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Your data matches 422 different statistics following compositions of up to 3 maps.
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Matching statistic: St001586
Mp00133: Integer compositions —delta morphism⟶ Integer compositions
Mp00180: Integer compositions —to ribbon⟶ Skew partitions
Mp00183: Skew partitions —inner shape⟶ Integer partitions
St001586: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00180: Integer compositions —to ribbon⟶ Skew partitions
Mp00183: Skew partitions —inner shape⟶ Integer partitions
St001586: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,1,2] => [2,1] => [[2,2],[1]]
=> [1]
=> 0
[1,1,1,2] => [3,1] => [[3,3],[2]]
=> [2]
=> 0
[1,1,2,1] => [2,1,1] => [[2,2,2],[1,1]]
=> [1,1]
=> 0
[1,1,3] => [2,1] => [[2,2],[1]]
=> [1]
=> 0
[2,2,1] => [2,1] => [[2,2],[1]]
=> [1]
=> 0
[1,1,1,1,2] => [4,1] => [[4,4],[3]]
=> [3]
=> 0
[1,1,1,2,1] => [3,1,1] => [[3,3,3],[2,2]]
=> [2,2]
=> 0
[1,1,1,3] => [3,1] => [[3,3],[2]]
=> [2]
=> 0
[1,1,2,1,1] => [2,1,2] => [[3,2,2],[1,1]]
=> [1,1]
=> 0
[1,1,2,2] => [2,2] => [[3,2],[1]]
=> [1]
=> 0
[1,1,3,1] => [2,1,1] => [[2,2,2],[1,1]]
=> [1,1]
=> 0
[1,1,4] => [2,1] => [[2,2],[1]]
=> [1]
=> 0
[1,2,2,1] => [1,2,1] => [[2,2,1],[1]]
=> [1]
=> 0
[2,1,1,2] => [1,2,1] => [[2,2,1],[1]]
=> [1]
=> 0
[2,2,1,1] => [2,2] => [[3,2],[1]]
=> [1]
=> 0
[1,1,1,1,1,2] => [5,1] => [[5,5],[4]]
=> [4]
=> 0
[1,1,1,1,2,1] => [4,1,1] => [[4,4,4],[3,3]]
=> [3,3]
=> 0
[1,1,1,1,3] => [4,1] => [[4,4],[3]]
=> [3]
=> 0
[1,1,1,2,1,1] => [3,1,2] => [[4,3,3],[2,2]]
=> [2,2]
=> 0
[1,1,1,2,2] => [3,2] => [[4,3],[2]]
=> [2]
=> 0
[1,1,1,3,1] => [3,1,1] => [[3,3,3],[2,2]]
=> [2,2]
=> 0
[1,1,1,4] => [3,1] => [[3,3],[2]]
=> [2]
=> 0
[1,1,2,1,1,1] => [2,1,3] => [[4,2,2],[1,1]]
=> [1,1]
=> 0
[1,1,2,1,2] => [2,1,1,1] => [[2,2,2,2],[1,1,1]]
=> [1,1,1]
=> 0
[1,1,2,2,1] => [2,2,1] => [[3,3,2],[2,1]]
=> [2,1]
=> 1
[1,1,2,3] => [2,1,1] => [[2,2,2],[1,1]]
=> [1,1]
=> 0
[1,1,3,1,1] => [2,1,2] => [[3,2,2],[1,1]]
=> [1,1]
=> 0
[1,1,3,2] => [2,1,1] => [[2,2,2],[1,1]]
=> [1,1]
=> 0
[1,1,4,1] => [2,1,1] => [[2,2,2],[1,1]]
=> [1,1]
=> 0
[1,1,5] => [2,1] => [[2,2],[1]]
=> [1]
=> 0
[1,2,1,1,2] => [1,1,2,1] => [[2,2,1,1],[1]]
=> [1]
=> 0
[1,2,2,1,1] => [1,2,2] => [[3,2,1],[1]]
=> [1]
=> 0
[2,1,1,1,2] => [1,3,1] => [[3,3,1],[2]]
=> [2]
=> 0
[2,1,1,2,1] => [1,2,1,1] => [[2,2,2,1],[1,1]]
=> [1,1]
=> 0
[2,1,1,3] => [1,2,1] => [[2,2,1],[1]]
=> [1]
=> 0
[2,2,1,1,1] => [2,3] => [[4,2],[1]]
=> [1]
=> 0
[2,2,1,2] => [2,1,1] => [[2,2,2],[1,1]]
=> [1,1]
=> 0
[2,2,2,1] => [3,1] => [[3,3],[2]]
=> [2]
=> 0
[2,2,3] => [2,1] => [[2,2],[1]]
=> [1]
=> 0
[3,1,1,2] => [1,2,1] => [[2,2,1],[1]]
=> [1]
=> 0
[3,3,1] => [2,1] => [[2,2],[1]]
=> [1]
=> 0
[1,1,1,1,1,1,2] => [6,1] => [[6,6],[5]]
=> [5]
=> 0
[1,1,1,1,1,2,1] => [5,1,1] => [[5,5,5],[4,4]]
=> [4,4]
=> 0
[1,1,1,1,1,3] => [5,1] => [[5,5],[4]]
=> [4]
=> 0
[1,1,1,1,2,1,1] => [4,1,2] => [[5,4,4],[3,3]]
=> [3,3]
=> 0
[1,1,1,1,2,2] => [4,2] => [[5,4],[3]]
=> [3]
=> 0
[1,1,1,1,3,1] => [4,1,1] => [[4,4,4],[3,3]]
=> [3,3]
=> 0
[1,1,1,1,4] => [4,1] => [[4,4],[3]]
=> [3]
=> 0
[1,1,1,2,1,1,1] => [3,1,3] => [[5,3,3],[2,2]]
=> [2,2]
=> 0
[1,1,1,2,1,2] => [3,1,1,1] => [[3,3,3,3],[2,2,2]]
=> [2,2,2]
=> 0
Description
The number of odd parts smaller than the largest even part in an integer partition.
Matching statistic: St000455
(load all 31 compositions to match this statistic)
(load all 31 compositions to match this statistic)
Mp00133: Integer compositions —delta morphism⟶ Integer compositions
Mp00039: Integer compositions —complement⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St000455: Graphs ⟶ ℤResult quality: 20% ●values known / values provided: 52%●distinct values known / distinct values provided: 20%
Mp00039: Integer compositions —complement⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St000455: Graphs ⟶ ℤResult quality: 20% ●values known / values provided: 52%●distinct values known / distinct values provided: 20%
Values
[1,1,2] => [2,1] => [1,2] => ([(1,2)],3)
=> 0
[1,1,1,2] => [3,1] => [1,1,2] => ([(1,2),(1,3),(2,3)],4)
=> 0
[1,1,2,1] => [2,1,1] => [1,3] => ([(2,3)],4)
=> 0
[1,1,3] => [2,1] => [1,2] => ([(1,2)],3)
=> 0
[2,2,1] => [2,1] => [1,2] => ([(1,2)],3)
=> 0
[1,1,1,1,2] => [4,1] => [1,1,1,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 0
[1,1,1,2,1] => [3,1,1] => [1,1,3] => ([(2,3),(2,4),(3,4)],5)
=> 0
[1,1,1,3] => [3,1] => [1,1,2] => ([(1,2),(1,3),(2,3)],4)
=> 0
[1,1,2,1,1] => [2,1,2] => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 0
[1,1,2,2] => [2,2] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> ? = 0
[1,1,3,1] => [2,1,1] => [1,3] => ([(2,3)],4)
=> 0
[1,1,4] => [2,1] => [1,2] => ([(1,2)],3)
=> 0
[1,2,2,1] => [1,2,1] => [2,2] => ([(1,3),(2,3)],4)
=> 0
[2,1,1,2] => [1,2,1] => [2,2] => ([(1,3),(2,3)],4)
=> 0
[2,2,1,1] => [2,2] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> ? = 0
[1,1,1,1,1,2] => [5,1] => [1,1,1,1,2] => ([(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 0
[1,1,1,1,2,1] => [4,1,1] => [1,1,1,3] => ([(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 0
[1,1,1,1,3] => [4,1] => [1,1,1,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 0
[1,1,1,2,1,1] => [3,1,2] => [1,1,3,1] => ([(0,5),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 0
[1,1,1,2,2] => [3,2] => [1,1,2,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 0
[1,1,1,3,1] => [3,1,1] => [1,1,3] => ([(2,3),(2,4),(3,4)],5)
=> 0
[1,1,1,4] => [3,1] => [1,1,2] => ([(1,2),(1,3),(2,3)],4)
=> 0
[1,1,2,1,1,1] => [2,1,3] => [1,3,1,1] => ([(0,4),(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 0
[1,1,2,1,2] => [2,1,1,1] => [1,4] => ([(3,4)],5)
=> 0
[1,1,2,2,1] => [2,2,1] => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 1
[1,1,2,3] => [2,1,1] => [1,3] => ([(2,3)],4)
=> 0
[1,1,3,1,1] => [2,1,2] => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 0
[1,1,3,2] => [2,1,1] => [1,3] => ([(2,3)],4)
=> 0
[1,1,4,1] => [2,1,1] => [1,3] => ([(2,3)],4)
=> 0
[1,1,5] => [2,1] => [1,2] => ([(1,2)],3)
=> 0
[1,2,1,1,2] => [1,1,2,1] => [3,2] => ([(1,4),(2,4),(3,4)],5)
=> 0
[1,2,2,1,1] => [1,2,2] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 0
[2,1,1,1,2] => [1,3,1] => [2,1,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 0
[2,1,1,2,1] => [1,2,1,1] => [2,3] => ([(2,4),(3,4)],5)
=> 0
[2,1,1,3] => [1,2,1] => [2,2] => ([(1,3),(2,3)],4)
=> 0
[2,2,1,1,1] => [2,3] => [1,2,1,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 0
[2,2,1,2] => [2,1,1] => [1,3] => ([(2,3)],4)
=> 0
[2,2,2,1] => [3,1] => [1,1,2] => ([(1,2),(1,3),(2,3)],4)
=> 0
[2,2,3] => [2,1] => [1,2] => ([(1,2)],3)
=> 0
[3,1,1,2] => [1,2,1] => [2,2] => ([(1,3),(2,3)],4)
=> 0
[3,3,1] => [2,1] => [1,2] => ([(1,2)],3)
=> 0
[1,1,1,1,1,1,2] => [6,1] => [1,1,1,1,1,2] => ([(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 0
[1,1,1,1,1,2,1] => [5,1,1] => [1,1,1,1,3] => ([(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 0
[1,1,1,1,1,3] => [5,1] => [1,1,1,1,2] => ([(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 0
[1,1,1,1,2,1,1] => [4,1,2] => [1,1,1,3,1] => ([(0,6),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 0
[1,1,1,1,2,2] => [4,2] => [1,1,1,2,1] => ([(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 0
[1,1,1,1,3,1] => [4,1,1] => [1,1,1,3] => ([(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 0
[1,1,1,1,4] => [4,1] => [1,1,1,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 0
[1,1,1,2,1,1,1] => [3,1,3] => [1,1,3,1,1] => ([(0,5),(0,6),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 0
[1,1,1,2,1,2] => [3,1,1,1] => [1,1,4] => ([(3,4),(3,5),(4,5)],6)
=> 0
[1,1,1,2,2,1] => [3,2,1] => [1,1,2,2] => ([(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 0
[1,1,1,2,3] => [3,1,1] => [1,1,3] => ([(2,3),(2,4),(3,4)],5)
=> 0
[1,1,1,3,1,1] => [3,1,2] => [1,1,3,1] => ([(0,5),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 0
[1,1,1,3,2] => [3,1,1] => [1,1,3] => ([(2,3),(2,4),(3,4)],5)
=> 0
[1,1,1,4,1] => [3,1,1] => [1,1,3] => ([(2,3),(2,4),(3,4)],5)
=> 0
[1,1,1,5] => [3,1] => [1,1,2] => ([(1,2),(1,3),(2,3)],4)
=> 0
[1,1,2,1,1,1,1] => [2,1,4] => [1,3,1,1,1] => ([(0,4),(0,5),(0,6),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 0
[1,1,2,1,1,2] => [2,1,2,1] => [1,3,2] => ([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 2
[1,1,2,1,2,1] => [2,1,1,1,1] => [1,5] => ([(4,5)],6)
=> 0
[1,1,2,1,3] => [2,1,1,1] => [1,4] => ([(3,4)],5)
=> 0
[1,1,2,2,1,1] => [2,2,2] => [1,2,2,1] => ([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 1
[1,1,2,2,2] => [2,3] => [1,2,1,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 0
[1,1,2,3,1] => [2,1,1,1] => [1,4] => ([(3,4)],5)
=> 0
[1,1,2,4] => [2,1,1] => [1,3] => ([(2,3)],4)
=> 0
[1,1,3,1,1,1] => [2,1,3] => [1,3,1,1] => ([(0,4),(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 0
[1,1,3,1,2] => [2,1,1,1] => [1,4] => ([(3,4)],5)
=> 0
[1,1,3,2,1] => [2,1,1,1] => [1,4] => ([(3,4)],5)
=> 0
[1,1,3,3] => [2,2] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> ? = 0
[1,1,4,1,1] => [2,1,2] => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 0
[1,1,4,2] => [2,1,1] => [1,3] => ([(2,3)],4)
=> 0
[1,1,5,1] => [2,1,1] => [1,3] => ([(2,3)],4)
=> 0
[1,1,6] => [2,1] => [1,2] => ([(1,2)],3)
=> 0
[1,2,2,1,1,1] => [1,2,3] => [2,2,1,1] => ([(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 0
[2,1,1,2,1,1] => [1,2,1,2] => [2,3,1] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 0
[2,1,1,2,2] => [1,2,2] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 0
[2,2,1,1,1,1] => [2,4] => [1,2,1,1,1] => ([(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 0
[2,2,1,1,2] => [2,2,1] => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 1
[2,2,2,1,1] => [3,2] => [1,1,2,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 0
[3,3,1,1] => [2,2] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> ? = 0
[1,1,1,1,1,1,1,2] => [7,1] => [1,1,1,1,1,1,2] => ([(1,2),(1,3),(1,4),(1,5),(1,6),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 0
[1,1,1,1,1,1,2,1] => [6,1,1] => [1,1,1,1,1,3] => ([(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 0
[1,1,1,1,1,2,2] => [5,2] => [1,1,1,1,2,1] => ([(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 0
[1,1,1,1,2,1,1,1] => [4,1,3] => [1,1,1,3,1,1] => ([(0,6),(0,7),(1,6),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 0
[1,1,1,1,2,2,1] => [4,2,1] => [1,1,1,2,2] => ([(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 1
[1,1,1,1,3,1,1] => [4,1,2] => [1,1,1,3,1] => ([(0,6),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 0
[1,1,1,2,1,1,2] => [3,1,2,1] => [1,1,3,2] => ([(1,6),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 0
[1,1,1,2,2,1,1] => [3,2,2] => [1,1,2,2,1] => ([(0,6),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 0
[1,1,1,2,2,2] => [3,3] => [1,1,2,1,1] => ([(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 0
[1,1,1,3,1,1,1] => [3,1,3] => [1,1,3,1,1] => ([(0,5),(0,6),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 0
[1,1,1,3,3] => [3,2] => [1,1,2,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 0
[1,1,1,4,1,1] => [3,1,2] => [1,1,3,1] => ([(0,5),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 0
[1,1,2,1,1,1,2] => [2,1,3,1] => [1,3,1,2] => ([(1,5),(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 0
[1,1,2,1,1,2,1] => [2,1,2,1,1] => [1,3,3] => ([(2,6),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 2
[1,1,2,1,1,3] => [2,1,2,1] => [1,3,2] => ([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 2
[1,1,2,1,2,1,1] => [2,1,1,1,2] => [1,5,1] => ([(0,6),(1,6),(2,6),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 0
[1,1,2,1,2,2] => [2,1,1,2] => [1,4,1] => ([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 0
[1,1,2,2,1,1,1] => [2,2,3] => [1,2,2,1,1] => ([(0,5),(0,6),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 1
[1,1,2,2,1,2] => [2,2,1,1] => [1,2,3] => ([(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 1
[1,1,2,2,2,1] => [2,3,1] => [1,2,1,2] => ([(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 0
[1,1,2,2,3] => [2,2,1] => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 1
Description
The second largest eigenvalue of a graph if it is integral.
This statistic is undefined if the second largest eigenvalue of the graph is not integral.
Chapter 4 of [1] provides lots of context.
Matching statistic: St000834
(load all 7 compositions to match this statistic)
(load all 7 compositions to match this statistic)
Mp00133: Integer compositions —delta morphism⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
Mp00201: Dyck paths —Ringel⟶ Permutations
St000834: Permutations ⟶ ℤResult quality: 34% ●values known / values provided: 34%●distinct values known / distinct values provided: 40%
Mp00231: Integer compositions —bounce path⟶ Dyck paths
Mp00201: Dyck paths —Ringel⟶ Permutations
St000834: Permutations ⟶ ℤResult quality: 34% ●values known / values provided: 34%●distinct values known / distinct values provided: 40%
Values
[1,1,2] => [2,1] => [1,1,0,0,1,0]
=> [2,4,1,3] => 2 = 0 + 2
[1,1,1,2] => [3,1] => [1,1,1,0,0,0,1,0]
=> [2,3,5,1,4] => 2 = 0 + 2
[1,1,2,1] => [2,1,1] => [1,1,0,0,1,0,1,0]
=> [2,5,1,3,4] => 2 = 0 + 2
[1,1,3] => [2,1] => [1,1,0,0,1,0]
=> [2,4,1,3] => 2 = 0 + 2
[2,2,1] => [2,1] => [1,1,0,0,1,0]
=> [2,4,1,3] => 2 = 0 + 2
[1,1,1,1,2] => [4,1] => [1,1,1,1,0,0,0,0,1,0]
=> [2,3,4,6,1,5] => 2 = 0 + 2
[1,1,1,2,1] => [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> [2,3,6,1,4,5] => 2 = 0 + 2
[1,1,1,3] => [3,1] => [1,1,1,0,0,0,1,0]
=> [2,3,5,1,4] => 2 = 0 + 2
[1,1,2,1,1] => [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
=> [2,5,1,3,6,4] => 2 = 0 + 2
[1,1,2,2] => [2,2] => [1,1,0,0,1,1,0,0]
=> [2,4,1,5,3] => 2 = 0 + 2
[1,1,3,1] => [2,1,1] => [1,1,0,0,1,0,1,0]
=> [2,5,1,3,4] => 2 = 0 + 2
[1,1,4] => [2,1] => [1,1,0,0,1,0]
=> [2,4,1,3] => 2 = 0 + 2
[1,2,2,1] => [1,2,1] => [1,0,1,1,0,0,1,0]
=> [3,1,5,2,4] => 2 = 0 + 2
[2,1,1,2] => [1,2,1] => [1,0,1,1,0,0,1,0]
=> [3,1,5,2,4] => 2 = 0 + 2
[2,2,1,1] => [2,2] => [1,1,0,0,1,1,0,0]
=> [2,4,1,5,3] => 2 = 0 + 2
[1,1,1,1,1,2] => [5,1] => [1,1,1,1,1,0,0,0,0,0,1,0]
=> [2,3,4,5,7,1,6] => 2 = 0 + 2
[1,1,1,1,2,1] => [4,1,1] => [1,1,1,1,0,0,0,0,1,0,1,0]
=> [2,3,4,7,1,5,6] => 2 = 0 + 2
[1,1,1,1,3] => [4,1] => [1,1,1,1,0,0,0,0,1,0]
=> [2,3,4,6,1,5] => 2 = 0 + 2
[1,1,1,2,1,1] => [3,1,2] => [1,1,1,0,0,0,1,0,1,1,0,0]
=> [2,3,6,1,4,7,5] => ? = 0 + 2
[1,1,1,2,2] => [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> [2,3,5,1,6,4] => 2 = 0 + 2
[1,1,1,3,1] => [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> [2,3,6,1,4,5] => 2 = 0 + 2
[1,1,1,4] => [3,1] => [1,1,1,0,0,0,1,0]
=> [2,3,5,1,4] => 2 = 0 + 2
[1,1,2,1,1,1] => [2,1,3] => [1,1,0,0,1,0,1,1,1,0,0,0]
=> [2,5,1,3,6,7,4] => ? = 0 + 2
[1,1,2,1,2] => [2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
=> [2,6,1,3,4,5] => 2 = 0 + 2
[1,1,2,2,1] => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> [2,4,1,6,3,5] => 3 = 1 + 2
[1,1,2,3] => [2,1,1] => [1,1,0,0,1,0,1,0]
=> [2,5,1,3,4] => 2 = 0 + 2
[1,1,3,1,1] => [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
=> [2,5,1,3,6,4] => 2 = 0 + 2
[1,1,3,2] => [2,1,1] => [1,1,0,0,1,0,1,0]
=> [2,5,1,3,4] => 2 = 0 + 2
[1,1,4,1] => [2,1,1] => [1,1,0,0,1,0,1,0]
=> [2,5,1,3,4] => 2 = 0 + 2
[1,1,5] => [2,1] => [1,1,0,0,1,0]
=> [2,4,1,3] => 2 = 0 + 2
[1,2,1,1,2] => [1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
=> [4,1,2,6,3,5] => 2 = 0 + 2
[1,2,2,1,1] => [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
=> [3,1,5,2,6,4] => 2 = 0 + 2
[2,1,1,1,2] => [1,3,1] => [1,0,1,1,1,0,0,0,1,0]
=> [3,1,4,6,2,5] => 2 = 0 + 2
[2,1,1,2,1] => [1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
=> [3,1,6,2,4,5] => 2 = 0 + 2
[2,1,1,3] => [1,2,1] => [1,0,1,1,0,0,1,0]
=> [3,1,5,2,4] => 2 = 0 + 2
[2,2,1,1,1] => [2,3] => [1,1,0,0,1,1,1,0,0,0]
=> [2,4,1,5,6,3] => 2 = 0 + 2
[2,2,1,2] => [2,1,1] => [1,1,0,0,1,0,1,0]
=> [2,5,1,3,4] => 2 = 0 + 2
[2,2,2,1] => [3,1] => [1,1,1,0,0,0,1,0]
=> [2,3,5,1,4] => 2 = 0 + 2
[2,2,3] => [2,1] => [1,1,0,0,1,0]
=> [2,4,1,3] => 2 = 0 + 2
[3,1,1,2] => [1,2,1] => [1,0,1,1,0,0,1,0]
=> [3,1,5,2,4] => 2 = 0 + 2
[3,3,1] => [2,1] => [1,1,0,0,1,0]
=> [2,4,1,3] => 2 = 0 + 2
[1,1,1,1,1,1,2] => [6,1] => [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> [2,3,4,5,6,8,1,7] => 2 = 0 + 2
[1,1,1,1,1,2,1] => [5,1,1] => [1,1,1,1,1,0,0,0,0,0,1,0,1,0]
=> [2,3,4,5,8,1,6,7] => ? = 0 + 2
[1,1,1,1,1,3] => [5,1] => [1,1,1,1,1,0,0,0,0,0,1,0]
=> [2,3,4,5,7,1,6] => 2 = 0 + 2
[1,1,1,1,2,1,1] => [4,1,2] => [1,1,1,1,0,0,0,0,1,0,1,1,0,0]
=> [2,3,4,7,1,5,8,6] => ? = 0 + 2
[1,1,1,1,2,2] => [4,2] => [1,1,1,1,0,0,0,0,1,1,0,0]
=> [2,3,4,6,1,7,5] => 2 = 0 + 2
[1,1,1,1,3,1] => [4,1,1] => [1,1,1,1,0,0,0,0,1,0,1,0]
=> [2,3,4,7,1,5,6] => 2 = 0 + 2
[1,1,1,1,4] => [4,1] => [1,1,1,1,0,0,0,0,1,0]
=> [2,3,4,6,1,5] => 2 = 0 + 2
[1,1,1,2,1,1,1] => [3,1,3] => [1,1,1,0,0,0,1,0,1,1,1,0,0,0]
=> [2,3,6,1,4,7,8,5] => ? = 0 + 2
[1,1,1,2,1,2] => [3,1,1,1] => [1,1,1,0,0,0,1,0,1,0,1,0]
=> [2,3,7,1,4,5,6] => ? = 0 + 2
[1,1,1,2,2,1] => [3,2,1] => [1,1,1,0,0,0,1,1,0,0,1,0]
=> [2,3,5,1,7,4,6] => ? = 0 + 2
[1,1,1,2,3] => [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> [2,3,6,1,4,5] => 2 = 0 + 2
[1,1,1,3,1,1] => [3,1,2] => [1,1,1,0,0,0,1,0,1,1,0,0]
=> [2,3,6,1,4,7,5] => ? = 0 + 2
[1,1,1,3,2] => [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> [2,3,6,1,4,5] => 2 = 0 + 2
[1,1,1,4,1] => [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> [2,3,6,1,4,5] => 2 = 0 + 2
[1,1,1,5] => [3,1] => [1,1,1,0,0,0,1,0]
=> [2,3,5,1,4] => 2 = 0 + 2
[1,1,2,1,1,1,1] => [2,1,4] => [1,1,0,0,1,0,1,1,1,1,0,0,0,0]
=> [2,5,1,3,6,7,8,4] => ? = 0 + 2
[1,1,2,1,1,2] => [2,1,2,1] => [1,1,0,0,1,0,1,1,0,0,1,0]
=> [2,5,1,3,7,4,6] => ? = 2 + 2
[1,1,2,1,2,1] => [2,1,1,1,1] => [1,1,0,0,1,0,1,0,1,0,1,0]
=> [2,7,1,3,4,5,6] => 2 = 0 + 2
[1,1,2,1,3] => [2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
=> [2,6,1,3,4,5] => 2 = 0 + 2
[1,1,2,2,1,1] => [2,2,2] => [1,1,0,0,1,1,0,0,1,1,0,0]
=> [2,4,1,6,3,7,5] => ? = 1 + 2
[1,1,3,1,1,1] => [2,1,3] => [1,1,0,0,1,0,1,1,1,0,0,0]
=> [2,5,1,3,6,7,4] => ? = 0 + 2
[1,2,1,1,1,2] => [1,1,3,1] => [1,0,1,0,1,1,1,0,0,0,1,0]
=> [4,1,2,5,7,3,6] => ? = 0 + 2
[1,2,1,1,2,1] => [1,1,2,1,1] => [1,0,1,0,1,1,0,0,1,0,1,0]
=> [4,1,2,7,3,5,6] => ? = 0 + 2
[1,2,2,1,1,1] => [1,2,3] => [1,0,1,1,0,0,1,1,1,0,0,0]
=> [3,1,5,2,6,7,4] => ? = 0 + 2
[2,1,1,1,1,2] => [1,4,1] => [1,0,1,1,1,1,0,0,0,0,1,0]
=> [3,1,4,5,7,2,6] => ? = 0 + 2
[2,1,1,1,2,1] => [1,3,1,1] => [1,0,1,1,1,0,0,0,1,0,1,0]
=> [3,1,4,7,2,5,6] => ? = 0 + 2
[2,1,1,2,1,1] => [1,2,1,2] => [1,0,1,1,0,0,1,0,1,1,0,0]
=> [3,1,6,2,4,7,5] => ? = 0 + 2
[2,2,1,1,1,1] => [2,4] => [1,1,0,0,1,1,1,1,0,0,0,0]
=> [2,4,1,5,6,7,3] => ? = 0 + 2
[1,1,1,1,1,1,2,1] => [6,1,1] => [1,1,1,1,1,1,0,0,0,0,0,0,1,0,1,0]
=> [2,3,4,5,6,9,1,7,8] => ? = 0 + 2
[1,1,1,1,1,2,2] => [5,2] => [1,1,1,1,1,0,0,0,0,0,1,1,0,0]
=> [2,3,4,5,7,1,8,6] => ? = 0 + 2
[1,1,1,1,1,3,1] => [5,1,1] => [1,1,1,1,1,0,0,0,0,0,1,0,1,0]
=> [2,3,4,5,8,1,6,7] => ? = 0 + 2
[1,1,1,1,2,1,1,1] => [4,1,3] => [1,1,1,1,0,0,0,0,1,0,1,1,1,0,0,0]
=> [2,3,4,7,1,5,8,9,6] => ? = 0 + 2
[1,1,1,1,2,1,2] => [4,1,1,1] => [1,1,1,1,0,0,0,0,1,0,1,0,1,0]
=> [2,3,4,8,1,5,6,7] => ? = 0 + 2
[1,1,1,1,2,2,1] => [4,2,1] => [1,1,1,1,0,0,0,0,1,1,0,0,1,0]
=> [2,3,4,6,1,8,5,7] => ? = 1 + 2
[1,1,1,1,3,1,1] => [4,1,2] => [1,1,1,1,0,0,0,0,1,0,1,1,0,0]
=> [2,3,4,7,1,5,8,6] => ? = 0 + 2
[1,1,1,2,1,1,2] => [3,1,2,1] => [1,1,1,0,0,0,1,0,1,1,0,0,1,0]
=> [2,3,6,1,4,8,5,7] => ? = 0 + 2
[1,1,1,2,1,2,1] => [3,1,1,1,1] => [1,1,1,0,0,0,1,0,1,0,1,0,1,0]
=> [2,3,8,1,4,5,6,7] => ? = 0 + 2
[1,1,1,2,1,3] => [3,1,1,1] => [1,1,1,0,0,0,1,0,1,0,1,0]
=> [2,3,7,1,4,5,6] => ? = 0 + 2
[1,1,1,2,2,1,1] => [3,2,2] => [1,1,1,0,0,0,1,1,0,0,1,1,0,0]
=> [2,3,5,1,7,4,8,6] => ? = 0 + 2
[1,1,1,2,2,2] => [3,3] => [1,1,1,0,0,0,1,1,1,0,0,0]
=> [2,3,5,1,6,7,4] => ? = 0 + 2
[1,1,1,2,3,1] => [3,1,1,1] => [1,1,1,0,0,0,1,0,1,0,1,0]
=> [2,3,7,1,4,5,6] => ? = 0 + 2
[1,1,1,3,1,1,1] => [3,1,3] => [1,1,1,0,0,0,1,0,1,1,1,0,0,0]
=> [2,3,6,1,4,7,8,5] => ? = 0 + 2
[1,1,1,3,1,2] => [3,1,1,1] => [1,1,1,0,0,0,1,0,1,0,1,0]
=> [2,3,7,1,4,5,6] => ? = 0 + 2
[1,1,1,3,2,1] => [3,1,1,1] => [1,1,1,0,0,0,1,0,1,0,1,0]
=> [2,3,7,1,4,5,6] => ? = 0 + 2
[1,1,1,4,1,1] => [3,1,2] => [1,1,1,0,0,0,1,0,1,1,0,0]
=> [2,3,6,1,4,7,5] => ? = 0 + 2
[1,1,2,1,1,1,2] => [2,1,3,1] => [1,1,0,0,1,0,1,1,1,0,0,0,1,0]
=> [2,5,1,3,6,8,4,7] => ? = 0 + 2
[1,1,2,1,1,2,1] => [2,1,2,1,1] => [1,1,0,0,1,0,1,1,0,0,1,0,1,0]
=> [2,5,1,3,8,4,6,7] => ? = 2 + 2
[1,1,2,1,1,3] => [2,1,2,1] => [1,1,0,0,1,0,1,1,0,0,1,0]
=> [2,5,1,3,7,4,6] => ? = 2 + 2
[1,1,2,1,2,1,1] => [2,1,1,1,2] => [1,1,0,0,1,0,1,0,1,0,1,1,0,0]
=> [2,7,1,3,4,5,8,6] => ? = 0 + 2
[1,1,2,1,2,2] => [2,1,1,2] => [1,1,0,0,1,0,1,0,1,1,0,0]
=> [2,6,1,3,4,7,5] => ? = 0 + 2
[1,1,2,2,1,1,1] => [2,2,3] => [1,1,0,0,1,1,0,0,1,1,1,0,0,0]
=> [2,4,1,6,3,7,8,5] => ? = 1 + 2
[1,1,2,2,1,2] => [2,2,1,1] => [1,1,0,0,1,1,0,0,1,0,1,0]
=> [2,4,1,7,3,5,6] => ? = 1 + 2
[1,1,2,2,2,1] => [2,3,1] => [1,1,0,0,1,1,1,0,0,0,1,0]
=> [2,4,1,5,7,3,6] => ? = 0 + 2
[1,1,2,3,1,1] => [2,1,1,2] => [1,1,0,0,1,0,1,0,1,1,0,0]
=> [2,6,1,3,4,7,5] => ? = 0 + 2
[1,1,3,1,1,1,1] => [2,1,4] => [1,1,0,0,1,0,1,1,1,1,0,0,0,0]
=> [2,5,1,3,6,7,8,4] => ? = 0 + 2
[1,1,3,1,1,2] => [2,1,2,1] => [1,1,0,0,1,0,1,1,0,0,1,0]
=> [2,5,1,3,7,4,6] => ? = 2 + 2
[1,1,3,2,1,1] => [2,1,1,2] => [1,1,0,0,1,0,1,0,1,1,0,0]
=> [2,6,1,3,4,7,5] => ? = 0 + 2
[1,1,4,1,1,1] => [2,1,3] => [1,1,0,0,1,0,1,1,1,0,0,0]
=> [2,5,1,3,6,7,4] => ? = 0 + 2
[1,2,1,1,1,1,2] => [1,1,4,1] => [1,0,1,0,1,1,1,1,0,0,0,0,1,0]
=> [4,1,2,5,6,8,3,7] => ? = 0 + 2
Description
The number of right outer peaks of a permutation.
A right outer peak in a permutation $w = [w_1,..., w_n]$ is either a position $i$ such that $w_{i-1} < w_i > w_{i+1}$ or $n$ if $w_n > w_{n-1}$.
In other words, it is a peak in the word $[w_1,..., w_n,0]$.
Matching statistic: St000256
(load all 6 compositions to match this statistic)
(load all 6 compositions to match this statistic)
Mp00133: Integer compositions —delta morphism⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
Mp00027: Dyck paths —to partition⟶ Integer partitions
St000256: Integer partitions ⟶ ℤResult quality: 31% ●values known / values provided: 31%●distinct values known / distinct values provided: 40%
Mp00231: Integer compositions —bounce path⟶ Dyck paths
Mp00027: Dyck paths —to partition⟶ Integer partitions
St000256: Integer partitions ⟶ ℤResult quality: 31% ●values known / values provided: 31%●distinct values known / distinct values provided: 40%
Values
[1,1,2] => [2,1] => [1,1,0,0,1,0]
=> [2]
=> 1 = 0 + 1
[1,1,1,2] => [3,1] => [1,1,1,0,0,0,1,0]
=> [3]
=> 1 = 0 + 1
[1,1,2,1] => [2,1,1] => [1,1,0,0,1,0,1,0]
=> [3,2]
=> 1 = 0 + 1
[1,1,3] => [2,1] => [1,1,0,0,1,0]
=> [2]
=> 1 = 0 + 1
[2,2,1] => [2,1] => [1,1,0,0,1,0]
=> [2]
=> 1 = 0 + 1
[1,1,1,1,2] => [4,1] => [1,1,1,1,0,0,0,0,1,0]
=> [4]
=> 1 = 0 + 1
[1,1,1,2,1] => [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> [4,3]
=> 1 = 0 + 1
[1,1,1,3] => [3,1] => [1,1,1,0,0,0,1,0]
=> [3]
=> 1 = 0 + 1
[1,1,2,1,1] => [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
=> [3,3,2]
=> 1 = 0 + 1
[1,1,2,2] => [2,2] => [1,1,0,0,1,1,0,0]
=> [2,2]
=> 1 = 0 + 1
[1,1,3,1] => [2,1,1] => [1,1,0,0,1,0,1,0]
=> [3,2]
=> 1 = 0 + 1
[1,1,4] => [2,1] => [1,1,0,0,1,0]
=> [2]
=> 1 = 0 + 1
[1,2,2,1] => [1,2,1] => [1,0,1,1,0,0,1,0]
=> [3,1,1]
=> 1 = 0 + 1
[2,1,1,2] => [1,2,1] => [1,0,1,1,0,0,1,0]
=> [3,1,1]
=> 1 = 0 + 1
[2,2,1,1] => [2,2] => [1,1,0,0,1,1,0,0]
=> [2,2]
=> 1 = 0 + 1
[1,1,1,1,1,2] => [5,1] => [1,1,1,1,1,0,0,0,0,0,1,0]
=> [5]
=> 1 = 0 + 1
[1,1,1,1,2,1] => [4,1,1] => [1,1,1,1,0,0,0,0,1,0,1,0]
=> [5,4]
=> 1 = 0 + 1
[1,1,1,1,3] => [4,1] => [1,1,1,1,0,0,0,0,1,0]
=> [4]
=> 1 = 0 + 1
[1,1,1,2,1,1] => [3,1,2] => [1,1,1,0,0,0,1,0,1,1,0,0]
=> [4,4,3]
=> ? = 0 + 1
[1,1,1,2,2] => [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> [3,3]
=> 1 = 0 + 1
[1,1,1,3,1] => [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> [4,3]
=> 1 = 0 + 1
[1,1,1,4] => [3,1] => [1,1,1,0,0,0,1,0]
=> [3]
=> 1 = 0 + 1
[1,1,2,1,1,1] => [2,1,3] => [1,1,0,0,1,0,1,1,1,0,0,0]
=> [3,3,3,2]
=> ? = 0 + 1
[1,1,2,1,2] => [2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
=> [4,3,2]
=> 1 = 0 + 1
[1,1,2,2,1] => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> [4,2,2]
=> 2 = 1 + 1
[1,1,2,3] => [2,1,1] => [1,1,0,0,1,0,1,0]
=> [3,2]
=> 1 = 0 + 1
[1,1,3,1,1] => [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
=> [3,3,2]
=> 1 = 0 + 1
[1,1,3,2] => [2,1,1] => [1,1,0,0,1,0,1,0]
=> [3,2]
=> 1 = 0 + 1
[1,1,4,1] => [2,1,1] => [1,1,0,0,1,0,1,0]
=> [3,2]
=> 1 = 0 + 1
[1,1,5] => [2,1] => [1,1,0,0,1,0]
=> [2]
=> 1 = 0 + 1
[1,2,1,1,2] => [1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
=> [4,2,2,1]
=> 1 = 0 + 1
[1,2,2,1,1] => [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
=> [3,3,1,1]
=> 1 = 0 + 1
[2,1,1,1,2] => [1,3,1] => [1,0,1,1,1,0,0,0,1,0]
=> [4,1,1,1]
=> 1 = 0 + 1
[2,1,1,2,1] => [1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
=> [4,3,1,1]
=> 1 = 0 + 1
[2,1,1,3] => [1,2,1] => [1,0,1,1,0,0,1,0]
=> [3,1,1]
=> 1 = 0 + 1
[2,2,1,1,1] => [2,3] => [1,1,0,0,1,1,1,0,0,0]
=> [2,2,2]
=> 1 = 0 + 1
[2,2,1,2] => [2,1,1] => [1,1,0,0,1,0,1,0]
=> [3,2]
=> 1 = 0 + 1
[2,2,2,1] => [3,1] => [1,1,1,0,0,0,1,0]
=> [3]
=> 1 = 0 + 1
[2,2,3] => [2,1] => [1,1,0,0,1,0]
=> [2]
=> 1 = 0 + 1
[3,1,1,2] => [1,2,1] => [1,0,1,1,0,0,1,0]
=> [3,1,1]
=> 1 = 0 + 1
[3,3,1] => [2,1] => [1,1,0,0,1,0]
=> [2]
=> 1 = 0 + 1
[1,1,1,1,1,1,2] => [6,1] => [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> [6]
=> 1 = 0 + 1
[1,1,1,1,1,2,1] => [5,1,1] => [1,1,1,1,1,0,0,0,0,0,1,0,1,0]
=> [6,5]
=> ? = 0 + 1
[1,1,1,1,1,3] => [5,1] => [1,1,1,1,1,0,0,0,0,0,1,0]
=> [5]
=> 1 = 0 + 1
[1,1,1,1,2,1,1] => [4,1,2] => [1,1,1,1,0,0,0,0,1,0,1,1,0,0]
=> [5,5,4]
=> ? = 0 + 1
[1,1,1,1,2,2] => [4,2] => [1,1,1,1,0,0,0,0,1,1,0,0]
=> [4,4]
=> 1 = 0 + 1
[1,1,1,1,3,1] => [4,1,1] => [1,1,1,1,0,0,0,0,1,0,1,0]
=> [5,4]
=> 1 = 0 + 1
[1,1,1,1,4] => [4,1] => [1,1,1,1,0,0,0,0,1,0]
=> [4]
=> 1 = 0 + 1
[1,1,1,2,1,1,1] => [3,1,3] => [1,1,1,0,0,0,1,0,1,1,1,0,0,0]
=> [4,4,4,3]
=> ? = 0 + 1
[1,1,1,2,1,2] => [3,1,1,1] => [1,1,1,0,0,0,1,0,1,0,1,0]
=> [5,4,3]
=> ? = 0 + 1
[1,1,1,2,2,1] => [3,2,1] => [1,1,1,0,0,0,1,1,0,0,1,0]
=> [5,3,3]
=> ? = 0 + 1
[1,1,1,2,3] => [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> [4,3]
=> 1 = 0 + 1
[1,1,1,3,1,1] => [3,1,2] => [1,1,1,0,0,0,1,0,1,1,0,0]
=> [4,4,3]
=> ? = 0 + 1
[1,1,1,3,2] => [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> [4,3]
=> 1 = 0 + 1
[1,1,1,4,1] => [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> [4,3]
=> 1 = 0 + 1
[1,1,1,5] => [3,1] => [1,1,1,0,0,0,1,0]
=> [3]
=> 1 = 0 + 1
[1,1,2,1,1,1,1] => [2,1,4] => [1,1,0,0,1,0,1,1,1,1,0,0,0,0]
=> [3,3,3,3,2]
=> ? = 0 + 1
[1,1,2,1,1,2] => [2,1,2,1] => [1,1,0,0,1,0,1,1,0,0,1,0]
=> [5,3,3,2]
=> ? = 2 + 1
[1,1,2,1,2,1] => [2,1,1,1,1] => [1,1,0,0,1,0,1,0,1,0,1,0]
=> [5,4,3,2]
=> ? = 0 + 1
[1,1,2,1,3] => [2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
=> [4,3,2]
=> 1 = 0 + 1
[1,1,2,2,1,1] => [2,2,2] => [1,1,0,0,1,1,0,0,1,1,0,0]
=> [4,4,2,2]
=> ? = 1 + 1
[1,1,2,2,2] => [2,3] => [1,1,0,0,1,1,1,0,0,0]
=> [2,2,2]
=> 1 = 0 + 1
[1,1,3,1,1,1] => [2,1,3] => [1,1,0,0,1,0,1,1,1,0,0,0]
=> [3,3,3,2]
=> ? = 0 + 1
[1,2,1,1,1,2] => [1,1,3,1] => [1,0,1,0,1,1,1,0,0,0,1,0]
=> [5,2,2,2,1]
=> ? = 0 + 1
[1,2,1,1,2,1] => [1,1,2,1,1] => [1,0,1,0,1,1,0,0,1,0,1,0]
=> [5,4,2,2,1]
=> ? = 0 + 1
[1,2,2,1,1,1] => [1,2,3] => [1,0,1,1,0,0,1,1,1,0,0,0]
=> [3,3,3,1,1]
=> ? = 0 + 1
[2,1,1,1,2,1] => [1,3,1,1] => [1,0,1,1,1,0,0,0,1,0,1,0]
=> [5,4,1,1,1]
=> ? = 0 + 1
[2,1,1,2,1,1] => [1,2,1,2] => [1,0,1,1,0,0,1,0,1,1,0,0]
=> [4,4,3,1,1]
=> ? = 0 + 1
[1,1,1,1,1,1,2,1] => [6,1,1] => [1,1,1,1,1,1,0,0,0,0,0,0,1,0,1,0]
=> [7,6]
=> ? = 0 + 1
[1,1,1,1,1,3,1] => [5,1,1] => [1,1,1,1,1,0,0,0,0,0,1,0,1,0]
=> [6,5]
=> ? = 0 + 1
[1,1,1,1,2,1,1,1] => [4,1,3] => [1,1,1,1,0,0,0,0,1,0,1,1,1,0,0,0]
=> [5,5,5,4]
=> ? = 0 + 1
[1,1,1,1,2,1,2] => [4,1,1,1] => [1,1,1,1,0,0,0,0,1,0,1,0,1,0]
=> [6,5,4]
=> ? = 0 + 1
[1,1,1,1,2,2,1] => [4,2,1] => [1,1,1,1,0,0,0,0,1,1,0,0,1,0]
=> [6,4,4]
=> ? = 1 + 1
[1,1,1,1,3,1,1] => [4,1,2] => [1,1,1,1,0,0,0,0,1,0,1,1,0,0]
=> [5,5,4]
=> ? = 0 + 1
[1,1,1,2,1,1,2] => [3,1,2,1] => [1,1,1,0,0,0,1,0,1,1,0,0,1,0]
=> [6,4,4,3]
=> ? = 0 + 1
[1,1,1,2,1,2,1] => [3,1,1,1,1] => [1,1,1,0,0,0,1,0,1,0,1,0,1,0]
=> [6,5,4,3]
=> ? = 0 + 1
[1,1,1,2,1,3] => [3,1,1,1] => [1,1,1,0,0,0,1,0,1,0,1,0]
=> [5,4,3]
=> ? = 0 + 1
[1,1,1,2,2,1,1] => [3,2,2] => [1,1,1,0,0,0,1,1,0,0,1,1,0,0]
=> [5,5,3,3]
=> ? = 0 + 1
[1,1,1,2,3,1] => [3,1,1,1] => [1,1,1,0,0,0,1,0,1,0,1,0]
=> [5,4,3]
=> ? = 0 + 1
[1,1,1,3,1,1,1] => [3,1,3] => [1,1,1,0,0,0,1,0,1,1,1,0,0,0]
=> [4,4,4,3]
=> ? = 0 + 1
[1,1,1,3,1,2] => [3,1,1,1] => [1,1,1,0,0,0,1,0,1,0,1,0]
=> [5,4,3]
=> ? = 0 + 1
[1,1,1,3,2,1] => [3,1,1,1] => [1,1,1,0,0,0,1,0,1,0,1,0]
=> [5,4,3]
=> ? = 0 + 1
[1,1,1,4,1,1] => [3,1,2] => [1,1,1,0,0,0,1,0,1,1,0,0]
=> [4,4,3]
=> ? = 0 + 1
[1,1,2,1,1,1,2] => [2,1,3,1] => [1,1,0,0,1,0,1,1,1,0,0,0,1,0]
=> [6,3,3,3,2]
=> ? = 0 + 1
[1,1,2,1,1,2,1] => [2,1,2,1,1] => [1,1,0,0,1,0,1,1,0,0,1,0,1,0]
=> [6,5,3,3,2]
=> ? = 2 + 1
[1,1,2,1,1,3] => [2,1,2,1] => [1,1,0,0,1,0,1,1,0,0,1,0]
=> [5,3,3,2]
=> ? = 2 + 1
[1,1,2,1,2,1,1] => [2,1,1,1,2] => [1,1,0,0,1,0,1,0,1,0,1,1,0,0]
=> [5,5,4,3,2]
=> ? = 0 + 1
[1,1,2,1,2,2] => [2,1,1,2] => [1,1,0,0,1,0,1,0,1,1,0,0]
=> [4,4,3,2]
=> ? = 0 + 1
[1,1,2,1,3,1] => [2,1,1,1,1] => [1,1,0,0,1,0,1,0,1,0,1,0]
=> [5,4,3,2]
=> ? = 0 + 1
[1,1,2,2,1,1,1] => [2,2,3] => [1,1,0,0,1,1,0,0,1,1,1,0,0,0]
=> [4,4,4,2,2]
=> ? = 1 + 1
[1,1,2,2,1,2] => [2,2,1,1] => [1,1,0,0,1,1,0,0,1,0,1,0]
=> [5,4,2,2]
=> ? = 1 + 1
[1,1,2,2,2,1] => [2,3,1] => [1,1,0,0,1,1,1,0,0,0,1,0]
=> [5,2,2,2]
=> ? = 0 + 1
[1,1,2,3,1,1] => [2,1,1,2] => [1,1,0,0,1,0,1,0,1,1,0,0]
=> [4,4,3,2]
=> ? = 0 + 1
[1,1,3,1,1,1,1] => [2,1,4] => [1,1,0,0,1,0,1,1,1,1,0,0,0,0]
=> [3,3,3,3,2]
=> ? = 0 + 1
[1,1,3,1,1,2] => [2,1,2,1] => [1,1,0,0,1,0,1,1,0,0,1,0]
=> [5,3,3,2]
=> ? = 2 + 1
[1,1,3,1,2,1] => [2,1,1,1,1] => [1,1,0,0,1,0,1,0,1,0,1,0]
=> [5,4,3,2]
=> ? = 0 + 1
[1,1,3,2,1,1] => [2,1,1,2] => [1,1,0,0,1,0,1,0,1,1,0,0]
=> [4,4,3,2]
=> ? = 0 + 1
[1,1,4,1,1,1] => [2,1,3] => [1,1,0,0,1,0,1,1,1,0,0,0]
=> [3,3,3,2]
=> ? = 0 + 1
[1,2,1,1,1,1,2] => [1,1,4,1] => [1,0,1,0,1,1,1,1,0,0,0,0,1,0]
=> [6,2,2,2,2,1]
=> ? = 0 + 1
[1,2,1,1,1,2,1] => [1,1,3,1,1] => [1,0,1,0,1,1,1,0,0,0,1,0,1,0]
=> [6,5,2,2,2,1]
=> ? = 0 + 1
Description
The number of parts from which one can substract 2 and still get an integer partition.
Matching statistic: St001487
(load all 4 compositions to match this statistic)
(load all 4 compositions to match this statistic)
Mp00133: Integer compositions —delta morphism⟶ Integer compositions
Mp00180: Integer compositions —to ribbon⟶ Skew partitions
St001487: Skew partitions ⟶ ℤResult quality: 27% ●values known / values provided: 27%●distinct values known / distinct values provided: 40%
Mp00180: Integer compositions —to ribbon⟶ Skew partitions
St001487: Skew partitions ⟶ ℤResult quality: 27% ●values known / values provided: 27%●distinct values known / distinct values provided: 40%
Values
[1,1,2] => [2,1] => [[2,2],[1]]
=> 2 = 0 + 2
[1,1,1,2] => [3,1] => [[3,3],[2]]
=> 2 = 0 + 2
[1,1,2,1] => [2,1,1] => [[2,2,2],[1,1]]
=> 2 = 0 + 2
[1,1,3] => [2,1] => [[2,2],[1]]
=> 2 = 0 + 2
[2,2,1] => [2,1] => [[2,2],[1]]
=> 2 = 0 + 2
[1,1,1,1,2] => [4,1] => [[4,4],[3]]
=> 2 = 0 + 2
[1,1,1,2,1] => [3,1,1] => [[3,3,3],[2,2]]
=> 2 = 0 + 2
[1,1,1,3] => [3,1] => [[3,3],[2]]
=> 2 = 0 + 2
[1,1,2,1,1] => [2,1,2] => [[3,2,2],[1,1]]
=> 2 = 0 + 2
[1,1,2,2] => [2,2] => [[3,2],[1]]
=> 2 = 0 + 2
[1,1,3,1] => [2,1,1] => [[2,2,2],[1,1]]
=> 2 = 0 + 2
[1,1,4] => [2,1] => [[2,2],[1]]
=> 2 = 0 + 2
[1,2,2,1] => [1,2,1] => [[2,2,1],[1]]
=> 2 = 0 + 2
[2,1,1,2] => [1,2,1] => [[2,2,1],[1]]
=> 2 = 0 + 2
[2,2,1,1] => [2,2] => [[3,2],[1]]
=> 2 = 0 + 2
[1,1,1,1,1,2] => [5,1] => [[5,5],[4]]
=> ? = 0 + 2
[1,1,1,1,2,1] => [4,1,1] => [[4,4,4],[3,3]]
=> ? = 0 + 2
[1,1,1,1,3] => [4,1] => [[4,4],[3]]
=> 2 = 0 + 2
[1,1,1,2,1,1] => [3,1,2] => [[4,3,3],[2,2]]
=> ? = 0 + 2
[1,1,1,2,2] => [3,2] => [[4,3],[2]]
=> 2 = 0 + 2
[1,1,1,3,1] => [3,1,1] => [[3,3,3],[2,2]]
=> 2 = 0 + 2
[1,1,1,4] => [3,1] => [[3,3],[2]]
=> 2 = 0 + 2
[1,1,2,1,1,1] => [2,1,3] => [[4,2,2],[1,1]]
=> ? = 0 + 2
[1,1,2,1,2] => [2,1,1,1] => [[2,2,2,2],[1,1,1]]
=> 2 = 0 + 2
[1,1,2,2,1] => [2,2,1] => [[3,3,2],[2,1]]
=> 3 = 1 + 2
[1,1,2,3] => [2,1,1] => [[2,2,2],[1,1]]
=> 2 = 0 + 2
[1,1,3,1,1] => [2,1,2] => [[3,2,2],[1,1]]
=> 2 = 0 + 2
[1,1,3,2] => [2,1,1] => [[2,2,2],[1,1]]
=> 2 = 0 + 2
[1,1,4,1] => [2,1,1] => [[2,2,2],[1,1]]
=> 2 = 0 + 2
[1,1,5] => [2,1] => [[2,2],[1]]
=> 2 = 0 + 2
[1,2,1,1,2] => [1,1,2,1] => [[2,2,1,1],[1]]
=> 2 = 0 + 2
[1,2,2,1,1] => [1,2,2] => [[3,2,1],[1]]
=> 2 = 0 + 2
[2,1,1,1,2] => [1,3,1] => [[3,3,1],[2]]
=> 2 = 0 + 2
[2,1,1,2,1] => [1,2,1,1] => [[2,2,2,1],[1,1]]
=> 2 = 0 + 2
[2,1,1,3] => [1,2,1] => [[2,2,1],[1]]
=> 2 = 0 + 2
[2,2,1,1,1] => [2,3] => [[4,2],[1]]
=> 2 = 0 + 2
[2,2,1,2] => [2,1,1] => [[2,2,2],[1,1]]
=> 2 = 0 + 2
[2,2,2,1] => [3,1] => [[3,3],[2]]
=> 2 = 0 + 2
[2,2,3] => [2,1] => [[2,2],[1]]
=> 2 = 0 + 2
[3,1,1,2] => [1,2,1] => [[2,2,1],[1]]
=> 2 = 0 + 2
[3,3,1] => [2,1] => [[2,2],[1]]
=> 2 = 0 + 2
[1,1,1,1,1,1,2] => [6,1] => [[6,6],[5]]
=> ? = 0 + 2
[1,1,1,1,1,2,1] => [5,1,1] => [[5,5,5],[4,4]]
=> ? = 0 + 2
[1,1,1,1,1,3] => [5,1] => [[5,5],[4]]
=> ? = 0 + 2
[1,1,1,1,2,1,1] => [4,1,2] => [[5,4,4],[3,3]]
=> ? = 0 + 2
[1,1,1,1,2,2] => [4,2] => [[5,4],[3]]
=> ? = 0 + 2
[1,1,1,1,3,1] => [4,1,1] => [[4,4,4],[3,3]]
=> ? = 0 + 2
[1,1,1,1,4] => [4,1] => [[4,4],[3]]
=> 2 = 0 + 2
[1,1,1,2,1,1,1] => [3,1,3] => [[5,3,3],[2,2]]
=> ? = 0 + 2
[1,1,1,2,1,2] => [3,1,1,1] => [[3,3,3,3],[2,2,2]]
=> ? = 0 + 2
[1,1,1,2,2,1] => [3,2,1] => [[4,4,3],[3,2]]
=> ? = 0 + 2
[1,1,1,2,3] => [3,1,1] => [[3,3,3],[2,2]]
=> 2 = 0 + 2
[1,1,1,3,1,1] => [3,1,2] => [[4,3,3],[2,2]]
=> ? = 0 + 2
[1,1,1,3,2] => [3,1,1] => [[3,3,3],[2,2]]
=> 2 = 0 + 2
[1,1,1,4,1] => [3,1,1] => [[3,3,3],[2,2]]
=> 2 = 0 + 2
[1,1,1,5] => [3,1] => [[3,3],[2]]
=> 2 = 0 + 2
[1,1,2,1,1,1,1] => [2,1,4] => [[5,2,2],[1,1]]
=> ? = 0 + 2
[1,1,2,1,1,2] => [2,1,2,1] => [[3,3,2,2],[2,1,1]]
=> ? = 2 + 2
[1,1,2,1,2,1] => [2,1,1,1,1] => [[2,2,2,2,2],[1,1,1,1]]
=> ? = 0 + 2
[1,1,2,1,3] => [2,1,1,1] => [[2,2,2,2],[1,1,1]]
=> 2 = 0 + 2
[1,1,2,2,1,1] => [2,2,2] => [[4,3,2],[2,1]]
=> ? = 1 + 2
[1,1,2,2,2] => [2,3] => [[4,2],[1]]
=> 2 = 0 + 2
[1,1,2,3,1] => [2,1,1,1] => [[2,2,2,2],[1,1,1]]
=> 2 = 0 + 2
[1,1,2,4] => [2,1,1] => [[2,2,2],[1,1]]
=> 2 = 0 + 2
[1,1,3,1,1,1] => [2,1,3] => [[4,2,2],[1,1]]
=> ? = 0 + 2
[1,1,3,1,2] => [2,1,1,1] => [[2,2,2,2],[1,1,1]]
=> 2 = 0 + 2
[1,1,3,2,1] => [2,1,1,1] => [[2,2,2,2],[1,1,1]]
=> 2 = 0 + 2
[1,1,3,3] => [2,2] => [[3,2],[1]]
=> 2 = 0 + 2
[1,1,4,1,1] => [2,1,2] => [[3,2,2],[1,1]]
=> 2 = 0 + 2
[1,2,1,1,1,2] => [1,1,3,1] => [[3,3,1,1],[2]]
=> ? = 0 + 2
[1,2,1,1,2,1] => [1,1,2,1,1] => [[2,2,2,1,1],[1,1]]
=> ? = 0 + 2
[1,2,2,1,1,1] => [1,2,3] => [[4,2,1],[1]]
=> ? = 0 + 2
[2,1,1,1,1,2] => [1,4,1] => [[4,4,1],[3]]
=> ? = 0 + 2
[2,1,1,1,2,1] => [1,3,1,1] => [[3,3,3,1],[2,2]]
=> ? = 0 + 2
[2,1,1,2,1,1] => [1,2,1,2] => [[3,2,2,1],[1,1]]
=> ? = 0 + 2
[2,2,1,1,1,1] => [2,4] => [[5,2],[1]]
=> ? = 0 + 2
[1,1,1,1,1,1,1,2] => [7,1] => [[7,7],[6]]
=> ? = 0 + 2
[1,1,1,1,1,1,2,1] => [6,1,1] => [[6,6,6],[5,5]]
=> ? = 0 + 2
[1,1,1,1,1,1,3] => [6,1] => [[6,6],[5]]
=> ? = 0 + 2
[1,1,1,1,1,2,2] => [5,2] => [[6,5],[4]]
=> ? = 0 + 2
[1,1,1,1,1,3,1] => [5,1,1] => [[5,5,5],[4,4]]
=> ? = 0 + 2
[1,1,1,1,1,4] => [5,1] => [[5,5],[4]]
=> ? = 0 + 2
[1,1,1,1,2,1,1,1] => [4,1,3] => [[6,4,4],[3,3]]
=> ? = 0 + 2
[1,1,1,1,2,1,2] => [4,1,1,1] => [[4,4,4,4],[3,3,3]]
=> ? = 0 + 2
[1,1,1,1,2,2,1] => [4,2,1] => [[5,5,4],[4,3]]
=> ? = 1 + 2
[1,1,1,1,2,3] => [4,1,1] => [[4,4,4],[3,3]]
=> ? = 0 + 2
[1,1,1,1,3,1,1] => [4,1,2] => [[5,4,4],[3,3]]
=> ? = 0 + 2
[1,1,1,1,3,2] => [4,1,1] => [[4,4,4],[3,3]]
=> ? = 0 + 2
[1,1,1,1,4,1] => [4,1,1] => [[4,4,4],[3,3]]
=> ? = 0 + 2
[1,1,1,2,1,1,2] => [3,1,2,1] => [[4,4,3,3],[3,2,2]]
=> ? = 0 + 2
[1,1,1,2,1,2,1] => [3,1,1,1,1] => [[3,3,3,3,3],[2,2,2,2]]
=> ? = 0 + 2
[1,1,1,2,1,3] => [3,1,1,1] => [[3,3,3,3],[2,2,2]]
=> ? = 0 + 2
[1,1,1,2,2,1,1] => [3,2,2] => [[5,4,3],[3,2]]
=> ? = 0 + 2
[1,1,1,2,2,2] => [3,3] => [[5,3],[2]]
=> ? = 0 + 2
[1,1,1,2,3,1] => [3,1,1,1] => [[3,3,3,3],[2,2,2]]
=> ? = 0 + 2
[1,1,1,3,1,1,1] => [3,1,3] => [[5,3,3],[2,2]]
=> ? = 0 + 2
[1,1,1,3,1,2] => [3,1,1,1] => [[3,3,3,3],[2,2,2]]
=> ? = 0 + 2
[1,1,1,3,2,1] => [3,1,1,1] => [[3,3,3,3],[2,2,2]]
=> ? = 0 + 2
[1,1,1,4,1,1] => [3,1,2] => [[4,3,3],[2,2]]
=> ? = 0 + 2
[1,1,2,1,1,1,2] => [2,1,3,1] => [[4,4,2,2],[3,1,1]]
=> ? = 0 + 2
Description
The number of inner corners of a skew partition.
Matching statistic: St001001
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Mp00133: Integer compositions —delta morphism⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
Mp00121: Dyck paths —Cori-Le Borgne involution⟶ Dyck paths
St001001: Dyck paths ⟶ ℤResult quality: 27% ●values known / values provided: 27%●distinct values known / distinct values provided: 40%
Mp00231: Integer compositions —bounce path⟶ Dyck paths
Mp00121: Dyck paths —Cori-Le Borgne involution⟶ Dyck paths
St001001: Dyck paths ⟶ ℤResult quality: 27% ●values known / values provided: 27%●distinct values known / distinct values provided: 40%
Values
[1,1,2] => [2,1] => [1,1,0,0,1,0]
=> [1,1,0,0,1,0]
=> 0
[1,1,1,2] => [3,1] => [1,1,1,0,0,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> 0
[1,1,2,1] => [2,1,1] => [1,1,0,0,1,0,1,0]
=> [1,1,0,0,1,0,1,0]
=> 0
[1,1,3] => [2,1] => [1,1,0,0,1,0]
=> [1,1,0,0,1,0]
=> 0
[2,2,1] => [2,1] => [1,1,0,0,1,0]
=> [1,1,0,0,1,0]
=> 0
[1,1,1,1,2] => [4,1] => [1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> 0
[1,1,1,2,1] => [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> 0
[1,1,1,3] => [3,1] => [1,1,1,0,0,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> 0
[1,1,2,1,1] => [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
=> [1,1,0,1,0,0,1,1,0,0]
=> 0
[1,1,2,2] => [2,2] => [1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> 0
[1,1,3,1] => [2,1,1] => [1,1,0,0,1,0,1,0]
=> [1,1,0,0,1,0,1,0]
=> 0
[1,1,4] => [2,1] => [1,1,0,0,1,0]
=> [1,1,0,0,1,0]
=> 0
[1,2,2,1] => [1,2,1] => [1,0,1,1,0,0,1,0]
=> [1,1,0,1,0,0,1,0]
=> 0
[2,1,1,2] => [1,2,1] => [1,0,1,1,0,0,1,0]
=> [1,1,0,1,0,0,1,0]
=> 0
[2,2,1,1] => [2,2] => [1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> 0
[1,1,1,1,1,2] => [5,1] => [1,1,1,1,1,0,0,0,0,0,1,0]
=> [1,1,1,1,1,0,0,1,0,0,0,0]
=> ? = 0
[1,1,1,1,2,1] => [4,1,1] => [1,1,1,1,0,0,0,0,1,0,1,0]
=> [1,1,1,1,0,0,1,0,1,0,0,0]
=> ? = 0
[1,1,1,1,3] => [4,1] => [1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> 0
[1,1,1,2,1,1] => [3,1,2] => [1,1,1,0,0,0,1,0,1,1,0,0]
=> [1,1,1,0,0,0,1,1,0,1,0,0]
=> ? = 0
[1,1,1,2,2] => [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> 0
[1,1,1,3,1] => [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> 0
[1,1,1,4] => [3,1] => [1,1,1,0,0,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> 0
[1,1,2,1,1,1] => [2,1,3] => [1,1,0,0,1,0,1,1,1,0,0,0]
=> [1,1,1,0,1,0,0,1,1,0,0,0]
=> ? = 0
[1,1,2,1,2] => [2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> 0
[1,1,2,2,1] => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> 1
[1,1,2,3] => [2,1,1] => [1,1,0,0,1,0,1,0]
=> [1,1,0,0,1,0,1,0]
=> 0
[1,1,3,1,1] => [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
=> [1,1,0,1,0,0,1,1,0,0]
=> 0
[1,1,3,2] => [2,1,1] => [1,1,0,0,1,0,1,0]
=> [1,1,0,0,1,0,1,0]
=> 0
[1,1,4,1] => [2,1,1] => [1,1,0,0,1,0,1,0]
=> [1,1,0,0,1,0,1,0]
=> 0
[1,1,5] => [2,1] => [1,1,0,0,1,0]
=> [1,1,0,0,1,0]
=> 0
[1,2,1,1,2] => [1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,0,1,0,0,1,0]
=> 0
[1,2,2,1,1] => [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,1,0,0]
=> 0
[2,1,1,1,2] => [1,3,1] => [1,0,1,1,1,0,0,0,1,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> 0
[2,1,1,2,1] => [1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
=> [1,1,0,1,0,0,1,0,1,0]
=> 0
[2,1,1,3] => [1,2,1] => [1,0,1,1,0,0,1,0]
=> [1,1,0,1,0,0,1,0]
=> 0
[2,2,1,1,1] => [2,3] => [1,1,0,0,1,1,1,0,0,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> 0
[2,2,1,2] => [2,1,1] => [1,1,0,0,1,0,1,0]
=> [1,1,0,0,1,0,1,0]
=> 0
[2,2,2,1] => [3,1] => [1,1,1,0,0,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> 0
[2,2,3] => [2,1] => [1,1,0,0,1,0]
=> [1,1,0,0,1,0]
=> 0
[3,1,1,2] => [1,2,1] => [1,0,1,1,0,0,1,0]
=> [1,1,0,1,0,0,1,0]
=> 0
[3,3,1] => [2,1] => [1,1,0,0,1,0]
=> [1,1,0,0,1,0]
=> 0
[1,1,1,1,1,1,2] => [6,1] => [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> [1,1,1,1,1,1,0,0,1,0,0,0,0,0]
=> ? = 0
[1,1,1,1,1,2,1] => [5,1,1] => [1,1,1,1,1,0,0,0,0,0,1,0,1,0]
=> [1,1,1,1,1,0,0,1,0,1,0,0,0,0]
=> ? = 0
[1,1,1,1,1,3] => [5,1] => [1,1,1,1,1,0,0,0,0,0,1,0]
=> [1,1,1,1,1,0,0,1,0,0,0,0]
=> ? = 0
[1,1,1,1,2,1,1] => [4,1,2] => [1,1,1,1,0,0,0,0,1,0,1,1,0,0]
=> [1,1,1,1,0,0,0,1,1,0,1,0,0,0]
=> ? = 0
[1,1,1,1,2,2] => [4,2] => [1,1,1,1,0,0,0,0,1,1,0,0]
=> [1,1,1,1,0,0,0,1,1,0,0,0]
=> ? = 0
[1,1,1,1,3,1] => [4,1,1] => [1,1,1,1,0,0,0,0,1,0,1,0]
=> [1,1,1,1,0,0,1,0,1,0,0,0]
=> ? = 0
[1,1,1,1,4] => [4,1] => [1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> 0
[1,1,1,2,1,1,1] => [3,1,3] => [1,1,1,0,0,0,1,0,1,1,1,0,0,0]
=> [1,1,1,0,1,0,0,0,1,1,1,0,0,0]
=> ? = 0
[1,1,1,2,1,2] => [3,1,1,1] => [1,1,1,0,0,0,1,0,1,0,1,0]
=> [1,1,1,0,0,1,0,1,0,1,0,0]
=> ? = 0
[1,1,1,2,2,1] => [3,2,1] => [1,1,1,0,0,0,1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0,1,1,0,0]
=> ? = 0
[1,1,1,2,3] => [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> 0
[1,1,1,3,1,1] => [3,1,2] => [1,1,1,0,0,0,1,0,1,1,0,0]
=> [1,1,1,0,0,0,1,1,0,1,0,0]
=> ? = 0
[1,1,1,3,2] => [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> 0
[1,1,1,4,1] => [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> 0
[1,1,1,5] => [3,1] => [1,1,1,0,0,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> 0
[1,1,2,1,1,1,1] => [2,1,4] => [1,1,0,0,1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,1,0,0,1,1,0,0,0,0]
=> ? = 0
[1,1,2,1,1,2] => [2,1,2,1] => [1,1,0,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,0,0,1,1,0,0,1,0]
=> ? = 2
[1,1,2,1,2,1] => [2,1,1,1,1] => [1,1,0,0,1,0,1,0,1,0,1,0]
=> [1,1,0,0,1,0,1,0,1,0,1,0]
=> ? = 0
[1,1,2,1,3] => [2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> 0
[1,1,2,2,1,1] => [2,2,2] => [1,1,0,0,1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,0,1,1,0,0]
=> ? = 1
[1,1,2,2,2] => [2,3] => [1,1,0,0,1,1,1,0,0,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> 0
[1,1,2,3,1] => [2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> 0
[1,1,2,4] => [2,1,1] => [1,1,0,0,1,0,1,0]
=> [1,1,0,0,1,0,1,0]
=> 0
[1,1,3,1,1,1] => [2,1,3] => [1,1,0,0,1,0,1,1,1,0,0,0]
=> [1,1,1,0,1,0,0,1,1,0,0,0]
=> ? = 0
[1,1,3,1,2] => [2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> 0
[1,1,3,2,1] => [2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> 0
[1,1,3,3] => [2,2] => [1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> 0
[1,1,4,1,1] => [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
=> [1,1,0,1,0,0,1,1,0,0]
=> 0
[1,2,1,1,1,2] => [1,1,3,1] => [1,0,1,0,1,1,1,0,0,0,1,0]
=> [1,1,1,0,1,0,1,0,0,1,0,0]
=> ? = 0
[1,2,1,1,2,1] => [1,1,2,1,1] => [1,0,1,0,1,1,0,0,1,0,1,0]
=> [1,1,0,1,0,1,0,0,1,0,1,0]
=> ? = 0
[1,2,2,1,1,1] => [1,2,3] => [1,0,1,1,0,0,1,1,1,0,0,0]
=> [1,1,1,0,0,1,1,0,1,0,0,0]
=> ? = 0
[2,1,1,1,1,2] => [1,4,1] => [1,0,1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,1,0,1,0,0,1,0,0,0]
=> ? = 0
[2,1,1,1,2,1] => [1,3,1,1] => [1,0,1,1,1,0,0,0,1,0,1,0]
=> [1,1,1,0,1,0,0,1,0,1,0,0]
=> ? = 0
[2,1,1,2,1,1] => [1,2,1,2] => [1,0,1,1,0,0,1,0,1,1,0,0]
=> [1,1,0,1,0,0,1,1,0,1,0,0]
=> ? = 0
[2,2,1,1,1,1] => [2,4] => [1,1,0,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,1,1,0,0,0,0]
=> ? = 0
[1,1,1,1,1,1,1,2] => [7,1] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> [1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0]
=> ? = 0
[1,1,1,1,1,1,2,1] => [6,1,1] => [1,1,1,1,1,1,0,0,0,0,0,0,1,0,1,0]
=> [1,1,1,1,1,1,0,0,1,0,1,0,0,0,0,0]
=> ? = 0
[1,1,1,1,1,1,3] => [6,1] => [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> [1,1,1,1,1,1,0,0,1,0,0,0,0,0]
=> ? = 0
[1,1,1,1,1,2,2] => [5,2] => [1,1,1,1,1,0,0,0,0,0,1,1,0,0]
=> [1,1,1,1,1,0,0,0,1,1,0,0,0,0]
=> ? = 0
[1,1,1,1,1,3,1] => [5,1,1] => [1,1,1,1,1,0,0,0,0,0,1,0,1,0]
=> [1,1,1,1,1,0,0,1,0,1,0,0,0,0]
=> ? = 0
[1,1,1,1,1,4] => [5,1] => [1,1,1,1,1,0,0,0,0,0,1,0]
=> [1,1,1,1,1,0,0,1,0,0,0,0]
=> ? = 0
[1,1,1,1,2,1,1,1] => [4,1,3] => [1,1,1,1,0,0,0,0,1,0,1,1,1,0,0,0]
=> [1,1,1,1,0,0,0,0,1,1,1,0,1,0,0,0]
=> ? = 0
[1,1,1,1,2,1,2] => [4,1,1,1] => [1,1,1,1,0,0,0,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,1,0,1,0,1,0,0,0]
=> ? = 0
[1,1,1,1,2,2,1] => [4,2,1] => [1,1,1,1,0,0,0,0,1,1,0,0,1,0]
=> [1,1,1,1,0,0,1,0,0,1,1,0,0,0]
=> ? = 1
[1,1,1,1,2,3] => [4,1,1] => [1,1,1,1,0,0,0,0,1,0,1,0]
=> [1,1,1,1,0,0,1,0,1,0,0,0]
=> ? = 0
[1,1,1,1,3,1,1] => [4,1,2] => [1,1,1,1,0,0,0,0,1,0,1,1,0,0]
=> [1,1,1,1,0,0,0,1,1,0,1,0,0,0]
=> ? = 0
[1,1,1,1,3,2] => [4,1,1] => [1,1,1,1,0,0,0,0,1,0,1,0]
=> [1,1,1,1,0,0,1,0,1,0,0,0]
=> ? = 0
[1,1,1,1,4,1] => [4,1,1] => [1,1,1,1,0,0,0,0,1,0,1,0]
=> [1,1,1,1,0,0,1,0,1,0,0,0]
=> ? = 0
[1,1,1,2,1,1,2] => [3,1,2,1] => [1,1,1,0,0,0,1,0,1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0,1,1,0,1,0,0]
=> ? = 0
[1,1,1,2,1,2,1] => [3,1,1,1,1] => [1,1,1,0,0,0,1,0,1,0,1,0,1,0]
=> [1,1,1,0,0,1,0,1,0,1,0,1,0,0]
=> ? = 0
[1,1,1,2,1,3] => [3,1,1,1] => [1,1,1,0,0,0,1,0,1,0,1,0]
=> [1,1,1,0,0,1,0,1,0,1,0,0]
=> ? = 0
[1,1,1,2,2,1,1] => [3,2,2] => [1,1,1,0,0,0,1,1,0,0,1,1,0,0]
=> [1,1,1,0,0,0,1,1,0,0,1,1,0,0]
=> ? = 0
[1,1,1,2,2,2] => [3,3] => [1,1,1,0,0,0,1,1,1,0,0,0]
=> [1,1,1,0,0,0,1,1,1,0,0,0]
=> ? = 0
[1,1,1,2,3,1] => [3,1,1,1] => [1,1,1,0,0,0,1,0,1,0,1,0]
=> [1,1,1,0,0,1,0,1,0,1,0,0]
=> ? = 0
[1,1,1,3,1,1,1] => [3,1,3] => [1,1,1,0,0,0,1,0,1,1,1,0,0,0]
=> [1,1,1,0,1,0,0,0,1,1,1,0,0,0]
=> ? = 0
[1,1,1,3,1,2] => [3,1,1,1] => [1,1,1,0,0,0,1,0,1,0,1,0]
=> [1,1,1,0,0,1,0,1,0,1,0,0]
=> ? = 0
[1,1,1,3,2,1] => [3,1,1,1] => [1,1,1,0,0,0,1,0,1,0,1,0]
=> [1,1,1,0,0,1,0,1,0,1,0,0]
=> ? = 0
[1,1,1,4,1,1] => [3,1,2] => [1,1,1,0,0,0,1,0,1,1,0,0]
=> [1,1,1,0,0,0,1,1,0,1,0,0]
=> ? = 0
[1,1,2,1,1,1,2] => [2,1,3,1] => [1,1,0,0,1,0,1,1,1,0,0,0,1,0]
=> [1,1,1,0,1,0,0,1,1,0,0,1,0,0]
=> ? = 0
Description
The number of indecomposable modules with projective and injective dimension equal to the global dimension of the Nakayama algebra corresponding to the Dyck path.
Matching statistic: St001960
Mp00133: Integer compositions —delta morphism⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
Mp00129: Dyck paths —to 321-avoiding permutation (Billey-Jockusch-Stanley)⟶ Permutations
St001960: Permutations ⟶ ℤResult quality: 27% ●values known / values provided: 27%●distinct values known / distinct values provided: 40%
Mp00231: Integer compositions —bounce path⟶ Dyck paths
Mp00129: Dyck paths —to 321-avoiding permutation (Billey-Jockusch-Stanley)⟶ Permutations
St001960: Permutations ⟶ ℤResult quality: 27% ●values known / values provided: 27%●distinct values known / distinct values provided: 40%
Values
[1,1,2] => [2,1] => [1,1,0,0,1,0]
=> [1,3,2] => 1 = 0 + 1
[1,1,1,2] => [3,1] => [1,1,1,0,0,0,1,0]
=> [1,2,4,3] => 1 = 0 + 1
[1,1,2,1] => [2,1,1] => [1,1,0,0,1,0,1,0]
=> [1,3,4,2] => 1 = 0 + 1
[1,1,3] => [2,1] => [1,1,0,0,1,0]
=> [1,3,2] => 1 = 0 + 1
[2,2,1] => [2,1] => [1,1,0,0,1,0]
=> [1,3,2] => 1 = 0 + 1
[1,1,1,1,2] => [4,1] => [1,1,1,1,0,0,0,0,1,0]
=> [1,2,3,5,4] => 1 = 0 + 1
[1,1,1,2,1] => [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> [1,2,4,5,3] => 1 = 0 + 1
[1,1,1,3] => [3,1] => [1,1,1,0,0,0,1,0]
=> [1,2,4,3] => 1 = 0 + 1
[1,1,2,1,1] => [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
=> [1,3,4,2,5] => 1 = 0 + 1
[1,1,2,2] => [2,2] => [1,1,0,0,1,1,0,0]
=> [1,3,2,4] => 1 = 0 + 1
[1,1,3,1] => [2,1,1] => [1,1,0,0,1,0,1,0]
=> [1,3,4,2] => 1 = 0 + 1
[1,1,4] => [2,1] => [1,1,0,0,1,0]
=> [1,3,2] => 1 = 0 + 1
[1,2,2,1] => [1,2,1] => [1,0,1,1,0,0,1,0]
=> [2,1,4,3] => 1 = 0 + 1
[2,1,1,2] => [1,2,1] => [1,0,1,1,0,0,1,0]
=> [2,1,4,3] => 1 = 0 + 1
[2,2,1,1] => [2,2] => [1,1,0,0,1,1,0,0]
=> [1,3,2,4] => 1 = 0 + 1
[1,1,1,1,1,2] => [5,1] => [1,1,1,1,1,0,0,0,0,0,1,0]
=> [1,2,3,4,6,5] => ? = 0 + 1
[1,1,1,1,2,1] => [4,1,1] => [1,1,1,1,0,0,0,0,1,0,1,0]
=> [1,2,3,5,6,4] => ? = 0 + 1
[1,1,1,1,3] => [4,1] => [1,1,1,1,0,0,0,0,1,0]
=> [1,2,3,5,4] => 1 = 0 + 1
[1,1,1,2,1,1] => [3,1,2] => [1,1,1,0,0,0,1,0,1,1,0,0]
=> [1,2,4,5,3,6] => ? = 0 + 1
[1,1,1,2,2] => [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> [1,2,4,3,5] => 1 = 0 + 1
[1,1,1,3,1] => [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> [1,2,4,5,3] => 1 = 0 + 1
[1,1,1,4] => [3,1] => [1,1,1,0,0,0,1,0]
=> [1,2,4,3] => 1 = 0 + 1
[1,1,2,1,1,1] => [2,1,3] => [1,1,0,0,1,0,1,1,1,0,0,0]
=> [1,3,4,2,5,6] => ? = 0 + 1
[1,1,2,1,2] => [2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
=> [1,3,4,5,2] => 1 = 0 + 1
[1,1,2,2,1] => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> [1,3,2,5,4] => 2 = 1 + 1
[1,1,2,3] => [2,1,1] => [1,1,0,0,1,0,1,0]
=> [1,3,4,2] => 1 = 0 + 1
[1,1,3,1,1] => [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
=> [1,3,4,2,5] => 1 = 0 + 1
[1,1,3,2] => [2,1,1] => [1,1,0,0,1,0,1,0]
=> [1,3,4,2] => 1 = 0 + 1
[1,1,4,1] => [2,1,1] => [1,1,0,0,1,0,1,0]
=> [1,3,4,2] => 1 = 0 + 1
[1,1,5] => [2,1] => [1,1,0,0,1,0]
=> [1,3,2] => 1 = 0 + 1
[1,2,1,1,2] => [1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
=> [2,3,1,5,4] => 1 = 0 + 1
[1,2,2,1,1] => [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
=> [2,1,4,3,5] => 1 = 0 + 1
[2,1,1,1,2] => [1,3,1] => [1,0,1,1,1,0,0,0,1,0]
=> [2,1,3,5,4] => 1 = 0 + 1
[2,1,1,2,1] => [1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
=> [2,1,4,5,3] => 1 = 0 + 1
[2,1,1,3] => [1,2,1] => [1,0,1,1,0,0,1,0]
=> [2,1,4,3] => 1 = 0 + 1
[2,2,1,1,1] => [2,3] => [1,1,0,0,1,1,1,0,0,0]
=> [1,3,2,4,5] => 1 = 0 + 1
[2,2,1,2] => [2,1,1] => [1,1,0,0,1,0,1,0]
=> [1,3,4,2] => 1 = 0 + 1
[2,2,2,1] => [3,1] => [1,1,1,0,0,0,1,0]
=> [1,2,4,3] => 1 = 0 + 1
[2,2,3] => [2,1] => [1,1,0,0,1,0]
=> [1,3,2] => 1 = 0 + 1
[3,1,1,2] => [1,2,1] => [1,0,1,1,0,0,1,0]
=> [2,1,4,3] => 1 = 0 + 1
[3,3,1] => [2,1] => [1,1,0,0,1,0]
=> [1,3,2] => 1 = 0 + 1
[1,1,1,1,1,1,2] => [6,1] => [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> [1,2,3,4,5,7,6] => ? = 0 + 1
[1,1,1,1,1,2,1] => [5,1,1] => [1,1,1,1,1,0,0,0,0,0,1,0,1,0]
=> [1,2,3,4,6,7,5] => ? = 0 + 1
[1,1,1,1,1,3] => [5,1] => [1,1,1,1,1,0,0,0,0,0,1,0]
=> [1,2,3,4,6,5] => ? = 0 + 1
[1,1,1,1,2,1,1] => [4,1,2] => [1,1,1,1,0,0,0,0,1,0,1,1,0,0]
=> [1,2,3,5,6,4,7] => ? = 0 + 1
[1,1,1,1,2,2] => [4,2] => [1,1,1,1,0,0,0,0,1,1,0,0]
=> [1,2,3,5,4,6] => ? = 0 + 1
[1,1,1,1,3,1] => [4,1,1] => [1,1,1,1,0,0,0,0,1,0,1,0]
=> [1,2,3,5,6,4] => ? = 0 + 1
[1,1,1,1,4] => [4,1] => [1,1,1,1,0,0,0,0,1,0]
=> [1,2,3,5,4] => 1 = 0 + 1
[1,1,1,2,1,1,1] => [3,1,3] => [1,1,1,0,0,0,1,0,1,1,1,0,0,0]
=> [1,2,4,5,3,6,7] => ? = 0 + 1
[1,1,1,2,1,2] => [3,1,1,1] => [1,1,1,0,0,0,1,0,1,0,1,0]
=> [1,2,4,5,6,3] => ? = 0 + 1
[1,1,1,2,2,1] => [3,2,1] => [1,1,1,0,0,0,1,1,0,0,1,0]
=> [1,2,4,3,6,5] => ? = 0 + 1
[1,1,1,2,3] => [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> [1,2,4,5,3] => 1 = 0 + 1
[1,1,1,3,1,1] => [3,1,2] => [1,1,1,0,0,0,1,0,1,1,0,0]
=> [1,2,4,5,3,6] => ? = 0 + 1
[1,1,1,3,2] => [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> [1,2,4,5,3] => 1 = 0 + 1
[1,1,1,4,1] => [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> [1,2,4,5,3] => 1 = 0 + 1
[1,1,1,5] => [3,1] => [1,1,1,0,0,0,1,0]
=> [1,2,4,3] => 1 = 0 + 1
[1,1,2,1,1,1,1] => [2,1,4] => [1,1,0,0,1,0,1,1,1,1,0,0,0,0]
=> [1,3,4,2,5,6,7] => ? = 0 + 1
[1,1,2,1,1,2] => [2,1,2,1] => [1,1,0,0,1,0,1,1,0,0,1,0]
=> [1,3,4,2,6,5] => ? = 2 + 1
[1,1,2,1,2,1] => [2,1,1,1,1] => [1,1,0,0,1,0,1,0,1,0,1,0]
=> [1,3,4,5,6,2] => ? = 0 + 1
[1,1,2,1,3] => [2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
=> [1,3,4,5,2] => 1 = 0 + 1
[1,1,2,2,1,1] => [2,2,2] => [1,1,0,0,1,1,0,0,1,1,0,0]
=> [1,3,2,5,4,6] => ? = 1 + 1
[1,1,2,2,2] => [2,3] => [1,1,0,0,1,1,1,0,0,0]
=> [1,3,2,4,5] => 1 = 0 + 1
[1,1,2,3,1] => [2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
=> [1,3,4,5,2] => 1 = 0 + 1
[1,1,2,4] => [2,1,1] => [1,1,0,0,1,0,1,0]
=> [1,3,4,2] => 1 = 0 + 1
[1,1,3,1,1,1] => [2,1,3] => [1,1,0,0,1,0,1,1,1,0,0,0]
=> [1,3,4,2,5,6] => ? = 0 + 1
[1,1,3,1,2] => [2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
=> [1,3,4,5,2] => 1 = 0 + 1
[1,1,3,2,1] => [2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
=> [1,3,4,5,2] => 1 = 0 + 1
[1,1,3,3] => [2,2] => [1,1,0,0,1,1,0,0]
=> [1,3,2,4] => 1 = 0 + 1
[1,1,4,1,1] => [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
=> [1,3,4,2,5] => 1 = 0 + 1
[1,2,1,1,1,2] => [1,1,3,1] => [1,0,1,0,1,1,1,0,0,0,1,0]
=> [2,3,1,4,6,5] => ? = 0 + 1
[1,2,1,1,2,1] => [1,1,2,1,1] => [1,0,1,0,1,1,0,0,1,0,1,0]
=> [2,3,1,5,6,4] => ? = 0 + 1
[1,2,2,1,1,1] => [1,2,3] => [1,0,1,1,0,0,1,1,1,0,0,0]
=> [2,1,4,3,5,6] => ? = 0 + 1
[2,1,1,1,1,2] => [1,4,1] => [1,0,1,1,1,1,0,0,0,0,1,0]
=> [2,1,3,4,6,5] => ? = 0 + 1
[2,1,1,1,2,1] => [1,3,1,1] => [1,0,1,1,1,0,0,0,1,0,1,0]
=> [2,1,3,5,6,4] => ? = 0 + 1
[2,1,1,2,1,1] => [1,2,1,2] => [1,0,1,1,0,0,1,0,1,1,0,0]
=> [2,1,4,5,3,6] => ? = 0 + 1
[2,2,1,1,1,1] => [2,4] => [1,1,0,0,1,1,1,1,0,0,0,0]
=> [1,3,2,4,5,6] => ? = 0 + 1
[1,1,1,1,1,1,1,2] => [7,1] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> [1,2,3,4,5,6,8,7] => ? = 0 + 1
[1,1,1,1,1,1,2,1] => [6,1,1] => [1,1,1,1,1,1,0,0,0,0,0,0,1,0,1,0]
=> [1,2,3,4,5,7,8,6] => ? = 0 + 1
[1,1,1,1,1,1,3] => [6,1] => [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> [1,2,3,4,5,7,6] => ? = 0 + 1
[1,1,1,1,1,2,2] => [5,2] => [1,1,1,1,1,0,0,0,0,0,1,1,0,0]
=> [1,2,3,4,6,5,7] => ? = 0 + 1
[1,1,1,1,1,3,1] => [5,1,1] => [1,1,1,1,1,0,0,0,0,0,1,0,1,0]
=> [1,2,3,4,6,7,5] => ? = 0 + 1
[1,1,1,1,1,4] => [5,1] => [1,1,1,1,1,0,0,0,0,0,1,0]
=> [1,2,3,4,6,5] => ? = 0 + 1
[1,1,1,1,2,1,1,1] => [4,1,3] => [1,1,1,1,0,0,0,0,1,0,1,1,1,0,0,0]
=> [1,2,3,5,6,4,7,8] => ? = 0 + 1
[1,1,1,1,2,1,2] => [4,1,1,1] => [1,1,1,1,0,0,0,0,1,0,1,0,1,0]
=> [1,2,3,5,6,7,4] => ? = 0 + 1
[1,1,1,1,2,2,1] => [4,2,1] => [1,1,1,1,0,0,0,0,1,1,0,0,1,0]
=> [1,2,3,5,4,7,6] => ? = 1 + 1
[1,1,1,1,2,3] => [4,1,1] => [1,1,1,1,0,0,0,0,1,0,1,0]
=> [1,2,3,5,6,4] => ? = 0 + 1
[1,1,1,1,3,1,1] => [4,1,2] => [1,1,1,1,0,0,0,0,1,0,1,1,0,0]
=> [1,2,3,5,6,4,7] => ? = 0 + 1
[1,1,1,1,3,2] => [4,1,1] => [1,1,1,1,0,0,0,0,1,0,1,0]
=> [1,2,3,5,6,4] => ? = 0 + 1
[1,1,1,1,4,1] => [4,1,1] => [1,1,1,1,0,0,0,0,1,0,1,0]
=> [1,2,3,5,6,4] => ? = 0 + 1
[1,1,1,2,1,1,2] => [3,1,2,1] => [1,1,1,0,0,0,1,0,1,1,0,0,1,0]
=> [1,2,4,5,3,7,6] => ? = 0 + 1
[1,1,1,2,1,2,1] => [3,1,1,1,1] => [1,1,1,0,0,0,1,0,1,0,1,0,1,0]
=> [1,2,4,5,6,7,3] => ? = 0 + 1
[1,1,1,2,1,3] => [3,1,1,1] => [1,1,1,0,0,0,1,0,1,0,1,0]
=> [1,2,4,5,6,3] => ? = 0 + 1
[1,1,1,2,2,1,1] => [3,2,2] => [1,1,1,0,0,0,1,1,0,0,1,1,0,0]
=> [1,2,4,3,6,5,7] => ? = 0 + 1
[1,1,1,2,2,2] => [3,3] => [1,1,1,0,0,0,1,1,1,0,0,0]
=> [1,2,4,3,5,6] => ? = 0 + 1
[1,1,1,2,3,1] => [3,1,1,1] => [1,1,1,0,0,0,1,0,1,0,1,0]
=> [1,2,4,5,6,3] => ? = 0 + 1
[1,1,1,3,1,1,1] => [3,1,3] => [1,1,1,0,0,0,1,0,1,1,1,0,0,0]
=> [1,2,4,5,3,6,7] => ? = 0 + 1
[1,1,1,3,1,2] => [3,1,1,1] => [1,1,1,0,0,0,1,0,1,0,1,0]
=> [1,2,4,5,6,3] => ? = 0 + 1
[1,1,1,3,2,1] => [3,1,1,1] => [1,1,1,0,0,0,1,0,1,0,1,0]
=> [1,2,4,5,6,3] => ? = 0 + 1
[1,1,1,4,1,1] => [3,1,2] => [1,1,1,0,0,0,1,0,1,1,0,0]
=> [1,2,4,5,3,6] => ? = 0 + 1
[1,1,2,1,1,1,2] => [2,1,3,1] => [1,1,0,0,1,0,1,1,1,0,0,0,1,0]
=> [1,3,4,2,5,7,6] => ? = 0 + 1
Description
The number of descents of a permutation minus one if its first entry is not one.
This statistic appears in [1, Theorem 2.3] in a gamma-positivity result, see also [2].
Matching statistic: St000243
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00133: Integer compositions —delta morphism⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
Mp00201: Dyck paths —Ringel⟶ Permutations
St000243: Permutations ⟶ ℤResult quality: 27% ●values known / values provided: 27%●distinct values known / distinct values provided: 40%
Mp00231: Integer compositions —bounce path⟶ Dyck paths
Mp00201: Dyck paths —Ringel⟶ Permutations
St000243: Permutations ⟶ ℤResult quality: 27% ●values known / values provided: 27%●distinct values known / distinct values provided: 40%
Values
[1,1,2] => [2,1] => [1,1,0,0,1,0]
=> [2,4,1,3] => 2 = 0 + 2
[1,1,1,2] => [3,1] => [1,1,1,0,0,0,1,0]
=> [2,3,5,1,4] => 2 = 0 + 2
[1,1,2,1] => [2,1,1] => [1,1,0,0,1,0,1,0]
=> [2,5,1,3,4] => 2 = 0 + 2
[1,1,3] => [2,1] => [1,1,0,0,1,0]
=> [2,4,1,3] => 2 = 0 + 2
[2,2,1] => [2,1] => [1,1,0,0,1,0]
=> [2,4,1,3] => 2 = 0 + 2
[1,1,1,1,2] => [4,1] => [1,1,1,1,0,0,0,0,1,0]
=> [2,3,4,6,1,5] => 2 = 0 + 2
[1,1,1,2,1] => [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> [2,3,6,1,4,5] => 2 = 0 + 2
[1,1,1,3] => [3,1] => [1,1,1,0,0,0,1,0]
=> [2,3,5,1,4] => 2 = 0 + 2
[1,1,2,1,1] => [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
=> [2,5,1,3,6,4] => 2 = 0 + 2
[1,1,2,2] => [2,2] => [1,1,0,0,1,1,0,0]
=> [2,4,1,5,3] => 2 = 0 + 2
[1,1,3,1] => [2,1,1] => [1,1,0,0,1,0,1,0]
=> [2,5,1,3,4] => 2 = 0 + 2
[1,1,4] => [2,1] => [1,1,0,0,1,0]
=> [2,4,1,3] => 2 = 0 + 2
[1,2,2,1] => [1,2,1] => [1,0,1,1,0,0,1,0]
=> [3,1,5,2,4] => 2 = 0 + 2
[2,1,1,2] => [1,2,1] => [1,0,1,1,0,0,1,0]
=> [3,1,5,2,4] => 2 = 0 + 2
[2,2,1,1] => [2,2] => [1,1,0,0,1,1,0,0]
=> [2,4,1,5,3] => 2 = 0 + 2
[1,1,1,1,1,2] => [5,1] => [1,1,1,1,1,0,0,0,0,0,1,0]
=> [2,3,4,5,7,1,6] => ? = 0 + 2
[1,1,1,1,2,1] => [4,1,1] => [1,1,1,1,0,0,0,0,1,0,1,0]
=> [2,3,4,7,1,5,6] => ? = 0 + 2
[1,1,1,1,3] => [4,1] => [1,1,1,1,0,0,0,0,1,0]
=> [2,3,4,6,1,5] => 2 = 0 + 2
[1,1,1,2,1,1] => [3,1,2] => [1,1,1,0,0,0,1,0,1,1,0,0]
=> [2,3,6,1,4,7,5] => ? = 0 + 2
[1,1,1,2,2] => [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> [2,3,5,1,6,4] => 2 = 0 + 2
[1,1,1,3,1] => [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> [2,3,6,1,4,5] => 2 = 0 + 2
[1,1,1,4] => [3,1] => [1,1,1,0,0,0,1,0]
=> [2,3,5,1,4] => 2 = 0 + 2
[1,1,2,1,1,1] => [2,1,3] => [1,1,0,0,1,0,1,1,1,0,0,0]
=> [2,5,1,3,6,7,4] => ? = 0 + 2
[1,1,2,1,2] => [2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
=> [2,6,1,3,4,5] => 2 = 0 + 2
[1,1,2,2,1] => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> [2,4,1,6,3,5] => 3 = 1 + 2
[1,1,2,3] => [2,1,1] => [1,1,0,0,1,0,1,0]
=> [2,5,1,3,4] => 2 = 0 + 2
[1,1,3,1,1] => [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
=> [2,5,1,3,6,4] => 2 = 0 + 2
[1,1,3,2] => [2,1,1] => [1,1,0,0,1,0,1,0]
=> [2,5,1,3,4] => 2 = 0 + 2
[1,1,4,1] => [2,1,1] => [1,1,0,0,1,0,1,0]
=> [2,5,1,3,4] => 2 = 0 + 2
[1,1,5] => [2,1] => [1,1,0,0,1,0]
=> [2,4,1,3] => 2 = 0 + 2
[1,2,1,1,2] => [1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
=> [4,1,2,6,3,5] => 2 = 0 + 2
[1,2,2,1,1] => [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
=> [3,1,5,2,6,4] => 2 = 0 + 2
[2,1,1,1,2] => [1,3,1] => [1,0,1,1,1,0,0,0,1,0]
=> [3,1,4,6,2,5] => 2 = 0 + 2
[2,1,1,2,1] => [1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
=> [3,1,6,2,4,5] => 2 = 0 + 2
[2,1,1,3] => [1,2,1] => [1,0,1,1,0,0,1,0]
=> [3,1,5,2,4] => 2 = 0 + 2
[2,2,1,1,1] => [2,3] => [1,1,0,0,1,1,1,0,0,0]
=> [2,4,1,5,6,3] => 2 = 0 + 2
[2,2,1,2] => [2,1,1] => [1,1,0,0,1,0,1,0]
=> [2,5,1,3,4] => 2 = 0 + 2
[2,2,2,1] => [3,1] => [1,1,1,0,0,0,1,0]
=> [2,3,5,1,4] => 2 = 0 + 2
[2,2,3] => [2,1] => [1,1,0,0,1,0]
=> [2,4,1,3] => 2 = 0 + 2
[3,1,1,2] => [1,2,1] => [1,0,1,1,0,0,1,0]
=> [3,1,5,2,4] => 2 = 0 + 2
[3,3,1] => [2,1] => [1,1,0,0,1,0]
=> [2,4,1,3] => 2 = 0 + 2
[1,1,1,1,1,1,2] => [6,1] => [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> [2,3,4,5,6,8,1,7] => ? = 0 + 2
[1,1,1,1,1,2,1] => [5,1,1] => [1,1,1,1,1,0,0,0,0,0,1,0,1,0]
=> [2,3,4,5,8,1,6,7] => ? = 0 + 2
[1,1,1,1,1,3] => [5,1] => [1,1,1,1,1,0,0,0,0,0,1,0]
=> [2,3,4,5,7,1,6] => ? = 0 + 2
[1,1,1,1,2,1,1] => [4,1,2] => [1,1,1,1,0,0,0,0,1,0,1,1,0,0]
=> [2,3,4,7,1,5,8,6] => ? = 0 + 2
[1,1,1,1,2,2] => [4,2] => [1,1,1,1,0,0,0,0,1,1,0,0]
=> [2,3,4,6,1,7,5] => ? = 0 + 2
[1,1,1,1,3,1] => [4,1,1] => [1,1,1,1,0,0,0,0,1,0,1,0]
=> [2,3,4,7,1,5,6] => ? = 0 + 2
[1,1,1,1,4] => [4,1] => [1,1,1,1,0,0,0,0,1,0]
=> [2,3,4,6,1,5] => 2 = 0 + 2
[1,1,1,2,1,1,1] => [3,1,3] => [1,1,1,0,0,0,1,0,1,1,1,0,0,0]
=> [2,3,6,1,4,7,8,5] => ? = 0 + 2
[1,1,1,2,1,2] => [3,1,1,1] => [1,1,1,0,0,0,1,0,1,0,1,0]
=> [2,3,7,1,4,5,6] => ? = 0 + 2
[1,1,1,2,2,1] => [3,2,1] => [1,1,1,0,0,0,1,1,0,0,1,0]
=> [2,3,5,1,7,4,6] => ? = 0 + 2
[1,1,1,2,3] => [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> [2,3,6,1,4,5] => 2 = 0 + 2
[1,1,1,3,1,1] => [3,1,2] => [1,1,1,0,0,0,1,0,1,1,0,0]
=> [2,3,6,1,4,7,5] => ? = 0 + 2
[1,1,1,3,2] => [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> [2,3,6,1,4,5] => 2 = 0 + 2
[1,1,1,4,1] => [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> [2,3,6,1,4,5] => 2 = 0 + 2
[1,1,1,5] => [3,1] => [1,1,1,0,0,0,1,0]
=> [2,3,5,1,4] => 2 = 0 + 2
[1,1,2,1,1,1,1] => [2,1,4] => [1,1,0,0,1,0,1,1,1,1,0,0,0,0]
=> [2,5,1,3,6,7,8,4] => ? = 0 + 2
[1,1,2,1,1,2] => [2,1,2,1] => [1,1,0,0,1,0,1,1,0,0,1,0]
=> [2,5,1,3,7,4,6] => ? = 2 + 2
[1,1,2,1,2,1] => [2,1,1,1,1] => [1,1,0,0,1,0,1,0,1,0,1,0]
=> [2,7,1,3,4,5,6] => ? = 0 + 2
[1,1,2,1,3] => [2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
=> [2,6,1,3,4,5] => 2 = 0 + 2
[1,1,2,2,1,1] => [2,2,2] => [1,1,0,0,1,1,0,0,1,1,0,0]
=> [2,4,1,6,3,7,5] => ? = 1 + 2
[1,1,2,2,2] => [2,3] => [1,1,0,0,1,1,1,0,0,0]
=> [2,4,1,5,6,3] => 2 = 0 + 2
[1,1,2,3,1] => [2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
=> [2,6,1,3,4,5] => 2 = 0 + 2
[1,1,2,4] => [2,1,1] => [1,1,0,0,1,0,1,0]
=> [2,5,1,3,4] => 2 = 0 + 2
[1,1,3,1,1,1] => [2,1,3] => [1,1,0,0,1,0,1,1,1,0,0,0]
=> [2,5,1,3,6,7,4] => ? = 0 + 2
[1,1,3,1,2] => [2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
=> [2,6,1,3,4,5] => 2 = 0 + 2
[1,1,3,2,1] => [2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
=> [2,6,1,3,4,5] => 2 = 0 + 2
[1,1,3,3] => [2,2] => [1,1,0,0,1,1,0,0]
=> [2,4,1,5,3] => 2 = 0 + 2
[1,1,4,1,1] => [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
=> [2,5,1,3,6,4] => 2 = 0 + 2
[1,2,1,1,1,2] => [1,1,3,1] => [1,0,1,0,1,1,1,0,0,0,1,0]
=> [4,1,2,5,7,3,6] => ? = 0 + 2
[1,2,1,1,2,1] => [1,1,2,1,1] => [1,0,1,0,1,1,0,0,1,0,1,0]
=> [4,1,2,7,3,5,6] => ? = 0 + 2
[1,2,2,1,1,1] => [1,2,3] => [1,0,1,1,0,0,1,1,1,0,0,0]
=> [3,1,5,2,6,7,4] => ? = 0 + 2
[2,1,1,1,1,2] => [1,4,1] => [1,0,1,1,1,1,0,0,0,0,1,0]
=> [3,1,4,5,7,2,6] => ? = 0 + 2
[2,1,1,1,2,1] => [1,3,1,1] => [1,0,1,1,1,0,0,0,1,0,1,0]
=> [3,1,4,7,2,5,6] => ? = 0 + 2
[2,1,1,2,1,1] => [1,2,1,2] => [1,0,1,1,0,0,1,0,1,1,0,0]
=> [3,1,6,2,4,7,5] => ? = 0 + 2
[2,2,1,1,1,1] => [2,4] => [1,1,0,0,1,1,1,1,0,0,0,0]
=> [2,4,1,5,6,7,3] => ? = 0 + 2
[1,1,1,1,1,1,1,2] => [7,1] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> [2,3,4,5,6,7,9,1,8] => ? = 0 + 2
[1,1,1,1,1,1,2,1] => [6,1,1] => [1,1,1,1,1,1,0,0,0,0,0,0,1,0,1,0]
=> [2,3,4,5,6,9,1,7,8] => ? = 0 + 2
[1,1,1,1,1,1,3] => [6,1] => [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> [2,3,4,5,6,8,1,7] => ? = 0 + 2
[1,1,1,1,1,2,2] => [5,2] => [1,1,1,1,1,0,0,0,0,0,1,1,0,0]
=> [2,3,4,5,7,1,8,6] => ? = 0 + 2
[1,1,1,1,1,3,1] => [5,1,1] => [1,1,1,1,1,0,0,0,0,0,1,0,1,0]
=> [2,3,4,5,8,1,6,7] => ? = 0 + 2
[1,1,1,1,1,4] => [5,1] => [1,1,1,1,1,0,0,0,0,0,1,0]
=> [2,3,4,5,7,1,6] => ? = 0 + 2
[1,1,1,1,2,1,1,1] => [4,1,3] => [1,1,1,1,0,0,0,0,1,0,1,1,1,0,0,0]
=> [2,3,4,7,1,5,8,9,6] => ? = 0 + 2
[1,1,1,1,2,1,2] => [4,1,1,1] => [1,1,1,1,0,0,0,0,1,0,1,0,1,0]
=> [2,3,4,8,1,5,6,7] => ? = 0 + 2
[1,1,1,1,2,2,1] => [4,2,1] => [1,1,1,1,0,0,0,0,1,1,0,0,1,0]
=> [2,3,4,6,1,8,5,7] => ? = 1 + 2
[1,1,1,1,2,3] => [4,1,1] => [1,1,1,1,0,0,0,0,1,0,1,0]
=> [2,3,4,7,1,5,6] => ? = 0 + 2
[1,1,1,1,3,1,1] => [4,1,2] => [1,1,1,1,0,0,0,0,1,0,1,1,0,0]
=> [2,3,4,7,1,5,8,6] => ? = 0 + 2
[1,1,1,1,3,2] => [4,1,1] => [1,1,1,1,0,0,0,0,1,0,1,0]
=> [2,3,4,7,1,5,6] => ? = 0 + 2
[1,1,1,1,4,1] => [4,1,1] => [1,1,1,1,0,0,0,0,1,0,1,0]
=> [2,3,4,7,1,5,6] => ? = 0 + 2
[1,1,1,2,1,1,2] => [3,1,2,1] => [1,1,1,0,0,0,1,0,1,1,0,0,1,0]
=> [2,3,6,1,4,8,5,7] => ? = 0 + 2
[1,1,1,2,1,2,1] => [3,1,1,1,1] => [1,1,1,0,0,0,1,0,1,0,1,0,1,0]
=> [2,3,8,1,4,5,6,7] => ? = 0 + 2
[1,1,1,2,1,3] => [3,1,1,1] => [1,1,1,0,0,0,1,0,1,0,1,0]
=> [2,3,7,1,4,5,6] => ? = 0 + 2
[1,1,1,2,2,1,1] => [3,2,2] => [1,1,1,0,0,0,1,1,0,0,1,1,0,0]
=> [2,3,5,1,7,4,8,6] => ? = 0 + 2
[1,1,1,2,2,2] => [3,3] => [1,1,1,0,0,0,1,1,1,0,0,0]
=> [2,3,5,1,6,7,4] => ? = 0 + 2
[1,1,1,2,3,1] => [3,1,1,1] => [1,1,1,0,0,0,1,0,1,0,1,0]
=> [2,3,7,1,4,5,6] => ? = 0 + 2
[1,1,1,3,1,1,1] => [3,1,3] => [1,1,1,0,0,0,1,0,1,1,1,0,0,0]
=> [2,3,6,1,4,7,8,5] => ? = 0 + 2
[1,1,1,3,1,2] => [3,1,1,1] => [1,1,1,0,0,0,1,0,1,0,1,0]
=> [2,3,7,1,4,5,6] => ? = 0 + 2
[1,1,1,3,2,1] => [3,1,1,1] => [1,1,1,0,0,0,1,0,1,0,1,0]
=> [2,3,7,1,4,5,6] => ? = 0 + 2
[1,1,1,4,1,1] => [3,1,2] => [1,1,1,0,0,0,1,0,1,1,0,0]
=> [2,3,6,1,4,7,5] => ? = 0 + 2
[1,1,2,1,1,1,2] => [2,1,3,1] => [1,1,0,0,1,0,1,1,1,0,0,0,1,0]
=> [2,5,1,3,6,8,4,7] => ? = 0 + 2
Description
The number of cyclic valleys and cyclic peaks of a permutation.
This is given by the number of indices $i$ such that $\pi_{i-1} > \pi_i < \pi_{i+1}$ with indices considered cyclically. Equivalently, this is the number of indices $i$ such that $\pi_{i-1} < \pi_i > \pi_{i+1}$ with indices considered cyclically.
Matching statistic: St000354
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Mp00133: Integer compositions —delta morphism⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
Mp00201: Dyck paths —Ringel⟶ Permutations
St000354: Permutations ⟶ ℤResult quality: 27% ●values known / values provided: 27%●distinct values known / distinct values provided: 40%
Mp00231: Integer compositions —bounce path⟶ Dyck paths
Mp00201: Dyck paths —Ringel⟶ Permutations
St000354: Permutations ⟶ ℤResult quality: 27% ●values known / values provided: 27%●distinct values known / distinct values provided: 40%
Values
[1,1,2] => [2,1] => [1,1,0,0,1,0]
=> [2,4,1,3] => 2 = 0 + 2
[1,1,1,2] => [3,1] => [1,1,1,0,0,0,1,0]
=> [2,3,5,1,4] => 2 = 0 + 2
[1,1,2,1] => [2,1,1] => [1,1,0,0,1,0,1,0]
=> [2,5,1,3,4] => 2 = 0 + 2
[1,1,3] => [2,1] => [1,1,0,0,1,0]
=> [2,4,1,3] => 2 = 0 + 2
[2,2,1] => [2,1] => [1,1,0,0,1,0]
=> [2,4,1,3] => 2 = 0 + 2
[1,1,1,1,2] => [4,1] => [1,1,1,1,0,0,0,0,1,0]
=> [2,3,4,6,1,5] => 2 = 0 + 2
[1,1,1,2,1] => [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> [2,3,6,1,4,5] => 2 = 0 + 2
[1,1,1,3] => [3,1] => [1,1,1,0,0,0,1,0]
=> [2,3,5,1,4] => 2 = 0 + 2
[1,1,2,1,1] => [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
=> [2,5,1,3,6,4] => 2 = 0 + 2
[1,1,2,2] => [2,2] => [1,1,0,0,1,1,0,0]
=> [2,4,1,5,3] => 2 = 0 + 2
[1,1,3,1] => [2,1,1] => [1,1,0,0,1,0,1,0]
=> [2,5,1,3,4] => 2 = 0 + 2
[1,1,4] => [2,1] => [1,1,0,0,1,0]
=> [2,4,1,3] => 2 = 0 + 2
[1,2,2,1] => [1,2,1] => [1,0,1,1,0,0,1,0]
=> [3,1,5,2,4] => 2 = 0 + 2
[2,1,1,2] => [1,2,1] => [1,0,1,1,0,0,1,0]
=> [3,1,5,2,4] => 2 = 0 + 2
[2,2,1,1] => [2,2] => [1,1,0,0,1,1,0,0]
=> [2,4,1,5,3] => 2 = 0 + 2
[1,1,1,1,1,2] => [5,1] => [1,1,1,1,1,0,0,0,0,0,1,0]
=> [2,3,4,5,7,1,6] => ? = 0 + 2
[1,1,1,1,2,1] => [4,1,1] => [1,1,1,1,0,0,0,0,1,0,1,0]
=> [2,3,4,7,1,5,6] => ? = 0 + 2
[1,1,1,1,3] => [4,1] => [1,1,1,1,0,0,0,0,1,0]
=> [2,3,4,6,1,5] => 2 = 0 + 2
[1,1,1,2,1,1] => [3,1,2] => [1,1,1,0,0,0,1,0,1,1,0,0]
=> [2,3,6,1,4,7,5] => ? = 0 + 2
[1,1,1,2,2] => [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> [2,3,5,1,6,4] => 2 = 0 + 2
[1,1,1,3,1] => [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> [2,3,6,1,4,5] => 2 = 0 + 2
[1,1,1,4] => [3,1] => [1,1,1,0,0,0,1,0]
=> [2,3,5,1,4] => 2 = 0 + 2
[1,1,2,1,1,1] => [2,1,3] => [1,1,0,0,1,0,1,1,1,0,0,0]
=> [2,5,1,3,6,7,4] => ? = 0 + 2
[1,1,2,1,2] => [2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
=> [2,6,1,3,4,5] => 2 = 0 + 2
[1,1,2,2,1] => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> [2,4,1,6,3,5] => 3 = 1 + 2
[1,1,2,3] => [2,1,1] => [1,1,0,0,1,0,1,0]
=> [2,5,1,3,4] => 2 = 0 + 2
[1,1,3,1,1] => [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
=> [2,5,1,3,6,4] => 2 = 0 + 2
[1,1,3,2] => [2,1,1] => [1,1,0,0,1,0,1,0]
=> [2,5,1,3,4] => 2 = 0 + 2
[1,1,4,1] => [2,1,1] => [1,1,0,0,1,0,1,0]
=> [2,5,1,3,4] => 2 = 0 + 2
[1,1,5] => [2,1] => [1,1,0,0,1,0]
=> [2,4,1,3] => 2 = 0 + 2
[1,2,1,1,2] => [1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
=> [4,1,2,6,3,5] => 2 = 0 + 2
[1,2,2,1,1] => [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
=> [3,1,5,2,6,4] => 2 = 0 + 2
[2,1,1,1,2] => [1,3,1] => [1,0,1,1,1,0,0,0,1,0]
=> [3,1,4,6,2,5] => 2 = 0 + 2
[2,1,1,2,1] => [1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
=> [3,1,6,2,4,5] => 2 = 0 + 2
[2,1,1,3] => [1,2,1] => [1,0,1,1,0,0,1,0]
=> [3,1,5,2,4] => 2 = 0 + 2
[2,2,1,1,1] => [2,3] => [1,1,0,0,1,1,1,0,0,0]
=> [2,4,1,5,6,3] => 2 = 0 + 2
[2,2,1,2] => [2,1,1] => [1,1,0,0,1,0,1,0]
=> [2,5,1,3,4] => 2 = 0 + 2
[2,2,2,1] => [3,1] => [1,1,1,0,0,0,1,0]
=> [2,3,5,1,4] => 2 = 0 + 2
[2,2,3] => [2,1] => [1,1,0,0,1,0]
=> [2,4,1,3] => 2 = 0 + 2
[3,1,1,2] => [1,2,1] => [1,0,1,1,0,0,1,0]
=> [3,1,5,2,4] => 2 = 0 + 2
[3,3,1] => [2,1] => [1,1,0,0,1,0]
=> [2,4,1,3] => 2 = 0 + 2
[1,1,1,1,1,1,2] => [6,1] => [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> [2,3,4,5,6,8,1,7] => ? = 0 + 2
[1,1,1,1,1,2,1] => [5,1,1] => [1,1,1,1,1,0,0,0,0,0,1,0,1,0]
=> [2,3,4,5,8,1,6,7] => ? = 0 + 2
[1,1,1,1,1,3] => [5,1] => [1,1,1,1,1,0,0,0,0,0,1,0]
=> [2,3,4,5,7,1,6] => ? = 0 + 2
[1,1,1,1,2,1,1] => [4,1,2] => [1,1,1,1,0,0,0,0,1,0,1,1,0,0]
=> [2,3,4,7,1,5,8,6] => ? = 0 + 2
[1,1,1,1,2,2] => [4,2] => [1,1,1,1,0,0,0,0,1,1,0,0]
=> [2,3,4,6,1,7,5] => ? = 0 + 2
[1,1,1,1,3,1] => [4,1,1] => [1,1,1,1,0,0,0,0,1,0,1,0]
=> [2,3,4,7,1,5,6] => ? = 0 + 2
[1,1,1,1,4] => [4,1] => [1,1,1,1,0,0,0,0,1,0]
=> [2,3,4,6,1,5] => 2 = 0 + 2
[1,1,1,2,1,1,1] => [3,1,3] => [1,1,1,0,0,0,1,0,1,1,1,0,0,0]
=> [2,3,6,1,4,7,8,5] => ? = 0 + 2
[1,1,1,2,1,2] => [3,1,1,1] => [1,1,1,0,0,0,1,0,1,0,1,0]
=> [2,3,7,1,4,5,6] => ? = 0 + 2
[1,1,1,2,2,1] => [3,2,1] => [1,1,1,0,0,0,1,1,0,0,1,0]
=> [2,3,5,1,7,4,6] => ? = 0 + 2
[1,1,1,2,3] => [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> [2,3,6,1,4,5] => 2 = 0 + 2
[1,1,1,3,1,1] => [3,1,2] => [1,1,1,0,0,0,1,0,1,1,0,0]
=> [2,3,6,1,4,7,5] => ? = 0 + 2
[1,1,1,3,2] => [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> [2,3,6,1,4,5] => 2 = 0 + 2
[1,1,1,4,1] => [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> [2,3,6,1,4,5] => 2 = 0 + 2
[1,1,1,5] => [3,1] => [1,1,1,0,0,0,1,0]
=> [2,3,5,1,4] => 2 = 0 + 2
[1,1,2,1,1,1,1] => [2,1,4] => [1,1,0,0,1,0,1,1,1,1,0,0,0,0]
=> [2,5,1,3,6,7,8,4] => ? = 0 + 2
[1,1,2,1,1,2] => [2,1,2,1] => [1,1,0,0,1,0,1,1,0,0,1,0]
=> [2,5,1,3,7,4,6] => ? = 2 + 2
[1,1,2,1,2,1] => [2,1,1,1,1] => [1,1,0,0,1,0,1,0,1,0,1,0]
=> [2,7,1,3,4,5,6] => ? = 0 + 2
[1,1,2,1,3] => [2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
=> [2,6,1,3,4,5] => 2 = 0 + 2
[1,1,2,2,1,1] => [2,2,2] => [1,1,0,0,1,1,0,0,1,1,0,0]
=> [2,4,1,6,3,7,5] => ? = 1 + 2
[1,1,2,2,2] => [2,3] => [1,1,0,0,1,1,1,0,0,0]
=> [2,4,1,5,6,3] => 2 = 0 + 2
[1,1,2,3,1] => [2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
=> [2,6,1,3,4,5] => 2 = 0 + 2
[1,1,2,4] => [2,1,1] => [1,1,0,0,1,0,1,0]
=> [2,5,1,3,4] => 2 = 0 + 2
[1,1,3,1,1,1] => [2,1,3] => [1,1,0,0,1,0,1,1,1,0,0,0]
=> [2,5,1,3,6,7,4] => ? = 0 + 2
[1,1,3,1,2] => [2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
=> [2,6,1,3,4,5] => 2 = 0 + 2
[1,1,3,2,1] => [2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
=> [2,6,1,3,4,5] => 2 = 0 + 2
[1,1,3,3] => [2,2] => [1,1,0,0,1,1,0,0]
=> [2,4,1,5,3] => 2 = 0 + 2
[1,1,4,1,1] => [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
=> [2,5,1,3,6,4] => 2 = 0 + 2
[1,2,1,1,1,2] => [1,1,3,1] => [1,0,1,0,1,1,1,0,0,0,1,0]
=> [4,1,2,5,7,3,6] => ? = 0 + 2
[1,2,1,1,2,1] => [1,1,2,1,1] => [1,0,1,0,1,1,0,0,1,0,1,0]
=> [4,1,2,7,3,5,6] => ? = 0 + 2
[1,2,2,1,1,1] => [1,2,3] => [1,0,1,1,0,0,1,1,1,0,0,0]
=> [3,1,5,2,6,7,4] => ? = 0 + 2
[2,1,1,1,1,2] => [1,4,1] => [1,0,1,1,1,1,0,0,0,0,1,0]
=> [3,1,4,5,7,2,6] => ? = 0 + 2
[2,1,1,1,2,1] => [1,3,1,1] => [1,0,1,1,1,0,0,0,1,0,1,0]
=> [3,1,4,7,2,5,6] => ? = 0 + 2
[2,1,1,2,1,1] => [1,2,1,2] => [1,0,1,1,0,0,1,0,1,1,0,0]
=> [3,1,6,2,4,7,5] => ? = 0 + 2
[2,2,1,1,1,1] => [2,4] => [1,1,0,0,1,1,1,1,0,0,0,0]
=> [2,4,1,5,6,7,3] => ? = 0 + 2
[1,1,1,1,1,1,1,2] => [7,1] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> [2,3,4,5,6,7,9,1,8] => ? = 0 + 2
[1,1,1,1,1,1,2,1] => [6,1,1] => [1,1,1,1,1,1,0,0,0,0,0,0,1,0,1,0]
=> [2,3,4,5,6,9,1,7,8] => ? = 0 + 2
[1,1,1,1,1,1,3] => [6,1] => [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> [2,3,4,5,6,8,1,7] => ? = 0 + 2
[1,1,1,1,1,2,2] => [5,2] => [1,1,1,1,1,0,0,0,0,0,1,1,0,0]
=> [2,3,4,5,7,1,8,6] => ? = 0 + 2
[1,1,1,1,1,3,1] => [5,1,1] => [1,1,1,1,1,0,0,0,0,0,1,0,1,0]
=> [2,3,4,5,8,1,6,7] => ? = 0 + 2
[1,1,1,1,1,4] => [5,1] => [1,1,1,1,1,0,0,0,0,0,1,0]
=> [2,3,4,5,7,1,6] => ? = 0 + 2
[1,1,1,1,2,1,1,1] => [4,1,3] => [1,1,1,1,0,0,0,0,1,0,1,1,1,0,0,0]
=> [2,3,4,7,1,5,8,9,6] => ? = 0 + 2
[1,1,1,1,2,1,2] => [4,1,1,1] => [1,1,1,1,0,0,0,0,1,0,1,0,1,0]
=> [2,3,4,8,1,5,6,7] => ? = 0 + 2
[1,1,1,1,2,2,1] => [4,2,1] => [1,1,1,1,0,0,0,0,1,1,0,0,1,0]
=> [2,3,4,6,1,8,5,7] => ? = 1 + 2
[1,1,1,1,2,3] => [4,1,1] => [1,1,1,1,0,0,0,0,1,0,1,0]
=> [2,3,4,7,1,5,6] => ? = 0 + 2
[1,1,1,1,3,1,1] => [4,1,2] => [1,1,1,1,0,0,0,0,1,0,1,1,0,0]
=> [2,3,4,7,1,5,8,6] => ? = 0 + 2
[1,1,1,1,3,2] => [4,1,1] => [1,1,1,1,0,0,0,0,1,0,1,0]
=> [2,3,4,7,1,5,6] => ? = 0 + 2
[1,1,1,1,4,1] => [4,1,1] => [1,1,1,1,0,0,0,0,1,0,1,0]
=> [2,3,4,7,1,5,6] => ? = 0 + 2
[1,1,1,2,1,1,2] => [3,1,2,1] => [1,1,1,0,0,0,1,0,1,1,0,0,1,0]
=> [2,3,6,1,4,8,5,7] => ? = 0 + 2
[1,1,1,2,1,2,1] => [3,1,1,1,1] => [1,1,1,0,0,0,1,0,1,0,1,0,1,0]
=> [2,3,8,1,4,5,6,7] => ? = 0 + 2
[1,1,1,2,1,3] => [3,1,1,1] => [1,1,1,0,0,0,1,0,1,0,1,0]
=> [2,3,7,1,4,5,6] => ? = 0 + 2
[1,1,1,2,2,1,1] => [3,2,2] => [1,1,1,0,0,0,1,1,0,0,1,1,0,0]
=> [2,3,5,1,7,4,8,6] => ? = 0 + 2
[1,1,1,2,2,2] => [3,3] => [1,1,1,0,0,0,1,1,1,0,0,0]
=> [2,3,5,1,6,7,4] => ? = 0 + 2
[1,1,1,2,3,1] => [3,1,1,1] => [1,1,1,0,0,0,1,0,1,0,1,0]
=> [2,3,7,1,4,5,6] => ? = 0 + 2
[1,1,1,3,1,1,1] => [3,1,3] => [1,1,1,0,0,0,1,0,1,1,1,0,0,0]
=> [2,3,6,1,4,7,8,5] => ? = 0 + 2
[1,1,1,3,1,2] => [3,1,1,1] => [1,1,1,0,0,0,1,0,1,0,1,0]
=> [2,3,7,1,4,5,6] => ? = 0 + 2
[1,1,1,3,2,1] => [3,1,1,1] => [1,1,1,0,0,0,1,0,1,0,1,0]
=> [2,3,7,1,4,5,6] => ? = 0 + 2
[1,1,1,4,1,1] => [3,1,2] => [1,1,1,0,0,0,1,0,1,1,0,0]
=> [2,3,6,1,4,7,5] => ? = 0 + 2
[1,1,2,1,1,1,2] => [2,1,3,1] => [1,1,0,0,1,0,1,1,1,0,0,0,1,0]
=> [2,5,1,3,6,8,4,7] => ? = 0 + 2
Description
The number of recoils of a permutation.
A '''recoil''', or '''inverse descent''' of a permutation $\pi$ is a value $i$ such that $i+1$ appears to the left of $i$ in $\pi_1,\pi_2,\dots,\pi_n$.
In other words, this is the number of descents of the inverse permutation. It can be also be described as the number of occurrences of the mesh pattern $([2,1], {(0,1),(1,1),(2,1)})$, i.e., the middle row is shaded.
Matching statistic: St001330
(load all 6 compositions to match this statistic)
(load all 6 compositions to match this statistic)
Mp00133: Integer compositions —delta morphism⟶ Integer compositions
Mp00041: Integer compositions —conjugate⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St001330: Graphs ⟶ ℤResult quality: 20% ●values known / values provided: 25%●distinct values known / distinct values provided: 20%
Mp00041: Integer compositions —conjugate⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St001330: Graphs ⟶ ℤResult quality: 20% ●values known / values provided: 25%●distinct values known / distinct values provided: 20%
Values
[1,1,2] => [2,1] => [2,1] => ([(0,2),(1,2)],3)
=> 2 = 0 + 2
[1,1,1,2] => [3,1] => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ? = 0 + 2
[1,1,2,1] => [2,1,1] => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 2 = 0 + 2
[1,1,3] => [2,1] => [2,1] => ([(0,2),(1,2)],3)
=> 2 = 0 + 2
[2,2,1] => [2,1] => [2,1] => ([(0,2),(1,2)],3)
=> 2 = 0 + 2
[1,1,1,1,2] => [4,1] => [2,1,1,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 0 + 2
[1,1,1,2,1] => [3,1,1] => [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 0 + 2
[1,1,1,3] => [3,1] => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ? = 0 + 2
[1,1,2,1,1] => [2,1,2] => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 0 + 2
[1,1,2,2] => [2,2] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> ? = 0 + 2
[1,1,3,1] => [2,1,1] => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 2 = 0 + 2
[1,1,4] => [2,1] => [2,1] => ([(0,2),(1,2)],3)
=> 2 = 0 + 2
[1,2,2,1] => [1,2,1] => [2,2] => ([(1,3),(2,3)],4)
=> 2 = 0 + 2
[2,1,1,2] => [1,2,1] => [2,2] => ([(1,3),(2,3)],4)
=> 2 = 0 + 2
[2,2,1,1] => [2,2] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> ? = 0 + 2
[1,1,1,1,1,2] => [5,1] => [2,1,1,1,1] => ([(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 0 + 2
[1,1,1,1,2,1] => [4,1,1] => [3,1,1,1] => ([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 0 + 2
[1,1,1,1,3] => [4,1] => [2,1,1,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 0 + 2
[1,1,1,2,1,1] => [3,1,2] => [1,3,1,1] => ([(0,4),(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 0 + 2
[1,1,1,2,2] => [3,2] => [1,2,1,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 0 + 2
[1,1,1,3,1] => [3,1,1] => [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 0 + 2
[1,1,1,4] => [3,1] => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ? = 0 + 2
[1,1,2,1,1,1] => [2,1,3] => [1,1,3,1] => ([(0,5),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 0 + 2
[1,1,2,1,2] => [2,1,1,1] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 2 = 0 + 2
[1,1,2,2,1] => [2,2,1] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 1 + 2
[1,1,2,3] => [2,1,1] => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 2 = 0 + 2
[1,1,3,1,1] => [2,1,2] => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 0 + 2
[1,1,3,2] => [2,1,1] => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 2 = 0 + 2
[1,1,4,1] => [2,1,1] => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 2 = 0 + 2
[1,1,5] => [2,1] => [2,1] => ([(0,2),(1,2)],3)
=> 2 = 0 + 2
[1,2,1,1,2] => [1,1,2,1] => [2,3] => ([(2,4),(3,4)],5)
=> 2 = 0 + 2
[1,2,2,1,1] => [1,2,2] => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 0 + 2
[2,1,1,1,2] => [1,3,1] => [2,1,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 0 + 2
[2,1,1,2,1] => [1,2,1,1] => [3,2] => ([(1,4),(2,4),(3,4)],5)
=> 2 = 0 + 2
[2,1,1,3] => [1,2,1] => [2,2] => ([(1,3),(2,3)],4)
=> 2 = 0 + 2
[2,2,1,1,1] => [2,3] => [1,1,2,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 0 + 2
[2,2,1,2] => [2,1,1] => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 2 = 0 + 2
[2,2,2,1] => [3,1] => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ? = 0 + 2
[2,2,3] => [2,1] => [2,1] => ([(0,2),(1,2)],3)
=> 2 = 0 + 2
[3,1,1,2] => [1,2,1] => [2,2] => ([(1,3),(2,3)],4)
=> 2 = 0 + 2
[3,3,1] => [2,1] => [2,1] => ([(0,2),(1,2)],3)
=> 2 = 0 + 2
[1,1,1,1,1,1,2] => [6,1] => [2,1,1,1,1,1] => ([(0,2),(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 0 + 2
[1,1,1,1,1,2,1] => [5,1,1] => [3,1,1,1,1] => ([(0,3),(0,4),(0,5),(0,6),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 0 + 2
[1,1,1,1,1,3] => [5,1] => [2,1,1,1,1] => ([(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 0 + 2
[1,1,1,1,2,1,1] => [4,1,2] => [1,3,1,1,1] => ([(0,4),(0,5),(0,6),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 0 + 2
[1,1,1,1,2,2] => [4,2] => [1,2,1,1,1] => ([(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 0 + 2
[1,1,1,1,3,1] => [4,1,1] => [3,1,1,1] => ([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 0 + 2
[1,1,1,1,4] => [4,1] => [2,1,1,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 0 + 2
[1,1,1,2,1,1,1] => [3,1,3] => [1,1,3,1,1] => ([(0,5),(0,6),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 0 + 2
[1,1,1,2,1,2] => [3,1,1,1] => [4,1,1] => ([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 0 + 2
[1,1,1,2,2,1] => [3,2,1] => [2,2,1,1] => ([(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 0 + 2
[1,1,1,2,3] => [3,1,1] => [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 0 + 2
[1,1,1,3,1,1] => [3,1,2] => [1,3,1,1] => ([(0,4),(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 0 + 2
[1,1,1,3,2] => [3,1,1] => [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 0 + 2
[1,1,1,4,1] => [3,1,1] => [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 0 + 2
[1,1,1,5] => [3,1] => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ? = 0 + 2
[1,1,2,1,1,1,1] => [2,1,4] => [1,1,1,3,1] => ([(0,6),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 0 + 2
[1,1,2,1,1,2] => [2,1,2,1] => [2,3,1] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 2 + 2
[1,1,2,1,2,1] => [2,1,1,1,1] => [5,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> 2 = 0 + 2
[1,1,2,1,3] => [2,1,1,1] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 2 = 0 + 2
[1,1,2,2,1,1] => [2,2,2] => [1,2,2,1] => ([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 1 + 2
[1,1,2,2,2] => [2,3] => [1,1,2,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 0 + 2
[1,1,2,3,1] => [2,1,1,1] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 2 = 0 + 2
[1,1,2,4] => [2,1,1] => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 2 = 0 + 2
[1,1,3,1,1,1] => [2,1,3] => [1,1,3,1] => ([(0,5),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 0 + 2
[1,1,3,1,2] => [2,1,1,1] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 2 = 0 + 2
[1,1,3,2,1] => [2,1,1,1] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 2 = 0 + 2
[1,1,3,3] => [2,2] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> ? = 0 + 2
[1,1,4,1,1] => [2,1,2] => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 0 + 2
[1,1,4,2] => [2,1,1] => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 2 = 0 + 2
[1,1,5,1] => [2,1,1] => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 2 = 0 + 2
[1,1,6] => [2,1] => [2,1] => ([(0,2),(1,2)],3)
=> 2 = 0 + 2
[1,2,1,1,1,2] => [1,1,3,1] => [2,1,3] => ([(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 0 + 2
[1,2,1,1,2,1] => [1,1,2,1,1] => [3,3] => ([(2,5),(3,5),(4,5)],6)
=> 2 = 0 + 2
[1,2,1,1,3] => [1,1,2,1] => [2,3] => ([(2,4),(3,4)],5)
=> 2 = 0 + 2
[1,2,2,1,1,1] => [1,2,3] => [1,1,2,2] => ([(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 0 + 2
[1,2,2,1,2] => [1,2,1,1] => [3,2] => ([(1,4),(2,4),(3,4)],5)
=> 2 = 0 + 2
[1,2,2,2,1] => [1,3,1] => [2,1,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 0 + 2
[1,2,2,3] => [1,2,1] => [2,2] => ([(1,3),(2,3)],4)
=> 2 = 0 + 2
[1,3,1,1,2] => [1,1,2,1] => [2,3] => ([(2,4),(3,4)],5)
=> 2 = 0 + 2
[1,3,3,1] => [1,2,1] => [2,2] => ([(1,3),(2,3)],4)
=> 2 = 0 + 2
[2,1,1,1,1,2] => [1,4,1] => [2,1,1,2] => ([(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 0 + 2
[2,1,1,1,2,1] => [1,3,1,1] => [3,1,2] => ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 0 + 2
[2,1,1,1,3] => [1,3,1] => [2,1,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 0 + 2
[2,1,1,2,1,1] => [1,2,1,2] => [1,3,2] => ([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 0 + 2
[2,1,1,3,1] => [1,2,1,1] => [3,2] => ([(1,4),(2,4),(3,4)],5)
=> 2 = 0 + 2
[2,1,1,4] => [1,2,1] => [2,2] => ([(1,3),(2,3)],4)
=> 2 = 0 + 2
[2,1,2,2,1] => [1,1,2,1] => [2,3] => ([(2,4),(3,4)],5)
=> 2 = 0 + 2
[2,2,1,2,1] => [2,1,1,1] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 2 = 0 + 2
[2,2,1,3] => [2,1,1] => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 2 = 0 + 2
[2,2,3,1] => [2,1,1] => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 2 = 0 + 2
[2,2,4] => [2,1] => [2,1] => ([(0,2),(1,2)],3)
=> 2 = 0 + 2
[3,1,1,2,1] => [1,2,1,1] => [3,2] => ([(1,4),(2,4),(3,4)],5)
=> 2 = 0 + 2
[3,1,1,3] => [1,2,1] => [2,2] => ([(1,3),(2,3)],4)
=> 2 = 0 + 2
[3,2,2,1] => [1,2,1] => [2,2] => ([(1,3),(2,3)],4)
=> 2 = 0 + 2
[3,3,2] => [2,1] => [2,1] => ([(0,2),(1,2)],3)
=> 2 = 0 + 2
[4,1,1,2] => [1,2,1] => [2,2] => ([(1,3),(2,3)],4)
=> 2 = 0 + 2
[1,1,2,1,3,1] => [2,1,1,1,1] => [5,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> 2 = 0 + 2
[1,1,2,1,4] => [2,1,1,1] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 2 = 0 + 2
[1,1,2,3,2] => [2,1,1,1] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 2 = 0 + 2
Description
The hat guessing number of a graph.
Suppose that each vertex of a graph corresponds to a player, wearing a hat whose color is arbitrarily chosen from a set of $q$ possible colors. Each player can see the hat colors of his neighbors, but not his own hat color. All of the players are asked to guess their own hat colors simultaneously, according to a predetermined guessing strategy and the hat colors they see, where no communication between them is allowed. The hat guessing number $HG(G)$ of a graph $G$ is the largest integer $q$ such that there exists a guessing strategy guaranteeing at least one correct guess for any hat assignment of $q$ possible colors.
Because it suffices that a single player guesses correctly, the hat guessing number of a graph is the maximum of the hat guessing numbers of its connected components.
The following 412 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000689The maximal n such that the minimal generator-cogenerator module in the LNakayama algebra of a Dyck path is n-rigid. St001371The length of the longest Yamanouchi prefix of a binary word. St001730The number of times the path corresponding to a binary word crosses the base line. St001803The maximal overlap of the cylindrical tableau associated with a tableau. St001195The global dimension of the algebra $A/AfA$ of the corresponding Nakayama algebra $A$ with minimal left faithful projective-injective module $Af$. St001208The number of connected components of the quiver of $A/T$ when $T$ is the 1-tilting module corresponding to the permutation in the Auslander algebra $A$ of $K[x]/(x^n)$. St001569The maximal modular displacement of a permutation. St001804The minimal height of the rectangular inner shape in a cylindrical tableau associated to a tableau. St001703The villainy of a graph. St001091The number of parts in an integer partition whose next smaller part has the same size. St000160The multiplicity of the smallest part of a partition. St000787The number of flips required to make a perfect matching noncrossing. St000788The number of nesting-similar perfect matchings of a perfect matching. St000929The constant term of the character polynomial of an integer partition. St000143The largest repeated part of a partition. St000510The number of invariant oriented cycles when acting with a permutation of given cycle type. St000706The product of the factorials of the multiplicities of an integer partition. St000897The number of different multiplicities of parts of an integer partition. St000993The multiplicity of the largest part of an integer partition. St000181The number of connected components of the Hasse diagram for the poset. St001283The number of finite solvable groups that are realised by the given partition over the complex numbers. St000306The bounce count of a Dyck path. St000119The number of occurrences of the pattern 321 in a permutation. St000123The difference in Coxeter length of a permutation and its image under the Simion-Schmidt map. St000223The number of nestings in the permutation. St000366The number of double descents of a permutation. St000371The number of mid points of decreasing subsequences of length 3 in a permutation. St000373The number of weak exceedences of a permutation that are also mid-points of a decreasing subsequence of length $3$. St000404The number of occurrences of the pattern 3241 or of the pattern 4231 in a permutation. 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. St000546The number of global descents of a permutation. St000647The number of big descents of a permutation. St000731The number of double exceedences of a permutation. St001086The number of occurrences of the consecutive pattern 132 in a permutation. St001394The genus of a permutation. St000035The number of left outer peaks of a permutation. St000884The number of isolated descents of a permutation. St001090The number of pop-stack-sorts needed to sort a permutation. St000451The length of the longest pattern of the form k 1 2. St000842The breadth of a permutation. St001292The injective dimension of the tensor product of two copies of the dual of the Nakayama algebra associated to a Dyck path. St000862The number of parts of the shifted shape of a permutation. St000022The number of fixed points of a permutation. St000153The number of adjacent cycles of a permutation. St001465The number of adjacent transpositions in the cycle decomposition of a permutation. St000359The number of occurrences of the pattern 23-1. St000028The number of stack-sorts needed to sort a permutation. St000007The number of saliances of the permutation. St000218The number of occurrences of the pattern 213 in a permutation. St000220The number of occurrences of the pattern 132 in a permutation. St000356The number of occurrences of the pattern 13-2. St000405The number of occurrences of the pattern 1324 in a permutation. St000534The number of 2-rises of a permutation. St000871The number of very big ascents of a permutation. St001083The number of boxed occurrences of 132 in a permutation. St001085The number of occurrences of the vincular pattern |21-3 in a permutation. St000651The maximal size of a rise in a permutation. St000742The number of big ascents of a permutation after prepending zero. St000463The number of admissible inversions of a permutation. St001084The number of occurrences of the vincular pattern |1-23 in a permutation. St000483The number of times a permutation switches from increasing to decreasing or decreasing to increasing. St000996The number of exclusive left-to-right maxima of a permutation. St001052The length of the exterior of a permutation. St001096The size of the overlap set of a permutation. St000097The order of the largest clique of the graph. St000441The number of successions of a permutation. St000665The number of rafts of a permutation. St001568The smallest positive integer that does not appear twice in the partition. St000214The number of adjacencies of a permutation. St000215The number of adjacencies of a permutation, zero appended. St000221The number of strong fixed points of a permutation. St000279The size of the preimage of the map 'cycle-as-one-line notation' from Permutations to Permutations. St000375The number of non weak exceedences of a permutation that are mid-points of a decreasing subsequence of length $3$. St000623The number of occurrences of the pattern 52341 in a permutation. St001204Call a CNakayama algebra (a Nakayama algebra with a cyclic quiver) with Kupisch series $L=[c_0,c_1,...,c_{n−1}]$ such that $n=c_0 < c_i$ for all $i > 0$ a special CNakayama algebra. St001314The number of tilting modules of arbitrary projective dimension that have no simple modules as a direct summand in the corresponding Nakayama algebra. St001381The fertility of a permutation. St001444The rank of the skew-symmetric form which is non-zero on crossing arcs of a perfect matching. St001466The number of transpositions swapping cyclically adjacent numbers in a permutation. St001549The number of restricted non-inversions between exceedances. St001551The number of restricted non-inversions between exceedances where the rightmost exceedance is linked. St001552The number of inversions between excedances and fixed points of a permutation. St001663The number of occurrences of the Hertzsprung pattern 132 in a permutation. St001810The number of fixed points of a permutation smaller than its largest moved point. St001811The Castelnuovo-Mumford regularity of a permutation. St001831The multiplicity of the non-nesting perfect matching in the chord expansion of a perfect matching. St001837The number of occurrences of a 312 pattern in the restricted growth word of a perfect matching. St001850The number of Hecke atoms of a permutation. St000056The decomposition (or block) number of a permutation. St000486The number of cycles of length at least 3 of a permutation. St000694The number of affine bounded permutations that project to a given permutation. St001174The Gorenstein dimension of the algebra $A/I$ when $I$ is the tilting module corresponding to the permutation in the Auslander algebra of $K[x]/(x^n)$. St001256Number of simple reflexive modules that are 2-stable reflexive. St001431Half of the Loewy length minus one of a modified stable Auslander algebra of the Nakayama algebra corresponding to the Dyck path. St001461The number of topologically connected components of the chord diagram of a permutation. St001590The crossing number of a perfect matching. St001830The chord expansion number of a perfect matching. St001832The number of non-crossing perfect matchings in the chord expansion of a perfect matching. St001859The number of factors of the Stanley symmetric function associated with a permutation. St001359The number of permutations in the equivalence class of a permutation obtained by taking inverses of cycles. 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 $eA$. St001290The first natural number n such that the tensor product of n copies of D(A) is zero for the corresponding Nakayama algebra A. St000648The number of 2-excedences of a permutation. St000662The staircase size of the code of a permutation. St000374The number of exclusive right-to-left minima of a permutation. St000478Another weight of a partition according to Alladi. St001890The maximum magnitude of the Möbius function of a poset. St001786The number of total orderings of the north steps of a Dyck path such that steps after the k-th east step are not among the first k positions in the order. St000011The number of touch points (or returns) of a Dyck path. St000036The evaluation at 1 of the Kazhdan-Lusztig polynomial with parameters given by the identity and the permutation. St000447The number of pairs of vertices of a graph with distance 3. St000449The number of pairs of vertices of a graph with distance 4. St001934The number of monotone factorisations of genus zero of a permutation of given cycle type. St000402Half the size of the symmetry class of a permutation. St001087The number of occurrences of the vincular pattern |12-3 in a permutation. St001305The number of induced cycles on four vertices in a graph. St001306The number of induced paths on four vertices in a graph. St001324The minimal number of occurrences of the chordal-pattern in a linear ordering of the vertices of the graph. St001325The minimal number of occurrences of the comparability-pattern in a linear ordering of the vertices of the graph. St001326The minimal number of occurrences of the interval-pattern in a linear ordering of the vertices of the graph. St001327The minimal number of occurrences of the split-pattern in a linear ordering of the vertices of the graph. St001353The number of prime nodes in the modular decomposition of a graph. St001356The number of vertices in prime modules of a graph. St001272The number of graphs with the same degree sequence. St001333The cardinality of a minimal edge-isolating set of a graph. St001393The induced matching number of a graph. St001496The number of graphs with the same Laplacian spectrum as the given graph. St001261The Castelnuovo-Mumford regularity of a graph. St000124The cardinality of the preimage of the Simion-Schmidt map. St000098The chromatic number of a graph. St001122The multiplicity of the sign representation in the Kronecker square corresponding to a partition. St001387Number of standard Young tableaux of the skew shape tracing the border of the given partition. St001780The order of promotion on the set of standard tableaux of given shape. St001490The number of connected components of a skew partition. St000058The order of a permutation. St000527The width of the poset. St000141The maximum drop size of a permutation. St000276The size of the preimage of the map 'to graph' from Ordered trees to Graphs. St000687The dimension of $Hom(I,P)$ for the LNakayama algebra of a Dyck path. St001025Number of simple modules with projective dimension 4 in the Nakayama algebra corresponding to the Dyck path. St001107The number of times one can erase the first up and the last down step in a Dyck path and still remain a Dyck path. St001577The minimal number of edges to add or remove to make a graph a cograph. St000048The multinomial of the parts of a partition. St000386The number of factors DDU in a Dyck path. St000655The length of the minimal rise of a Dyck path. St001063Numbers of 3-torsionfree simple modules in the corresponding Nakayama algebra. St001064Number of simple modules in the corresponding Nakayama algebra that are 3-syzygy modules. St001066The number of simple reflexive modules in the corresponding Nakayama algebra. St001518The number of graphs with the same ordinary spectrum as the given graph. St000322The skewness of a graph. St001631The number of simple modules $S$ with $dim Ext^1(S,A)=1$ in the incidence algebra $A$ of the poset. St001633The number of simple modules with projective dimension two in the incidence algebra of the poset. St000183The side length of the Durfee square of an integer partition. St000257The number of distinct parts of a partition that occur at least twice. St000298The order dimension or Dushnik-Miller dimension of a poset. St001734The lettericity of a graph. St000274The number of perfect matchings of a graph. St001783The number of odd automorphisms of a graph. St001871The number of triconnected components of a graph. St001908The number of semistandard tableaux of distinct weight whose maximal entry is the length of the partition. St000920The logarithmic height of a Dyck path. St000013The height of a Dyck path. St000212The number of standard Young tableaux for an integer partition such that no two consecutive entries appear in the same row. St000475The number of parts equal to 1 in a partition. St000481The number of upper covers of a partition in dominance order. St000506The number of standard desarrangement tableaux of shape equal to the given partition. St000513The number of invariant subsets of size 2 when acting with a permutation of given cycle type. St000567The sum of the products of all pairs of parts. 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. St000936The number of even values of the symmetric group character corresponding to the partition. St000938The number of zeros of the symmetric group character corresponding to the partition. St001097The coefficient of the monomial symmetric function indexed by the partition in the formal group law for linear orders. St001098The 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 vertex labelled trees. 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. 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. 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. St001123The multiplicity of the dual of the standard representation in the Kronecker square corresponding to a partition. St000003The number of standard Young tableaux of the partition. St000159The number of distinct parts of the integer partition. St000278The size of the preimage of the map 'to partition' from Integer compositions to Integer partitions. St000284The Plancherel distribution on integer partitions. St000346The number of coarsenings of a partition. St000473The number of parts of a partition that are strictly bigger than the number of ones. St000480The number of lower covers of a partition in dominance order. St000533The minimum of the number of parts and the size of the first part of an integer partition. St000548The number of different non-empty partial sums of an integer partition. St000618The number of self-evacuating tableaux of given shape. St000704The number of semistandard tableaux on a given integer partition with minimal maximal entry. St000759The smallest missing part in an integer partition. St000810The sum of the entries in the column specified by the partition of the change of basis matrix from powersum symmetric functions to monomial symmetric functions. St000813The number of zero-one matrices with weakly decreasing column sums and row sums given by the partition. St000814The sum of the entries in the column specified by the partition of the change of basis matrix from elementary symmetric functions to Schur symmetric functions. St000901The cube of the number of standard Young tableaux with shape given by the partition. St001121The multiplicity of the irreducible representation indexed by the partition in the Kronecker square corresponding to the partition. St001128The exponens consonantiae of a partition. St001432The order dimension of the partition. St000511The number of invariant subsets when acting with a permutation of given cycle type. St000002The number of occurrences of the pattern 123 in a permutation. St000210Minimum over maximum difference of elements in cycles. St000281The size of the preimage of the map 'to poset' from Binary trees to Posets. St000282The size of the preimage of the map 'to poset' from Ordered trees to Posets. St000358The number of occurrences of the pattern 31-2. St000360The number of occurrences of the pattern 32-1. St000367The number of simsun double descents of a permutation. St000406The number of occurrences of the pattern 3241 in a permutation. St000432The number of occurrences of the pattern 231 or of the pattern 312 in a permutation. St000436The number of occurrences of the pattern 231 or of the pattern 321 in a permutation. St000566The number of ways to select a row of a Ferrers shape and two cells in this row. St000621The number of standard tableaux of shape equal to the given partition such that the minimal cyclic descent is even. St000622The number of occurrences of the patterns 2143 or 4231 in a permutation. St000666The number of right tethers of a permutation. St000710The number of big deficiencies of a permutation. St000713The dimension of the irreducible representation of Sp(4) labelled by an integer partition. St000714The number of semistandard Young tableau of given shape, with entries at most 2. St000732The number of double deficiencies of a permutation. St000750The number of occurrences of the pattern 4213 in a permutation. St000752The Grundy value for the game 'Couples are forever' on an integer partition. St000779The tier of a permutation. St000800The number of occurrences of the vincular pattern |231 in a permutation. St000801The number of occurrences of the vincular pattern |312 in a permutation. St000802The number of occurrences of the vincular pattern |321 in a permutation. St000872The number of very big descents of a permutation. St000934The 2-degree of an integer partition. St000940The number of characters of the symmetric group whose value on the partition is zero. St000962The 3-shifted major index of a permutation. St000963The 2-shifted major index of a permutation. St000995The largest even part of an integer partition. St001095The number of non-isomorphic posets with precisely one further covering relation. St001301The first Betti number of the order complex associated with the poset. St001411The number of patterns 321 or 3412 in a permutation. St001513The number of nested exceedences of a permutation. St001534The alternating sum of the coefficients of the Poincare polynomial of the poset cone. St001537The number of cyclic crossings of a permutation. St001550The number of inversions between exceedances where the greater exceedance is linked. St001559The number of transpositions that are smaller or equal to a permutation in Bruhat order while not being inversions. St001715The number of non-records in a permutation. St001728The number of invisible descents of a permutation. St001744The number of occurrences of the arrow pattern 1-2 with an arrow from 1 to 2 in a permutation. St001847The number of occurrences of the pattern 1432 in a permutation. St001906Half of the difference between the total displacement and the number of inversions and the reflection length of a permutation. St000345The number of refinements of a partition. St000487The length of the shortest cycle of a permutation. St000620The number of standard tableaux of shape equal to the given partition such that the minimal cyclic descent is odd. St000667The greatest common divisor of the parts of the partition. St000668The least common multiple of the parts of the partition. St000707The product of the factorials of the parts. St000708The product of the parts of an integer partition. St000755The number of real roots of the characteristic polynomial of a linear recurrence associated with an integer partition. 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. St000886The number of permutations with the same antidiagonal sums. St000913The number of ways to refine the partition into singletons. St000933The number of multipartitions of sizes given by an integer partition. St001162The minimum jump of a permutation. St001344The neighbouring number of a permutation. St001761The maximal multiplicity of a letter in a reduced word of a permutation. St000485The length of the longest cycle of a permutation. St000642The size of the smallest orbit of antichains under Panyushev complementation. St001741The largest integer such that all patterns of this size are contained in the permutation. St000437The number of occurrences of the pattern 312 or of the pattern 321 in a permutation. St000649The number of 3-excedences of a permutation. St000709The number of occurrences of 14-2-3 or 14-3-2. St000711The number of big exceedences of a permutation. St001727The number of invisible inversions of a permutation. St000042The number of crossings of a perfect matching. St000234The number of global ascents of a permutation. St000296The length of the symmetric border of a binary word. St000379The number of Hamiltonian cycles in a graph. St000629The defect of a binary word. St000877The depth of the binary word interpreted as a path. St000878The number of ones minus the number of zeros of a binary word. St000885The number of critical steps in the Catalan decomposition of a binary word. St001047The maximal number of arcs crossing a given arc of a perfect matching. St000069The number of maximal elements of a poset. St000260The radius of a connected graph. St000326The position of the first one in a binary word after appending a 1 at the end. St000352The Elizalde-Pak rank of a permutation. St000627The exponent of a binary word. St000660The number of rises of length at least 3 of a Dyck path. St000876The number of factors in the Catalan decomposition of a binary word. St000259The diameter of a connected graph. St000733The row containing the largest entry of a standard tableau. St000845The maximal number of elements covered by an element in a poset. St001367The smallest number which does not occur as degree of a vertex in a graph. St001479The number of bridges of a graph. St001593This is the number of standard Young tableaux of the given shifted shape. St001785The number of ways to obtain a partition as the multiset of antidiagonal lengths of the Ferrers diagram of a partition. St000255The number of reduced Kogan faces with the permutation as type. St001340The cardinality of a minimal non-edge isolating set of a graph. St001354The number of series nodes in the modular decomposition of a graph. St000961The shifted major index of a permutation. St000052The number of valleys of a Dyck path not on the x-axis. St001139The number of occurrences of hills of size 2 in a Dyck path. St000632The jump number of the poset. St000703The number of deficiencies of a permutation. St000745The index of the last row whose first entry is the row number in a standard Young tableau. St000907The number of maximal antichains of minimal length in a poset. St000994The number of cycle peaks and the number of cycle valleys of a permutation. St001571The Cartan determinant of the integer partition. St000846The maximal number of elements covering an element of a poset. St001141The number of occurrences of hills of size 3 in a Dyck path. St001396Number of triples of incomparable elements in a finite poset. St001695The natural comajor index of a standard Young tableau. St001698The comajor index of a standard tableau minus the weighted size of its shape. St001699The major index of a standard tableau minus the weighted size of its shape. St001712The number of natural descents of a standard Young tableau. St001877Number of indecomposable injective modules with projective dimension 2. St000396The register function (or Horton-Strahler number) of a binary tree. St001630The global dimension of the incidence algebra of the lattice over the rational numbers. St000891The number of distinct diagonal sums of a permutation matrix. St000353The number of inner valleys of a permutation. St000355The number of occurrences of the pattern 21-3. St000407The number of occurrences of the pattern 2143 in a permutation. St000434The number of occurrences of the pattern 213 or of the pattern 312 in a permutation. St000461The rix statistic of a permutation. St000516The number of stretching pairs of a permutation. St000726The normalized sum of the leaf labels of the increasing binary tree associated to a permutation. St000751The number of occurrences of either of the pattern 2143 or 2143 in a permutation. St000799The number of occurrences of the vincular pattern |213 in a permutation. St000866The number of admissible inversions of a permutation in the sense of Shareshian-Wachs. St000989The number of final rises of a permutation. St001284The number of finite groups that are realised by the given partition over the complex numbers. St001687The number of distinct positions of the pattern letter 2 in occurrences of 213 in a permutation. St001745The number of occurrences of the arrow pattern 13 with an arrow from 1 to 2 in a permutation. St000990The first ascent of a permutation. St001665The number of pure excedances of a permutation. St001729The number of visible descents of a permutation. St001737The number of descents of type 2 in a permutation. St001928The number of non-overlapping descents in a permutation. St000470The number of runs in a permutation. St000619The number of cyclic descents of a permutation. St001735The number of permutations with the same set of runs. St000699The toughness times the least common multiple of 1,. St000512The number of invariant subsets of size 3 when acting with a permutation of given cycle type. St001061The number of indices that are both descents and recoils of a permutation. St001498The normalised height of a Nakayama algebra with magnitude 1. St001640The number of ascent tops in the permutation such that all smaller elements appear before. St000078The number of alternating sign matrices whose left key is the permutation. St000701The protection number of a binary tree. St001199The dominant dimension of $eAe$ for the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St000777The number of distinct eigenvalues of the distance Laplacian of a connected graph. St000185The weighted size of a partition. St000225Difference between largest and smallest parts in a partition. St000312The number of leaves in a graph. St001124The multiplicity of the standard representation in the Kronecker square corresponding to a partition. St001214The aft of an integer partition. St001307The number of induced stars on four vertices in a graph. St001308The number of induced paths on three vertices in a graph. St001310The number of induced diamond graphs in a graph. St001323The independence gap of a graph. St001350Half of the Albertson index of a graph. St001351The Albertson index of a graph. St001374The Padmakar-Ivan index of a graph. St001440The number of standard Young tableaux whose major index is congruent one modulo the size of a given integer partition. St000049The number of set partitions whose sorted block sizes correspond to the partition. St000146The Andrews-Garvan crank of a partition. St000275Number of permutations whose sorted list of non zero multiplicities of the Lehmer code is the given partition. St000781The number of proper colouring schemes of a Ferrers diagram. St000783The side length of the largest staircase partition fitting into a partition. St001282The number of graphs with the same chromatic polynomial. St001349The number of different graphs obtained from the given graph by removing an edge. St001442The number of standard Young tableaux whose major index is divisible by the size of a given integer partition. St000318The number of addable cells of the Ferrers diagram of an integer partition. St000283The size of the preimage of the map 'to graph' from Binary trees to Graphs. St000319The spin of an integer partition. St000320The dinv adjustment of an integer partition. St000552The number of cut vertices of a graph. St000944The 3-degree of an integer partition. St001398Number of subsets of size 3 of elements in a poset that form a "v". St001793The difference between the clique number and the chromatic number of a graph. St001826The maximal number of leaves on a vertex of a graph. St001957The number of Hasse diagrams with a given underlying undirected graph. St000273The domination number of a graph. St000321The number of integer partitions of n that are dominated by an integer partition. St000544The cop number of a graph. St000553The number of blocks of a graph. St000640The rank of the largest boolean interval in a poset. St000775The multiplicity of the largest eigenvalue in a graph. St000785The number of distinct colouring schemes of a graph. St000908The length of the shortest maximal antichain in a poset. St000914The sum of the values of the Möbius function of a poset. St000916The packing number of a graph. St000917The open packing number of a graph. St001385The number of conjugacy classes of subgroups with connected subgroups of sizes prescribed by an integer partition. St001469The holeyness of a permutation. St001562The value of the complete homogeneous symmetric function evaluated at 1. St001563The value of the power-sum symmetric function evaluated at 1. St001564The value of the forgotten symmetric functions when all variables set to 1. St001634The trace of the Coxeter matrix of the incidence algebra of a poset. St001672The restrained domination number of a graph. St001739The number of graphs with the same edge polytope as the given graph. St001740The number of graphs with the same symmetric edge polytope as the given graph. St001776The degree of the minimal polynomial of the largest Laplacian eigenvalue of a graph. St001829The common independence number of a graph. St001942The number of loops of the quiver corresponding to the reduced incidence algebra of a poset. St000258The burning number of a graph. St000918The 2-limited packing number of a graph. St000264The girth of a graph, which is not a tree. St001536The number of cyclic misalignments of a permutation. St000782The indicator function of whether a given perfect matching is an L & P matching. St001722The number of minimal chains with small intervals between a binary word and the top element. St001816Eigenvalues of the top-to-random operator acting on a simple module. St000043The number of crossings plus two-nestings of a perfect matching. St001207The Lowey length of the algebra $A/T$ when $T$ is the 1-tilting module corresponding to the permutation in the Auslander algebra of $K[x]/(x^n)$. St001582The grades of the simple modules corresponding to the points in the poset of the symmetric group under the Bruhat order. St000075The orbit size of a standard tableau under promotion. St000793The length of the longest partition in the vacillating tableau corresponding to a set partition. St001583The projective dimension of the simple module corresponding to the point in the poset of the symmetric group under bruhat order. St001171The vector space dimension of $Ext_A^1(I_o,A)$ when $I_o$ is the tilting module corresponding to the permutation $o$ in the Auslander algebra $A$ of $K[x]/(x^n)$. St001168The vector space dimension of the tilting module corresponding to the permutation in the Auslander algebra of $K[x]/(x^n)$. St000529The number of permutations whose descent word is the given binary word.
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