(1,7),(2,3),(2,4),(3,5),(4,6),(5,7),(6,8)],9)
 => ([(0,1),(0,3),(0,4),(0,5),(0,6),(0,8),(1,2),(1,4),(1,5),(1,6),(1,7),(2,3),(2,4),(2,6),(2,7),(2,8),(3,5),(3,6),(3,7),(3,8),(4,5),(4,7),(4,8),(5,7),(5,8),(6,7),(6,8),(7,8)],9)
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4}
[2,4,1,6,5,3] => ([(0,4),(1,2),(1,5),(2,5),(3,4),(3,5)],6)
 => ([(0,6),(1,5),(2,3),(2,4),(3,5),(4,6)],7)
 => ([(0,1),(0,3),(0,4),(0,6),(1,2),(1,4),(1,5),(2,3),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4}
[2,4,3,6,1,5] => ([(0,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)
 => ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
 => ([(0,2),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6)
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4}
[2,4,5,1,6,3] => ([(0,5),(1,4),(2,4),(2,5),(3,4),(3,5)],6)
 => ([(0,4),(1,3),(2,3),(2,4)],5)
 => ([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4}

Description

The dimension of a graph. The dimension of a graph is the least integer $n$ such that there exists a representation of the graph in the Euclidean space of dimension $n$ with all vertices distinct and all edges having unit length. Edges are allowed to intersect, however.

Matching statistic: St001176
(load all 7 compositions to match this statistic)

Mapping

Mp00108: Permutations —cycle type⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St001176: Integer partitions ⟶ ℤResult quality: 82% ●values known / values provided: 82%●distinct values known / distinct values provided: 100%

Values

[1] => [1]
 => []
 => ? = 0
[1,2] => [1,1]
 => [1]
 => 0
[2,1] => [2]
 => []
 => ? = 0
[1,2,3] => [1,1,1]
 => [1,1]
 => 1
[1,3,2] => [2,1]
 => [1]
 => 0
[2,1,3] => [2,1]
 => [1]
 => 0
[2,3,1] => [3]
 => []
 => ? ∊ {0,1}
[3,1,2] => [3]
 => []
 => ? ∊ {0,1}
[3,2,1] => [2,1]
 => [1]
 => 0
[1,2,3,4] => [1,1,1,1]
 => [1,1,1]
 => 2
[1,2,4,3] => [2,1,1]
 => [1,1]
 => 1
[1,3,2,4] => [2,1,1]
 => [1,1]
 => 1
[1,3,4,2] => [3,1]
 => [1]
 => 0
[1,4,2,3] => [3,1]
 => [1]
 => 0
[1,4,3,2] => [2,1,1]
 => [1,1]
 => 1
[2,1,3,4] => [2,1,1]
 => [1,1]
 => 1
[2,1,4,3] => [2,2]
 => [2]
 => 0
[2,3,1,4] => [3,1]
 => [1]
 => 0
[2,3,4,1] => [4]
 => []
 => ? ∊ {0,0,1,1,2,2}
[2,4,1,3] => [4]
 => []
 => ? ∊ {0,0,1,1,2,2}
[2,4,3,1] => [3,1]
 => [1]
 => 0
[3,1,2,4] => [3,1]
 => [1]
 => 0
[3,1,4,2] => [4]
 => []
 => ? ∊ {0,0,1,1,2,2}
[3,2,1,4] => [2,1,1]
 => [1,1]
 => 1
[3,2,4,1] => [3,1]
 => [1]
 => 0
[3,4,1,2] => [2,2]
 => [2]
 => 0
[3,4,2,1] => [4]
 => []
 => ? ∊ {0,0,1,1,2,2}
[4,1,2,3] => [4]
 => []
 => ? ∊ {0,0,1,1,2,2}
[4,1,3,2] => [3,1]
 => [1]
 => 0
[4,2,1,3] => [3,1]
 => [1]
 => 0
[4,2,3,1] => [2,1,1]
 => [1,1]
 => 1
[4,3,1,2] => [4]
 => []
 => ? ∊ {0,0,1,1,2,2}
[4,3,2,1] => [2,2]
 => [2]
 => 0
[1,2,3,4,5] => [1,1,1,1,1]
 => [1,1,1,1]
 => 3
[1,2,3,5,4] => [2,1,1,1]
 => [1,1,1]
 => 2
[1,2,4,3,5] => [2,1,1,1]
 => [1,1,1]
 => 2
[1,2,4,5,3] => [3,1,1]
 => [1,1]
 => 1
[1,2,5,3,4] => [3,1,1]
 => [1,1]
 => 1
[1,2,5,4,3] => [2,1,1,1]
 => [1,1,1]
 => 2
[1,3,2,4,5] => [2,1,1,1]
 => [1,1,1]
 => 2
[1,3,2,5,4] => [2,2,1]
 => [2,1]
 => 1
[1,3,4,2,5] => [3,1,1]
 => [1,1]
 => 1
[1,3,4,5,2] => [4,1]
 => [1]
 => 0
[1,3,5,2,4] => [4,1]
 => [1]
 => 0
[1,3,5,4,2] => [3,1,1]
 => [1,1]
 => 1
[1,4,2,3,5] => [3,1,1]
 => [1,1]
 => 1
[1,4,2,5,3] => [4,1]
 => [1]
 => 0
[1,4,3,2,5] => [2,1,1,1]
 => [1,1,1]
 => 2
[1,4,3,5,2] => [3,1,1]
 => [1,1]
 => 1
[1,4,5,2,3] => [2,2,1]
 => [2,1]
 => 1
[1,4,5,3,2] => [4,1]
 => [1]
 => 0
[1,5,2,3,4] => [4,1]
 => [1]
 => 0
[1,5,2,4,3] => [3,1,1]
 => [1,1]
 => 1
[1,5,3,2,4] => [3,1,1]
 => [1,1]
 => 1
[1,5,3,4,2] => [2,1,1,1]
 => [1,1,1]
 => 2
[1,5,4,2,3] => [4,1]
 => [1]
 => 0
[1,5,4,3,2] => [2,2,1]
 => [2,1]
 => 1
[2,1,3,4,5] => [2,1,1,1]
 => [1,1,1]
 => 2
[2,1,3,5,4] => [2,2,1]
 => [2,1]
 => 1
[2,1,4,3,5] => [2,2,1]
 => [2,1]
 => 1
[2,3,4,5,1] => [5]
 => []
 => ? ∊ {0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,2,2,2,3,3,3}
[2,3,5,1,4] => [5]
 => []
 => ? ∊ {0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,2,2,2,3,3,3}
[2,4,1,5,3] => [5]
 => []
 => ? ∊ {0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,2,2,2,3,3,3}
[2,4,5,3,1] => [5]
 => []
 => ? ∊ {0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,2,2,2,3,3,3}
[2,5,1,3,4] => [5]
 => []
 => ? ∊ {0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,2,2,2,3,3,3}
[2,5,4,1,3] => [5]
 => []
 => ? ∊ {0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,2,2,2,3,3,3}
[3,1,4,5,2] => [5]
 => []
 => ? ∊ {0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,2,2,2,3,3,3}
[3,1,5,2,4] => [5]
 => []
 => ? ∊ {0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,2,2,2,3,3,3}
[3,4,2,5,1] => [5]
 => []
 => ? ∊ {0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,2,2,2,3,3,3}
[3,4,5,1,2] => [5]
 => []
 => ? ∊ {0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,2,2,2,3,3,3}
[3,5,2,1,4] => [5]
 => []
 => ? ∊ {0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,2,2,2,3,3,3}
[3,5,4,2,1] => [5]
 => []
 => ? ∊ {0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,2,2,2,3,3,3}
[4,1,2,5,3] => [5]
 => []
 => ? ∊ {0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,2,2,2,3,3,3}
[4,1,5,3,2] => [5]
 => []
 => ? ∊ {0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,2,2,2,3,3,3}
[4,3,1,5,2] => [5]
 => []
 => ? ∊ {0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,2,2,2,3,3,3}
[4,3,5,2,1] => [5]
 => []
 => ? ∊ {0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,2,2,2,3,3,3}
[4,5,1,2,3] => [5]
 => []
 => ? ∊ {0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,2,2,2,3,3,3}
[4,5,2,3,1] => [5]
 => []
 => ? ∊ {0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,2,2,2,3,3,3}
[5,1,2,3,4] => [5]
 => []
 => ? ∊ {0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,2,2,2,3,3,3}
[5,1,4,2,3] => [5]
 => []
 => ? ∊ {0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,2,2,2,3,3,3}
[5,3,1,2,4] => [5]
 => []
 => ? ∊ {0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,2,2,2,3,3,3}
[5,3,4,1,2] => [5]
 => []
 => ? ∊ {0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,2,2,2,3,3,3}
[5,4,1,3,2] => [5]
 => []
 => ? ∊ {0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,2,2,2,3,3,3}
[5,4,2,1,3] => [5]
 => []
 => ? ∊ {0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,2,2,2,3,3,3}
[2,3,4,5,6,1] => [6]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,3,3,3,3,3,3,3,3,3,4,4,4,4}
[2,3,4,6,1,5] => [6]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,3,3,3,3,3,3,3,3,3,4,4,4,4}
[2,3,5,1,6,4] => [6]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,3,3,3,3,3,3,3,3,3,4,4,4,4}
[2,3,5,6,4,1] => [6]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,3,3,3,3,3,3,3,3,3,4,4,4,4}
[2,3,6,1,4,5] => [6]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,3,3,3,3,3,3,3,3,3,4,4,4,4}
[2,3,6,5,1,4] => [6]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,3,3,3,3,3,3,3,3,3,4,4,4,4}
[2,4,1,5,6,3] => [6]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,3,3,3,3,3,3,3,3,3,4,4,4,4}
[2,4,1,6,3,5] => [6]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,3,3,3,3,3,3,3,3,3,4,4,4,4}
[2,4,5,3,6,1] => [6]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,3,3,3,3,3,3,3,3,3,4,4,4,4}
[2,4,5,6,1,3] => [6]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,3,3,3,3,3,3,3,3,3,4,4,4,4}
[2,4,6,3,1,5] => [6]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,3,3,3,3,3,3,3,3,3,4,4,4,4}
[2,4,6,5,3,1] => [6]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,3,3,3,3,3,3,3,3,3,4,4,4,4}
[2,5,1,3,6,4] => [6]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,3,3,3,3,3,3,3,3,3,4,4,4,4}
[2,5,1,6,4,3] => [6]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,3,3,3,3,3,3,3,3,3,4,4,4,4}
[2,5,4,1,6,3] => [6]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,3,3,3,3,3,3,3,3,3,4,4,4,4}
[2,5,4,6,3,1] => [6]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,3,3,3,3,3,3,3,3,3,4,4,4,4}

Description

The size of a partition minus its first part. This is the number of boxes in its diagram that are not in the first row.

Matching statistic: St000010

Mapping

Mp00108: Permutations —cycle type⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000010: Integer partitions ⟶ ℤResult quality: 82% ●values known / values provided: 82%●distinct values known / distinct values provided: 100%

Values

[1] => [1]
 => []
 => ?
 => ? = 0
[1,2] => [1,1]
 => [1]
 => []
 => 0
[2,1] => [2]
 => []
 => ?
 => ? = 0
[1,2,3] => [1,1,1]
 => [1,1]
 => [1]
 => 1
[1,3,2] => [2,1]
 => [1]
 => []
 => 0
[2,1,3] => [2,1]
 => [1]
 => []
 => 0
[2,3,1] => [3]
 => []
 => ?
 => ? ∊ {0,1}
[3,1,2] => [3]
 => []
 => ?
 => ? ∊ {0,1}
[3,2,1] => [2,1]
 => [1]
 => []
 => 0
[1,2,3,4] => [1,1,1,1]
 => [1,1,1]
 => [1,1]
 => 2
[1,2,4,3] => [2,1,1]
 => [1,1]
 => [1]
 => 1
[1,3,2,4] => [2,1,1]
 => [1,1]
 => [1]
 => 1
[1,3,4,2] => [3,1]
 => [1]
 => []
 => 0
[1,4,2,3] => [3,1]
 => [1]
 => []
 => 0
[1,4,3,2] => [2,1,1]
 => [1,1]
 => [1]
 => 1
[2,1,3,4] => [2,1,1]
 => [1,1]
 => [1]
 => 1
[2,1,4,3] => [2,2]
 => [2]
 => []
 => 0
[2,3,1,4] => [3,1]
 => [1]
 => []
 => 0
[2,3,4,1] => [4]
 => []
 => ?
 => ? ∊ {0,0,1,1,2,2}
[2,4,1,3] => [4]
 => []
 => ?
 => ? ∊ {0,0,1,1,2,2}
[2,4,3,1] => [3,1]
 => [1]
 => []
 => 0
[3,1,2,4] => [3,1]
 => [1]
 => []
 => 0
[3,1,4,2] => [4]
 => []
 => ?
 => ? ∊ {0,0,1,1,2,2}
[3,2,1,4] => [2,1,1]
 => [1,1]
 => [1]
 => 1
[3,2,4,1] => [3,1]
 => [1]
 => []
 => 0
[3,4,1,2] => [2,2]
 => [2]
 => []
 => 0
[3,4,2,1] => [4]
 => []
 => ?
 => ? ∊ {0,0,1,1,2,2}
[4,1,2,3] => [4]
 => []
 => ?
 => ? ∊ {0,0,1,1,2,2}
[4,1,3,2] => [3,1]
 => [1]
 => []
 => 0
[4,2,1,3] => [3,1]
 => [1]
 => []
 => 0
[4,2,3,1] => [2,1,1]
 => [1,1]
 => [1]
 => 1
[4,3,1,2] => [4]
 => []
 => ?
 => ? ∊ {0,0,1,1,2,2}
[4,3,2,1] => [2,2]
 => [2]
 => []
 => 0
[1,2,3,4,5] => [1,1,1,1,1]
 => [1,1,1,1]
 => [1,1,1]
 => 3
[1,2,3,5,4] => [2,1,1,1]
 => [1,1,1]
 => [1,1]
 => 2
[1,2,4,3,5] => [2,1,1,1]
 => [1,1,1]
 => [1,1]
 => 2
[1,2,4,5,3] => [3,1,1]
 => [1,1]
 => [1]
 => 1
[1,2,5,3,4] => [3,1,1]
 => [1,1]
 => [1]
 => 1
[1,2,5,4,3] => [2,1,1,1]
 => [1,1,1]
 => [1,1]
 => 2
[1,3,2,4,5] => [2,1,1,1]
 => [1,1,1]
 => [1,1]
 => 2
[1,3,2,5,4] => [2,2,1]
 => [2,1]
 => [1]
 => 1
[1,3,4,2,5] => [3,1,1]
 => [1,1]
 => [1]
 => 1
[1,3,4,5,2] => [4,1]
 => [1]
 => []
 => 0
[1,3,5,2,4] => [4,1]
 => [1]
 => []
 => 0
[1,3,5,4,2] => [3,1,1]
 => [1,1]
 => [1]
 => 1
[1,4,2,3,5] => [3,1,1]
 => [1,1]
 => [1]
 => 1
[1,4,2,5,3] => [4,1]
 => [1]
 => []
 => 0
[1,4,3,2,5] => [2,1,1,1]
 => [1,1,1]
 => [1,1]
 => 2
[1,4,3,5,2] => [3,1,1]
 => [1,1]
 => [1]
 => 1
[1,4,5,2,3] => [2,2,1]
 => [2,1]
 => [1]
 => 1
[1,4,5,3,2] => [4,1]
 => [1]
 => []
 => 0
[1,5,2,3,4] => [4,1]
 => [1]
 => []
 => 0
[1,5,2,4,3] => [3,1,1]
 => [1,1]
 => [1]
 => 1
[1,5,3,2,4] => [3,1,1]
 => [1,1]
 => [1]
 => 1
[1,5,3,4,2] => [2,1,1,1]
 => [1,1,1]
 => [1,1]
 => 2
[1,5,4,2,3] => [4,1]
 => [1]
 => []
 => 0
[1,5,4,3,2] => [2,2,1]
 => [2,1]
 => [1]
 => 1
[2,1,3,4,5] => [2,1,1,1]
 => [1,1,1]
 => [1,1]
 => 2
[2,1,3,5,4] => [2,2,1]
 => [2,1]
 => [1]
 => 1
[2,1,4,3,5] => [2,2,1]
 => [2,1]
 => [1]
 => 1
[2,3,4,5,1] => [5]
 => []
 => ?
 => ? ∊ {0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,2,2,2,3,3,3}
[2,3,5,1,4] => [5]
 => []
 => ?
 => ? ∊ {0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,2,2,2,3,3,3}
[2,4,1,5,3] => [5]
 => []
 => ?
 => ? ∊ {0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,2,2,2,3,3,3}
[2,4,5,3,1] => [5]
 => []
 => ?
 => ? ∊ {0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,2,2,2,3,3,3}
[2,5,1,3,4] => [5]
 => []
 => ?
 => ? ∊ {0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,2,2,2,3,3,3}
[2,5,4,1,3] => [5]
 => []
 => ?
 => ? ∊ {0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,2,2,2,3,3,3}
[3,1,4,5,2] => [5]
 => []
 => ?
 => ? ∊ {0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,2,2,2,3,3,3}
[3,1,5,2,4] => [5]
 => []
 => ?
 => ? ∊ {0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,2,2,2,3,3,3}
[3,4,2,5,1] => [5]
 => []
 => ?
 => ? ∊ {0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,2,2,2,3,3,3}
[3,4,5,1,2] => [5]
 => []
 => ?
 => ? ∊ {0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,2,2,2,3,3,3}
[3,5,2,1,4] => [5]
 => []
 => ?
 => ? ∊ {0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,2,2,2,3,3,3}
[3,5,4,2,1] => [5]
 => []
 => ?
 => ? ∊ {0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,2,2,2,3,3,3}
[4,1,2,5,3] => [5]
 => []
 => ?
 => ? ∊ {0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,2,2,2,3,3,3}
[4,1,5,3,2] => [5]
 => []
 => ?
 => ? ∊ {0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,2,2,2,3,3,3}
[4,3,1,5,2] => [5]
 => []
 => ?
 => ? ∊ {0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,2,2,2,3,3,3}
[4,3,5,2,1] => [5]
 => []
 => ?
 => ? ∊ {0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,2,2,2,3,3,3}
[4,5,1,2,3] => [5]
 => []
 => ?
 => ? ∊ {0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,2,2,2,3,3,3}
[4,5,2,3,1] => [5]
 => []
 => ?
 => ? ∊ {0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,2,2,2,3,3,3}
[5,1,2,3,4] => [5]
 => []
 => ?
 => ? ∊ {0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,2,2,2,3,3,3}
[5,1,4,2,3] => [5]
 => []
 => ?
 => ? ∊ {0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,2,2,2,3,3,3}
[5,3,1,2,4] => [5]
 => []
 => ?
 => ? ∊ {0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,2,2,2,3,3,3}
[5,3,4,1,2] => [5]
 => []
 => ?
 => ? ∊ {0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,2,2,2,3,3,3}
[5,4,1,3,2] => [5]
 => []
 => ?
 => ? ∊ {0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,2,2,2,3,3,3}
[5,4,2,1,3] => [5]
 => []
 => ?
 => ? ∊ {0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,2,2,2,3,3,3}
[2,3,4,5,6,1] => [6]
 => []
 => ?
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,4,4,4,4}
[2,3,4,6,1,5] => [6]
 => []
 => ?
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,4,4,4,4}
[2,3,5,1,6,4] => [6]
 => []
 => ?
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,4,4,4,4}
[2,3,5,6,4,1] => [6]
 => []
 => ?
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,4,4,4,4}
[2,3,6,1,4,5] => [6]
 => []
 => ?
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,4,4,4,4}
[2,3,6,5,1,4] => [6]
 => []
 => ?
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,4,4,4,4}
[2,4,1,5,6,3] => [6]
 => []
 => ?
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,4,4,4,4}
[2,4,1,6,3,5] => [6]
 => []
 => ?
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,4,4,4,4}
[2,4,5,3,6,1] => [6]
 => []
 => ?
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,4,4,4,4}
[2,4,5,6,1,3] => [6]
 => []
 => ?
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,4,4,4,4}
[2,4,6,3,1,5] => [6]
 => []
 => ?
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,4,4,4,4}
[2,4,6,5,3,1] => [6]
 => []
 => ?
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,4,4,4,4}
[2,5,1,3,6,4] => [6]
 => []
 => ?
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,4,4,4,4}
[2,5,1,6,4,3] => [6]
 => []
 => ?
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,4,4,4,4}
[2,5,4,1,6,3] => [6]
 => []
 => ?
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,4,4,4,4}
[2,5,4,6,3,1] => [6]
 => []
 => ?
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,4,4,4,4}

Description

The length of the partition.

Matching statistic: St000148

Mapping

Mp00108: Permutations —cycle type⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000148: Integer partitions ⟶ ℤResult quality: 82% ●values known / values provided: 82%●distinct values known / distinct values provided: 100%

Values

[1] => [1]
 => []
 => ?
 => ? = 0
[1,2] => [1,1]
 => [1]
 => []
 => 0
[2,1] => [2]
 => []
 => ?
 => ? = 0
[1,2,3] => [1,1,1]
 => [1,1]
 => [1]
 => 1
[1,3,2] => [2,1]
 => [1]
 => []
 => 0
[2,1,3] => [2,1]
 => [1]
 => []
 => 0
[2,3,1] => [3]
 => []
 => ?
 => ? ∊ {0,1}
[3,1,2] => [3]
 => []
 => ?
 => ? ∊ {0,1}
[3,2,1] => [2,1]
 => [1]
 => []
 => 0
[1,2,3,4] => [1,1,1,1]
 => [1,1,1]
 => [1,1]
 => 2
[1,2,4,3] => [2,1,1]
 => [1,1]
 => [1]
 => 1
[1,3,2,4] => [2,1,1]
 => [1,1]
 => [1]
 => 1
[1,3,4,2] => [3,1]
 => [1]
 => []
 => 0
[1,4,2,3] => [3,1]
 => [1]
 => []
 => 0
[1,4,3,2] => [2,1,1]
 => [1,1]
 => [1]
 => 1
[2,1,3,4] => [2,1,1]
 => [1,1]
 => [1]
 => 1
[2,1,4,3] => [2,2]
 => [2]
 => []
 => 0
[2,3,1,4] => [3,1]
 => [1]
 => []
 => 0
[2,3,4,1] => [4]
 => []
 => ?
 => ? ∊ {0,0,1,1,2,2}
[2,4,1,3] => [4]
 => []
 => ?
 => ? ∊ {0,0,1,1,2,2}
[2,4,3,1] => [3,1]
 => [1]
 => []
 => 0
[3,1,2,4] => [3,1]
 => [1]
 => []
 => 0
[3,1,4,2] => [4]
 => []
 => ?
 => ? ∊ {0,0,1,1,2,2}
[3,2,1,4] => [2,1,1]
 => [1,1]
 => [1]
 => 1
[3,2,4,1] => [3,1]
 => [1]
 => []
 => 0
[3,4,1,2] => [2,2]
 => [2]
 => []
 => 0
[3,4,2,1] => [4]
 => []
 => ?
 => ? ∊ {0,0,1,1,2,2}
[4,1,2,3] => [4]
 => []
 => ?
 => ? ∊ {0,0,1,1,2,2}
[4,1,3,2] => [3,1]
 => [1]
 => []
 => 0
[4,2,1,3] => [3,1]
 => [1]
 => []
 => 0
[4,2,3,1] => [2,1,1]
 => [1,1]
 => [1]
 => 1
[4,3,1,2] => [4]
 => []
 => ?
 => ? ∊ {0,0,1,1,2,2}
[4,3,2,1] => [2,2]
 => [2]
 => []
 => 0
[1,2,3,4,5] => [1,1,1,1,1]
 => [1,1,1,1]
 => [1,1,1]
 => 3
[1,2,3,5,4] => [2,1,1,1]
 => [1,1,1]
 => [1,1]
 => 2
[1,2,4,3,5] => [2,1,1,1]
 => [1,1,1]
 => [1,1]
 => 2
[1,2,4,5,3] => [3,1,1]
 => [1,1]
 => [1]
 => 1
[1,2,5,3,4] => [3,1,1]
 => [1,1]
 => [1]
 => 1
[1,2,5,4,3] => [2,1,1,1]
 => [1,1,1]
 => [1,1]
 => 2
[1,3,2,4,5] => [2,1,1,1]
 => [1,1,1]
 => [1,1]
 => 2
[1,3,2,5,4] => [2,2,1]
 => [2,1]
 => [1]
 => 1
[1,3,4,2,5] => [3,1,1]
 => [1,1]
 => [1]
 => 1
[1,3,4,5,2] => [4,1]
 => [1]
 => []
 => 0
[1,3,5,2,4] => [4,1]
 => [1]
 => []
 => 0
[1,3,5,4,2] => [3,1,1]
 => [1,1]
 => [1]
 => 1
[1,4,2,3,5] => [3,1,1]
 => [1,1]
 => [1]
 => 1
[1,4,2,5,3] => [4,1]
 => [1]
 => []
 => 0
[1,4,3,2,5] => [2,1,1,1]
 => [1,1,1]
 => [1,1]
 => 2
[1,4,3,5,2] => [3,1,1]
 => [1,1]
 => [1]
 => 1
[1,4,5,2,3] => [2,2,1]
 => [2,1]
 => [1]
 => 1
[1,4,5,3,2] => [4,1]
 => [1]
 => []
 => 0
[1,5,2,3,4] => [4,1]
 => [1]
 => []
 => 0
[1,5,2,4,3] => [3,1,1]
 => [1,1]
 => [1]
 => 1
[1,5,3,2,4] => [3,1,1]
 => [1,1]
 => [1]
 => 1
[1,5,3,4,2] => [2,1,1,1]
 => [1,1,1]
 => [1,1]
 => 2
[1,5,4,2,3] => [4,1]
 => [1]
 => []
 => 0
[1,5,4,3,2] => [2,2,1]
 => [2,1]
 => [1]
 => 1
[2,1,3,4,5] => [2,1,1,1]
 => [1,1,1]
 => [1,1]
 => 2
[2,1,3,5,4] => [2,2,1]
 => [2,1]
 => [1]
 => 1
[2,1,4,3,5] => [2,2,1]
 => [2,1]
 => [1]
 => 1
[2,3,4,5,1] => [5]
 => []
 => ?
 => ? ∊ {0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,2,2,2,3,3,3}
[2,3,5,1,4] => [5]
 => []
 => ?
 => ? ∊ {0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,2,2,2,3,3,3}
[2,4,1,5,3] => [5]
 => []
 => ?
 => ? ∊ {0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,2,2,2,3,3,3}
[2,4,5,3,1] => [5]
 => []
 => ?
 => ? ∊ {0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,2,2,2,3,3,3}
[2,5,1,3,4] => [5]
 => []
 => ?
 => ? ∊ {0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,2,2,2,3,3,3}
[2,5,4,1,3] => [5]
 => []
 => ?
 => ? ∊ {0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,2,2,2,3,3,3}
[3,1,4,5,2] => [5]
 => []
 => ?
 => ? ∊ {0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,2,2,2,3,3,3}
[3,1,5,2,4] => [5]
 => []
 => ?
 => ? ∊ {0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,2,2,2,3,3,3}
[3,4,2,5,1] => [5]
 => []
 => ?
 => ? ∊ {0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,2,2,2,3,3,3}
[3,4,5,1,2] => [5]
 => []
 => ?
 => ? ∊ {0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,2,2,2,3,3,3}
[3,5,2,1,4] => [5]
 => []
 => ?
 => ? ∊ {0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,2,2,2,3,3,3}
[3,5,4,2,1] => [5]
 => []
 => ?
 => ? ∊ {0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,2,2,2,3,3,3}
[4,1,2,5,3] => [5]
 => []
 => ?
 => ? ∊ {0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,2,2,2,3,3,3}
[4,1,5,3,2] => [5]
 => []
 => ?
 => ? ∊ {0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,2,2,2,3,3,3}
[4,3,1,5,2] => [5]
 => []
 => ?
 => ? ∊ {0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,2,2,2,3,3,3}
[4,3,5,2,1] => [5]
 => []
 => ?
 => ? ∊ {0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,2,2,2,3,3,3}
[4,5,1,2,3] => [5]
 => []
 => ?
 => ? ∊ {0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,2,2,2,3,3,3}
[4,5,2,3,1] => [5]
 => []
 => ?
 => ? ∊ {0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,2,2,2,3,3,3}
[5,1,2,3,4] => [5]
 => []
 => ?
 => ? ∊ {0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,2,2,2,3,3,3}
[5,1,4,2,3] => [5]
 => []
 => ?
 => ? ∊ {0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,2,2,2,3,3,3}
[5,3,1,2,4] => [5]
 => []
 => ?
 => ? ∊ {0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,2,2,2,3,3,3}
[5,3,4,1,2] => [5]
 => []
 => ?
 => ? ∊ {0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,2,2,2,3,3,3}
[5,4,1,3,2] => [5]
 => []
 => ?
 => ? ∊ {0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,2,2,2,3,3,3}
[5,4,2,1,3] => [5]
 => []
 => ?
 => ? ∊ {0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,2,2,2,3,3,3}
[2,3,4,5,6,1] => [6]
 => []
 => ?
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,4,4,4,4}
[2,3,4,6,1,5] => [6]
 => []
 => ?
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,4,4,4,4}
[2,3,5,1,6,4] => [6]
 => []
 => ?
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,4,4,4,4}
[2,3,5,6,4,1] => [6]
 => []
 => ?
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,4,4,4,4}
[2,3,6,1,4,5] => [6]
 => []
 => ?
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,4,4,4,4}
[2,3,6,5,1,4] => [6]
 => []
 => ?
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,4,4,4,4}
[2,4,1,5,6,3] => [6]
 => []
 => ?
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,4,4,4,4}
[2,4,1,6,3,5] => [6]
 => []
 => ?
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,4,4,4,4}
[2,4,5,3,6,1] => [6]
 => []
 => ?
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,4,4,4,4}
[2,4,5,6,1,3] => [6]
 => []
 => ?
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,4,4,4,4}
[2,4,6,3,1,5] => [6]
 => []
 => ?
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,4,4,4,4}
[2,4,6,5,3,1] => [6]
 => []
 => ?
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,4,4,4,4}
[2,5,1,3,6,4] => [6]
 => []
 => ?
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,4,4,4,4}
[2,5,1,6,4,3] => [6]
 => []
 => ?
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,4,4,4,4}
[2,5,4,1,6,3] => [6]
 => []
 => ?
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,4,4,4,4}
[2,5,4,6,3,1] => [6]
 => []
 => ?
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,4,4,4,4}

Description

The number of odd parts of a partition.

Matching statistic: St000150

Mapping

Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
Mp00027: Dyck paths —to partition⟶ Integer partitions
Mp00321: Integer partitions —2-conjugate⟶ Integer partitions
St000150: Integer partitions ⟶ ℤResult quality: 80% ●values known / values provided: 82%●distinct values known / distinct values provided: 80%

Values

[1] => [1,0]
 => []
 => ?
 => ? = 0
[1,2] => [1,0,1,0]
 => [1]
 => [1]
 => 0
[2,1] => [1,1,0,0]
 => []
 => ?
 => ? = 0
[1,2,3] => [1,0,1,0,1,0]
 => [2,1]
 => [3]
 => 0
[1,3,2] => [1,0,1,1,0,0]
 => [1,1]
 => [1,1]
 => 1
[2,1,3] => [1,1,0,0,1,0]
 => [2]
 => [2]
 => 0
[2,3,1] => [1,1,0,1,0,0]
 => [1]
 => [1]
 => 0
[3,1,2] => [1,1,1,0,0,0]
 => []
 => ?
 => ? ∊ {0,1}
[3,2,1] => [1,1,1,0,0,0]
 => []
 => ?
 => ? ∊ {0,1}
[1,2,3,4] => [1,0,1,0,1,0,1,0]
 => [3,2,1]
 => [3,3]
 => 1
[1,2,4,3] => [1,0,1,0,1,1,0,0]
 => [2,2,1]
 => [5]
 => 0
[1,3,2,4] => [1,0,1,1,0,0,1,0]
 => [3,1,1]
 => [2,1,1,1]
 => 1
[1,3,4,2] => [1,0,1,1,0,1,0,0]
 => [2,1,1]
 => [3,1]
 => 0
[1,4,2,3] => [1,0,1,1,1,0,0,0]
 => [1,1,1]
 => [1,1,1]
 => 1
[1,4,3,2] => [1,0,1,1,1,0,0,0]
 => [1,1,1]
 => [1,1,1]
 => 1
[2,1,3,4] => [1,1,0,0,1,0,1,0]
 => [3,2]
 => [4,1]
 => 0
[2,1,4,3] => [1,1,0,0,1,1,0,0]
 => [2,2]
 => [4]
 => 0
[2,3,1,4] => [1,1,0,1,0,0,1,0]
 => [3,1]
 => [2,1,1]
 => 1
[2,3,4,1] => [1,1,0,1,0,1,0,0]
 => [2,1]
 => [3]
 => 0
[2,4,1,3] => [1,1,0,1,1,0,0,0]
 => [1,1]
 => [1,1]
 => 1
[2,4,3,1] => [1,1,0,1,1,0,0,0]
 => [1,1]
 => [1,1]
 => 1
[3,1,2,4] => [1,1,1,0,0,0,1,0]
 => [3]
 => [2,1]
 => 0
[3,1,4,2] => [1,1,1,0,0,1,0,0]
 => [2]
 => [2]
 => 0
[3,2,1,4] => [1,1,1,0,0,0,1,0]
 => [3]
 => [2,1]
 => 0
[3,2,4,1] => [1,1,1,0,0,1,0,0]
 => [2]
 => [2]
 => 0
[3,4,1,2] => [1,1,1,0,1,0,0,0]
 => [1]
 => [1]
 => 0
[3,4,2,1] => [1,1,1,0,1,0,0,0]
 => [1]
 => [1]
 => 0
[4,1,2,3] => [1,1,1,1,0,0,0,0]
 => []
 => ?
 => ? ∊ {0,0,1,2,2,2}
[4,1,3,2] => [1,1,1,1,0,0,0,0]
 => []
 => ?
 => ? ∊ {0,0,1,2,2,2}
[4,2,1,3] => [1,1,1,1,0,0,0,0]
 => []
 => ?
 => ? ∊ {0,0,1,2,2,2}
[4,2,3,1] => [1,1,1,1,0,0,0,0]
 => []
 => ?
 => ? ∊ {0,0,1,2,2,2}
[4,3,1,2] => [1,1,1,1,0,0,0,0]
 => []
 => ?
 => ? ∊ {0,0,1,2,2,2}
[4,3,2,1] => [1,1,1,1,0,0,0,0]
 => []
 => ?
 => ? ∊ {0,0,1,2,2,2}
[1,2,3,4,5] => [1,0,1,0,1,0,1,0,1,0]
 => [4,3,2,1]
 => [5,5]
 => 1
[1,2,3,5,4] => [1,0,1,0,1,0,1,1,0,0]
 => [3,3,2,1]
 => [3,3,3]
 => 1
[1,2,4,3,5] => [1,0,1,0,1,1,0,0,1,0]
 => [4,2,2,1]
 => [7,2]
 => 0
[1,2,4,5,3] => [1,0,1,0,1,1,0,1,0,0]
 => [3,2,2,1]
 => [5,3]
 => 0
[1,2,5,3,4] => [1,0,1,0,1,1,1,0,0,0]
 => [2,2,2,1]
 => [7]
 => 0
[1,2,5,4,3] => [1,0,1,0,1,1,1,0,0,0]
 => [2,2,2,1]
 => [7]
 => 0
[1,3,2,4,5] => [1,0,1,1,0,0,1,0,1,0]
 => [4,3,1,1]
 => [5,2,1,1]
 => 1
[1,3,2,5,4] => [1,0,1,1,0,0,1,1,0,0]
 => [3,3,1,1]
 => [3,2,1,1,1]
 => 1
[1,3,4,2,5] => [1,0,1,1,0,1,0,0,1,0]
 => [4,2,1,1]
 => [6,1,1]
 => 1
[1,3,4,5,2] => [1,0,1,1,0,1,0,1,0,0]
 => [3,2,1,1]
 => [3,3,1]
 => 1
[1,3,5,2,4] => [1,0,1,1,0,1,1,0,0,0]
 => [2,2,1,1]
 => [5,1]
 => 0
[1,3,5,4,2] => [1,0,1,1,0,1,1,0,0,0]
 => [2,2,1,1]
 => [5,1]
 => 0
[1,4,2,3,5] => [1,0,1,1,1,0,0,0,1,0]
 => [4,1,1,1]
 => [4,1,1,1]
 => 1
[1,4,2,5,3] => [1,0,1,1,1,0,0,1,0,0]
 => [3,1,1,1]
 => [2,1,1,1,1]
 => 2
[1,4,3,2,5] => [1,0,1,1,1,0,0,0,1,0]
 => [4,1,1,1]
 => [4,1,1,1]
 => 1
[1,4,3,5,2] => [1,0,1,1,1,0,0,1,0,0]
 => [3,1,1,1]
 => [2,1,1,1,1]
 => 2
[1,4,5,2,3] => [1,0,1,1,1,0,1,0,0,0]
 => [2,1,1,1]
 => [3,1,1]
 => 1
[1,4,5,3,2] => [1,0,1,1,1,0,1,0,0,0]
 => [2,1,1,1]
 => [3,1,1]
 => 1
[1,5,2,3,4] => [1,0,1,1,1,1,0,0,0,0]
 => [1,1,1,1]
 => [1,1,1,1]
 => 2
[1,5,2,4,3] => [1,0,1,1,1,1,0,0,0,0]
 => [1,1,1,1]
 => [1,1,1,1]
 => 2
[1,5,3,2,4] => [1,0,1,1,1,1,0,0,0,0]
 => [1,1,1,1]
 => [1,1,1,1]
 => 2
[1,5,3,4,2] => [1,0,1,1,1,1,0,0,0,0]
 => [1,1,1,1]
 => [1,1,1,1]
 => 2
[1,5,4,2,3] => [1,0,1,1,1,1,0,0,0,0]
 => [1,1,1,1]
 => [1,1,1,1]
 => 2
[1,5,4,3,2] => [1,0,1,1,1,1,0,0,0,0]
 => [1,1,1,1]
 => [1,1,1,1]
 => 2
[2,1,3,4,5] => [1,1,0,0,1,0,1,0,1,0]
 => [4,3,2]
 => [6,3]
 => 0
[2,1,3,5,4] => [1,1,0,0,1,0,1,1,0,0]
 => [3,3,2]
 => [5,2,1]
 => 0
[2,1,4,3,5] => [1,1,0,0,1,1,0,0,1,0]
 => [4,2,2]
 => [6,2]
 => 0
[5,1,2,3,4] => [1,1,1,1,1,0,0,0,0,0]
 => []
 => ?
 => ? ∊ {0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3}
[5,1,2,4,3] => [1,1,1,1,1,0,0,0,0,0]
 => []
 => ?
 => ? ∊ {0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3}
[5,1,3,2,4] => [1,1,1,1,1,0,0,0,0,0]
 => []
 => ?
 => ? ∊ {0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3}
[5,1,3,4,2] => [1,1,1,1,1,0,0,0,0,0]
 => []
 => ?
 => ? ∊ {0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3}
[5,1,4,2,3] => [1,1,1,1,1,0,0,0,0,0]
 => []
 => ?
 => ? ∊ {0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3}
[5,1,4,3,2] => [1,1,1,1,1,0,0,0,0,0]
 => []
 => ?
 => ? ∊ {0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3}
[5,2,1,3,4] => [1,1,1,1,1,0,0,0,0,0]
 => []
 => ?
 => ? ∊ {0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3}
[5,2,1,4,3] => [1,1,1,1,1,0,0,0,0,0]
 => []
 => ?
 => ? ∊ {0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3}
[5,2,3,1,4] => [1,1,1,1,1,0,0,0,0,0]
 => []
 => ?
 => ? ∊ {0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3}
[5,2,3,4,1] => [1,1,1,1,1,0,0,0,0,0]
 => []
 => ?
 => ? ∊ {0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3}
[5,2,4,1,3] => [1,1,1,1,1,0,0,0,0,0]
 => []
 => ?
 => ? ∊ {0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3}
[5,2,4,3,1] => [1,1,1,1,1,0,0,0,0,0]
 => []
 => ?
 => ? ∊ {0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3}
[5,3,1,2,4] => [1,1,1,1,1,0,0,0,0,0]
 => []
 => ?
 => ? ∊ {0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3}
[5,3,1,4,2] => [1,1,1,1,1,0,0,0,0,0]
 => []
 => ?
 => ? ∊ {0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3}
[5,3,2,1,4] => [1,1,1,1,1,0,0,0,0,0]
 => []
 => ?
 => ? ∊ {0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3}
[5,3,2,4,1] => [1,1,1,1,1,0,0,0,0,0]
 => []
 => ?
 => ? ∊ {0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3}
[5,3,4,1,2] => [1,1,1,1,1,0,0,0,0,0]
 => []
 => ?
 => ? ∊ {0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3}
[5,3,4,2,1] => [1,1,1,1,1,0,0,0,0,0]
 => []
 => ?
 => ? ∊ {0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3}
[5,4,1,2,3] => [1,1,1,1,1,0,0,0,0,0]
 => []
 => ?
 => ? ∊ {0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3}
[5,4,1,3,2] => [1,1,1,1,1,0,0,0,0,0]
 => []
 => ?
 => ? ∊ {0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3}
[5,4,2,1,3] => [1,1,1,1,1,0,0,0,0,0]
 => []
 => ?
 => ? ∊ {0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3}
[5,4,2,3,1] => [1,1,1,1,1,0,0,0,0,0]
 => []
 => ?
 => ? ∊ {0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3}
[5,4,3,1,2] => [1,1,1,1,1,0,0,0,0,0]
 => []
 => ?
 => ? ∊ {0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3}
[5,4,3,2,1] => [1,1,1,1,1,0,0,0,0,0]
 => []
 => ?
 => ? ∊ {0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3}
[6,1,2,3,4,5] => [1,1,1,1,1,1,0,0,0,0,0,0]
 => []
 => ?
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,4,4,4}
[6,1,2,3,5,4] => [1,1,1,1,1,1,0,0,0,0,0,0]
 => []
 => ?
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,4,4,4}
[6,1,2,4,3,5] => [1,1,1,1,1,1,0,0,0,0,0,0]
 => []
 => ?
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,4,4,4}
[6,1,2,4,5,3] => [1,1,1,1,1,1,0,0,0,0,0,0]
 => []
 => ?
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,4,4,4}
[6,1,2,5,3,4] => [1,1,1,1,1,1,0,0,0,0,0,0]
 => []
 => ?
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,4,4,4}
[6,1,2,5,4,3] => [1,1,1,1,1,1,0,0,0,0,0,0]
 => []
 => ?
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,4,4,4}
[6,1,3,2,4,5] => [1,1,1,1,1,1,0,0,0,0,0,0]
 => []
 => ?
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,4,4,4}
[6,1,3,2,5,4] => [1,1,1,1,1,1,0,0,0,0,0,0]
 => []
 => ?
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,4,4,4}
[6,1,3,4,2,5] => [1,1,1,1,1,1,0,0,0,0,0,0]
 => []
 => ?
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,4,4,4}
[6,1,3,4,5,2] => [1,1,1,1,1,1,0,0,0,0,0,0]
 => []
 => ?
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,4,4,4}
[6,1,3,5,2,4] => [1,1,1,1,1,1,0,0,0,0,0,0]
 => []
 => ?
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,4,4,4}
[6,1,3,5,4,2] => [1,1,1,1,1,1,0,0,0,0,0,0]
 => []
 => ?
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,4,4,4}
[6,1,4,2,3,5] => [1,1,1,1,1,1,0,0,0,0,0,0]
 => []
 => ?
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,4,4,4}
[6,1,4,2,5,3] => [1,1,1,1,1,1,0,0,0,0,0,0]
 => []
 => ?
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,4,4,4}
[6,1,4,3,2,5] => [1,1,1,1,1,1,0,0,0,0,0,0]
 => []
 => ?
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,4,4,4}
[6,1,4,3,5,2] => [1,1,1,1,1,1,0,0,0,0,0,0]
 => []
 => ?
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,4,4,4}

Description

The floored half-sum of the multiplicities of a partition. This statistic is equidistributed with [[St000143]] and [[St000149]], see [1].

Matching statistic: St000160

Mapping

Mp00108: Permutations —cycle type⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000160: Integer partitions ⟶ ℤResult quality: 82% ●values known / values provided: 82%●distinct values known / distinct values provided: 100%

Values

[1] => [1]
 => []
 => ?
 => ? = 0
[1,2] => [1,1]
 => [1]
 => []
 => 0
[2,1] => [2]
 => []
 => ?
 => ? = 0
[1,2,3] => [1,1,1]
 => [1,1]
 => [1]
 => 1
[1,3,2] => [2,1]
 => [1]
 => []
 => 0
[2,1,3] => [2,1]
 => [1]
 => []
 => 0
[2,3,1] => [3]
 => []
 => ?
 => ? ∊ {0,1}
[3,1,2] => [3]
 => []
 => ?
 => ? ∊ {0,1}
[3,2,1] => [2,1]
 => [1]
 => []
 => 0
[1,2,3,4] => [1,1,1,1]
 => [1,1,1]
 => [1,1]
 => 2
[1,2,4,3] => [2,1,1]
 => [1,1]
 => [1]
 => 1
[1,3,2,4] => [2,1,1]
 => [1,1]
 => [1]
 => 1
[1,3,4,2] => [3,1]
 => [1]
 => []
 => 0
[1,4,2,3] => [3,1]
 => [1]
 => []
 => 0
[1,4,3,2] => [2,1,1]
 => [1,1]
 => [1]
 => 1
[2,1,3,4] => [2,1,1]
 => [1,1]
 => [1]
 => 1
[2,1,4,3] => [2,2]
 => [2]
 => []
 => 0
[2,3,1,4] => [3,1]
 => [1]
 => []
 => 0
[2,3,4,1] => [4]
 => []
 => ?
 => ? ∊ {0,0,1,1,2,2}
[2,4,1,3] => [4]
 => []
 => ?
 => ? ∊ {0,0,1,1,2,2}
[2,4,3,1] => [3,1]
 => [1]
 => []
 => 0
[3,1,2,4] => [3,1]
 => [1]
 => []
 => 0
[3,1,4,2] => [4]
 => []
 => ?
 => ? ∊ {0,0,1,1,2,2}
[3,2,1,4] => [2,1,1]
 => [1,1]
 => [1]
 => 1
[3,2,4,1] => [3,1]
 => [1]
 => []
 => 0
[3,4,1,2] => [2,2]
 => [2]
 => []
 => 0
[3,4,2,1] => [4]
 => []
 => ?
 => ? ∊ {0,0,1,1,2,2}
[4,1,2,3] => [4]
 => []
 => ?
 => ? ∊ {0,0,1,1,2,2}
[4,1,3,2] => [3,1]
 => [1]
 => []
 => 0
[4,2,1,3] => [3,1]
 => [1]
 => []
 => 0
[4,2,3,1] => [2,1,1]
 => [1,1]
 => [1]
 => 1
[4,3,1,2] => [4]
 => []
 => ?
 => ? ∊ {0,0,1,1,2,2}
[4,3,2,1] => [2,2]
 => [2]
 => []
 => 0
[1,2,3,4,5] => [1,1,1,1,1]
 => [1,1,1,1]
 => [1,1,1]
 => 3
[1,2,3,5,4] => [2,1,1,1]
 => [1,1,1]
 => [1,1]
 => 2
[1,2,4,3,5] => [2,1,1,1]
 => [1,1,1]
 => [1,1]
 => 2
[1,2,4,5,3] => [3,1,1]
 => [1,1]
 => [1]
 => 1
[1,2,5,3,4] => [3,1,1]
 => [1,1]
 => [1]
 => 1
[1,2,5,4,3] => [2,1,1,1]
 => [1,1,1]
 => [1,1]
 => 2
[1,3,2,4,5] => [2,1,1,1]
 => [1,1,1]
 => [1,1]
 => 2
[1,3,2,5,4] => [2,2,1]
 => [2,1]
 => [1]
 => 1
[1,3,4,2,5] => [3,1,1]
 => [1,1]
 => [1]
 => 1
[1,3,4,5,2] => [4,1]
 => [1]
 => []
 => 0
[1,3,5,2,4] => [4,1]
 => [1]
 => []
 => 0
[1,3,5,4,2] => [3,1,1]
 => [1,1]
 => [1]
 => 1
[1,4,2,3,5] => [3,1,1]
 => [1,1]
 => [1]
 => 1
[1,4,2,5,3] => [4,1]
 => [1]
 => []
 => 0
[1,4,3,2,5] => [2,1,1,1]
 => [1,1,1]
 => [1,1]
 => 2
[1,4,3,5,2] => [3,1,1]
 => [1,1]
 => [1]
 => 1
[1,4,5,2,3] => [2,2,1]
 => [2,1]
 => [1]
 => 1
[1,4,5,3,2] => [4,1]
 => [1]
 => []
 => 0
[1,5,2,3,4] => [4,1]
 => [1]
 => []
 => 0
[1,5,2,4,3] => [3,1,1]
 => [1,1]
 => [1]
 => 1
[1,5,3,2,4] => [3,1,1]
 => [1,1]
 => [1]
 => 1
[1,5,3,4,2] => [2,1,1,1]
 => [1,1,1]
 => [1,1]
 => 2
[1,5,4,2,3] => [4,1]
 => [1]
 => []
 => 0
[1,5,4,3,2] => [2,2,1]
 => [2,1]
 => [1]
 => 1
[2,1,3,4,5] => [2,1,1,1]
 => [1,1,1]
 => [1,1]
 => 2
[2,1,3,5,4] => [2,2,1]
 => [2,1]
 => [1]
 => 1
[2,1,4,3,5] => [2,2,1]
 => [2,1]
 => [1]
 => 1
[2,3,4,5,1] => [5]
 => []
 => ?
 => ? ∊ {0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,2,2,2,3,3,3}
[2,3,5,1,4] => [5]
 => []
 => ?
 => ? ∊ {0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,2,2,2,3,3,3}
[2,4,1,5,3] => [5]
 => []
 => ?
 => ? ∊ {0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,2,2,2,3,3,3}
[2,4,5,3,1] => [5]
 => []
 => ?
 => ? ∊ {0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,2,2,2,3,3,3}
[2,5,1,3,4] => [5]
 => []
 => ?
 => ? ∊ {0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,2,2,2,3,3,3}
[2,5,4,1,3] => [5]
 => []
 => ?
 => ? ∊ {0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,2,2,2,3,3,3}
[3,1,4,5,2] => [5]
 => []
 => ?
 => ? ∊ {0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,2,2,2,3,3,3}
[3,1,5,2,4] => [5]
 => []
 => ?
 => ? ∊ {0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,2,2,2,3,3,3}
[3,4,2,5,1] => [5]
 => []
 => ?
 => ? ∊ {0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,2,2,2,3,3,3}
[3,4,5,1,2] => [5]
 => []
 => ?
 => ? ∊ {0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,2,2,2,3,3,3}
[3,5,2,1,4] => [5]
 => []
 => ?
 => ? ∊ {0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,2,2,2,3,3,3}
[3,5,4,2,1] => [5]
 => []
 => ?
 => ? ∊ {0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,2,2,2,3,3,3}
[4,1,2,5,3] => [5]
 => []
 => ?
 => ? ∊ {0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,2,2,2,3,3,3}
[4,1,5,3,2] => [5]
 => []
 => ?
 => ? ∊ {0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,2,2,2,3,3,3}
[4,3,1,5,2] => [5]
 => []
 => ?
 => ? ∊ {0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,2,2,2,3,3,3}
[4,3,5,2,1] => [5]
 => []
 => ?
 => ? ∊ {0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,2,2,2,3,3,3}
[4,5,1,2,3] => [5]
 => []
 => ?
 => ? ∊ {0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,2,2,2,3,3,3}
[4,5,2,3,1] => [5]
 => []
 => ?
 => ? ∊ {0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,2,2,2,3,3,3}
[5,1,2,3,4] => [5]
 => []
 => ?
 => ? ∊ {0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,2,2,2,3,3,3}
[5,1,4,2,3] => [5]
 => []
 => ?
 => ? ∊ {0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,2,2,2,3,3,3}
[5,3,1,2,4] => [5]
 => []
 => ?
 => ? ∊ {0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,2,2,2,3,3,3}
[5,3,4,1,2] => [5]
 => []
 => ?
 => ? ∊ {0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,2,2,2,3,3,3}
[5,4,1,3,2] => [5]
 => []
 => ?
 => ? ∊ {0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,2,2,2,3,3,3}
[5,4,2,1,3] => [5]
 => []
 => ?
 => ? ∊ {0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,2,2,2,3,3,3}
[2,3,4,5,6,1] => [6]
 => []
 => ?
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,4,4,4,4}
[2,3,4,6,1,5] => [6]
 => []
 => ?
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,4,4,4,4}
[2,3,5,1,6,4] => [6]
 => []
 => ?
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,4,4,4,4}
[2,3,5,6,4,1] => [6]
 => []
 => ?
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,4,4,4,4}
[2,3,6,1,4,5] => [6]
 => []
 => ?
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,4,4,4,4}
[2,3,6,5,1,4] => [6]
 => []
 => ?
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,4,4,4,4}
[2,4,1,5,6,3] => [6]
 => []
 => ?
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,4,4,4,4}
[2,4,1,6,3,5] => [6]
 => []
 => ?
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,4,4,4,4}
[2,4,5,3,6,1] => [6]
 => []
 => ?
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,4,4,4,4}
[2,4,5,6,1,3] => [6]
 => []
 => ?
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,4,4,4,4}
[2,4,6,3,1,5] => [6]
 => []
 => ?
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,4,4,4,4}
[2,4,6,5,3,1] => [6]
 => []
 => ?
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,4,4,4,4}
[2,5,1,3,6,4] => [6]
 => []
 => ?
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,4,4,4,4}
[2,5,1,6,4,3] => [6]
 => []
 => ?
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,4,4,4,4}
[2,5,4,1,6,3] => [6]
 => []
 => ?
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,4,4,4,4}
[2,5,4,6,3,1] => [6]
 => []
 => ?
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,4,4,4,4}

Description

The multiplicity of the smallest part of a partition. This counts the number of occurrences of the smallest part $spt(\lambda)$ of a partition $\lambda$. The sum $spt(n) = \sum_{\lambda \vdash n} spt(\lambda)$ satisfies the congruences \begin{align*} spt(5n+4) &\equiv 0\quad \pmod{5}\\\ spt(7n+5) &\equiv 0\quad \pmod{7}\\\ spt(13n+6) &\equiv 0\quad \pmod{13}, \end{align*} analogous to those of the counting function of partitions, see [1] and [2].

Matching statistic: St000228

Mapping

Mp00108: Permutations —cycle type⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000228: Integer partitions ⟶ ℤResult quality: 82% ●values known / values provided: 82%●distinct values known / distinct values provided: 100%

Values

[1] => [1]
 => []
 => ?
 => ? = 0
[1,2] => [1,1]
 => [1]
 => []
 => 0
[2,1] => [2]
 => []
 => ?
 => ? = 0
[1,2,3] => [1,1,1]
 => [1,1]
 => [1]
 => 1
[1,3,2] => [2,1]
 => [1]
 => []
 => 0
[2,1,3] => [2,1]
 => [1]
 => []
 => 0
[2,3,1] => [3]
 => []
 => ?
 => ? ∊ {0,1}
[3,1,2] => [3]
 => []
 => ?
 => ? ∊ {0,1}
[3,2,1] => [2,1]
 => [1]
 => []
 => 0
[1,2,3,4] => [1,1,1,1]
 => [1,1,1]
 => [1,1]
 => 2
[1,2,4,3] => [2,1,1]
 => [1,1]
 => [1]
 => 1
[1,3,2,4] => [2,1,1]
 => [1,1]
 => [1]
 => 1
[1,3,4,2] => [3,1]
 => [1]
 => []
 => 0
[1,4,2,3] => [3,1]
 => [1]
 => []
 => 0
[1,4,3,2] => [2,1,1]
 => [1,1]
 => [1]
 => 1
[2,1,3,4] => [2,1,1]
 => [1,1]
 => [1]
 => 1
[2,1,4,3] => [2,2]
 => [2]
 => []
 => 0
[2,3,1,4] => [3,1]
 => [1]
 => []
 => 0
[2,3,4,1] => [4]
 => []
 => ?
 => ? ∊ {0,0,1,1,2,2}
[2,4,1,3] => [4]
 => []
 => ?
 => ? ∊ {0,0,1,1,2,2}
[2,4,3,1] => [3,1]
 => [1]
 => []
 => 0
[3,1,2,4] => [3,1]
 => [1]
 => []
 => 0
[3,1,4,2] => [4]
 => []
 => ?
 => ? ∊ {0,0,1,1,2,2}
[3,2,1,4] => [2,1,1]
 => [1,1]
 => [1]
 => 1
[3,2,4,1] => [3,1]
 => [1]
 => []
 => 0
[3,4,1,2] => [2,2]
 => [2]
 => []
 => 0
[3,4,2,1] => [4]
 => []
 => ?
 => ? ∊ {0,0,1,1,2,2}
[4,1,2,3] => [4]
 => []
 => ?
 => ? ∊ {0,0,1,1,2,2}
[4,1,3,2] => [3,1]
 => [1]
 => []
 => 0
[4,2,1,3] => [3,1]
 => [1]
 => []
 => 0
[4,2,3,1] => [2,1,1]
 => [1,1]
 => [1]
 => 1
[4,3,1,2] => [4]
 => []
 => ?
 => ? ∊ {0,0,1,1,2,2}
[4,3,2,1] => [2,2]
 => [2]
 => []
 => 0
[1,2,3,4,5] => [1,1,1,1,1]
 => [1,1,1,1]
 => [1,1,1]
 => 3
[1,2,3,5,4] => [2,1,1,1]
 => [1,1,1]
 => [1,1]
 => 2
[1,2,4,3,5] => [2,1,1,1]
 => [1,1,1]
 => [1,1]
 => 2
[1,2,4,5,3] => [3,1,1]
 => [1,1]
 => [1]
 => 1
[1,2,5,3,4] => [3,1,1]
 => [1,1]
 => [1]
 => 1
[1,2,5,4,3] => [2,1,1,1]
 => [1,1,1]
 => [1,1]
 => 2
[1,3,2,4,5] => [2,1,1,1]
 => [1,1,1]
 => [1,1]
 => 2
[1,3,2,5,4] => [2,2,1]
 => [2,1]
 => [1]
 => 1
[1,3,4,2,5] => [3,1,1]
 => [1,1]
 => [1]
 => 1
[1,3,4,5,2] => [4,1]
 => [1]
 => []
 => 0
[1,3,5,2,4] => [4,1]
 => [1]
 => []
 => 0
[1,3,5,4,2] => [3,1,1]
 => [1,1]
 => [1]
 => 1
[1,4,2,3,5] => [3,1,1]
 => [1,1]
 => [1]
 => 1
[1,4,2,5,3] => [4,1]
 => [1]
 => []
 => 0
[1,4,3,2,5] => [2,1,1,1]
 => [1,1,1]
 => [1,1]
 => 2
[1,4,3,5,2] => [3,1,1]
 => [1,1]
 => [1]
 => 1
[1,4,5,2,3] => [2,2,1]
 => [2,1]
 => [1]
 => 1
[1,4,5,3,2] => [4,1]
 => [1]
 => []
 => 0
[1,5,2,3,4] => [4,1]
 => [1]
 => []
 => 0
[1,5,2,4,3] => [3,1,1]
 => [1,1]
 => [1]
 => 1
[1,5,3,2,4] => [3,1,1]
 => [1,1]
 => [1]
 => 1
[1,5,3,4,2] => [2,1,1,1]
 => [1,1,1]
 => [1,1]
 => 2
[1,5,4,2,3] => [4,1]
 => [1]
 => []
 => 0
[1,5,4,3,2] => [2,2,1]
 => [2,1]
 => [1]
 => 1
[2,1,3,4,5] => [2,1,1,1]
 => [1,1,1]
 => [1,1]
 => 2
[2,1,3,5,4] => [2,2,1]
 => [2,1]
 => [1]
 => 1
[2,1,4,3,5] => [2,2,1]
 => [2,1]
 => [1]
 => 1
[2,3,4,5,1] => [5]
 => []
 => ?
 => ? ∊ {0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,2,2,2,3,3,3}
[2,3,5,1,4] => [5]
 => []
 => ?
 => ? ∊ {0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,2,2,2,3,3,3}
[2,4,1,5,3] => [5]
 => []
 => ?
 => ? ∊ {0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,2,2,2,3,3,3}
[2,4,5,3,1] => [5]
 => []
 => ?
 => ? ∊ {0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,2,2,2,3,3,3}
[2,5,1,3,4] => [5]
 => []
 => ?
 => ? ∊ {0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,2,2,2,3,3,3}
[2,5,4,1,3] => [5]
 => []
 => ?
 => ? ∊ {0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,2,2,2,3,3,3}
[3,1,4,5,2] => [5]
 => []
 => ?
 => ? ∊ {0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,2,2,2,3,3,3}
[3,1,5,2,4] => [5]
 => []
 => ?
 => ? ∊ {0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,2,2,2,3,3,3}
[3,4,2,5,1] => [5]
 => []
 => ?
 => ? ∊ {0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,2,2,2,3,3,3}
[3,4,5,1,2] => [5]
 => []
 => ?
 => ? ∊ {0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,2,2,2,3,3,3}
[3,5,2,1,4] => [5]
 => []
 => ?
 => ? ∊ {0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,2,2,2,3,3,3}
[3,5,4,2,1] => [5]
 => []
 => ?
 => ? ∊ {0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,2,2,2,3,3,3}
[4,1,2,5,3] => [5]
 => []
 => ?
 => ? ∊ {0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,2,2,2,3,3,3}
[4,1,5,3,2] => [5]
 => []
 => ?
 => ? ∊ {0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,2,2,2,3,3,3}
[4,3,1,5,2] => [5]
 => []
 => ?
 => ? ∊ {0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,2,2,2,3,3,3}
[4,3,5,2,1] => [5]
 => []
 => ?
 => ? ∊ {0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,2,2,2,3,3,3}
[4,5,1,2,3] => [5]
 => []
 => ?
 => ? ∊ {0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,2,2,2,3,3,3}
[4,5,2,3,1] => [5]
 => []
 => ?
 => ? ∊ {0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,2,2,2,3,3,3}
[5,1,2,3,4] => [5]
 => []
 => ?
 => ? ∊ {0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,2,2,2,3,3,3}
[5,1,4,2,3] => [5]
 => []
 => ?
 => ? ∊ {0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,2,2,2,3,3,3}
[5,3,1,2,4] => [5]
 => []
 => ?
 => ? ∊ {0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,2,2,2,3,3,3}
[5,3,4,1,2] => [5]
 => []
 => ?
 => ? ∊ {0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,2,2,2,3,3,3}
[5,4,1,3,2] => [5]
 => []
 => ?
 => ? ∊ {0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,2,2,2,3,3,3}
[5,4,2,1,3] => [5]
 => []
 => ?
 => ? ∊ {0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,2,2,2,3,3,3}
[2,3,4,5,6,1] => [6]
 => []
 => ?
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,3,3,3,3,3,3,3,3,3,4,4,4,4}
[2,3,4,6,1,5] => [6]
 => []
 => ?
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,3,3,3,3,3,3,3,3,3,4,4,4,4}
[2,3,5,1,6,4] => [6]
 => []
 => ?
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,3,3,3,3,3,3,3,3,3,4,4,4,4}
[2,3,5,6,4,1] => [6]
 => []
 => ?
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,3,3,3,3,3,3,3,3,3,4,4,4,4}
[2,3,6,1,4,5] => [6]
 => []
 => ?
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,3,3,3,3,3,3,3,3,3,4,4,4,4}
[2,3,6,5,1,4] => [6]
 => []
 => ?
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,3,3,3,3,3,3,3,3,3,4,4,4,4}
[2,4,1,5,6,3] => [6]
 => []
 => ?
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,3,3,3,3,3,3,3,3,3,4,4,4,4}
[2,4,1,6,3,5] => [6]
 => []
 => ?
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,3,3,3,3,3,3,3,3,3,4,4,4,4}
[2,4,5,3,6,1] => [6]
 => []
 => ?
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,3,3,3,3,3,3,3,3,3,4,4,4,4}
[2,4,5,6,1,3] => [6]
 => []
 => ?
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,3,3,3,3,3,3,3,3,3,4,4,4,4}
[2,4,6,3,1,5] => [6]
 => []
 => ?
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,3,3,3,3,3,3,3,3,3,4,4,4,4}
[2,4,6,5,3,1] => [6]
 => []
 => ?
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,3,3,3,3,3,3,3,3,3,4,4,4,4}
[2,5,1,3,6,4] => [6]
 => []
 => ?
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,3,3,3,3,3,3,3,3,3,4,4,4,4}
[2,5,1,6,4,3] => [6]
 => []
 => ?
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,3,3,3,3,3,3,3,3,3,4,4,4,4}
[2,5,4,1,6,3] => [6]
 => []
 => ?
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,3,3,3,3,3,3,3,3,3,4,4,4,4}
[2,5,4,6,3,1] => [6]
 => []
 => ?
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,3,3,3,3,3,3,3,3,3,4,4,4,4}

Description

The size of a partition. This statistic is the constant statistic of the level sets.

Matching statistic: St000292

Mapping

Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
Mp00027: Dyck paths —to partition⟶ Integer partitions
Mp00095: Integer partitions —to binary word⟶ Binary words
St000292: Binary words ⟶ ℤResult quality: 82% ●values known / values provided: 82%●distinct values known / distinct values provided: 100%

Values

[1] => [1,0]
 => []
 =>  => ? = 0
[1,2] => [1,0,1,0]
 => [1]
 => 10 => 0
[2,1] => [1,1,0,0]
 => []
 =>  => ? = 0
[1,2,3] => [1,0,1,0,1,0]
 => [2,1]
 => 1010 => 1
[1,3,2] => [1,0,1,1,0,0]
 => [1,1]
 => 110 => 0
[2,1,3] => [1,1,0,0,1,0]
 => [2]
 => 100 => 0
[2,3,1] => [1,1,0,1,0,0]
 => [1]
 => 10 => 0
[3,1,2] => [1,1,1,0,0,0]
 => []
 =>  => ? ∊ {0,1}
[3,2,1] => [1,1,1,0,0,0]
 => []
 =>  => ? ∊ {0,1}
[1,2,3,4] => [1,0,1,0,1,0,1,0]
 => [3,2,1]
 => 101010 => 2
[1,2,4,3] => [1,0,1,0,1,1,0,0]
 => [2,2,1]
 => 11010 => 1
[1,3,2,4] => [1,0,1,1,0,0,1,0]
 => [3,1,1]
 => 100110 => 1
[1,3,4,2] => [1,0,1,1,0,1,0,0]
 => [2,1,1]
 => 10110 => 1
[1,4,2,3] => [1,0,1,1,1,0,0,0]
 => [1,1,1]
 => 1110 => 0
[1,4,3,2] => [1,0,1,1,1,0,0,0]
 => [1,1,1]
 => 1110 => 0
[2,1,3,4] => [1,1,0,0,1,0,1,0]
 => [3,2]
 => 10100 => 1
[2,1,4,3] => [1,1,0,0,1,1,0,0]
 => [2,2]
 => 1100 => 0
[2,3,1,4] => [1,1,0,1,0,0,1,0]
 => [3,1]
 => 10010 => 1
[2,3,4,1] => [1,1,0,1,0,1,0,0]
 => [2,1]
 => 1010 => 1
[2,4,1,3] => [1,1,0,1,1,0,0,0]
 => [1,1]
 => 110 => 0
[2,4,3,1] => [1,1,0,1,1,0,0,0]
 => [1,1]
 => 110 => 0
[3,1,2,4] => [1,1,1,0,0,0,1,0]
 => [3]
 => 1000 => 0
[3,1,4,2] => [1,1,1,0,0,1,0,0]
 => [2]
 => 100 => 0
[3,2,1,4] => [1,1,1,0,0,0,1,0]
 => [3]
 => 1000 => 0
[3,2,4,1] => [1,1,1,0,0,1,0,0]
 => [2]
 => 100 => 0
[3,4,1,2] => [1,1,1,0,1,0,0,0]
 => [1]
 => 10 => 0
[3,4,2,1] => [1,1,1,0,1,0,0,0]
 => [1]
 => 10 => 0
[4,1,2,3] => [1,1,1,1,0,0,0,0]
 => []
 =>  => ? ∊ {0,0,1,1,2,2}
[4,1,3,2] => [1,1,1,1,0,0,0,0]
 => []
 =>  => ? ∊ {0,0,1,1,2,2}
[4,2,1,3] => [1,1,1,1,0,0,0,0]
 => []
 =>  => ? ∊ {0,0,1,1,2,2}
[4,2,3,1] => [1,1,1,1,0,0,0,0]
 => []
 =>  => ? ∊ {0,0,1,1,2,2}
[4,3,1,2] => [1,1,1,1,0,0,0,0]
 => []
 =>  => ? ∊ {0,0,1,1,2,2}
[4,3,2,1] => [1,1,1,1,0,0,0,0]
 => []
 =>  => ? ∊ {0,0,1,1,2,2}
[1,2,3,4,5] => [1,0,1,0,1,0,1,0,1,0]
 => [4,3,2,1]
 => 10101010 => 3
[1,2,3,5,4] => [1,0,1,0,1,0,1,1,0,0]
 => [3,3,2,1]
 => 1101010 => 2
[1,2,4,3,5] => [1,0,1,0,1,1,0,0,1,0]
 => [4,2,2,1]
 => 10011010 => 2
[1,2,4,5,3] => [1,0,1,0,1,1,0,1,0,0]
 => [3,2,2,1]
 => 1011010 => 2
[1,2,5,3,4] => [1,0,1,0,1,1,1,0,0,0]
 => [2,2,2,1]
 => 111010 => 1
[1,2,5,4,3] => [1,0,1,0,1,1,1,0,0,0]
 => [2,2,2,1]
 => 111010 => 1
[1,3,2,4,5] => [1,0,1,1,0,0,1,0,1,0]
 => [4,3,1,1]
 => 10100110 => 2
[1,3,2,5,4] => [1,0,1,1,0,0,1,1,0,0]
 => [3,3,1,1]
 => 1100110 => 1
[1,3,4,2,5] => [1,0,1,1,0,1,0,0,1,0]
 => [4,2,1,1]
 => 10010110 => 2
[1,3,4,5,2] => [1,0,1,1,0,1,0,1,0,0]
 => [3,2,1,1]
 => 1010110 => 2
[1,3,5,2,4] => [1,0,1,1,0,1,1,0,0,0]
 => [2,2,1,1]
 => 110110 => 1
[1,3,5,4,2] => [1,0,1,1,0,1,1,0,0,0]
 => [2,2,1,1]
 => 110110 => 1
[1,4,2,3,5] => [1,0,1,1,1,0,0,0,1,0]
 => [4,1,1,1]
 => 10001110 => 1
[1,4,2,5,3] => [1,0,1,1,1,0,0,1,0,0]
 => [3,1,1,1]
 => 1001110 => 1
[1,4,3,2,5] => [1,0,1,1,1,0,0,0,1,0]
 => [4,1,1,1]
 => 10001110 => 1
[1,4,3,5,2] => [1,0,1,1,1,0,0,1,0,0]
 => [3,1,1,1]
 => 1001110 => 1
[1,4,5,2,3] => [1,0,1,1,1,0,1,0,0,0]
 => [2,1,1,1]
 => 101110 => 1
[1,4,5,3,2] => [1,0,1,1,1,0,1,0,0,0]
 => [2,1,1,1]
 => 101110 => 1
[1,5,2,3,4] => [1,0,1,1,1,1,0,0,0,0]
 => [1,1,1,1]
 => 11110 => 0
[1,5,2,4,3] => [1,0,1,1,1,1,0,0,0,0]
 => [1,1,1,1]
 => 11110 => 0
[1,5,3,2,4] => [1,0,1,1,1,1,0,0,0,0]
 => [1,1,1,1]
 => 11110 => 0
[1,5,3,4,2] => [1,0,1,1,1,1,0,0,0,0]
 => [1,1,1,1]
 => 11110 => 0
[1,5,4,2,3] => [1,0,1,1,1,1,0,0,0,0]
 => [1,1,1,1]
 => 11110 => 0
[1,5,4,3,2] => [1,0,1,1,1,1,0,0,0,0]
 => [1,1,1,1]
 => 11110 => 0
[2,1,3,4,5] => [1,1,0,0,1,0,1,0,1,0]
 => [4,3,2]
 => 1010100 => 2
[2,1,3,5,4] => [1,1,0,0,1,0,1,1,0,0]
 => [3,3,2]
 => 110100 => 1
[2,1,4,3,5] => [1,1,0,0,1,1,0,0,1,0]
 => [4,2,2]
 => 1001100 => 1
[5,1,2,3,4] => [1,1,1,1,1,0,0,0,0,0]
 => []
 =>  => ? ∊ {0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,2,2,2,3,3,3}
[5,1,2,4,3] => [1,1,1,1,1,0,0,0,0,0]
 => []
 =>  => ? ∊ {0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,2,2,2,3,3,3}
[5,1,3,2,4] => [1,1,1,1,1,0,0,0,0,0]
 => []
 =>  => ? ∊ {0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,2,2,2,3,3,3}
[5,1,3,4,2] => [1,1,1,1,1,0,0,0,0,0]
 => []
 =>  => ? ∊ {0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,2,2,2,3,3,3}
[5,1,4,2,3] => [1,1,1,1,1,0,0,0,0,0]
 => []
 =>  => ? ∊ {0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,2,2,2,3,3,3}
[5,1,4,3,2] => [1,1,1,1,1,0,0,0,0,0]
 => []
 =>  => ? ∊ {0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,2,2,2,3,3,3}
[5,2,1,3,4] => [1,1,1,1,1,0,0,0,0,0]
 => []
 =>  => ? ∊ {0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,2,2,2,3,3,3}
[5,2,1,4,3] => [1,1,1,1,1,0,0,0,0,0]
 => []
 =>  => ? ∊ {0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,2,2,2,3,3,3}
[5,2,3,1,4] => [1,1,1,1,1,0,0,0,0,0]
 => []
 =>  => ? ∊ {0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,2,2,2,3,3,3}
[5,2,3,4,1] => [1,1,1,1,1,0,0,0,0,0]
 => []
 =>  => ? ∊ {0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,2,2,2,3,3,3}
[5,2,4,1,3] => [1,1,1,1,1,0,0,0,0,0]
 => []
 =>  => ? ∊ {0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,2,2,2,3,3,3}
[5,2,4,3,1] => [1,1,1,1,1,0,0,0,0,0]
 => []
 =>  => ? ∊ {0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,2,2,2,3,3,3}
[5,3,1,2,4] => [1,1,1,1,1,0,0,0,0,0]
 => []
 =>  => ? ∊ {0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,2,2,2,3,3,3}
[5,3,1,4,2] => [1,1,1,1,1,0,0,0,0,0]
 => []
 =>  => ? ∊ {0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,2,2,2,3,3,3}
[5,3,2,1,4] => [1,1,1,1,1,0,0,0,0,0]
 => []
 =>  => ? ∊ {0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,2,2,2,3,3,3}
[5,3,2,4,1] => [1,1,1,1,1,0,0,0,0,0]
 => []
 =>  => ? ∊ {0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,2,2,2,3,3,3}
[5,3,4,1,2] => [1,1,1,1,1,0,0,0,0,0]
 => []
 =>  => ? ∊ {0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,2,2,2,3,3,3}
[5,3,4,2,1] => [1,1,1,1,1,0,0,0,0,0]
 => []
 =>  => ? ∊ {0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,2,2,2,3,3,3}
[5,4,1,2,3] => [1,1,1,1,1,0,0,0,0,0]
 => []
 =>  => ? ∊ {0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,2,2,2,3,3,3}
[5,4,1,3,2] => [1,1,1,1,1,0,0,0,0,0]
 => []
 =>  => ? ∊ {0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,2,2,2,3,3,3}
[5,4,2,1,3] => [1,1,1,1,1,0,0,0,0,0]
 => []
 =>  => ? ∊ {0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,2,2,2,3,3,3}
[5,4,2,3,1] => [1,1,1,1,1,0,0,0,0,0]
 => []
 =>  => ? ∊ {0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,2,2,2,3,3,3}
[5,4,3,1,2] => [1,1,1,1,1,0,0,0,0,0]
 => []
 =>  => ? ∊ {0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,2,2,2,3,3,3}
[5,4,3,2,1] => [1,1,1,1,1,0,0,0,0,0]
 => []
 =>  => ? ∊ {0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,2,2,2,3,3,3}
[6,1,2,3,4,5] => [1,1,1,1,1,1,0,0,0,0,0,0]
 => []
 =>  => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,4,4,4,4}
[6,1,2,3,5,4] => [1,1,1,1,1,1,0,0,0,0,0,0]
 => []
 =>  => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,4,4,4,4}
[6,1,2,4,3,5] => [1,1,1,1,1,1,0,0,0,0,0,0]
 => []
 =>  => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,4,4,4,4}
[6,1,2,4,5,3] => [1,1,1,1,1,1,0,0,0,0,0,0]
 => []
 =>  => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,4,4,4,4}
[6,1,2,5,3,4] => [1,1,1,1,1,1,0,0,0,0,0,0]
 => []
 =>  => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,4,4,4,4}
[6,1,2,5,4,3] => [1,1,1,1,1,1,0,0,0,0,0,0]
 => []
 =>  => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,4,4,4,4}
[6,1,3,2,4,5] => [1,1,1,1,1,1,0,0,0,0,0,0]
 => []
 =>  => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,4,4,4,4}
[6,1,3,2,5,4] => [1,1,1,1,1,1,0,0,0,0,0,0]
 => []
 =>  => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,4,4,4,4}
[6,1,3,4,2,5] => [1,1,1,1,1,1,0,0,0,0,0,0]
 => []
 =>  => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,4,4,4,4}
[6,1,3,4,5,2] => [1,1,1,1,1,1,0,0,0,0,0,0]
 => []
 =>  => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,4,4,4,4}
[6,1,3,5,2,4] => [1,1,1,1,1,1,0,0,0,0,0,0]
 => []
 =>  => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,4,4,4,4}
[6,1,3,5,4,2] => [1,1,1,1,1,1,0,0,0,0,0,0]
 => []
 =>  => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,4,4,4,4}
[6,1,4,2,3,5] => [1,1,1,1,1,1,0,0,0,0,0,0]
 => []
 =>  => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,4,4,4,4}
[6,1,4,2,5,3] => [1,1,1,1,1,1,0,0,0,0,0,0]
 => []
 =>  => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,4,4,4,4}
[6,1,4,3,2,5] => [1,1,1,1,1,1,0,0,0,0,0,0]
 => []
 =>  => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,4,4,4,4}
[6,1,4,3,5,2] => [1,1,1,1,1,1,0,0,0,0,0,0]
 => []
 =>  => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,4,4,4,4}

Description

The number of ascents of a binary word.

Matching statistic: St000319
(load all 4 compositions to match this statistic)

Mapping

Mp00108: Permutations —cycle type⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00044: Integer partitions —conjugate⟶ Integer partitions
St000319: Integer partitions ⟶ ℤResult quality: 82% ●values known / values provided: 82%●distinct values known / distinct values provided: 100%

Values

[1] => [1]
 => []
 => []
 => ? = 0
[1,2] => [1,1]
 => [1]
 => [1]
 => 0
[2,1] => [2]
 => []
 => []
 => ? = 0
[1,2,3] => [1,1,1]
 => [1,1]
 => [2]
 => 1
[1,3,2] => [2,1]
 => [1]
 => [1]
 => 0
[2,1,3] => [2,1]
 => [1]
 => [1]
 => 0
[2,3,1] => [3]
 => []
 => []
 => ? ∊ {0,1}
[3,1,2] => [3]
 => []
 => []
 => ? ∊ {0,1}
[3,2,1] => [2,1]
 => [1]
 => [1]
 => 0
[1,2,3,4] => [1,1,1,1]
 => [1,1,1]
 => [3]
 => 2
[1,2,4,3] => [2,1,1]
 => [1,1]
 => [2]
 => 1
[1,3,2,4] => [2,1,1]
 => [1,1]
 => [2]
 => 1
[1,3,4,2] => [3,1]
 => [1]
 => [1]
 => 0
[1,4,2,3] => [3,1]
 => [1]
 => [1]
 => 0
[1,4,3,2] => [2,1,1]
 => [1,1]
 => [2]
 => 1
[2,1,3,4] => [2,1,1]
 => [1,1]
 => [2]
 => 1
[2,1,4,3] => [2,2]
 => [2]
 => [1,1]
 => 0
[2,3,1,4] => [3,1]
 => [1]
 => [1]
 => 0
[2,3,4,1] => [4]
 => []
 => []
 => ? ∊ {0,0,1,1,2,2}
[2,4,1,3] => [4]
 => []
 => []
 => ? ∊ {0,0,1,1,2,2}
[2,4,3,1] => [3,1]
 => [1]
 => [1]
 => 0
[3,1,2,4] => [3,1]
 => [1]
 => [1]
 => 0
[3,1,4,2] => [4]
 => []
 => []
 => ? ∊ {0,0,1,1,2,2}
[3,2,1,4] => [2,1,1]
 => [1,1]
 => [2]
 => 1
[3,2,4,1] => [3,1]
 => [1]
 => [1]
 => 0
[3,4,1,2] => [2,2]
 => [2]
 => [1,1]
 => 0
[3,4,2,1] => [4]
 => []
 => []
 => ? ∊ {0,0,1,1,2,2}
[4,1,2,3] => [4]
 => []
 => []
 => ? ∊ {0,0,1,1,2,2}
[4,1,3,2] => [3,1]
 => [1]
 => [1]
 => 0
[4,2,1,3] => [3,1]
 => [1]
 => [1]
 => 0
[4,2,3,1] => [2,1,1]
 => [1,1]
 => [2]
 => 1
[4,3,1,2] => [4]
 => []
 => []
 => ? ∊ {0,0,1,1,2,2}
[4,3,2,1] => [2,2]
 => [2]
 => [1,1]
 => 0
[1,2,3,4,5] => [1,1,1,1,1]
 => [1,1,1,1]
 => [4]
 => 3
[1,2,3,5,4] => [2,1,1,1]
 => [1,1,1]
 => [3]
 => 2
[1,2,4,3,5] => [2,1,1,1]
 => [1,1,1]
 => [3]
 => 2
[1,2,4,5,3] => [3,1,1]
 => [1,1]
 => [2]
 => 1
[1,2,5,3,4] => [3,1,1]
 => [1,1]
 => [2]
 => 1
[1,2,5,4,3] => [2,1,1,1]
 => [1,1,1]
 => [3]
 => 2
[1,3,2,4,5] => [2,1,1,1]
 => [1,1,1]
 => [3]
 => 2
[1,3,2,5,4] => [2,2,1]
 => [2,1]
 => [2,1]
 => 1
[1,3,4,2,5] => [3,1,1]
 => [1,1]
 => [2]
 => 1
[1,3,4,5,2] => [4,1]
 => [1]
 => [1]
 => 0
[1,3,5,2,4] => [4,1]
 => [1]
 => [1]
 => 0
[1,3,5,4,2] => [3,1,1]
 => [1,1]
 => [2]
 => 1
[1,4,2,3,5] => [3,1,1]
 => [1,1]
 => [2]
 => 1
[1,4,2,5,3] => [4,1]
 => [1]
 => [1]
 => 0
[1,4,3,2,5] => [2,1,1,1]
 => [1,1,1]
 => [3]
 => 2
[1,4,3,5,2] => [3,1,1]
 => [1,1]
 => [2]
 => 1
[1,4,5,2,3] => [2,2,1]
 => [2,1]
 => [2,1]
 => 1
[1,4,5,3,2] => [4,1]
 => [1]
 => [1]
 => 0
[1,5,2,3,4] => [4,1]
 => [1]
 => [1]
 => 0
[1,5,2,4,3] => [3,1,1]
 => [1,1]
 => [2]
 => 1
[1,5,3,2,4] => [3,1,1]
 => [1,1]
 => [2]
 => 1
[1,5,3,4,2] => [2,1,1,1]
 => [1,1,1]
 => [3]
 => 2
[1,5,4,2,3] => [4,1]
 => [1]
 => [1]
 => 0
[1,5,4,3,2] => [2,2,1]
 => [2,1]
 => [2,1]
 => 1
[2,1,3,4,5] => [2,1,1,1]
 => [1,1,1]
 => [3]
 => 2
[2,1,3,5,4] => [2,2,1]
 => [2,1]
 => [2,1]
 => 1
[2,1,4,3,5] => [2,2,1]
 => [2,1]
 => [2,1]
 => 1
[2,3,4,5,1] => [5]
 => []
 => []
 => ? ∊ {0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,2,2,2,3,3,3}
[2,3,5,1,4] => [5]
 => []
 => []
 => ? ∊ {0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,2,2,2,3,3,3}
[2,4,1,5,3] => [5]
 => []
 => []
 => ? ∊ {0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,2,2,2,3,3,3}
[2,4,5,3,1] => [5]
 => []
 => []
 => ? ∊ {0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,2,2,2,3,3,3}
[2,5,1,3,4] => [5]
 => []
 => []
 => ? ∊ {0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,2,2,2,3,3,3}
[2,5,4,1,3] => [5]
 => []
 => []
 => ? ∊ {0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,2,2,2,3,3,3}
[3,1,4,5,2] => [5]
 => []
 => []
 => ? ∊ {0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,2,2,2,3,3,3}
[3,1,5,2,4] => [5]
 => []
 => []
 => ? ∊ {0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,2,2,2,3,3,3}
[3,4,2,5,1] => [5]
 => []
 => []
 => ? ∊ {0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,2,2,2,3,3,3}
[3,4,5,1,2] => [5]
 => []
 => []
 => ? ∊ {0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,2,2,2,3,3,3}
[3,5,2,1,4] => [5]
 => []
 => []
 => ? ∊ {0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,2,2,2,3,3,3}
[3,5,4,2,1] => [5]
 => []
 => []
 => ? ∊ {0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,2,2,2,3,3,3}
[4,1,2,5,3] => [5]
 => []
 => []
 => ? ∊ {0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,2,2,2,3,3,3}
[4,1,5,3,2] => [5]
 => []
 => []
 => ? ∊ {0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,2,2,2,3,3,3}
[4,3,1,5,2] => [5]
 => []
 => []
 => ? ∊ {0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,2,2,2,3,3,3}
[4,3,5,2,1] => [5]
 => []
 => []
 => ? ∊ {0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,2,2,2,3,3,3}
[4,5,1,2,3] => [5]
 => []
 => []
 => ? ∊ {0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,2,2,2,3,3,3}
[4,5,2,3,1] => [5]
 => []
 => []
 => ? ∊ {0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,2,2,2,3,3,3}
[5,1,2,3,4] => [5]
 => []
 => []
 => ? ∊ {0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,2,2,2,3,3,3}
[5,1,4,2,3] => [5]
 => []
 => []
 => ? ∊ {0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,2,2,2,3,3,3}
[5,3,1,2,4] => [5]
 => []
 => []
 => ? ∊ {0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,2,2,2,3,3,3}
[5,3,4,1,2] => [5]
 => []
 => []
 => ? ∊ {0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,2,2,2,3,3,3}
[5,4,1,3,2] => [5]
 => []
 => []
 => ? ∊ {0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,2,2,2,3,3,3}
[5,4,2,1,3] => [5]
 => []
 => []
 => ? ∊ {0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,2,2,2,3,3,3}
[2,3,4,5,6,1] => [6]
 => []
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,4,4,4,4}
[2,3,4,6,1,5] => [6]
 => []
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,4,4,4,4}
[2,3,5,1,6,4] => [6]
 => []
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,4,4,4,4}
[2,3,5,6,4,1] => [6]
 => []
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,4,4,4,4}
[2,3,6,1,4,5] => [6]
 => []
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,4,4,4,4}
[2,3,6,5,1,4] => [6]
 => []
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,4,4,4,4}
[2,4,1,5,6,3] => [6]
 => []
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,4,4,4,4}
[2,4,1,6,3,5] => [6]
 => []
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,4,4,4,4}
[2,4,5,3,6,1] => [6]
 => []
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,4,4,4,4}
[2,4,5,6,1,3] => [6]
 => []
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,4,4,4,4}
[2,4,6,3,1,5] => [6]
 => []
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,4,4,4,4}
[2,4,6,5,3,1] => [6]
 => []
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,4,4,4,4}
[2,5,1,3,6,4] => [6]
 => []
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,4,4,4,4}
[2,5,1,6,4,3] => [6]
 => []
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,4,4,4,4}
[2,5,4,1,6,3] => [6]
 => []
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,4,4,4,4}
[2,5,4,6,3,1] => [6]
 => []
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,4,4,4,4}

Description

The spin of an integer partition. The Ferrers shape of an integer partition $\lambda$ can be decomposed into border strips. The spin is then defined to be the total number of crossings of border strips of $\lambda$ with the vertical lines in the Ferrers shape. The following example is taken from Appendix B in [1]: Let $\lambda = (5,5,4,4,2,1)$. Removing the border strips successively yields the sequence of partitions $$(5,5,4,4,2,1), (4,3,3,1), (2,2), (1), ().$$ The first strip $(5,5,4,4,2,1) \setminus (4,3,3,1)$ crosses $4$ times, the second strip $(4,3,3,1) \setminus (2,2)$ crosses $3$ times, the strip $(2,2) \setminus (1)$ crosses $1$ time, and the remaining strip $(1) \setminus ()$ does not cross. This yields the spin of $(5,5,4,4,2,1)$ to be $4+3+1 = 8$.

The following 161 statistics, ordered by result quality, also match your data. Click on any of them to see the details.

St000320The dinv adjustment of an integer partition. St000329The number of evenly positioned ascents of the Dyck path, with the initial position equal to 1. St000377The dinv defect of an integer partition. St000384The maximal part of the shifted composition of an integer partition. St000394The sum of the heights of the peaks of a Dyck path minus the number of peaks. St000459The hook length of the base cell of a partition. St000475The number of parts equal to 1 in a partition. St000519The largest length of a factor maximising the subword complexity. St000548The number of different non-empty partial sums of an integer partition. St000784The maximum of the length and the largest part of the integer partition. St001508The degree of the standard monomial associated to a Dyck path relative to the diagonal boundary. St001553The number of indecomposable summands of the square of the Jacobson radical as a bimodule in the Nakayama algebra corresponding to the Dyck path. St001918The degree of the cyclic sieving polynomial corresponding to an integer partition. St000934The 2-degree of an integer partition. St001414Half the length of the longest odd length palindromic prefix of a binary word. St001730The number of times the path corresponding to a binary word crosses the base line. St001384The number of boxes in the diagram of a partition that do not lie in the largest triangle it contains. St001036The number of inner corners of the parallelogram polyomino associated with the Dyck path. St001037The number of inner corners of the upper path of the parallelogram polyomino associated with the Dyck path. St001124The multiplicity of the standard representation in the Kronecker square corresponding to a partition. St000771The largest multiplicity of a distance Laplacian eigenvalue in a connected graph. St000512The number of invariant subsets of size 3 when acting with a permutation of given cycle type. St000940The number of characters of the symmetric group whose value on the partition is zero. St000931The number of occurrences of the pattern UUU in a Dyck path. St001276The number of 2-regular indecomposable modules in the corresponding Nakayama algebra. St001629The coefficient of the integer composition in the quasisymmetric expansion of the relabelling action of the symmetric group on cycles. St000938The number of zeros of the symmetric group character corresponding to the partition. St001498The normalised height of a Nakayama algebra with magnitude 1. St000478Another weight of a partition according to Alladi. St000567The sum of the products of all pairs of parts. St000621The number of standard tableaux of shape equal to the given partition such that the minimal cyclic descent is even. St000929The constant term of the character polynomial of an integer partition. St001099The coefficient times the product of the factorials of the parts of the monomial symmetric function indexed by the partition in the formal group law for leaf labelled binary trees. St001604The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on polygons. St001089Number of indecomposable projective non-injective modules minus the number of indecomposable projective non-injective modules with dominant dimension equal to the injective dimension in the corresponding Nakayama algebra. St001113Number of indecomposable projective non-injective modules with reflexive Auslander-Reiten sequences in the corresponding Nakayama algebra. St001222Number of simple modules in the corresponding LNakayama algebra that have a unique 2-extension with the regular module. St001266The largest vector space dimension of an indecomposable non-projective module that is reflexive in the corresponding Nakayama algebra. St000137The Grundy value of an integer partition. St000225Difference between largest and smallest parts in a partition. St001283The number of finite solvable groups that are realised by the given partition over the complex numbers. St001284The number of finite groups that are realised by the given partition over the complex numbers. St001525The number of symmetric hooks on the diagonal of a partition. St001587Half of the largest even part of an integer partition. St001632The number of indecomposable injective modules $I$ with $dim Ext^1(I,A)=1$ for the incidence algebra A of a poset. St001657The number of twos in an integer partition. St001939The number of parts that are equal to their multiplicity in the integer partition. St001940The number of distinct parts that are equal to their multiplicity in the integer partition. St000772The multiplicity of the largest distance Laplacian eigenvalue in a connected graph. St000260The radius of a connected graph. St000566The number of ways to select a row of a Ferrers shape and two cells in this row. St001651The Frankl number of a lattice. St000941The number of characters of the symmetric group whose value on the partition is even. St000369The dinv deficit of a Dyck path. St000376The bounce deficit of a Dyck path. St000683The number of points below the Dyck path such that the diagonal to the north-east hits the path between two down steps, and the diagonal to the north-west hits the path between two up steps. St001035The convexity degree of the parallelogram polyomino associated with the Dyck path. St001101The coefficient times the product of the factorials of the parts of the monomial symmetric function indexed by the partition in the formal group law for increasing trees. St001139The number of occurrences of hills of size 2 in a Dyck path. St001418Half of the global dimension of the stable Auslander algebra of the Nakayama algebra corresponding to the Dyck path. St001431Half of the Loewy length minus one of a modified stable Auslander algebra of the Nakayama algebra corresponding to the Dyck path. St001480The number of simple summands of the module J^2/J^3. St001502The global dimension minus the dominant dimension of magnitude 1 Nakayama algebras. St001195The global dimension of the algebra $A/AfA$ of the corresponding Nakayama algebra $A$ with minimal left faithful projective-injective module $Af$. St001877Number of indecomposable injective modules with projective dimension 2. St001876The number of 2-regular simple modules in the incidence algebra of the lattice. St001964The interval resolution global dimension of a poset. St001123The multiplicity of the dual of the standard representation in the Kronecker square corresponding to a partition. St000506The number of standard desarrangement tableaux of shape equal to the given partition. St001122The multiplicity of the sign representation in the Kronecker square corresponding to a partition. St001247The number of parts of a partition that are not congruent 2 modulo 3. St001280The number of parts of an integer partition that are at least two. St001392The largest nonnegative integer which is not a part and is smaller than the largest part of the partition. St001440The number of standard Young tableaux whose major index is congruent one modulo the size of a given integer partition. St001714The number of subpartitions of an integer partition that do not dominate the conjugate subpartition. St001767The largest minimal number of arrows pointing to a cell in the Ferrers diagram in any assignment. St001785The number of ways to obtain a partition as the multiset of antidiagonal lengths of the Ferrers diagram of a partition. St000936The number of even values of the symmetric group character corresponding to the partition. St000455The second largest eigenvalue of a graph if it is integral. St000620The number of standard tableaux of shape equal to the given partition such that the minimal cyclic descent is odd. St001097The coefficient of the monomial symmetric function indexed by the partition in the formal group law for linear orders. St001100The coefficient times the product of the factorials of the parts of the monomial symmetric function indexed by the partition in the formal group law for leaf labelled trees. St000668The least common multiple of the parts of the partition. St000681The Grundy value of Chomp on Ferrers diagrams. St000708The product of the parts of an integer partition. St000933The number of multipartitions of sizes given by an integer partition. St000454The largest eigenvalue of a graph if it is integral. St000460The hook length of the last cell along the main diagonal of an integer partition. St000870The product of the hook lengths of the diagonal cells in an integer partition. St001248Sum of the even parts of a partition. St001249Sum of the odd parts of a partition. St001250The number of parts of a partition that are not congruent 0 modulo 3. St001279The sum of the parts of an integer partition that are at least two. St001360The number of covering relations in Young's lattice below a partition. St001383The BG-rank of an integer partition. St001442The number of standard Young tableaux whose major index is divisible by the size of a given integer partition. St001541The Gini index of an integer partition. St001593This is the number of standard Young tableaux of the given shifted shape. St001600The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on simple graphs. St001601The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on trees. St001606The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on set partitions. St001628The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on simple connected graphs. St001912The length of the preperiod in Bulgarian solitaire corresponding to an integer partition. St001933The largest multiplicity of a part in an integer partition. St001878The projective dimension of the simple modules corresponding to the minimum of L in the incidence algebra of the lattice L. St001570The minimal number of edges to add to make a graph Hamiltonian. St000175Degree of the polynomial counting the number of semistandard Young tableaux when stretching the shape. St000205Number of non-integral Gelfand-Tsetlin polytopes with prescribed top row and partition weight. St000206Number of non-integral Gelfand-Tsetlin polytopes with prescribed top row and integer composition weight. St000749The smallest integer d such that the restriction of the representation corresponding to a partition of n to the symmetric group on n-d letters has a constituent of odd degree. St000944The 3-degree of an integer partition. St001175The size of a partition minus the hook length of the base cell. St001586The number of odd parts smaller than the largest even part in an integer partition. St001199The dominant dimension of $eAe$ for the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St001603The number of colourings of a polygon such that the multiplicities of a colour are given by a partition. St001605The number of colourings of a cycle such that the multiplicities of colours are given by a partition. St000456The monochromatic index of a connected graph. St000698The number of 2-rim hooks removed from an integer partition to obtain its associated 2-core. St000704The number of semistandard tableaux on a given integer partition with minimal maximal entry. St000993The multiplicity of the largest part of an integer partition. St001568The smallest positive integer that does not appear twice in the partition. St001630The global dimension of the incidence algebra of the lattice over the rational numbers. St001875The number of simple modules with projective dimension at most 1. St000284The Plancherel distribution on integer partitions. St000707The product of the factorials of the parts. St000770The major index of an integer partition when read from bottom to top. St000815The number of semistandard Young tableaux of partition weight of given shape. St000901The cube of the number of standard Young tableaux with shape given by the partition. St000937The number of positive values of the symmetric group character corresponding to the partition. St001128The exponens consonantiae of a partition. St001491The number of indecomposable projective-injective modules in the algebra corresponding to a subset. St000307The number of rowmotion orbits of a poset. St000632The jump number of the poset. St000714The number of semistandard Young tableau of given shape, with entries at most 2. St001881The number of factors of a lattice as a Cartesian product of lattices. St001820The size of the image of the pop stack sorting operator. St001200The number of simple modules in $eAe$ with projective dimension at most 2 in the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St001624The breadth of a lattice. St000298The order dimension or Dushnik-Miller dimension of a poset. St000640The rank of the largest boolean interval in a poset. St001719The number of shortest chains of small intervals from the bottom to the top in a lattice. St000534The number of 2-rises of a permutation. St000451The length of the longest pattern of the form k 1 2. St000842The breadth of a permutation. St000259The diameter of a connected graph. St000302The determinant of the distance matrix of a connected graph. St000466The Gutman (or modified Schultz) index of a connected graph. St000467The hyper-Wiener index of a connected graph. St000777The number of distinct eigenvalues of the distance Laplacian of a connected graph. St001645The pebbling number of a connected graph. St001330The hat guessing number of a graph. St001864The number of excedances of a signed permutation. St000848The balance constant multiplied with the number of linear extensions of a poset. St000849The number of 1/3-balanced pairs in a poset. St000850The number of 1/2-balanced pairs in a poset. St001095The number of non-isomorphic posets with precisely one further covering relation. St001526The Loewy length of the Auslander-Reiten translate of the regular module as a bimodule of the Nakayama algebra corresponding to the Dyck path. St000408The number of occurrences of the pattern 4231 in a permutation. St000440The number of occurrences of the pattern 4132 or of the pattern 4231 in a permutation. St000036The evaluation at 1 of the Kazhdan-Lusztig polynomial with parameters given by the identity and the permutation. St001060The distinguishing index of a graph.