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Your data matches 30 different statistics following compositions of up to 3 maps.
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Matching statistic: St000480
Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
Mp00037: Graphs —to partition of connected components⟶ Integer partitions
St000480: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00160: Permutations —graph of inversions⟶ Graphs
Mp00037: Graphs —to partition of connected components⟶ Integer partitions
St000480: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1] => ([],1)
=> [1]
=> 0
[1,2] => [1,2] => ([],2)
=> [1,1]
=> 0
[2,1] => [1,2] => ([],2)
=> [1,1]
=> 0
[1,2,3] => [1,2,3] => ([],3)
=> [1,1,1]
=> 0
[1,3,2] => [1,2,3] => ([],3)
=> [1,1,1]
=> 0
[2,1,3] => [1,2,3] => ([],3)
=> [1,1,1]
=> 0
[2,3,1] => [1,2,3] => ([],3)
=> [1,1,1]
=> 0
[3,1,2] => [1,3,2] => ([(1,2)],3)
=> [2,1]
=> 1
[3,2,1] => [1,3,2] => ([(1,2)],3)
=> [2,1]
=> 1
[1,2,3,4] => [1,2,3,4] => ([],4)
=> [1,1,1,1]
=> 0
[1,2,4,3] => [1,2,3,4] => ([],4)
=> [1,1,1,1]
=> 0
[1,3,2,4] => [1,2,3,4] => ([],4)
=> [1,1,1,1]
=> 0
[1,3,4,2] => [1,2,3,4] => ([],4)
=> [1,1,1,1]
=> 0
[1,4,2,3] => [1,2,4,3] => ([(2,3)],4)
=> [2,1,1]
=> 1
[1,4,3,2] => [1,2,4,3] => ([(2,3)],4)
=> [2,1,1]
=> 1
[2,1,3,4] => [1,2,3,4] => ([],4)
=> [1,1,1,1]
=> 0
[2,1,4,3] => [1,2,3,4] => ([],4)
=> [1,1,1,1]
=> 0
[2,3,1,4] => [1,2,3,4] => ([],4)
=> [1,1,1,1]
=> 0
[2,3,4,1] => [1,2,3,4] => ([],4)
=> [1,1,1,1]
=> 0
[2,4,1,3] => [1,2,4,3] => ([(2,3)],4)
=> [2,1,1]
=> 1
[2,4,3,1] => [1,2,4,3] => ([(2,3)],4)
=> [2,1,1]
=> 1
[3,1,2,4] => [1,3,2,4] => ([(2,3)],4)
=> [2,1,1]
=> 1
[3,1,4,2] => [1,3,4,2] => ([(1,3),(2,3)],4)
=> [3,1]
=> 1
[3,2,1,4] => [1,3,2,4] => ([(2,3)],4)
=> [2,1,1]
=> 1
[3,2,4,1] => [1,3,4,2] => ([(1,3),(2,3)],4)
=> [3,1]
=> 1
[3,4,1,2] => [1,3,2,4] => ([(2,3)],4)
=> [2,1,1]
=> 1
[3,4,2,1] => [1,3,2,4] => ([(2,3)],4)
=> [2,1,1]
=> 1
[4,1,2,3] => [1,4,3,2] => ([(1,2),(1,3),(2,3)],4)
=> [3,1]
=> 1
[4,1,3,2] => [1,4,2,3] => ([(1,3),(2,3)],4)
=> [3,1]
=> 1
[4,2,1,3] => [1,4,3,2] => ([(1,2),(1,3),(2,3)],4)
=> [3,1]
=> 1
[4,2,3,1] => [1,4,2,3] => ([(1,3),(2,3)],4)
=> [3,1]
=> 1
[4,3,1,2] => [1,4,2,3] => ([(1,3),(2,3)],4)
=> [3,1]
=> 1
[4,3,2,1] => [1,4,2,3] => ([(1,3),(2,3)],4)
=> [3,1]
=> 1
[1,2,3,4,5] => [1,2,3,4,5] => ([],5)
=> [1,1,1,1,1]
=> 0
[1,2,3,5,4] => [1,2,3,4,5] => ([],5)
=> [1,1,1,1,1]
=> 0
[1,2,4,3,5] => [1,2,3,4,5] => ([],5)
=> [1,1,1,1,1]
=> 0
[1,2,4,5,3] => [1,2,3,4,5] => ([],5)
=> [1,1,1,1,1]
=> 0
[1,2,5,3,4] => [1,2,3,5,4] => ([(3,4)],5)
=> [2,1,1,1]
=> 1
[1,2,5,4,3] => [1,2,3,5,4] => ([(3,4)],5)
=> [2,1,1,1]
=> 1
[1,3,2,4,5] => [1,2,3,4,5] => ([],5)
=> [1,1,1,1,1]
=> 0
[1,3,2,5,4] => [1,2,3,4,5] => ([],5)
=> [1,1,1,1,1]
=> 0
[1,3,4,2,5] => [1,2,3,4,5] => ([],5)
=> [1,1,1,1,1]
=> 0
[1,3,4,5,2] => [1,2,3,4,5] => ([],5)
=> [1,1,1,1,1]
=> 0
[1,3,5,2,4] => [1,2,3,5,4] => ([(3,4)],5)
=> [2,1,1,1]
=> 1
[1,3,5,4,2] => [1,2,3,5,4] => ([(3,4)],5)
=> [2,1,1,1]
=> 1
[1,4,2,3,5] => [1,2,4,3,5] => ([(3,4)],5)
=> [2,1,1,1]
=> 1
[1,4,2,5,3] => [1,2,4,5,3] => ([(2,4),(3,4)],5)
=> [3,1,1]
=> 1
[1,4,3,2,5] => [1,2,4,3,5] => ([(3,4)],5)
=> [2,1,1,1]
=> 1
[1,4,3,5,2] => [1,2,4,5,3] => ([(2,4),(3,4)],5)
=> [3,1,1]
=> 1
[1,4,5,2,3] => [1,2,4,3,5] => ([(3,4)],5)
=> [2,1,1,1]
=> 1
Description
The number of lower covers of a partition in dominance order.
According to [1], Corollary 2.4, the maximum number of elements one element (apparently for n≠2) can cover is
12(√1+8n−3)
and an element which covers this number of elements is given by (c+i,c,c−1,…,3,2,1), where 1≤i≤c+2.
Matching statistic: St000455
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
Mp00071: Permutations —descent composition⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St000455: Graphs ⟶ ℤResult quality: 24% ●values known / values provided: 24%●distinct values known / distinct values provided: 33%
Mp00071: Permutations —descent composition⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St000455: Graphs ⟶ ℤResult quality: 24% ●values known / values provided: 24%●distinct values known / distinct values provided: 33%
Values
[1] => [1] => [1] => ([],1)
=> ? = 0 - 1
[1,2] => [1,2] => [2] => ([],2)
=> ? = 0 - 1
[2,1] => [1,2] => [2] => ([],2)
=> ? = 0 - 1
[1,2,3] => [1,2,3] => [3] => ([],3)
=> ? = 0 - 1
[1,3,2] => [1,2,3] => [3] => ([],3)
=> ? = 0 - 1
[2,1,3] => [1,2,3] => [3] => ([],3)
=> ? = 0 - 1
[2,3,1] => [1,2,3] => [3] => ([],3)
=> ? = 0 - 1
[3,1,2] => [1,3,2] => [2,1] => ([(0,2),(1,2)],3)
=> 0 = 1 - 1
[3,2,1] => [1,3,2] => [2,1] => ([(0,2),(1,2)],3)
=> 0 = 1 - 1
[1,2,3,4] => [1,2,3,4] => [4] => ([],4)
=> ? = 0 - 1
[1,2,4,3] => [1,2,3,4] => [4] => ([],4)
=> ? = 0 - 1
[1,3,2,4] => [1,2,3,4] => [4] => ([],4)
=> ? = 0 - 1
[1,3,4,2] => [1,2,3,4] => [4] => ([],4)
=> ? = 0 - 1
[1,4,2,3] => [1,2,4,3] => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 0 = 1 - 1
[1,4,3,2] => [1,2,4,3] => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 0 = 1 - 1
[2,1,3,4] => [1,2,3,4] => [4] => ([],4)
=> ? = 0 - 1
[2,1,4,3] => [1,2,3,4] => [4] => ([],4)
=> ? = 0 - 1
[2,3,1,4] => [1,2,3,4] => [4] => ([],4)
=> ? = 0 - 1
[2,3,4,1] => [1,2,3,4] => [4] => ([],4)
=> ? = 0 - 1
[2,4,1,3] => [1,2,4,3] => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 0 = 1 - 1
[2,4,3,1] => [1,2,4,3] => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 0 = 1 - 1
[3,1,2,4] => [1,3,2,4] => [2,2] => ([(1,3),(2,3)],4)
=> 0 = 1 - 1
[3,1,4,2] => [1,3,4,2] => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 0 = 1 - 1
[3,2,1,4] => [1,3,2,4] => [2,2] => ([(1,3),(2,3)],4)
=> 0 = 1 - 1
[3,2,4,1] => [1,3,4,2] => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 0 = 1 - 1
[3,4,1,2] => [1,3,2,4] => [2,2] => ([(1,3),(2,3)],4)
=> 0 = 1 - 1
[3,4,2,1] => [1,3,2,4] => [2,2] => ([(1,3),(2,3)],4)
=> 0 = 1 - 1
[4,1,2,3] => [1,4,3,2] => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 0 = 1 - 1
[4,1,3,2] => [1,4,2,3] => [2,2] => ([(1,3),(2,3)],4)
=> 0 = 1 - 1
[4,2,1,3] => [1,4,3,2] => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 0 = 1 - 1
[4,2,3,1] => [1,4,2,3] => [2,2] => ([(1,3),(2,3)],4)
=> 0 = 1 - 1
[4,3,1,2] => [1,4,2,3] => [2,2] => ([(1,3),(2,3)],4)
=> 0 = 1 - 1
[4,3,2,1] => [1,4,2,3] => [2,2] => ([(1,3),(2,3)],4)
=> 0 = 1 - 1
[1,2,3,4,5] => [1,2,3,4,5] => [5] => ([],5)
=> ? = 0 - 1
[1,2,3,5,4] => [1,2,3,4,5] => [5] => ([],5)
=> ? = 0 - 1
[1,2,4,3,5] => [1,2,3,4,5] => [5] => ([],5)
=> ? = 0 - 1
[1,2,4,5,3] => [1,2,3,4,5] => [5] => ([],5)
=> ? = 0 - 1
[1,2,5,3,4] => [1,2,3,5,4] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 0 = 1 - 1
[1,2,5,4,3] => [1,2,3,5,4] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 0 = 1 - 1
[1,3,2,4,5] => [1,2,3,4,5] => [5] => ([],5)
=> ? = 0 - 1
[1,3,2,5,4] => [1,2,3,4,5] => [5] => ([],5)
=> ? = 0 - 1
[1,3,4,2,5] => [1,2,3,4,5] => [5] => ([],5)
=> ? = 0 - 1
[1,3,4,5,2] => [1,2,3,4,5] => [5] => ([],5)
=> ? = 0 - 1
[1,3,5,2,4] => [1,2,3,5,4] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 0 = 1 - 1
[1,3,5,4,2] => [1,2,3,5,4] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 0 = 1 - 1
[1,4,2,3,5] => [1,2,4,3,5] => [3,2] => ([(1,4),(2,4),(3,4)],5)
=> 0 = 1 - 1
[1,4,2,5,3] => [1,2,4,5,3] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 0 = 1 - 1
[1,4,3,2,5] => [1,2,4,3,5] => [3,2] => ([(1,4),(2,4),(3,4)],5)
=> 0 = 1 - 1
[1,4,3,5,2] => [1,2,4,5,3] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 0 = 1 - 1
[1,4,5,2,3] => [1,2,4,3,5] => [3,2] => ([(1,4),(2,4),(3,4)],5)
=> 0 = 1 - 1
[1,4,5,3,2] => [1,2,4,3,5] => [3,2] => ([(1,4),(2,4),(3,4)],5)
=> 0 = 1 - 1
[1,5,2,3,4] => [1,2,5,4,3] => [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 0 = 1 - 1
[1,5,2,4,3] => [1,2,5,3,4] => [3,2] => ([(1,4),(2,4),(3,4)],5)
=> 0 = 1 - 1
[1,5,3,2,4] => [1,2,5,4,3] => [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 0 = 1 - 1
[1,5,3,4,2] => [1,2,5,3,4] => [3,2] => ([(1,4),(2,4),(3,4)],5)
=> 0 = 1 - 1
[1,5,4,2,3] => [1,2,5,3,4] => [3,2] => ([(1,4),(2,4),(3,4)],5)
=> 0 = 1 - 1
[1,5,4,3,2] => [1,2,5,3,4] => [3,2] => ([(1,4),(2,4),(3,4)],5)
=> 0 = 1 - 1
[2,1,3,4,5] => [1,2,3,4,5] => [5] => ([],5)
=> ? = 0 - 1
[2,1,3,5,4] => [1,2,3,4,5] => [5] => ([],5)
=> ? = 0 - 1
[2,1,4,3,5] => [1,2,3,4,5] => [5] => ([],5)
=> ? = 0 - 1
[2,1,4,5,3] => [1,2,3,4,5] => [5] => ([],5)
=> ? = 0 - 1
[2,1,5,3,4] => [1,2,3,5,4] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 0 = 1 - 1
[2,1,5,4,3] => [1,2,3,5,4] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 0 = 1 - 1
[2,3,1,4,5] => [1,2,3,4,5] => [5] => ([],5)
=> ? = 0 - 1
[2,3,1,5,4] => [1,2,3,4,5] => [5] => ([],5)
=> ? = 0 - 1
[2,3,4,1,5] => [1,2,3,4,5] => [5] => ([],5)
=> ? = 0 - 1
[2,3,4,5,1] => [1,2,3,4,5] => [5] => ([],5)
=> ? = 0 - 1
[2,3,5,1,4] => [1,2,3,5,4] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 0 = 1 - 1
[2,3,5,4,1] => [1,2,3,5,4] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 0 = 1 - 1
[2,4,1,3,5] => [1,2,4,3,5] => [3,2] => ([(1,4),(2,4),(3,4)],5)
=> 0 = 1 - 1
[2,4,1,5,3] => [1,2,4,5,3] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 0 = 1 - 1
[2,4,3,1,5] => [1,2,4,3,5] => [3,2] => ([(1,4),(2,4),(3,4)],5)
=> 0 = 1 - 1
[2,4,3,5,1] => [1,2,4,5,3] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 0 = 1 - 1
[2,4,5,1,3] => [1,2,4,3,5] => [3,2] => ([(1,4),(2,4),(3,4)],5)
=> 0 = 1 - 1
[2,4,5,3,1] => [1,2,4,3,5] => [3,2] => ([(1,4),(2,4),(3,4)],5)
=> 0 = 1 - 1
[2,5,1,3,4] => [1,2,5,4,3] => [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 0 = 1 - 1
[2,5,1,4,3] => [1,2,5,3,4] => [3,2] => ([(1,4),(2,4),(3,4)],5)
=> 0 = 1 - 1
[2,5,3,1,4] => [1,2,5,4,3] => [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 0 = 1 - 1
[2,5,3,4,1] => [1,2,5,3,4] => [3,2] => ([(1,4),(2,4),(3,4)],5)
=> 0 = 1 - 1
[2,5,4,1,3] => [1,2,5,3,4] => [3,2] => ([(1,4),(2,4),(3,4)],5)
=> 0 = 1 - 1
[2,5,4,3,1] => [1,2,5,3,4] => [3,2] => ([(1,4),(2,4),(3,4)],5)
=> 0 = 1 - 1
[3,5,1,2,4] => [1,3,2,5,4] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 1 - 1
[3,5,1,4,2] => [1,3,2,5,4] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 1 - 1
[3,5,2,1,4] => [1,3,2,5,4] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 1 - 1
[3,5,2,4,1] => [1,3,2,5,4] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 1 - 1
[4,1,5,3,2] => [1,4,3,5,2] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 1 - 1
[4,2,5,3,1] => [1,4,3,5,2] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 1 - 1
[4,5,1,2,3] => [1,4,2,5,3] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 1 - 1
[4,5,2,1,3] => [1,4,2,5,3] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 1 - 1
[4,5,3,1,2] => [1,4,2,5,3] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 1 - 1
[4,5,3,2,1] => [1,4,2,5,3] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 1 - 1
[5,1,4,2,3] => [1,5,3,4,2] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 1 - 1
[5,2,4,1,3] => [1,5,3,4,2] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 1 - 1
[5,4,1,3,2] => [1,5,2,4,3] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 1 - 1
[5,4,2,3,1] => [1,5,2,4,3] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 1 - 1
[5,4,3,1,2] => [1,5,2,4,3] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 1 - 1
[5,4,3,2,1] => [1,5,2,4,3] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 1 - 1
[1,2,3,4,5,6] => [1,2,3,4,5,6] => [6] => ([],6)
=> ? = 0 - 1
[1,2,3,4,6,5] => [1,2,3,4,5,6] => [6] => ([],6)
=> ? = 0 - 1
[1,2,3,5,4,6] => [1,2,3,4,5,6] => [6] => ([],6)
=> ? = 0 - 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: St001330
Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
Mp00071: Permutations —descent composition⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St001330: Graphs ⟶ ℤResult quality: 5% ●values known / values provided: 5%●distinct values known / distinct values provided: 67%
Mp00071: Permutations —descent composition⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St001330: Graphs ⟶ ℤResult quality: 5% ●values known / values provided: 5%●distinct values known / distinct values provided: 67%
Values
[1] => [1] => [1] => ([],1)
=> 1 = 0 + 1
[1,2] => [1,2] => [2] => ([],2)
=> 1 = 0 + 1
[2,1] => [1,2] => [2] => ([],2)
=> 1 = 0 + 1
[1,2,3] => [1,2,3] => [3] => ([],3)
=> 1 = 0 + 1
[1,3,2] => [1,2,3] => [3] => ([],3)
=> 1 = 0 + 1
[2,1,3] => [1,2,3] => [3] => ([],3)
=> 1 = 0 + 1
[2,3,1] => [1,2,3] => [3] => ([],3)
=> 1 = 0 + 1
[3,1,2] => [1,3,2] => [2,1] => ([(0,2),(1,2)],3)
=> 2 = 1 + 1
[3,2,1] => [1,3,2] => [2,1] => ([(0,2),(1,2)],3)
=> 2 = 1 + 1
[1,2,3,4] => [1,2,3,4] => [4] => ([],4)
=> 1 = 0 + 1
[1,2,4,3] => [1,2,3,4] => [4] => ([],4)
=> 1 = 0 + 1
[1,3,2,4] => [1,2,3,4] => [4] => ([],4)
=> 1 = 0 + 1
[1,3,4,2] => [1,2,3,4] => [4] => ([],4)
=> 1 = 0 + 1
[1,4,2,3] => [1,2,4,3] => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 2 = 1 + 1
[1,4,3,2] => [1,2,4,3] => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 2 = 1 + 1
[2,1,3,4] => [1,2,3,4] => [4] => ([],4)
=> 1 = 0 + 1
[2,1,4,3] => [1,2,3,4] => [4] => ([],4)
=> 1 = 0 + 1
[2,3,1,4] => [1,2,3,4] => [4] => ([],4)
=> 1 = 0 + 1
[2,3,4,1] => [1,2,3,4] => [4] => ([],4)
=> 1 = 0 + 1
[2,4,1,3] => [1,2,4,3] => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 2 = 1 + 1
[2,4,3,1] => [1,2,4,3] => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 2 = 1 + 1
[3,1,2,4] => [1,3,2,4] => [2,2] => ([(1,3),(2,3)],4)
=> 2 = 1 + 1
[3,1,4,2] => [1,3,4,2] => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 2 = 1 + 1
[3,2,1,4] => [1,3,2,4] => [2,2] => ([(1,3),(2,3)],4)
=> 2 = 1 + 1
[3,2,4,1] => [1,3,4,2] => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 2 = 1 + 1
[3,4,1,2] => [1,3,2,4] => [2,2] => ([(1,3),(2,3)],4)
=> 2 = 1 + 1
[3,4,2,1] => [1,3,2,4] => [2,2] => ([(1,3),(2,3)],4)
=> 2 = 1 + 1
[4,1,2,3] => [1,4,3,2] => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ? = 1 + 1
[4,1,3,2] => [1,4,2,3] => [2,2] => ([(1,3),(2,3)],4)
=> 2 = 1 + 1
[4,2,1,3] => [1,4,3,2] => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ? = 1 + 1
[4,2,3,1] => [1,4,2,3] => [2,2] => ([(1,3),(2,3)],4)
=> 2 = 1 + 1
[4,3,1,2] => [1,4,2,3] => [2,2] => ([(1,3),(2,3)],4)
=> 2 = 1 + 1
[4,3,2,1] => [1,4,2,3] => [2,2] => ([(1,3),(2,3)],4)
=> 2 = 1 + 1
[1,2,3,4,5] => [1,2,3,4,5] => [5] => ([],5)
=> 1 = 0 + 1
[1,2,3,5,4] => [1,2,3,4,5] => [5] => ([],5)
=> 1 = 0 + 1
[1,2,4,3,5] => [1,2,3,4,5] => [5] => ([],5)
=> 1 = 0 + 1
[1,2,4,5,3] => [1,2,3,4,5] => [5] => ([],5)
=> 1 = 0 + 1
[1,2,5,3,4] => [1,2,3,5,4] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 2 = 1 + 1
[1,2,5,4,3] => [1,2,3,5,4] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 2 = 1 + 1
[1,3,2,4,5] => [1,2,3,4,5] => [5] => ([],5)
=> 1 = 0 + 1
[1,3,2,5,4] => [1,2,3,4,5] => [5] => ([],5)
=> 1 = 0 + 1
[1,3,4,2,5] => [1,2,3,4,5] => [5] => ([],5)
=> 1 = 0 + 1
[1,3,4,5,2] => [1,2,3,4,5] => [5] => ([],5)
=> 1 = 0 + 1
[1,3,5,2,4] => [1,2,3,5,4] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 2 = 1 + 1
[1,3,5,4,2] => [1,2,3,5,4] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 2 = 1 + 1
[1,4,2,3,5] => [1,2,4,3,5] => [3,2] => ([(1,4),(2,4),(3,4)],5)
=> 2 = 1 + 1
[1,4,2,5,3] => [1,2,4,5,3] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 2 = 1 + 1
[1,4,3,2,5] => [1,2,4,3,5] => [3,2] => ([(1,4),(2,4),(3,4)],5)
=> 2 = 1 + 1
[1,4,3,5,2] => [1,2,4,5,3] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 2 = 1 + 1
[1,4,5,2,3] => [1,2,4,3,5] => [3,2] => ([(1,4),(2,4),(3,4)],5)
=> 2 = 1 + 1
[1,4,5,3,2] => [1,2,4,3,5] => [3,2] => ([(1,4),(2,4),(3,4)],5)
=> 2 = 1 + 1
[1,5,2,3,4] => [1,2,5,4,3] => [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 1 + 1
[1,5,2,4,3] => [1,2,5,3,4] => [3,2] => ([(1,4),(2,4),(3,4)],5)
=> 2 = 1 + 1
[1,5,3,2,4] => [1,2,5,4,3] => [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 1 + 1
[2,5,1,3,4] => [1,2,5,4,3] => [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 1 + 1
[2,5,3,1,4] => [1,2,5,4,3] => [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 1 + 1
[3,1,5,2,4] => [1,3,5,4,2] => [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 1 + 1
[3,2,5,1,4] => [1,3,5,4,2] => [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 1 + 1
[3,5,1,2,4] => [1,3,2,5,4] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 1 + 1
[3,5,1,4,2] => [1,3,2,5,4] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 1 + 1
[3,5,2,1,4] => [1,3,2,5,4] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 1 + 1
[3,5,2,4,1] => [1,3,2,5,4] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 1 + 1
[4,1,2,3,5] => [1,4,3,2,5] => [2,1,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 1 + 1
[4,1,2,5,3] => [1,4,5,3,2] => [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 1 + 1
[4,1,5,3,2] => [1,4,3,5,2] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 1 + 1
[4,2,1,3,5] => [1,4,3,2,5] => [2,1,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 1 + 1
[4,2,1,5,3] => [1,4,5,3,2] => [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 1 + 1
[4,2,5,3,1] => [1,4,3,5,2] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 1 + 1
[4,5,1,2,3] => [1,4,2,5,3] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 1 + 1
[4,5,1,3,2] => [1,4,3,2,5] => [2,1,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 1 + 1
[4,5,2,1,3] => [1,4,2,5,3] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 1 + 1
[4,5,2,3,1] => [1,4,3,2,5] => [2,1,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 1 + 1
[4,5,3,1,2] => [1,4,2,5,3] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 1 + 1
[4,5,3,2,1] => [1,4,2,5,3] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 1 + 1
[5,1,2,3,4] => [1,5,4,3,2] => [2,1,1,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 1 + 1
[5,1,2,4,3] => [1,5,3,2,4] => [2,1,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 1 + 1
[5,1,3,2,4] => [1,5,4,2,3] => [2,1,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 1 + 1
[5,1,4,2,3] => [1,5,3,4,2] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 1 + 1
[5,2,1,3,4] => [1,5,4,3,2] => [2,1,1,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 1 + 1
[5,2,1,4,3] => [1,5,3,2,4] => [2,1,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 1 + 1
[5,2,3,1,4] => [1,5,4,2,3] => [2,1,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 1 + 1
[5,2,4,1,3] => [1,5,3,4,2] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 1 + 1
[5,3,1,2,4] => [1,5,4,2,3] => [2,1,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 1 + 1
[5,3,2,1,4] => [1,5,4,2,3] => [2,1,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 1 + 1
[5,4,1,2,3] => [1,5,3,2,4] => [2,1,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 1 + 1
[5,4,1,3,2] => [1,5,2,4,3] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 1 + 1
[5,4,2,1,3] => [1,5,3,2,4] => [2,1,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 1 + 1
[5,4,2,3,1] => [1,5,2,4,3] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 1 + 1
[5,4,3,1,2] => [1,5,2,4,3] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 1 + 1
[5,4,3,2,1] => [1,5,2,4,3] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 1 + 1
[1,2,6,3,4,5] => [1,2,3,6,5,4] => [4,1,1] => ([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 1 + 1
[1,2,6,4,3,5] => [1,2,3,6,5,4] => [4,1,1] => ([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 1 + 1
[1,3,6,2,4,5] => [1,2,3,6,5,4] => [4,1,1] => ([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 1 + 1
[1,3,6,4,2,5] => [1,2,3,6,5,4] => [4,1,1] => ([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 1 + 1
[1,4,2,6,3,5] => [1,2,4,6,5,3] => [4,1,1] => ([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 1 + 1
[1,4,3,6,2,5] => [1,2,4,6,5,3] => [4,1,1] => ([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 1 + 1
[1,4,6,2,3,5] => [1,2,4,3,6,5] => [3,2,1] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 1 + 1
[1,4,6,2,5,3] => [1,2,4,3,6,5] => [3,2,1] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 1 + 1
[1,4,6,3,2,5] => [1,2,4,3,6,5] => [3,2,1] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 1 + 1
[1,4,6,3,5,2] => [1,2,4,3,6,5] => [3,2,1] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 1 + 1
Description
The hat guessing number of a graph.
Suppose that each vertex of a graph corresponds to a player, wearing a hat whose color is arbitrarily chosen from a set of q possible colors. Each player can see the hat colors of his neighbors, but not his own hat color. All of the players are asked to guess their own hat colors simultaneously, according to a predetermined guessing strategy and the hat colors they see, where no communication between them is allowed. The hat guessing number HG(G) of a graph G is the largest integer q such that there exists a guessing strategy guaranteeing at least one correct guess for any hat assignment of q possible colors.
Because it suffices that a single player guesses correctly, the hat guessing number of a graph is the maximum of the hat guessing numbers of its connected components.
Matching statistic: St000298
(load all 10 compositions to match this statistic)
(load all 10 compositions to match this statistic)
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
Mp00121: Dyck paths —Cori-Le Borgne involution⟶ Dyck paths
Mp00232: Dyck paths —parallelogram poset⟶ Posets
St000298: Posets ⟶ ℤResult quality: 2% ●values known / values provided: 2%●distinct values known / distinct values provided: 67%
Mp00121: Dyck paths —Cori-Le Borgne involution⟶ Dyck paths
Mp00232: Dyck paths —parallelogram poset⟶ Posets
St000298: Posets ⟶ ℤResult quality: 2% ●values known / values provided: 2%●distinct values known / distinct values provided: 67%
Values
[1] => [1,0]
=> [1,0]
=> ([],1)
=> 1 = 0 + 1
[1,2] => [1,0,1,0]
=> [1,0,1,0]
=> ([(0,1)],2)
=> 1 = 0 + 1
[2,1] => [1,1,0,0]
=> [1,1,0,0]
=> ([(0,1)],2)
=> 1 = 0 + 1
[1,2,3] => [1,0,1,0,1,0]
=> [1,0,1,0,1,0]
=> ([(0,2),(2,1)],3)
=> 1 = 0 + 1
[1,3,2] => [1,0,1,1,0,0]
=> [1,1,0,1,0,0]
=> ([(0,2),(2,1)],3)
=> 1 = 0 + 1
[2,1,3] => [1,1,0,0,1,0]
=> [1,1,0,0,1,0]
=> ([(0,2),(2,1)],3)
=> 1 = 0 + 1
[2,3,1] => [1,1,0,1,0,0]
=> [1,0,1,1,0,0]
=> ([(0,2),(2,1)],3)
=> 1 = 0 + 1
[3,1,2] => [1,1,1,0,0,0]
=> [1,1,1,0,0,0]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2 = 1 + 1
[3,2,1] => [1,1,1,0,0,0]
=> [1,1,1,0,0,0]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2 = 1 + 1
[1,2,3,4] => [1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0]
=> ([(0,3),(2,1),(3,2)],4)
=> 1 = 0 + 1
[1,2,4,3] => [1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,0]
=> ([(0,3),(2,1),(3,2)],4)
=> 1 = 0 + 1
[1,3,2,4] => [1,0,1,1,0,0,1,0]
=> [1,1,0,1,0,0,1,0]
=> ([(0,3),(2,1),(3,2)],4)
=> 1 = 0 + 1
[1,3,4,2] => [1,0,1,1,0,1,0,0]
=> [1,0,1,1,0,1,0,0]
=> ([(0,3),(2,1),(3,2)],4)
=> 1 = 0 + 1
[1,4,2,3] => [1,0,1,1,1,0,0,0]
=> [1,1,1,0,1,0,0,0]
=> ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 2 = 1 + 1
[1,4,3,2] => [1,0,1,1,1,0,0,0]
=> [1,1,1,0,1,0,0,0]
=> ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 2 = 1 + 1
[2,1,3,4] => [1,1,0,0,1,0,1,0]
=> [1,1,0,0,1,0,1,0]
=> ([(0,3),(2,1),(3,2)],4)
=> 1 = 0 + 1
[2,1,4,3] => [1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> ([(0,3),(2,1),(3,2)],4)
=> 1 = 0 + 1
[2,3,1,4] => [1,1,0,1,0,0,1,0]
=> [1,0,1,1,0,0,1,0]
=> ([(0,3),(2,1),(3,2)],4)
=> 1 = 0 + 1
[2,3,4,1] => [1,1,0,1,0,1,0,0]
=> [1,0,1,0,1,1,0,0]
=> ([(0,3),(2,1),(3,2)],4)
=> 1 = 0 + 1
[2,4,1,3] => [1,1,0,1,1,0,0,0]
=> [1,1,0,1,1,0,0,0]
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> 2 = 1 + 1
[2,4,3,1] => [1,1,0,1,1,0,0,0]
=> [1,1,0,1,1,0,0,0]
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> 2 = 1 + 1
[3,1,2,4] => [1,1,1,0,0,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> 2 = 1 + 1
[3,1,4,2] => [1,1,1,0,0,1,0,0]
=> [1,1,1,0,0,0,1,0]
=> ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> 2 = 1 + 1
[3,2,1,4] => [1,1,1,0,0,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> 2 = 1 + 1
[3,2,4,1] => [1,1,1,0,0,1,0,0]
=> [1,1,1,0,0,0,1,0]
=> ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> 2 = 1 + 1
[3,4,1,2] => [1,1,1,0,1,0,0,0]
=> [1,0,1,1,1,0,0,0]
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> 2 = 1 + 1
[3,4,2,1] => [1,1,1,0,1,0,0,0]
=> [1,0,1,1,1,0,0,0]
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> 2 = 1 + 1
[4,1,2,3] => [1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 2 = 1 + 1
[4,1,3,2] => [1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 2 = 1 + 1
[4,2,1,3] => [1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 2 = 1 + 1
[4,2,3,1] => [1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 2 = 1 + 1
[4,3,1,2] => [1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 2 = 1 + 1
[4,3,2,1] => [1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 2 = 1 + 1
[1,2,3,4,5] => [1,0,1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1 = 0 + 1
[1,2,3,5,4] => [1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1 = 0 + 1
[1,2,4,3,5] => [1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,0,1,0,0,1,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1 = 0 + 1
[1,2,4,5,3] => [1,0,1,0,1,1,0,1,0,0]
=> [1,0,1,1,0,1,0,1,0,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1 = 0 + 1
[1,2,5,3,4] => [1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? = 1 + 1
[1,2,5,4,3] => [1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? = 1 + 1
[1,3,2,4,5] => [1,0,1,1,0,0,1,0,1,0]
=> [1,1,0,1,0,0,1,0,1,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1 = 0 + 1
[1,3,2,5,4] => [1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,1,0,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1 = 0 + 1
[1,3,4,2,5] => [1,0,1,1,0,1,0,0,1,0]
=> [1,0,1,1,0,1,0,0,1,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1 = 0 + 1
[1,3,4,5,2] => [1,0,1,1,0,1,0,1,0,0]
=> [1,0,1,0,1,1,0,1,0,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1 = 0 + 1
[1,3,5,2,4] => [1,0,1,1,0,1,1,0,0,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> ([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
=> ? = 1 + 1
[1,3,5,4,2] => [1,0,1,1,0,1,1,0,0,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> ([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
=> ? = 1 + 1
[1,4,2,3,5] => [1,0,1,1,1,0,0,0,1,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> ([(0,3),(0,4),(2,6),(3,5),(4,2),(4,5),(5,6),(6,1)],7)
=> ? = 1 + 1
[1,4,2,5,3] => [1,0,1,1,1,0,0,1,0,0]
=> [1,1,1,0,1,0,0,0,1,0]
=> ([(0,3),(0,4),(2,6),(3,5),(4,2),(4,5),(5,6),(6,1)],7)
=> ? = 1 + 1
[1,4,3,2,5] => [1,0,1,1,1,0,0,0,1,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> ([(0,3),(0,4),(2,6),(3,5),(4,2),(4,5),(5,6),(6,1)],7)
=> ? = 1 + 1
[1,4,3,5,2] => [1,0,1,1,1,0,0,1,0,0]
=> [1,1,1,0,1,0,0,0,1,0]
=> ([(0,3),(0,4),(2,6),(3,5),(4,2),(4,5),(5,6),(6,1)],7)
=> ? = 1 + 1
[1,4,5,2,3] => [1,0,1,1,1,0,1,0,0,0]
=> [1,0,1,1,1,0,1,0,0,0]
=> ([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
=> ? = 1 + 1
[1,4,5,3,2] => [1,0,1,1,1,0,1,0,0,0]
=> [1,0,1,1,1,0,1,0,0,0]
=> ([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
=> ? = 1 + 1
[1,5,2,3,4] => [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? = 1 + 1
[1,5,2,4,3] => [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? = 1 + 1
[1,5,3,2,4] => [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? = 1 + 1
[1,5,3,4,2] => [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? = 1 + 1
[1,5,4,2,3] => [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? = 1 + 1
[1,5,4,3,2] => [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? = 1 + 1
[2,1,3,4,5] => [1,1,0,0,1,0,1,0,1,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1 = 0 + 1
[2,1,3,5,4] => [1,1,0,0,1,0,1,1,0,0]
=> [1,1,0,1,0,0,1,1,0,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1 = 0 + 1
[2,1,4,3,5] => [1,1,0,0,1,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1 = 0 + 1
[2,1,4,5,3] => [1,1,0,0,1,1,0,1,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1 = 0 + 1
[2,1,5,3,4] => [1,1,0,0,1,1,1,0,0,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> ([(0,3),(0,4),(1,5),(2,5),(3,6),(4,6),(6,1),(6,2)],7)
=> 2 = 1 + 1
[2,1,5,4,3] => [1,1,0,0,1,1,1,0,0,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> ([(0,3),(0,4),(1,5),(2,5),(3,6),(4,6),(6,1),(6,2)],7)
=> 2 = 1 + 1
[2,3,1,4,5] => [1,1,0,1,0,0,1,0,1,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1 = 0 + 1
[2,3,1,5,4] => [1,1,0,1,0,0,1,1,0,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1 = 0 + 1
[2,3,4,1,5] => [1,1,0,1,0,1,0,0,1,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1 = 0 + 1
[2,5,1,3,4] => [1,1,0,1,1,1,0,0,0,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> ([(0,3),(0,4),(1,6),(2,5),(3,7),(4,2),(4,7),(5,6),(7,1),(7,5)],8)
=> ? = 1 + 1
[2,5,1,4,3] => [1,1,0,1,1,1,0,0,0,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> ([(0,3),(0,4),(1,6),(2,5),(3,7),(4,2),(4,7),(5,6),(7,1),(7,5)],8)
=> ? = 1 + 1
[2,5,3,1,4] => [1,1,0,1,1,1,0,0,0,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> ([(0,3),(0,4),(1,6),(2,5),(3,7),(4,2),(4,7),(5,6),(7,1),(7,5)],8)
=> ? = 1 + 1
[2,5,3,4,1] => [1,1,0,1,1,1,0,0,0,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> ([(0,3),(0,4),(1,6),(2,5),(3,7),(4,2),(4,7),(5,6),(7,1),(7,5)],8)
=> ? = 1 + 1
[2,5,4,1,3] => [1,1,0,1,1,1,0,0,0,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> ([(0,3),(0,4),(1,6),(2,5),(3,7),(4,2),(4,7),(5,6),(7,1),(7,5)],8)
=> ? = 1 + 1
[2,5,4,3,1] => [1,1,0,1,1,1,0,0,0,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> ([(0,3),(0,4),(1,6),(2,5),(3,7),(4,2),(4,7),(5,6),(7,1),(7,5)],8)
=> ? = 1 + 1
[3,5,1,2,4] => [1,1,1,0,1,1,0,0,0,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> ([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
=> ? = 1 + 1
[3,5,1,4,2] => [1,1,1,0,1,1,0,0,0,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> ([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
=> ? = 1 + 1
[3,5,2,1,4] => [1,1,1,0,1,1,0,0,0,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> ([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
=> ? = 1 + 1
[3,5,2,4,1] => [1,1,1,0,1,1,0,0,0,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> ([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
=> ? = 1 + 1
[3,5,4,1,2] => [1,1,1,0,1,1,0,0,0,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> ([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
=> ? = 1 + 1
[3,5,4,2,1] => [1,1,1,0,1,1,0,0,0,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> ([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
=> ? = 1 + 1
[4,1,2,3,5] => [1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> ([(0,3),(0,4),(1,6),(2,5),(3,7),(4,2),(4,7),(5,6),(7,1),(7,5)],8)
=> ? = 1 + 1
[4,1,2,5,3] => [1,1,1,1,0,0,0,1,0,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> ([(0,3),(0,4),(2,6),(3,5),(4,2),(4,5),(5,6),(6,1)],7)
=> ? = 1 + 1
[4,1,3,2,5] => [1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> ([(0,3),(0,4),(1,6),(2,5),(3,7),(4,2),(4,7),(5,6),(7,1),(7,5)],8)
=> ? = 1 + 1
[4,1,3,5,2] => [1,1,1,1,0,0,0,1,0,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> ([(0,3),(0,4),(2,6),(3,5),(4,2),(4,5),(5,6),(6,1)],7)
=> ? = 1 + 1
[4,1,5,2,3] => [1,1,1,1,0,0,1,0,0,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> ([(0,3),(0,4),(2,6),(3,5),(4,2),(4,5),(5,6),(6,1)],7)
=> ? = 1 + 1
[4,1,5,3,2] => [1,1,1,1,0,0,1,0,0,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> ([(0,3),(0,4),(2,6),(3,5),(4,2),(4,5),(5,6),(6,1)],7)
=> ? = 1 + 1
[4,2,1,3,5] => [1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> ([(0,3),(0,4),(1,6),(2,5),(3,7),(4,2),(4,7),(5,6),(7,1),(7,5)],8)
=> ? = 1 + 1
[4,2,1,5,3] => [1,1,1,1,0,0,0,1,0,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> ([(0,3),(0,4),(2,6),(3,5),(4,2),(4,5),(5,6),(6,1)],7)
=> ? = 1 + 1
[4,2,3,1,5] => [1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> ([(0,3),(0,4),(1,6),(2,5),(3,7),(4,2),(4,7),(5,6),(7,1),(7,5)],8)
=> ? = 1 + 1
[4,2,3,5,1] => [1,1,1,1,0,0,0,1,0,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> ([(0,3),(0,4),(2,6),(3,5),(4,2),(4,5),(5,6),(6,1)],7)
=> ? = 1 + 1
[4,2,5,1,3] => [1,1,1,1,0,0,1,0,0,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> ([(0,3),(0,4),(2,6),(3,5),(4,2),(4,5),(5,6),(6,1)],7)
=> ? = 1 + 1
[4,2,5,3,1] => [1,1,1,1,0,0,1,0,0,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> ([(0,3),(0,4),(2,6),(3,5),(4,2),(4,5),(5,6),(6,1)],7)
=> ? = 1 + 1
[4,3,1,2,5] => [1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> ([(0,3),(0,4),(1,6),(2,5),(3,7),(4,2),(4,7),(5,6),(7,1),(7,5)],8)
=> ? = 1 + 1
[4,3,1,5,2] => [1,1,1,1,0,0,0,1,0,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> ([(0,3),(0,4),(2,6),(3,5),(4,2),(4,5),(5,6),(6,1)],7)
=> ? = 1 + 1
[4,3,2,1,5] => [1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> ([(0,3),(0,4),(1,6),(2,5),(3,7),(4,2),(4,7),(5,6),(7,1),(7,5)],8)
=> ? = 1 + 1
[4,3,2,5,1] => [1,1,1,1,0,0,0,1,0,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> ([(0,3),(0,4),(2,6),(3,5),(4,2),(4,5),(5,6),(6,1)],7)
=> ? = 1 + 1
[4,3,5,1,2] => [1,1,1,1,0,0,1,0,0,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> ([(0,3),(0,4),(2,6),(3,5),(4,2),(4,5),(5,6),(6,1)],7)
=> ? = 1 + 1
[4,3,5,2,1] => [1,1,1,1,0,0,1,0,0,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> ([(0,3),(0,4),(2,6),(3,5),(4,2),(4,5),(5,6),(6,1)],7)
=> ? = 1 + 1
[4,5,1,2,3] => [1,1,1,1,0,1,0,0,0,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> ([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
=> ? = 1 + 1
[4,5,1,3,2] => [1,1,1,1,0,1,0,0,0,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> ([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
=> ? = 1 + 1
[4,5,2,1,3] => [1,1,1,1,0,1,0,0,0,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> ([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
=> ? = 1 + 1
[4,5,2,3,1] => [1,1,1,1,0,1,0,0,0,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> ([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
=> ? = 1 + 1
Description
The order dimension or Dushnik-Miller dimension of a poset.
This is the minimal number of linear orderings whose intersection is the given poset.
Matching statistic: St001431
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
Mp00175: Permutations —inverse Foata bijection⟶ Permutations
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
St001431: Dyck paths ⟶ ℤResult quality: 2% ●values known / values provided: 2%●distinct values known / distinct values provided: 67%
Mp00175: Permutations —inverse Foata bijection⟶ Permutations
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
St001431: Dyck paths ⟶ ℤResult quality: 2% ●values known / values provided: 2%●distinct values known / distinct values provided: 67%
Values
[1] => [1] => [1] => [1,0]
=> ? = 0
[1,2] => [1,2] => [1,2] => [1,0,1,0]
=> 0
[2,1] => [1,2] => [1,2] => [1,0,1,0]
=> 0
[1,2,3] => [1,2,3] => [1,2,3] => [1,0,1,0,1,0]
=> 0
[1,3,2] => [1,2,3] => [1,2,3] => [1,0,1,0,1,0]
=> 0
[2,1,3] => [1,2,3] => [1,2,3] => [1,0,1,0,1,0]
=> 0
[2,3,1] => [1,2,3] => [1,2,3] => [1,0,1,0,1,0]
=> 0
[3,1,2] => [1,3,2] => [3,1,2] => [1,1,1,0,0,0]
=> 1
[3,2,1] => [1,3,2] => [3,1,2] => [1,1,1,0,0,0]
=> 1
[1,2,3,4] => [1,2,3,4] => [1,2,3,4] => [1,0,1,0,1,0,1,0]
=> 0
[1,2,4,3] => [1,2,3,4] => [1,2,3,4] => [1,0,1,0,1,0,1,0]
=> 0
[1,3,2,4] => [1,2,3,4] => [1,2,3,4] => [1,0,1,0,1,0,1,0]
=> 0
[1,3,4,2] => [1,2,3,4] => [1,2,3,4] => [1,0,1,0,1,0,1,0]
=> 0
[1,4,2,3] => [1,2,4,3] => [4,1,2,3] => [1,1,1,1,0,0,0,0]
=> 1
[1,4,3,2] => [1,2,4,3] => [4,1,2,3] => [1,1,1,1,0,0,0,0]
=> 1
[2,1,3,4] => [1,2,3,4] => [1,2,3,4] => [1,0,1,0,1,0,1,0]
=> 0
[2,1,4,3] => [1,2,3,4] => [1,2,3,4] => [1,0,1,0,1,0,1,0]
=> 0
[2,3,1,4] => [1,2,3,4] => [1,2,3,4] => [1,0,1,0,1,0,1,0]
=> 0
[2,3,4,1] => [1,2,3,4] => [1,2,3,4] => [1,0,1,0,1,0,1,0]
=> 0
[2,4,1,3] => [1,2,4,3] => [4,1,2,3] => [1,1,1,1,0,0,0,0]
=> 1
[2,4,3,1] => [1,2,4,3] => [4,1,2,3] => [1,1,1,1,0,0,0,0]
=> 1
[3,1,2,4] => [1,3,2,4] => [3,1,2,4] => [1,1,1,0,0,0,1,0]
=> 1
[3,1,4,2] => [1,3,4,2] => [3,4,1,2] => [1,1,1,0,1,0,0,0]
=> 1
[3,2,1,4] => [1,3,2,4] => [3,1,2,4] => [1,1,1,0,0,0,1,0]
=> 1
[3,2,4,1] => [1,3,4,2] => [3,4,1,2] => [1,1,1,0,1,0,0,0]
=> 1
[3,4,1,2] => [1,3,2,4] => [3,1,2,4] => [1,1,1,0,0,0,1,0]
=> 1
[3,4,2,1] => [1,3,2,4] => [3,1,2,4] => [1,1,1,0,0,0,1,0]
=> 1
[4,1,2,3] => [1,4,3,2] => [4,3,1,2] => [1,1,1,1,0,0,0,0]
=> 1
[4,1,3,2] => [1,4,2,3] => [1,4,2,3] => [1,0,1,1,1,0,0,0]
=> 1
[4,2,1,3] => [1,4,3,2] => [4,3,1,2] => [1,1,1,1,0,0,0,0]
=> 1
[4,2,3,1] => [1,4,2,3] => [1,4,2,3] => [1,0,1,1,1,0,0,0]
=> 1
[4,3,1,2] => [1,4,2,3] => [1,4,2,3] => [1,0,1,1,1,0,0,0]
=> 1
[4,3,2,1] => [1,4,2,3] => [1,4,2,3] => [1,0,1,1,1,0,0,0]
=> 1
[1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => [1,0,1,0,1,0,1,0,1,0]
=> 0
[1,2,3,5,4] => [1,2,3,4,5] => [1,2,3,4,5] => [1,0,1,0,1,0,1,0,1,0]
=> 0
[1,2,4,3,5] => [1,2,3,4,5] => [1,2,3,4,5] => [1,0,1,0,1,0,1,0,1,0]
=> 0
[1,2,4,5,3] => [1,2,3,4,5] => [1,2,3,4,5] => [1,0,1,0,1,0,1,0,1,0]
=> 0
[1,2,5,3,4] => [1,2,3,5,4] => [5,1,2,3,4] => [1,1,1,1,1,0,0,0,0,0]
=> 1
[1,2,5,4,3] => [1,2,3,5,4] => [5,1,2,3,4] => [1,1,1,1,1,0,0,0,0,0]
=> 1
[1,3,2,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => [1,0,1,0,1,0,1,0,1,0]
=> 0
[1,3,2,5,4] => [1,2,3,4,5] => [1,2,3,4,5] => [1,0,1,0,1,0,1,0,1,0]
=> 0
[1,3,4,2,5] => [1,2,3,4,5] => [1,2,3,4,5] => [1,0,1,0,1,0,1,0,1,0]
=> 0
[1,3,4,5,2] => [1,2,3,4,5] => [1,2,3,4,5] => [1,0,1,0,1,0,1,0,1,0]
=> 0
[1,3,5,2,4] => [1,2,3,5,4] => [5,1,2,3,4] => [1,1,1,1,1,0,0,0,0,0]
=> 1
[1,3,5,4,2] => [1,2,3,5,4] => [5,1,2,3,4] => [1,1,1,1,1,0,0,0,0,0]
=> 1
[1,4,2,3,5] => [1,2,4,3,5] => [4,1,2,3,5] => [1,1,1,1,0,0,0,0,1,0]
=> 1
[1,4,2,5,3] => [1,2,4,5,3] => [4,5,1,2,3] => [1,1,1,1,0,1,0,0,0,0]
=> 1
[1,4,3,2,5] => [1,2,4,3,5] => [4,1,2,3,5] => [1,1,1,1,0,0,0,0,1,0]
=> 1
[1,4,3,5,2] => [1,2,4,5,3] => [4,5,1,2,3] => [1,1,1,1,0,1,0,0,0,0]
=> 1
[1,4,5,2,3] => [1,2,4,3,5] => [4,1,2,3,5] => [1,1,1,1,0,0,0,0,1,0]
=> 1
[1,4,5,3,2] => [1,2,4,3,5] => [4,1,2,3,5] => [1,1,1,1,0,0,0,0,1,0]
=> 1
[1,2,3,6,4,5] => [1,2,3,4,6,5] => [6,1,2,3,4,5] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> ? = 1
[1,2,3,6,5,4] => [1,2,3,4,6,5] => [6,1,2,3,4,5] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> ? = 1
[1,2,4,6,3,5] => [1,2,3,4,6,5] => [6,1,2,3,4,5] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> ? = 1
[1,2,4,6,5,3] => [1,2,3,4,6,5] => [6,1,2,3,4,5] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> ? = 1
[1,2,5,3,4,6] => [1,2,3,5,4,6] => [5,1,2,3,4,6] => [1,1,1,1,1,0,0,0,0,0,1,0]
=> ? = 1
[1,2,5,3,6,4] => [1,2,3,5,6,4] => [5,6,1,2,3,4] => [1,1,1,1,1,0,1,0,0,0,0,0]
=> ? = 1
[1,2,5,4,3,6] => [1,2,3,5,4,6] => [5,1,2,3,4,6] => [1,1,1,1,1,0,0,0,0,0,1,0]
=> ? = 1
[1,2,5,4,6,3] => [1,2,3,5,6,4] => [5,6,1,2,3,4] => [1,1,1,1,1,0,1,0,0,0,0,0]
=> ? = 1
[1,2,5,6,3,4] => [1,2,3,5,4,6] => [5,1,2,3,4,6] => [1,1,1,1,1,0,0,0,0,0,1,0]
=> ? = 1
[1,2,5,6,4,3] => [1,2,3,5,4,6] => [5,1,2,3,4,6] => [1,1,1,1,1,0,0,0,0,0,1,0]
=> ? = 1
[1,2,6,3,4,5] => [1,2,3,6,5,4] => [6,5,1,2,3,4] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> ? = 1
[1,2,6,3,5,4] => [1,2,3,6,4,5] => [1,6,2,3,4,5] => [1,0,1,1,1,1,1,0,0,0,0,0]
=> ? = 1
[1,2,6,4,3,5] => [1,2,3,6,5,4] => [6,5,1,2,3,4] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> ? = 1
[1,2,6,4,5,3] => [1,2,3,6,4,5] => [1,6,2,3,4,5] => [1,0,1,1,1,1,1,0,0,0,0,0]
=> ? = 1
[1,2,6,5,3,4] => [1,2,3,6,4,5] => [1,6,2,3,4,5] => [1,0,1,1,1,1,1,0,0,0,0,0]
=> ? = 1
[1,2,6,5,4,3] => [1,2,3,6,4,5] => [1,6,2,3,4,5] => [1,0,1,1,1,1,1,0,0,0,0,0]
=> ? = 1
[1,3,2,6,4,5] => [1,2,3,4,6,5] => [6,1,2,3,4,5] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> ? = 1
[1,3,2,6,5,4] => [1,2,3,4,6,5] => [6,1,2,3,4,5] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> ? = 1
[1,3,4,6,2,5] => [1,2,3,4,6,5] => [6,1,2,3,4,5] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> ? = 1
[1,3,4,6,5,2] => [1,2,3,4,6,5] => [6,1,2,3,4,5] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> ? = 1
[1,3,5,2,4,6] => [1,2,3,5,4,6] => [5,1,2,3,4,6] => [1,1,1,1,1,0,0,0,0,0,1,0]
=> ? = 1
[1,3,5,2,6,4] => [1,2,3,5,6,4] => [5,6,1,2,3,4] => [1,1,1,1,1,0,1,0,0,0,0,0]
=> ? = 1
[1,3,5,4,2,6] => [1,2,3,5,4,6] => [5,1,2,3,4,6] => [1,1,1,1,1,0,0,0,0,0,1,0]
=> ? = 1
[1,3,5,4,6,2] => [1,2,3,5,6,4] => [5,6,1,2,3,4] => [1,1,1,1,1,0,1,0,0,0,0,0]
=> ? = 1
[1,3,5,6,2,4] => [1,2,3,5,4,6] => [5,1,2,3,4,6] => [1,1,1,1,1,0,0,0,0,0,1,0]
=> ? = 1
[1,3,5,6,4,2] => [1,2,3,5,4,6] => [5,1,2,3,4,6] => [1,1,1,1,1,0,0,0,0,0,1,0]
=> ? = 1
[1,3,6,2,4,5] => [1,2,3,6,5,4] => [6,5,1,2,3,4] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> ? = 1
[1,3,6,2,5,4] => [1,2,3,6,4,5] => [1,6,2,3,4,5] => [1,0,1,1,1,1,1,0,0,0,0,0]
=> ? = 1
[1,3,6,4,2,5] => [1,2,3,6,5,4] => [6,5,1,2,3,4] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> ? = 1
[1,3,6,4,5,2] => [1,2,3,6,4,5] => [1,6,2,3,4,5] => [1,0,1,1,1,1,1,0,0,0,0,0]
=> ? = 1
[1,3,6,5,2,4] => [1,2,3,6,4,5] => [1,6,2,3,4,5] => [1,0,1,1,1,1,1,0,0,0,0,0]
=> ? = 1
[1,3,6,5,4,2] => [1,2,3,6,4,5] => [1,6,2,3,4,5] => [1,0,1,1,1,1,1,0,0,0,0,0]
=> ? = 1
[1,4,2,3,5,6] => [1,2,4,3,5,6] => [4,1,2,3,5,6] => [1,1,1,1,0,0,0,0,1,0,1,0]
=> ? = 1
[1,4,2,3,6,5] => [1,2,4,3,5,6] => [4,1,2,3,5,6] => [1,1,1,1,0,0,0,0,1,0,1,0]
=> ? = 1
[1,4,2,5,3,6] => [1,2,4,5,3,6] => [4,5,1,2,3,6] => [1,1,1,1,0,1,0,0,0,0,1,0]
=> ? = 1
[1,4,2,5,6,3] => [1,2,4,5,6,3] => [4,5,6,1,2,3] => [1,1,1,1,0,1,0,1,0,0,0,0]
=> ? = 1
[1,4,2,6,3,5] => [1,2,4,6,5,3] => [6,4,5,1,2,3] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> ? = 1
[1,4,2,6,5,3] => [1,2,4,6,3,5] => [6,4,1,2,3,5] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> ? = 1
[1,4,3,2,5,6] => [1,2,4,3,5,6] => [4,1,2,3,5,6] => [1,1,1,1,0,0,0,0,1,0,1,0]
=> ? = 1
[1,4,3,2,6,5] => [1,2,4,3,5,6] => [4,1,2,3,5,6] => [1,1,1,1,0,0,0,0,1,0,1,0]
=> ? = 1
[1,4,3,5,2,6] => [1,2,4,5,3,6] => [4,5,1,2,3,6] => [1,1,1,1,0,1,0,0,0,0,1,0]
=> ? = 1
[1,4,3,5,6,2] => [1,2,4,5,6,3] => [4,5,6,1,2,3] => [1,1,1,1,0,1,0,1,0,0,0,0]
=> ? = 1
[1,4,3,6,2,5] => [1,2,4,6,5,3] => [6,4,5,1,2,3] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> ? = 1
[1,4,3,6,5,2] => [1,2,4,6,3,5] => [6,4,1,2,3,5] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> ? = 1
[1,4,5,2,3,6] => [1,2,4,3,5,6] => [4,1,2,3,5,6] => [1,1,1,1,0,0,0,0,1,0,1,0]
=> ? = 1
[1,4,5,2,6,3] => [1,2,4,3,5,6] => [4,1,2,3,5,6] => [1,1,1,1,0,0,0,0,1,0,1,0]
=> ? = 1
[1,4,5,3,2,6] => [1,2,4,3,5,6] => [4,1,2,3,5,6] => [1,1,1,1,0,0,0,0,1,0,1,0]
=> ? = 1
[1,4,5,3,6,2] => [1,2,4,3,5,6] => [4,1,2,3,5,6] => [1,1,1,1,0,0,0,0,1,0,1,0]
=> ? = 1
[1,4,5,6,2,3] => [1,2,4,6,3,5] => [6,4,1,2,3,5] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> ? = 1
Description
Half of the Loewy length minus one of a modified stable Auslander algebra of the Nakayama algebra corresponding to the Dyck path.
The modified algebra B is obtained from the stable Auslander algebra kQ/I by deleting all relations which contain walks of length at least three (conjectural this step of deletion is not necessary as the stable higher Auslander algebras might be quadratic) and taking as B then the algebra kQ^(op)/J when J is the quadratic perp of the ideal I.
See http://www.findstat.org/DyckPaths/NakayamaAlgebras for the definition of Loewy length and Nakayama algebras associated to Dyck paths.
Matching statistic: St000307
(load all 9 compositions to match this statistic)
(load all 9 compositions to match this statistic)
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
Mp00121: Dyck paths —Cori-Le Borgne involution⟶ Dyck paths
Mp00232: Dyck paths —parallelogram poset⟶ Posets
St000307: Posets ⟶ ℤResult quality: 2% ●values known / values provided: 2%●distinct values known / distinct values provided: 67%
Mp00121: Dyck paths —Cori-Le Borgne involution⟶ Dyck paths
Mp00232: Dyck paths —parallelogram poset⟶ Posets
St000307: Posets ⟶ ℤResult quality: 2% ●values known / values provided: 2%●distinct values known / distinct values provided: 67%
Values
[1] => [1,0]
=> [1,0]
=> ([],1)
=> 1 = 0 + 1
[1,2] => [1,0,1,0]
=> [1,0,1,0]
=> ([(0,1)],2)
=> 1 = 0 + 1
[2,1] => [1,1,0,0]
=> [1,1,0,0]
=> ([(0,1)],2)
=> 1 = 0 + 1
[1,2,3] => [1,0,1,0,1,0]
=> [1,0,1,0,1,0]
=> ([(0,2),(2,1)],3)
=> 1 = 0 + 1
[1,3,2] => [1,0,1,1,0,0]
=> [1,1,0,1,0,0]
=> ([(0,2),(2,1)],3)
=> 1 = 0 + 1
[2,1,3] => [1,1,0,0,1,0]
=> [1,1,0,0,1,0]
=> ([(0,2),(2,1)],3)
=> 1 = 0 + 1
[2,3,1] => [1,1,0,1,0,0]
=> [1,0,1,1,0,0]
=> ([(0,2),(2,1)],3)
=> 1 = 0 + 1
[3,1,2] => [1,1,1,0,0,0]
=> [1,1,1,0,0,0]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2 = 1 + 1
[3,2,1] => [1,1,1,0,0,0]
=> [1,1,1,0,0,0]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2 = 1 + 1
[1,2,3,4] => [1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0]
=> ([(0,3),(2,1),(3,2)],4)
=> 1 = 0 + 1
[1,2,4,3] => [1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,0]
=> ([(0,3),(2,1),(3,2)],4)
=> 1 = 0 + 1
[1,3,2,4] => [1,0,1,1,0,0,1,0]
=> [1,1,0,1,0,0,1,0]
=> ([(0,3),(2,1),(3,2)],4)
=> 1 = 0 + 1
[1,3,4,2] => [1,0,1,1,0,1,0,0]
=> [1,0,1,1,0,1,0,0]
=> ([(0,3),(2,1),(3,2)],4)
=> 1 = 0 + 1
[1,4,2,3] => [1,0,1,1,1,0,0,0]
=> [1,1,1,0,1,0,0,0]
=> ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 2 = 1 + 1
[1,4,3,2] => [1,0,1,1,1,0,0,0]
=> [1,1,1,0,1,0,0,0]
=> ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 2 = 1 + 1
[2,1,3,4] => [1,1,0,0,1,0,1,0]
=> [1,1,0,0,1,0,1,0]
=> ([(0,3),(2,1),(3,2)],4)
=> 1 = 0 + 1
[2,1,4,3] => [1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> ([(0,3),(2,1),(3,2)],4)
=> 1 = 0 + 1
[2,3,1,4] => [1,1,0,1,0,0,1,0]
=> [1,0,1,1,0,0,1,0]
=> ([(0,3),(2,1),(3,2)],4)
=> 1 = 0 + 1
[2,3,4,1] => [1,1,0,1,0,1,0,0]
=> [1,0,1,0,1,1,0,0]
=> ([(0,3),(2,1),(3,2)],4)
=> 1 = 0 + 1
[2,4,1,3] => [1,1,0,1,1,0,0,0]
=> [1,1,0,1,1,0,0,0]
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> 2 = 1 + 1
[2,4,3,1] => [1,1,0,1,1,0,0,0]
=> [1,1,0,1,1,0,0,0]
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> 2 = 1 + 1
[3,1,2,4] => [1,1,1,0,0,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> 2 = 1 + 1
[3,1,4,2] => [1,1,1,0,0,1,0,0]
=> [1,1,1,0,0,0,1,0]
=> ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> 2 = 1 + 1
[3,2,1,4] => [1,1,1,0,0,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> 2 = 1 + 1
[3,2,4,1] => [1,1,1,0,0,1,0,0]
=> [1,1,1,0,0,0,1,0]
=> ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> 2 = 1 + 1
[3,4,1,2] => [1,1,1,0,1,0,0,0]
=> [1,0,1,1,1,0,0,0]
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> 2 = 1 + 1
[3,4,2,1] => [1,1,1,0,1,0,0,0]
=> [1,0,1,1,1,0,0,0]
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> 2 = 1 + 1
[4,1,2,3] => [1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 2 = 1 + 1
[4,1,3,2] => [1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 2 = 1 + 1
[4,2,1,3] => [1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 2 = 1 + 1
[4,2,3,1] => [1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 2 = 1 + 1
[4,3,1,2] => [1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 2 = 1 + 1
[4,3,2,1] => [1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 2 = 1 + 1
[1,2,3,4,5] => [1,0,1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1 = 0 + 1
[1,2,3,5,4] => [1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1 = 0 + 1
[1,2,4,3,5] => [1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,0,1,0,0,1,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1 = 0 + 1
[1,2,4,5,3] => [1,0,1,0,1,1,0,1,0,0]
=> [1,0,1,1,0,1,0,1,0,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1 = 0 + 1
[1,2,5,3,4] => [1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? = 1 + 1
[1,2,5,4,3] => [1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? = 1 + 1
[1,3,2,4,5] => [1,0,1,1,0,0,1,0,1,0]
=> [1,1,0,1,0,0,1,0,1,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1 = 0 + 1
[1,3,2,5,4] => [1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,1,0,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1 = 0 + 1
[1,3,4,2,5] => [1,0,1,1,0,1,0,0,1,0]
=> [1,0,1,1,0,1,0,0,1,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1 = 0 + 1
[1,3,4,5,2] => [1,0,1,1,0,1,0,1,0,0]
=> [1,0,1,0,1,1,0,1,0,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1 = 0 + 1
[1,3,5,2,4] => [1,0,1,1,0,1,1,0,0,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> ([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
=> ? = 1 + 1
[1,3,5,4,2] => [1,0,1,1,0,1,1,0,0,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> ([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
=> ? = 1 + 1
[1,4,2,3,5] => [1,0,1,1,1,0,0,0,1,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> ([(0,3),(0,4),(2,6),(3,5),(4,2),(4,5),(5,6),(6,1)],7)
=> ? = 1 + 1
[1,4,2,5,3] => [1,0,1,1,1,0,0,1,0,0]
=> [1,1,1,0,1,0,0,0,1,0]
=> ([(0,3),(0,4),(2,6),(3,5),(4,2),(4,5),(5,6),(6,1)],7)
=> ? = 1 + 1
[1,4,3,2,5] => [1,0,1,1,1,0,0,0,1,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> ([(0,3),(0,4),(2,6),(3,5),(4,2),(4,5),(5,6),(6,1)],7)
=> ? = 1 + 1
[1,4,3,5,2] => [1,0,1,1,1,0,0,1,0,0]
=> [1,1,1,0,1,0,0,0,1,0]
=> ([(0,3),(0,4),(2,6),(3,5),(4,2),(4,5),(5,6),(6,1)],7)
=> ? = 1 + 1
[1,4,5,2,3] => [1,0,1,1,1,0,1,0,0,0]
=> [1,0,1,1,1,0,1,0,0,0]
=> ([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
=> ? = 1 + 1
[1,4,5,3,2] => [1,0,1,1,1,0,1,0,0,0]
=> [1,0,1,1,1,0,1,0,0,0]
=> ([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
=> ? = 1 + 1
[1,5,2,3,4] => [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? = 1 + 1
[1,5,2,4,3] => [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? = 1 + 1
[1,5,3,2,4] => [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? = 1 + 1
[1,5,3,4,2] => [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? = 1 + 1
[1,5,4,2,3] => [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? = 1 + 1
[1,5,4,3,2] => [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? = 1 + 1
[2,1,3,4,5] => [1,1,0,0,1,0,1,0,1,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1 = 0 + 1
[2,1,3,5,4] => [1,1,0,0,1,0,1,1,0,0]
=> [1,1,0,1,0,0,1,1,0,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1 = 0 + 1
[2,1,4,3,5] => [1,1,0,0,1,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1 = 0 + 1
[2,1,4,5,3] => [1,1,0,0,1,1,0,1,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1 = 0 + 1
[2,1,5,3,4] => [1,1,0,0,1,1,1,0,0,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> ([(0,3),(0,4),(1,5),(2,5),(3,6),(4,6),(6,1),(6,2)],7)
=> ? = 1 + 1
[2,1,5,4,3] => [1,1,0,0,1,1,1,0,0,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> ([(0,3),(0,4),(1,5),(2,5),(3,6),(4,6),(6,1),(6,2)],7)
=> ? = 1 + 1
[2,3,1,4,5] => [1,1,0,1,0,0,1,0,1,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1 = 0 + 1
[2,3,1,5,4] => [1,1,0,1,0,0,1,1,0,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1 = 0 + 1
[2,3,4,1,5] => [1,1,0,1,0,1,0,0,1,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1 = 0 + 1
[2,3,4,5,1] => [1,1,0,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1 = 0 + 1
[2,3,5,1,4] => [1,1,0,1,0,1,1,0,0,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> ([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6)
=> 2 = 1 + 1
[2,5,1,3,4] => [1,1,0,1,1,1,0,0,0,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> ([(0,3),(0,4),(1,6),(2,5),(3,7),(4,2),(4,7),(5,6),(7,1),(7,5)],8)
=> ? = 1 + 1
[2,5,1,4,3] => [1,1,0,1,1,1,0,0,0,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> ([(0,3),(0,4),(1,6),(2,5),(3,7),(4,2),(4,7),(5,6),(7,1),(7,5)],8)
=> ? = 1 + 1
[2,5,3,1,4] => [1,1,0,1,1,1,0,0,0,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> ([(0,3),(0,4),(1,6),(2,5),(3,7),(4,2),(4,7),(5,6),(7,1),(7,5)],8)
=> ? = 1 + 1
[2,5,3,4,1] => [1,1,0,1,1,1,0,0,0,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> ([(0,3),(0,4),(1,6),(2,5),(3,7),(4,2),(4,7),(5,6),(7,1),(7,5)],8)
=> ? = 1 + 1
[2,5,4,1,3] => [1,1,0,1,1,1,0,0,0,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> ([(0,3),(0,4),(1,6),(2,5),(3,7),(4,2),(4,7),(5,6),(7,1),(7,5)],8)
=> ? = 1 + 1
[2,5,4,3,1] => [1,1,0,1,1,1,0,0,0,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> ([(0,3),(0,4),(1,6),(2,5),(3,7),(4,2),(4,7),(5,6),(7,1),(7,5)],8)
=> ? = 1 + 1
[3,5,1,2,4] => [1,1,1,0,1,1,0,0,0,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> ([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
=> ? = 1 + 1
[3,5,1,4,2] => [1,1,1,0,1,1,0,0,0,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> ([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
=> ? = 1 + 1
[3,5,2,1,4] => [1,1,1,0,1,1,0,0,0,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> ([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
=> ? = 1 + 1
[3,5,2,4,1] => [1,1,1,0,1,1,0,0,0,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> ([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
=> ? = 1 + 1
[3,5,4,1,2] => [1,1,1,0,1,1,0,0,0,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> ([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
=> ? = 1 + 1
[3,5,4,2,1] => [1,1,1,0,1,1,0,0,0,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> ([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
=> ? = 1 + 1
[4,1,2,3,5] => [1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> ([(0,3),(0,4),(1,6),(2,5),(3,7),(4,2),(4,7),(5,6),(7,1),(7,5)],8)
=> ? = 1 + 1
[4,1,2,5,3] => [1,1,1,1,0,0,0,1,0,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> ([(0,3),(0,4),(2,6),(3,5),(4,2),(4,5),(5,6),(6,1)],7)
=> ? = 1 + 1
[4,1,3,2,5] => [1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> ([(0,3),(0,4),(1,6),(2,5),(3,7),(4,2),(4,7),(5,6),(7,1),(7,5)],8)
=> ? = 1 + 1
[4,1,3,5,2] => [1,1,1,1,0,0,0,1,0,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> ([(0,3),(0,4),(2,6),(3,5),(4,2),(4,5),(5,6),(6,1)],7)
=> ? = 1 + 1
[4,1,5,2,3] => [1,1,1,1,0,0,1,0,0,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> ([(0,3),(0,4),(2,6),(3,5),(4,2),(4,5),(5,6),(6,1)],7)
=> ? = 1 + 1
[4,1,5,3,2] => [1,1,1,1,0,0,1,0,0,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> ([(0,3),(0,4),(2,6),(3,5),(4,2),(4,5),(5,6),(6,1)],7)
=> ? = 1 + 1
[4,2,1,3,5] => [1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> ([(0,3),(0,4),(1,6),(2,5),(3,7),(4,2),(4,7),(5,6),(7,1),(7,5)],8)
=> ? = 1 + 1
[4,2,1,5,3] => [1,1,1,1,0,0,0,1,0,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> ([(0,3),(0,4),(2,6),(3,5),(4,2),(4,5),(5,6),(6,1)],7)
=> ? = 1 + 1
[4,2,3,1,5] => [1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> ([(0,3),(0,4),(1,6),(2,5),(3,7),(4,2),(4,7),(5,6),(7,1),(7,5)],8)
=> ? = 1 + 1
[4,2,3,5,1] => [1,1,1,1,0,0,0,1,0,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> ([(0,3),(0,4),(2,6),(3,5),(4,2),(4,5),(5,6),(6,1)],7)
=> ? = 1 + 1
[4,2,5,1,3] => [1,1,1,1,0,0,1,0,0,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> ([(0,3),(0,4),(2,6),(3,5),(4,2),(4,5),(5,6),(6,1)],7)
=> ? = 1 + 1
[4,2,5,3,1] => [1,1,1,1,0,0,1,0,0,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> ([(0,3),(0,4),(2,6),(3,5),(4,2),(4,5),(5,6),(6,1)],7)
=> ? = 1 + 1
[4,3,1,2,5] => [1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> ([(0,3),(0,4),(1,6),(2,5),(3,7),(4,2),(4,7),(5,6),(7,1),(7,5)],8)
=> ? = 1 + 1
[4,3,1,5,2] => [1,1,1,1,0,0,0,1,0,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> ([(0,3),(0,4),(2,6),(3,5),(4,2),(4,5),(5,6),(6,1)],7)
=> ? = 1 + 1
[4,3,2,1,5] => [1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> ([(0,3),(0,4),(1,6),(2,5),(3,7),(4,2),(4,7),(5,6),(7,1),(7,5)],8)
=> ? = 1 + 1
[4,3,2,5,1] => [1,1,1,1,0,0,0,1,0,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> ([(0,3),(0,4),(2,6),(3,5),(4,2),(4,5),(5,6),(6,1)],7)
=> ? = 1 + 1
[4,3,5,1,2] => [1,1,1,1,0,0,1,0,0,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> ([(0,3),(0,4),(2,6),(3,5),(4,2),(4,5),(5,6),(6,1)],7)
=> ? = 1 + 1
[4,3,5,2,1] => [1,1,1,1,0,0,1,0,0,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> ([(0,3),(0,4),(2,6),(3,5),(4,2),(4,5),(5,6),(6,1)],7)
=> ? = 1 + 1
[4,5,1,2,3] => [1,1,1,1,0,1,0,0,0,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> ([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
=> ? = 1 + 1
[4,5,1,3,2] => [1,1,1,1,0,1,0,0,0,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> ([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
=> ? = 1 + 1
Description
The number of rowmotion orbits of a poset.
Rowmotion is an operation on order ideals in a poset P. It sends an order ideal I to the order ideal generated by the minimal antichain of P∖I.
Matching statistic: St001553
Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
Mp00175: Permutations —inverse Foata bijection⟶ Permutations
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
St001553: Dyck paths ⟶ ℤResult quality: 2% ●values known / values provided: 2%●distinct values known / distinct values provided: 67%
Mp00175: Permutations —inverse Foata bijection⟶ Permutations
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
St001553: Dyck paths ⟶ ℤResult quality: 2% ●values known / values provided: 2%●distinct values known / distinct values provided: 67%
Values
[1] => [1] => [1] => [1,0]
=> 0
[1,2] => [1,2] => [1,2] => [1,0,1,0]
=> 0
[2,1] => [1,2] => [1,2] => [1,0,1,0]
=> 0
[1,2,3] => [1,2,3] => [1,2,3] => [1,0,1,0,1,0]
=> 0
[1,3,2] => [1,2,3] => [1,2,3] => [1,0,1,0,1,0]
=> 0
[2,1,3] => [1,2,3] => [1,2,3] => [1,0,1,0,1,0]
=> 0
[2,3,1] => [1,2,3] => [1,2,3] => [1,0,1,0,1,0]
=> 0
[3,1,2] => [1,3,2] => [3,1,2] => [1,1,1,0,0,0]
=> 1
[3,2,1] => [1,3,2] => [3,1,2] => [1,1,1,0,0,0]
=> 1
[1,2,3,4] => [1,2,3,4] => [1,2,3,4] => [1,0,1,0,1,0,1,0]
=> 0
[1,2,4,3] => [1,2,3,4] => [1,2,3,4] => [1,0,1,0,1,0,1,0]
=> 0
[1,3,2,4] => [1,2,3,4] => [1,2,3,4] => [1,0,1,0,1,0,1,0]
=> 0
[1,3,4,2] => [1,2,3,4] => [1,2,3,4] => [1,0,1,0,1,0,1,0]
=> 0
[1,4,2,3] => [1,2,4,3] => [4,1,2,3] => [1,1,1,1,0,0,0,0]
=> 1
[1,4,3,2] => [1,2,4,3] => [4,1,2,3] => [1,1,1,1,0,0,0,0]
=> 1
[2,1,3,4] => [1,2,3,4] => [1,2,3,4] => [1,0,1,0,1,0,1,0]
=> 0
[2,1,4,3] => [1,2,3,4] => [1,2,3,4] => [1,0,1,0,1,0,1,0]
=> 0
[2,3,1,4] => [1,2,3,4] => [1,2,3,4] => [1,0,1,0,1,0,1,0]
=> 0
[2,3,4,1] => [1,2,3,4] => [1,2,3,4] => [1,0,1,0,1,0,1,0]
=> 0
[2,4,1,3] => [1,2,4,3] => [4,1,2,3] => [1,1,1,1,0,0,0,0]
=> 1
[2,4,3,1] => [1,2,4,3] => [4,1,2,3] => [1,1,1,1,0,0,0,0]
=> 1
[3,1,2,4] => [1,3,2,4] => [3,1,2,4] => [1,1,1,0,0,0,1,0]
=> 1
[3,1,4,2] => [1,3,4,2] => [3,4,1,2] => [1,1,1,0,1,0,0,0]
=> 1
[3,2,1,4] => [1,3,2,4] => [3,1,2,4] => [1,1,1,0,0,0,1,0]
=> 1
[3,2,4,1] => [1,3,4,2] => [3,4,1,2] => [1,1,1,0,1,0,0,0]
=> 1
[3,4,1,2] => [1,3,2,4] => [3,1,2,4] => [1,1,1,0,0,0,1,0]
=> 1
[3,4,2,1] => [1,3,2,4] => [3,1,2,4] => [1,1,1,0,0,0,1,0]
=> 1
[4,1,2,3] => [1,4,3,2] => [4,3,1,2] => [1,1,1,1,0,0,0,0]
=> 1
[4,1,3,2] => [1,4,2,3] => [1,4,2,3] => [1,0,1,1,1,0,0,0]
=> 1
[4,2,1,3] => [1,4,3,2] => [4,3,1,2] => [1,1,1,1,0,0,0,0]
=> 1
[4,2,3,1] => [1,4,2,3] => [1,4,2,3] => [1,0,1,1,1,0,0,0]
=> 1
[4,3,1,2] => [1,4,2,3] => [1,4,2,3] => [1,0,1,1,1,0,0,0]
=> 1
[4,3,2,1] => [1,4,2,3] => [1,4,2,3] => [1,0,1,1,1,0,0,0]
=> 1
[1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => [1,0,1,0,1,0,1,0,1,0]
=> 0
[1,2,3,5,4] => [1,2,3,4,5] => [1,2,3,4,5] => [1,0,1,0,1,0,1,0,1,0]
=> 0
[1,2,4,3,5] => [1,2,3,4,5] => [1,2,3,4,5] => [1,0,1,0,1,0,1,0,1,0]
=> 0
[1,2,4,5,3] => [1,2,3,4,5] => [1,2,3,4,5] => [1,0,1,0,1,0,1,0,1,0]
=> 0
[1,2,5,3,4] => [1,2,3,5,4] => [5,1,2,3,4] => [1,1,1,1,1,0,0,0,0,0]
=> 1
[1,2,5,4,3] => [1,2,3,5,4] => [5,1,2,3,4] => [1,1,1,1,1,0,0,0,0,0]
=> 1
[1,3,2,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => [1,0,1,0,1,0,1,0,1,0]
=> 0
[1,3,2,5,4] => [1,2,3,4,5] => [1,2,3,4,5] => [1,0,1,0,1,0,1,0,1,0]
=> 0
[1,3,4,2,5] => [1,2,3,4,5] => [1,2,3,4,5] => [1,0,1,0,1,0,1,0,1,0]
=> 0
[1,3,4,5,2] => [1,2,3,4,5] => [1,2,3,4,5] => [1,0,1,0,1,0,1,0,1,0]
=> 0
[1,3,5,2,4] => [1,2,3,5,4] => [5,1,2,3,4] => [1,1,1,1,1,0,0,0,0,0]
=> 1
[1,3,5,4,2] => [1,2,3,5,4] => [5,1,2,3,4] => [1,1,1,1,1,0,0,0,0,0]
=> 1
[1,4,2,3,5] => [1,2,4,3,5] => [4,1,2,3,5] => [1,1,1,1,0,0,0,0,1,0]
=> 1
[1,4,2,5,3] => [1,2,4,5,3] => [4,5,1,2,3] => [1,1,1,1,0,1,0,0,0,0]
=> 1
[1,4,3,2,5] => [1,2,4,3,5] => [4,1,2,3,5] => [1,1,1,1,0,0,0,0,1,0]
=> 1
[1,4,3,5,2] => [1,2,4,5,3] => [4,5,1,2,3] => [1,1,1,1,0,1,0,0,0,0]
=> 1
[1,4,5,2,3] => [1,2,4,3,5] => [4,1,2,3,5] => [1,1,1,1,0,0,0,0,1,0]
=> 1
[1,2,3,4,5,6] => [1,2,3,4,5,6] => [1,2,3,4,5,6] => [1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 0
[1,2,3,4,6,5] => [1,2,3,4,5,6] => [1,2,3,4,5,6] => [1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 0
[1,2,3,5,4,6] => [1,2,3,4,5,6] => [1,2,3,4,5,6] => [1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 0
[1,2,3,5,6,4] => [1,2,3,4,5,6] => [1,2,3,4,5,6] => [1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 0
[1,2,3,6,4,5] => [1,2,3,4,6,5] => [6,1,2,3,4,5] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> ? = 1
[1,2,3,6,5,4] => [1,2,3,4,6,5] => [6,1,2,3,4,5] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> ? = 1
[1,2,4,3,5,6] => [1,2,3,4,5,6] => [1,2,3,4,5,6] => [1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 0
[1,2,4,3,6,5] => [1,2,3,4,5,6] => [1,2,3,4,5,6] => [1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 0
[1,2,4,5,3,6] => [1,2,3,4,5,6] => [1,2,3,4,5,6] => [1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 0
[1,2,4,5,6,3] => [1,2,3,4,5,6] => [1,2,3,4,5,6] => [1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 0
[1,2,4,6,3,5] => [1,2,3,4,6,5] => [6,1,2,3,4,5] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> ? = 1
[1,2,4,6,5,3] => [1,2,3,4,6,5] => [6,1,2,3,4,5] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> ? = 1
[1,2,5,3,4,6] => [1,2,3,5,4,6] => [5,1,2,3,4,6] => [1,1,1,1,1,0,0,0,0,0,1,0]
=> ? = 1
[1,2,5,3,6,4] => [1,2,3,5,6,4] => [5,6,1,2,3,4] => [1,1,1,1,1,0,1,0,0,0,0,0]
=> ? = 1
[1,2,5,4,3,6] => [1,2,3,5,4,6] => [5,1,2,3,4,6] => [1,1,1,1,1,0,0,0,0,0,1,0]
=> ? = 1
[1,2,5,4,6,3] => [1,2,3,5,6,4] => [5,6,1,2,3,4] => [1,1,1,1,1,0,1,0,0,0,0,0]
=> ? = 1
[1,2,5,6,3,4] => [1,2,3,5,4,6] => [5,1,2,3,4,6] => [1,1,1,1,1,0,0,0,0,0,1,0]
=> ? = 1
[1,2,5,6,4,3] => [1,2,3,5,4,6] => [5,1,2,3,4,6] => [1,1,1,1,1,0,0,0,0,0,1,0]
=> ? = 1
[1,2,6,3,4,5] => [1,2,3,6,5,4] => [6,5,1,2,3,4] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> ? = 1
[1,2,6,3,5,4] => [1,2,3,6,4,5] => [1,6,2,3,4,5] => [1,0,1,1,1,1,1,0,0,0,0,0]
=> ? = 1
[1,2,6,4,3,5] => [1,2,3,6,5,4] => [6,5,1,2,3,4] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> ? = 1
[1,2,6,4,5,3] => [1,2,3,6,4,5] => [1,6,2,3,4,5] => [1,0,1,1,1,1,1,0,0,0,0,0]
=> ? = 1
[1,2,6,5,3,4] => [1,2,3,6,4,5] => [1,6,2,3,4,5] => [1,0,1,1,1,1,1,0,0,0,0,0]
=> ? = 1
[1,2,6,5,4,3] => [1,2,3,6,4,5] => [1,6,2,3,4,5] => [1,0,1,1,1,1,1,0,0,0,0,0]
=> ? = 1
[1,3,2,4,5,6] => [1,2,3,4,5,6] => [1,2,3,4,5,6] => [1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 0
[1,3,2,4,6,5] => [1,2,3,4,5,6] => [1,2,3,4,5,6] => [1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 0
[1,3,2,5,4,6] => [1,2,3,4,5,6] => [1,2,3,4,5,6] => [1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 0
[1,3,2,5,6,4] => [1,2,3,4,5,6] => [1,2,3,4,5,6] => [1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 0
[1,3,2,6,4,5] => [1,2,3,4,6,5] => [6,1,2,3,4,5] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> ? = 1
[1,3,2,6,5,4] => [1,2,3,4,6,5] => [6,1,2,3,4,5] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> ? = 1
[1,3,4,2,5,6] => [1,2,3,4,5,6] => [1,2,3,4,5,6] => [1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 0
[1,3,4,2,6,5] => [1,2,3,4,5,6] => [1,2,3,4,5,6] => [1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 0
[1,3,4,5,2,6] => [1,2,3,4,5,6] => [1,2,3,4,5,6] => [1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 0
[1,3,4,5,6,2] => [1,2,3,4,5,6] => [1,2,3,4,5,6] => [1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 0
[1,3,4,6,2,5] => [1,2,3,4,6,5] => [6,1,2,3,4,5] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> ? = 1
[1,3,4,6,5,2] => [1,2,3,4,6,5] => [6,1,2,3,4,5] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> ? = 1
[1,3,5,2,4,6] => [1,2,3,5,4,6] => [5,1,2,3,4,6] => [1,1,1,1,1,0,0,0,0,0,1,0]
=> ? = 1
[1,3,5,2,6,4] => [1,2,3,5,6,4] => [5,6,1,2,3,4] => [1,1,1,1,1,0,1,0,0,0,0,0]
=> ? = 1
[1,3,5,4,2,6] => [1,2,3,5,4,6] => [5,1,2,3,4,6] => [1,1,1,1,1,0,0,0,0,0,1,0]
=> ? = 1
[1,3,5,4,6,2] => [1,2,3,5,6,4] => [5,6,1,2,3,4] => [1,1,1,1,1,0,1,0,0,0,0,0]
=> ? = 1
[1,3,5,6,2,4] => [1,2,3,5,4,6] => [5,1,2,3,4,6] => [1,1,1,1,1,0,0,0,0,0,1,0]
=> ? = 1
[1,3,5,6,4,2] => [1,2,3,5,4,6] => [5,1,2,3,4,6] => [1,1,1,1,1,0,0,0,0,0,1,0]
=> ? = 1
[1,3,6,2,4,5] => [1,2,3,6,5,4] => [6,5,1,2,3,4] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> ? = 1
[1,3,6,2,5,4] => [1,2,3,6,4,5] => [1,6,2,3,4,5] => [1,0,1,1,1,1,1,0,0,0,0,0]
=> ? = 1
[1,3,6,4,2,5] => [1,2,3,6,5,4] => [6,5,1,2,3,4] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> ? = 1
[1,3,6,4,5,2] => [1,2,3,6,4,5] => [1,6,2,3,4,5] => [1,0,1,1,1,1,1,0,0,0,0,0]
=> ? = 1
[1,3,6,5,2,4] => [1,2,3,6,4,5] => [1,6,2,3,4,5] => [1,0,1,1,1,1,1,0,0,0,0,0]
=> ? = 1
[1,3,6,5,4,2] => [1,2,3,6,4,5] => [1,6,2,3,4,5] => [1,0,1,1,1,1,1,0,0,0,0,0]
=> ? = 1
[1,4,2,3,5,6] => [1,2,4,3,5,6] => [4,1,2,3,5,6] => [1,1,1,1,0,0,0,0,1,0,1,0]
=> ? = 1
[1,4,2,3,6,5] => [1,2,4,3,5,6] => [4,1,2,3,5,6] => [1,1,1,1,0,0,0,0,1,0,1,0]
=> ? = 1
Description
The number of indecomposable summands of the square of the Jacobson radical as a bimodule in the Nakayama algebra corresponding to the Dyck path.
The statistic returns zero in case that bimodule is the zero module.
Matching statistic: St001555
Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
Mp00159: Permutations —Demazure product with inverse⟶ Permutations
Mp00170: Permutations —to signed permutation⟶ Signed permutations
St001555: Signed permutations ⟶ ℤResult quality: 2% ●values known / values provided: 2%●distinct values known / distinct values provided: 67%
Mp00159: Permutations —Demazure product with inverse⟶ Permutations
Mp00170: Permutations —to signed permutation⟶ Signed permutations
St001555: Signed permutations ⟶ ℤResult quality: 2% ●values known / values provided: 2%●distinct values known / distinct values provided: 67%
Values
[1] => [1] => [1] => [1] => 1 = 0 + 1
[1,2] => [1,2] => [1,2] => [1,2] => 1 = 0 + 1
[2,1] => [1,2] => [1,2] => [1,2] => 1 = 0 + 1
[1,2,3] => [1,2,3] => [1,2,3] => [1,2,3] => 1 = 0 + 1
[1,3,2] => [1,2,3] => [1,2,3] => [1,2,3] => 1 = 0 + 1
[2,1,3] => [1,2,3] => [1,2,3] => [1,2,3] => 1 = 0 + 1
[2,3,1] => [1,2,3] => [1,2,3] => [1,2,3] => 1 = 0 + 1
[3,1,2] => [1,3,2] => [1,3,2] => [1,3,2] => 2 = 1 + 1
[3,2,1] => [1,3,2] => [1,3,2] => [1,3,2] => 2 = 1 + 1
[1,2,3,4] => [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 1 = 0 + 1
[1,2,4,3] => [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 1 = 0 + 1
[1,3,2,4] => [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 1 = 0 + 1
[1,3,4,2] => [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 1 = 0 + 1
[1,4,2,3] => [1,2,4,3] => [1,2,4,3] => [1,2,4,3] => 2 = 1 + 1
[1,4,3,2] => [1,2,4,3] => [1,2,4,3] => [1,2,4,3] => 2 = 1 + 1
[2,1,3,4] => [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 1 = 0 + 1
[2,1,4,3] => [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 1 = 0 + 1
[2,3,1,4] => [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 1 = 0 + 1
[2,3,4,1] => [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 1 = 0 + 1
[2,4,1,3] => [1,2,4,3] => [1,2,4,3] => [1,2,4,3] => 2 = 1 + 1
[2,4,3,1] => [1,2,4,3] => [1,2,4,3] => [1,2,4,3] => 2 = 1 + 1
[3,1,2,4] => [1,3,2,4] => [1,3,2,4] => [1,3,2,4] => 2 = 1 + 1
[3,1,4,2] => [1,3,4,2] => [1,4,3,2] => [1,4,3,2] => 2 = 1 + 1
[3,2,1,4] => [1,3,2,4] => [1,3,2,4] => [1,3,2,4] => 2 = 1 + 1
[3,2,4,1] => [1,3,4,2] => [1,4,3,2] => [1,4,3,2] => 2 = 1 + 1
[3,4,1,2] => [1,3,2,4] => [1,3,2,4] => [1,3,2,4] => 2 = 1 + 1
[3,4,2,1] => [1,3,2,4] => [1,3,2,4] => [1,3,2,4] => 2 = 1 + 1
[4,1,2,3] => [1,4,3,2] => [1,4,3,2] => [1,4,3,2] => 2 = 1 + 1
[4,1,3,2] => [1,4,2,3] => [1,4,3,2] => [1,4,3,2] => 2 = 1 + 1
[4,2,1,3] => [1,4,3,2] => [1,4,3,2] => [1,4,3,2] => 2 = 1 + 1
[4,2,3,1] => [1,4,2,3] => [1,4,3,2] => [1,4,3,2] => 2 = 1 + 1
[4,3,1,2] => [1,4,2,3] => [1,4,3,2] => [1,4,3,2] => 2 = 1 + 1
[4,3,2,1] => [1,4,2,3] => [1,4,3,2] => [1,4,3,2] => 2 = 1 + 1
[1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => 1 = 0 + 1
[1,2,3,5,4] => [1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => 1 = 0 + 1
[1,2,4,3,5] => [1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => 1 = 0 + 1
[1,2,4,5,3] => [1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => 1 = 0 + 1
[1,2,5,3,4] => [1,2,3,5,4] => [1,2,3,5,4] => [1,2,3,5,4] => 2 = 1 + 1
[1,2,5,4,3] => [1,2,3,5,4] => [1,2,3,5,4] => [1,2,3,5,4] => 2 = 1 + 1
[1,3,2,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => 1 = 0 + 1
[1,3,2,5,4] => [1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => 1 = 0 + 1
[1,3,4,2,5] => [1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => 1 = 0 + 1
[1,3,4,5,2] => [1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => 1 = 0 + 1
[1,3,5,2,4] => [1,2,3,5,4] => [1,2,3,5,4] => [1,2,3,5,4] => 2 = 1 + 1
[1,3,5,4,2] => [1,2,3,5,4] => [1,2,3,5,4] => [1,2,3,5,4] => 2 = 1 + 1
[1,4,2,3,5] => [1,2,4,3,5] => [1,2,4,3,5] => [1,2,4,3,5] => 2 = 1 + 1
[1,4,2,5,3] => [1,2,4,5,3] => [1,2,5,4,3] => [1,2,5,4,3] => 2 = 1 + 1
[1,4,3,2,5] => [1,2,4,3,5] => [1,2,4,3,5] => [1,2,4,3,5] => 2 = 1 + 1
[1,4,3,5,2] => [1,2,4,5,3] => [1,2,5,4,3] => [1,2,5,4,3] => 2 = 1 + 1
[1,4,5,2,3] => [1,2,4,3,5] => [1,2,4,3,5] => [1,2,4,3,5] => 2 = 1 + 1
[1,2,3,4,5,6] => [1,2,3,4,5,6] => [1,2,3,4,5,6] => [1,2,3,4,5,6] => ? = 0 + 1
[1,2,3,4,6,5] => [1,2,3,4,5,6] => [1,2,3,4,5,6] => [1,2,3,4,5,6] => ? = 0 + 1
[1,2,3,5,4,6] => [1,2,3,4,5,6] => [1,2,3,4,5,6] => [1,2,3,4,5,6] => ? = 0 + 1
[1,2,3,5,6,4] => [1,2,3,4,5,6] => [1,2,3,4,5,6] => [1,2,3,4,5,6] => ? = 0 + 1
[1,2,3,6,4,5] => [1,2,3,4,6,5] => [1,2,3,4,6,5] => [1,2,3,4,6,5] => ? = 1 + 1
[1,2,3,6,5,4] => [1,2,3,4,6,5] => [1,2,3,4,6,5] => [1,2,3,4,6,5] => ? = 1 + 1
[1,2,4,3,5,6] => [1,2,3,4,5,6] => [1,2,3,4,5,6] => [1,2,3,4,5,6] => ? = 0 + 1
[1,2,4,3,6,5] => [1,2,3,4,5,6] => [1,2,3,4,5,6] => [1,2,3,4,5,6] => ? = 0 + 1
[1,2,4,5,3,6] => [1,2,3,4,5,6] => [1,2,3,4,5,6] => [1,2,3,4,5,6] => ? = 0 + 1
[1,2,4,5,6,3] => [1,2,3,4,5,6] => [1,2,3,4,5,6] => [1,2,3,4,5,6] => ? = 0 + 1
[1,2,4,6,3,5] => [1,2,3,4,6,5] => [1,2,3,4,6,5] => [1,2,3,4,6,5] => ? = 1 + 1
[1,2,4,6,5,3] => [1,2,3,4,6,5] => [1,2,3,4,6,5] => [1,2,3,4,6,5] => ? = 1 + 1
[1,2,5,3,4,6] => [1,2,3,5,4,6] => [1,2,3,5,4,6] => [1,2,3,5,4,6] => ? = 1 + 1
[1,2,5,3,6,4] => [1,2,3,5,6,4] => [1,2,3,6,5,4] => [1,2,3,6,5,4] => ? = 1 + 1
[1,2,5,4,3,6] => [1,2,3,5,4,6] => [1,2,3,5,4,6] => [1,2,3,5,4,6] => ? = 1 + 1
[1,2,5,4,6,3] => [1,2,3,5,6,4] => [1,2,3,6,5,4] => [1,2,3,6,5,4] => ? = 1 + 1
[1,2,5,6,3,4] => [1,2,3,5,4,6] => [1,2,3,5,4,6] => [1,2,3,5,4,6] => ? = 1 + 1
[1,2,5,6,4,3] => [1,2,3,5,4,6] => [1,2,3,5,4,6] => [1,2,3,5,4,6] => ? = 1 + 1
[1,2,6,3,4,5] => [1,2,3,6,5,4] => [1,2,3,6,5,4] => [1,2,3,6,5,4] => ? = 1 + 1
[1,2,6,3,5,4] => [1,2,3,6,4,5] => [1,2,3,6,5,4] => [1,2,3,6,5,4] => ? = 1 + 1
[1,2,6,4,3,5] => [1,2,3,6,5,4] => [1,2,3,6,5,4] => [1,2,3,6,5,4] => ? = 1 + 1
[1,2,6,4,5,3] => [1,2,3,6,4,5] => [1,2,3,6,5,4] => [1,2,3,6,5,4] => ? = 1 + 1
[1,2,6,5,3,4] => [1,2,3,6,4,5] => [1,2,3,6,5,4] => [1,2,3,6,5,4] => ? = 1 + 1
[1,2,6,5,4,3] => [1,2,3,6,4,5] => [1,2,3,6,5,4] => [1,2,3,6,5,4] => ? = 1 + 1
[1,3,2,4,5,6] => [1,2,3,4,5,6] => [1,2,3,4,5,6] => [1,2,3,4,5,6] => ? = 0 + 1
[1,3,2,4,6,5] => [1,2,3,4,5,6] => [1,2,3,4,5,6] => [1,2,3,4,5,6] => ? = 0 + 1
[1,3,2,5,4,6] => [1,2,3,4,5,6] => [1,2,3,4,5,6] => [1,2,3,4,5,6] => ? = 0 + 1
[1,3,2,5,6,4] => [1,2,3,4,5,6] => [1,2,3,4,5,6] => [1,2,3,4,5,6] => ? = 0 + 1
[1,3,2,6,4,5] => [1,2,3,4,6,5] => [1,2,3,4,6,5] => [1,2,3,4,6,5] => ? = 1 + 1
[1,3,2,6,5,4] => [1,2,3,4,6,5] => [1,2,3,4,6,5] => [1,2,3,4,6,5] => ? = 1 + 1
[1,3,4,2,5,6] => [1,2,3,4,5,6] => [1,2,3,4,5,6] => [1,2,3,4,5,6] => ? = 0 + 1
[1,3,4,2,6,5] => [1,2,3,4,5,6] => [1,2,3,4,5,6] => [1,2,3,4,5,6] => ? = 0 + 1
[1,3,4,5,2,6] => [1,2,3,4,5,6] => [1,2,3,4,5,6] => [1,2,3,4,5,6] => ? = 0 + 1
[1,3,4,5,6,2] => [1,2,3,4,5,6] => [1,2,3,4,5,6] => [1,2,3,4,5,6] => ? = 0 + 1
[1,3,4,6,2,5] => [1,2,3,4,6,5] => [1,2,3,4,6,5] => [1,2,3,4,6,5] => ? = 1 + 1
[1,3,4,6,5,2] => [1,2,3,4,6,5] => [1,2,3,4,6,5] => [1,2,3,4,6,5] => ? = 1 + 1
[1,3,5,2,4,6] => [1,2,3,5,4,6] => [1,2,3,5,4,6] => [1,2,3,5,4,6] => ? = 1 + 1
[1,3,5,2,6,4] => [1,2,3,5,6,4] => [1,2,3,6,5,4] => [1,2,3,6,5,4] => ? = 1 + 1
[1,3,5,4,2,6] => [1,2,3,5,4,6] => [1,2,3,5,4,6] => [1,2,3,5,4,6] => ? = 1 + 1
[1,3,5,4,6,2] => [1,2,3,5,6,4] => [1,2,3,6,5,4] => [1,2,3,6,5,4] => ? = 1 + 1
[1,3,5,6,2,4] => [1,2,3,5,4,6] => [1,2,3,5,4,6] => [1,2,3,5,4,6] => ? = 1 + 1
[1,3,5,6,4,2] => [1,2,3,5,4,6] => [1,2,3,5,4,6] => [1,2,3,5,4,6] => ? = 1 + 1
[1,3,6,2,4,5] => [1,2,3,6,5,4] => [1,2,3,6,5,4] => [1,2,3,6,5,4] => ? = 1 + 1
[1,3,6,2,5,4] => [1,2,3,6,4,5] => [1,2,3,6,5,4] => [1,2,3,6,5,4] => ? = 1 + 1
[1,3,6,4,2,5] => [1,2,3,6,5,4] => [1,2,3,6,5,4] => [1,2,3,6,5,4] => ? = 1 + 1
[1,3,6,4,5,2] => [1,2,3,6,4,5] => [1,2,3,6,5,4] => [1,2,3,6,5,4] => ? = 1 + 1
[1,3,6,5,2,4] => [1,2,3,6,4,5] => [1,2,3,6,5,4] => [1,2,3,6,5,4] => ? = 1 + 1
[1,3,6,5,4,2] => [1,2,3,6,4,5] => [1,2,3,6,5,4] => [1,2,3,6,5,4] => ? = 1 + 1
[1,4,2,3,5,6] => [1,2,4,3,5,6] => [1,2,4,3,5,6] => [1,2,4,3,5,6] => ? = 1 + 1
[1,4,2,3,6,5] => [1,2,4,3,5,6] => [1,2,4,3,5,6] => [1,2,4,3,5,6] => ? = 1 + 1
Description
The order of a signed permutation.
Matching statistic: St001195
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
Mp00175: Permutations —inverse Foata bijection⟶ Permutations
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
St001195: Dyck paths ⟶ ℤResult quality: 2% ●values known / values provided: 2%●distinct values known / distinct values provided: 67%
Mp00175: Permutations —inverse Foata bijection⟶ Permutations
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
St001195: Dyck paths ⟶ ℤResult quality: 2% ●values known / values provided: 2%●distinct values known / distinct values provided: 67%
Values
[1] => [1] => [1] => [1,0]
=> ? = 0
[1,2] => [1,2] => [1,2] => [1,0,1,0]
=> ? = 0
[2,1] => [1,2] => [1,2] => [1,0,1,0]
=> ? = 0
[1,2,3] => [1,2,3] => [1,2,3] => [1,0,1,0,1,0]
=> 0
[1,3,2] => [1,2,3] => [1,2,3] => [1,0,1,0,1,0]
=> 0
[2,1,3] => [1,2,3] => [1,2,3] => [1,0,1,0,1,0]
=> 0
[2,3,1] => [1,2,3] => [1,2,3] => [1,0,1,0,1,0]
=> 0
[3,1,2] => [1,3,2] => [3,1,2] => [1,1,1,0,0,0]
=> 1
[3,2,1] => [1,3,2] => [3,1,2] => [1,1,1,0,0,0]
=> 1
[1,2,3,4] => [1,2,3,4] => [1,2,3,4] => [1,0,1,0,1,0,1,0]
=> 0
[1,2,4,3] => [1,2,3,4] => [1,2,3,4] => [1,0,1,0,1,0,1,0]
=> 0
[1,3,2,4] => [1,2,3,4] => [1,2,3,4] => [1,0,1,0,1,0,1,0]
=> 0
[1,3,4,2] => [1,2,3,4] => [1,2,3,4] => [1,0,1,0,1,0,1,0]
=> 0
[1,4,2,3] => [1,2,4,3] => [4,1,2,3] => [1,1,1,1,0,0,0,0]
=> 1
[1,4,3,2] => [1,2,4,3] => [4,1,2,3] => [1,1,1,1,0,0,0,0]
=> 1
[2,1,3,4] => [1,2,3,4] => [1,2,3,4] => [1,0,1,0,1,0,1,0]
=> 0
[2,1,4,3] => [1,2,3,4] => [1,2,3,4] => [1,0,1,0,1,0,1,0]
=> 0
[2,3,1,4] => [1,2,3,4] => [1,2,3,4] => [1,0,1,0,1,0,1,0]
=> 0
[2,3,4,1] => [1,2,3,4] => [1,2,3,4] => [1,0,1,0,1,0,1,0]
=> 0
[2,4,1,3] => [1,2,4,3] => [4,1,2,3] => [1,1,1,1,0,0,0,0]
=> 1
[2,4,3,1] => [1,2,4,3] => [4,1,2,3] => [1,1,1,1,0,0,0,0]
=> 1
[3,1,2,4] => [1,3,2,4] => [3,1,2,4] => [1,1,1,0,0,0,1,0]
=> 1
[3,1,4,2] => [1,3,4,2] => [3,4,1,2] => [1,1,1,0,1,0,0,0]
=> 1
[3,2,1,4] => [1,3,2,4] => [3,1,2,4] => [1,1,1,0,0,0,1,0]
=> 1
[3,2,4,1] => [1,3,4,2] => [3,4,1,2] => [1,1,1,0,1,0,0,0]
=> 1
[3,4,1,2] => [1,3,2,4] => [3,1,2,4] => [1,1,1,0,0,0,1,0]
=> 1
[3,4,2,1] => [1,3,2,4] => [3,1,2,4] => [1,1,1,0,0,0,1,0]
=> 1
[4,1,2,3] => [1,4,3,2] => [4,3,1,2] => [1,1,1,1,0,0,0,0]
=> 1
[4,1,3,2] => [1,4,2,3] => [1,4,2,3] => [1,0,1,1,1,0,0,0]
=> 1
[4,2,1,3] => [1,4,3,2] => [4,3,1,2] => [1,1,1,1,0,0,0,0]
=> 1
[4,2,3,1] => [1,4,2,3] => [1,4,2,3] => [1,0,1,1,1,0,0,0]
=> 1
[4,3,1,2] => [1,4,2,3] => [1,4,2,3] => [1,0,1,1,1,0,0,0]
=> 1
[4,3,2,1] => [1,4,2,3] => [1,4,2,3] => [1,0,1,1,1,0,0,0]
=> 1
[1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => [1,0,1,0,1,0,1,0,1,0]
=> 0
[1,2,3,5,4] => [1,2,3,4,5] => [1,2,3,4,5] => [1,0,1,0,1,0,1,0,1,0]
=> 0
[1,2,4,3,5] => [1,2,3,4,5] => [1,2,3,4,5] => [1,0,1,0,1,0,1,0,1,0]
=> 0
[1,2,4,5,3] => [1,2,3,4,5] => [1,2,3,4,5] => [1,0,1,0,1,0,1,0,1,0]
=> 0
[1,2,5,3,4] => [1,2,3,5,4] => [5,1,2,3,4] => [1,1,1,1,1,0,0,0,0,0]
=> 1
[1,2,5,4,3] => [1,2,3,5,4] => [5,1,2,3,4] => [1,1,1,1,1,0,0,0,0,0]
=> 1
[1,3,2,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => [1,0,1,0,1,0,1,0,1,0]
=> 0
[1,3,2,5,4] => [1,2,3,4,5] => [1,2,3,4,5] => [1,0,1,0,1,0,1,0,1,0]
=> 0
[1,3,4,2,5] => [1,2,3,4,5] => [1,2,3,4,5] => [1,0,1,0,1,0,1,0,1,0]
=> 0
[1,3,4,5,2] => [1,2,3,4,5] => [1,2,3,4,5] => [1,0,1,0,1,0,1,0,1,0]
=> 0
[1,3,5,2,4] => [1,2,3,5,4] => [5,1,2,3,4] => [1,1,1,1,1,0,0,0,0,0]
=> 1
[1,3,5,4,2] => [1,2,3,5,4] => [5,1,2,3,4] => [1,1,1,1,1,0,0,0,0,0]
=> 1
[1,4,2,3,5] => [1,2,4,3,5] => [4,1,2,3,5] => [1,1,1,1,0,0,0,0,1,0]
=> 1
[1,4,2,5,3] => [1,2,4,5,3] => [4,5,1,2,3] => [1,1,1,1,0,1,0,0,0,0]
=> 1
[1,4,3,2,5] => [1,2,4,3,5] => [4,1,2,3,5] => [1,1,1,1,0,0,0,0,1,0]
=> 1
[1,4,3,5,2] => [1,2,4,5,3] => [4,5,1,2,3] => [1,1,1,1,0,1,0,0,0,0]
=> 1
[1,4,5,2,3] => [1,2,4,3,5] => [4,1,2,3,5] => [1,1,1,1,0,0,0,0,1,0]
=> 1
[1,4,5,3,2] => [1,2,4,3,5] => [4,1,2,3,5] => [1,1,1,1,0,0,0,0,1,0]
=> 1
[1,5,2,3,4] => [1,2,5,4,3] => [5,4,1,2,3] => [1,1,1,1,1,0,0,0,0,0]
=> 1
[1,5,2,4,3] => [1,2,5,3,4] => [1,5,2,3,4] => [1,0,1,1,1,1,0,0,0,0]
=> 1
[1,2,3,4,5,6] => [1,2,3,4,5,6] => [1,2,3,4,5,6] => [1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 0
[1,2,3,4,6,5] => [1,2,3,4,5,6] => [1,2,3,4,5,6] => [1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 0
[1,2,3,5,4,6] => [1,2,3,4,5,6] => [1,2,3,4,5,6] => [1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 0
[1,2,3,5,6,4] => [1,2,3,4,5,6] => [1,2,3,4,5,6] => [1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 0
[1,2,3,6,4,5] => [1,2,3,4,6,5] => [6,1,2,3,4,5] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> ? = 1
[1,2,3,6,5,4] => [1,2,3,4,6,5] => [6,1,2,3,4,5] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> ? = 1
[1,2,4,3,5,6] => [1,2,3,4,5,6] => [1,2,3,4,5,6] => [1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 0
[1,2,4,3,6,5] => [1,2,3,4,5,6] => [1,2,3,4,5,6] => [1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 0
[1,2,4,5,3,6] => [1,2,3,4,5,6] => [1,2,3,4,5,6] => [1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 0
[1,2,4,5,6,3] => [1,2,3,4,5,6] => [1,2,3,4,5,6] => [1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 0
[1,2,4,6,3,5] => [1,2,3,4,6,5] => [6,1,2,3,4,5] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> ? = 1
[1,2,4,6,5,3] => [1,2,3,4,6,5] => [6,1,2,3,4,5] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> ? = 1
[1,2,5,3,4,6] => [1,2,3,5,4,6] => [5,1,2,3,4,6] => [1,1,1,1,1,0,0,0,0,0,1,0]
=> ? = 1
[1,2,5,3,6,4] => [1,2,3,5,6,4] => [5,6,1,2,3,4] => [1,1,1,1,1,0,1,0,0,0,0,0]
=> ? = 1
[1,2,5,4,3,6] => [1,2,3,5,4,6] => [5,1,2,3,4,6] => [1,1,1,1,1,0,0,0,0,0,1,0]
=> ? = 1
[1,2,5,4,6,3] => [1,2,3,5,6,4] => [5,6,1,2,3,4] => [1,1,1,1,1,0,1,0,0,0,0,0]
=> ? = 1
[1,2,5,6,3,4] => [1,2,3,5,4,6] => [5,1,2,3,4,6] => [1,1,1,1,1,0,0,0,0,0,1,0]
=> ? = 1
[1,2,5,6,4,3] => [1,2,3,5,4,6] => [5,1,2,3,4,6] => [1,1,1,1,1,0,0,0,0,0,1,0]
=> ? = 1
[1,2,6,3,4,5] => [1,2,3,6,5,4] => [6,5,1,2,3,4] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> ? = 1
[1,2,6,3,5,4] => [1,2,3,6,4,5] => [1,6,2,3,4,5] => [1,0,1,1,1,1,1,0,0,0,0,0]
=> ? = 1
[1,2,6,4,3,5] => [1,2,3,6,5,4] => [6,5,1,2,3,4] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> ? = 1
[1,2,6,4,5,3] => [1,2,3,6,4,5] => [1,6,2,3,4,5] => [1,0,1,1,1,1,1,0,0,0,0,0]
=> ? = 1
[1,2,6,5,3,4] => [1,2,3,6,4,5] => [1,6,2,3,4,5] => [1,0,1,1,1,1,1,0,0,0,0,0]
=> ? = 1
[1,2,6,5,4,3] => [1,2,3,6,4,5] => [1,6,2,3,4,5] => [1,0,1,1,1,1,1,0,0,0,0,0]
=> ? = 1
[1,3,2,4,5,6] => [1,2,3,4,5,6] => [1,2,3,4,5,6] => [1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 0
[1,3,2,4,6,5] => [1,2,3,4,5,6] => [1,2,3,4,5,6] => [1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 0
[1,3,2,5,4,6] => [1,2,3,4,5,6] => [1,2,3,4,5,6] => [1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 0
[1,3,2,5,6,4] => [1,2,3,4,5,6] => [1,2,3,4,5,6] => [1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 0
[1,3,2,6,4,5] => [1,2,3,4,6,5] => [6,1,2,3,4,5] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> ? = 1
[1,3,2,6,5,4] => [1,2,3,4,6,5] => [6,1,2,3,4,5] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> ? = 1
[1,3,4,2,5,6] => [1,2,3,4,5,6] => [1,2,3,4,5,6] => [1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 0
[1,3,4,2,6,5] => [1,2,3,4,5,6] => [1,2,3,4,5,6] => [1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 0
[1,3,4,5,2,6] => [1,2,3,4,5,6] => [1,2,3,4,5,6] => [1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 0
[1,3,4,5,6,2] => [1,2,3,4,5,6] => [1,2,3,4,5,6] => [1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 0
[1,3,4,6,2,5] => [1,2,3,4,6,5] => [6,1,2,3,4,5] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> ? = 1
[1,3,4,6,5,2] => [1,2,3,4,6,5] => [6,1,2,3,4,5] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> ? = 1
[1,3,5,2,4,6] => [1,2,3,5,4,6] => [5,1,2,3,4,6] => [1,1,1,1,1,0,0,0,0,0,1,0]
=> ? = 1
[1,3,5,2,6,4] => [1,2,3,5,6,4] => [5,6,1,2,3,4] => [1,1,1,1,1,0,1,0,0,0,0,0]
=> ? = 1
[1,3,5,4,2,6] => [1,2,3,5,4,6] => [5,1,2,3,4,6] => [1,1,1,1,1,0,0,0,0,0,1,0]
=> ? = 1
[1,3,5,4,6,2] => [1,2,3,5,6,4] => [5,6,1,2,3,4] => [1,1,1,1,1,0,1,0,0,0,0,0]
=> ? = 1
[1,3,5,6,2,4] => [1,2,3,5,4,6] => [5,1,2,3,4,6] => [1,1,1,1,1,0,0,0,0,0,1,0]
=> ? = 1
[1,3,5,6,4,2] => [1,2,3,5,4,6] => [5,1,2,3,4,6] => [1,1,1,1,1,0,0,0,0,0,1,0]
=> ? = 1
[1,3,6,2,4,5] => [1,2,3,6,5,4] => [6,5,1,2,3,4] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> ? = 1
[1,3,6,2,5,4] => [1,2,3,6,4,5] => [1,6,2,3,4,5] => [1,0,1,1,1,1,1,0,0,0,0,0]
=> ? = 1
[1,3,6,4,2,5] => [1,2,3,6,5,4] => [6,5,1,2,3,4] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> ? = 1
[1,3,6,4,5,2] => [1,2,3,6,4,5] => [1,6,2,3,4,5] => [1,0,1,1,1,1,1,0,0,0,0,0]
=> ? = 1
[1,3,6,5,2,4] => [1,2,3,6,4,5] => [1,6,2,3,4,5] => [1,0,1,1,1,1,1,0,0,0,0,0]
=> ? = 1
Description
The global dimension of the algebra A/AfA of the corresponding Nakayama algebra A with minimal left faithful projective-injective module Af.
Matching statistic: St001200
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
Mp00030: Dyck paths —zeta map⟶ Dyck paths
Mp00099: Dyck paths —bounce path⟶ Dyck paths
St001200: Dyck paths ⟶ ℤResult quality: 2% ●values known / values provided: 2%●distinct values known / distinct values provided: 67%
Mp00030: Dyck paths —zeta map⟶ Dyck paths
Mp00099: Dyck paths —bounce path⟶ Dyck paths
St001200: Dyck paths ⟶ ℤResult quality: 2% ●values known / values provided: 2%●distinct values known / distinct values provided: 67%
Values
[1] => [1,0]
=> [1,0]
=> [1,0]
=> ? = 0 + 2
[1,2] => [1,0,1,0]
=> [1,1,0,0]
=> [1,1,0,0]
=> ? = 0 + 2
[2,1] => [1,1,0,0]
=> [1,0,1,0]
=> [1,0,1,0]
=> 2 = 0 + 2
[1,2,3] => [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> [1,1,1,0,0,0]
=> ? = 0 + 2
[1,3,2] => [1,0,1,1,0,0]
=> [1,0,1,1,0,0]
=> [1,0,1,1,0,0]
=> 2 = 0 + 2
[2,1,3] => [1,1,0,0,1,0]
=> [1,1,0,1,0,0]
=> [1,0,1,1,0,0]
=> 2 = 0 + 2
[2,3,1] => [1,1,0,1,0,0]
=> [1,1,0,0,1,0]
=> [1,1,0,0,1,0]
=> 2 = 0 + 2
[3,1,2] => [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> [1,0,1,0,1,0]
=> 3 = 1 + 2
[3,2,1] => [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> [1,0,1,0,1,0]
=> 3 = 1 + 2
[1,2,3,4] => [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> ? = 0 + 2
[1,2,4,3] => [1,0,1,0,1,1,0,0]
=> [1,0,1,1,1,0,0,0]
=> [1,0,1,1,1,0,0,0]
=> 2 = 0 + 2
[1,3,2,4] => [1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,0]
=> [1,0,1,1,1,0,0,0]
=> 2 = 0 + 2
[1,3,4,2] => [1,0,1,1,0,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> 2 = 0 + 2
[1,4,2,3] => [1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,0,0]
=> [1,0,1,0,1,1,0,0]
=> 3 = 1 + 2
[1,4,3,2] => [1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,0,0]
=> [1,0,1,0,1,1,0,0]
=> 3 = 1 + 2
[2,1,3,4] => [1,1,0,0,1,0,1,0]
=> [1,1,1,0,1,0,0,0]
=> [1,0,1,1,1,0,0,0]
=> 2 = 0 + 2
[2,1,4,3] => [1,1,0,0,1,1,0,0]
=> [1,1,0,1,0,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> 2 = 0 + 2
[2,3,1,4] => [1,1,0,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> 2 = 0 + 2
[2,3,4,1] => [1,1,0,1,0,1,0,0]
=> [1,1,1,0,0,0,1,0]
=> [1,1,1,0,0,0,1,0]
=> 2 = 0 + 2
[2,4,1,3] => [1,1,0,1,1,0,0,0]
=> [1,0,1,1,0,0,1,0]
=> [1,0,1,1,0,0,1,0]
=> 3 = 1 + 2
[2,4,3,1] => [1,1,0,1,1,0,0,0]
=> [1,0,1,1,0,0,1,0]
=> [1,0,1,1,0,0,1,0]
=> 3 = 1 + 2
[3,1,2,4] => [1,1,1,0,0,0,1,0]
=> [1,0,1,1,0,1,0,0]
=> [1,0,1,0,1,1,0,0]
=> 3 = 1 + 2
[3,1,4,2] => [1,1,1,0,0,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> [1,0,1,1,0,0,1,0]
=> 3 = 1 + 2
[3,2,1,4] => [1,1,1,0,0,0,1,0]
=> [1,0,1,1,0,1,0,0]
=> [1,0,1,0,1,1,0,0]
=> 3 = 1 + 2
[3,2,4,1] => [1,1,1,0,0,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> [1,0,1,1,0,0,1,0]
=> 3 = 1 + 2
[3,4,1,2] => [1,1,1,0,1,0,0,0]
=> [1,1,0,0,1,0,1,0]
=> [1,1,0,0,1,0,1,0]
=> 3 = 1 + 2
[3,4,2,1] => [1,1,1,0,1,0,0,0]
=> [1,1,0,0,1,0,1,0]
=> [1,1,0,0,1,0,1,0]
=> 3 = 1 + 2
[4,1,2,3] => [1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0]
=> 3 = 1 + 2
[4,1,3,2] => [1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0]
=> 3 = 1 + 2
[4,2,1,3] => [1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0]
=> 3 = 1 + 2
[4,2,3,1] => [1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0]
=> 3 = 1 + 2
[4,3,1,2] => [1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0]
=> 3 = 1 + 2
[4,3,2,1] => [1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0]
=> 3 = 1 + 2
[1,2,3,4,5] => [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> ? = 0 + 2
[1,2,3,5,4] => [1,0,1,0,1,0,1,1,0,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> 2 = 0 + 2
[1,2,4,3,5] => [1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> 2 = 0 + 2
[1,2,4,5,3] => [1,0,1,0,1,1,0,1,0,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> 2 = 0 + 2
[1,2,5,3,4] => [1,0,1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> 3 = 1 + 2
[1,2,5,4,3] => [1,0,1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> 3 = 1 + 2
[1,3,2,4,5] => [1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> 2 = 0 + 2
[1,3,2,5,4] => [1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> 2 = 0 + 2
[1,3,4,2,5] => [1,0,1,1,0,1,0,0,1,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> 2 = 0 + 2
[1,3,4,5,2] => [1,0,1,1,0,1,0,1,0,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> 2 = 0 + 2
[1,3,5,2,4] => [1,0,1,1,0,1,1,0,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> 3 = 1 + 2
[1,3,5,4,2] => [1,0,1,1,0,1,1,0,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> 3 = 1 + 2
[1,4,2,3,5] => [1,0,1,1,1,0,0,0,1,0]
=> [1,0,1,1,0,1,1,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> 3 = 1 + 2
[1,4,2,5,3] => [1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,1,0,0,1,1,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> 3 = 1 + 2
[1,4,3,2,5] => [1,0,1,1,1,0,0,0,1,0]
=> [1,0,1,1,0,1,1,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> 3 = 1 + 2
[1,4,3,5,2] => [1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,1,0,0,1,1,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> 3 = 1 + 2
[1,4,5,2,3] => [1,0,1,1,1,0,1,0,0,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> 3 = 1 + 2
[1,4,5,3,2] => [1,0,1,1,1,0,1,0,0,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> 3 = 1 + 2
[1,5,2,3,4] => [1,0,1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> 3 = 1 + 2
[1,5,2,4,3] => [1,0,1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> 3 = 1 + 2
[1,5,3,2,4] => [1,0,1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> 3 = 1 + 2
[1,5,3,4,2] => [1,0,1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> 3 = 1 + 2
[1,2,3,4,5,6] => [1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0]
=> ? = 0 + 2
[1,2,3,4,6,5] => [1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> ? = 0 + 2
[1,2,3,5,4,6] => [1,0,1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,1,1,0,0,0,0,0]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> ? = 0 + 2
[1,2,3,5,6,4] => [1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,0,0,1,1,1,1,0,0,0,0]
=> [1,1,0,0,1,1,1,1,0,0,0,0]
=> ? = 0 + 2
[1,2,3,6,4,5] => [1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,1,1,1,0,0,0,0]
=> ? = 1 + 2
[1,2,3,6,5,4] => [1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,1,1,1,0,0,0,0]
=> ? = 1 + 2
[1,2,4,3,5,6] => [1,0,1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,0,1,1,1,0,0,0,0,0]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> ? = 0 + 2
[1,2,4,3,6,5] => [1,0,1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,1,0,1,1,1,0,0,0,0]
=> [1,1,0,0,1,1,1,1,0,0,0,0]
=> ? = 0 + 2
[1,2,4,5,3,6] => [1,0,1,0,1,1,0,1,0,0,1,0]
=> [1,1,1,0,0,1,1,1,0,0,0,0]
=> [1,1,0,0,1,1,1,1,0,0,0,0]
=> ? = 0 + 2
[1,2,4,5,6,3] => [1,0,1,0,1,1,0,1,0,1,0,0]
=> [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 + 2
[1,2,4,6,3,5] => [1,0,1,0,1,1,0,1,1,0,0,0]
=> [1,0,1,1,0,0,1,1,1,0,0,0]
=> [1,0,1,1,0,0,1,1,1,0,0,0]
=> ? = 1 + 2
[1,2,4,6,5,3] => [1,0,1,0,1,1,0,1,1,0,0,0]
=> [1,0,1,1,0,0,1,1,1,0,0,0]
=> [1,0,1,1,0,0,1,1,1,0,0,0]
=> ? = 1 + 2
[1,2,5,3,4,6] => [1,0,1,0,1,1,1,0,0,0,1,0]
=> [1,0,1,1,0,1,1,1,0,0,0,0]
=> [1,0,1,0,1,1,1,1,0,0,0,0]
=> ? = 1 + 2
[1,2,5,3,6,4] => [1,0,1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,1,0,0,1,1,1,0,0,0]
=> [1,0,1,1,0,0,1,1,1,0,0,0]
=> ? = 1 + 2
[1,2,5,4,3,6] => [1,0,1,0,1,1,1,0,0,0,1,0]
=> [1,0,1,1,0,1,1,1,0,0,0,0]
=> [1,0,1,0,1,1,1,1,0,0,0,0]
=> ? = 1 + 2
[1,2,5,4,6,3] => [1,0,1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,1,0,0,1,1,1,0,0,0]
=> [1,0,1,1,0,0,1,1,1,0,0,0]
=> ? = 1 + 2
[1,2,5,6,3,4] => [1,0,1,0,1,1,1,0,1,0,0,0]
=> [1,1,0,0,1,0,1,1,1,0,0,0]
=> [1,1,0,0,1,0,1,1,1,0,0,0]
=> ? = 1 + 2
[1,2,5,6,4,3] => [1,0,1,0,1,1,1,0,1,0,0,0]
=> [1,1,0,0,1,0,1,1,1,0,0,0]
=> [1,1,0,0,1,0,1,1,1,0,0,0]
=> ? = 1 + 2
[1,2,6,3,4,5] => [1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,0,1,1,1,0,0,0]
=> ? = 1 + 2
[1,2,6,3,5,4] => [1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,0,1,1,1,0,0,0]
=> ? = 1 + 2
[1,2,6,4,3,5] => [1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,0,1,1,1,0,0,0]
=> ? = 1 + 2
[1,2,6,4,5,3] => [1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,0,1,1,1,0,0,0]
=> ? = 1 + 2
[1,2,6,5,3,4] => [1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,0,1,1,1,0,0,0]
=> ? = 1 + 2
[1,2,6,5,4,3] => [1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,0,1,1,1,0,0,0]
=> ? = 1 + 2
[1,3,2,4,5,6] => [1,0,1,1,0,0,1,0,1,0,1,0]
=> [1,1,1,1,0,1,1,0,0,0,0,0]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> ? = 0 + 2
[1,3,2,4,6,5] => [1,0,1,1,0,0,1,0,1,1,0,0]
=> [1,1,0,1,1,0,1,1,0,0,0,0]
=> [1,1,0,0,1,1,1,1,0,0,0,0]
=> ? = 0 + 2
[1,3,2,5,4,6] => [1,0,1,1,0,0,1,1,0,0,1,0]
=> [1,1,1,0,1,0,1,1,0,0,0,0]
=> [1,1,0,0,1,1,1,1,0,0,0,0]
=> ? = 0 + 2
[1,3,2,5,6,4] => [1,0,1,1,0,0,1,1,0,1,0,0]
=> [1,1,1,0,0,1,0,1,1,0,0,0]
=> [1,1,1,0,0,0,1,1,1,0,0,0]
=> ? = 0 + 2
[1,3,2,6,4,5] => [1,0,1,1,0,0,1,1,1,0,0,0]
=> [1,0,1,1,0,1,0,1,1,0,0,0]
=> [1,0,1,1,0,0,1,1,1,0,0,0]
=> ? = 1 + 2
[1,3,2,6,5,4] => [1,0,1,1,0,0,1,1,1,0,0,0]
=> [1,0,1,1,0,1,0,1,1,0,0,0]
=> [1,0,1,1,0,0,1,1,1,0,0,0]
=> ? = 1 + 2
[1,3,4,2,5,6] => [1,0,1,1,0,1,0,0,1,0,1,0]
=> [1,1,1,1,0,0,1,1,0,0,0,0]
=> [1,1,0,0,1,1,1,1,0,0,0,0]
=> ? = 0 + 2
[1,3,4,2,6,5] => [1,0,1,1,0,1,0,0,1,1,0,0]
=> [1,1,1,0,1,0,0,1,1,0,0,0]
=> [1,1,1,0,0,0,1,1,1,0,0,0]
=> ? = 0 + 2
[1,3,4,5,2,6] => [1,0,1,1,0,1,0,1,0,0,1,0]
=> [1,1,1,1,0,0,0,1,1,0,0,0]
=> [1,1,1,0,0,0,1,1,1,0,0,0]
=> ? = 0 + 2
[1,3,4,5,6,2] => [1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,1,1,1,0,0,0,0,1,1,0,0]
=> [1,1,1,1,0,0,0,0,1,1,0,0]
=> ? = 0 + 2
[1,3,4,6,2,5] => [1,0,1,1,0,1,0,1,1,0,0,0]
=> [1,0,1,1,1,0,0,0,1,1,0,0]
=> [1,0,1,1,1,0,0,0,1,1,0,0]
=> ? = 1 + 2
[1,3,4,6,5,2] => [1,0,1,1,0,1,0,1,1,0,0,0]
=> [1,0,1,1,1,0,0,0,1,1,0,0]
=> [1,0,1,1,1,0,0,0,1,1,0,0]
=> ? = 1 + 2
[1,3,5,2,4,6] => [1,0,1,1,0,1,1,0,0,0,1,0]
=> [1,0,1,1,1,0,0,1,1,0,0,0]
=> [1,0,1,1,0,0,1,1,1,0,0,0]
=> ? = 1 + 2
[1,3,5,2,6,4] => [1,0,1,1,0,1,1,0,0,1,0,0]
=> [1,1,0,1,1,0,0,0,1,1,0,0]
=> [1,0,1,1,1,0,0,0,1,1,0,0]
=> ? = 1 + 2
[1,3,5,4,2,6] => [1,0,1,1,0,1,1,0,0,0,1,0]
=> [1,0,1,1,1,0,0,1,1,0,0,0]
=> [1,0,1,1,0,0,1,1,1,0,0,0]
=> ? = 1 + 2
[1,3,5,4,6,2] => [1,0,1,1,0,1,1,0,0,1,0,0]
=> [1,1,0,1,1,0,0,0,1,1,0,0]
=> [1,0,1,1,1,0,0,0,1,1,0,0]
=> ? = 1 + 2
[1,3,5,6,2,4] => [1,0,1,1,0,1,1,0,1,0,0,0]
=> [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 + 2
[1,3,5,6,4,2] => [1,0,1,1,0,1,1,0,1,0,0,0]
=> [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 + 2
[1,3,6,2,4,5] => [1,0,1,1,0,1,1,1,0,0,0,0]
=> [1,0,1,0,1,1,0,0,1,1,0,0]
=> [1,0,1,0,1,1,0,0,1,1,0,0]
=> ? = 1 + 2
[1,3,6,2,5,4] => [1,0,1,1,0,1,1,1,0,0,0,0]
=> [1,0,1,0,1,1,0,0,1,1,0,0]
=> [1,0,1,0,1,1,0,0,1,1,0,0]
=> ? = 1 + 2
[1,3,6,4,2,5] => [1,0,1,1,0,1,1,1,0,0,0,0]
=> [1,0,1,0,1,1,0,0,1,1,0,0]
=> [1,0,1,0,1,1,0,0,1,1,0,0]
=> ? = 1 + 2
Description
The 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.
The following 20 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St001632The number of indecomposable injective modules I with dimExt1(I,A)=1 for the incidence algebra A of a poset. St000632The jump number of the poset. St000640The rank of the largest boolean interval in a poset. St000259The diameter of a connected graph. St000260The radius of a connected graph. St000302The determinant of the distance matrix of a connected graph. St000466The Gutman (or modified Schultz) index of a connected graph. St000467The hyper-Wiener index of a connected graph. St000771The largest multiplicity of a distance Laplacian eigenvalue in a connected graph. St000772The multiplicity of the largest distance Laplacian eigenvalue in a connected graph. St000777The number of distinct eigenvalues of the distance Laplacian of a connected graph. St001645The pebbling number of a connected graph. St001624The breadth of a lattice. St001491The number of indecomposable projective-injective modules in the algebra corresponding to a subset. St001823The Stasinski-Voll length of a signed permutation. St001905The number of preferred parking spots in a parking function less than the index of the car. St001960The number of descents of a permutation minus one if its first entry is not one. St001569The maximal modular displacement of a permutation. St001582The grades of the simple modules corresponding to the points in the poset of the symmetric group under the Bruhat order. St001860The number of factors of the Stanley symmetric function associated with a signed permutation.
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