Your data matches 10 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)
Matching statistic: St001060
Mp00159: Permutations Demazure product with inversePermutations
Mp00310: Permutations toric promotionPermutations
Mp00160: Permutations graph of inversionsGraphs
St001060: Graphs ⟶ ℤResult quality: 100% values known / values provided: 100%distinct values known / distinct values provided: 100%
Values
[1,2,3] => [1,2,3] => [3,2,1] => ([(0,1),(0,2),(1,2)],3)
=> 3
[1,3,2] => [1,3,2] => [2,3,1] => ([(0,2),(1,2)],3)
=> 2
[2,1,3] => [2,1,3] => [3,1,2] => ([(0,2),(1,2)],3)
=> 2
[1,2,3,4] => [1,2,3,4] => [4,2,3,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 2
[1,2,4,3] => [1,2,4,3] => [4,3,1,2] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 2
[1,3,2,4] => [1,3,2,4] => [2,4,1,3] => ([(0,3),(1,2),(2,3)],4)
=> 2
[1,3,4,2] => [1,4,3,2] => [3,1,2,4] => ([(1,3),(2,3)],4)
=> 2
[1,4,2,3] => [1,4,3,2] => [3,1,2,4] => ([(1,3),(2,3)],4)
=> 2
[1,4,3,2] => [1,4,3,2] => [3,1,2,4] => ([(1,3),(2,3)],4)
=> 2
[2,1,3,4] => [2,1,3,4] => [4,1,2,3] => ([(0,3),(1,3),(2,3)],4)
=> 3
[2,1,4,3] => [2,1,4,3] => [4,1,3,2] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 2
[2,3,4,1] => [4,2,3,1] => [3,1,4,2] => ([(0,3),(1,2),(2,3)],4)
=> 2
[2,4,1,3] => [3,4,1,2] => [2,3,1,4] => ([(1,3),(2,3)],4)
=> 2
[3,1,4,2] => [4,2,3,1] => [3,1,4,2] => ([(0,3),(1,2),(2,3)],4)
=> 2
[3,2,4,1] => [4,2,3,1] => [3,1,4,2] => ([(0,3),(1,2),(2,3)],4)
=> 2
[4,1,2,3] => [4,2,3,1] => [3,1,4,2] => ([(0,3),(1,2),(2,3)],4)
=> 2
[4,1,3,2] => [4,2,3,1] => [3,1,4,2] => ([(0,3),(1,2),(2,3)],4)
=> 2
[1,2,3,4,5] => [1,2,3,4,5] => [5,2,3,4,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
[1,2,3,5,4] => [1,2,3,5,4] => [5,2,4,1,3] => ([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
[1,2,4,3,5] => [1,2,4,3,5] => [5,3,1,2,4] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
[1,2,4,5,3] => [1,2,5,4,3] => [5,4,1,3,2] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
[1,2,5,3,4] => [1,2,5,4,3] => [5,4,1,3,2] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
[1,2,5,4,3] => [1,2,5,4,3] => [5,4,1,3,2] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
[1,3,2,4,5] => [1,3,2,4,5] => [2,5,1,3,4] => ([(0,4),(1,4),(2,3),(3,4)],5)
=> 2
[1,3,2,5,4] => [1,3,2,5,4] => [2,5,1,4,3] => ([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> 2
[1,3,4,5,2] => [1,5,3,4,2] => [4,1,2,3,5] => ([(1,4),(2,4),(3,4)],5)
=> 3
[1,3,5,2,4] => [1,4,5,2,3] => [3,4,5,1,2] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> 2
[1,3,5,4,2] => [1,5,4,3,2] => [4,1,3,2,5] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> 2
[1,4,2,5,3] => [1,5,3,4,2] => [4,1,2,3,5] => ([(1,4),(2,4),(3,4)],5)
=> 3
[1,4,3,5,2] => [1,5,3,4,2] => [4,1,2,3,5] => ([(1,4),(2,4),(3,4)],5)
=> 3
[1,4,5,2,3] => [1,5,4,3,2] => [4,1,3,2,5] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> 2
[1,4,5,3,2] => [1,5,4,3,2] => [4,1,3,2,5] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> 2
[1,5,2,3,4] => [1,5,3,4,2] => [4,1,2,3,5] => ([(1,4),(2,4),(3,4)],5)
=> 3
[1,5,2,4,3] => [1,5,3,4,2] => [4,1,2,3,5] => ([(1,4),(2,4),(3,4)],5)
=> 3
[1,5,3,2,4] => [1,5,4,3,2] => [4,1,3,2,5] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> 2
[1,5,3,4,2] => [1,5,4,3,2] => [4,1,3,2,5] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> 2
[1,5,4,2,3] => [1,5,4,3,2] => [4,1,3,2,5] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> 2
[1,5,4,3,2] => [1,5,4,3,2] => [4,1,3,2,5] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> 2
[2,1,3,4,5] => [2,1,3,4,5] => [5,1,2,3,4] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 4
[2,1,3,5,4] => [2,1,3,5,4] => [5,1,2,4,3] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
[2,1,4,3,5] => [2,1,4,3,5] => [5,1,3,2,4] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
[2,1,4,5,3] => [2,1,5,4,3] => [5,1,4,3,2] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
[2,1,5,3,4] => [2,1,5,4,3] => [5,1,4,3,2] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
[2,1,5,4,3] => [2,1,5,4,3] => [5,1,4,3,2] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
[2,3,4,1,5] => [4,2,3,1,5] => [3,1,5,2,4] => ([(0,4),(1,3),(2,3),(2,4)],5)
=> 2
[2,3,4,5,1] => [5,2,3,4,1] => [4,5,2,1,3] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
=> 2
[2,3,5,1,4] => [4,2,5,1,3] => [3,5,4,2,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
[2,3,5,4,1] => [5,2,4,3,1] => [4,1,5,3,2] => ([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
=> 2
[2,4,1,3,5] => [3,4,1,2,5] => [2,3,5,4,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
[2,4,1,5,3] => [3,5,1,4,2] => [2,4,3,5,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
Description
The distinguishing index of a graph. This is the smallest number of colours such that there is a colouring of the edges which is not preserved by any automorphism. If the graph has a connected component which is a single edge, or at least two isolated vertices, this statistic is undefined.
Matching statistic: St000259
Mp00067: Permutations Foata bijectionPermutations
Mp00248: Permutations DEX compositionInteger compositions
Mp00184: Integer compositions to threshold graphGraphs
St000259: Graphs ⟶ ℤResult quality: 20% values known / values provided: 26%distinct values known / distinct values provided: 20%
Values
[1,2,3] => [1,2,3] => [3] => ([],3)
=> ? = 3
[1,3,2] => [3,1,2] => [3] => ([],3)
=> ? = 2
[2,1,3] => [2,1,3] => [3] => ([],3)
=> ? = 2
[1,2,3,4] => [1,2,3,4] => [4] => ([],4)
=> ? = 2
[1,2,4,3] => [4,1,2,3] => [4] => ([],4)
=> ? = 2
[1,3,2,4] => [3,1,2,4] => [4] => ([],4)
=> ? = 2
[1,3,4,2] => [3,1,4,2] => [2,2] => ([(1,3),(2,3)],4)
=> ? = 2
[1,4,2,3] => [1,4,2,3] => [1,3] => ([(2,3)],4)
=> ? = 2
[1,4,3,2] => [4,3,1,2] => [1,3] => ([(2,3)],4)
=> ? = 2
[2,1,3,4] => [2,1,3,4] => [4] => ([],4)
=> ? = 3
[2,1,4,3] => [4,2,1,3] => [2,2] => ([(1,3),(2,3)],4)
=> ? = 2
[2,3,4,1] => [2,3,4,1] => [4] => ([],4)
=> ? = 2
[2,4,1,3] => [2,1,4,3] => [2,2] => ([(1,3),(2,3)],4)
=> ? = 2
[3,1,4,2] => [3,4,1,2] => [4] => ([],4)
=> ? = 2
[3,2,4,1] => [3,2,4,1] => [2,2] => ([(1,3),(2,3)],4)
=> ? = 2
[4,1,2,3] => [1,2,4,3] => [2,2] => ([(1,3),(2,3)],4)
=> ? = 2
[4,1,3,2] => [4,1,3,2] => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 2
[1,2,3,4,5] => [1,2,3,4,5] => [5] => ([],5)
=> ? = 2
[1,2,3,5,4] => [5,1,2,3,4] => [5] => ([],5)
=> ? = 2
[1,2,4,3,5] => [4,1,2,3,5] => [5] => ([],5)
=> ? = 2
[1,2,4,5,3] => [4,1,2,5,3] => [3,2] => ([(1,4),(2,4),(3,4)],5)
=> ? = 2
[1,2,5,3,4] => [1,5,2,3,4] => [1,4] => ([(3,4)],5)
=> ? = 2
[1,2,5,4,3] => [5,4,1,2,3] => [1,4] => ([(3,4)],5)
=> ? = 2
[1,3,2,4,5] => [3,1,2,4,5] => [5] => ([],5)
=> ? = 2
[1,3,2,5,4] => [5,3,1,2,4] => [1,4] => ([(3,4)],5)
=> ? = 2
[1,3,4,5,2] => [3,1,4,5,2] => [2,3] => ([(2,4),(3,4)],5)
=> ? = 3
[1,3,5,2,4] => [3,1,2,5,4] => [3,2] => ([(1,4),(2,4),(3,4)],5)
=> ? = 2
[1,3,5,4,2] => [5,3,1,4,2] => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
[1,4,2,5,3] => [4,5,1,2,3] => [5] => ([],5)
=> ? = 3
[1,4,3,5,2] => [4,3,1,5,2] => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 3
[1,4,5,2,3] => [1,4,2,5,3] => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 2
[1,4,5,3,2] => [4,5,1,3,2] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 2
[1,5,2,3,4] => [1,2,5,3,4] => [2,3] => ([(2,4),(3,4)],5)
=> ? = 3
[1,5,2,4,3] => [5,1,4,2,3] => [2,3] => ([(2,4),(3,4)],5)
=> ? = 3
[1,5,3,2,4] => [3,5,1,2,4] => [5] => ([],5)
=> ? = 2
[1,5,3,4,2] => [5,1,3,4,2] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 2
[1,5,4,2,3] => [1,5,4,2,3] => [1,1,3] => ([(2,3),(2,4),(3,4)],5)
=> ? = 2
[1,5,4,3,2] => [5,4,3,1,2] => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 2
[2,1,3,4,5] => [2,1,3,4,5] => [5] => ([],5)
=> ? = 4
[2,1,3,5,4] => [5,2,1,3,4] => [2,3] => ([(2,4),(3,4)],5)
=> ? = 2
[2,1,4,3,5] => [4,2,1,3,5] => [2,3] => ([(2,4),(3,4)],5)
=> ? = 2
[2,1,4,5,3] => [4,2,1,5,3] => [2,1,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 2
[2,1,5,3,4] => [2,5,1,3,4] => [5] => ([],5)
=> ? = 2
[2,1,5,4,3] => [5,4,2,1,3] => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 2
[2,3,4,1,5] => [2,3,4,1,5] => [5] => ([],5)
=> ? = 2
[2,3,4,5,1] => [2,3,4,5,1] => [5] => ([],5)
=> ? = 2
[2,3,5,1,4] => [2,3,1,5,4] => [3,2] => ([(1,4),(2,4),(3,4)],5)
=> ? = 2
[2,3,5,4,1] => [5,2,3,4,1] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 2
[2,4,1,3,5] => [2,1,4,3,5] => [2,3] => ([(2,4),(3,4)],5)
=> ? = 2
[2,4,1,5,3] => [4,2,5,1,3] => [2,3] => ([(2,4),(3,4)],5)
=> ? = 2
[2,4,3,5,1] => [4,2,3,5,1] => [3,2] => ([(1,4),(2,4),(3,4)],5)
=> ? = 2
[2,4,5,1,3] => [2,1,4,5,3] => [2,3] => ([(2,4),(3,4)],5)
=> ? = 2
[2,5,1,3,4] => [2,1,3,5,4] => [3,2] => ([(1,4),(2,4),(3,4)],5)
=> ? = 2
[2,5,1,4,3] => [5,2,1,4,3] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
[2,5,3,1,4] => [2,5,3,1,4] => [3,2] => ([(1,4),(2,4),(3,4)],5)
=> ? = 2
[2,5,4,1,3] => [2,5,1,4,3] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 2
[3,1,4,2,5] => [3,4,1,2,5] => [5] => ([],5)
=> ? = 2
[3,2,5,4,1] => [5,3,2,4,1] => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
[5,1,2,4,3] => [5,1,2,4,3] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 2
[5,1,3,4,2] => [3,1,5,4,2] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
[5,1,4,2,3] => [1,5,2,4,3] => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
[5,1,4,3,2] => [5,4,1,3,2] => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
[1,2,4,6,5,3] => [6,4,1,2,5,3] => [1,4,1] => ([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> 2
[1,2,5,6,4,3] => [5,6,1,2,4,3] => [5,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> 2
[1,2,6,4,5,3] => [6,1,4,2,5,3] => [2,3,1] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 2
[1,3,4,6,5,2] => [6,3,1,4,5,2] => [1,4,1] => ([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> 2
[1,3,5,6,4,2] => [5,3,1,6,4,2] => [1,2,2,1] => ([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 2
[1,3,6,2,5,4] => [6,3,1,2,5,4] => [1,4,1] => ([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> 2
[1,3,6,4,5,2] => [6,1,3,4,5,2] => [5,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> 2
[1,3,6,5,2,4] => [3,6,1,2,5,4] => [5,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> 2
[1,3,6,5,4,2] => [6,5,3,1,4,2] => [1,2,2,1] => ([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 2
[1,4,3,6,5,2] => [6,4,3,1,5,2] => [1,2,2,1] => ([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 2
[1,4,5,6,3,2] => [4,5,1,6,3,2] => [3,2,1] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 2
[1,4,6,2,5,3] => [4,6,1,2,5,3] => [5,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> 2
[1,4,6,3,5,2] => [4,3,1,6,5,2] => [1,2,2,1] => ([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 2
[1,4,6,5,2,3] => [1,6,4,2,5,3] => [1,1,3,1] => ([(0,5),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 2
[1,4,6,5,3,2] => [6,4,5,1,3,2] => [1,4,1] => ([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> 2
[1,5,3,6,4,2] => [5,6,3,1,4,2] => [3,2,1] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 2
[1,5,4,6,3,2] => [5,4,6,1,3,2] => [1,4,1] => ([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> 2
[1,5,6,2,4,3] => [5,1,6,2,4,3] => [2,3,1] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 2
[1,5,6,3,4,2] => [5,1,3,6,4,2] => [3,2,1] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 2
[1,5,6,4,2,3] => [1,5,6,2,4,3] => [1,4,1] => ([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> 2
[1,5,6,4,3,2] => [5,6,4,1,3,2] => [2,3,1] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 2
[1,6,2,4,5,3] => [6,1,2,4,5,3] => [5,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> 2
[1,6,3,4,5,2] => [3,1,6,4,5,2] => [2,3,1] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 2
[1,6,3,5,2,4] => [1,6,3,2,5,4] => [1,2,2,1] => ([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 2
[1,6,3,5,4,2] => [6,5,1,3,4,2] => [1,4,1] => ([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> 2
[1,6,4,3,5,2] => [4,6,3,1,5,2] => [3,2,1] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 2
[1,6,4,5,2,3] => [1,6,2,4,5,3] => [1,4,1] => ([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> 2
[1,6,4,5,3,2] => [6,4,1,5,3,2] => [1,2,2,1] => ([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 2
[1,6,5,3,4,2] => [6,1,5,3,4,2] => [2,3,1] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 2
[2,1,4,6,5,3] => [6,4,2,1,5,3] => [1,2,2,1] => ([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 2
[2,1,5,6,4,3] => [5,6,2,1,4,3] => [3,2,1] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 2
[2,1,6,4,5,3] => [6,2,4,1,5,3] => [2,3,1] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 2
[2,3,4,6,5,1] => [6,2,3,4,5,1] => [5,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> 2
[2,3,5,6,4,1] => [5,2,3,6,4,1] => [3,2,1] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 2
[2,3,6,1,5,4] => [6,2,3,1,5,4] => [3,2,1] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 2
[2,3,6,4,5,1] => [2,6,3,4,5,1] => [5,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> 2
[2,3,6,5,4,1] => [6,5,2,3,4,1] => [1,4,1] => ([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> 2
[2,4,3,6,5,1] => [6,4,2,3,5,1] => [1,4,1] => ([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> 2
Description
The diameter of a connected graph. This is the greatest distance between any pair of vertices.
Matching statistic: St000260
Mp00067: Permutations Foata bijectionPermutations
Mp00248: Permutations DEX compositionInteger compositions
Mp00184: Integer compositions to threshold graphGraphs
St000260: Graphs ⟶ ℤResult quality: 20% values known / values provided: 26%distinct values known / distinct values provided: 20%
Values
[1,2,3] => [1,2,3] => [3] => ([],3)
=> ? = 3 - 1
[1,3,2] => [3,1,2] => [3] => ([],3)
=> ? = 2 - 1
[2,1,3] => [2,1,3] => [3] => ([],3)
=> ? = 2 - 1
[1,2,3,4] => [1,2,3,4] => [4] => ([],4)
=> ? = 2 - 1
[1,2,4,3] => [4,1,2,3] => [4] => ([],4)
=> ? = 2 - 1
[1,3,2,4] => [3,1,2,4] => [4] => ([],4)
=> ? = 2 - 1
[1,3,4,2] => [3,1,4,2] => [2,2] => ([(1,3),(2,3)],4)
=> ? = 2 - 1
[1,4,2,3] => [1,4,2,3] => [1,3] => ([(2,3)],4)
=> ? = 2 - 1
[1,4,3,2] => [4,3,1,2] => [1,3] => ([(2,3)],4)
=> ? = 2 - 1
[2,1,3,4] => [2,1,3,4] => [4] => ([],4)
=> ? = 3 - 1
[2,1,4,3] => [4,2,1,3] => [2,2] => ([(1,3),(2,3)],4)
=> ? = 2 - 1
[2,3,4,1] => [2,3,4,1] => [4] => ([],4)
=> ? = 2 - 1
[2,4,1,3] => [2,1,4,3] => [2,2] => ([(1,3),(2,3)],4)
=> ? = 2 - 1
[3,1,4,2] => [3,4,1,2] => [4] => ([],4)
=> ? = 2 - 1
[3,2,4,1] => [3,2,4,1] => [2,2] => ([(1,3),(2,3)],4)
=> ? = 2 - 1
[4,1,2,3] => [1,2,4,3] => [2,2] => ([(1,3),(2,3)],4)
=> ? = 2 - 1
[4,1,3,2] => [4,1,3,2] => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 1 = 2 - 1
[1,2,3,4,5] => [1,2,3,4,5] => [5] => ([],5)
=> ? = 2 - 1
[1,2,3,5,4] => [5,1,2,3,4] => [5] => ([],5)
=> ? = 2 - 1
[1,2,4,3,5] => [4,1,2,3,5] => [5] => ([],5)
=> ? = 2 - 1
[1,2,4,5,3] => [4,1,2,5,3] => [3,2] => ([(1,4),(2,4),(3,4)],5)
=> ? = 2 - 1
[1,2,5,3,4] => [1,5,2,3,4] => [1,4] => ([(3,4)],5)
=> ? = 2 - 1
[1,2,5,4,3] => [5,4,1,2,3] => [1,4] => ([(3,4)],5)
=> ? = 2 - 1
[1,3,2,4,5] => [3,1,2,4,5] => [5] => ([],5)
=> ? = 2 - 1
[1,3,2,5,4] => [5,3,1,2,4] => [1,4] => ([(3,4)],5)
=> ? = 2 - 1
[1,3,4,5,2] => [3,1,4,5,2] => [2,3] => ([(2,4),(3,4)],5)
=> ? = 3 - 1
[1,3,5,2,4] => [3,1,2,5,4] => [3,2] => ([(1,4),(2,4),(3,4)],5)
=> ? = 2 - 1
[1,3,5,4,2] => [5,3,1,4,2] => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> 1 = 2 - 1
[1,4,2,5,3] => [4,5,1,2,3] => [5] => ([],5)
=> ? = 3 - 1
[1,4,3,5,2] => [4,3,1,5,2] => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 3 - 1
[1,4,5,2,3] => [1,4,2,5,3] => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 2 - 1
[1,4,5,3,2] => [4,5,1,3,2] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 1 = 2 - 1
[1,5,2,3,4] => [1,2,5,3,4] => [2,3] => ([(2,4),(3,4)],5)
=> ? = 3 - 1
[1,5,2,4,3] => [5,1,4,2,3] => [2,3] => ([(2,4),(3,4)],5)
=> ? = 3 - 1
[1,5,3,2,4] => [3,5,1,2,4] => [5] => ([],5)
=> ? = 2 - 1
[1,5,3,4,2] => [5,1,3,4,2] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 1 = 2 - 1
[1,5,4,2,3] => [1,5,4,2,3] => [1,1,3] => ([(2,3),(2,4),(3,4)],5)
=> ? = 2 - 1
[1,5,4,3,2] => [5,4,3,1,2] => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 2 - 1
[2,1,3,4,5] => [2,1,3,4,5] => [5] => ([],5)
=> ? = 4 - 1
[2,1,3,5,4] => [5,2,1,3,4] => [2,3] => ([(2,4),(3,4)],5)
=> ? = 2 - 1
[2,1,4,3,5] => [4,2,1,3,5] => [2,3] => ([(2,4),(3,4)],5)
=> ? = 2 - 1
[2,1,4,5,3] => [4,2,1,5,3] => [2,1,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 2 - 1
[2,1,5,3,4] => [2,5,1,3,4] => [5] => ([],5)
=> ? = 2 - 1
[2,1,5,4,3] => [5,4,2,1,3] => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 2 - 1
[2,3,4,1,5] => [2,3,4,1,5] => [5] => ([],5)
=> ? = 2 - 1
[2,3,4,5,1] => [2,3,4,5,1] => [5] => ([],5)
=> ? = 2 - 1
[2,3,5,1,4] => [2,3,1,5,4] => [3,2] => ([(1,4),(2,4),(3,4)],5)
=> ? = 2 - 1
[2,3,5,4,1] => [5,2,3,4,1] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 1 = 2 - 1
[2,4,1,3,5] => [2,1,4,3,5] => [2,3] => ([(2,4),(3,4)],5)
=> ? = 2 - 1
[2,4,1,5,3] => [4,2,5,1,3] => [2,3] => ([(2,4),(3,4)],5)
=> ? = 2 - 1
[2,4,3,5,1] => [4,2,3,5,1] => [3,2] => ([(1,4),(2,4),(3,4)],5)
=> ? = 2 - 1
[2,4,5,1,3] => [2,1,4,5,3] => [2,3] => ([(2,4),(3,4)],5)
=> ? = 2 - 1
[2,5,1,3,4] => [2,1,3,5,4] => [3,2] => ([(1,4),(2,4),(3,4)],5)
=> ? = 2 - 1
[2,5,1,4,3] => [5,2,1,4,3] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1 = 2 - 1
[2,5,3,1,4] => [2,5,3,1,4] => [3,2] => ([(1,4),(2,4),(3,4)],5)
=> ? = 2 - 1
[2,5,4,1,3] => [2,5,1,4,3] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 1 = 2 - 1
[3,1,4,2,5] => [3,4,1,2,5] => [5] => ([],5)
=> ? = 2 - 1
[3,2,5,4,1] => [5,3,2,4,1] => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> 1 = 2 - 1
[5,1,2,4,3] => [5,1,2,4,3] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 1 = 2 - 1
[5,1,3,4,2] => [3,1,5,4,2] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1 = 2 - 1
[5,1,4,2,3] => [1,5,2,4,3] => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> 1 = 2 - 1
[5,1,4,3,2] => [5,4,1,3,2] => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> 1 = 2 - 1
[1,2,4,6,5,3] => [6,4,1,2,5,3] => [1,4,1] => ([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> 1 = 2 - 1
[1,2,5,6,4,3] => [5,6,1,2,4,3] => [5,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> 1 = 2 - 1
[1,2,6,4,5,3] => [6,1,4,2,5,3] => [2,3,1] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 1 = 2 - 1
[1,3,4,6,5,2] => [6,3,1,4,5,2] => [1,4,1] => ([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> 1 = 2 - 1
[1,3,5,6,4,2] => [5,3,1,6,4,2] => [1,2,2,1] => ([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 1 = 2 - 1
[1,3,6,2,5,4] => [6,3,1,2,5,4] => [1,4,1] => ([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> 1 = 2 - 1
[1,3,6,4,5,2] => [6,1,3,4,5,2] => [5,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> 1 = 2 - 1
[1,3,6,5,2,4] => [3,6,1,2,5,4] => [5,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> 1 = 2 - 1
[1,3,6,5,4,2] => [6,5,3,1,4,2] => [1,2,2,1] => ([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 1 = 2 - 1
[1,4,3,6,5,2] => [6,4,3,1,5,2] => [1,2,2,1] => ([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 1 = 2 - 1
[1,4,5,6,3,2] => [4,5,1,6,3,2] => [3,2,1] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 1 = 2 - 1
[1,4,6,2,5,3] => [4,6,1,2,5,3] => [5,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> 1 = 2 - 1
[1,4,6,3,5,2] => [4,3,1,6,5,2] => [1,2,2,1] => ([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 1 = 2 - 1
[1,4,6,5,2,3] => [1,6,4,2,5,3] => [1,1,3,1] => ([(0,5),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 1 = 2 - 1
[1,4,6,5,3,2] => [6,4,5,1,3,2] => [1,4,1] => ([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> 1 = 2 - 1
[1,5,3,6,4,2] => [5,6,3,1,4,2] => [3,2,1] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 1 = 2 - 1
[1,5,4,6,3,2] => [5,4,6,1,3,2] => [1,4,1] => ([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> 1 = 2 - 1
[1,5,6,2,4,3] => [5,1,6,2,4,3] => [2,3,1] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 1 = 2 - 1
[1,5,6,3,4,2] => [5,1,3,6,4,2] => [3,2,1] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 1 = 2 - 1
[1,5,6,4,2,3] => [1,5,6,2,4,3] => [1,4,1] => ([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> 1 = 2 - 1
[1,5,6,4,3,2] => [5,6,4,1,3,2] => [2,3,1] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 1 = 2 - 1
[1,6,2,4,5,3] => [6,1,2,4,5,3] => [5,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> 1 = 2 - 1
[1,6,3,4,5,2] => [3,1,6,4,5,2] => [2,3,1] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 1 = 2 - 1
[1,6,3,5,2,4] => [1,6,3,2,5,4] => [1,2,2,1] => ([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 1 = 2 - 1
[1,6,3,5,4,2] => [6,5,1,3,4,2] => [1,4,1] => ([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> 1 = 2 - 1
[1,6,4,3,5,2] => [4,6,3,1,5,2] => [3,2,1] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 1 = 2 - 1
[1,6,4,5,2,3] => [1,6,2,4,5,3] => [1,4,1] => ([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> 1 = 2 - 1
[1,6,4,5,3,2] => [6,4,1,5,3,2] => [1,2,2,1] => ([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 1 = 2 - 1
[1,6,5,3,4,2] => [6,1,5,3,4,2] => [2,3,1] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 1 = 2 - 1
[2,1,4,6,5,3] => [6,4,2,1,5,3] => [1,2,2,1] => ([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 1 = 2 - 1
[2,1,5,6,4,3] => [5,6,2,1,4,3] => [3,2,1] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 1 = 2 - 1
[2,1,6,4,5,3] => [6,2,4,1,5,3] => [2,3,1] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 1 = 2 - 1
[2,3,4,6,5,1] => [6,2,3,4,5,1] => [5,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> 1 = 2 - 1
[2,3,5,6,4,1] => [5,2,3,6,4,1] => [3,2,1] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 1 = 2 - 1
[2,3,6,1,5,4] => [6,2,3,1,5,4] => [3,2,1] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 1 = 2 - 1
[2,3,6,4,5,1] => [2,6,3,4,5,1] => [5,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> 1 = 2 - 1
[2,3,6,5,4,1] => [6,5,2,3,4,1] => [1,4,1] => ([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> 1 = 2 - 1
[2,4,3,6,5,1] => [6,4,2,3,5,1] => [1,4,1] => ([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> 1 = 2 - 1
Description
The radius of a connected graph. This is the minimum eccentricity of any vertex.
Mp00160: Permutations graph of inversionsGraphs
Mp00203: Graphs coneGraphs
St001645: Graphs ⟶ ℤResult quality: 15% values known / values provided: 15%distinct values known / distinct values provided: 20%
Values
[1,2,3] => ([],3)
=> ([(0,3),(1,3),(2,3)],4)
=> ? = 3 + 5
[1,3,2] => ([(1,2)],3)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> ? = 2 + 5
[2,1,3] => ([(1,2)],3)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> ? = 2 + 5
[1,2,3,4] => ([],4)
=> ([(0,4),(1,4),(2,4),(3,4)],5)
=> ? = 2 + 5
[1,2,4,3] => ([(2,3)],4)
=> ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 2 + 5
[1,3,2,4] => ([(2,3)],4)
=> ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 2 + 5
[1,3,4,2] => ([(1,3),(2,3)],4)
=> ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 2 + 5
[1,4,2,3] => ([(1,3),(2,3)],4)
=> ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 2 + 5
[1,4,3,2] => ([(1,2),(1,3),(2,3)],4)
=> ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 2 + 5
[2,1,3,4] => ([(2,3)],4)
=> ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 3 + 5
[2,1,4,3] => ([(0,3),(1,2)],4)
=> ([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
=> ? = 2 + 5
[2,3,4,1] => ([(0,3),(1,3),(2,3)],4)
=> ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 2 + 5
[2,4,1,3] => ([(0,3),(1,2),(2,3)],4)
=> ([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 2 + 5
[3,1,4,2] => ([(0,3),(1,2),(2,3)],4)
=> ([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 2 + 5
[3,2,4,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 2 + 5
[4,1,2,3] => ([(0,3),(1,3),(2,3)],4)
=> ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 2 + 5
[4,1,3,2] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 2 + 5
[1,2,3,4,5] => ([],5)
=> ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> ? = 2 + 5
[1,2,3,5,4] => ([(3,4)],5)
=> ([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 2 + 5
[1,2,4,3,5] => ([(3,4)],5)
=> ([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 2 + 5
[1,2,4,5,3] => ([(2,4),(3,4)],5)
=> ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 2 + 5
[1,2,5,3,4] => ([(2,4),(3,4)],5)
=> ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 2 + 5
[1,2,5,4,3] => ([(2,3),(2,4),(3,4)],5)
=> ([(0,5),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 2 + 5
[1,3,2,4,5] => ([(3,4)],5)
=> ([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 2 + 5
[1,3,2,5,4] => ([(1,4),(2,3)],5)
=> ([(0,5),(1,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)
=> ? = 2 + 5
[1,3,4,5,2] => ([(1,4),(2,4),(3,4)],5)
=> ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 3 + 5
[1,3,5,2,4] => ([(1,4),(2,3),(3,4)],5)
=> ([(0,5),(1,4),(1,5),(2,3),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 2 + 5
[1,3,5,4,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 2 + 5
[1,4,2,5,3] => ([(1,4),(2,3),(3,4)],5)
=> ([(0,5),(1,4),(1,5),(2,3),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 3 + 5
[1,4,3,5,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 3 + 5
[1,4,5,2,3] => ([(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
=> ? = 2 + 5
[1,4,5,3,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 2 + 5
[1,5,2,3,4] => ([(1,4),(2,4),(3,4)],5)
=> ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 3 + 5
[1,5,2,4,3] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 3 + 5
[1,5,3,2,4] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 2 + 5
[1,5,3,4,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 2 + 5
[1,5,4,2,3] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 2 + 5
[1,5,4,3,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 2 + 5
[2,1,3,4,5] => ([(3,4)],5)
=> ([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 4 + 5
[2,1,3,5,4] => ([(1,4),(2,3)],5)
=> ([(0,5),(1,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)
=> ? = 2 + 5
[2,1,4,3,5] => ([(1,4),(2,3)],5)
=> ([(0,5),(1,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)
=> ? = 2 + 5
[2,1,4,5,3] => ([(0,1),(2,4),(3,4)],5)
=> ([(0,1),(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 2 + 5
[2,1,5,3,4] => ([(0,1),(2,4),(3,4)],5)
=> ([(0,1),(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 2 + 5
[2,1,5,4,3] => ([(0,1),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,5),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 2 + 5
[2,3,4,1,5] => ([(1,4),(2,4),(3,4)],5)
=> ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 2 + 5
[2,3,4,5,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> ([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 2 + 5
[2,3,5,1,4] => ([(0,4),(1,4),(2,3),(3,4)],5)
=> ([(0,4),(0,5),(1,4),(1,5),(2,3),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 2 + 5
[2,3,5,4,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 2 + 5
[2,4,1,3,5] => ([(1,4),(2,3),(3,4)],5)
=> ([(0,5),(1,4),(1,5),(2,3),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 2 + 5
[2,4,1,5,3] => ([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,4),(0,5),(1,3),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
=> ? = 2 + 5
[3,4,5,6,1,2] => ([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> ([(0,4),(0,5),(0,6),(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,6),(5,6)],7)
=> 7 = 2 + 5
[3,4,5,6,2,1] => ([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,4),(0,5),(0,6),(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 7 = 2 + 5
[3,4,6,1,5,2] => ([(0,3),(0,5),(1,3),(1,5),(2,4),(2,5),(3,4),(4,5)],6)
=> ([(0,3),(0,5),(0,6),(1,3),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,6),(4,5),(4,6),(5,6)],7)
=> 7 = 2 + 5
[3,4,6,2,5,1] => ([(0,4),(0,5),(1,4),(1,5),(2,3),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,4),(0,5),(0,6),(1,4),(1,5),(1,6),(2,3),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 7 = 2 + 5
[3,5,1,6,2,4] => ([(0,3),(0,5),(1,2),(1,5),(2,4),(3,4),(4,5)],6)
=> ([(0,3),(0,5),(0,6),(1,2),(1,5),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(4,6),(5,6)],7)
=> 7 = 2 + 5
[3,5,1,6,4,2] => ([(0,1),(0,5),(1,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,5),(0,6),(1,3),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 7 = 2 + 5
[3,5,2,6,1,4] => ([(0,1),(0,5),(1,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,5),(0,6),(1,3),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 7 = 2 + 5
[3,5,2,6,4,1] => ([(0,4),(0,5),(1,3),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
=> ([(0,4),(0,5),(0,6),(1,3),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 7 = 2 + 5
[3,5,4,6,1,2] => ([(0,4),(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5)],6)
=> ([(0,4),(0,5),(0,6),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,6),(5,6)],7)
=> 7 = 2 + 5
[3,5,4,6,2,1] => ([(0,4),(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,4),(0,5),(0,6),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 7 = 2 + 5
[3,5,6,1,2,4] => ([(0,4),(0,5),(1,2),(1,3),(2,4),(2,5),(3,4),(3,5)],6)
=> ([(0,4),(0,5),(0,6),(1,2),(1,3),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,6),(5,6)],7)
=> 7 = 2 + 5
[3,5,6,2,1,4] => ([(0,4),(0,5),(1,2),(1,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,4),(0,5),(0,6),(1,2),(1,3),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 7 = 2 + 5
[3,6,1,4,5,2] => ([(0,1),(0,5),(1,4),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,5),(0,6),(1,4),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 7 = 2 + 5
[3,6,1,5,2,4] => ([(0,1),(0,5),(1,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,5),(0,6),(1,3),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 7 = 2 + 5
[3,6,1,5,4,2] => ([(0,1),(0,5),(1,4),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,5),(0,6),(1,4),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 7 = 2 + 5
[3,6,2,4,5,1] => ([(0,4),(0,5),(1,4),(1,5),(2,3),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,4),(0,5),(0,6),(1,4),(1,5),(1,6),(2,3),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 7 = 2 + 5
[3,6,2,5,1,4] => ([(0,3),(0,5),(1,2),(1,4),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,3),(0,5),(0,6),(1,2),(1,4),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 7 = 2 + 5
[3,6,2,5,4,1] => ([(0,3),(0,5),(1,2),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,3),(0,5),(0,6),(1,2),(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 7 = 2 + 5
[3,6,5,1,2,4] => ([(0,4),(0,5),(1,2),(1,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,4),(0,5),(0,6),(1,2),(1,3),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 7 = 2 + 5
[3,6,5,2,1,4] => ([(0,4),(0,5),(1,2),(1,3),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,4),(0,5),(0,6),(1,2),(1,3),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 7 = 2 + 5
[4,2,5,6,1,3] => ([(0,3),(0,5),(1,3),(1,5),(2,4),(2,5),(3,4),(4,5)],6)
=> ([(0,3),(0,5),(0,6),(1,3),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,6),(4,5),(4,6),(5,6)],7)
=> 7 = 2 + 5
[4,2,5,6,3,1] => ([(0,4),(0,5),(1,4),(1,5),(2,3),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,4),(0,5),(0,6),(1,4),(1,5),(1,6),(2,3),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 7 = 2 + 5
[4,2,6,1,5,3] => ([(0,4),(0,5),(1,2),(1,3),(2,3),(2,5),(3,4),(4,5)],6)
=> ([(0,4),(0,5),(0,6),(1,2),(1,3),(1,6),(2,3),(2,5),(2,6),(3,4),(3,6),(4,5),(4,6),(5,6)],7)
=> 7 = 2 + 5
[4,2,6,3,5,1] => ([(0,4),(0,5),(1,3),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
=> ([(0,4),(0,5),(0,6),(1,3),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 7 = 2 + 5
[4,3,5,6,1,2] => ([(0,4),(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5)],6)
=> ([(0,4),(0,5),(0,6),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,6),(5,6)],7)
=> 7 = 2 + 5
[4,3,5,6,2,1] => ([(0,4),(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,4),(0,5),(0,6),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 7 = 2 + 5
[4,3,6,1,5,2] => ([(0,4),(0,5),(1,2),(1,3),(1,4),(2,3),(2,5),(3,5),(4,5)],6)
=> ([(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,6),(2,3),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 7 = 2 + 5
[4,3,6,2,5,1] => ([(0,3),(0,5),(1,2),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,3),(0,5),(0,6),(1,2),(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 7 = 2 + 5
[4,5,1,6,2,3] => ([(0,4),(0,5),(1,2),(1,3),(2,4),(2,5),(3,4),(3,5)],6)
=> ([(0,4),(0,5),(0,6),(1,2),(1,3),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,6),(5,6)],7)
=> 7 = 2 + 5
[4,5,1,6,3,2] => ([(0,4),(0,5),(1,2),(1,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,4),(0,5),(0,6),(1,2),(1,3),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 7 = 2 + 5
[4,5,2,6,1,3] => ([(0,4),(0,5),(1,2),(1,3),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> ([(0,4),(0,5),(0,6),(1,2),(1,3),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,6),(5,6)],7)
=> 7 = 2 + 5
[4,5,2,6,3,1] => ([(0,4),(0,5),(1,2),(1,3),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,4),(0,5),(0,6),(1,2),(1,3),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 7 = 2 + 5
[4,5,3,6,1,2] => ([(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5)],6)
=> ([(0,4),(0,5),(0,6),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,6),(5,6)],7)
=> 7 = 2 + 5
[4,5,3,6,2,1] => ([(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,4),(0,5),(0,6),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 7 = 2 + 5
[4,6,1,2,5,3] => ([(0,3),(0,5),(1,3),(1,5),(2,4),(2,5),(3,4),(4,5)],6)
=> ([(0,3),(0,5),(0,6),(1,3),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,6),(4,5),(4,6),(5,6)],7)
=> 7 = 2 + 5
[4,6,1,3,5,2] => ([(0,4),(0,5),(1,2),(1,4),(2,3),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,4),(0,5),(0,6),(1,2),(1,4),(1,6),(2,3),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 7 = 2 + 5
[4,6,1,5,2,3] => ([(0,4),(0,5),(1,2),(1,3),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> ([(0,4),(0,5),(0,6),(1,2),(1,3),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,6),(5,6)],7)
=> 7 = 2 + 5
[4,6,1,5,3,2] => ([(0,2),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6)
=> ([(0,2),(0,5),(0,6),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 7 = 2 + 5
[5,2,3,6,1,4] => ([(0,1),(0,5),(1,4),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,5),(0,6),(1,4),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 7 = 2 + 5
[5,2,3,6,4,1] => ([(0,4),(0,5),(1,4),(1,5),(2,3),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,4),(0,5),(0,6),(1,4),(1,5),(1,6),(2,3),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 7 = 2 + 5
[5,2,4,6,1,3] => ([(0,4),(0,5),(1,2),(1,4),(2,3),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,4),(0,5),(0,6),(1,2),(1,4),(1,6),(2,3),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 7 = 2 + 5
[5,2,4,6,3,1] => ([(0,4),(0,5),(1,3),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,4),(0,5),(0,6),(1,3),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 7 = 2 + 5
[5,2,6,1,3,4] => ([(0,3),(0,5),(1,3),(1,5),(2,4),(2,5),(3,4),(4,5)],6)
=> ([(0,3),(0,5),(0,6),(1,3),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,6),(4,5),(4,6),(5,6)],7)
=> 7 = 2 + 5
[5,2,6,1,4,3] => ([(0,4),(0,5),(1,2),(1,3),(1,4),(2,3),(2,5),(3,5),(4,5)],6)
=> ([(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,6),(2,3),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 7 = 2 + 5
[5,3,1,6,2,4] => ([(0,1),(0,5),(1,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,5),(0,6),(1,3),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 7 = 2 + 5
[5,3,1,6,4,2] => ([(0,3),(0,5),(1,2),(1,4),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,3),(0,5),(0,6),(1,2),(1,4),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 7 = 2 + 5
[5,3,2,6,1,4] => ([(0,1),(0,5),(1,4),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,5),(0,6),(1,4),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 7 = 2 + 5
[5,3,2,6,4,1] => ([(0,3),(0,5),(1,2),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,3),(0,5),(0,6),(1,2),(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 7 = 2 + 5
[5,3,4,6,1,2] => ([(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5)],6)
=> ([(0,4),(0,5),(0,6),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,6),(5,6)],7)
=> 7 = 2 + 5
[5,3,4,6,2,1] => ([(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,4),(0,5),(0,6),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 7 = 2 + 5
Description
The pebbling number of a connected graph.
Matching statistic: St000665
Mp00063: Permutations to alternating sign matrixAlternating sign matrices
Mp00098: Alternating sign matrices link patternPerfect matchings
Mp00283: Perfect matchings non-nesting-exceedence permutationPermutations
St000665: Permutations ⟶ ℤResult quality: 7% values known / values provided: 7%distinct values known / distinct values provided: 40%
Values
[1,2,3] => [[1,0,0],[0,1,0],[0,0,1]]
=> [(1,6),(2,5),(3,4)]
=> [4,5,6,3,2,1] => 1 = 3 - 2
[1,3,2] => [[1,0,0],[0,0,1],[0,1,0]]
=> [(1,6),(2,3),(4,5)]
=> [3,5,2,6,4,1] => 0 = 2 - 2
[2,1,3] => [[0,1,0],[1,0,0],[0,0,1]]
=> [(1,2),(3,4),(5,6)]
=> [2,1,4,3,6,5] => 0 = 2 - 2
[1,2,3,4] => [[1,0,0,0],[0,1,0,0],[0,0,1,0],[0,0,0,1]]
=> [(1,8),(2,7),(3,6),(4,5)]
=> [5,6,7,8,4,3,2,1] => ? = 2 - 2
[1,2,4,3] => [[1,0,0,0],[0,1,0,0],[0,0,0,1],[0,0,1,0]]
=> [(1,8),(2,7),(3,4),(5,6)]
=> [4,6,7,3,8,5,2,1] => ? = 2 - 2
[1,3,2,4] => [[1,0,0,0],[0,0,1,0],[0,1,0,0],[0,0,0,1]]
=> [(1,8),(2,3),(4,5),(6,7)]
=> [3,5,2,7,4,8,6,1] => ? = 2 - 2
[1,3,4,2] => [[1,0,0,0],[0,0,0,1],[0,1,0,0],[0,0,1,0]]
=> [(1,8),(2,7),(3,4),(5,6)]
=> [4,6,7,3,8,5,2,1] => ? = 2 - 2
[1,4,2,3] => [[1,0,0,0],[0,0,1,0],[0,0,0,1],[0,1,0,0]]
=> [(1,8),(2,3),(4,5),(6,7)]
=> [3,5,2,7,4,8,6,1] => ? = 2 - 2
[1,4,3,2] => [[1,0,0,0],[0,0,0,1],[0,0,1,0],[0,1,0,0]]
=> [(1,8),(2,5),(3,4),(6,7)]
=> [4,5,7,3,2,8,6,1] => ? = 2 - 2
[2,1,3,4] => [[0,1,0,0],[1,0,0,0],[0,0,1,0],[0,0,0,1]]
=> [(1,2),(3,6),(4,5),(7,8)]
=> [2,1,5,6,4,3,8,7] => ? = 3 - 2
[2,1,4,3] => [[0,1,0,0],[1,0,0,0],[0,0,0,1],[0,0,1,0]]
=> [(1,2),(3,4),(5,6),(7,8)]
=> [2,1,4,3,6,5,8,7] => 0 = 2 - 2
[2,3,4,1] => [[0,0,0,1],[1,0,0,0],[0,1,0,0],[0,0,1,0]]
=> [(1,8),(2,3),(4,7),(5,6)]
=> [3,6,2,7,8,5,4,1] => ? = 2 - 2
[2,4,1,3] => [[0,0,1,0],[1,0,0,0],[0,0,0,1],[0,1,0,0]]
=> [(1,8),(2,3),(4,5),(6,7)]
=> [3,5,2,7,4,8,6,1] => ? = 2 - 2
[3,1,4,2] => [[0,1,0,0],[0,0,0,1],[1,0,0,0],[0,0,1,0]]
=> [(1,2),(3,4),(5,6),(7,8)]
=> [2,1,4,3,6,5,8,7] => 0 = 2 - 2
[3,2,4,1] => [[0,0,0,1],[0,1,0,0],[1,0,0,0],[0,0,1,0]]
=> [(1,4),(2,3),(5,6),(7,8)]
=> [3,4,2,1,6,5,8,7] => ? = 2 - 2
[4,1,2,3] => [[0,1,0,0],[0,0,1,0],[0,0,0,1],[1,0,0,0]]
=> [(1,2),(3,8),(4,5),(6,7)]
=> [2,1,5,7,4,8,6,3] => ? = 2 - 2
[4,1,3,2] => [[0,1,0,0],[0,0,0,1],[0,0,1,0],[1,0,0,0]]
=> [(1,2),(3,4),(5,8),(6,7)]
=> [2,1,4,3,7,8,6,5] => ? = 2 - 2
[1,2,3,4,5] => [[1,0,0,0,0],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,1,0],[0,0,0,0,1]]
=> [(1,10),(2,9),(3,8),(4,7),(5,6)]
=> [6,7,8,9,10,5,4,3,2,1] => ? = 2 - 2
[1,2,3,5,4] => [[1,0,0,0,0],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,0,1],[0,0,0,1,0]]
=> [(1,10),(2,9),(3,8),(4,5),(6,7)]
=> [5,7,8,9,4,10,6,3,2,1] => ? = 2 - 2
[1,2,4,3,5] => [[1,0,0,0,0],[0,1,0,0,0],[0,0,0,1,0],[0,0,1,0,0],[0,0,0,0,1]]
=> [(1,10),(2,9),(3,4),(5,6),(7,8)]
=> [4,6,8,3,9,5,10,7,2,1] => ? = 2 - 2
[1,2,4,5,3] => [[1,0,0,0,0],[0,1,0,0,0],[0,0,0,0,1],[0,0,1,0,0],[0,0,0,1,0]]
=> [(1,10),(2,9),(3,8),(4,5),(6,7)]
=> [5,7,8,9,4,10,6,3,2,1] => ? = 2 - 2
[1,2,5,3,4] => [[1,0,0,0,0],[0,1,0,0,0],[0,0,0,1,0],[0,0,0,0,1],[0,0,1,0,0]]
=> [(1,10),(2,9),(3,4),(5,6),(7,8)]
=> [4,6,8,3,9,5,10,7,2,1] => ? = 2 - 2
[1,2,5,4,3] => [[1,0,0,0,0],[0,1,0,0,0],[0,0,0,0,1],[0,0,0,1,0],[0,0,1,0,0]]
=> [(1,10),(2,9),(3,6),(4,5),(7,8)]
=> [5,6,8,9,4,3,10,7,2,1] => ? = 2 - 2
[1,3,2,4,5] => [[1,0,0,0,0],[0,0,1,0,0],[0,1,0,0,0],[0,0,0,1,0],[0,0,0,0,1]]
=> [(1,10),(2,3),(4,7),(5,6),(8,9)]
=> [3,6,2,7,9,5,4,10,8,1] => ? = 2 - 2
[1,3,2,5,4] => [[1,0,0,0,0],[0,0,1,0,0],[0,1,0,0,0],[0,0,0,0,1],[0,0,0,1,0]]
=> [(1,10),(2,3),(4,5),(6,7),(8,9)]
=> [3,5,2,7,4,9,6,10,8,1] => ? = 2 - 2
[1,3,4,5,2] => [[1,0,0,0,0],[0,0,0,0,1],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,1,0]]
=> [(1,10),(2,9),(3,4),(5,8),(6,7)]
=> [4,7,8,3,9,10,6,5,2,1] => ? = 3 - 2
[1,3,5,2,4] => [[1,0,0,0,0],[0,0,0,1,0],[0,1,0,0,0],[0,0,0,0,1],[0,0,1,0,0]]
=> [(1,10),(2,9),(3,4),(5,6),(7,8)]
=> [4,6,8,3,9,5,10,7,2,1] => ? = 2 - 2
[1,3,5,4,2] => [[1,0,0,0,0],[0,0,0,0,1],[0,1,0,0,0],[0,0,0,1,0],[0,0,1,0,0]]
=> [(1,10),(2,9),(3,4),(5,6),(7,8)]
=> [4,6,8,3,9,5,10,7,2,1] => ? = 2 - 2
[1,4,2,5,3] => [[1,0,0,0,0],[0,0,1,0,0],[0,0,0,0,1],[0,1,0,0,0],[0,0,0,1,0]]
=> [(1,10),(2,3),(4,5),(6,7),(8,9)]
=> [3,5,2,7,4,9,6,10,8,1] => ? = 3 - 2
[1,4,3,5,2] => [[1,0,0,0,0],[0,0,0,0,1],[0,0,1,0,0],[0,1,0,0,0],[0,0,0,1,0]]
=> [(1,10),(2,5),(3,4),(6,7),(8,9)]
=> [4,5,7,3,2,9,6,10,8,1] => ? = 3 - 2
[1,4,5,2,3] => [[1,0,0,0,0],[0,0,0,1,0],[0,0,0,0,1],[0,1,0,0,0],[0,0,1,0,0]]
=> [(1,10),(2,3),(4,5),(6,7),(8,9)]
=> [3,5,2,7,4,9,6,10,8,1] => ? = 2 - 2
[1,4,5,3,2] => [[1,0,0,0,0],[0,0,0,0,1],[0,0,0,1,0],[0,1,0,0,0],[0,0,1,0,0]]
=> [(1,10),(2,5),(3,4),(6,7),(8,9)]
=> [4,5,7,3,2,9,6,10,8,1] => ? = 2 - 2
[1,5,2,3,4] => [[1,0,0,0,0],[0,0,1,0,0],[0,0,0,1,0],[0,0,0,0,1],[0,1,0,0,0]]
=> [(1,10),(2,3),(4,9),(5,6),(7,8)]
=> [3,6,2,8,9,5,10,7,4,1] => ? = 3 - 2
[1,5,2,4,3] => [[1,0,0,0,0],[0,0,1,0,0],[0,0,0,0,1],[0,0,0,1,0],[0,1,0,0,0]]
=> [(1,10),(2,3),(4,5),(6,9),(7,8)]
=> [3,5,2,8,4,9,10,7,6,1] => ? = 3 - 2
[1,5,3,2,4] => [[1,0,0,0,0],[0,0,0,1,0],[0,0,1,0,0],[0,0,0,0,1],[0,1,0,0,0]]
=> [(1,10),(2,9),(3,4),(5,6),(7,8)]
=> [4,6,8,3,9,5,10,7,2,1] => ? = 2 - 2
[1,5,3,4,2] => [[1,0,0,0,0],[0,0,0,0,1],[0,0,1,0,0],[0,0,0,1,0],[0,1,0,0,0]]
=> [(1,10),(2,9),(3,4),(5,6),(7,8)]
=> [4,6,8,3,9,5,10,7,2,1] => ? = 2 - 2
[1,5,4,2,3] => [[1,0,0,0,0],[0,0,0,1,0],[0,0,0,0,1],[0,0,1,0,0],[0,1,0,0,0]]
=> [(1,10),(2,3),(4,5),(6,9),(7,8)]
=> [3,5,2,8,4,9,10,7,6,1] => ? = 2 - 2
[1,5,4,3,2] => [[1,0,0,0,0],[0,0,0,0,1],[0,0,0,1,0],[0,0,1,0,0],[0,1,0,0,0]]
=> [(1,10),(2,5),(3,4),(6,9),(7,8)]
=> [4,5,8,3,2,9,10,7,6,1] => ? = 2 - 2
[2,1,3,4,5] => [[0,1,0,0,0],[1,0,0,0,0],[0,0,1,0,0],[0,0,0,1,0],[0,0,0,0,1]]
=> [(1,2),(3,8),(4,7),(5,6),(9,10)]
=> [2,1,6,7,8,5,4,3,10,9] => ? = 4 - 2
[2,1,3,5,4] => [[0,1,0,0,0],[1,0,0,0,0],[0,0,1,0,0],[0,0,0,0,1],[0,0,0,1,0]]
=> [(1,2),(3,8),(4,5),(6,7),(9,10)]
=> [2,1,5,7,4,8,6,3,10,9] => ? = 2 - 2
[2,1,4,3,5] => [[0,1,0,0,0],[1,0,0,0,0],[0,0,0,1,0],[0,0,1,0,0],[0,0,0,0,1]]
=> [(1,2),(3,4),(5,6),(7,8),(9,10)]
=> [2,1,4,3,6,5,8,7,10,9] => 0 = 2 - 2
[2,1,4,5,3] => [[0,1,0,0,0],[1,0,0,0,0],[0,0,0,0,1],[0,0,1,0,0],[0,0,0,1,0]]
=> [(1,2),(3,8),(4,5),(6,7),(9,10)]
=> [2,1,5,7,4,8,6,3,10,9] => ? = 2 - 2
[2,1,5,3,4] => [[0,1,0,0,0],[1,0,0,0,0],[0,0,0,1,0],[0,0,0,0,1],[0,0,1,0,0]]
=> [(1,2),(3,4),(5,6),(7,8),(9,10)]
=> [2,1,4,3,6,5,8,7,10,9] => 0 = 2 - 2
[2,1,5,4,3] => [[0,1,0,0,0],[1,0,0,0,0],[0,0,0,0,1],[0,0,0,1,0],[0,0,1,0,0]]
=> [(1,2),(3,6),(4,5),(7,8),(9,10)]
=> [2,1,5,6,4,3,8,7,10,9] => ? = 2 - 2
[2,3,4,1,5] => [[0,0,0,1,0],[1,0,0,0,0],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,0,1]]
=> [(1,10),(2,3),(4,9),(5,6),(7,8)]
=> [3,6,2,8,9,5,10,7,4,1] => ? = 2 - 2
[2,3,4,5,1] => [[0,0,0,0,1],[1,0,0,0,0],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,1,0]]
=> [(1,10),(2,9),(3,4),(5,8),(6,7)]
=> [4,7,8,3,9,10,6,5,2,1] => ? = 2 - 2
[2,3,5,1,4] => [[0,0,0,1,0],[1,0,0,0,0],[0,1,0,0,0],[0,0,0,0,1],[0,0,1,0,0]]
=> [(1,10),(2,3),(4,9),(5,6),(7,8)]
=> [3,6,2,8,9,5,10,7,4,1] => ? = 2 - 2
[2,3,5,4,1] => [[0,0,0,0,1],[1,0,0,0,0],[0,1,0,0,0],[0,0,0,1,0],[0,0,1,0,0]]
=> [(1,10),(2,9),(3,4),(5,6),(7,8)]
=> [4,6,8,3,9,5,10,7,2,1] => ? = 2 - 2
[2,4,1,3,5] => [[0,0,1,0,0],[1,0,0,0,0],[0,0,0,1,0],[0,1,0,0,0],[0,0,0,0,1]]
=> [(1,10),(2,3),(4,7),(5,6),(8,9)]
=> [3,6,2,7,9,5,4,10,8,1] => ? = 2 - 2
[2,4,1,5,3] => [[0,0,1,0,0],[1,0,0,0,0],[0,0,0,0,1],[0,1,0,0,0],[0,0,0,1,0]]
=> [(1,10),(2,3),(4,5),(6,7),(8,9)]
=> [3,5,2,7,4,9,6,10,8,1] => ? = 2 - 2
[2,4,3,5,1] => [[0,0,0,0,1],[1,0,0,0,0],[0,0,1,0,0],[0,1,0,0,0],[0,0,0,1,0]]
=> [(1,10),(2,5),(3,4),(6,7),(8,9)]
=> [4,5,7,3,2,9,6,10,8,1] => ? = 2 - 2
[2,4,5,1,3] => [[0,0,0,1,0],[1,0,0,0,0],[0,0,0,0,1],[0,1,0,0,0],[0,0,1,0,0]]
=> [(1,10),(2,3),(4,5),(6,7),(8,9)]
=> [3,5,2,7,4,9,6,10,8,1] => ? = 2 - 2
[2,5,1,3,4] => [[0,0,1,0,0],[1,0,0,0,0],[0,0,0,1,0],[0,0,0,0,1],[0,1,0,0,0]]
=> [(1,10),(2,3),(4,9),(5,6),(7,8)]
=> [3,6,2,8,9,5,10,7,4,1] => ? = 2 - 2
[2,5,1,4,3] => [[0,0,1,0,0],[1,0,0,0,0],[0,0,0,0,1],[0,0,0,1,0],[0,1,0,0,0]]
=> [(1,10),(2,3),(4,5),(6,9),(7,8)]
=> [3,5,2,8,4,9,10,7,6,1] => ? = 2 - 2
[2,5,3,1,4] => [[0,0,0,1,0],[1,0,0,0,0],[0,0,1,0,0],[0,0,0,0,1],[0,1,0,0,0]]
=> [(1,10),(2,3),(4,9),(5,6),(7,8)]
=> [3,6,2,8,9,5,10,7,4,1] => ? = 2 - 2
[2,5,4,1,3] => [[0,0,0,1,0],[1,0,0,0,0],[0,0,0,0,1],[0,0,1,0,0],[0,1,0,0,0]]
=> [(1,10),(2,3),(4,5),(6,9),(7,8)]
=> [3,5,2,8,4,9,10,7,6,1] => ? = 2 - 2
[3,1,4,2,5] => [[0,1,0,0,0],[0,0,0,1,0],[1,0,0,0,0],[0,0,1,0,0],[0,0,0,0,1]]
=> [(1,2),(3,4),(5,6),(7,8),(9,10)]
=> [2,1,4,3,6,5,8,7,10,9] => 0 = 2 - 2
[3,1,4,5,2] => [[0,1,0,0,0],[0,0,0,0,1],[1,0,0,0,0],[0,0,1,0,0],[0,0,0,1,0]]
=> [(1,2),(3,4),(5,8),(6,7),(9,10)]
=> [2,1,4,3,7,8,6,5,10,9] => ? = 2 - 2
[3,1,5,2,4] => [[0,1,0,0,0],[0,0,0,1,0],[1,0,0,0,0],[0,0,0,0,1],[0,0,1,0,0]]
=> [(1,2),(3,4),(5,6),(7,8),(9,10)]
=> [2,1,4,3,6,5,8,7,10,9] => 0 = 2 - 2
[3,1,5,4,2] => [[0,1,0,0,0],[0,0,0,0,1],[1,0,0,0,0],[0,0,0,1,0],[0,0,1,0,0]]
=> [(1,2),(3,4),(5,6),(7,8),(9,10)]
=> [2,1,4,3,6,5,8,7,10,9] => 0 = 2 - 2
[3,2,5,4,1] => [[0,0,0,0,1],[0,1,0,0,0],[1,0,0,0,0],[0,0,0,1,0],[0,0,1,0,0]]
=> [(1,2),(3,4),(5,6),(7,8),(9,10)]
=> [2,1,4,3,6,5,8,7,10,9] => 0 = 2 - 2
[3,5,1,2,4] => [[0,0,1,0,0],[0,0,0,1,0],[1,0,0,0,0],[0,0,0,0,1],[0,1,0,0,0]]
=> [(1,2),(3,4),(5,6),(7,8),(9,10)]
=> [2,1,4,3,6,5,8,7,10,9] => 0 = 2 - 2
[2,1,4,3,6,5] => [[0,1,0,0,0,0],[1,0,0,0,0,0],[0,0,0,1,0,0],[0,0,1,0,0,0],[0,0,0,0,0,1],[0,0,0,0,1,0]]
=> [(1,2),(3,4),(5,6),(7,8),(9,10),(11,12)]
=> [2,1,4,3,6,5,8,7,10,9,12,11] => 0 = 2 - 2
[2,1,5,3,6,4] => [[0,1,0,0,0,0],[1,0,0,0,0,0],[0,0,0,1,0,0],[0,0,0,0,0,1],[0,0,1,0,0,0],[0,0,0,0,1,0]]
=> [(1,2),(3,4),(5,6),(7,8),(9,10),(11,12)]
=> [2,1,4,3,6,5,8,7,10,9,12,11] => 0 = 2 - 2
[2,1,5,6,3,4] => [[0,1,0,0,0,0],[1,0,0,0,0,0],[0,0,0,0,1,0],[0,0,0,0,0,1],[0,0,1,0,0,0],[0,0,0,1,0,0]]
=> [(1,2),(3,4),(5,6),(7,8),(9,10),(11,12)]
=> [2,1,4,3,6,5,8,7,10,9,12,11] => 0 = 2 - 2
[3,1,4,2,6,5] => [[0,1,0,0,0,0],[0,0,0,1,0,0],[1,0,0,0,0,0],[0,0,1,0,0,0],[0,0,0,0,0,1],[0,0,0,0,1,0]]
=> [(1,2),(3,4),(5,6),(7,8),(9,10),(11,12)]
=> [2,1,4,3,6,5,8,7,10,9,12,11] => 0 = 2 - 2
[3,1,5,2,6,4] => [[0,1,0,0,0,0],[0,0,0,1,0,0],[1,0,0,0,0,0],[0,0,0,0,0,1],[0,0,1,0,0,0],[0,0,0,0,1,0]]
=> [(1,2),(3,4),(5,6),(7,8),(9,10),(11,12)]
=> [2,1,4,3,6,5,8,7,10,9,12,11] => 0 = 2 - 2
[3,2,5,4,6,1] => [[0,0,0,0,0,1],[0,1,0,0,0,0],[1,0,0,0,0,0],[0,0,0,1,0,0],[0,0,1,0,0,0],[0,0,0,0,1,0]]
=> [(1,2),(3,4),(5,6),(7,8),(9,10),(11,12)]
=> [2,1,4,3,6,5,8,7,10,9,12,11] => 0 = 2 - 2
[3,2,5,6,4,1] => [[0,0,0,0,0,1],[0,1,0,0,0,0],[1,0,0,0,0,0],[0,0,0,0,1,0],[0,0,1,0,0,0],[0,0,0,1,0,0]]
=> [(1,2),(3,4),(5,6),(7,8),(9,10),(11,12)]
=> [2,1,4,3,6,5,8,7,10,9,12,11] => 0 = 2 - 2
[3,5,1,2,6,4] => [[0,0,1,0,0,0],[0,0,0,1,0,0],[1,0,0,0,0,0],[0,0,0,0,0,1],[0,1,0,0,0,0],[0,0,0,0,1,0]]
=> [(1,2),(3,4),(5,6),(7,8),(9,10),(11,12)]
=> [2,1,4,3,6,5,8,7,10,9,12,11] => 0 = 2 - 2
[3,5,1,6,2,4] => [[0,0,1,0,0,0],[0,0,0,0,1,0],[1,0,0,0,0,0],[0,0,0,0,0,1],[0,1,0,0,0,0],[0,0,0,1,0,0]]
=> [(1,2),(3,4),(5,6),(7,8),(9,10),(11,12)]
=> [2,1,4,3,6,5,8,7,10,9,12,11] => 0 = 2 - 2
[3,5,2,4,6,1] => [[0,0,0,0,0,1],[0,0,1,0,0,0],[1,0,0,0,0,0],[0,0,0,1,0,0],[0,1,0,0,0,0],[0,0,0,0,1,0]]
=> [(1,2),(3,4),(5,6),(7,8),(9,10),(11,12)]
=> [2,1,4,3,6,5,8,7,10,9,12,11] => 0 = 2 - 2
[3,5,2,6,1,4] => [[0,0,0,0,1,0],[0,0,1,0,0,0],[1,0,0,0,0,0],[0,0,0,0,0,1],[0,1,0,0,0,0],[0,0,0,1,0,0]]
=> [(1,2),(3,4),(5,6),(7,8),(9,10),(11,12)]
=> [2,1,4,3,6,5,8,7,10,9,12,11] => 0 = 2 - 2
[3,5,2,6,4,1] => [[0,0,0,0,0,1],[0,0,1,0,0,0],[1,0,0,0,0,0],[0,0,0,0,1,0],[0,1,0,0,0,0],[0,0,0,1,0,0]]
=> [(1,2),(3,4),(5,6),(7,8),(9,10),(11,12)]
=> [2,1,4,3,6,5,8,7,10,9,12,11] => 0 = 2 - 2
[3,6,1,2,5,4] => [[0,0,1,0,0,0],[0,0,0,1,0,0],[1,0,0,0,0,0],[0,0,0,0,0,1],[0,0,0,0,1,0],[0,1,0,0,0,0]]
=> [(1,2),(3,4),(5,6),(7,8),(9,10),(11,12)]
=> [2,1,4,3,6,5,8,7,10,9,12,11] => 0 = 2 - 2
[3,6,1,5,2,4] => [[0,0,1,0,0,0],[0,0,0,0,1,0],[1,0,0,0,0,0],[0,0,0,0,0,1],[0,0,0,1,0,0],[0,1,0,0,0,0]]
=> [(1,2),(3,4),(5,6),(7,8),(9,10),(11,12)]
=> [2,1,4,3,6,5,8,7,10,9,12,11] => 0 = 2 - 2
[3,6,2,5,1,4] => [[0,0,0,0,1,0],[0,0,1,0,0,0],[1,0,0,0,0,0],[0,0,0,0,0,1],[0,0,0,1,0,0],[0,1,0,0,0,0]]
=> [(1,2),(3,4),(5,6),(7,8),(9,10),(11,12)]
=> [2,1,4,3,6,5,8,7,10,9,12,11] => 0 = 2 - 2
[3,6,2,5,4,1] => [[0,0,0,0,0,1],[0,0,1,0,0,0],[1,0,0,0,0,0],[0,0,0,0,1,0],[0,0,0,1,0,0],[0,1,0,0,0,0]]
=> [(1,2),(3,4),(5,6),(7,8),(9,10),(11,12)]
=> [2,1,4,3,6,5,8,7,10,9,12,11] => 0 = 2 - 2
[4,5,1,6,2,3] => [[0,0,1,0,0,0],[0,0,0,0,1,0],[0,0,0,0,0,1],[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,0,1,0,0]]
=> [(1,2),(3,4),(5,6),(7,8),(9,10),(11,12)]
=> [2,1,4,3,6,5,8,7,10,9,12,11] => 0 = 2 - 2
[4,5,2,3,6,1] => [[0,0,0,0,0,1],[0,0,1,0,0,0],[0,0,0,1,0,0],[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,0,0,1,0]]
=> [(1,2),(3,4),(5,6),(7,8),(9,10),(11,12)]
=> [2,1,4,3,6,5,8,7,10,9,12,11] => 0 = 2 - 2
[4,5,2,6,1,3] => [[0,0,0,0,1,0],[0,0,1,0,0,0],[0,0,0,0,0,1],[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,0,1,0,0]]
=> [(1,2),(3,4),(5,6),(7,8),(9,10),(11,12)]
=> [2,1,4,3,6,5,8,7,10,9,12,11] => 0 = 2 - 2
[4,5,2,6,3,1] => [[0,0,0,0,0,1],[0,0,1,0,0,0],[0,0,0,0,1,0],[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,0,1,0,0]]
=> [(1,2),(3,4),(5,6),(7,8),(9,10),(11,12)]
=> [2,1,4,3,6,5,8,7,10,9,12,11] => 0 = 2 - 2
[4,6,1,5,2,3] => [[0,0,1,0,0,0],[0,0,0,0,1,0],[0,0,0,0,0,1],[1,0,0,0,0,0],[0,0,0,1,0,0],[0,1,0,0,0,0]]
=> [(1,2),(3,4),(5,6),(7,8),(9,10),(11,12)]
=> [2,1,4,3,6,5,8,7,10,9,12,11] => 0 = 2 - 2
[6,1,3,2,5,4] => [[0,1,0,0,0,0],[0,0,0,1,0,0],[0,0,1,0,0,0],[0,0,0,0,0,1],[0,0,0,0,1,0],[1,0,0,0,0,0]]
=> [(1,2),(3,4),(5,6),(7,8),(9,10),(11,12)]
=> [2,1,4,3,6,5,8,7,10,9,12,11] => 0 = 2 - 2
[6,1,3,5,2,4] => [[0,1,0,0,0,0],[0,0,0,0,1,0],[0,0,1,0,0,0],[0,0,0,0,0,1],[0,0,0,1,0,0],[1,0,0,0,0,0]]
=> [(1,2),(3,4),(5,6),(7,8),(9,10),(11,12)]
=> [2,1,4,3,6,5,8,7,10,9,12,11] => 0 = 2 - 2
[6,1,4,5,2,3] => [[0,1,0,0,0,0],[0,0,0,0,1,0],[0,0,0,0,0,1],[0,0,1,0,0,0],[0,0,0,1,0,0],[1,0,0,0,0,0]]
=> [(1,2),(3,4),(5,6),(7,8),(9,10),(11,12)]
=> [2,1,4,3,6,5,8,7,10,9,12,11] => 0 = 2 - 2
[6,3,1,2,5,4] => [[0,0,1,0,0,0],[0,0,0,1,0,0],[0,1,0,0,0,0],[0,0,0,0,0,1],[0,0,0,0,1,0],[1,0,0,0,0,0]]
=> [(1,2),(3,4),(5,6),(7,8),(9,10),(11,12)]
=> [2,1,4,3,6,5,8,7,10,9,12,11] => 0 = 2 - 2
[6,3,1,5,2,4] => [[0,0,1,0,0,0],[0,0,0,0,1,0],[0,1,0,0,0,0],[0,0,0,0,0,1],[0,0,0,1,0,0],[1,0,0,0,0,0]]
=> [(1,2),(3,4),(5,6),(7,8),(9,10),(11,12)]
=> [2,1,4,3,6,5,8,7,10,9,12,11] => 0 = 2 - 2
[6,3,2,5,1,4] => [[0,0,0,0,1,0],[0,0,1,0,0,0],[0,1,0,0,0,0],[0,0,0,0,0,1],[0,0,0,1,0,0],[1,0,0,0,0,0]]
=> [(1,2),(3,4),(5,6),(7,8),(9,10),(11,12)]
=> [2,1,4,3,6,5,8,7,10,9,12,11] => 0 = 2 - 2
[6,3,2,5,4,1] => [[0,0,0,0,0,1],[0,0,1,0,0,0],[0,1,0,0,0,0],[0,0,0,0,1,0],[0,0,0,1,0,0],[1,0,0,0,0,0]]
=> [(1,2),(3,4),(5,6),(7,8),(9,10),(11,12)]
=> [2,1,4,3,6,5,8,7,10,9,12,11] => 0 = 2 - 2
[6,4,1,5,2,3] => [[0,0,1,0,0,0],[0,0,0,0,1,0],[0,0,0,0,0,1],[0,1,0,0,0,0],[0,0,0,1,0,0],[1,0,0,0,0,0]]
=> [(1,2),(3,4),(5,6),(7,8),(9,10),(11,12)]
=> [2,1,4,3,6,5,8,7,10,9,12,11] => 0 = 2 - 2
Description
The number of rafts of a permutation. Let $\pi$ be a permutation of length $n$. A small ascent of $\pi$ is an index $i$ such that $\pi(i+1)= \pi(i)+1$, see [[St000441]], and a raft of $\pi$ is a non-empty maximal sequence of consecutive small ascents.
Mp00072: Permutations binary search tree: left to rightBinary trees
Mp00019: Binary trees right rotateBinary trees
Mp00011: Binary trees to graphGraphs
St000454: Graphs ⟶ ℤResult quality: 6% values known / values provided: 6%distinct values known / distinct values provided: 20%
Values
[1,2,3] => [.,[.,[.,.]]]
=> [[.,[.,.]],.]
=> ([(0,2),(1,2)],3)
=> ? = 3
[1,3,2] => [.,[[.,.],.]]
=> [[[.,.],.],.]
=> ([(0,2),(1,2)],3)
=> ? = 2
[2,1,3] => [[.,.],[.,.]]
=> [.,[.,[.,.]]]
=> ([(0,2),(1,2)],3)
=> ? = 2
[1,2,3,4] => [.,[.,[.,[.,.]]]]
=> [[.,[.,[.,.]]],.]
=> ([(0,3),(1,2),(2,3)],4)
=> ? = 2
[1,2,4,3] => [.,[.,[[.,.],.]]]
=> [[.,[[.,.],.]],.]
=> ([(0,3),(1,2),(2,3)],4)
=> ? = 2
[1,3,2,4] => [.,[[.,.],[.,.]]]
=> [[[.,.],[.,.]],.]
=> ([(0,3),(1,3),(2,3)],4)
=> ? = 2
[1,3,4,2] => [.,[[.,.],[.,.]]]
=> [[[.,.],[.,.]],.]
=> ([(0,3),(1,3),(2,3)],4)
=> ? = 2
[1,4,2,3] => [.,[[.,[.,.]],.]]
=> [[[.,[.,.]],.],.]
=> ([(0,3),(1,2),(2,3)],4)
=> ? = 2
[1,4,3,2] => [.,[[[.,.],.],.]]
=> [[[[.,.],.],.],.]
=> ([(0,3),(1,2),(2,3)],4)
=> ? = 2
[2,1,3,4] => [[.,.],[.,[.,.]]]
=> [.,[.,[.,[.,.]]]]
=> ([(0,3),(1,2),(2,3)],4)
=> ? = 3
[2,1,4,3] => [[.,.],[[.,.],.]]
=> [.,[.,[[.,.],.]]]
=> ([(0,3),(1,2),(2,3)],4)
=> ? = 2
[2,3,4,1] => [[.,.],[.,[.,.]]]
=> [.,[.,[.,[.,.]]]]
=> ([(0,3),(1,2),(2,3)],4)
=> ? = 2
[2,4,1,3] => [[.,.],[[.,.],.]]
=> [.,[.,[[.,.],.]]]
=> ([(0,3),(1,2),(2,3)],4)
=> ? = 2
[3,1,4,2] => [[.,[.,.]],[.,.]]
=> [.,[[.,.],[.,.]]]
=> ([(0,3),(1,3),(2,3)],4)
=> ? = 2
[3,2,4,1] => [[[.,.],.],[.,.]]
=> [[.,.],[.,[.,.]]]
=> ([(0,3),(1,2),(2,3)],4)
=> ? = 2
[4,1,2,3] => [[.,[.,[.,.]]],.]
=> [.,[[.,[.,.]],.]]
=> ([(0,3),(1,2),(2,3)],4)
=> ? = 2
[4,1,3,2] => [[.,[[.,.],.]],.]
=> [.,[[[.,.],.],.]]
=> ([(0,3),(1,2),(2,3)],4)
=> ? = 2
[1,2,3,4,5] => [.,[.,[.,[.,[.,.]]]]]
=> [[.,[.,[.,[.,.]]]],.]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ? = 2
[1,2,3,5,4] => [.,[.,[.,[[.,.],.]]]]
=> [[.,[.,[[.,.],.]]],.]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ? = 2
[1,2,4,3,5] => [.,[.,[[.,.],[.,.]]]]
=> [[.,[[.,.],[.,.]]],.]
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> ? = 2
[1,2,4,5,3] => [.,[.,[[.,.],[.,.]]]]
=> [[.,[[.,.],[.,.]]],.]
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> ? = 2
[1,2,5,3,4] => [.,[.,[[.,[.,.]],.]]]
=> [[.,[[.,[.,.]],.]],.]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ? = 2
[1,2,5,4,3] => [.,[.,[[[.,.],.],.]]]
=> [[.,[[[.,.],.],.]],.]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ? = 2
[1,3,2,4,5] => [.,[[.,.],[.,[.,.]]]]
=> [[[.,.],[.,[.,.]]],.]
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> ? = 2
[1,3,2,5,4] => [.,[[.,.],[[.,.],.]]]
=> [[[.,.],[[.,.],.]],.]
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> ? = 2
[1,3,4,5,2] => [.,[[.,.],[.,[.,.]]]]
=> [[[.,.],[.,[.,.]]],.]
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> ? = 3
[1,3,5,2,4] => [.,[[.,.],[[.,.],.]]]
=> [[[.,.],[[.,.],.]],.]
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> ? = 2
[1,3,5,4,2] => [.,[[.,.],[[.,.],.]]]
=> [[[.,.],[[.,.],.]],.]
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> ? = 2
[1,4,2,5,3] => [.,[[.,[.,.]],[.,.]]]
=> [[[.,[.,.]],[.,.]],.]
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> ? = 3
[1,4,3,5,2] => [.,[[[.,.],.],[.,.]]]
=> [[[[.,.],.],[.,.]],.]
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> ? = 3
[1,4,5,2,3] => [.,[[.,[.,.]],[.,.]]]
=> [[[.,[.,.]],[.,.]],.]
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> ? = 2
[1,4,5,3,2] => [.,[[[.,.],.],[.,.]]]
=> [[[[.,.],.],[.,.]],.]
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> ? = 2
[1,5,2,3,4] => [.,[[.,[.,[.,.]]],.]]
=> [[[.,[.,[.,.]]],.],.]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ? = 3
[1,5,2,4,3] => [.,[[.,[[.,.],.]],.]]
=> [[[.,[[.,.],.]],.],.]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ? = 3
[1,5,3,2,4] => [.,[[[.,.],[.,.]],.]]
=> [[[[.,.],[.,.]],.],.]
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> ? = 2
[1,5,3,4,2] => [.,[[[.,.],[.,.]],.]]
=> [[[[.,.],[.,.]],.],.]
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> ? = 2
[1,5,4,2,3] => [.,[[[.,[.,.]],.],.]]
=> [[[[.,[.,.]],.],.],.]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ? = 2
[1,5,4,3,2] => [.,[[[[.,.],.],.],.]]
=> [[[[[.,.],.],.],.],.]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ? = 2
[2,1,3,4,5] => [[.,.],[.,[.,[.,.]]]]
=> [.,[.,[.,[.,[.,.]]]]]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ? = 4
[2,1,3,5,4] => [[.,.],[.,[[.,.],.]]]
=> [.,[.,[.,[[.,.],.]]]]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ? = 2
[2,1,4,3,5] => [[.,.],[[.,.],[.,.]]]
=> [.,[.,[[.,.],[.,.]]]]
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> ? = 2
[2,1,4,5,3] => [[.,.],[[.,.],[.,.]]]
=> [.,[.,[[.,.],[.,.]]]]
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> ? = 2
[2,1,5,3,4] => [[.,.],[[.,[.,.]],.]]
=> [.,[.,[[.,[.,.]],.]]]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ? = 2
[2,1,5,4,3] => [[.,.],[[[.,.],.],.]]
=> [.,[.,[[[.,.],.],.]]]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ? = 2
[2,3,4,1,5] => [[.,.],[.,[.,[.,.]]]]
=> [.,[.,[.,[.,[.,.]]]]]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ? = 2
[2,3,4,5,1] => [[.,.],[.,[.,[.,.]]]]
=> [.,[.,[.,[.,[.,.]]]]]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ? = 2
[2,3,5,1,4] => [[.,.],[.,[[.,.],.]]]
=> [.,[.,[.,[[.,.],.]]]]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ? = 2
[2,3,5,4,1] => [[.,.],[.,[[.,.],.]]]
=> [.,[.,[.,[[.,.],.]]]]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ? = 2
[2,4,1,3,5] => [[.,.],[[.,.],[.,.]]]
=> [.,[.,[[.,.],[.,.]]]]
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> ? = 2
[2,4,1,5,3] => [[.,.],[[.,.],[.,.]]]
=> [.,[.,[[.,.],[.,.]]]]
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> ? = 2
[1,3,2,5,4,6] => [.,[[.,.],[[.,.],[.,.]]]]
=> [[[.,.],[[.,.],[.,.]]],.]
=> ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> 2
[1,3,2,5,6,4] => [.,[[.,.],[[.,.],[.,.]]]]
=> [[[.,.],[[.,.],[.,.]]],.]
=> ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> 2
[1,3,5,2,4,6] => [.,[[.,.],[[.,.],[.,.]]]]
=> [[[.,.],[[.,.],[.,.]]],.]
=> ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> 2
[1,3,5,2,6,4] => [.,[[.,.],[[.,.],[.,.]]]]
=> [[[.,.],[[.,.],[.,.]]],.]
=> ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> 2
[1,3,5,4,6,2] => [.,[[.,.],[[.,.],[.,.]]]]
=> [[[.,.],[[.,.],[.,.]]],.]
=> ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> 2
[1,3,5,6,2,4] => [.,[[.,.],[[.,.],[.,.]]]]
=> [[[.,.],[[.,.],[.,.]]],.]
=> ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> 2
[1,3,5,6,4,2] => [.,[[.,.],[[.,.],[.,.]]]]
=> [[[.,.],[[.,.],[.,.]]],.]
=> ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> 2
[1,5,3,2,6,4] => [.,[[[.,.],[.,.]],[.,.]]]
=> [[[[.,.],[.,.]],[.,.]],.]
=> ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> 2
[1,5,3,4,6,2] => [.,[[[.,.],[.,.]],[.,.]]]
=> [[[[.,.],[.,.]],[.,.]],.]
=> ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> 2
[1,5,3,6,2,4] => [.,[[[.,.],[.,.]],[.,.]]]
=> [[[[.,.],[.,.]],[.,.]],.]
=> ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> 2
[1,5,3,6,4,2] => [.,[[[.,.],[.,.]],[.,.]]]
=> [[[[.,.],[.,.]],[.,.]],.]
=> ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> 2
[1,5,6,3,2,4] => [.,[[[.,.],[.,.]],[.,.]]]
=> [[[[.,.],[.,.]],[.,.]],.]
=> ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> 2
[1,5,6,3,4,2] => [.,[[[.,.],[.,.]],[.,.]]]
=> [[[[.,.],[.,.]],[.,.]],.]
=> ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> 2
[3,1,5,2,4,6] => [[.,[.,.]],[[.,.],[.,.]]]
=> [.,[[.,.],[[.,.],[.,.]]]]
=> ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> 2
[3,1,5,2,6,4] => [[.,[.,.]],[[.,.],[.,.]]]
=> [.,[[.,.],[[.,.],[.,.]]]]
=> ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> 2
[3,1,5,4,2,6] => [[.,[.,.]],[[.,.],[.,.]]]
=> [.,[[.,.],[[.,.],[.,.]]]]
=> ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> 2
[3,1,5,4,6,2] => [[.,[.,.]],[[.,.],[.,.]]]
=> [.,[[.,.],[[.,.],[.,.]]]]
=> ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> 2
[3,1,5,6,4,2] => [[.,[.,.]],[[.,.],[.,.]]]
=> [.,[[.,.],[[.,.],[.,.]]]]
=> ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> 2
[3,5,1,2,4,6] => [[.,[.,.]],[[.,.],[.,.]]]
=> [.,[[.,.],[[.,.],[.,.]]]]
=> ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> 2
[3,5,1,2,6,4] => [[.,[.,.]],[[.,.],[.,.]]]
=> [.,[[.,.],[[.,.],[.,.]]]]
=> ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> 2
[3,5,1,4,6,2] => [[.,[.,.]],[[.,.],[.,.]]]
=> [.,[[.,.],[[.,.],[.,.]]]]
=> ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> 2
[3,5,1,6,2,4] => [[.,[.,.]],[[.,.],[.,.]]]
=> [.,[[.,.],[[.,.],[.,.]]]]
=> ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> 2
[3,5,1,6,4,2] => [[.,[.,.]],[[.,.],[.,.]]]
=> [.,[[.,.],[[.,.],[.,.]]]]
=> ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> 2
[3,5,4,1,6,2] => [[.,[.,.]],[[.,.],[.,.]]]
=> [.,[[.,.],[[.,.],[.,.]]]]
=> ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> 2
[3,5,4,6,1,2] => [[.,[.,.]],[[.,.],[.,.]]]
=> [.,[[.,.],[[.,.],[.,.]]]]
=> ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> 2
[3,5,6,1,2,4] => [[.,[.,.]],[[.,.],[.,.]]]
=> [.,[[.,.],[[.,.],[.,.]]]]
=> ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> 2
[5,1,3,2,4,6] => [[.,[[.,.],[.,.]]],[.,.]]
=> [.,[[[.,.],[.,.]],[.,.]]]
=> ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> 2
[5,1,3,2,6,4] => [[.,[[.,.],[.,.]]],[.,.]]
=> [.,[[[.,.],[.,.]],[.,.]]]
=> ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> 2
[5,1,3,4,2,6] => [[.,[[.,.],[.,.]]],[.,.]]
=> [.,[[[.,.],[.,.]],[.,.]]]
=> ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> 2
[5,1,3,4,6,2] => [[.,[[.,.],[.,.]]],[.,.]]
=> [.,[[[.,.],[.,.]],[.,.]]]
=> ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> 2
[5,1,3,6,2,4] => [[.,[[.,.],[.,.]]],[.,.]]
=> [.,[[[.,.],[.,.]],[.,.]]]
=> ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> 2
[5,1,3,6,4,2] => [[.,[[.,.],[.,.]]],[.,.]]
=> [.,[[[.,.],[.,.]],[.,.]]]
=> ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> 2
[5,1,6,3,2,4] => [[.,[[.,.],[.,.]]],[.,.]]
=> [.,[[[.,.],[.,.]],[.,.]]]
=> ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> 2
[5,1,6,3,4,2] => [[.,[[.,.],[.,.]]],[.,.]]
=> [.,[[[.,.],[.,.]],[.,.]]]
=> ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> 2
[5,6,1,3,2,4] => [[.,[[.,.],[.,.]]],[.,.]]
=> [.,[[[.,.],[.,.]],[.,.]]]
=> ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> 2
[5,6,1,3,4,2] => [[.,[[.,.],[.,.]]],[.,.]]
=> [.,[[[.,.],[.,.]],[.,.]]]
=> ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> 2
Description
The largest eigenvalue of a graph if it is integral. If a graph is $d$-regular, then its largest eigenvalue equals $d$. One can show that the largest eigenvalue always lies between the average degree and the maximal degree. This statistic is undefined if the largest eigenvalue of the graph is not integral.
Mp00072: Permutations binary search tree: left to rightBinary trees
Mp00019: Binary trees right rotateBinary trees
Mp00011: Binary trees to graphGraphs
St000422: Graphs ⟶ ℤResult quality: 6% values known / values provided: 6%distinct values known / distinct values provided: 20%
Values
[1,2,3] => [.,[.,[.,.]]]
=> [[.,[.,.]],.]
=> ([(0,2),(1,2)],3)
=> ? = 3 + 4
[1,3,2] => [.,[[.,.],.]]
=> [[[.,.],.],.]
=> ([(0,2),(1,2)],3)
=> ? = 2 + 4
[2,1,3] => [[.,.],[.,.]]
=> [.,[.,[.,.]]]
=> ([(0,2),(1,2)],3)
=> ? = 2 + 4
[1,2,3,4] => [.,[.,[.,[.,.]]]]
=> [[.,[.,[.,.]]],.]
=> ([(0,3),(1,2),(2,3)],4)
=> ? = 2 + 4
[1,2,4,3] => [.,[.,[[.,.],.]]]
=> [[.,[[.,.],.]],.]
=> ([(0,3),(1,2),(2,3)],4)
=> ? = 2 + 4
[1,3,2,4] => [.,[[.,.],[.,.]]]
=> [[[.,.],[.,.]],.]
=> ([(0,3),(1,3),(2,3)],4)
=> ? = 2 + 4
[1,3,4,2] => [.,[[.,.],[.,.]]]
=> [[[.,.],[.,.]],.]
=> ([(0,3),(1,3),(2,3)],4)
=> ? = 2 + 4
[1,4,2,3] => [.,[[.,[.,.]],.]]
=> [[[.,[.,.]],.],.]
=> ([(0,3),(1,2),(2,3)],4)
=> ? = 2 + 4
[1,4,3,2] => [.,[[[.,.],.],.]]
=> [[[[.,.],.],.],.]
=> ([(0,3),(1,2),(2,3)],4)
=> ? = 2 + 4
[2,1,3,4] => [[.,.],[.,[.,.]]]
=> [.,[.,[.,[.,.]]]]
=> ([(0,3),(1,2),(2,3)],4)
=> ? = 3 + 4
[2,1,4,3] => [[.,.],[[.,.],.]]
=> [.,[.,[[.,.],.]]]
=> ([(0,3),(1,2),(2,3)],4)
=> ? = 2 + 4
[2,3,4,1] => [[.,.],[.,[.,.]]]
=> [.,[.,[.,[.,.]]]]
=> ([(0,3),(1,2),(2,3)],4)
=> ? = 2 + 4
[2,4,1,3] => [[.,.],[[.,.],.]]
=> [.,[.,[[.,.],.]]]
=> ([(0,3),(1,2),(2,3)],4)
=> ? = 2 + 4
[3,1,4,2] => [[.,[.,.]],[.,.]]
=> [.,[[.,.],[.,.]]]
=> ([(0,3),(1,3),(2,3)],4)
=> ? = 2 + 4
[3,2,4,1] => [[[.,.],.],[.,.]]
=> [[.,.],[.,[.,.]]]
=> ([(0,3),(1,2),(2,3)],4)
=> ? = 2 + 4
[4,1,2,3] => [[.,[.,[.,.]]],.]
=> [.,[[.,[.,.]],.]]
=> ([(0,3),(1,2),(2,3)],4)
=> ? = 2 + 4
[4,1,3,2] => [[.,[[.,.],.]],.]
=> [.,[[[.,.],.],.]]
=> ([(0,3),(1,2),(2,3)],4)
=> ? = 2 + 4
[1,2,3,4,5] => [.,[.,[.,[.,[.,.]]]]]
=> [[.,[.,[.,[.,.]]]],.]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ? = 2 + 4
[1,2,3,5,4] => [.,[.,[.,[[.,.],.]]]]
=> [[.,[.,[[.,.],.]]],.]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ? = 2 + 4
[1,2,4,3,5] => [.,[.,[[.,.],[.,.]]]]
=> [[.,[[.,.],[.,.]]],.]
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> ? = 2 + 4
[1,2,4,5,3] => [.,[.,[[.,.],[.,.]]]]
=> [[.,[[.,.],[.,.]]],.]
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> ? = 2 + 4
[1,2,5,3,4] => [.,[.,[[.,[.,.]],.]]]
=> [[.,[[.,[.,.]],.]],.]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ? = 2 + 4
[1,2,5,4,3] => [.,[.,[[[.,.],.],.]]]
=> [[.,[[[.,.],.],.]],.]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ? = 2 + 4
[1,3,2,4,5] => [.,[[.,.],[.,[.,.]]]]
=> [[[.,.],[.,[.,.]]],.]
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> ? = 2 + 4
[1,3,2,5,4] => [.,[[.,.],[[.,.],.]]]
=> [[[.,.],[[.,.],.]],.]
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> ? = 2 + 4
[1,3,4,5,2] => [.,[[.,.],[.,[.,.]]]]
=> [[[.,.],[.,[.,.]]],.]
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> ? = 3 + 4
[1,3,5,2,4] => [.,[[.,.],[[.,.],.]]]
=> [[[.,.],[[.,.],.]],.]
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> ? = 2 + 4
[1,3,5,4,2] => [.,[[.,.],[[.,.],.]]]
=> [[[.,.],[[.,.],.]],.]
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> ? = 2 + 4
[1,4,2,5,3] => [.,[[.,[.,.]],[.,.]]]
=> [[[.,[.,.]],[.,.]],.]
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> ? = 3 + 4
[1,4,3,5,2] => [.,[[[.,.],.],[.,.]]]
=> [[[[.,.],.],[.,.]],.]
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> ? = 3 + 4
[1,4,5,2,3] => [.,[[.,[.,.]],[.,.]]]
=> [[[.,[.,.]],[.,.]],.]
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> ? = 2 + 4
[1,4,5,3,2] => [.,[[[.,.],.],[.,.]]]
=> [[[[.,.],.],[.,.]],.]
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> ? = 2 + 4
[1,5,2,3,4] => [.,[[.,[.,[.,.]]],.]]
=> [[[.,[.,[.,.]]],.],.]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ? = 3 + 4
[1,5,2,4,3] => [.,[[.,[[.,.],.]],.]]
=> [[[.,[[.,.],.]],.],.]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ? = 3 + 4
[1,5,3,2,4] => [.,[[[.,.],[.,.]],.]]
=> [[[[.,.],[.,.]],.],.]
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> ? = 2 + 4
[1,5,3,4,2] => [.,[[[.,.],[.,.]],.]]
=> [[[[.,.],[.,.]],.],.]
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> ? = 2 + 4
[1,5,4,2,3] => [.,[[[.,[.,.]],.],.]]
=> [[[[.,[.,.]],.],.],.]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ? = 2 + 4
[1,5,4,3,2] => [.,[[[[.,.],.],.],.]]
=> [[[[[.,.],.],.],.],.]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ? = 2 + 4
[2,1,3,4,5] => [[.,.],[.,[.,[.,.]]]]
=> [.,[.,[.,[.,[.,.]]]]]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ? = 4 + 4
[2,1,3,5,4] => [[.,.],[.,[[.,.],.]]]
=> [.,[.,[.,[[.,.],.]]]]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ? = 2 + 4
[2,1,4,3,5] => [[.,.],[[.,.],[.,.]]]
=> [.,[.,[[.,.],[.,.]]]]
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> ? = 2 + 4
[2,1,4,5,3] => [[.,.],[[.,.],[.,.]]]
=> [.,[.,[[.,.],[.,.]]]]
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> ? = 2 + 4
[2,1,5,3,4] => [[.,.],[[.,[.,.]],.]]
=> [.,[.,[[.,[.,.]],.]]]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ? = 2 + 4
[2,1,5,4,3] => [[.,.],[[[.,.],.],.]]
=> [.,[.,[[[.,.],.],.]]]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ? = 2 + 4
[2,3,4,1,5] => [[.,.],[.,[.,[.,.]]]]
=> [.,[.,[.,[.,[.,.]]]]]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ? = 2 + 4
[2,3,4,5,1] => [[.,.],[.,[.,[.,.]]]]
=> [.,[.,[.,[.,[.,.]]]]]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ? = 2 + 4
[2,3,5,1,4] => [[.,.],[.,[[.,.],.]]]
=> [.,[.,[.,[[.,.],.]]]]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ? = 2 + 4
[2,3,5,4,1] => [[.,.],[.,[[.,.],.]]]
=> [.,[.,[.,[[.,.],.]]]]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ? = 2 + 4
[2,4,1,3,5] => [[.,.],[[.,.],[.,.]]]
=> [.,[.,[[.,.],[.,.]]]]
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> ? = 2 + 4
[2,4,1,5,3] => [[.,.],[[.,.],[.,.]]]
=> [.,[.,[[.,.],[.,.]]]]
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> ? = 2 + 4
[1,3,2,5,4,6] => [.,[[.,.],[[.,.],[.,.]]]]
=> [[[.,.],[[.,.],[.,.]]],.]
=> ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> 6 = 2 + 4
[1,3,2,5,6,4] => [.,[[.,.],[[.,.],[.,.]]]]
=> [[[.,.],[[.,.],[.,.]]],.]
=> ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> 6 = 2 + 4
[1,3,5,2,4,6] => [.,[[.,.],[[.,.],[.,.]]]]
=> [[[.,.],[[.,.],[.,.]]],.]
=> ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> 6 = 2 + 4
[1,3,5,2,6,4] => [.,[[.,.],[[.,.],[.,.]]]]
=> [[[.,.],[[.,.],[.,.]]],.]
=> ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> 6 = 2 + 4
[1,3,5,4,6,2] => [.,[[.,.],[[.,.],[.,.]]]]
=> [[[.,.],[[.,.],[.,.]]],.]
=> ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> 6 = 2 + 4
[1,3,5,6,2,4] => [.,[[.,.],[[.,.],[.,.]]]]
=> [[[.,.],[[.,.],[.,.]]],.]
=> ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> 6 = 2 + 4
[1,3,5,6,4,2] => [.,[[.,.],[[.,.],[.,.]]]]
=> [[[.,.],[[.,.],[.,.]]],.]
=> ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> 6 = 2 + 4
[1,5,3,2,6,4] => [.,[[[.,.],[.,.]],[.,.]]]
=> [[[[.,.],[.,.]],[.,.]],.]
=> ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> 6 = 2 + 4
[1,5,3,4,6,2] => [.,[[[.,.],[.,.]],[.,.]]]
=> [[[[.,.],[.,.]],[.,.]],.]
=> ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> 6 = 2 + 4
[1,5,3,6,2,4] => [.,[[[.,.],[.,.]],[.,.]]]
=> [[[[.,.],[.,.]],[.,.]],.]
=> ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> 6 = 2 + 4
[1,5,3,6,4,2] => [.,[[[.,.],[.,.]],[.,.]]]
=> [[[[.,.],[.,.]],[.,.]],.]
=> ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> 6 = 2 + 4
[1,5,6,3,2,4] => [.,[[[.,.],[.,.]],[.,.]]]
=> [[[[.,.],[.,.]],[.,.]],.]
=> ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> 6 = 2 + 4
[1,5,6,3,4,2] => [.,[[[.,.],[.,.]],[.,.]]]
=> [[[[.,.],[.,.]],[.,.]],.]
=> ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> 6 = 2 + 4
[3,1,5,2,4,6] => [[.,[.,.]],[[.,.],[.,.]]]
=> [.,[[.,.],[[.,.],[.,.]]]]
=> ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> 6 = 2 + 4
[3,1,5,2,6,4] => [[.,[.,.]],[[.,.],[.,.]]]
=> [.,[[.,.],[[.,.],[.,.]]]]
=> ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> 6 = 2 + 4
[3,1,5,4,2,6] => [[.,[.,.]],[[.,.],[.,.]]]
=> [.,[[.,.],[[.,.],[.,.]]]]
=> ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> 6 = 2 + 4
[3,1,5,4,6,2] => [[.,[.,.]],[[.,.],[.,.]]]
=> [.,[[.,.],[[.,.],[.,.]]]]
=> ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> 6 = 2 + 4
[3,1,5,6,4,2] => [[.,[.,.]],[[.,.],[.,.]]]
=> [.,[[.,.],[[.,.],[.,.]]]]
=> ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> 6 = 2 + 4
[3,5,1,2,4,6] => [[.,[.,.]],[[.,.],[.,.]]]
=> [.,[[.,.],[[.,.],[.,.]]]]
=> ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> 6 = 2 + 4
[3,5,1,2,6,4] => [[.,[.,.]],[[.,.],[.,.]]]
=> [.,[[.,.],[[.,.],[.,.]]]]
=> ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> 6 = 2 + 4
[3,5,1,4,6,2] => [[.,[.,.]],[[.,.],[.,.]]]
=> [.,[[.,.],[[.,.],[.,.]]]]
=> ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> 6 = 2 + 4
[3,5,1,6,2,4] => [[.,[.,.]],[[.,.],[.,.]]]
=> [.,[[.,.],[[.,.],[.,.]]]]
=> ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> 6 = 2 + 4
[3,5,1,6,4,2] => [[.,[.,.]],[[.,.],[.,.]]]
=> [.,[[.,.],[[.,.],[.,.]]]]
=> ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> 6 = 2 + 4
[3,5,4,1,6,2] => [[.,[.,.]],[[.,.],[.,.]]]
=> [.,[[.,.],[[.,.],[.,.]]]]
=> ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> 6 = 2 + 4
[3,5,4,6,1,2] => [[.,[.,.]],[[.,.],[.,.]]]
=> [.,[[.,.],[[.,.],[.,.]]]]
=> ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> 6 = 2 + 4
[3,5,6,1,2,4] => [[.,[.,.]],[[.,.],[.,.]]]
=> [.,[[.,.],[[.,.],[.,.]]]]
=> ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> 6 = 2 + 4
[5,1,3,2,4,6] => [[.,[[.,.],[.,.]]],[.,.]]
=> [.,[[[.,.],[.,.]],[.,.]]]
=> ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> 6 = 2 + 4
[5,1,3,2,6,4] => [[.,[[.,.],[.,.]]],[.,.]]
=> [.,[[[.,.],[.,.]],[.,.]]]
=> ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> 6 = 2 + 4
[5,1,3,4,2,6] => [[.,[[.,.],[.,.]]],[.,.]]
=> [.,[[[.,.],[.,.]],[.,.]]]
=> ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> 6 = 2 + 4
[5,1,3,4,6,2] => [[.,[[.,.],[.,.]]],[.,.]]
=> [.,[[[.,.],[.,.]],[.,.]]]
=> ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> 6 = 2 + 4
[5,1,3,6,2,4] => [[.,[[.,.],[.,.]]],[.,.]]
=> [.,[[[.,.],[.,.]],[.,.]]]
=> ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> 6 = 2 + 4
[5,1,3,6,4,2] => [[.,[[.,.],[.,.]]],[.,.]]
=> [.,[[[.,.],[.,.]],[.,.]]]
=> ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> 6 = 2 + 4
[5,1,6,3,2,4] => [[.,[[.,.],[.,.]]],[.,.]]
=> [.,[[[.,.],[.,.]],[.,.]]]
=> ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> 6 = 2 + 4
[5,1,6,3,4,2] => [[.,[[.,.],[.,.]]],[.,.]]
=> [.,[[[.,.],[.,.]],[.,.]]]
=> ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> 6 = 2 + 4
[5,6,1,3,2,4] => [[.,[[.,.],[.,.]]],[.,.]]
=> [.,[[[.,.],[.,.]],[.,.]]]
=> ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> 6 = 2 + 4
[5,6,1,3,4,2] => [[.,[[.,.],[.,.]]],[.,.]]
=> [.,[[[.,.],[.,.]],[.,.]]]
=> ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> 6 = 2 + 4
Description
The energy of a graph, if it is integral. The energy of a graph is the sum of the absolute values of its eigenvalues. This statistic is only defined for graphs with integral energy. It is known, that the energy is never an odd integer [2]. In fact, it is never the square root of an odd integer [3]. The energy of a graph is the sum of the energies of the connected components of a graph. The energy of the complete graph $K_n$ equals $2n-2$. For this reason, we do not define the energy of the empty graph.
Mp00064: Permutations reversePermutations
Mp00159: Permutations Demazure product with inversePermutations
Mp00127: Permutations left-to-right-maxima to Dyck pathDyck paths
St001200: Dyck paths ⟶ ℤResult quality: 5% values known / values provided: 5%distinct values known / distinct values provided: 20%
Values
[1,2,3] => [3,2,1] => [3,2,1] => [1,1,1,0,0,0]
=> ? = 3
[1,3,2] => [2,3,1] => [3,2,1] => [1,1,1,0,0,0]
=> ? = 2
[2,1,3] => [3,1,2] => [3,2,1] => [1,1,1,0,0,0]
=> ? = 2
[1,2,3,4] => [4,3,2,1] => [4,3,2,1] => [1,1,1,1,0,0,0,0]
=> ? = 2
[1,2,4,3] => [3,4,2,1] => [4,3,2,1] => [1,1,1,1,0,0,0,0]
=> ? = 2
[1,3,2,4] => [4,2,3,1] => [4,3,2,1] => [1,1,1,1,0,0,0,0]
=> ? = 2
[1,3,4,2] => [2,4,3,1] => [4,3,2,1] => [1,1,1,1,0,0,0,0]
=> ? = 2
[1,4,2,3] => [3,2,4,1] => [4,2,3,1] => [1,1,1,1,0,0,0,0]
=> ? = 2
[1,4,3,2] => [2,3,4,1] => [4,2,3,1] => [1,1,1,1,0,0,0,0]
=> ? = 2
[2,1,3,4] => [4,3,1,2] => [4,3,2,1] => [1,1,1,1,0,0,0,0]
=> ? = 3
[2,1,4,3] => [3,4,1,2] => [4,3,2,1] => [1,1,1,1,0,0,0,0]
=> ? = 2
[2,3,4,1] => [1,4,3,2] => [1,4,3,2] => [1,0,1,1,1,0,0,0]
=> 2
[2,4,1,3] => [3,1,4,2] => [4,2,3,1] => [1,1,1,1,0,0,0,0]
=> ? = 2
[3,1,4,2] => [2,4,1,3] => [3,4,1,2] => [1,1,1,0,1,0,0,0]
=> 2
[3,2,4,1] => [1,4,2,3] => [1,4,3,2] => [1,0,1,1,1,0,0,0]
=> 2
[4,1,2,3] => [3,2,1,4] => [3,2,1,4] => [1,1,1,0,0,0,1,0]
=> 2
[4,1,3,2] => [2,3,1,4] => [3,2,1,4] => [1,1,1,0,0,0,1,0]
=> 2
[1,2,3,4,5] => [5,4,3,2,1] => [5,4,3,2,1] => [1,1,1,1,1,0,0,0,0,0]
=> ? = 2
[1,2,3,5,4] => [4,5,3,2,1] => [5,4,3,2,1] => [1,1,1,1,1,0,0,0,0,0]
=> ? = 2
[1,2,4,3,5] => [5,3,4,2,1] => [5,4,3,2,1] => [1,1,1,1,1,0,0,0,0,0]
=> ? = 2
[1,2,4,5,3] => [3,5,4,2,1] => [5,4,3,2,1] => [1,1,1,1,1,0,0,0,0,0]
=> ? = 2
[1,2,5,3,4] => [4,3,5,2,1] => [5,4,3,2,1] => [1,1,1,1,1,0,0,0,0,0]
=> ? = 2
[1,2,5,4,3] => [3,4,5,2,1] => [5,4,3,2,1] => [1,1,1,1,1,0,0,0,0,0]
=> ? = 2
[1,3,2,4,5] => [5,4,2,3,1] => [5,4,3,2,1] => [1,1,1,1,1,0,0,0,0,0]
=> ? = 2
[1,3,2,5,4] => [4,5,2,3,1] => [5,4,3,2,1] => [1,1,1,1,1,0,0,0,0,0]
=> ? = 2
[1,3,4,5,2] => [2,5,4,3,1] => [5,4,3,2,1] => [1,1,1,1,1,0,0,0,0,0]
=> ? = 3
[1,3,5,2,4] => [4,2,5,3,1] => [5,4,3,2,1] => [1,1,1,1,1,0,0,0,0,0]
=> ? = 2
[1,3,5,4,2] => [2,4,5,3,1] => [5,4,3,2,1] => [1,1,1,1,1,0,0,0,0,0]
=> ? = 2
[1,4,2,5,3] => [3,5,2,4,1] => [5,4,3,2,1] => [1,1,1,1,1,0,0,0,0,0]
=> ? = 3
[1,4,3,5,2] => [2,5,3,4,1] => [5,4,3,2,1] => [1,1,1,1,1,0,0,0,0,0]
=> ? = 3
[1,4,5,2,3] => [3,2,5,4,1] => [5,2,4,3,1] => [1,1,1,1,1,0,0,0,0,0]
=> ? = 2
[1,4,5,3,2] => [2,3,5,4,1] => [5,2,4,3,1] => [1,1,1,1,1,0,0,0,0,0]
=> ? = 2
[1,5,2,3,4] => [4,3,2,5,1] => [5,3,2,4,1] => [1,1,1,1,1,0,0,0,0,0]
=> ? = 3
[1,5,2,4,3] => [3,4,2,5,1] => [5,3,2,4,1] => [1,1,1,1,1,0,0,0,0,0]
=> ? = 3
[1,5,3,2,4] => [4,2,3,5,1] => [5,3,2,4,1] => [1,1,1,1,1,0,0,0,0,0]
=> ? = 2
[1,5,3,4,2] => [2,4,3,5,1] => [5,3,2,4,1] => [1,1,1,1,1,0,0,0,0,0]
=> ? = 2
[1,5,4,2,3] => [3,2,4,5,1] => [5,2,3,4,1] => [1,1,1,1,1,0,0,0,0,0]
=> ? = 2
[1,5,4,3,2] => [2,3,4,5,1] => [5,2,3,4,1] => [1,1,1,1,1,0,0,0,0,0]
=> ? = 2
[2,1,3,4,5] => [5,4,3,1,2] => [5,4,3,2,1] => [1,1,1,1,1,0,0,0,0,0]
=> ? = 4
[2,1,3,5,4] => [4,5,3,1,2] => [5,4,3,2,1] => [1,1,1,1,1,0,0,0,0,0]
=> ? = 2
[2,1,4,3,5] => [5,3,4,1,2] => [5,4,3,2,1] => [1,1,1,1,1,0,0,0,0,0]
=> ? = 2
[2,1,4,5,3] => [3,5,4,1,2] => [5,4,3,2,1] => [1,1,1,1,1,0,0,0,0,0]
=> ? = 2
[2,1,5,3,4] => [4,3,5,1,2] => [5,4,3,2,1] => [1,1,1,1,1,0,0,0,0,0]
=> ? = 2
[2,1,5,4,3] => [3,4,5,1,2] => [5,4,3,2,1] => [1,1,1,1,1,0,0,0,0,0]
=> ? = 2
[2,3,4,1,5] => [5,1,4,3,2] => [5,2,4,3,1] => [1,1,1,1,1,0,0,0,0,0]
=> ? = 2
[2,3,4,5,1] => [1,5,4,3,2] => [1,5,4,3,2] => [1,0,1,1,1,1,0,0,0,0]
=> 2
[2,3,5,1,4] => [4,1,5,3,2] => [5,2,4,3,1] => [1,1,1,1,1,0,0,0,0,0]
=> ? = 2
[2,3,5,4,1] => [1,4,5,3,2] => [1,5,4,3,2] => [1,0,1,1,1,1,0,0,0,0]
=> 2
[2,4,1,3,5] => [5,3,1,4,2] => [5,4,3,2,1] => [1,1,1,1,1,0,0,0,0,0]
=> ? = 2
[2,4,1,5,3] => [3,5,1,4,2] => [5,4,3,2,1] => [1,1,1,1,1,0,0,0,0,0]
=> ? = 2
[2,4,3,5,1] => [1,5,3,4,2] => [1,5,4,3,2] => [1,0,1,1,1,1,0,0,0,0]
=> 2
[2,4,5,1,3] => [3,1,5,4,2] => [5,2,4,3,1] => [1,1,1,1,1,0,0,0,0,0]
=> ? = 2
[2,5,1,3,4] => [4,3,1,5,2] => [5,3,2,4,1] => [1,1,1,1,1,0,0,0,0,0]
=> ? = 2
[2,5,1,4,3] => [3,4,1,5,2] => [5,3,2,4,1] => [1,1,1,1,1,0,0,0,0,0]
=> ? = 2
[2,5,3,1,4] => [4,1,3,5,2] => [5,2,3,4,1] => [1,1,1,1,1,0,0,0,0,0]
=> ? = 2
[2,5,4,1,3] => [3,1,4,5,2] => [5,2,3,4,1] => [1,1,1,1,1,0,0,0,0,0]
=> ? = 2
[3,1,4,2,5] => [5,2,4,1,3] => [5,4,3,2,1] => [1,1,1,1,1,0,0,0,0,0]
=> ? = 2
[3,1,4,5,2] => [2,5,4,1,3] => [4,5,3,1,2] => [1,1,1,1,0,1,0,0,0,0]
=> 2
[3,1,5,2,4] => [4,2,5,1,3] => [5,4,3,2,1] => [1,1,1,1,1,0,0,0,0,0]
=> ? = 2
[3,1,5,4,2] => [2,4,5,1,3] => [4,5,3,1,2] => [1,1,1,1,0,1,0,0,0,0]
=> 2
[3,2,4,5,1] => [1,5,4,2,3] => [1,5,4,3,2] => [1,0,1,1,1,1,0,0,0,0]
=> 2
[3,2,5,4,1] => [1,4,5,2,3] => [1,5,4,3,2] => [1,0,1,1,1,1,0,0,0,0]
=> 2
[3,4,1,5,2] => [2,5,1,4,3] => [3,5,1,4,2] => [1,1,1,0,1,1,0,0,0,0]
=> 2
[3,4,2,5,1] => [1,5,2,4,3] => [1,5,3,4,2] => [1,0,1,1,1,1,0,0,0,0]
=> 2
[4,1,2,5,3] => [3,5,2,1,4] => [4,5,3,1,2] => [1,1,1,1,0,1,0,0,0,0]
=> 2
[4,1,3,5,2] => [2,5,3,1,4] => [4,5,3,1,2] => [1,1,1,1,0,1,0,0,0,0]
=> 2
[4,1,5,2,3] => [3,2,5,1,4] => [4,2,5,1,3] => [1,1,1,1,0,0,1,0,0,0]
=> 2
[4,1,5,3,2] => [2,3,5,1,4] => [4,2,5,1,3] => [1,1,1,1,0,0,1,0,0,0]
=> 2
[4,2,1,5,3] => [3,5,1,2,4] => [4,5,3,1,2] => [1,1,1,1,0,1,0,0,0,0]
=> 2
[4,2,3,5,1] => [1,5,3,2,4] => [1,5,4,3,2] => [1,0,1,1,1,1,0,0,0,0]
=> 2
[4,3,1,5,2] => [2,5,1,3,4] => [3,5,1,4,2] => [1,1,1,0,1,1,0,0,0,0]
=> 2
[4,3,2,5,1] => [1,5,2,3,4] => [1,5,3,4,2] => [1,0,1,1,1,1,0,0,0,0]
=> 2
[5,1,2,3,4] => [4,3,2,1,5] => [4,3,2,1,5] => [1,1,1,1,0,0,0,0,1,0]
=> 2
[5,1,2,4,3] => [3,4,2,1,5] => [4,3,2,1,5] => [1,1,1,1,0,0,0,0,1,0]
=> 2
[5,1,3,2,4] => [4,2,3,1,5] => [4,3,2,1,5] => [1,1,1,1,0,0,0,0,1,0]
=> 2
[5,1,3,4,2] => [2,4,3,1,5] => [4,3,2,1,5] => [1,1,1,1,0,0,0,0,1,0]
=> 2
[5,1,4,2,3] => [3,2,4,1,5] => [4,2,3,1,5] => [1,1,1,1,0,0,0,0,1,0]
=> 2
[5,1,4,3,2] => [2,3,4,1,5] => [4,2,3,1,5] => [1,1,1,1,0,0,0,0,1,0]
=> 2
[5,2,1,3,4] => [4,3,1,2,5] => [4,3,2,1,5] => [1,1,1,1,0,0,0,0,1,0]
=> 2
[5,2,1,4,3] => [3,4,1,2,5] => [4,3,2,1,5] => [1,1,1,1,0,0,0,0,1,0]
=> 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 $eA$.
Matching statistic: St000264
Mp00068: Permutations Simion-Schmidt mapPermutations
Mp00223: Permutations runsortPermutations
Mp00160: Permutations graph of inversionsGraphs
St000264: Graphs ⟶ ℤResult quality: 5% values known / values provided: 5%distinct values known / distinct values provided: 20%
Values
[1,2,3] => [1,3,2] => [1,3,2] => ([(1,2)],3)
=> ? = 3 + 2
[1,3,2] => [1,3,2] => [1,3,2] => ([(1,2)],3)
=> ? = 2 + 2
[2,1,3] => [2,1,3] => [1,3,2] => ([(1,2)],3)
=> ? = 2 + 2
[1,2,3,4] => [1,4,3,2] => [1,4,2,3] => ([(1,3),(2,3)],4)
=> ? = 2 + 2
[1,2,4,3] => [1,4,3,2] => [1,4,2,3] => ([(1,3),(2,3)],4)
=> ? = 2 + 2
[1,3,2,4] => [1,4,3,2] => [1,4,2,3] => ([(1,3),(2,3)],4)
=> ? = 2 + 2
[1,3,4,2] => [1,4,3,2] => [1,4,2,3] => ([(1,3),(2,3)],4)
=> ? = 2 + 2
[1,4,2,3] => [1,4,3,2] => [1,4,2,3] => ([(1,3),(2,3)],4)
=> ? = 2 + 2
[1,4,3,2] => [1,4,3,2] => [1,4,2,3] => ([(1,3),(2,3)],4)
=> ? = 2 + 2
[2,1,3,4] => [2,1,4,3] => [1,4,2,3] => ([(1,3),(2,3)],4)
=> ? = 3 + 2
[2,1,4,3] => [2,1,4,3] => [1,4,2,3] => ([(1,3),(2,3)],4)
=> ? = 2 + 2
[2,3,4,1] => [2,4,3,1] => [1,2,4,3] => ([(2,3)],4)
=> ? = 2 + 2
[2,4,1,3] => [2,4,1,3] => [1,3,2,4] => ([(2,3)],4)
=> ? = 2 + 2
[3,1,4,2] => [3,1,4,2] => [1,4,2,3] => ([(1,3),(2,3)],4)
=> ? = 2 + 2
[3,2,4,1] => [3,2,4,1] => [1,2,4,3] => ([(2,3)],4)
=> ? = 2 + 2
[4,1,2,3] => [4,1,3,2] => [1,3,2,4] => ([(2,3)],4)
=> ? = 2 + 2
[4,1,3,2] => [4,1,3,2] => [1,3,2,4] => ([(2,3)],4)
=> ? = 2 + 2
[1,2,3,4,5] => [1,5,4,3,2] => [1,5,2,3,4] => ([(1,4),(2,4),(3,4)],5)
=> ? = 2 + 2
[1,2,3,5,4] => [1,5,4,3,2] => [1,5,2,3,4] => ([(1,4),(2,4),(3,4)],5)
=> ? = 2 + 2
[1,2,4,3,5] => [1,5,4,3,2] => [1,5,2,3,4] => ([(1,4),(2,4),(3,4)],5)
=> ? = 2 + 2
[1,2,4,5,3] => [1,5,4,3,2] => [1,5,2,3,4] => ([(1,4),(2,4),(3,4)],5)
=> ? = 2 + 2
[1,2,5,3,4] => [1,5,4,3,2] => [1,5,2,3,4] => ([(1,4),(2,4),(3,4)],5)
=> ? = 2 + 2
[1,2,5,4,3] => [1,5,4,3,2] => [1,5,2,3,4] => ([(1,4),(2,4),(3,4)],5)
=> ? = 2 + 2
[1,3,2,4,5] => [1,5,4,3,2] => [1,5,2,3,4] => ([(1,4),(2,4),(3,4)],5)
=> ? = 2 + 2
[1,3,2,5,4] => [1,5,4,3,2] => [1,5,2,3,4] => ([(1,4),(2,4),(3,4)],5)
=> ? = 2 + 2
[1,3,4,5,2] => [1,5,4,3,2] => [1,5,2,3,4] => ([(1,4),(2,4),(3,4)],5)
=> ? = 3 + 2
[1,3,5,2,4] => [1,5,4,3,2] => [1,5,2,3,4] => ([(1,4),(2,4),(3,4)],5)
=> ? = 2 + 2
[1,3,5,4,2] => [1,5,4,3,2] => [1,5,2,3,4] => ([(1,4),(2,4),(3,4)],5)
=> ? = 2 + 2
[1,4,2,5,3] => [1,5,4,3,2] => [1,5,2,3,4] => ([(1,4),(2,4),(3,4)],5)
=> ? = 3 + 2
[1,4,3,5,2] => [1,5,4,3,2] => [1,5,2,3,4] => ([(1,4),(2,4),(3,4)],5)
=> ? = 3 + 2
[1,4,5,2,3] => [1,5,4,3,2] => [1,5,2,3,4] => ([(1,4),(2,4),(3,4)],5)
=> ? = 2 + 2
[1,4,5,3,2] => [1,5,4,3,2] => [1,5,2,3,4] => ([(1,4),(2,4),(3,4)],5)
=> ? = 2 + 2
[1,5,2,3,4] => [1,5,4,3,2] => [1,5,2,3,4] => ([(1,4),(2,4),(3,4)],5)
=> ? = 3 + 2
[1,5,2,4,3] => [1,5,4,3,2] => [1,5,2,3,4] => ([(1,4),(2,4),(3,4)],5)
=> ? = 3 + 2
[1,5,3,2,4] => [1,5,4,3,2] => [1,5,2,3,4] => ([(1,4),(2,4),(3,4)],5)
=> ? = 2 + 2
[1,5,3,4,2] => [1,5,4,3,2] => [1,5,2,3,4] => ([(1,4),(2,4),(3,4)],5)
=> ? = 2 + 2
[1,5,4,2,3] => [1,5,4,3,2] => [1,5,2,3,4] => ([(1,4),(2,4),(3,4)],5)
=> ? = 2 + 2
[1,5,4,3,2] => [1,5,4,3,2] => [1,5,2,3,4] => ([(1,4),(2,4),(3,4)],5)
=> ? = 2 + 2
[2,1,3,4,5] => [2,1,5,4,3] => [1,5,2,3,4] => ([(1,4),(2,4),(3,4)],5)
=> ? = 4 + 2
[2,1,3,5,4] => [2,1,5,4,3] => [1,5,2,3,4] => ([(1,4),(2,4),(3,4)],5)
=> ? = 2 + 2
[2,1,4,3,5] => [2,1,5,4,3] => [1,5,2,3,4] => ([(1,4),(2,4),(3,4)],5)
=> ? = 2 + 2
[2,1,4,5,3] => [2,1,5,4,3] => [1,5,2,3,4] => ([(1,4),(2,4),(3,4)],5)
=> ? = 2 + 2
[2,1,5,3,4] => [2,1,5,4,3] => [1,5,2,3,4] => ([(1,4),(2,4),(3,4)],5)
=> ? = 2 + 2
[2,1,5,4,3] => [2,1,5,4,3] => [1,5,2,3,4] => ([(1,4),(2,4),(3,4)],5)
=> ? = 2 + 2
[2,3,4,1,5] => [2,5,4,1,3] => [1,3,2,5,4] => ([(1,4),(2,3)],5)
=> ? = 2 + 2
[2,3,4,5,1] => [2,5,4,3,1] => [1,2,5,3,4] => ([(2,4),(3,4)],5)
=> ? = 2 + 2
[2,3,5,1,4] => [2,5,4,1,3] => [1,3,2,5,4] => ([(1,4),(2,3)],5)
=> ? = 2 + 2
[2,3,5,4,1] => [2,5,4,3,1] => [1,2,5,3,4] => ([(2,4),(3,4)],5)
=> ? = 2 + 2
[2,4,1,3,5] => [2,5,1,4,3] => [1,4,2,5,3] => ([(1,4),(2,3),(3,4)],5)
=> ? = 2 + 2
[2,4,1,5,3] => [2,5,1,4,3] => [1,4,2,5,3] => ([(1,4),(2,3),(3,4)],5)
=> ? = 2 + 2
[2,4,1,3,5,6] => [2,6,1,5,4,3] => [1,5,2,6,3,4] => ([(1,5),(2,3),(2,4),(3,5),(4,5)],6)
=> 4 = 2 + 2
[2,4,1,3,6,5] => [2,6,1,5,4,3] => [1,5,2,6,3,4] => ([(1,5),(2,3),(2,4),(3,5),(4,5)],6)
=> 4 = 2 + 2
[2,4,1,5,3,6] => [2,6,1,5,4,3] => [1,5,2,6,3,4] => ([(1,5),(2,3),(2,4),(3,5),(4,5)],6)
=> 4 = 2 + 2
[2,4,1,5,6,3] => [2,6,1,5,4,3] => [1,5,2,6,3,4] => ([(1,5),(2,3),(2,4),(3,5),(4,5)],6)
=> 4 = 2 + 2
[2,4,1,6,5,3] => [2,6,1,5,4,3] => [1,5,2,6,3,4] => ([(1,5),(2,3),(2,4),(3,5),(4,5)],6)
=> 4 = 2 + 2
[2,5,1,3,4,6] => [2,6,1,5,4,3] => [1,5,2,6,3,4] => ([(1,5),(2,3),(2,4),(3,5),(4,5)],6)
=> 4 = 2 + 2
[2,5,1,3,6,4] => [2,6,1,5,4,3] => [1,5,2,6,3,4] => ([(1,5),(2,3),(2,4),(3,5),(4,5)],6)
=> 4 = 2 + 2
[2,5,1,4,3,6] => [2,6,1,5,4,3] => [1,5,2,6,3,4] => ([(1,5),(2,3),(2,4),(3,5),(4,5)],6)
=> 4 = 2 + 2
[2,5,1,4,6,3] => [2,6,1,5,4,3] => [1,5,2,6,3,4] => ([(1,5),(2,3),(2,4),(3,5),(4,5)],6)
=> 4 = 2 + 2
[2,5,1,6,3,4] => [2,6,1,5,4,3] => [1,5,2,6,3,4] => ([(1,5),(2,3),(2,4),(3,5),(4,5)],6)
=> 4 = 2 + 2
[2,5,1,6,4,3] => [2,6,1,5,4,3] => [1,5,2,6,3,4] => ([(1,5),(2,3),(2,4),(3,5),(4,5)],6)
=> 4 = 2 + 2
[2,6,1,3,4,5] => [2,6,1,5,4,3] => [1,5,2,6,3,4] => ([(1,5),(2,3),(2,4),(3,5),(4,5)],6)
=> 4 = 2 + 2
[2,6,1,3,5,4] => [2,6,1,5,4,3] => [1,5,2,6,3,4] => ([(1,5),(2,3),(2,4),(3,5),(4,5)],6)
=> 4 = 2 + 2
[2,6,1,4,3,5] => [2,6,1,5,4,3] => [1,5,2,6,3,4] => ([(1,5),(2,3),(2,4),(3,5),(4,5)],6)
=> 4 = 2 + 2
[2,6,1,4,5,3] => [2,6,1,5,4,3] => [1,5,2,6,3,4] => ([(1,5),(2,3),(2,4),(3,5),(4,5)],6)
=> 4 = 2 + 2
[2,6,1,5,3,4] => [2,6,1,5,4,3] => [1,5,2,6,3,4] => ([(1,5),(2,3),(2,4),(3,5),(4,5)],6)
=> 4 = 2 + 2
[2,6,1,5,4,3] => [2,6,1,5,4,3] => [1,5,2,6,3,4] => ([(1,5),(2,3),(2,4),(3,5),(4,5)],6)
=> 4 = 2 + 2
[3,2,4,1,5,6] => [3,2,6,1,5,4] => [1,5,2,6,3,4] => ([(1,5),(2,3),(2,4),(3,5),(4,5)],6)
=> 4 = 2 + 2
[3,2,4,1,6,5] => [3,2,6,1,5,4] => [1,5,2,6,3,4] => ([(1,5),(2,3),(2,4),(3,5),(4,5)],6)
=> 4 = 2 + 2
[3,2,5,1,4,6] => [3,2,6,1,5,4] => [1,5,2,6,3,4] => ([(1,5),(2,3),(2,4),(3,5),(4,5)],6)
=> 4 = 2 + 2
[3,2,5,1,6,4] => [3,2,6,1,5,4] => [1,5,2,6,3,4] => ([(1,5),(2,3),(2,4),(3,5),(4,5)],6)
=> 4 = 2 + 2
[3,2,6,1,4,5] => [3,2,6,1,5,4] => [1,5,2,6,3,4] => ([(1,5),(2,3),(2,4),(3,5),(4,5)],6)
=> 4 = 2 + 2
[3,2,6,1,5,4] => [3,2,6,1,5,4] => [1,5,2,6,3,4] => ([(1,5),(2,3),(2,4),(3,5),(4,5)],6)
=> 4 = 2 + 2
[4,2,5,1,6,3] => [4,2,6,1,5,3] => [1,5,2,6,3,4] => ([(1,5),(2,3),(2,4),(3,5),(4,5)],6)
=> 4 = 2 + 2
[4,2,6,1,3,5] => [4,2,6,1,5,3] => [1,5,2,6,3,4] => ([(1,5),(2,3),(2,4),(3,5),(4,5)],6)
=> 4 = 2 + 2
[4,2,6,1,5,3] => [4,2,6,1,5,3] => [1,5,2,6,3,4] => ([(1,5),(2,3),(2,4),(3,5),(4,5)],6)
=> 4 = 2 + 2
[4,3,2,5,1,6] => [4,3,2,6,1,5] => [1,5,2,6,3,4] => ([(1,5),(2,3),(2,4),(3,5),(4,5)],6)
=> 4 = 2 + 2
[4,3,2,6,1,5] => [4,3,2,6,1,5] => [1,5,2,6,3,4] => ([(1,5),(2,3),(2,4),(3,5),(4,5)],6)
=> 4 = 2 + 2
Description
The girth of a graph, which is not a tree. This is the length of the shortest cycle in the graph.
Mp00064: Permutations reversePermutations
Mp00114: Permutations connectivity setBinary words
St001491: Binary words ⟶ ℤResult quality: 3% values known / values provided: 3%distinct values known / distinct values provided: 20%
Values
[1,2,3] => [3,2,1] => 00 => ? = 3 - 1
[1,3,2] => [2,3,1] => 00 => ? = 2 - 1
[2,1,3] => [3,1,2] => 00 => ? = 2 - 1
[1,2,3,4] => [4,3,2,1] => 000 => ? = 2 - 1
[1,2,4,3] => [3,4,2,1] => 000 => ? = 2 - 1
[1,3,2,4] => [4,2,3,1] => 000 => ? = 2 - 1
[1,3,4,2] => [2,4,3,1] => 000 => ? = 2 - 1
[1,4,2,3] => [3,2,4,1] => 000 => ? = 2 - 1
[1,4,3,2] => [2,3,4,1] => 000 => ? = 2 - 1
[2,1,3,4] => [4,3,1,2] => 000 => ? = 3 - 1
[2,1,4,3] => [3,4,1,2] => 000 => ? = 2 - 1
[2,3,4,1] => [1,4,3,2] => 100 => 1 = 2 - 1
[2,4,1,3] => [3,1,4,2] => 000 => ? = 2 - 1
[3,1,4,2] => [2,4,1,3] => 000 => ? = 2 - 1
[3,2,4,1] => [1,4,2,3] => 100 => 1 = 2 - 1
[4,1,2,3] => [3,2,1,4] => 001 => 1 = 2 - 1
[4,1,3,2] => [2,3,1,4] => 001 => 1 = 2 - 1
[1,2,3,4,5] => [5,4,3,2,1] => 0000 => ? = 2 - 1
[1,2,3,5,4] => [4,5,3,2,1] => 0000 => ? = 2 - 1
[1,2,4,3,5] => [5,3,4,2,1] => 0000 => ? = 2 - 1
[1,2,4,5,3] => [3,5,4,2,1] => 0000 => ? = 2 - 1
[1,2,5,3,4] => [4,3,5,2,1] => 0000 => ? = 2 - 1
[1,2,5,4,3] => [3,4,5,2,1] => 0000 => ? = 2 - 1
[1,3,2,4,5] => [5,4,2,3,1] => 0000 => ? = 2 - 1
[1,3,2,5,4] => [4,5,2,3,1] => 0000 => ? = 2 - 1
[1,3,4,5,2] => [2,5,4,3,1] => 0000 => ? = 3 - 1
[1,3,5,2,4] => [4,2,5,3,1] => 0000 => ? = 2 - 1
[1,3,5,4,2] => [2,4,5,3,1] => 0000 => ? = 2 - 1
[1,4,2,5,3] => [3,5,2,4,1] => 0000 => ? = 3 - 1
[1,4,3,5,2] => [2,5,3,4,1] => 0000 => ? = 3 - 1
[1,4,5,2,3] => [3,2,5,4,1] => 0000 => ? = 2 - 1
[1,4,5,3,2] => [2,3,5,4,1] => 0000 => ? = 2 - 1
[1,5,2,3,4] => [4,3,2,5,1] => 0000 => ? = 3 - 1
[1,5,2,4,3] => [3,4,2,5,1] => 0000 => ? = 3 - 1
[1,5,3,2,4] => [4,2,3,5,1] => 0000 => ? = 2 - 1
[1,5,3,4,2] => [2,4,3,5,1] => 0000 => ? = 2 - 1
[1,5,4,2,3] => [3,2,4,5,1] => 0000 => ? = 2 - 1
[1,5,4,3,2] => [2,3,4,5,1] => 0000 => ? = 2 - 1
[2,1,3,4,5] => [5,4,3,1,2] => 0000 => ? = 4 - 1
[2,1,3,5,4] => [4,5,3,1,2] => 0000 => ? = 2 - 1
[2,1,4,3,5] => [5,3,4,1,2] => 0000 => ? = 2 - 1
[2,1,4,5,3] => [3,5,4,1,2] => 0000 => ? = 2 - 1
[2,1,5,3,4] => [4,3,5,1,2] => 0000 => ? = 2 - 1
[2,1,5,4,3] => [3,4,5,1,2] => 0000 => ? = 2 - 1
[2,3,4,1,5] => [5,1,4,3,2] => 0000 => ? = 2 - 1
[2,3,4,5,1] => [1,5,4,3,2] => 1000 => 1 = 2 - 1
[2,3,5,1,4] => [4,1,5,3,2] => 0000 => ? = 2 - 1
[2,3,5,4,1] => [1,4,5,3,2] => 1000 => 1 = 2 - 1
[2,4,1,3,5] => [5,3,1,4,2] => 0000 => ? = 2 - 1
[2,4,1,5,3] => [3,5,1,4,2] => 0000 => ? = 2 - 1
[2,4,3,5,1] => [1,5,3,4,2] => 1000 => 1 = 2 - 1
[2,4,5,1,3] => [3,1,5,4,2] => 0000 => ? = 2 - 1
[2,5,1,3,4] => [4,3,1,5,2] => 0000 => ? = 2 - 1
[2,5,1,4,3] => [3,4,1,5,2] => 0000 => ? = 2 - 1
[2,5,3,1,4] => [4,1,3,5,2] => 0000 => ? = 2 - 1
[2,5,4,1,3] => [3,1,4,5,2] => 0000 => ? = 2 - 1
[3,1,4,2,5] => [5,2,4,1,3] => 0000 => ? = 2 - 1
[3,2,4,5,1] => [1,5,4,2,3] => 1000 => 1 = 2 - 1
[3,2,5,4,1] => [1,4,5,2,3] => 1000 => 1 = 2 - 1
[3,4,2,5,1] => [1,5,2,4,3] => 1000 => 1 = 2 - 1
[4,2,3,5,1] => [1,5,3,2,4] => 1000 => 1 = 2 - 1
[4,3,2,5,1] => [1,5,2,3,4] => 1000 => 1 = 2 - 1
[5,1,2,3,4] => [4,3,2,1,5] => 0001 => 1 = 2 - 1
[5,1,2,4,3] => [3,4,2,1,5] => 0001 => 1 = 2 - 1
[5,1,3,2,4] => [4,2,3,1,5] => 0001 => 1 = 2 - 1
[5,1,3,4,2] => [2,4,3,1,5] => 0001 => 1 = 2 - 1
[5,1,4,2,3] => [3,2,4,1,5] => 0001 => 1 = 2 - 1
[5,1,4,3,2] => [2,3,4,1,5] => 0001 => 1 = 2 - 1
[5,2,1,3,4] => [4,3,1,2,5] => 0001 => 1 = 2 - 1
[5,2,1,4,3] => [3,4,1,2,5] => 0001 => 1 = 2 - 1
Description
The number of indecomposable projective-injective modules in the algebra corresponding to a subset. Let $A_n=K[x]/(x^n)$. We associate to a nonempty subset S of an (n-1)-set the module $M_S$, which is the direct sum of $A_n$-modules with indecomposable non-projective direct summands of dimension $i$ when $i$ is in $S$ (note that such modules have vector space dimension at most n-1). Then the corresponding algebra associated to S is the stable endomorphism ring of $M_S$. We decode the subset as a binary word so that for example the subset $S=\{1,3 \} $ of $\{1,2,3 \}$ is decoded as 101.