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Your data matches 22 different statistics following compositions of up to 3 maps.
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Matching statistic: St001741
(load all 4 compositions to match this statistic)
(load all 4 compositions to match this statistic)
Mp00223: Permutations —runsort⟶ Permutations
Mp00067: Permutations —Foata bijection⟶ Permutations
St001741: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00067: Permutations —Foata bijection⟶ Permutations
St001741: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1] => [1] => 1
[1,2] => [1,2] => [1,2] => 1
[2,1] => [1,2] => [1,2] => 1
[1,2,3] => [1,2,3] => [1,2,3] => 1
[1,3,2] => [1,3,2] => [3,1,2] => 2
[2,1,3] => [1,3,2] => [3,1,2] => 2
[2,3,1] => [1,2,3] => [1,2,3] => 1
[3,1,2] => [1,2,3] => [1,2,3] => 1
[3,2,1] => [1,2,3] => [1,2,3] => 1
[1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 1
[1,2,4,3] => [1,2,4,3] => [4,1,2,3] => 2
[1,3,2,4] => [1,3,2,4] => [3,1,2,4] => 2
[1,3,4,2] => [1,3,4,2] => [3,1,4,2] => 2
[1,4,2,3] => [1,4,2,3] => [1,4,2,3] => 2
[1,4,3,2] => [1,4,2,3] => [1,4,2,3] => 2
[2,1,3,4] => [1,3,4,2] => [3,1,4,2] => 2
[2,1,4,3] => [1,4,2,3] => [1,4,2,3] => 2
[2,3,1,4] => [1,4,2,3] => [1,4,2,3] => 2
[2,3,4,1] => [1,2,3,4] => [1,2,3,4] => 1
[2,4,1,3] => [1,3,2,4] => [3,1,2,4] => 2
[2,4,3,1] => [1,2,4,3] => [4,1,2,3] => 2
[3,1,2,4] => [1,2,4,3] => [4,1,2,3] => 2
[3,1,4,2] => [1,4,2,3] => [1,4,2,3] => 2
[3,2,1,4] => [1,4,2,3] => [1,4,2,3] => 2
[3,2,4,1] => [1,2,4,3] => [4,1,2,3] => 2
[3,4,1,2] => [1,2,3,4] => [1,2,3,4] => 1
[3,4,2,1] => [1,2,3,4] => [1,2,3,4] => 1
[4,1,2,3] => [1,2,3,4] => [1,2,3,4] => 1
[4,1,3,2] => [1,3,2,4] => [3,1,2,4] => 2
[4,2,1,3] => [1,3,2,4] => [3,1,2,4] => 2
[4,2,3,1] => [1,2,3,4] => [1,2,3,4] => 1
[4,3,1,2] => [1,2,3,4] => [1,2,3,4] => 1
[4,3,2,1] => [1,2,3,4] => [1,2,3,4] => 1
[1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => 1
[1,2,3,5,4] => [1,2,3,5,4] => [5,1,2,3,4] => 2
[1,2,4,3,5] => [1,2,4,3,5] => [4,1,2,3,5] => 2
[1,2,4,5,3] => [1,2,4,5,3] => [4,1,2,5,3] => 2
[1,2,5,3,4] => [1,2,5,3,4] => [1,5,2,3,4] => 2
[1,2,5,4,3] => [1,2,5,3,4] => [1,5,2,3,4] => 2
[1,3,2,4,5] => [1,3,2,4,5] => [3,1,2,4,5] => 2
[1,3,2,5,4] => [1,3,2,5,4] => [5,3,1,2,4] => 2
[1,3,4,2,5] => [1,3,4,2,5] => [3,1,4,2,5] => 2
[1,3,4,5,2] => [1,3,4,5,2] => [3,1,4,5,2] => 2
[1,3,5,2,4] => [1,3,5,2,4] => [3,1,2,5,4] => 2
[1,3,5,4,2] => [1,3,5,2,4] => [3,1,2,5,4] => 2
[1,4,2,3,5] => [1,4,2,3,5] => [1,4,2,3,5] => 2
[1,4,2,5,3] => [1,4,2,5,3] => [4,5,1,2,3] => 2
[1,4,3,2,5] => [1,4,2,5,3] => [4,5,1,2,3] => 2
[1,4,3,5,2] => [1,4,2,3,5] => [1,4,2,3,5] => 2
[1,4,5,2,3] => [1,4,5,2,3] => [1,4,2,5,3] => 2
Description
The largest integer such that all patterns of this size are contained in the permutation.
Matching statistic: St000455
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00223: Permutations —runsort⟶ Permutations
Mp00071: Permutations —descent composition⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St000455: Graphs ⟶ ℤResult quality: 33% ●values known / values provided: 69%●distinct values known / distinct values provided: 33%
Mp00071: Permutations —descent composition⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St000455: Graphs ⟶ ℤResult quality: 33% ●values known / values provided: 69%●distinct values known / distinct values provided: 33%
Values
[1] => [1] => [1] => ([],1)
=> ? = 1 - 2
[1,2] => [1,2] => [2] => ([],2)
=> ? = 1 - 2
[2,1] => [1,2] => [2] => ([],2)
=> ? = 1 - 2
[1,2,3] => [1,2,3] => [3] => ([],3)
=> ? = 1 - 2
[1,3,2] => [1,3,2] => [2,1] => ([(0,2),(1,2)],3)
=> 0 = 2 - 2
[2,1,3] => [1,3,2] => [2,1] => ([(0,2),(1,2)],3)
=> 0 = 2 - 2
[2,3,1] => [1,2,3] => [3] => ([],3)
=> ? = 1 - 2
[3,1,2] => [1,2,3] => [3] => ([],3)
=> ? = 1 - 2
[3,2,1] => [1,2,3] => [3] => ([],3)
=> ? = 1 - 2
[1,2,3,4] => [1,2,3,4] => [4] => ([],4)
=> ? = 1 - 2
[1,2,4,3] => [1,2,4,3] => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 0 = 2 - 2
[1,3,2,4] => [1,3,2,4] => [2,2] => ([(1,3),(2,3)],4)
=> 0 = 2 - 2
[1,3,4,2] => [1,3,4,2] => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 0 = 2 - 2
[1,4,2,3] => [1,4,2,3] => [2,2] => ([(1,3),(2,3)],4)
=> 0 = 2 - 2
[1,4,3,2] => [1,4,2,3] => [2,2] => ([(1,3),(2,3)],4)
=> 0 = 2 - 2
[2,1,3,4] => [1,3,4,2] => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 0 = 2 - 2
[2,1,4,3] => [1,4,2,3] => [2,2] => ([(1,3),(2,3)],4)
=> 0 = 2 - 2
[2,3,1,4] => [1,4,2,3] => [2,2] => ([(1,3),(2,3)],4)
=> 0 = 2 - 2
[2,3,4,1] => [1,2,3,4] => [4] => ([],4)
=> ? = 1 - 2
[2,4,1,3] => [1,3,2,4] => [2,2] => ([(1,3),(2,3)],4)
=> 0 = 2 - 2
[2,4,3,1] => [1,2,4,3] => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 0 = 2 - 2
[3,1,2,4] => [1,2,4,3] => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 0 = 2 - 2
[3,1,4,2] => [1,4,2,3] => [2,2] => ([(1,3),(2,3)],4)
=> 0 = 2 - 2
[3,2,1,4] => [1,4,2,3] => [2,2] => ([(1,3),(2,3)],4)
=> 0 = 2 - 2
[3,2,4,1] => [1,2,4,3] => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 0 = 2 - 2
[3,4,1,2] => [1,2,3,4] => [4] => ([],4)
=> ? = 1 - 2
[3,4,2,1] => [1,2,3,4] => [4] => ([],4)
=> ? = 1 - 2
[4,1,2,3] => [1,2,3,4] => [4] => ([],4)
=> ? = 1 - 2
[4,1,3,2] => [1,3,2,4] => [2,2] => ([(1,3),(2,3)],4)
=> 0 = 2 - 2
[4,2,1,3] => [1,3,2,4] => [2,2] => ([(1,3),(2,3)],4)
=> 0 = 2 - 2
[4,2,3,1] => [1,2,3,4] => [4] => ([],4)
=> ? = 1 - 2
[4,3,1,2] => [1,2,3,4] => [4] => ([],4)
=> ? = 1 - 2
[4,3,2,1] => [1,2,3,4] => [4] => ([],4)
=> ? = 1 - 2
[1,2,3,4,5] => [1,2,3,4,5] => [5] => ([],5)
=> ? = 1 - 2
[1,2,3,5,4] => [1,2,3,5,4] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 0 = 2 - 2
[1,2,4,3,5] => [1,2,4,3,5] => [3,2] => ([(1,4),(2,4),(3,4)],5)
=> 0 = 2 - 2
[1,2,4,5,3] => [1,2,4,5,3] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 0 = 2 - 2
[1,2,5,3,4] => [1,2,5,3,4] => [3,2] => ([(1,4),(2,4),(3,4)],5)
=> 0 = 2 - 2
[1,2,5,4,3] => [1,2,5,3,4] => [3,2] => ([(1,4),(2,4),(3,4)],5)
=> 0 = 2 - 2
[1,3,2,4,5] => [1,3,2,4,5] => [2,3] => ([(2,4),(3,4)],5)
=> 0 = 2 - 2
[1,3,2,5,4] => [1,3,2,5,4] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 2 - 2
[1,3,4,2,5] => [1,3,4,2,5] => [3,2] => ([(1,4),(2,4),(3,4)],5)
=> 0 = 2 - 2
[1,3,4,5,2] => [1,3,4,5,2] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 0 = 2 - 2
[1,3,5,2,4] => [1,3,5,2,4] => [3,2] => ([(1,4),(2,4),(3,4)],5)
=> 0 = 2 - 2
[1,3,5,4,2] => [1,3,5,2,4] => [3,2] => ([(1,4),(2,4),(3,4)],5)
=> 0 = 2 - 2
[1,4,2,3,5] => [1,4,2,3,5] => [2,3] => ([(2,4),(3,4)],5)
=> 0 = 2 - 2
[1,4,2,5,3] => [1,4,2,5,3] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 2 - 2
[1,4,3,2,5] => [1,4,2,5,3] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 2 - 2
[1,4,3,5,2] => [1,4,2,3,5] => [2,3] => ([(2,4),(3,4)],5)
=> 0 = 2 - 2
[1,4,5,2,3] => [1,4,5,2,3] => [3,2] => ([(1,4),(2,4),(3,4)],5)
=> 0 = 2 - 2
[1,4,5,3,2] => [1,4,5,2,3] => [3,2] => ([(1,4),(2,4),(3,4)],5)
=> 0 = 2 - 2
[1,5,2,3,4] => [1,5,2,3,4] => [2,3] => ([(2,4),(3,4)],5)
=> 0 = 2 - 2
[1,5,2,4,3] => [1,5,2,4,3] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 2 - 2
[1,5,3,2,4] => [1,5,2,4,3] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 2 - 2
[1,5,3,4,2] => [1,5,2,3,4] => [2,3] => ([(2,4),(3,4)],5)
=> 0 = 2 - 2
[1,5,4,2,3] => [1,5,2,3,4] => [2,3] => ([(2,4),(3,4)],5)
=> 0 = 2 - 2
[1,5,4,3,2] => [1,5,2,3,4] => [2,3] => ([(2,4),(3,4)],5)
=> 0 = 2 - 2
[2,1,3,4,5] => [1,3,4,5,2] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 0 = 2 - 2
[2,1,3,5,4] => [1,3,5,2,4] => [3,2] => ([(1,4),(2,4),(3,4)],5)
=> 0 = 2 - 2
[2,1,4,3,5] => [1,4,2,3,5] => [2,3] => ([(2,4),(3,4)],5)
=> 0 = 2 - 2
[2,1,4,5,3] => [1,4,5,2,3] => [3,2] => ([(1,4),(2,4),(3,4)],5)
=> 0 = 2 - 2
[2,1,5,3,4] => [1,5,2,3,4] => [2,3] => ([(2,4),(3,4)],5)
=> 0 = 2 - 2
[2,1,5,4,3] => [1,5,2,3,4] => [2,3] => ([(2,4),(3,4)],5)
=> 0 = 2 - 2
[2,3,1,4,5] => [1,4,5,2,3] => [3,2] => ([(1,4),(2,4),(3,4)],5)
=> 0 = 2 - 2
[2,3,1,5,4] => [1,5,2,3,4] => [2,3] => ([(2,4),(3,4)],5)
=> 0 = 2 - 2
[2,3,4,1,5] => [1,5,2,3,4] => [2,3] => ([(2,4),(3,4)],5)
=> 0 = 2 - 2
[2,3,4,5,1] => [1,2,3,4,5] => [5] => ([],5)
=> ? = 1 - 2
[2,3,5,1,4] => [1,4,2,3,5] => [2,3] => ([(2,4),(3,4)],5)
=> 0 = 2 - 2
[2,3,5,4,1] => [1,2,3,5,4] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 0 = 2 - 2
[2,4,1,3,5] => [1,3,5,2,4] => [3,2] => ([(1,4),(2,4),(3,4)],5)
=> 0 = 2 - 2
[2,4,1,5,3] => [1,5,2,4,3] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 2 - 2
[2,4,3,1,5] => [1,5,2,4,3] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 2 - 2
[2,4,3,5,1] => [1,2,4,3,5] => [3,2] => ([(1,4),(2,4),(3,4)],5)
=> 0 = 2 - 2
[2,4,5,1,3] => [1,3,2,4,5] => [2,3] => ([(2,4),(3,4)],5)
=> 0 = 2 - 2
[2,5,1,4,3] => [1,4,2,5,3] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 2 - 2
[2,5,3,1,4] => [1,4,2,5,3] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 2 - 2
[2,5,4,1,3] => [1,3,2,5,4] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 2 - 2
[3,1,4,2,5] => [1,4,2,5,3] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 2 - 2
[3,1,5,2,4] => [1,5,2,4,3] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 2 - 2
[3,2,4,1,5] => [1,5,2,4,3] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 2 - 2
[3,2,5,1,4] => [1,4,2,5,3] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 2 - 2
[3,4,5,1,2] => [1,2,3,4,5] => [5] => ([],5)
=> ? = 1 - 2
[3,4,5,2,1] => [1,2,3,4,5] => [5] => ([],5)
=> ? = 1 - 2
[4,1,3,2,5] => [1,3,2,5,4] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 2 - 2
[4,2,5,1,3] => [1,3,2,5,4] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 2 - 2
[4,5,1,2,3] => [1,2,3,4,5] => [5] => ([],5)
=> ? = 1 - 2
[4,5,2,3,1] => [1,2,3,4,5] => [5] => ([],5)
=> ? = 1 - 2
[4,5,3,1,2] => [1,2,3,4,5] => [5] => ([],5)
=> ? = 1 - 2
[4,5,3,2,1] => [1,2,3,4,5] => [5] => ([],5)
=> ? = 1 - 2
[5,1,2,3,4] => [1,2,3,4,5] => [5] => ([],5)
=> ? = 1 - 2
[5,2,3,4,1] => [1,2,3,4,5] => [5] => ([],5)
=> ? = 1 - 2
[5,3,4,1,2] => [1,2,3,4,5] => [5] => ([],5)
=> ? = 1 - 2
[5,3,4,2,1] => [1,2,3,4,5] => [5] => ([],5)
=> ? = 1 - 2
[5,4,1,2,3] => [1,2,3,4,5] => [5] => ([],5)
=> ? = 1 - 2
[5,4,2,3,1] => [1,2,3,4,5] => [5] => ([],5)
=> ? = 1 - 2
[5,4,3,1,2] => [1,2,3,4,5] => [5] => ([],5)
=> ? = 1 - 2
[5,4,3,2,1] => [1,2,3,4,5] => [5] => ([],5)
=> ? = 1 - 2
[1,2,3,4,5,6] => [1,2,3,4,5,6] => [6] => ([],6)
=> ? = 1 - 2
[1,2,4,3,6,5] => [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)
=> ? = 2 - 2
[1,2,5,3,6,4] => [1,2,5,3,6,4] => [3,2,1] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 2 - 2
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
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00223: Permutations —runsort⟶ Permutations
Mp00067: Permutations —Foata bijection⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
St001330: Graphs ⟶ ℤResult quality: 26% ●values known / values provided: 26%●distinct values known / distinct values provided: 67%
Mp00067: Permutations —Foata bijection⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
St001330: Graphs ⟶ ℤResult quality: 26% ●values known / values provided: 26%●distinct values known / distinct values provided: 67%
Values
[1] => [1] => [1] => ([],1)
=> 1
[1,2] => [1,2] => [1,2] => ([],2)
=> 1
[2,1] => [1,2] => [1,2] => ([],2)
=> 1
[1,2,3] => [1,2,3] => [1,2,3] => ([],3)
=> 1
[1,3,2] => [1,3,2] => [3,1,2] => ([(0,2),(1,2)],3)
=> 2
[2,1,3] => [1,3,2] => [3,1,2] => ([(0,2),(1,2)],3)
=> 2
[2,3,1] => [1,2,3] => [1,2,3] => ([],3)
=> 1
[3,1,2] => [1,2,3] => [1,2,3] => ([],3)
=> 1
[3,2,1] => [1,2,3] => [1,2,3] => ([],3)
=> 1
[1,2,3,4] => [1,2,3,4] => [1,2,3,4] => ([],4)
=> 1
[1,2,4,3] => [1,2,4,3] => [4,1,2,3] => ([(0,3),(1,3),(2,3)],4)
=> 2
[1,3,2,4] => [1,3,2,4] => [3,1,2,4] => ([(1,3),(2,3)],4)
=> 2
[1,3,4,2] => [1,3,4,2] => [3,1,4,2] => ([(0,3),(1,2),(2,3)],4)
=> 2
[1,4,2,3] => [1,4,2,3] => [1,4,2,3] => ([(1,3),(2,3)],4)
=> 2
[1,4,3,2] => [1,4,2,3] => [1,4,2,3] => ([(1,3),(2,3)],4)
=> 2
[2,1,3,4] => [1,3,4,2] => [3,1,4,2] => ([(0,3),(1,2),(2,3)],4)
=> 2
[2,1,4,3] => [1,4,2,3] => [1,4,2,3] => ([(1,3),(2,3)],4)
=> 2
[2,3,1,4] => [1,4,2,3] => [1,4,2,3] => ([(1,3),(2,3)],4)
=> 2
[2,3,4,1] => [1,2,3,4] => [1,2,3,4] => ([],4)
=> 1
[2,4,1,3] => [1,3,2,4] => [3,1,2,4] => ([(1,3),(2,3)],4)
=> 2
[2,4,3,1] => [1,2,4,3] => [4,1,2,3] => ([(0,3),(1,3),(2,3)],4)
=> 2
[3,1,2,4] => [1,2,4,3] => [4,1,2,3] => ([(0,3),(1,3),(2,3)],4)
=> 2
[3,1,4,2] => [1,4,2,3] => [1,4,2,3] => ([(1,3),(2,3)],4)
=> 2
[3,2,1,4] => [1,4,2,3] => [1,4,2,3] => ([(1,3),(2,3)],4)
=> 2
[3,2,4,1] => [1,2,4,3] => [4,1,2,3] => ([(0,3),(1,3),(2,3)],4)
=> 2
[3,4,1,2] => [1,2,3,4] => [1,2,3,4] => ([],4)
=> 1
[3,4,2,1] => [1,2,3,4] => [1,2,3,4] => ([],4)
=> 1
[4,1,2,3] => [1,2,3,4] => [1,2,3,4] => ([],4)
=> 1
[4,1,3,2] => [1,3,2,4] => [3,1,2,4] => ([(1,3),(2,3)],4)
=> 2
[4,2,1,3] => [1,3,2,4] => [3,1,2,4] => ([(1,3),(2,3)],4)
=> 2
[4,2,3,1] => [1,2,3,4] => [1,2,3,4] => ([],4)
=> 1
[4,3,1,2] => [1,2,3,4] => [1,2,3,4] => ([],4)
=> 1
[4,3,2,1] => [1,2,3,4] => [1,2,3,4] => ([],4)
=> 1
[1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => ([],5)
=> 1
[1,2,3,5,4] => [1,2,3,5,4] => [5,1,2,3,4] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 2
[1,2,4,3,5] => [1,2,4,3,5] => [4,1,2,3,5] => ([(1,4),(2,4),(3,4)],5)
=> 2
[1,2,4,5,3] => [1,2,4,5,3] => [4,1,2,5,3] => ([(0,4),(1,4),(2,3),(3,4)],5)
=> 2
[1,2,5,3,4] => [1,2,5,3,4] => [1,5,2,3,4] => ([(1,4),(2,4),(3,4)],5)
=> 2
[1,2,5,4,3] => [1,2,5,3,4] => [1,5,2,3,4] => ([(1,4),(2,4),(3,4)],5)
=> 2
[1,3,2,4,5] => [1,3,2,4,5] => [3,1,2,4,5] => ([(2,4),(3,4)],5)
=> 2
[1,3,2,5,4] => [1,3,2,5,4] => [5,3,1,2,4] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 2
[1,3,4,2,5] => [1,3,4,2,5] => [3,1,4,2,5] => ([(1,4),(2,3),(3,4)],5)
=> 2
[1,3,4,5,2] => [1,3,4,5,2] => [3,1,4,5,2] => ([(0,4),(1,4),(2,3),(3,4)],5)
=> 2
[1,3,5,2,4] => [1,3,5,2,4] => [3,1,2,5,4] => ([(0,1),(2,4),(3,4)],5)
=> 2
[1,3,5,4,2] => [1,3,5,2,4] => [3,1,2,5,4] => ([(0,1),(2,4),(3,4)],5)
=> 2
[1,4,2,3,5] => [1,4,2,3,5] => [1,4,2,3,5] => ([(2,4),(3,4)],5)
=> 2
[1,4,2,5,3] => [1,4,2,5,3] => [4,5,1,2,3] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> ? = 2
[1,4,3,2,5] => [1,4,2,5,3] => [4,5,1,2,3] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> ? = 2
[1,4,3,5,2] => [1,4,2,3,5] => [1,4,2,3,5] => ([(2,4),(3,4)],5)
=> 2
[1,4,5,2,3] => [1,4,5,2,3] => [1,4,2,5,3] => ([(1,4),(2,3),(3,4)],5)
=> 2
[1,4,5,3,2] => [1,4,5,2,3] => [1,4,2,5,3] => ([(1,4),(2,3),(3,4)],5)
=> 2
[1,5,2,3,4] => [1,5,2,3,4] => [1,2,5,3,4] => ([(2,4),(3,4)],5)
=> 2
[1,5,2,4,3] => [1,5,2,4,3] => [5,1,4,2,3] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 2
[1,5,3,2,4] => [1,5,2,4,3] => [5,1,4,2,3] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 2
[1,5,3,4,2] => [1,5,2,3,4] => [1,2,5,3,4] => ([(2,4),(3,4)],5)
=> 2
[2,4,1,5,3] => [1,5,2,4,3] => [5,1,4,2,3] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 2
[2,4,3,1,5] => [1,5,2,4,3] => [5,1,4,2,3] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 2
[2,5,1,4,3] => [1,4,2,5,3] => [4,5,1,2,3] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> ? = 2
[2,5,3,1,4] => [1,4,2,5,3] => [4,5,1,2,3] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> ? = 2
[2,5,4,1,3] => [1,3,2,5,4] => [5,3,1,2,4] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 2
[3,1,4,2,5] => [1,4,2,5,3] => [4,5,1,2,3] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> ? = 2
[3,1,5,2,4] => [1,5,2,4,3] => [5,1,4,2,3] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 2
[3,2,4,1,5] => [1,5,2,4,3] => [5,1,4,2,3] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 2
[3,2,5,1,4] => [1,4,2,5,3] => [4,5,1,2,3] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> ? = 2
[4,1,3,2,5] => [1,3,2,5,4] => [5,3,1,2,4] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 2
[4,2,5,1,3] => [1,3,2,5,4] => [5,3,1,2,4] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 2
[1,2,4,3,6,5] => [1,2,4,3,6,5] => [6,4,1,2,3,5] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 2
[1,2,5,3,6,4] => [1,2,5,3,6,4] => [5,6,1,2,3,4] => ([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> ? = 2
[1,2,5,4,3,6] => [1,2,5,3,6,4] => [5,6,1,2,3,4] => ([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> ? = 2
[1,2,6,3,5,4] => [1,2,6,3,5,4] => [6,1,5,2,3,4] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 2
[1,2,6,4,3,5] => [1,2,6,3,5,4] => [6,1,5,2,3,4] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 2
[1,3,2,4,6,5] => [1,3,2,4,6,5] => [6,3,1,2,4,5] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 2
[1,3,2,5,4,6] => [1,3,2,5,4,6] => [5,3,1,2,4,6] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 2
[1,3,2,5,6,4] => [1,3,2,5,6,4] => [5,3,1,2,6,4] => ([(0,3),(1,4),(1,5),(2,4),(2,5),(3,5),(4,5)],6)
=> ? = 3
[1,3,2,6,4,5] => [1,3,2,6,4,5] => [3,6,1,2,4,5] => ([(0,5),(1,5),(2,3),(2,4),(3,5),(4,5)],6)
=> ? = 2
[1,3,2,6,5,4] => [1,3,2,6,4,5] => [3,6,1,2,4,5] => ([(0,5),(1,5),(2,3),(2,4),(3,5),(4,5)],6)
=> ? = 2
[1,3,4,2,6,5] => [1,3,4,2,6,5] => [6,3,1,4,2,5] => ([(0,5),(1,4),(1,5),(2,3),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 3
[1,3,5,2,6,4] => [1,3,5,2,6,4] => [5,3,1,6,2,4] => ([(0,1),(0,5),(1,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 3
[1,3,5,4,2,6] => [1,3,5,2,6,4] => [5,3,1,6,2,4] => ([(0,1),(0,5),(1,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 3
[1,3,6,2,5,4] => [1,3,6,2,5,4] => [6,3,1,2,5,4] => ([(0,1),(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 2
[1,3,6,4,2,5] => [1,3,6,2,5,4] => [6,3,1,2,5,4] => ([(0,1),(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 2
[1,4,2,3,6,5] => [1,4,2,3,6,5] => [6,1,4,2,3,5] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 2
[1,4,2,5,3,6] => [1,4,2,5,3,6] => [4,5,1,2,3,6] => ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> ? = 2
[1,4,2,5,6,3] => [1,4,2,5,6,3] => [4,5,1,2,6,3] => ([(0,5),(1,3),(1,4),(2,3),(2,4),(3,5),(4,5)],6)
=> ? = 2
[1,4,2,6,3,5] => [1,4,2,6,3,5] => [4,1,6,2,3,5] => ([(0,5),(1,4),(2,4),(2,5),(3,4),(3,5)],6)
=> ? = 2
[1,4,2,6,5,3] => [1,4,2,6,3,5] => [4,1,6,2,3,5] => ([(0,5),(1,4),(2,4),(2,5),(3,4),(3,5)],6)
=> ? = 2
[1,4,3,2,5,6] => [1,4,2,5,6,3] => [4,5,1,2,6,3] => ([(0,5),(1,3),(1,4),(2,3),(2,4),(3,5),(4,5)],6)
=> ? = 2
[1,4,3,2,6,5] => [1,4,2,6,3,5] => [4,1,6,2,3,5] => ([(0,5),(1,4),(2,4),(2,5),(3,4),(3,5)],6)
=> ? = 2
[1,4,3,5,2,6] => [1,4,2,6,3,5] => [4,1,6,2,3,5] => ([(0,5),(1,4),(2,4),(2,5),(3,4),(3,5)],6)
=> ? = 2
[1,4,3,6,2,5] => [1,4,2,5,3,6] => [4,5,1,2,3,6] => ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> ? = 2
[1,4,3,6,5,2] => [1,4,2,3,6,5] => [6,1,4,2,3,5] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 2
[1,4,5,2,6,3] => [1,4,5,2,6,3] => [4,5,1,6,2,3] => ([(0,4),(0,5),(1,2),(1,3),(2,4),(2,5),(3,4),(3,5)],6)
=> ? = 2
[1,4,5,3,2,6] => [1,4,5,2,6,3] => [4,5,1,6,2,3] => ([(0,4),(0,5),(1,2),(1,3),(2,4),(2,5),(3,4),(3,5)],6)
=> ? = 2
[1,4,6,2,5,3] => [1,4,6,2,5,3] => [4,6,1,2,5,3] => ([(0,3),(0,5),(1,3),(1,5),(2,4),(2,5),(3,4),(4,5)],6)
=> ? = 3
[1,4,6,3,2,5] => [1,4,6,2,5,3] => [4,6,1,2,5,3] => ([(0,3),(0,5),(1,3),(1,5),(2,4),(2,5),(3,4),(4,5)],6)
=> ? = 3
[1,5,2,3,6,4] => [1,5,2,3,6,4] => [5,1,6,2,3,4] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> ? = 2
[1,5,2,4,3,6] => [1,5,2,4,3,6] => [5,1,4,2,3,6] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 2
[1,5,2,4,6,3] => [1,5,2,4,6,3] => [5,1,4,2,6,3] => ([(0,5),(1,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 3
[1,5,2,6,3,4] => [1,5,2,6,3,4] => [1,5,6,2,3,4] => ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> ? = 2
[1,5,2,6,4,3] => [1,5,2,6,3,4] => [1,5,6,2,3,4] => ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> ? = 2
Description
The hat guessing number of a graph.
Suppose that each vertex of a graph corresponds to a player, wearing a hat whose color is arbitrarily chosen from a set of $q$ possible colors. Each player can see the hat colors of his neighbors, but not his own hat color. All of the players are asked to guess their own hat colors simultaneously, according to a predetermined guessing strategy and the hat colors they see, where no communication between them is allowed. The hat guessing number $HG(G)$ of a graph $G$ is the largest integer $q$ such that there exists a guessing strategy guaranteeing at least one correct guess for any hat assignment of $q$ possible colors.
Because it suffices that a single player guesses correctly, the hat guessing number of a graph is the maximum of the hat guessing numbers of its connected components.
Matching statistic: St001431
(load all 6 compositions to match this statistic)
(load all 6 compositions to match this statistic)
Mp00223: Permutations —runsort⟶ Permutations
Mp00175: Permutations —inverse Foata bijection⟶ Permutations
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
St001431: Dyck paths ⟶ ℤResult quality: 8% ●values known / values provided: 8%●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: 8% ●values known / values provided: 8%●distinct values known / distinct values provided: 67%
Values
[1] => [1] => [1] => [1,0]
=> ? = 1 - 1
[1,2] => [1,2] => [1,2] => [1,0,1,0]
=> 0 = 1 - 1
[2,1] => [1,2] => [1,2] => [1,0,1,0]
=> 0 = 1 - 1
[1,2,3] => [1,2,3] => [1,2,3] => [1,0,1,0,1,0]
=> 0 = 1 - 1
[1,3,2] => [1,3,2] => [3,1,2] => [1,1,1,0,0,0]
=> 1 = 2 - 1
[2,1,3] => [1,3,2] => [3,1,2] => [1,1,1,0,0,0]
=> 1 = 2 - 1
[2,3,1] => [1,2,3] => [1,2,3] => [1,0,1,0,1,0]
=> 0 = 1 - 1
[3,1,2] => [1,2,3] => [1,2,3] => [1,0,1,0,1,0]
=> 0 = 1 - 1
[3,2,1] => [1,2,3] => [1,2,3] => [1,0,1,0,1,0]
=> 0 = 1 - 1
[1,2,3,4] => [1,2,3,4] => [1,2,3,4] => [1,0,1,0,1,0,1,0]
=> 0 = 1 - 1
[1,2,4,3] => [1,2,4,3] => [4,1,2,3] => [1,1,1,1,0,0,0,0]
=> 1 = 2 - 1
[1,3,2,4] => [1,3,2,4] => [3,1,2,4] => [1,1,1,0,0,0,1,0]
=> 1 = 2 - 1
[1,3,4,2] => [1,3,4,2] => [3,4,1,2] => [1,1,1,0,1,0,0,0]
=> 1 = 2 - 1
[1,4,2,3] => [1,4,2,3] => [1,4,2,3] => [1,0,1,1,1,0,0,0]
=> 1 = 2 - 1
[1,4,3,2] => [1,4,2,3] => [1,4,2,3] => [1,0,1,1,1,0,0,0]
=> 1 = 2 - 1
[2,1,3,4] => [1,3,4,2] => [3,4,1,2] => [1,1,1,0,1,0,0,0]
=> 1 = 2 - 1
[2,1,4,3] => [1,4,2,3] => [1,4,2,3] => [1,0,1,1,1,0,0,0]
=> 1 = 2 - 1
[2,3,1,4] => [1,4,2,3] => [1,4,2,3] => [1,0,1,1,1,0,0,0]
=> 1 = 2 - 1
[2,3,4,1] => [1,2,3,4] => [1,2,3,4] => [1,0,1,0,1,0,1,0]
=> 0 = 1 - 1
[2,4,1,3] => [1,3,2,4] => [3,1,2,4] => [1,1,1,0,0,0,1,0]
=> 1 = 2 - 1
[2,4,3,1] => [1,2,4,3] => [4,1,2,3] => [1,1,1,1,0,0,0,0]
=> 1 = 2 - 1
[3,1,2,4] => [1,2,4,3] => [4,1,2,3] => [1,1,1,1,0,0,0,0]
=> 1 = 2 - 1
[3,1,4,2] => [1,4,2,3] => [1,4,2,3] => [1,0,1,1,1,0,0,0]
=> 1 = 2 - 1
[3,2,1,4] => [1,4,2,3] => [1,4,2,3] => [1,0,1,1,1,0,0,0]
=> 1 = 2 - 1
[3,2,4,1] => [1,2,4,3] => [4,1,2,3] => [1,1,1,1,0,0,0,0]
=> 1 = 2 - 1
[3,4,1,2] => [1,2,3,4] => [1,2,3,4] => [1,0,1,0,1,0,1,0]
=> 0 = 1 - 1
[3,4,2,1] => [1,2,3,4] => [1,2,3,4] => [1,0,1,0,1,0,1,0]
=> 0 = 1 - 1
[4,1,2,3] => [1,2,3,4] => [1,2,3,4] => [1,0,1,0,1,0,1,0]
=> 0 = 1 - 1
[4,1,3,2] => [1,3,2,4] => [3,1,2,4] => [1,1,1,0,0,0,1,0]
=> 1 = 2 - 1
[4,2,1,3] => [1,3,2,4] => [3,1,2,4] => [1,1,1,0,0,0,1,0]
=> 1 = 2 - 1
[4,2,3,1] => [1,2,3,4] => [1,2,3,4] => [1,0,1,0,1,0,1,0]
=> 0 = 1 - 1
[4,3,1,2] => [1,2,3,4] => [1,2,3,4] => [1,0,1,0,1,0,1,0]
=> 0 = 1 - 1
[4,3,2,1] => [1,2,3,4] => [1,2,3,4] => [1,0,1,0,1,0,1,0]
=> 0 = 1 - 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 - 1
[1,2,3,5,4] => [1,2,3,5,4] => [5,1,2,3,4] => [1,1,1,1,1,0,0,0,0,0]
=> 1 = 2 - 1
[1,2,4,3,5] => [1,2,4,3,5] => [4,1,2,3,5] => [1,1,1,1,0,0,0,0,1,0]
=> 1 = 2 - 1
[1,2,4,5,3] => [1,2,4,5,3] => [4,5,1,2,3] => [1,1,1,1,0,1,0,0,0,0]
=> 1 = 2 - 1
[1,2,5,3,4] => [1,2,5,3,4] => [1,5,2,3,4] => [1,0,1,1,1,1,0,0,0,0]
=> 1 = 2 - 1
[1,2,5,4,3] => [1,2,5,3,4] => [1,5,2,3,4] => [1,0,1,1,1,1,0,0,0,0]
=> 1 = 2 - 1
[1,3,2,4,5] => [1,3,2,4,5] => [3,1,2,4,5] => [1,1,1,0,0,0,1,0,1,0]
=> 1 = 2 - 1
[1,3,2,5,4] => [1,3,2,5,4] => [3,5,1,2,4] => [1,1,1,0,1,1,0,0,0,0]
=> 1 = 2 - 1
[1,3,4,2,5] => [1,3,4,2,5] => [3,4,1,2,5] => [1,1,1,0,1,0,0,0,1,0]
=> 1 = 2 - 1
[1,3,4,5,2] => [1,3,4,5,2] => [3,4,5,1,2] => [1,1,1,0,1,0,1,0,0,0]
=> 1 = 2 - 1
[1,3,5,2,4] => [1,3,5,2,4] => [5,3,1,2,4] => [1,1,1,1,1,0,0,0,0,0]
=> 1 = 2 - 1
[1,3,5,4,2] => [1,3,5,2,4] => [5,3,1,2,4] => [1,1,1,1,1,0,0,0,0,0]
=> 1 = 2 - 1
[1,4,2,3,5] => [1,4,2,3,5] => [1,4,2,3,5] => [1,0,1,1,1,0,0,0,1,0]
=> 1 = 2 - 1
[1,4,2,5,3] => [1,4,2,5,3] => [1,4,5,2,3] => [1,0,1,1,1,0,1,0,0,0]
=> 1 = 2 - 1
[1,4,3,2,5] => [1,4,2,5,3] => [1,4,5,2,3] => [1,0,1,1,1,0,1,0,0,0]
=> 1 = 2 - 1
[1,4,3,5,2] => [1,4,2,3,5] => [1,4,2,3,5] => [1,0,1,1,1,0,0,0,1,0]
=> 1 = 2 - 1
[1,4,5,2,3] => [1,4,5,2,3] => [4,1,5,2,3] => [1,1,1,1,0,0,1,0,0,0]
=> 1 = 2 - 1
[1,4,5,3,2] => [1,4,5,2,3] => [4,1,5,2,3] => [1,1,1,1,0,0,1,0,0,0]
=> 1 = 2 - 1
[1,2,3,4,6,5] => [1,2,3,4,6,5] => [6,1,2,3,4,5] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> ? = 2 - 1
[1,2,3,5,4,6] => [1,2,3,5,4,6] => [5,1,2,3,4,6] => [1,1,1,1,1,0,0,0,0,0,1,0]
=> ? = 2 - 1
[1,2,3,5,6,4] => [1,2,3,5,6,4] => [5,6,1,2,3,4] => [1,1,1,1,1,0,1,0,0,0,0,0]
=> ? = 2 - 1
[1,2,3,6,4,5] => [1,2,3,6,4,5] => [1,6,2,3,4,5] => [1,0,1,1,1,1,1,0,0,0,0,0]
=> ? = 2 - 1
[1,2,3,6,5,4] => [1,2,3,6,4,5] => [1,6,2,3,4,5] => [1,0,1,1,1,1,1,0,0,0,0,0]
=> ? = 2 - 1
[1,2,4,3,5,6] => [1,2,4,3,5,6] => [4,1,2,3,5,6] => [1,1,1,1,0,0,0,0,1,0,1,0]
=> ? = 2 - 1
[1,2,4,3,6,5] => [1,2,4,3,6,5] => [4,6,1,2,3,5] => [1,1,1,1,0,1,1,0,0,0,0,0]
=> ? = 2 - 1
[1,2,4,5,3,6] => [1,2,4,5,3,6] => [4,5,1,2,3,6] => [1,1,1,1,0,1,0,0,0,0,1,0]
=> ? = 2 - 1
[1,2,4,5,6,3] => [1,2,4,5,6,3] => [4,5,6,1,2,3] => [1,1,1,1,0,1,0,1,0,0,0,0]
=> ? = 2 - 1
[1,2,4,6,3,5] => [1,2,4,6,3,5] => [6,4,1,2,3,5] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> ? = 2 - 1
[1,2,4,6,5,3] => [1,2,4,6,3,5] => [6,4,1,2,3,5] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> ? = 2 - 1
[1,2,5,3,4,6] => [1,2,5,3,4,6] => [1,5,2,3,4,6] => [1,0,1,1,1,1,0,0,0,0,1,0]
=> ? = 2 - 1
[1,2,5,3,6,4] => [1,2,5,3,6,4] => [1,5,6,2,3,4] => [1,0,1,1,1,1,0,1,0,0,0,0]
=> ? = 2 - 1
[1,2,5,4,3,6] => [1,2,5,3,6,4] => [1,5,6,2,3,4] => [1,0,1,1,1,1,0,1,0,0,0,0]
=> ? = 2 - 1
[1,2,5,4,6,3] => [1,2,5,3,4,6] => [1,5,2,3,4,6] => [1,0,1,1,1,1,0,0,0,0,1,0]
=> ? = 2 - 1
[1,2,5,6,3,4] => [1,2,5,6,3,4] => [5,1,6,2,3,4] => [1,1,1,1,1,0,0,1,0,0,0,0]
=> ? = 2 - 1
[1,2,5,6,4,3] => [1,2,5,6,3,4] => [5,1,6,2,3,4] => [1,1,1,1,1,0,0,1,0,0,0,0]
=> ? = 2 - 1
[1,2,6,3,4,5] => [1,2,6,3,4,5] => [1,2,6,3,4,5] => [1,0,1,0,1,1,1,1,0,0,0,0]
=> ? = 2 - 1
[1,2,6,3,5,4] => [1,2,6,3,5,4] => [6,1,5,2,3,4] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> ? = 2 - 1
[1,2,6,4,3,5] => [1,2,6,3,5,4] => [6,1,5,2,3,4] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> ? = 2 - 1
[1,2,6,4,5,3] => [1,2,6,3,4,5] => [1,2,6,3,4,5] => [1,0,1,0,1,1,1,1,0,0,0,0]
=> ? = 2 - 1
[1,2,6,5,3,4] => [1,2,6,3,4,5] => [1,2,6,3,4,5] => [1,0,1,0,1,1,1,1,0,0,0,0]
=> ? = 2 - 1
[1,2,6,5,4,3] => [1,2,6,3,4,5] => [1,2,6,3,4,5] => [1,0,1,0,1,1,1,1,0,0,0,0]
=> ? = 2 - 1
[1,3,2,4,5,6] => [1,3,2,4,5,6] => [3,1,2,4,5,6] => [1,1,1,0,0,0,1,0,1,0,1,0]
=> ? = 2 - 1
[1,3,2,4,6,5] => [1,3,2,4,6,5] => [3,6,1,2,4,5] => [1,1,1,0,1,1,1,0,0,0,0,0]
=> ? = 2 - 1
[1,3,2,5,4,6] => [1,3,2,5,4,6] => [3,5,1,2,4,6] => [1,1,1,0,1,1,0,0,0,0,1,0]
=> ? = 2 - 1
[1,3,2,5,6,4] => [1,3,2,5,6,4] => [3,5,6,1,2,4] => [1,1,1,0,1,1,0,1,0,0,0,0]
=> ? = 3 - 1
[1,3,2,6,4,5] => [1,3,2,6,4,5] => [6,3,1,2,4,5] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> ? = 2 - 1
[1,3,2,6,5,4] => [1,3,2,6,4,5] => [6,3,1,2,4,5] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> ? = 2 - 1
[1,3,4,2,5,6] => [1,3,4,2,5,6] => [3,4,1,2,5,6] => [1,1,1,0,1,0,0,0,1,0,1,0]
=> ? = 2 - 1
[1,3,4,2,6,5] => [1,3,4,2,6,5] => [3,4,6,1,2,5] => [1,1,1,0,1,0,1,1,0,0,0,0]
=> ? = 3 - 1
[1,3,4,5,2,6] => [1,3,4,5,2,6] => [3,4,5,1,2,6] => [1,1,1,0,1,0,1,0,0,0,1,0]
=> ? = 2 - 1
[1,3,4,5,6,2] => [1,3,4,5,6,2] => [3,4,5,6,1,2] => [1,1,1,0,1,0,1,0,1,0,0,0]
=> ? = 2 - 1
[1,3,4,6,2,5] => [1,3,4,6,2,5] => [6,3,4,1,2,5] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> ? = 2 - 1
[1,3,4,6,5,2] => [1,3,4,6,2,5] => [6,3,4,1,2,5] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> ? = 2 - 1
[1,3,5,2,4,6] => [1,3,5,2,4,6] => [5,3,1,2,4,6] => [1,1,1,1,1,0,0,0,0,0,1,0]
=> ? = 2 - 1
[1,3,5,2,6,4] => [1,3,5,2,6,4] => [5,3,6,1,2,4] => [1,1,1,1,1,0,0,1,0,0,0,0]
=> ? = 3 - 1
[1,3,5,4,2,6] => [1,3,5,2,6,4] => [5,3,6,1,2,4] => [1,1,1,1,1,0,0,1,0,0,0,0]
=> ? = 3 - 1
[1,3,5,4,6,2] => [1,3,5,2,4,6] => [5,3,1,2,4,6] => [1,1,1,1,1,0,0,0,0,0,1,0]
=> ? = 2 - 1
[1,3,5,6,2,4] => [1,3,5,6,2,4] => [5,6,3,1,2,4] => [1,1,1,1,1,0,1,0,0,0,0,0]
=> ? = 2 - 1
[1,3,5,6,4,2] => [1,3,5,6,2,4] => [5,6,3,1,2,4] => [1,1,1,1,1,0,1,0,0,0,0,0]
=> ? = 2 - 1
[1,3,6,2,4,5] => [1,3,6,2,4,5] => [6,1,3,2,4,5] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> ? = 2 - 1
[1,3,6,2,5,4] => [1,3,6,2,5,4] => [3,6,5,1,2,4] => [1,1,1,0,1,1,1,0,0,0,0,0]
=> ? = 2 - 1
[1,3,6,4,2,5] => [1,3,6,2,5,4] => [3,6,5,1,2,4] => [1,1,1,0,1,1,1,0,0,0,0,0]
=> ? = 2 - 1
[1,3,6,4,5,2] => [1,3,6,2,4,5] => [6,1,3,2,4,5] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> ? = 2 - 1
[1,3,6,5,2,4] => [1,3,6,2,4,5] => [6,1,3,2,4,5] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> ? = 2 - 1
[1,3,6,5,4,2] => [1,3,6,2,4,5] => [6,1,3,2,4,5] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> ? = 2 - 1
[1,4,2,3,5,6] => [1,4,2,3,5,6] => [1,4,2,3,5,6] => [1,0,1,1,1,0,0,0,1,0,1,0]
=> ? = 2 - 1
[1,4,2,3,6,5] => [1,4,2,3,6,5] => [1,4,6,2,3,5] => [1,0,1,1,1,0,1,1,0,0,0,0]
=> ? = 2 - 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: St001555
Mp00223: Permutations —runsort⟶ Permutations
Mp00159: Permutations —Demazure product with inverse⟶ Permutations
Mp00170: Permutations —to signed permutation⟶ Signed permutations
St001555: Signed permutations ⟶ ℤResult quality: 7% ●values known / values provided: 7%●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: 7% ●values known / values provided: 7%●distinct values known / distinct values provided: 67%
Values
[1] => [1] => [1] => [1] => 1
[1,2] => [1,2] => [1,2] => [1,2] => 1
[2,1] => [1,2] => [1,2] => [1,2] => 1
[1,2,3] => [1,2,3] => [1,2,3] => [1,2,3] => 1
[1,3,2] => [1,3,2] => [1,3,2] => [1,3,2] => 2
[2,1,3] => [1,3,2] => [1,3,2] => [1,3,2] => 2
[2,3,1] => [1,2,3] => [1,2,3] => [1,2,3] => 1
[3,1,2] => [1,2,3] => [1,2,3] => [1,2,3] => 1
[3,2,1] => [1,2,3] => [1,2,3] => [1,2,3] => 1
[1,2,3,4] => [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 1
[1,2,4,3] => [1,2,4,3] => [1,2,4,3] => [1,2,4,3] => 2
[1,3,2,4] => [1,3,2,4] => [1,3,2,4] => [1,3,2,4] => 2
[1,3,4,2] => [1,3,4,2] => [1,4,3,2] => [1,4,3,2] => 2
[1,4,2,3] => [1,4,2,3] => [1,4,3,2] => [1,4,3,2] => 2
[1,4,3,2] => [1,4,2,3] => [1,4,3,2] => [1,4,3,2] => 2
[2,1,3,4] => [1,3,4,2] => [1,4,3,2] => [1,4,3,2] => 2
[2,1,4,3] => [1,4,2,3] => [1,4,3,2] => [1,4,3,2] => 2
[2,3,1,4] => [1,4,2,3] => [1,4,3,2] => [1,4,3,2] => 2
[2,3,4,1] => [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 1
[2,4,1,3] => [1,3,2,4] => [1,3,2,4] => [1,3,2,4] => 2
[2,4,3,1] => [1,2,4,3] => [1,2,4,3] => [1,2,4,3] => 2
[3,1,2,4] => [1,2,4,3] => [1,2,4,3] => [1,2,4,3] => 2
[3,1,4,2] => [1,4,2,3] => [1,4,3,2] => [1,4,3,2] => 2
[3,2,1,4] => [1,4,2,3] => [1,4,3,2] => [1,4,3,2] => 2
[3,2,4,1] => [1,2,4,3] => [1,2,4,3] => [1,2,4,3] => 2
[3,4,1,2] => [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 1
[3,4,2,1] => [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 1
[4,1,2,3] => [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 1
[4,1,3,2] => [1,3,2,4] => [1,3,2,4] => [1,3,2,4] => 2
[4,2,1,3] => [1,3,2,4] => [1,3,2,4] => [1,3,2,4] => 2
[4,2,3,1] => [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 1
[4,3,1,2] => [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 1
[4,3,2,1] => [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 1
[1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => 1
[1,2,3,5,4] => [1,2,3,5,4] => [1,2,3,5,4] => [1,2,3,5,4] => 2
[1,2,4,3,5] => [1,2,4,3,5] => [1,2,4,3,5] => [1,2,4,3,5] => 2
[1,2,4,5,3] => [1,2,4,5,3] => [1,2,5,4,3] => [1,2,5,4,3] => 2
[1,2,5,3,4] => [1,2,5,3,4] => [1,2,5,4,3] => [1,2,5,4,3] => 2
[1,2,5,4,3] => [1,2,5,3,4] => [1,2,5,4,3] => [1,2,5,4,3] => 2
[1,3,2,4,5] => [1,3,2,4,5] => [1,3,2,4,5] => [1,3,2,4,5] => 2
[1,3,2,5,4] => [1,3,2,5,4] => [1,3,2,5,4] => [1,3,2,5,4] => 2
[1,3,4,2,5] => [1,3,4,2,5] => [1,4,3,2,5] => [1,4,3,2,5] => 2
[1,3,4,5,2] => [1,3,4,5,2] => [1,5,3,4,2] => [1,5,3,4,2] => 2
[1,3,5,2,4] => [1,3,5,2,4] => [1,4,5,2,3] => [1,4,5,2,3] => 2
[1,3,5,4,2] => [1,3,5,2,4] => [1,4,5,2,3] => [1,4,5,2,3] => 2
[1,4,2,3,5] => [1,4,2,3,5] => [1,4,3,2,5] => [1,4,3,2,5] => 2
[1,4,2,5,3] => [1,4,2,5,3] => [1,5,3,4,2] => [1,5,3,4,2] => 2
[1,4,3,2,5] => [1,4,2,5,3] => [1,5,3,4,2] => [1,5,3,4,2] => 2
[1,4,3,5,2] => [1,4,2,3,5] => [1,4,3,2,5] => [1,4,3,2,5] => 2
[1,4,5,2,3] => [1,4,5,2,3] => [1,5,4,3,2] => [1,5,4,3,2] => 2
[1,2,3,4,5,6] => [1,2,3,4,5,6] => [1,2,3,4,5,6] => [1,2,3,4,5,6] => ? = 1
[1,2,3,4,6,5] => [1,2,3,4,6,5] => [1,2,3,4,6,5] => [1,2,3,4,6,5] => ? = 2
[1,2,3,5,4,6] => [1,2,3,5,4,6] => [1,2,3,5,4,6] => [1,2,3,5,4,6] => ? = 2
[1,2,3,5,6,4] => [1,2,3,5,6,4] => [1,2,3,6,5,4] => [1,2,3,6,5,4] => ? = 2
[1,2,3,6,4,5] => [1,2,3,6,4,5] => [1,2,3,6,5,4] => [1,2,3,6,5,4] => ? = 2
[1,2,3,6,5,4] => [1,2,3,6,4,5] => [1,2,3,6,5,4] => [1,2,3,6,5,4] => ? = 2
[1,2,4,3,5,6] => [1,2,4,3,5,6] => [1,2,4,3,5,6] => [1,2,4,3,5,6] => ? = 2
[1,2,4,3,6,5] => [1,2,4,3,6,5] => [1,2,4,3,6,5] => [1,2,4,3,6,5] => ? = 2
[1,2,4,5,3,6] => [1,2,4,5,3,6] => [1,2,5,4,3,6] => [1,2,5,4,3,6] => ? = 2
[1,2,4,5,6,3] => [1,2,4,5,6,3] => [1,2,6,4,5,3] => [1,2,6,4,5,3] => ? = 2
[1,2,4,6,3,5] => [1,2,4,6,3,5] => [1,2,5,6,3,4] => [1,2,5,6,3,4] => ? = 2
[1,2,4,6,5,3] => [1,2,4,6,3,5] => [1,2,5,6,3,4] => [1,2,5,6,3,4] => ? = 2
[1,2,5,3,4,6] => [1,2,5,3,4,6] => [1,2,5,4,3,6] => [1,2,5,4,3,6] => ? = 2
[1,2,5,3,6,4] => [1,2,5,3,6,4] => [1,2,6,4,5,3] => [1,2,6,4,5,3] => ? = 2
[1,2,5,4,3,6] => [1,2,5,3,6,4] => [1,2,6,4,5,3] => [1,2,6,4,5,3] => ? = 2
[1,2,5,4,6,3] => [1,2,5,3,4,6] => [1,2,5,4,3,6] => [1,2,5,4,3,6] => ? = 2
[1,2,5,6,3,4] => [1,2,5,6,3,4] => [1,2,6,5,4,3] => [1,2,6,5,4,3] => ? = 2
[1,2,5,6,4,3] => [1,2,5,6,3,4] => [1,2,6,5,4,3] => [1,2,6,5,4,3] => ? = 2
[1,2,6,3,4,5] => [1,2,6,3,4,5] => [1,2,6,4,5,3] => [1,2,6,4,5,3] => ? = 2
[1,2,6,3,5,4] => [1,2,6,3,5,4] => [1,2,6,4,5,3] => [1,2,6,4,5,3] => ? = 2
[1,2,6,4,3,5] => [1,2,6,3,5,4] => [1,2,6,4,5,3] => [1,2,6,4,5,3] => ? = 2
[1,2,6,4,5,3] => [1,2,6,3,4,5] => [1,2,6,4,5,3] => [1,2,6,4,5,3] => ? = 2
[1,2,6,5,3,4] => [1,2,6,3,4,5] => [1,2,6,4,5,3] => [1,2,6,4,5,3] => ? = 2
[1,2,6,5,4,3] => [1,2,6,3,4,5] => [1,2,6,4,5,3] => [1,2,6,4,5,3] => ? = 2
[1,3,2,4,5,6] => [1,3,2,4,5,6] => [1,3,2,4,5,6] => [1,3,2,4,5,6] => ? = 2
[1,3,2,4,6,5] => [1,3,2,4,6,5] => [1,3,2,4,6,5] => [1,3,2,4,6,5] => ? = 2
[1,3,2,5,4,6] => [1,3,2,5,4,6] => [1,3,2,5,4,6] => [1,3,2,5,4,6] => ? = 2
[1,3,2,5,6,4] => [1,3,2,5,6,4] => [1,3,2,6,5,4] => [1,3,2,6,5,4] => ? = 3
[1,3,2,6,4,5] => [1,3,2,6,4,5] => [1,3,2,6,5,4] => [1,3,2,6,5,4] => ? = 2
[1,3,2,6,5,4] => [1,3,2,6,4,5] => [1,3,2,6,5,4] => [1,3,2,6,5,4] => ? = 2
[1,3,4,2,5,6] => [1,3,4,2,5,6] => [1,4,3,2,5,6] => [1,4,3,2,5,6] => ? = 2
[1,3,4,2,6,5] => [1,3,4,2,6,5] => [1,4,3,2,6,5] => [1,4,3,2,6,5] => ? = 3
[1,3,4,5,2,6] => [1,3,4,5,2,6] => [1,5,3,4,2,6] => [1,5,3,4,2,6] => ? = 2
[1,3,4,5,6,2] => [1,3,4,5,6,2] => [1,6,3,4,5,2] => [1,6,3,4,5,2] => ? = 2
[1,3,4,6,2,5] => [1,3,4,6,2,5] => [1,5,3,6,2,4] => [1,5,3,6,2,4] => ? = 2
[1,3,4,6,5,2] => [1,3,4,6,2,5] => [1,5,3,6,2,4] => [1,5,3,6,2,4] => ? = 2
[1,3,5,2,4,6] => [1,3,5,2,4,6] => [1,4,5,2,3,6] => [1,4,5,2,3,6] => ? = 2
[1,3,5,2,6,4] => [1,3,5,2,6,4] => [1,4,6,2,5,3] => [1,4,6,2,5,3] => ? = 3
[1,3,5,4,2,6] => [1,3,5,2,6,4] => [1,4,6,2,5,3] => [1,4,6,2,5,3] => ? = 3
[1,3,5,4,6,2] => [1,3,5,2,4,6] => [1,4,5,2,3,6] => [1,4,5,2,3,6] => ? = 2
[1,3,5,6,2,4] => [1,3,5,6,2,4] => [1,5,6,4,2,3] => [1,5,6,4,2,3] => ? = 2
[1,3,5,6,4,2] => [1,3,5,6,2,4] => [1,5,6,4,2,3] => [1,5,6,4,2,3] => ? = 2
[1,3,6,2,4,5] => [1,3,6,2,4,5] => [1,4,6,2,5,3] => [1,4,6,2,5,3] => ? = 2
[1,3,6,2,5,4] => [1,3,6,2,5,4] => [1,4,6,2,5,3] => [1,4,6,2,5,3] => ? = 2
[1,3,6,4,2,5] => [1,3,6,2,5,4] => [1,4,6,2,5,3] => [1,4,6,2,5,3] => ? = 2
[1,3,6,4,5,2] => [1,3,6,2,4,5] => [1,4,6,2,5,3] => [1,4,6,2,5,3] => ? = 2
[1,3,6,5,2,4] => [1,3,6,2,4,5] => [1,4,6,2,5,3] => [1,4,6,2,5,3] => ? = 2
[1,3,6,5,4,2] => [1,3,6,2,4,5] => [1,4,6,2,5,3] => [1,4,6,2,5,3] => ? = 2
[1,4,2,3,5,6] => [1,4,2,3,5,6] => [1,4,3,2,5,6] => [1,4,3,2,5,6] => ? = 2
[1,4,2,3,6,5] => [1,4,2,3,6,5] => [1,4,3,2,6,5] => [1,4,3,2,6,5] => ? = 2
Description
The order of a signed permutation.
Matching statistic: St001553
Mp00223: Permutations —runsort⟶ Permutations
Mp00175: Permutations —inverse Foata bijection⟶ Permutations
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
St001553: Dyck paths ⟶ ℤResult quality: 7% ●values known / values provided: 7%●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: 7% ●values known / values provided: 7%●distinct values known / distinct values provided: 67%
Values
[1] => [1] => [1] => [1,0]
=> 0 = 1 - 1
[1,2] => [1,2] => [1,2] => [1,0,1,0]
=> 0 = 1 - 1
[2,1] => [1,2] => [1,2] => [1,0,1,0]
=> 0 = 1 - 1
[1,2,3] => [1,2,3] => [1,2,3] => [1,0,1,0,1,0]
=> 0 = 1 - 1
[1,3,2] => [1,3,2] => [3,1,2] => [1,1,1,0,0,0]
=> 1 = 2 - 1
[2,1,3] => [1,3,2] => [3,1,2] => [1,1,1,0,0,0]
=> 1 = 2 - 1
[2,3,1] => [1,2,3] => [1,2,3] => [1,0,1,0,1,0]
=> 0 = 1 - 1
[3,1,2] => [1,2,3] => [1,2,3] => [1,0,1,0,1,0]
=> 0 = 1 - 1
[3,2,1] => [1,2,3] => [1,2,3] => [1,0,1,0,1,0]
=> 0 = 1 - 1
[1,2,3,4] => [1,2,3,4] => [1,2,3,4] => [1,0,1,0,1,0,1,0]
=> 0 = 1 - 1
[1,2,4,3] => [1,2,4,3] => [4,1,2,3] => [1,1,1,1,0,0,0,0]
=> 1 = 2 - 1
[1,3,2,4] => [1,3,2,4] => [3,1,2,4] => [1,1,1,0,0,0,1,0]
=> 1 = 2 - 1
[1,3,4,2] => [1,3,4,2] => [3,4,1,2] => [1,1,1,0,1,0,0,0]
=> 1 = 2 - 1
[1,4,2,3] => [1,4,2,3] => [1,4,2,3] => [1,0,1,1,1,0,0,0]
=> 1 = 2 - 1
[1,4,3,2] => [1,4,2,3] => [1,4,2,3] => [1,0,1,1,1,0,0,0]
=> 1 = 2 - 1
[2,1,3,4] => [1,3,4,2] => [3,4,1,2] => [1,1,1,0,1,0,0,0]
=> 1 = 2 - 1
[2,1,4,3] => [1,4,2,3] => [1,4,2,3] => [1,0,1,1,1,0,0,0]
=> 1 = 2 - 1
[2,3,1,4] => [1,4,2,3] => [1,4,2,3] => [1,0,1,1,1,0,0,0]
=> 1 = 2 - 1
[2,3,4,1] => [1,2,3,4] => [1,2,3,4] => [1,0,1,0,1,0,1,0]
=> 0 = 1 - 1
[2,4,1,3] => [1,3,2,4] => [3,1,2,4] => [1,1,1,0,0,0,1,0]
=> 1 = 2 - 1
[2,4,3,1] => [1,2,4,3] => [4,1,2,3] => [1,1,1,1,0,0,0,0]
=> 1 = 2 - 1
[3,1,2,4] => [1,2,4,3] => [4,1,2,3] => [1,1,1,1,0,0,0,0]
=> 1 = 2 - 1
[3,1,4,2] => [1,4,2,3] => [1,4,2,3] => [1,0,1,1,1,0,0,0]
=> 1 = 2 - 1
[3,2,1,4] => [1,4,2,3] => [1,4,2,3] => [1,0,1,1,1,0,0,0]
=> 1 = 2 - 1
[3,2,4,1] => [1,2,4,3] => [4,1,2,3] => [1,1,1,1,0,0,0,0]
=> 1 = 2 - 1
[3,4,1,2] => [1,2,3,4] => [1,2,3,4] => [1,0,1,0,1,0,1,0]
=> 0 = 1 - 1
[3,4,2,1] => [1,2,3,4] => [1,2,3,4] => [1,0,1,0,1,0,1,0]
=> 0 = 1 - 1
[4,1,2,3] => [1,2,3,4] => [1,2,3,4] => [1,0,1,0,1,0,1,0]
=> 0 = 1 - 1
[4,1,3,2] => [1,3,2,4] => [3,1,2,4] => [1,1,1,0,0,0,1,0]
=> 1 = 2 - 1
[4,2,1,3] => [1,3,2,4] => [3,1,2,4] => [1,1,1,0,0,0,1,0]
=> 1 = 2 - 1
[4,2,3,1] => [1,2,3,4] => [1,2,3,4] => [1,0,1,0,1,0,1,0]
=> 0 = 1 - 1
[4,3,1,2] => [1,2,3,4] => [1,2,3,4] => [1,0,1,0,1,0,1,0]
=> 0 = 1 - 1
[4,3,2,1] => [1,2,3,4] => [1,2,3,4] => [1,0,1,0,1,0,1,0]
=> 0 = 1 - 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 - 1
[1,2,3,5,4] => [1,2,3,5,4] => [5,1,2,3,4] => [1,1,1,1,1,0,0,0,0,0]
=> 1 = 2 - 1
[1,2,4,3,5] => [1,2,4,3,5] => [4,1,2,3,5] => [1,1,1,1,0,0,0,0,1,0]
=> 1 = 2 - 1
[1,2,4,5,3] => [1,2,4,5,3] => [4,5,1,2,3] => [1,1,1,1,0,1,0,0,0,0]
=> 1 = 2 - 1
[1,2,5,3,4] => [1,2,5,3,4] => [1,5,2,3,4] => [1,0,1,1,1,1,0,0,0,0]
=> 1 = 2 - 1
[1,2,5,4,3] => [1,2,5,3,4] => [1,5,2,3,4] => [1,0,1,1,1,1,0,0,0,0]
=> 1 = 2 - 1
[1,3,2,4,5] => [1,3,2,4,5] => [3,1,2,4,5] => [1,1,1,0,0,0,1,0,1,0]
=> 1 = 2 - 1
[1,3,2,5,4] => [1,3,2,5,4] => [3,5,1,2,4] => [1,1,1,0,1,1,0,0,0,0]
=> 1 = 2 - 1
[1,3,4,2,5] => [1,3,4,2,5] => [3,4,1,2,5] => [1,1,1,0,1,0,0,0,1,0]
=> 1 = 2 - 1
[1,3,4,5,2] => [1,3,4,5,2] => [3,4,5,1,2] => [1,1,1,0,1,0,1,0,0,0]
=> 1 = 2 - 1
[1,3,5,2,4] => [1,3,5,2,4] => [5,3,1,2,4] => [1,1,1,1,1,0,0,0,0,0]
=> 1 = 2 - 1
[1,3,5,4,2] => [1,3,5,2,4] => [5,3,1,2,4] => [1,1,1,1,1,0,0,0,0,0]
=> 1 = 2 - 1
[1,4,2,3,5] => [1,4,2,3,5] => [1,4,2,3,5] => [1,0,1,1,1,0,0,0,1,0]
=> 1 = 2 - 1
[1,4,2,5,3] => [1,4,2,5,3] => [1,4,5,2,3] => [1,0,1,1,1,0,1,0,0,0]
=> 1 = 2 - 1
[1,4,3,2,5] => [1,4,2,5,3] => [1,4,5,2,3] => [1,0,1,1,1,0,1,0,0,0]
=> 1 = 2 - 1
[1,4,3,5,2] => [1,4,2,3,5] => [1,4,2,3,5] => [1,0,1,1,1,0,0,0,1,0]
=> 1 = 2 - 1
[1,4,5,2,3] => [1,4,5,2,3] => [4,1,5,2,3] => [1,1,1,1,0,0,1,0,0,0]
=> 1 = 2 - 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]
=> ? = 1 - 1
[1,2,3,4,6,5] => [1,2,3,4,6,5] => [6,1,2,3,4,5] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> ? = 2 - 1
[1,2,3,5,4,6] => [1,2,3,5,4,6] => [5,1,2,3,4,6] => [1,1,1,1,1,0,0,0,0,0,1,0]
=> ? = 2 - 1
[1,2,3,5,6,4] => [1,2,3,5,6,4] => [5,6,1,2,3,4] => [1,1,1,1,1,0,1,0,0,0,0,0]
=> ? = 2 - 1
[1,2,3,6,4,5] => [1,2,3,6,4,5] => [1,6,2,3,4,5] => [1,0,1,1,1,1,1,0,0,0,0,0]
=> ? = 2 - 1
[1,2,3,6,5,4] => [1,2,3,6,4,5] => [1,6,2,3,4,5] => [1,0,1,1,1,1,1,0,0,0,0,0]
=> ? = 2 - 1
[1,2,4,3,5,6] => [1,2,4,3,5,6] => [4,1,2,3,5,6] => [1,1,1,1,0,0,0,0,1,0,1,0]
=> ? = 2 - 1
[1,2,4,3,6,5] => [1,2,4,3,6,5] => [4,6,1,2,3,5] => [1,1,1,1,0,1,1,0,0,0,0,0]
=> ? = 2 - 1
[1,2,4,5,3,6] => [1,2,4,5,3,6] => [4,5,1,2,3,6] => [1,1,1,1,0,1,0,0,0,0,1,0]
=> ? = 2 - 1
[1,2,4,5,6,3] => [1,2,4,5,6,3] => [4,5,6,1,2,3] => [1,1,1,1,0,1,0,1,0,0,0,0]
=> ? = 2 - 1
[1,2,4,6,3,5] => [1,2,4,6,3,5] => [6,4,1,2,3,5] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> ? = 2 - 1
[1,2,4,6,5,3] => [1,2,4,6,3,5] => [6,4,1,2,3,5] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> ? = 2 - 1
[1,2,5,3,4,6] => [1,2,5,3,4,6] => [1,5,2,3,4,6] => [1,0,1,1,1,1,0,0,0,0,1,0]
=> ? = 2 - 1
[1,2,5,3,6,4] => [1,2,5,3,6,4] => [1,5,6,2,3,4] => [1,0,1,1,1,1,0,1,0,0,0,0]
=> ? = 2 - 1
[1,2,5,4,3,6] => [1,2,5,3,6,4] => [1,5,6,2,3,4] => [1,0,1,1,1,1,0,1,0,0,0,0]
=> ? = 2 - 1
[1,2,5,4,6,3] => [1,2,5,3,4,6] => [1,5,2,3,4,6] => [1,0,1,1,1,1,0,0,0,0,1,0]
=> ? = 2 - 1
[1,2,5,6,3,4] => [1,2,5,6,3,4] => [5,1,6,2,3,4] => [1,1,1,1,1,0,0,1,0,0,0,0]
=> ? = 2 - 1
[1,2,5,6,4,3] => [1,2,5,6,3,4] => [5,1,6,2,3,4] => [1,1,1,1,1,0,0,1,0,0,0,0]
=> ? = 2 - 1
[1,2,6,3,4,5] => [1,2,6,3,4,5] => [1,2,6,3,4,5] => [1,0,1,0,1,1,1,1,0,0,0,0]
=> ? = 2 - 1
[1,2,6,3,5,4] => [1,2,6,3,5,4] => [6,1,5,2,3,4] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> ? = 2 - 1
[1,2,6,4,3,5] => [1,2,6,3,5,4] => [6,1,5,2,3,4] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> ? = 2 - 1
[1,2,6,4,5,3] => [1,2,6,3,4,5] => [1,2,6,3,4,5] => [1,0,1,0,1,1,1,1,0,0,0,0]
=> ? = 2 - 1
[1,2,6,5,3,4] => [1,2,6,3,4,5] => [1,2,6,3,4,5] => [1,0,1,0,1,1,1,1,0,0,0,0]
=> ? = 2 - 1
[1,2,6,5,4,3] => [1,2,6,3,4,5] => [1,2,6,3,4,5] => [1,0,1,0,1,1,1,1,0,0,0,0]
=> ? = 2 - 1
[1,3,2,4,5,6] => [1,3,2,4,5,6] => [3,1,2,4,5,6] => [1,1,1,0,0,0,1,0,1,0,1,0]
=> ? = 2 - 1
[1,3,2,4,6,5] => [1,3,2,4,6,5] => [3,6,1,2,4,5] => [1,1,1,0,1,1,1,0,0,0,0,0]
=> ? = 2 - 1
[1,3,2,5,4,6] => [1,3,2,5,4,6] => [3,5,1,2,4,6] => [1,1,1,0,1,1,0,0,0,0,1,0]
=> ? = 2 - 1
[1,3,2,5,6,4] => [1,3,2,5,6,4] => [3,5,6,1,2,4] => [1,1,1,0,1,1,0,1,0,0,0,0]
=> ? = 3 - 1
[1,3,2,6,4,5] => [1,3,2,6,4,5] => [6,3,1,2,4,5] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> ? = 2 - 1
[1,3,2,6,5,4] => [1,3,2,6,4,5] => [6,3,1,2,4,5] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> ? = 2 - 1
[1,3,4,2,5,6] => [1,3,4,2,5,6] => [3,4,1,2,5,6] => [1,1,1,0,1,0,0,0,1,0,1,0]
=> ? = 2 - 1
[1,3,4,2,6,5] => [1,3,4,2,6,5] => [3,4,6,1,2,5] => [1,1,1,0,1,0,1,1,0,0,0,0]
=> ? = 3 - 1
[1,3,4,5,2,6] => [1,3,4,5,2,6] => [3,4,5,1,2,6] => [1,1,1,0,1,0,1,0,0,0,1,0]
=> ? = 2 - 1
[1,3,4,5,6,2] => [1,3,4,5,6,2] => [3,4,5,6,1,2] => [1,1,1,0,1,0,1,0,1,0,0,0]
=> ? = 2 - 1
[1,3,4,6,2,5] => [1,3,4,6,2,5] => [6,3,4,1,2,5] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> ? = 2 - 1
[1,3,4,6,5,2] => [1,3,4,6,2,5] => [6,3,4,1,2,5] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> ? = 2 - 1
[1,3,5,2,4,6] => [1,3,5,2,4,6] => [5,3,1,2,4,6] => [1,1,1,1,1,0,0,0,0,0,1,0]
=> ? = 2 - 1
[1,3,5,2,6,4] => [1,3,5,2,6,4] => [5,3,6,1,2,4] => [1,1,1,1,1,0,0,1,0,0,0,0]
=> ? = 3 - 1
[1,3,5,4,2,6] => [1,3,5,2,6,4] => [5,3,6,1,2,4] => [1,1,1,1,1,0,0,1,0,0,0,0]
=> ? = 3 - 1
[1,3,5,4,6,2] => [1,3,5,2,4,6] => [5,3,1,2,4,6] => [1,1,1,1,1,0,0,0,0,0,1,0]
=> ? = 2 - 1
[1,3,5,6,2,4] => [1,3,5,6,2,4] => [5,6,3,1,2,4] => [1,1,1,1,1,0,1,0,0,0,0,0]
=> ? = 2 - 1
[1,3,5,6,4,2] => [1,3,5,6,2,4] => [5,6,3,1,2,4] => [1,1,1,1,1,0,1,0,0,0,0,0]
=> ? = 2 - 1
[1,3,6,2,4,5] => [1,3,6,2,4,5] => [6,1,3,2,4,5] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> ? = 2 - 1
[1,3,6,2,5,4] => [1,3,6,2,5,4] => [3,6,5,1,2,4] => [1,1,1,0,1,1,1,0,0,0,0,0]
=> ? = 2 - 1
[1,3,6,4,2,5] => [1,3,6,2,5,4] => [3,6,5,1,2,4] => [1,1,1,0,1,1,1,0,0,0,0,0]
=> ? = 2 - 1
[1,3,6,4,5,2] => [1,3,6,2,4,5] => [6,1,3,2,4,5] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> ? = 2 - 1
[1,3,6,5,2,4] => [1,3,6,2,4,5] => [6,1,3,2,4,5] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> ? = 2 - 1
[1,3,6,5,4,2] => [1,3,6,2,4,5] => [6,1,3,2,4,5] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> ? = 2 - 1
[1,4,2,3,5,6] => [1,4,2,3,5,6] => [1,4,2,3,5,6] => [1,0,1,1,1,0,0,0,1,0,1,0]
=> ? = 2 - 1
[1,4,2,3,6,5] => [1,4,2,3,6,5] => [1,4,6,2,3,5] => [1,0,1,1,1,0,1,1,0,0,0,0]
=> ? = 2 - 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: St001195
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00223: Permutations —runsort⟶ Permutations
Mp00062: Permutations —Lehmer-code to major-code bijection⟶ Permutations
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
St001195: Dyck paths ⟶ ℤResult quality: 7% ●values known / values provided: 7%●distinct values known / distinct values provided: 67%
Mp00062: Permutations —Lehmer-code to major-code bijection⟶ Permutations
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
St001195: Dyck paths ⟶ ℤResult quality: 7% ●values known / values provided: 7%●distinct values known / distinct values provided: 67%
Values
[1] => [1] => [1] => [1,0]
=> ? = 1 - 1
[1,2] => [1,2] => [1,2] => [1,0,1,0]
=> ? = 1 - 1
[2,1] => [1,2] => [1,2] => [1,0,1,0]
=> ? = 1 - 1
[1,2,3] => [1,2,3] => [1,2,3] => [1,0,1,0,1,0]
=> 0 = 1 - 1
[1,3,2] => [1,3,2] => [3,1,2] => [1,1,1,0,0,0]
=> 1 = 2 - 1
[2,1,3] => [1,3,2] => [3,1,2] => [1,1,1,0,0,0]
=> 1 = 2 - 1
[2,3,1] => [1,2,3] => [1,2,3] => [1,0,1,0,1,0]
=> 0 = 1 - 1
[3,1,2] => [1,2,3] => [1,2,3] => [1,0,1,0,1,0]
=> 0 = 1 - 1
[3,2,1] => [1,2,3] => [1,2,3] => [1,0,1,0,1,0]
=> 0 = 1 - 1
[1,2,3,4] => [1,2,3,4] => [1,2,3,4] => [1,0,1,0,1,0,1,0]
=> 0 = 1 - 1
[1,2,4,3] => [1,2,4,3] => [4,1,2,3] => [1,1,1,1,0,0,0,0]
=> 1 = 2 - 1
[1,3,2,4] => [1,3,2,4] => [3,1,2,4] => [1,1,1,0,0,0,1,0]
=> 1 = 2 - 1
[1,3,4,2] => [1,3,4,2] => [2,4,1,3] => [1,1,0,1,1,0,0,0]
=> 1 = 2 - 1
[1,4,2,3] => [1,4,2,3] => [3,4,1,2] => [1,1,1,0,1,0,0,0]
=> 1 = 2 - 1
[1,4,3,2] => [1,4,2,3] => [3,4,1,2] => [1,1,1,0,1,0,0,0]
=> 1 = 2 - 1
[2,1,3,4] => [1,3,4,2] => [2,4,1,3] => [1,1,0,1,1,0,0,0]
=> 1 = 2 - 1
[2,1,4,3] => [1,4,2,3] => [3,4,1,2] => [1,1,1,0,1,0,0,0]
=> 1 = 2 - 1
[2,3,1,4] => [1,4,2,3] => [3,4,1,2] => [1,1,1,0,1,0,0,0]
=> 1 = 2 - 1
[2,3,4,1] => [1,2,3,4] => [1,2,3,4] => [1,0,1,0,1,0,1,0]
=> 0 = 1 - 1
[2,4,1,3] => [1,3,2,4] => [3,1,2,4] => [1,1,1,0,0,0,1,0]
=> 1 = 2 - 1
[2,4,3,1] => [1,2,4,3] => [4,1,2,3] => [1,1,1,1,0,0,0,0]
=> 1 = 2 - 1
[3,1,2,4] => [1,2,4,3] => [4,1,2,3] => [1,1,1,1,0,0,0,0]
=> 1 = 2 - 1
[3,1,4,2] => [1,4,2,3] => [3,4,1,2] => [1,1,1,0,1,0,0,0]
=> 1 = 2 - 1
[3,2,1,4] => [1,4,2,3] => [3,4,1,2] => [1,1,1,0,1,0,0,0]
=> 1 = 2 - 1
[3,2,4,1] => [1,2,4,3] => [4,1,2,3] => [1,1,1,1,0,0,0,0]
=> 1 = 2 - 1
[3,4,1,2] => [1,2,3,4] => [1,2,3,4] => [1,0,1,0,1,0,1,0]
=> 0 = 1 - 1
[3,4,2,1] => [1,2,3,4] => [1,2,3,4] => [1,0,1,0,1,0,1,0]
=> 0 = 1 - 1
[4,1,2,3] => [1,2,3,4] => [1,2,3,4] => [1,0,1,0,1,0,1,0]
=> 0 = 1 - 1
[4,1,3,2] => [1,3,2,4] => [3,1,2,4] => [1,1,1,0,0,0,1,0]
=> 1 = 2 - 1
[4,2,1,3] => [1,3,2,4] => [3,1,2,4] => [1,1,1,0,0,0,1,0]
=> 1 = 2 - 1
[4,2,3,1] => [1,2,3,4] => [1,2,3,4] => [1,0,1,0,1,0,1,0]
=> 0 = 1 - 1
[4,3,1,2] => [1,2,3,4] => [1,2,3,4] => [1,0,1,0,1,0,1,0]
=> 0 = 1 - 1
[4,3,2,1] => [1,2,3,4] => [1,2,3,4] => [1,0,1,0,1,0,1,0]
=> 0 = 1 - 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 - 1
[1,2,3,5,4] => [1,2,3,5,4] => [5,1,2,3,4] => [1,1,1,1,1,0,0,0,0,0]
=> 1 = 2 - 1
[1,2,4,3,5] => [1,2,4,3,5] => [4,1,2,3,5] => [1,1,1,1,0,0,0,0,1,0]
=> 1 = 2 - 1
[1,2,4,5,3] => [1,2,4,5,3] => [3,5,1,2,4] => [1,1,1,0,1,1,0,0,0,0]
=> 1 = 2 - 1
[1,2,5,3,4] => [1,2,5,3,4] => [4,5,1,2,3] => [1,1,1,1,0,1,0,0,0,0]
=> 1 = 2 - 1
[1,2,5,4,3] => [1,2,5,3,4] => [4,5,1,2,3] => [1,1,1,1,0,1,0,0,0,0]
=> 1 = 2 - 1
[1,3,2,4,5] => [1,3,2,4,5] => [3,1,2,4,5] => [1,1,1,0,0,0,1,0,1,0]
=> 1 = 2 - 1
[1,3,2,5,4] => [1,3,2,5,4] => [2,5,1,3,4] => [1,1,0,1,1,1,0,0,0,0]
=> 1 = 2 - 1
[1,3,4,2,5] => [1,3,4,2,5] => [2,4,1,3,5] => [1,1,0,1,1,0,0,0,1,0]
=> 1 = 2 - 1
[1,3,4,5,2] => [1,3,4,5,2] => [2,3,5,1,4] => [1,1,0,1,0,1,1,0,0,0]
=> 1 = 2 - 1
[1,3,5,2,4] => [1,3,5,2,4] => [2,4,5,1,3] => [1,1,0,1,1,0,1,0,0,0]
=> 1 = 2 - 1
[1,3,5,4,2] => [1,3,5,2,4] => [2,4,5,1,3] => [1,1,0,1,1,0,1,0,0,0]
=> 1 = 2 - 1
[1,4,2,3,5] => [1,4,2,3,5] => [3,4,1,2,5] => [1,1,1,0,1,0,0,0,1,0]
=> 1 = 2 - 1
[1,4,2,5,3] => [1,4,2,5,3] => [5,3,1,2,4] => [1,1,1,1,1,0,0,0,0,0]
=> 1 = 2 - 1
[1,4,3,2,5] => [1,4,2,5,3] => [5,3,1,2,4] => [1,1,1,1,1,0,0,0,0,0]
=> 1 = 2 - 1
[1,4,3,5,2] => [1,4,2,3,5] => [3,4,1,2,5] => [1,1,1,0,1,0,0,0,1,0]
=> 1 = 2 - 1
[1,4,5,2,3] => [1,4,5,2,3] => [4,2,5,1,3] => [1,1,1,1,0,0,1,0,0,0]
=> 1 = 2 - 1
[1,4,5,3,2] => [1,4,5,2,3] => [4,2,5,1,3] => [1,1,1,1,0,0,1,0,0,0]
=> 1 = 2 - 1
[1,5,2,3,4] => [1,5,2,3,4] => [3,4,5,1,2] => [1,1,1,0,1,0,1,0,0,0]
=> 1 = 2 - 1
[1,5,2,4,3] => [1,5,2,4,3] => [5,3,4,1,2] => [1,1,1,1,1,0,0,0,0,0]
=> 1 = 2 - 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]
=> ? = 1 - 1
[1,2,3,4,6,5] => [1,2,3,4,6,5] => [6,1,2,3,4,5] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> ? = 2 - 1
[1,2,3,5,4,6] => [1,2,3,5,4,6] => [5,1,2,3,4,6] => [1,1,1,1,1,0,0,0,0,0,1,0]
=> ? = 2 - 1
[1,2,3,5,6,4] => [1,2,3,5,6,4] => [4,6,1,2,3,5] => [1,1,1,1,0,1,1,0,0,0,0,0]
=> ? = 2 - 1
[1,2,3,6,4,5] => [1,2,3,6,4,5] => [5,6,1,2,3,4] => [1,1,1,1,1,0,1,0,0,0,0,0]
=> ? = 2 - 1
[1,2,3,6,5,4] => [1,2,3,6,4,5] => [5,6,1,2,3,4] => [1,1,1,1,1,0,1,0,0,0,0,0]
=> ? = 2 - 1
[1,2,4,3,5,6] => [1,2,4,3,5,6] => [4,1,2,3,5,6] => [1,1,1,1,0,0,0,0,1,0,1,0]
=> ? = 2 - 1
[1,2,4,3,6,5] => [1,2,4,3,6,5] => [3,6,1,2,4,5] => [1,1,1,0,1,1,1,0,0,0,0,0]
=> ? = 2 - 1
[1,2,4,5,3,6] => [1,2,4,5,3,6] => [3,5,1,2,4,6] => [1,1,1,0,1,1,0,0,0,0,1,0]
=> ? = 2 - 1
[1,2,4,5,6,3] => [1,2,4,5,6,3] => [3,4,6,1,2,5] => [1,1,1,0,1,0,1,1,0,0,0,0]
=> ? = 2 - 1
[1,2,4,6,3,5] => [1,2,4,6,3,5] => [3,5,6,1,2,4] => [1,1,1,0,1,1,0,1,0,0,0,0]
=> ? = 2 - 1
[1,2,4,6,5,3] => [1,2,4,6,3,5] => [3,5,6,1,2,4] => [1,1,1,0,1,1,0,1,0,0,0,0]
=> ? = 2 - 1
[1,2,5,3,4,6] => [1,2,5,3,4,6] => [4,5,1,2,3,6] => [1,1,1,1,0,1,0,0,0,0,1,0]
=> ? = 2 - 1
[1,2,5,3,6,4] => [1,2,5,3,6,4] => [6,4,1,2,3,5] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> ? = 2 - 1
[1,2,5,4,3,6] => [1,2,5,3,6,4] => [6,4,1,2,3,5] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> ? = 2 - 1
[1,2,5,4,6,3] => [1,2,5,3,4,6] => [4,5,1,2,3,6] => [1,1,1,1,0,1,0,0,0,0,1,0]
=> ? = 2 - 1
[1,2,5,6,3,4] => [1,2,5,6,3,4] => [5,3,6,1,2,4] => [1,1,1,1,1,0,0,1,0,0,0,0]
=> ? = 2 - 1
[1,2,5,6,4,3] => [1,2,5,6,3,4] => [5,3,6,1,2,4] => [1,1,1,1,1,0,0,1,0,0,0,0]
=> ? = 2 - 1
[1,2,6,3,4,5] => [1,2,6,3,4,5] => [4,5,6,1,2,3] => [1,1,1,1,0,1,0,1,0,0,0,0]
=> ? = 2 - 1
[1,2,6,3,5,4] => [1,2,6,3,5,4] => [6,4,5,1,2,3] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> ? = 2 - 1
[1,2,6,4,3,5] => [1,2,6,3,5,4] => [6,4,5,1,2,3] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> ? = 2 - 1
[1,2,6,4,5,3] => [1,2,6,3,4,5] => [4,5,6,1,2,3] => [1,1,1,1,0,1,0,1,0,0,0,0]
=> ? = 2 - 1
[1,2,6,5,3,4] => [1,2,6,3,4,5] => [4,5,6,1,2,3] => [1,1,1,1,0,1,0,1,0,0,0,0]
=> ? = 2 - 1
[1,2,6,5,4,3] => [1,2,6,3,4,5] => [4,5,6,1,2,3] => [1,1,1,1,0,1,0,1,0,0,0,0]
=> ? = 2 - 1
[1,3,2,4,5,6] => [1,3,2,4,5,6] => [3,1,2,4,5,6] => [1,1,1,0,0,0,1,0,1,0,1,0]
=> ? = 2 - 1
[1,3,2,4,6,5] => [1,3,2,4,6,5] => [2,6,1,3,4,5] => [1,1,0,1,1,1,1,0,0,0,0,0]
=> ? = 2 - 1
[1,3,2,5,4,6] => [1,3,2,5,4,6] => [2,5,1,3,4,6] => [1,1,0,1,1,1,0,0,0,0,1,0]
=> ? = 2 - 1
[1,3,2,5,6,4] => [1,3,2,5,6,4] => [2,4,6,1,3,5] => [1,1,0,1,1,0,1,1,0,0,0,0]
=> ? = 3 - 1
[1,3,2,6,4,5] => [1,3,2,6,4,5] => [2,5,6,1,3,4] => [1,1,0,1,1,1,0,1,0,0,0,0]
=> ? = 2 - 1
[1,3,2,6,5,4] => [1,3,2,6,4,5] => [2,5,6,1,3,4] => [1,1,0,1,1,1,0,1,0,0,0,0]
=> ? = 2 - 1
[1,3,4,2,5,6] => [1,3,4,2,5,6] => [2,4,1,3,5,6] => [1,1,0,1,1,0,0,0,1,0,1,0]
=> ? = 2 - 1
[1,3,4,2,6,5] => [1,3,4,2,6,5] => [2,3,6,1,4,5] => [1,1,0,1,0,1,1,1,0,0,0,0]
=> ? = 3 - 1
[1,3,4,5,2,6] => [1,3,4,5,2,6] => [2,3,5,1,4,6] => [1,1,0,1,0,1,1,0,0,0,1,0]
=> ? = 2 - 1
[1,3,4,5,6,2] => [1,3,4,5,6,2] => [2,3,4,6,1,5] => [1,1,0,1,0,1,0,1,1,0,0,0]
=> ? = 2 - 1
[1,3,4,6,2,5] => [1,3,4,6,2,5] => [2,3,5,6,1,4] => [1,1,0,1,0,1,1,0,1,0,0,0]
=> ? = 2 - 1
[1,3,4,6,5,2] => [1,3,4,6,2,5] => [2,3,5,6,1,4] => [1,1,0,1,0,1,1,0,1,0,0,0]
=> ? = 2 - 1
[1,3,5,2,4,6] => [1,3,5,2,4,6] => [2,4,5,1,3,6] => [1,1,0,1,1,0,1,0,0,0,1,0]
=> ? = 2 - 1
[1,3,5,2,6,4] => [1,3,5,2,6,4] => [6,2,4,1,3,5] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> ? = 3 - 1
[1,3,5,4,2,6] => [1,3,5,2,6,4] => [6,2,4,1,3,5] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> ? = 3 - 1
[1,3,5,4,6,2] => [1,3,5,2,4,6] => [2,4,5,1,3,6] => [1,1,0,1,1,0,1,0,0,0,1,0]
=> ? = 2 - 1
[1,3,5,6,2,4] => [1,3,5,6,2,4] => [5,2,3,6,1,4] => [1,1,1,1,1,0,0,0,1,0,0,0]
=> ? = 2 - 1
[1,3,5,6,4,2] => [1,3,5,6,2,4] => [5,2,3,6,1,4] => [1,1,1,1,1,0,0,0,1,0,0,0]
=> ? = 2 - 1
[1,3,6,2,4,5] => [1,3,6,2,4,5] => [2,4,5,6,1,3] => [1,1,0,1,1,0,1,0,1,0,0,0]
=> ? = 2 - 1
[1,3,6,2,5,4] => [1,3,6,2,5,4] => [6,2,4,5,1,3] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> ? = 2 - 1
[1,3,6,4,2,5] => [1,3,6,2,5,4] => [6,2,4,5,1,3] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> ? = 2 - 1
[1,3,6,4,5,2] => [1,3,6,2,4,5] => [2,4,5,6,1,3] => [1,1,0,1,1,0,1,0,1,0,0,0]
=> ? = 2 - 1
[1,3,6,5,2,4] => [1,3,6,2,4,5] => [2,4,5,6,1,3] => [1,1,0,1,1,0,1,0,1,0,0,0]
=> ? = 2 - 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: St000298
Mp00223: Permutations —runsort⟶ Permutations
Mp00209: Permutations —pattern poset⟶ Posets
St000298: Posets ⟶ ℤResult quality: 7% ●values known / values provided: 7%●distinct values known / distinct values provided: 67%
Mp00209: Permutations —pattern poset⟶ Posets
St000298: Posets ⟶ ℤResult quality: 7% ●values known / values provided: 7%●distinct values known / distinct values provided: 67%
Values
[1] => [1] => ([],1)
=> 1
[1,2] => [1,2] => ([(0,1)],2)
=> 1
[2,1] => [1,2] => ([(0,1)],2)
=> 1
[1,2,3] => [1,2,3] => ([(0,2),(2,1)],3)
=> 1
[1,3,2] => [1,3,2] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
[2,1,3] => [1,3,2] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
[2,3,1] => [1,2,3] => ([(0,2),(2,1)],3)
=> 1
[3,1,2] => [1,2,3] => ([(0,2),(2,1)],3)
=> 1
[3,2,1] => [1,2,3] => ([(0,2),(2,1)],3)
=> 1
[1,2,3,4] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> 1
[1,2,4,3] => [1,2,4,3] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 2
[1,3,2,4] => [1,3,2,4] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
=> 2
[1,3,4,2] => [1,3,4,2] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
=> 2
[1,4,2,3] => [1,4,2,3] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
=> 2
[1,4,3,2] => [1,4,2,3] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
=> 2
[2,1,3,4] => [1,3,4,2] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
=> 2
[2,1,4,3] => [1,4,2,3] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
=> 2
[2,3,1,4] => [1,4,2,3] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
=> 2
[2,3,4,1] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> 1
[2,4,1,3] => [1,3,2,4] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
=> 2
[2,4,3,1] => [1,2,4,3] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 2
[3,1,2,4] => [1,2,4,3] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 2
[3,1,4,2] => [1,4,2,3] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
=> 2
[3,2,1,4] => [1,4,2,3] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
=> 2
[3,2,4,1] => [1,2,4,3] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 2
[3,4,1,2] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> 1
[3,4,2,1] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> 1
[4,1,2,3] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> 1
[4,1,3,2] => [1,3,2,4] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
=> 2
[4,2,1,3] => [1,3,2,4] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
=> 2
[4,2,3,1] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> 1
[4,3,1,2] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> 1
[4,3,2,1] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> 1
[1,2,3,4,5] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1
[1,2,3,5,4] => [1,2,3,5,4] => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? = 2
[1,2,4,3,5] => [1,2,4,3,5] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
=> ? = 2
[1,2,4,5,3] => [1,2,4,5,3] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
=> ? = 2
[1,2,5,3,4] => [1,2,5,3,4] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
=> ? = 2
[1,2,5,4,3] => [1,2,5,3,4] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
=> ? = 2
[1,3,2,4,5] => [1,3,2,4,5] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
=> ? = 2
[1,3,2,5,4] => [1,3,2,5,4] => ([(0,1),(0,2),(0,3),(1,7),(1,8),(2,5),(2,8),(3,5),(3,7),(3,8),(5,9),(6,4),(7,6),(7,9),(8,6),(8,9),(9,4)],10)
=> ? = 2
[1,3,4,2,5] => [1,3,4,2,5] => ([(0,1),(0,2),(0,3),(0,4),(1,10),(2,6),(2,7),(2,10),(3,5),(3,7),(3,10),(4,5),(4,6),(4,10),(5,9),(5,11),(6,9),(6,11),(7,9),(7,11),(9,8),(10,11),(11,8)],12)
=> ? = 2
[1,3,4,5,2] => [1,3,4,5,2] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
=> ? = 2
[1,3,5,2,4] => [1,3,5,2,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,10),(1,12),(2,8),(2,10),(2,12),(3,7),(3,10),(3,12),(4,6),(4,10),(4,12),(5,6),(5,7),(5,8),(5,12),(6,11),(6,13),(7,11),(7,13),(8,11),(8,13),(10,13),(11,9),(12,11),(12,13),(13,9)],14)
=> ? = 2
[1,3,5,4,2] => [1,3,5,2,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,10),(1,12),(2,8),(2,10),(2,12),(3,7),(3,10),(3,12),(4,6),(4,10),(4,12),(5,6),(5,7),(5,8),(5,12),(6,11),(6,13),(7,11),(7,13),(8,11),(8,13),(10,13),(11,9),(12,11),(12,13),(13,9)],14)
=> ? = 2
[1,4,2,3,5] => [1,4,2,3,5] => ([(0,1),(0,2),(0,3),(0,4),(1,10),(2,6),(2,7),(2,10),(3,5),(3,7),(3,10),(4,5),(4,6),(4,10),(5,9),(5,11),(6,9),(6,11),(7,9),(7,11),(9,8),(10,11),(11,8)],12)
=> ? = 2
[1,4,2,5,3] => [1,4,2,5,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,10),(1,12),(2,8),(2,10),(2,12),(3,7),(3,10),(3,12),(4,6),(4,10),(4,12),(5,6),(5,7),(5,8),(5,12),(6,11),(6,13),(7,11),(7,13),(8,11),(8,13),(10,13),(11,9),(12,11),(12,13),(13,9)],14)
=> ? = 2
[1,4,3,2,5] => [1,4,2,5,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,10),(1,12),(2,8),(2,10),(2,12),(3,7),(3,10),(3,12),(4,6),(4,10),(4,12),(5,6),(5,7),(5,8),(5,12),(6,11),(6,13),(7,11),(7,13),(8,11),(8,13),(10,13),(11,9),(12,11),(12,13),(13,9)],14)
=> ? = 2
[1,4,3,5,2] => [1,4,2,3,5] => ([(0,1),(0,2),(0,3),(0,4),(1,10),(2,6),(2,7),(2,10),(3,5),(3,7),(3,10),(4,5),(4,6),(4,10),(5,9),(5,11),(6,9),(6,11),(7,9),(7,11),(9,8),(10,11),(11,8)],12)
=> ? = 2
[1,4,5,2,3] => [1,4,5,2,3] => ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,6),(2,7),(2,8),(3,5),(3,7),(3,8),(5,9),(5,10),(6,9),(6,10),(7,10),(8,9),(8,10),(9,4),(10,4)],11)
=> ? = 2
[1,4,5,3,2] => [1,4,5,2,3] => ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,6),(2,7),(2,8),(3,5),(3,7),(3,8),(5,9),(5,10),(6,9),(6,10),(7,10),(8,9),(8,10),(9,4),(10,4)],11)
=> ? = 2
[1,5,2,3,4] => [1,5,2,3,4] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
=> ? = 2
[1,5,2,4,3] => [1,5,2,4,3] => ([(0,1),(0,2),(0,3),(0,4),(1,6),(1,11),(2,5),(2,11),(3,5),(3,7),(3,11),(4,6),(4,7),(4,11),(5,9),(6,10),(7,9),(7,10),(9,8),(10,8),(11,9),(11,10)],12)
=> ? = 2
[1,5,3,2,4] => [1,5,2,4,3] => ([(0,1),(0,2),(0,3),(0,4),(1,6),(1,11),(2,5),(2,11),(3,5),(3,7),(3,11),(4,6),(4,7),(4,11),(5,9),(6,10),(7,9),(7,10),(9,8),(10,8),(11,9),(11,10)],12)
=> ? = 2
[1,5,3,4,2] => [1,5,2,3,4] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
=> ? = 2
[1,5,4,2,3] => [1,5,2,3,4] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
=> ? = 2
[1,5,4,3,2] => [1,5,2,3,4] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
=> ? = 2
[2,1,3,4,5] => [1,3,4,5,2] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
=> ? = 2
[2,1,3,5,4] => [1,3,5,2,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,10),(1,12),(2,8),(2,10),(2,12),(3,7),(3,10),(3,12),(4,6),(4,10),(4,12),(5,6),(5,7),(5,8),(5,12),(6,11),(6,13),(7,11),(7,13),(8,11),(8,13),(10,13),(11,9),(12,11),(12,13),(13,9)],14)
=> ? = 2
[2,1,4,3,5] => [1,4,2,3,5] => ([(0,1),(0,2),(0,3),(0,4),(1,10),(2,6),(2,7),(2,10),(3,5),(3,7),(3,10),(4,5),(4,6),(4,10),(5,9),(5,11),(6,9),(6,11),(7,9),(7,11),(9,8),(10,11),(11,8)],12)
=> ? = 2
[2,1,4,5,3] => [1,4,5,2,3] => ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,6),(2,7),(2,8),(3,5),(3,7),(3,8),(5,9),(5,10),(6,9),(6,10),(7,10),(8,9),(8,10),(9,4),(10,4)],11)
=> ? = 2
[2,1,5,3,4] => [1,5,2,3,4] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
=> ? = 2
[2,1,5,4,3] => [1,5,2,3,4] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
=> ? = 2
[2,3,1,4,5] => [1,4,5,2,3] => ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,6),(2,7),(2,8),(3,5),(3,7),(3,8),(5,9),(5,10),(6,9),(6,10),(7,10),(8,9),(8,10),(9,4),(10,4)],11)
=> ? = 2
[2,3,1,5,4] => [1,5,2,3,4] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
=> ? = 2
[2,3,4,1,5] => [1,5,2,3,4] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
=> ? = 2
[2,3,4,5,1] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1
[2,3,5,1,4] => [1,4,2,3,5] => ([(0,1),(0,2),(0,3),(0,4),(1,10),(2,6),(2,7),(2,10),(3,5),(3,7),(3,10),(4,5),(4,6),(4,10),(5,9),(5,11),(6,9),(6,11),(7,9),(7,11),(9,8),(10,11),(11,8)],12)
=> ? = 2
[2,3,5,4,1] => [1,2,3,5,4] => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? = 2
[2,4,1,3,5] => [1,3,5,2,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,10),(1,12),(2,8),(2,10),(2,12),(3,7),(3,10),(3,12),(4,6),(4,10),(4,12),(5,6),(5,7),(5,8),(5,12),(6,11),(6,13),(7,11),(7,13),(8,11),(8,13),(10,13),(11,9),(12,11),(12,13),(13,9)],14)
=> ? = 2
[2,4,1,5,3] => [1,5,2,4,3] => ([(0,1),(0,2),(0,3),(0,4),(1,6),(1,11),(2,5),(2,11),(3,5),(3,7),(3,11),(4,6),(4,7),(4,11),(5,9),(6,10),(7,9),(7,10),(9,8),(10,8),(11,9),(11,10)],12)
=> ? = 2
[2,4,3,1,5] => [1,5,2,4,3] => ([(0,1),(0,2),(0,3),(0,4),(1,6),(1,11),(2,5),(2,11),(3,5),(3,7),(3,11),(4,6),(4,7),(4,11),(5,9),(6,10),(7,9),(7,10),(9,8),(10,8),(11,9),(11,10)],12)
=> ? = 2
[2,4,3,5,1] => [1,2,4,3,5] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
=> ? = 2
[2,4,5,1,3] => [1,3,2,4,5] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
=> ? = 2
[2,4,5,3,1] => [1,2,4,5,3] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
=> ? = 2
[2,5,1,3,4] => [1,3,4,2,5] => ([(0,1),(0,2),(0,3),(0,4),(1,10),(2,6),(2,7),(2,10),(3,5),(3,7),(3,10),(4,5),(4,6),(4,10),(5,9),(5,11),(6,9),(6,11),(7,9),(7,11),(9,8),(10,11),(11,8)],12)
=> ? = 2
[2,5,1,4,3] => [1,4,2,5,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,10),(1,12),(2,8),(2,10),(2,12),(3,7),(3,10),(3,12),(4,6),(4,10),(4,12),(5,6),(5,7),(5,8),(5,12),(6,11),(6,13),(7,11),(7,13),(8,11),(8,13),(10,13),(11,9),(12,11),(12,13),(13,9)],14)
=> ? = 2
[2,5,3,1,4] => [1,4,2,5,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,10),(1,12),(2,8),(2,10),(2,12),(3,7),(3,10),(3,12),(4,6),(4,10),(4,12),(5,6),(5,7),(5,8),(5,12),(6,11),(6,13),(7,11),(7,13),(8,11),(8,13),(10,13),(11,9),(12,11),(12,13),(13,9)],14)
=> ? = 2
[2,5,3,4,1] => [1,2,5,3,4] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
=> ? = 2
[2,5,4,1,3] => [1,3,2,5,4] => ([(0,1),(0,2),(0,3),(1,7),(1,8),(2,5),(2,8),(3,5),(3,7),(3,8),(5,9),(6,4),(7,6),(7,9),(8,6),(8,9),(9,4)],10)
=> ? = 2
[2,5,4,3,1] => [1,2,5,3,4] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
=> ? = 2
[3,1,2,4,5] => [1,2,4,5,3] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
=> ? = 2
[3,1,2,5,4] => [1,2,5,3,4] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
=> ? = 2
[3,1,4,2,5] => [1,4,2,5,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,10),(1,12),(2,8),(2,10),(2,12),(3,7),(3,10),(3,12),(4,6),(4,10),(4,12),(5,6),(5,7),(5,8),(5,12),(6,11),(6,13),(7,11),(7,13),(8,11),(8,13),(10,13),(11,9),(12,11),(12,13),(13,9)],14)
=> ? = 2
[3,1,4,5,2] => [1,4,5,2,3] => ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,6),(2,7),(2,8),(3,5),(3,7),(3,8),(5,9),(5,10),(6,9),(6,10),(7,10),(8,9),(8,10),(9,4),(10,4)],11)
=> ? = 2
[3,4,5,1,2] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1
[3,4,5,2,1] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1
[4,5,1,2,3] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1
[4,5,2,3,1] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1
[4,5,3,1,2] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1
[4,5,3,2,1] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1
[5,1,2,3,4] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1
[5,2,3,4,1] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1
[5,3,4,1,2] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1
[5,3,4,2,1] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1
[5,4,1,2,3] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1
[5,4,2,3,1] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1
[5,4,3,1,2] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1
[5,4,3,2,1] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1
[1,2,3,4,5,6] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 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: St000307
Mp00223: Permutations —runsort⟶ Permutations
Mp00209: Permutations —pattern poset⟶ Posets
St000307: Posets ⟶ ℤResult quality: 6% ●values known / values provided: 6%●distinct values known / distinct values provided: 67%
Mp00209: Permutations —pattern poset⟶ Posets
St000307: Posets ⟶ ℤResult quality: 6% ●values known / values provided: 6%●distinct values known / distinct values provided: 67%
Values
[1] => [1] => ([],1)
=> 1
[1,2] => [1,2] => ([(0,1)],2)
=> 1
[2,1] => [1,2] => ([(0,1)],2)
=> 1
[1,2,3] => [1,2,3] => ([(0,2),(2,1)],3)
=> 1
[1,3,2] => [1,3,2] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
[2,1,3] => [1,3,2] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
[2,3,1] => [1,2,3] => ([(0,2),(2,1)],3)
=> 1
[3,1,2] => [1,2,3] => ([(0,2),(2,1)],3)
=> 1
[3,2,1] => [1,2,3] => ([(0,2),(2,1)],3)
=> 1
[1,2,3,4] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> 1
[1,2,4,3] => [1,2,4,3] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 2
[1,3,2,4] => [1,3,2,4] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
=> ? = 2
[1,3,4,2] => [1,3,4,2] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
=> ? = 2
[1,4,2,3] => [1,4,2,3] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
=> ? = 2
[1,4,3,2] => [1,4,2,3] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
=> ? = 2
[2,1,3,4] => [1,3,4,2] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
=> ? = 2
[2,1,4,3] => [1,4,2,3] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
=> ? = 2
[2,3,1,4] => [1,4,2,3] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
=> ? = 2
[2,3,4,1] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> 1
[2,4,1,3] => [1,3,2,4] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
=> ? = 2
[2,4,3,1] => [1,2,4,3] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 2
[3,1,2,4] => [1,2,4,3] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 2
[3,1,4,2] => [1,4,2,3] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
=> ? = 2
[3,2,1,4] => [1,4,2,3] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
=> ? = 2
[3,2,4,1] => [1,2,4,3] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 2
[3,4,1,2] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> 1
[3,4,2,1] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> 1
[4,1,2,3] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> 1
[4,1,3,2] => [1,3,2,4] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
=> ? = 2
[4,2,1,3] => [1,3,2,4] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
=> ? = 2
[4,2,3,1] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> 1
[4,3,1,2] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> 1
[4,3,2,1] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> 1
[1,2,3,4,5] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1
[1,2,3,5,4] => [1,2,3,5,4] => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? = 2
[1,2,4,3,5] => [1,2,4,3,5] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
=> ? = 2
[1,2,4,5,3] => [1,2,4,5,3] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
=> ? = 2
[1,2,5,3,4] => [1,2,5,3,4] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
=> ? = 2
[1,2,5,4,3] => [1,2,5,3,4] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
=> ? = 2
[1,3,2,4,5] => [1,3,2,4,5] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
=> ? = 2
[1,3,2,5,4] => [1,3,2,5,4] => ([(0,1),(0,2),(0,3),(1,7),(1,8),(2,5),(2,8),(3,5),(3,7),(3,8),(5,9),(6,4),(7,6),(7,9),(8,6),(8,9),(9,4)],10)
=> ? = 2
[1,3,4,2,5] => [1,3,4,2,5] => ([(0,1),(0,2),(0,3),(0,4),(1,10),(2,6),(2,7),(2,10),(3,5),(3,7),(3,10),(4,5),(4,6),(4,10),(5,9),(5,11),(6,9),(6,11),(7,9),(7,11),(9,8),(10,11),(11,8)],12)
=> ? = 2
[1,3,4,5,2] => [1,3,4,5,2] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
=> ? = 2
[1,3,5,2,4] => [1,3,5,2,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,10),(1,12),(2,8),(2,10),(2,12),(3,7),(3,10),(3,12),(4,6),(4,10),(4,12),(5,6),(5,7),(5,8),(5,12),(6,11),(6,13),(7,11),(7,13),(8,11),(8,13),(10,13),(11,9),(12,11),(12,13),(13,9)],14)
=> ? = 2
[1,3,5,4,2] => [1,3,5,2,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,10),(1,12),(2,8),(2,10),(2,12),(3,7),(3,10),(3,12),(4,6),(4,10),(4,12),(5,6),(5,7),(5,8),(5,12),(6,11),(6,13),(7,11),(7,13),(8,11),(8,13),(10,13),(11,9),(12,11),(12,13),(13,9)],14)
=> ? = 2
[1,4,2,3,5] => [1,4,2,3,5] => ([(0,1),(0,2),(0,3),(0,4),(1,10),(2,6),(2,7),(2,10),(3,5),(3,7),(3,10),(4,5),(4,6),(4,10),(5,9),(5,11),(6,9),(6,11),(7,9),(7,11),(9,8),(10,11),(11,8)],12)
=> ? = 2
[1,4,2,5,3] => [1,4,2,5,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,10),(1,12),(2,8),(2,10),(2,12),(3,7),(3,10),(3,12),(4,6),(4,10),(4,12),(5,6),(5,7),(5,8),(5,12),(6,11),(6,13),(7,11),(7,13),(8,11),(8,13),(10,13),(11,9),(12,11),(12,13),(13,9)],14)
=> ? = 2
[1,4,3,2,5] => [1,4,2,5,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,10),(1,12),(2,8),(2,10),(2,12),(3,7),(3,10),(3,12),(4,6),(4,10),(4,12),(5,6),(5,7),(5,8),(5,12),(6,11),(6,13),(7,11),(7,13),(8,11),(8,13),(10,13),(11,9),(12,11),(12,13),(13,9)],14)
=> ? = 2
[1,4,3,5,2] => [1,4,2,3,5] => ([(0,1),(0,2),(0,3),(0,4),(1,10),(2,6),(2,7),(2,10),(3,5),(3,7),(3,10),(4,5),(4,6),(4,10),(5,9),(5,11),(6,9),(6,11),(7,9),(7,11),(9,8),(10,11),(11,8)],12)
=> ? = 2
[1,4,5,2,3] => [1,4,5,2,3] => ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,6),(2,7),(2,8),(3,5),(3,7),(3,8),(5,9),(5,10),(6,9),(6,10),(7,10),(8,9),(8,10),(9,4),(10,4)],11)
=> ? = 2
[1,4,5,3,2] => [1,4,5,2,3] => ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,6),(2,7),(2,8),(3,5),(3,7),(3,8),(5,9),(5,10),(6,9),(6,10),(7,10),(8,9),(8,10),(9,4),(10,4)],11)
=> ? = 2
[1,5,2,3,4] => [1,5,2,3,4] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
=> ? = 2
[1,5,2,4,3] => [1,5,2,4,3] => ([(0,1),(0,2),(0,3),(0,4),(1,6),(1,11),(2,5),(2,11),(3,5),(3,7),(3,11),(4,6),(4,7),(4,11),(5,9),(6,10),(7,9),(7,10),(9,8),(10,8),(11,9),(11,10)],12)
=> ? = 2
[1,5,3,2,4] => [1,5,2,4,3] => ([(0,1),(0,2),(0,3),(0,4),(1,6),(1,11),(2,5),(2,11),(3,5),(3,7),(3,11),(4,6),(4,7),(4,11),(5,9),(6,10),(7,9),(7,10),(9,8),(10,8),(11,9),(11,10)],12)
=> ? = 2
[1,5,3,4,2] => [1,5,2,3,4] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
=> ? = 2
[1,5,4,2,3] => [1,5,2,3,4] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
=> ? = 2
[1,5,4,3,2] => [1,5,2,3,4] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
=> ? = 2
[2,1,3,4,5] => [1,3,4,5,2] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
=> ? = 2
[2,1,3,5,4] => [1,3,5,2,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,10),(1,12),(2,8),(2,10),(2,12),(3,7),(3,10),(3,12),(4,6),(4,10),(4,12),(5,6),(5,7),(5,8),(5,12),(6,11),(6,13),(7,11),(7,13),(8,11),(8,13),(10,13),(11,9),(12,11),(12,13),(13,9)],14)
=> ? = 2
[2,1,4,3,5] => [1,4,2,3,5] => ([(0,1),(0,2),(0,3),(0,4),(1,10),(2,6),(2,7),(2,10),(3,5),(3,7),(3,10),(4,5),(4,6),(4,10),(5,9),(5,11),(6,9),(6,11),(7,9),(7,11),(9,8),(10,11),(11,8)],12)
=> ? = 2
[2,1,4,5,3] => [1,4,5,2,3] => ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,6),(2,7),(2,8),(3,5),(3,7),(3,8),(5,9),(5,10),(6,9),(6,10),(7,10),(8,9),(8,10),(9,4),(10,4)],11)
=> ? = 2
[2,1,5,3,4] => [1,5,2,3,4] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
=> ? = 2
[2,1,5,4,3] => [1,5,2,3,4] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
=> ? = 2
[2,3,1,4,5] => [1,4,5,2,3] => ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,6),(2,7),(2,8),(3,5),(3,7),(3,8),(5,9),(5,10),(6,9),(6,10),(7,10),(8,9),(8,10),(9,4),(10,4)],11)
=> ? = 2
[2,3,1,5,4] => [1,5,2,3,4] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
=> ? = 2
[2,3,4,1,5] => [1,5,2,3,4] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
=> ? = 2
[2,3,4,5,1] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1
[2,3,5,1,4] => [1,4,2,3,5] => ([(0,1),(0,2),(0,3),(0,4),(1,10),(2,6),(2,7),(2,10),(3,5),(3,7),(3,10),(4,5),(4,6),(4,10),(5,9),(5,11),(6,9),(6,11),(7,9),(7,11),(9,8),(10,11),(11,8)],12)
=> ? = 2
[2,3,5,4,1] => [1,2,3,5,4] => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? = 2
[2,4,1,3,5] => [1,3,5,2,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,10),(1,12),(2,8),(2,10),(2,12),(3,7),(3,10),(3,12),(4,6),(4,10),(4,12),(5,6),(5,7),(5,8),(5,12),(6,11),(6,13),(7,11),(7,13),(8,11),(8,13),(10,13),(11,9),(12,11),(12,13),(13,9)],14)
=> ? = 2
[2,4,1,5,3] => [1,5,2,4,3] => ([(0,1),(0,2),(0,3),(0,4),(1,6),(1,11),(2,5),(2,11),(3,5),(3,7),(3,11),(4,6),(4,7),(4,11),(5,9),(6,10),(7,9),(7,10),(9,8),(10,8),(11,9),(11,10)],12)
=> ? = 2
[2,4,3,1,5] => [1,5,2,4,3] => ([(0,1),(0,2),(0,3),(0,4),(1,6),(1,11),(2,5),(2,11),(3,5),(3,7),(3,11),(4,6),(4,7),(4,11),(5,9),(6,10),(7,9),(7,10),(9,8),(10,8),(11,9),(11,10)],12)
=> ? = 2
[2,4,3,5,1] => [1,2,4,3,5] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
=> ? = 2
[3,4,5,1,2] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1
[3,4,5,2,1] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1
[4,5,1,2,3] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1
[4,5,2,3,1] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1
[4,5,3,1,2] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1
[4,5,3,2,1] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1
[5,1,2,3,4] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1
[5,2,3,4,1] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1
[5,3,4,1,2] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1
[5,3,4,2,1] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1
[5,4,1,2,3] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1
[5,4,2,3,1] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1
[5,4,3,1,2] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1
[5,4,3,2,1] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1
[1,2,3,4,5,6] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 1
[2,3,4,5,6,1] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 1
[3,4,5,6,1,2] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 1
[3,4,5,6,2,1] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 1
[4,5,6,1,2,3] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 1
[4,5,6,2,3,1] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 1
[4,5,6,3,1,2] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 1
[4,5,6,3,2,1] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 1
[5,6,1,2,3,4] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 1
[5,6,2,3,4,1] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 1
[5,6,3,4,1,2] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 1
[5,6,3,4,2,1] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 1
[5,6,4,1,2,3] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 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 $P \setminus I$.
Matching statistic: St001632
Mp00223: Permutations —runsort⟶ Permutations
Mp00209: Permutations —pattern poset⟶ Posets
St001632: Posets ⟶ ℤResult quality: 6% ●values known / values provided: 6%●distinct values known / distinct values provided: 67%
Mp00209: Permutations —pattern poset⟶ Posets
St001632: Posets ⟶ ℤResult quality: 6% ●values known / values provided: 6%●distinct values known / distinct values provided: 67%
Values
[1] => [1] => ([],1)
=> ? = 1
[1,2] => [1,2] => ([(0,1)],2)
=> 1
[2,1] => [1,2] => ([(0,1)],2)
=> 1
[1,2,3] => [1,2,3] => ([(0,2),(2,1)],3)
=> 1
[1,3,2] => [1,3,2] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
[2,1,3] => [1,3,2] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
[2,3,1] => [1,2,3] => ([(0,2),(2,1)],3)
=> 1
[3,1,2] => [1,2,3] => ([(0,2),(2,1)],3)
=> 1
[3,2,1] => [1,2,3] => ([(0,2),(2,1)],3)
=> 1
[1,2,3,4] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> 1
[1,2,4,3] => [1,2,4,3] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 2
[1,3,2,4] => [1,3,2,4] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
=> ? = 2
[1,3,4,2] => [1,3,4,2] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
=> ? = 2
[1,4,2,3] => [1,4,2,3] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
=> ? = 2
[1,4,3,2] => [1,4,2,3] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
=> ? = 2
[2,1,3,4] => [1,3,4,2] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
=> ? = 2
[2,1,4,3] => [1,4,2,3] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
=> ? = 2
[2,3,1,4] => [1,4,2,3] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
=> ? = 2
[2,3,4,1] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> 1
[2,4,1,3] => [1,3,2,4] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
=> ? = 2
[2,4,3,1] => [1,2,4,3] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 2
[3,1,2,4] => [1,2,4,3] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 2
[3,1,4,2] => [1,4,2,3] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
=> ? = 2
[3,2,1,4] => [1,4,2,3] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
=> ? = 2
[3,2,4,1] => [1,2,4,3] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 2
[3,4,1,2] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> 1
[3,4,2,1] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> 1
[4,1,2,3] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> 1
[4,1,3,2] => [1,3,2,4] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
=> ? = 2
[4,2,1,3] => [1,3,2,4] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
=> ? = 2
[4,2,3,1] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> 1
[4,3,1,2] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> 1
[4,3,2,1] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> 1
[1,2,3,4,5] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1
[1,2,3,5,4] => [1,2,3,5,4] => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? = 2
[1,2,4,3,5] => [1,2,4,3,5] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
=> ? = 2
[1,2,4,5,3] => [1,2,4,5,3] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
=> ? = 2
[1,2,5,3,4] => [1,2,5,3,4] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
=> ? = 2
[1,2,5,4,3] => [1,2,5,3,4] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
=> ? = 2
[1,3,2,4,5] => [1,3,2,4,5] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
=> ? = 2
[1,3,2,5,4] => [1,3,2,5,4] => ([(0,1),(0,2),(0,3),(1,7),(1,8),(2,5),(2,8),(3,5),(3,7),(3,8),(5,9),(6,4),(7,6),(7,9),(8,6),(8,9),(9,4)],10)
=> ? = 2
[1,3,4,2,5] => [1,3,4,2,5] => ([(0,1),(0,2),(0,3),(0,4),(1,10),(2,6),(2,7),(2,10),(3,5),(3,7),(3,10),(4,5),(4,6),(4,10),(5,9),(5,11),(6,9),(6,11),(7,9),(7,11),(9,8),(10,11),(11,8)],12)
=> ? = 2
[1,3,4,5,2] => [1,3,4,5,2] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
=> ? = 2
[1,3,5,2,4] => [1,3,5,2,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,10),(1,12),(2,8),(2,10),(2,12),(3,7),(3,10),(3,12),(4,6),(4,10),(4,12),(5,6),(5,7),(5,8),(5,12),(6,11),(6,13),(7,11),(7,13),(8,11),(8,13),(10,13),(11,9),(12,11),(12,13),(13,9)],14)
=> ? = 2
[1,3,5,4,2] => [1,3,5,2,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,10),(1,12),(2,8),(2,10),(2,12),(3,7),(3,10),(3,12),(4,6),(4,10),(4,12),(5,6),(5,7),(5,8),(5,12),(6,11),(6,13),(7,11),(7,13),(8,11),(8,13),(10,13),(11,9),(12,11),(12,13),(13,9)],14)
=> ? = 2
[1,4,2,3,5] => [1,4,2,3,5] => ([(0,1),(0,2),(0,3),(0,4),(1,10),(2,6),(2,7),(2,10),(3,5),(3,7),(3,10),(4,5),(4,6),(4,10),(5,9),(5,11),(6,9),(6,11),(7,9),(7,11),(9,8),(10,11),(11,8)],12)
=> ? = 2
[1,4,2,5,3] => [1,4,2,5,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,10),(1,12),(2,8),(2,10),(2,12),(3,7),(3,10),(3,12),(4,6),(4,10),(4,12),(5,6),(5,7),(5,8),(5,12),(6,11),(6,13),(7,11),(7,13),(8,11),(8,13),(10,13),(11,9),(12,11),(12,13),(13,9)],14)
=> ? = 2
[1,4,3,2,5] => [1,4,2,5,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,10),(1,12),(2,8),(2,10),(2,12),(3,7),(3,10),(3,12),(4,6),(4,10),(4,12),(5,6),(5,7),(5,8),(5,12),(6,11),(6,13),(7,11),(7,13),(8,11),(8,13),(10,13),(11,9),(12,11),(12,13),(13,9)],14)
=> ? = 2
[1,4,3,5,2] => [1,4,2,3,5] => ([(0,1),(0,2),(0,3),(0,4),(1,10),(2,6),(2,7),(2,10),(3,5),(3,7),(3,10),(4,5),(4,6),(4,10),(5,9),(5,11),(6,9),(6,11),(7,9),(7,11),(9,8),(10,11),(11,8)],12)
=> ? = 2
[1,4,5,2,3] => [1,4,5,2,3] => ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,6),(2,7),(2,8),(3,5),(3,7),(3,8),(5,9),(5,10),(6,9),(6,10),(7,10),(8,9),(8,10),(9,4),(10,4)],11)
=> ? = 2
[1,4,5,3,2] => [1,4,5,2,3] => ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,6),(2,7),(2,8),(3,5),(3,7),(3,8),(5,9),(5,10),(6,9),(6,10),(7,10),(8,9),(8,10),(9,4),(10,4)],11)
=> ? = 2
[1,5,2,3,4] => [1,5,2,3,4] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
=> ? = 2
[1,5,2,4,3] => [1,5,2,4,3] => ([(0,1),(0,2),(0,3),(0,4),(1,6),(1,11),(2,5),(2,11),(3,5),(3,7),(3,11),(4,6),(4,7),(4,11),(5,9),(6,10),(7,9),(7,10),(9,8),(10,8),(11,9),(11,10)],12)
=> ? = 2
[1,5,3,2,4] => [1,5,2,4,3] => ([(0,1),(0,2),(0,3),(0,4),(1,6),(1,11),(2,5),(2,11),(3,5),(3,7),(3,11),(4,6),(4,7),(4,11),(5,9),(6,10),(7,9),(7,10),(9,8),(10,8),(11,9),(11,10)],12)
=> ? = 2
[1,5,3,4,2] => [1,5,2,3,4] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
=> ? = 2
[1,5,4,2,3] => [1,5,2,3,4] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
=> ? = 2
[1,5,4,3,2] => [1,5,2,3,4] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
=> ? = 2
[2,1,3,4,5] => [1,3,4,5,2] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
=> ? = 2
[2,1,3,5,4] => [1,3,5,2,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,10),(1,12),(2,8),(2,10),(2,12),(3,7),(3,10),(3,12),(4,6),(4,10),(4,12),(5,6),(5,7),(5,8),(5,12),(6,11),(6,13),(7,11),(7,13),(8,11),(8,13),(10,13),(11,9),(12,11),(12,13),(13,9)],14)
=> ? = 2
[2,1,4,3,5] => [1,4,2,3,5] => ([(0,1),(0,2),(0,3),(0,4),(1,10),(2,6),(2,7),(2,10),(3,5),(3,7),(3,10),(4,5),(4,6),(4,10),(5,9),(5,11),(6,9),(6,11),(7,9),(7,11),(9,8),(10,11),(11,8)],12)
=> ? = 2
[2,1,4,5,3] => [1,4,5,2,3] => ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,6),(2,7),(2,8),(3,5),(3,7),(3,8),(5,9),(5,10),(6,9),(6,10),(7,10),(8,9),(8,10),(9,4),(10,4)],11)
=> ? = 2
[2,1,5,3,4] => [1,5,2,3,4] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
=> ? = 2
[2,1,5,4,3] => [1,5,2,3,4] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
=> ? = 2
[2,3,1,4,5] => [1,4,5,2,3] => ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,6),(2,7),(2,8),(3,5),(3,7),(3,8),(5,9),(5,10),(6,9),(6,10),(7,10),(8,9),(8,10),(9,4),(10,4)],11)
=> ? = 2
[2,3,1,5,4] => [1,5,2,3,4] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
=> ? = 2
[2,3,4,1,5] => [1,5,2,3,4] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
=> ? = 2
[2,3,4,5,1] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1
[2,3,5,1,4] => [1,4,2,3,5] => ([(0,1),(0,2),(0,3),(0,4),(1,10),(2,6),(2,7),(2,10),(3,5),(3,7),(3,10),(4,5),(4,6),(4,10),(5,9),(5,11),(6,9),(6,11),(7,9),(7,11),(9,8),(10,11),(11,8)],12)
=> ? = 2
[2,3,5,4,1] => [1,2,3,5,4] => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? = 2
[2,4,1,3,5] => [1,3,5,2,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,10),(1,12),(2,8),(2,10),(2,12),(3,7),(3,10),(3,12),(4,6),(4,10),(4,12),(5,6),(5,7),(5,8),(5,12),(6,11),(6,13),(7,11),(7,13),(8,11),(8,13),(10,13),(11,9),(12,11),(12,13),(13,9)],14)
=> ? = 2
[2,4,1,5,3] => [1,5,2,4,3] => ([(0,1),(0,2),(0,3),(0,4),(1,6),(1,11),(2,5),(2,11),(3,5),(3,7),(3,11),(4,6),(4,7),(4,11),(5,9),(6,10),(7,9),(7,10),(9,8),(10,8),(11,9),(11,10)],12)
=> ? = 2
[2,4,3,1,5] => [1,5,2,4,3] => ([(0,1),(0,2),(0,3),(0,4),(1,6),(1,11),(2,5),(2,11),(3,5),(3,7),(3,11),(4,6),(4,7),(4,11),(5,9),(6,10),(7,9),(7,10),(9,8),(10,8),(11,9),(11,10)],12)
=> ? = 2
[3,4,5,1,2] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1
[3,4,5,2,1] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1
[4,5,1,2,3] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1
[4,5,2,3,1] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1
[4,5,3,1,2] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1
[4,5,3,2,1] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1
[5,1,2,3,4] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1
[5,2,3,4,1] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1
[5,3,4,1,2] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1
[5,3,4,2,1] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1
[5,4,1,2,3] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1
[5,4,2,3,1] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1
[5,4,3,1,2] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1
[5,4,3,2,1] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1
[1,2,3,4,5,6] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 1
[2,3,4,5,6,1] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 1
[3,4,5,6,1,2] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 1
[3,4,5,6,2,1] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 1
[4,5,6,1,2,3] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 1
[4,5,6,2,3,1] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 1
[4,5,6,3,1,2] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 1
[4,5,6,3,2,1] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 1
[5,6,1,2,3,4] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 1
[5,6,2,3,4,1] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 1
[5,6,3,4,1,2] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 1
[5,6,3,4,2,1] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 1
[5,6,4,1,2,3] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 1
[5,6,4,2,3,1] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 1
Description
The number of indecomposable injective modules $I$ with $dim Ext^1(I,A)=1$ for the incidence algebra A of a poset.
The following 12 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St001491The number of indecomposable projective-injective modules in the algebra corresponding to a subset. St001896The number of right descents of a signed permutations. St001624The breadth of a lattice. St000640The rank of the largest boolean interval in a poset. St001823The Stasinski-Voll length of a signed permutation. St001946The number of descents in a parking function. St001569The maximal modular displacement of a permutation. St001773The number of minimal elements in Bruhat order not less than the signed permutation. St001935The number of ascents in a parking function. St001960The number of descents of a permutation minus one if its first entry is not one. 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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