Processing math: 100%

Your data matches 15 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)
Matching statistic: St001116
Mp00065: Permutations permutation posetPosets
Mp00074: Posets to graphGraphs
St001116: Graphs ⟶ ℤResult quality: 100% values known / values provided: 100%distinct values known / distinct values provided: 100%
Values
[1] => ([],1)
=> ([],1)
=> 1 = 0 + 1
[1,2] => ([(0,1)],2)
=> ([(0,1)],2)
=> 2 = 1 + 1
[1,2,3] => ([(0,2),(2,1)],3)
=> ([(0,2),(1,2)],3)
=> 2 = 1 + 1
[1,3,2] => ([(0,1),(0,2)],3)
=> ([(0,2),(1,2)],3)
=> 2 = 1 + 1
[2,1,3] => ([(0,2),(1,2)],3)
=> ([(0,2),(1,2)],3)
=> 2 = 1 + 1
[1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> 3 = 2 + 1
[1,2,4,3] => ([(0,3),(3,1),(3,2)],4)
=> ([(0,3),(1,3),(2,3)],4)
=> 2 = 1 + 1
[1,3,2,4] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(0,2),(0,3),(1,2),(1,3)],4)
=> 3 = 2 + 1
[1,3,4,2] => ([(0,2),(0,3),(3,1)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> 3 = 2 + 1
[1,4,2,3] => ([(0,2),(0,3),(3,1)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> 3 = 2 + 1
[1,4,3,2] => ([(0,1),(0,2),(0,3)],4)
=> ([(0,3),(1,3),(2,3)],4)
=> 2 = 1 + 1
[2,1,3,4] => ([(0,3),(1,3),(3,2)],4)
=> ([(0,3),(1,3),(2,3)],4)
=> 2 = 1 + 1
[2,1,4,3] => ([(0,2),(0,3),(1,2),(1,3)],4)
=> ([(0,2),(0,3),(1,2),(1,3)],4)
=> 3 = 2 + 1
[2,3,1,4] => ([(0,3),(1,2),(2,3)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> 3 = 2 + 1
[2,4,1,3] => ([(0,3),(1,2),(1,3)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> 3 = 2 + 1
[3,1,2,4] => ([(0,3),(1,2),(2,3)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> 3 = 2 + 1
[3,1,4,2] => ([(0,3),(1,2),(1,3)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> 3 = 2 + 1
[3,2,1,4] => ([(0,3),(1,3),(2,3)],4)
=> ([(0,3),(1,3),(2,3)],4)
=> 2 = 1 + 1
[1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 3 = 2 + 1
[1,2,3,5,4] => ([(0,3),(3,4),(4,1),(4,2)],5)
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> 3 = 2 + 1
[1,2,4,3,5] => ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> 3 = 2 + 1
[1,2,4,5,3] => ([(0,4),(3,2),(4,1),(4,3)],5)
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> 3 = 2 + 1
[1,2,5,3,4] => ([(0,4),(3,2),(4,1),(4,3)],5)
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> 3 = 2 + 1
[1,2,5,4,3] => ([(0,4),(4,1),(4,2),(4,3)],5)
=> ([(0,4),(1,4),(2,4),(3,4)],5)
=> 2 = 1 + 1
[1,3,2,4,5] => ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> 3 = 2 + 1
[1,3,2,5,4] => ([(0,1),(0,2),(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> 3 = 2 + 1
[1,3,4,2,5] => ([(0,2),(0,3),(1,4),(2,4),(3,1)],5)
=> ([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
=> 3 = 2 + 1
[1,3,4,5,2] => ([(0,2),(0,4),(3,1),(4,3)],5)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 3 = 2 + 1
[1,3,5,2,4] => ([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
=> ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> 3 = 2 + 1
[1,3,5,4,2] => ([(0,3),(0,4),(4,1),(4,2)],5)
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> 3 = 2 + 1
[1,4,2,3,5] => ([(0,2),(0,3),(1,4),(2,4),(3,1)],5)
=> ([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
=> 3 = 2 + 1
[1,4,2,5,3] => ([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
=> ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> 3 = 2 + 1
[1,4,3,2,5] => ([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
=> ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> 3 = 2 + 1
[1,4,3,5,2] => ([(0,1),(0,2),(0,3),(2,4),(3,4)],5)
=> ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> 3 = 2 + 1
[1,4,5,2,3] => ([(0,3),(0,4),(3,2),(4,1)],5)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 3 = 2 + 1
[1,4,5,3,2] => ([(0,2),(0,3),(0,4),(4,1)],5)
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> 3 = 2 + 1
[1,5,2,3,4] => ([(0,2),(0,4),(3,1),(4,3)],5)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 3 = 2 + 1
[1,5,2,4,3] => ([(0,3),(0,4),(4,1),(4,2)],5)
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> 3 = 2 + 1
[1,5,3,2,4] => ([(0,1),(0,2),(0,3),(2,4),(3,4)],5)
=> ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> 3 = 2 + 1
[1,5,3,4,2] => ([(0,2),(0,3),(0,4),(4,1)],5)
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> 3 = 2 + 1
[1,5,4,2,3] => ([(0,2),(0,3),(0,4),(4,1)],5)
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> 3 = 2 + 1
[1,5,4,3,2] => ([(0,1),(0,2),(0,3),(0,4)],5)
=> ([(0,4),(1,4),(2,4),(3,4)],5)
=> 2 = 1 + 1
[2,1,3,4,5] => ([(0,4),(1,4),(2,3),(4,2)],5)
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> 3 = 2 + 1
[2,1,3,5,4] => ([(0,4),(1,4),(4,2),(4,3)],5)
=> ([(0,4),(1,4),(2,4),(3,4)],5)
=> 2 = 1 + 1
[2,1,4,3,5] => ([(0,3),(0,4),(1,3),(1,4),(3,2),(4,2)],5)
=> ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> 3 = 2 + 1
[2,1,4,5,3] => ([(0,3),(0,4),(1,3),(1,4),(4,2)],5)
=> ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> 3 = 2 + 1
[2,1,5,3,4] => ([(0,3),(0,4),(1,3),(1,4),(4,2)],5)
=> ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> 3 = 2 + 1
[2,1,5,4,3] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4)],5)
=> ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> 3 = 2 + 1
[2,3,1,4,5] => ([(0,4),(1,2),(2,4),(4,3)],5)
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> 3 = 2 + 1
[2,3,1,5,4] => ([(0,3),(0,4),(1,2),(2,3),(2,4)],5)
=> ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> 3 = 2 + 1
Description
The game chromatic number of a graph. Two players, Alice and Bob, take turns colouring properly any uncolored vertex of the graph. Alice begins. If it is not possible for either player to colour a vertex, then Bob wins. If the graph is completely colored, Alice wins. The game chromatic number is the smallest number of colours such that Alice has a winning strategy.
Mp00065: Permutations permutation posetPosets
Mp00074: Posets to graphGraphs
Mp00157: Graphs connected complementGraphs
St000260: Graphs ⟶ ℤResult quality: 100% values known / values provided: 100%distinct values known / distinct values provided: 100%
Values
[1] => ([],1)
=> ([],1)
=> ([],1)
=> 0
[1,2] => ([(0,1)],2)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 1
[1,2,3] => ([(0,2),(2,1)],3)
=> ([(0,2),(1,2)],3)
=> ([(0,2),(1,2)],3)
=> 1
[1,3,2] => ([(0,1),(0,2)],3)
=> ([(0,2),(1,2)],3)
=> ([(0,2),(1,2)],3)
=> 1
[2,1,3] => ([(0,2),(1,2)],3)
=> ([(0,2),(1,2)],3)
=> ([(0,2),(1,2)],3)
=> 1
[1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> 2
[1,2,4,3] => ([(0,3),(3,1),(3,2)],4)
=> ([(0,3),(1,3),(2,3)],4)
=> ([(0,3),(1,3),(2,3)],4)
=> 1
[1,3,2,4] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(0,2),(0,3),(1,2),(1,3)],4)
=> ([(0,2),(0,3),(1,2),(1,3)],4)
=> 2
[1,3,4,2] => ([(0,2),(0,3),(3,1)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> 2
[1,4,2,3] => ([(0,2),(0,3),(3,1)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> 2
[1,4,3,2] => ([(0,1),(0,2),(0,3)],4)
=> ([(0,3),(1,3),(2,3)],4)
=> ([(0,3),(1,3),(2,3)],4)
=> 1
[2,1,3,4] => ([(0,3),(1,3),(3,2)],4)
=> ([(0,3),(1,3),(2,3)],4)
=> ([(0,3),(1,3),(2,3)],4)
=> 1
[2,1,4,3] => ([(0,2),(0,3),(1,2),(1,3)],4)
=> ([(0,2),(0,3),(1,2),(1,3)],4)
=> ([(0,2),(0,3),(1,2),(1,3)],4)
=> 2
[2,3,1,4] => ([(0,3),(1,2),(2,3)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> 2
[2,4,1,3] => ([(0,3),(1,2),(1,3)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> 2
[3,1,2,4] => ([(0,3),(1,2),(2,3)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> 2
[3,1,4,2] => ([(0,3),(1,2),(1,3)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> 2
[3,2,1,4] => ([(0,3),(1,3),(2,3)],4)
=> ([(0,3),(1,3),(2,3)],4)
=> ([(0,3),(1,3),(2,3)],4)
=> 1
[1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> 2
[1,2,3,5,4] => ([(0,3),(3,4),(4,1),(4,2)],5)
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> ([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
=> 2
[1,2,4,3,5] => ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> ([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> 2
[1,2,4,5,3] => ([(0,4),(3,2),(4,1),(4,3)],5)
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> ([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
=> 2
[1,2,5,3,4] => ([(0,4),(3,2),(4,1),(4,3)],5)
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> ([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
=> 2
[1,2,5,4,3] => ([(0,4),(4,1),(4,2),(4,3)],5)
=> ([(0,4),(1,4),(2,4),(3,4)],5)
=> ([(0,4),(1,4),(2,4),(3,4)],5)
=> 1
[1,3,2,4,5] => ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> ([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> 2
[1,3,2,5,4] => ([(0,1),(0,2),(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> 2
[1,3,4,2,5] => ([(0,2),(0,3),(1,4),(2,4),(3,1)],5)
=> ([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
=> ([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
=> 2
[1,3,4,5,2] => ([(0,2),(0,4),(3,1),(4,3)],5)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> 2
[1,3,5,2,4] => ([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
=> ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> ([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> 2
[1,3,5,4,2] => ([(0,3),(0,4),(4,1),(4,2)],5)
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> ([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
=> 2
[1,4,2,3,5] => ([(0,2),(0,3),(1,4),(2,4),(3,1)],5)
=> ([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
=> ([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
=> 2
[1,4,2,5,3] => ([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
=> ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> ([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> 2
[1,4,3,2,5] => ([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
=> ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> 2
[1,4,3,5,2] => ([(0,1),(0,2),(0,3),(2,4),(3,4)],5)
=> ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> ([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> 2
[1,4,5,2,3] => ([(0,3),(0,4),(3,2),(4,1)],5)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> 2
[1,4,5,3,2] => ([(0,2),(0,3),(0,4),(4,1)],5)
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> ([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
=> 2
[1,5,2,3,4] => ([(0,2),(0,4),(3,1),(4,3)],5)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> 2
[1,5,2,4,3] => ([(0,3),(0,4),(4,1),(4,2)],5)
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> ([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
=> 2
[1,5,3,2,4] => ([(0,1),(0,2),(0,3),(2,4),(3,4)],5)
=> ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> ([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> 2
[1,5,3,4,2] => ([(0,2),(0,3),(0,4),(4,1)],5)
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> ([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
=> 2
[1,5,4,2,3] => ([(0,2),(0,3),(0,4),(4,1)],5)
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> ([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
=> 2
[1,5,4,3,2] => ([(0,1),(0,2),(0,3),(0,4)],5)
=> ([(0,4),(1,4),(2,4),(3,4)],5)
=> ([(0,4),(1,4),(2,4),(3,4)],5)
=> 1
[2,1,3,4,5] => ([(0,4),(1,4),(2,3),(4,2)],5)
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> ([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
=> 2
[2,1,3,5,4] => ([(0,4),(1,4),(4,2),(4,3)],5)
=> ([(0,4),(1,4),(2,4),(3,4)],5)
=> ([(0,4),(1,4),(2,4),(3,4)],5)
=> 1
[2,1,4,3,5] => ([(0,3),(0,4),(1,3),(1,4),(3,2),(4,2)],5)
=> ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> 2
[2,1,4,5,3] => ([(0,3),(0,4),(1,3),(1,4),(4,2)],5)
=> ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> ([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> 2
[2,1,5,3,4] => ([(0,3),(0,4),(1,3),(1,4),(4,2)],5)
=> ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> ([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> 2
[2,1,5,4,3] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4)],5)
=> ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> 2
[2,3,1,4,5] => ([(0,4),(1,2),(2,4),(4,3)],5)
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> ([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
=> 2
[2,3,1,5,4] => ([(0,3),(0,4),(1,2),(2,3),(2,4)],5)
=> ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> ([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> 2
Description
The radius of a connected graph. This is the minimum eccentricity of any vertex.
Mp00239: Permutations CorteelPermutations
Mp00248: Permutations DEX compositionInteger compositions
Mp00184: Integer compositions to threshold graphGraphs
St000264: Graphs ⟶ ℤResult quality: 33% values known / values provided: 80%distinct values known / distinct values provided: 33%
Values
[1] => [1] => [1] => ([],1)
=> ? = 0 + 1
[1,2] => [1,2] => [2] => ([],2)
=> ? = 1 + 1
[1,2,3] => [1,2,3] => [3] => ([],3)
=> ? = 1 + 1
[1,3,2] => [1,3,2] => [1,2] => ([(1,2)],3)
=> ? = 1 + 1
[2,1,3] => [2,1,3] => [3] => ([],3)
=> ? = 1 + 1
[1,2,3,4] => [1,2,3,4] => [4] => ([],4)
=> ? = 2 + 1
[1,2,4,3] => [1,2,4,3] => [2,2] => ([(1,3),(2,3)],4)
=> ? = 1 + 1
[1,3,2,4] => [1,3,2,4] => [1,3] => ([(2,3)],4)
=> ? = 2 + 1
[1,3,4,2] => [1,4,3,2] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 3 = 2 + 1
[1,4,2,3] => [1,4,2,3] => [1,3] => ([(2,3)],4)
=> ? = 2 + 1
[1,4,3,2] => [1,3,4,2] => [1,3] => ([(2,3)],4)
=> ? = 1 + 1
[2,1,3,4] => [2,1,3,4] => [4] => ([],4)
=> ? = 1 + 1
[2,1,4,3] => [2,1,4,3] => [2,2] => ([(1,3),(2,3)],4)
=> ? = 2 + 1
[2,3,1,4] => [3,2,1,4] => [2,2] => ([(1,3),(2,3)],4)
=> ? = 2 + 1
[2,4,1,3] => [4,2,1,3] => [2,2] => ([(1,3),(2,3)],4)
=> ? = 2 + 1
[3,1,2,4] => [3,1,2,4] => [4] => ([],4)
=> ? = 2 + 1
[3,1,4,2] => [4,1,3,2] => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> ? = 2 + 1
[3,2,1,4] => [2,3,1,4] => [4] => ([],4)
=> ? = 1 + 1
[1,2,3,4,5] => [1,2,3,4,5] => [5] => ([],5)
=> ? = 2 + 1
[1,2,3,5,4] => [1,2,3,5,4] => [3,2] => ([(1,4),(2,4),(3,4)],5)
=> ? = 2 + 1
[1,2,4,3,5] => [1,2,4,3,5] => [2,3] => ([(2,4),(3,4)],5)
=> ? = 2 + 1
[1,2,4,5,3] => [1,2,5,4,3] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = 2 + 1
[1,2,5,3,4] => [1,2,5,3,4] => [2,3] => ([(2,4),(3,4)],5)
=> ? = 2 + 1
[1,2,5,4,3] => [1,2,4,5,3] => [2,3] => ([(2,4),(3,4)],5)
=> ? = 1 + 1
[1,3,2,4,5] => [1,3,2,4,5] => [1,4] => ([(3,4)],5)
=> ? = 2 + 1
[1,3,2,5,4] => [1,3,2,5,4] => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = 2 + 1
[1,3,4,2,5] => [1,4,3,2,5] => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = 2 + 1
[1,3,4,5,2] => [1,5,3,4,2] => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = 2 + 1
[1,3,5,2,4] => [1,5,3,2,4] => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = 2 + 1
[1,3,5,4,2] => [1,4,3,5,2] => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = 2 + 1
[1,4,2,3,5] => [1,4,2,3,5] => [1,4] => ([(3,4)],5)
=> ? = 2 + 1
[1,4,2,5,3] => [1,5,2,4,3] => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = 2 + 1
[1,4,3,2,5] => [1,3,4,2,5] => [1,4] => ([(3,4)],5)
=> ? = 2 + 1
[1,4,3,5,2] => [1,3,5,4,2] => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = 2 + 1
[1,4,5,2,3] => [1,5,4,3,2] => [1,1,2,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = 2 + 1
[1,4,5,3,2] => [1,5,4,2,3] => [1,1,3] => ([(2,3),(2,4),(3,4)],5)
=> 3 = 2 + 1
[1,5,2,3,4] => [1,5,2,3,4] => [1,4] => ([(3,4)],5)
=> ? = 2 + 1
[1,5,2,4,3] => [1,4,2,5,3] => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = 2 + 1
[1,5,3,2,4] => [1,3,5,2,4] => [1,4] => ([(3,4)],5)
=> ? = 2 + 1
[1,5,3,4,2] => [1,3,4,5,2] => [1,4] => ([(3,4)],5)
=> ? = 2 + 1
[1,5,4,2,3] => [1,4,5,3,2] => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = 2 + 1
[1,5,4,3,2] => [1,4,5,2,3] => [1,4] => ([(3,4)],5)
=> ? = 1 + 1
[2,1,3,4,5] => [2,1,3,4,5] => [5] => ([],5)
=> ? = 2 + 1
[2,1,3,5,4] => [2,1,3,5,4] => [3,2] => ([(1,4),(2,4),(3,4)],5)
=> ? = 1 + 1
[2,1,4,3,5] => [2,1,4,3,5] => [2,3] => ([(2,4),(3,4)],5)
=> ? = 2 + 1
[2,1,4,5,3] => [2,1,5,4,3] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = 2 + 1
[2,1,5,3,4] => [2,1,5,3,4] => [2,3] => ([(2,4),(3,4)],5)
=> ? = 2 + 1
[2,1,5,4,3] => [2,1,4,5,3] => [2,3] => ([(2,4),(3,4)],5)
=> ? = 2 + 1
[2,3,1,4,5] => [3,2,1,4,5] => [2,3] => ([(2,4),(3,4)],5)
=> ? = 2 + 1
[2,3,1,5,4] => [3,2,1,5,4] => [2,1,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = 2 + 1
[2,3,4,1,5] => [4,2,3,1,5] => [3,2] => ([(1,4),(2,4),(3,4)],5)
=> ? = 2 + 1
[2,3,5,1,4] => [5,2,3,1,4] => [3,2] => ([(1,4),(2,4),(3,4)],5)
=> ? = 2 + 1
[2,4,1,3,5] => [4,2,1,3,5] => [2,3] => ([(2,4),(3,4)],5)
=> ? = 2 + 1
[2,4,1,5,3] => [5,2,1,4,3] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = 2 + 1
[2,4,3,1,5] => [3,2,4,1,5] => [2,3] => ([(2,4),(3,4)],5)
=> ? = 2 + 1
[2,4,5,1,3] => [5,2,4,3,1] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = 2 + 1
[2,5,1,3,4] => [5,2,1,3,4] => [2,3] => ([(2,4),(3,4)],5)
=> ? = 2 + 1
[2,5,1,4,3] => [4,2,1,5,3] => [2,1,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = 2 + 1
[2,5,3,1,4] => [3,2,5,1,4] => [2,3] => ([(2,4),(3,4)],5)
=> ? = 2 + 1
[2,5,4,1,3] => [4,2,5,3,1] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = 2 + 1
[3,1,2,4,5] => [3,1,2,4,5] => [5] => ([],5)
=> ? = 2 + 1
[3,1,2,5,4] => [3,1,2,5,4] => [3,2] => ([(1,4),(2,4),(3,4)],5)
=> ? = 2 + 1
[3,1,4,2,5] => [4,1,3,2,5] => [3,2] => ([(1,4),(2,4),(3,4)],5)
=> ? = 2 + 1
[3,1,4,5,2] => [5,1,3,4,2] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> ? = 2 + 1
[3,1,5,2,4] => [5,1,3,2,4] => [3,2] => ([(1,4),(2,4),(3,4)],5)
=> ? = 2 + 1
[3,1,5,4,2] => [4,1,3,5,2] => [3,2] => ([(1,4),(2,4),(3,4)],5)
=> ? = 2 + 1
[3,2,1,4,5] => [2,3,1,4,5] => [5] => ([],5)
=> ? = 1 + 1
[3,2,1,5,4] => [2,3,1,5,4] => [3,2] => ([(1,4),(2,4),(3,4)],5)
=> ? = 2 + 1
[3,2,4,1,5] => [2,4,3,1,5] => [3,2] => ([(1,4),(2,4),(3,4)],5)
=> ? = 2 + 1
[3,4,1,2,5] => [4,3,2,1,5] => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = 2 + 1
[3,4,1,5,2] => [5,3,2,4,1] => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = 2 + 1
[3,5,1,2,4] => [5,3,2,1,4] => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = 2 + 1
[3,5,1,4,2] => [4,3,2,5,1] => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = 2 + 1
[4,1,3,5,2] => [3,1,5,4,2] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = 2 + 1
[4,1,5,2,3] => [5,1,4,3,2] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = 2 + 1
[4,2,5,1,3] => [2,5,4,3,1] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = 2 + 1
[1,2,3,5,6,4] => [1,2,3,6,5,4] => [3,2,1] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 3 = 2 + 1
[1,2,4,3,6,5] => [1,2,4,3,6,5] => [2,2,2] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 3 = 2 + 1
[1,2,4,5,3,6] => [1,2,5,4,3,6] => [2,2,2] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 3 = 2 + 1
[1,2,4,5,6,3] => [1,2,6,4,5,3] => [2,3,1] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 3 = 2 + 1
[1,2,4,6,3,5] => [1,2,6,4,3,5] => [2,2,2] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 3 = 2 + 1
[1,2,4,6,5,3] => [1,2,5,4,6,3] => [2,2,2] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 3 = 2 + 1
[1,2,5,3,6,4] => [1,2,6,3,5,4] => [2,3,1] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 3 = 2 + 1
[1,2,5,4,6,3] => [1,2,4,6,5,3] => [2,3,1] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 3 = 2 + 1
[1,2,5,6,3,4] => [1,2,6,5,4,3] => [2,1,2,1] => ([(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 3 = 2 + 1
[1,2,5,6,4,3] => [1,2,6,5,3,4] => [2,1,3] => ([(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 3 = 2 + 1
[1,2,6,3,5,4] => [1,2,5,3,6,4] => [2,2,2] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 3 = 2 + 1
[1,2,6,5,3,4] => [1,2,5,6,4,3] => [2,3,1] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 3 = 2 + 1
[1,3,2,4,6,5] => [1,3,2,4,6,5] => [1,3,2] => ([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> 3 = 2 + 1
[1,3,2,5,4,6] => [1,3,2,5,4,6] => [1,2,3] => ([(2,5),(3,4),(3,5),(4,5)],6)
=> 3 = 2 + 1
[1,3,2,5,6,4] => [1,3,2,6,5,4] => [1,2,2,1] => ([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 3 = 2 + 1
[1,3,2,6,4,5] => [1,3,2,6,4,5] => [1,2,3] => ([(2,5),(3,4),(3,5),(4,5)],6)
=> 3 = 2 + 1
[1,3,2,6,5,4] => [1,3,2,5,6,4] => [1,2,3] => ([(2,5),(3,4),(3,5),(4,5)],6)
=> 3 = 2 + 1
[1,3,4,2,5,6] => [1,4,3,2,5,6] => [1,2,3] => ([(2,5),(3,4),(3,5),(4,5)],6)
=> 3 = 2 + 1
[1,3,4,2,6,5] => [1,4,3,2,6,5] => [1,2,1,2] => ([(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 3 = 2 + 1
[1,3,4,5,2,6] => [1,5,3,4,2,6] => [1,3,2] => ([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> 3 = 2 + 1
[1,3,4,5,6,2] => [1,6,3,4,5,2] => [1,4,1] => ([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> 3 = 2 + 1
[1,3,4,6,2,5] => [1,6,3,4,2,5] => [1,3,2] => ([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> 3 = 2 + 1
[1,3,4,6,5,2] => [1,5,3,4,6,2] => [1,3,2] => ([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> 3 = 2 + 1
[1,3,5,2,4,6] => [1,5,3,2,4,6] => [1,2,3] => ([(2,5),(3,4),(3,5),(4,5)],6)
=> 3 = 2 + 1
Description
The girth of a graph, which is not a tree. This is the length of the shortest cycle in the graph.
Mp00127: Permutations left-to-right-maxima to Dyck pathDyck paths
Mp00023: Dyck paths to non-crossing permutationPermutations
Mp00160: Permutations graph of inversionsGraphs
St000259: Graphs ⟶ ℤResult quality: 52% values known / values provided: 52%distinct values known / distinct values provided: 67%
Values
[1] => [1,0]
=> [1] => ([],1)
=> 0
[1,2] => [1,0,1,0]
=> [1,2] => ([],2)
=> ? = 1
[1,2,3] => [1,0,1,0,1,0]
=> [1,2,3] => ([],3)
=> ? = 1
[1,3,2] => [1,0,1,1,0,0]
=> [1,3,2] => ([(1,2)],3)
=> ? = 1
[2,1,3] => [1,1,0,0,1,0]
=> [2,1,3] => ([(1,2)],3)
=> ? = 1
[1,2,3,4] => [1,0,1,0,1,0,1,0]
=> [1,2,3,4] => ([],4)
=> ? = 2
[1,2,4,3] => [1,0,1,0,1,1,0,0]
=> [1,2,4,3] => ([(2,3)],4)
=> ? = 1
[1,3,2,4] => [1,0,1,1,0,0,1,0]
=> [1,3,2,4] => ([(2,3)],4)
=> ? = 2
[1,3,4,2] => [1,0,1,1,0,1,0,0]
=> [1,3,4,2] => ([(1,3),(2,3)],4)
=> ? = 2
[1,4,2,3] => [1,0,1,1,1,0,0,0]
=> [1,4,3,2] => ([(1,2),(1,3),(2,3)],4)
=> ? = 2
[1,4,3,2] => [1,0,1,1,1,0,0,0]
=> [1,4,3,2] => ([(1,2),(1,3),(2,3)],4)
=> ? = 1
[2,1,3,4] => [1,1,0,0,1,0,1,0]
=> [2,1,3,4] => ([(2,3)],4)
=> ? = 1
[2,1,4,3] => [1,1,0,0,1,1,0,0]
=> [2,1,4,3] => ([(0,3),(1,2)],4)
=> ? = 2
[2,3,1,4] => [1,1,0,1,0,0,1,0]
=> [2,3,1,4] => ([(1,3),(2,3)],4)
=> ? = 2
[2,4,1,3] => [1,1,0,1,1,0,0,0]
=> [2,4,3,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 2
[3,1,2,4] => [1,1,1,0,0,0,1,0]
=> [3,2,1,4] => ([(1,2),(1,3),(2,3)],4)
=> ? = 2
[3,1,4,2] => [1,1,1,0,0,1,0,0]
=> [3,2,4,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 2
[3,2,1,4] => [1,1,1,0,0,0,1,0]
=> [3,2,1,4] => ([(1,2),(1,3),(2,3)],4)
=> ? = 1
[1,2,3,4,5] => [1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5] => ([],5)
=> ? = 2
[1,2,3,5,4] => [1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,5,4] => ([(3,4)],5)
=> ? = 2
[1,2,4,3,5] => [1,0,1,0,1,1,0,0,1,0]
=> [1,2,4,3,5] => ([(3,4)],5)
=> ? = 2
[1,2,4,5,3] => [1,0,1,0,1,1,0,1,0,0]
=> [1,2,4,5,3] => ([(2,4),(3,4)],5)
=> ? = 2
[1,2,5,3,4] => [1,0,1,0,1,1,1,0,0,0]
=> [1,2,5,4,3] => ([(2,3),(2,4),(3,4)],5)
=> ? = 2
[1,2,5,4,3] => [1,0,1,0,1,1,1,0,0,0]
=> [1,2,5,4,3] => ([(2,3),(2,4),(3,4)],5)
=> ? = 1
[1,3,2,4,5] => [1,0,1,1,0,0,1,0,1,0]
=> [1,3,2,4,5] => ([(3,4)],5)
=> ? = 2
[1,3,2,5,4] => [1,0,1,1,0,0,1,1,0,0]
=> [1,3,2,5,4] => ([(1,4),(2,3)],5)
=> ? = 2
[1,3,4,2,5] => [1,0,1,1,0,1,0,0,1,0]
=> [1,3,4,2,5] => ([(2,4),(3,4)],5)
=> ? = 2
[1,3,4,5,2] => [1,0,1,1,0,1,0,1,0,0]
=> [1,3,4,5,2] => ([(1,4),(2,4),(3,4)],5)
=> ? = 2
[1,3,5,2,4] => [1,0,1,1,0,1,1,0,0,0]
=> [1,3,5,4,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 2
[1,3,5,4,2] => [1,0,1,1,0,1,1,0,0,0]
=> [1,3,5,4,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 2
[1,4,2,3,5] => [1,0,1,1,1,0,0,0,1,0]
=> [1,4,3,2,5] => ([(2,3),(2,4),(3,4)],5)
=> ? = 2
[1,4,2,5,3] => [1,0,1,1,1,0,0,1,0,0]
=> [1,4,3,5,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 2
[1,4,3,2,5] => [1,0,1,1,1,0,0,0,1,0]
=> [1,4,3,2,5] => ([(2,3),(2,4),(3,4)],5)
=> ? = 2
[1,4,3,5,2] => [1,0,1,1,1,0,0,1,0,0]
=> [1,4,3,5,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 2
[1,4,5,2,3] => [1,0,1,1,1,0,1,0,0,0]
=> [1,5,3,4,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 2
[1,4,5,3,2] => [1,0,1,1,1,0,1,0,0,0]
=> [1,5,3,4,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 2
[1,5,2,3,4] => [1,0,1,1,1,1,0,0,0,0]
=> [1,5,4,3,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 2
[1,5,2,4,3] => [1,0,1,1,1,1,0,0,0,0]
=> [1,5,4,3,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 2
[1,5,3,2,4] => [1,0,1,1,1,1,0,0,0,0]
=> [1,5,4,3,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 2
[1,5,3,4,2] => [1,0,1,1,1,1,0,0,0,0]
=> [1,5,4,3,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 2
[1,5,4,2,3] => [1,0,1,1,1,1,0,0,0,0]
=> [1,5,4,3,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 2
[1,5,4,3,2] => [1,0,1,1,1,1,0,0,0,0]
=> [1,5,4,3,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 1
[2,1,3,4,5] => [1,1,0,0,1,0,1,0,1,0]
=> [2,1,3,4,5] => ([(3,4)],5)
=> ? = 2
[2,1,3,5,4] => [1,1,0,0,1,0,1,1,0,0]
=> [2,1,3,5,4] => ([(1,4),(2,3)],5)
=> ? = 1
[2,1,4,3,5] => [1,1,0,0,1,1,0,0,1,0]
=> [2,1,4,3,5] => ([(1,4),(2,3)],5)
=> ? = 2
[2,1,4,5,3] => [1,1,0,0,1,1,0,1,0,0]
=> [2,1,4,5,3] => ([(0,1),(2,4),(3,4)],5)
=> ? = 2
[2,1,5,3,4] => [1,1,0,0,1,1,1,0,0,0]
=> [2,1,5,4,3] => ([(0,1),(2,3),(2,4),(3,4)],5)
=> ? = 2
[2,1,5,4,3] => [1,1,0,0,1,1,1,0,0,0]
=> [2,1,5,4,3] => ([(0,1),(2,3),(2,4),(3,4)],5)
=> ? = 2
[2,3,1,4,5] => [1,1,0,1,0,0,1,0,1,0]
=> [2,3,1,4,5] => ([(2,4),(3,4)],5)
=> ? = 2
[2,3,1,5,4] => [1,1,0,1,0,0,1,1,0,0]
=> [2,3,1,5,4] => ([(0,1),(2,4),(3,4)],5)
=> ? = 2
[2,3,4,1,5] => [1,1,0,1,0,1,0,0,1,0]
=> [2,3,4,1,5] => ([(1,4),(2,4),(3,4)],5)
=> ? = 2
[2,3,5,1,4] => [1,1,0,1,0,1,1,0,0,0]
=> [2,3,5,4,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
[2,4,1,3,5] => [1,1,0,1,1,0,0,0,1,0]
=> [2,4,3,1,5] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 2
[2,4,1,5,3] => [1,1,0,1,1,0,0,1,0,0]
=> [2,4,3,5,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
[2,4,3,1,5] => [1,1,0,1,1,0,0,0,1,0]
=> [2,4,3,1,5] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 2
[2,4,5,1,3] => [1,1,0,1,1,0,1,0,0,0]
=> [2,5,3,4,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
[2,5,1,3,4] => [1,1,0,1,1,1,0,0,0,0]
=> [2,5,4,3,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
[2,5,1,4,3] => [1,1,0,1,1,1,0,0,0,0]
=> [2,5,4,3,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
[2,5,3,1,4] => [1,1,0,1,1,1,0,0,0,0]
=> [2,5,4,3,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
[2,5,4,1,3] => [1,1,0,1,1,1,0,0,0,0]
=> [2,5,4,3,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
[3,1,4,5,2] => [1,1,1,0,0,1,0,1,0,0]
=> [3,2,4,5,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
[3,1,5,2,4] => [1,1,1,0,0,1,1,0,0,0]
=> [3,2,5,4,1] => ([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
=> 2
[3,1,5,4,2] => [1,1,1,0,0,1,1,0,0,0]
=> [3,2,5,4,1] => ([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
=> 2
[3,2,5,1,4] => [1,1,1,0,0,1,1,0,0,0]
=> [3,2,5,4,1] => ([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
=> 2
[3,4,1,5,2] => [1,1,1,0,1,0,0,1,0,0]
=> [4,2,3,5,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
[3,5,1,2,4] => [1,1,1,0,1,1,0,0,0,0]
=> [5,2,4,3,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
[3,5,1,4,2] => [1,1,1,0,1,1,0,0,0,0]
=> [5,2,4,3,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
[3,5,2,1,4] => [1,1,1,0,1,1,0,0,0,0]
=> [5,2,4,3,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
[4,1,2,5,3] => [1,1,1,1,0,0,0,1,0,0]
=> [4,3,2,5,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
[4,1,3,5,2] => [1,1,1,1,0,0,0,1,0,0]
=> [4,3,2,5,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
[4,1,5,2,3] => [1,1,1,1,0,0,1,0,0,0]
=> [5,3,2,4,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
[4,1,5,3,2] => [1,1,1,1,0,0,1,0,0,0]
=> [5,3,2,4,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
[4,2,1,5,3] => [1,1,1,1,0,0,0,1,0,0]
=> [4,3,2,5,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
[4,2,5,1,3] => [1,1,1,1,0,0,1,0,0,0]
=> [5,3,2,4,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
[4,3,1,5,2] => [1,1,1,1,0,0,0,1,0,0]
=> [4,3,2,5,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
[2,3,4,6,1,5] => [1,1,0,1,0,1,0,1,1,0,0,0]
=> [2,3,4,6,5,1] => ([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> 2
[2,3,5,1,6,4] => [1,1,0,1,0,1,1,0,0,1,0,0]
=> [2,3,5,4,6,1] => ([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> 2
[2,3,5,6,1,4] => [1,1,0,1,0,1,1,0,1,0,0,0]
=> [2,3,6,4,5,1] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 2
[2,3,6,1,4,5] => [1,1,0,1,0,1,1,1,0,0,0,0]
=> [2,3,6,5,4,1] => ([(0,5),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 2
[2,3,6,1,5,4] => [1,1,0,1,0,1,1,1,0,0,0,0]
=> [2,3,6,5,4,1] => ([(0,5),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 2
[2,3,6,4,1,5] => [1,1,0,1,0,1,1,1,0,0,0,0]
=> [2,3,6,5,4,1] => ([(0,5),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 2
[2,3,6,5,1,4] => [1,1,0,1,0,1,1,1,0,0,0,0]
=> [2,3,6,5,4,1] => ([(0,5),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 2
[2,4,1,5,6,3] => [1,1,0,1,1,0,0,1,0,1,0,0]
=> [2,4,3,5,6,1] => ([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> 2
[2,4,1,6,3,5] => [1,1,0,1,1,0,0,1,1,0,0,0]
=> [2,4,3,6,5,1] => ([(0,5),(1,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)
=> 2
[2,4,1,6,5,3] => [1,1,0,1,1,0,0,1,1,0,0,0]
=> [2,4,3,6,5,1] => ([(0,5),(1,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)
=> 2
[2,4,3,6,1,5] => [1,1,0,1,1,0,0,1,1,0,0,0]
=> [2,4,3,6,5,1] => ([(0,5),(1,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)
=> 2
[2,4,5,1,6,3] => [1,1,0,1,1,0,1,0,0,1,0,0]
=> [2,5,3,4,6,1] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 2
[2,4,5,6,1,3] => [1,1,0,1,1,0,1,0,1,0,0,0]
=> [2,6,3,4,5,1] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 2
[2,4,6,1,3,5] => [1,1,0,1,1,0,1,1,0,0,0,0]
=> [2,6,3,5,4,1] => ([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 2
[2,4,6,1,5,3] => [1,1,0,1,1,0,1,1,0,0,0,0]
=> [2,6,3,5,4,1] => ([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 2
[2,4,6,3,1,5] => [1,1,0,1,1,0,1,1,0,0,0,0]
=> [2,6,3,5,4,1] => ([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 2
[2,4,6,5,1,3] => [1,1,0,1,1,0,1,1,0,0,0,0]
=> [2,6,3,5,4,1] => ([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 2
[2,5,1,3,6,4] => [1,1,0,1,1,1,0,0,0,1,0,0]
=> [2,5,4,3,6,1] => ([(0,5),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 2
[2,5,1,4,6,3] => [1,1,0,1,1,1,0,0,0,1,0,0]
=> [2,5,4,3,6,1] => ([(0,5),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 2
[2,5,1,6,3,4] => [1,1,0,1,1,1,0,0,1,0,0,0]
=> [2,6,4,3,5,1] => ([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 2
[2,5,1,6,4,3] => [1,1,0,1,1,1,0,0,1,0,0,0]
=> [2,6,4,3,5,1] => ([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 2
[2,5,3,1,6,4] => [1,1,0,1,1,1,0,0,0,1,0,0]
=> [2,5,4,3,6,1] => ([(0,5),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 2
[2,5,3,6,1,4] => [1,1,0,1,1,1,0,0,1,0,0,0]
=> [2,6,4,3,5,1] => ([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 2
[2,5,4,1,6,3] => [1,1,0,1,1,1,0,0,0,1,0,0]
=> [2,5,4,3,6,1] => ([(0,5),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 2
[2,5,4,6,1,3] => [1,1,0,1,1,1,0,0,1,0,0,0]
=> [2,6,4,3,5,1] => ([(0,5),(1,4),(1,5),(2,3),(2,4),(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: St000455
Mp00160: Permutations graph of inversionsGraphs
Mp00274: Graphs block-cut treeGraphs
Mp00157: Graphs connected complementGraphs
St000455: Graphs ⟶ ℤResult quality: 33% values known / values provided: 41%distinct values known / distinct values provided: 33%
Values
[1] => ([],1)
=> ([],1)
=> ([],1)
=> ? = 0 - 2
[1,2] => ([],2)
=> ([],2)
=> ([],2)
=> ? = 1 - 2
[1,2,3] => ([],3)
=> ([],3)
=> ([],3)
=> ? = 1 - 2
[1,3,2] => ([(1,2)],3)
=> ([],2)
=> ([],2)
=> ? = 1 - 2
[2,1,3] => ([(1,2)],3)
=> ([],2)
=> ([],2)
=> ? = 1 - 2
[1,2,3,4] => ([],4)
=> ([],4)
=> ([],4)
=> ? = 2 - 2
[1,2,4,3] => ([(2,3)],4)
=> ([],3)
=> ([],3)
=> ? = 1 - 2
[1,3,2,4] => ([(2,3)],4)
=> ([],3)
=> ([],3)
=> ? = 2 - 2
[1,3,4,2] => ([(1,3),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> 0 = 2 - 2
[1,4,2,3] => ([(1,3),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> 0 = 2 - 2
[1,4,3,2] => ([(1,2),(1,3),(2,3)],4)
=> ([],2)
=> ([],2)
=> ? = 1 - 2
[2,1,3,4] => ([(2,3)],4)
=> ([],3)
=> ([],3)
=> ? = 1 - 2
[2,1,4,3] => ([(0,3),(1,2)],4)
=> ([],2)
=> ([],2)
=> ? = 2 - 2
[2,3,1,4] => ([(1,3),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> 0 = 2 - 2
[2,4,1,3] => ([(0,3),(1,2),(2,3)],4)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> ? = 2 - 2
[3,1,2,4] => ([(1,3),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> 0 = 2 - 2
[3,1,4,2] => ([(0,3),(1,2),(2,3)],4)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> ? = 2 - 2
[3,2,1,4] => ([(1,2),(1,3),(2,3)],4)
=> ([],2)
=> ([],2)
=> ? = 1 - 2
[1,2,3,4,5] => ([],5)
=> ([],5)
=> ([],5)
=> ? = 2 - 2
[1,2,3,5,4] => ([(3,4)],5)
=> ([],4)
=> ([],4)
=> ? = 2 - 2
[1,2,4,3,5] => ([(3,4)],5)
=> ([],4)
=> ([],4)
=> ? = 2 - 2
[1,2,4,5,3] => ([(2,4),(3,4)],5)
=> ([(2,4),(3,4)],5)
=> ([(2,4),(3,4)],5)
=> 0 = 2 - 2
[1,2,5,3,4] => ([(2,4),(3,4)],5)
=> ([(2,4),(3,4)],5)
=> ([(2,4),(3,4)],5)
=> 0 = 2 - 2
[1,2,5,4,3] => ([(2,3),(2,4),(3,4)],5)
=> ([],3)
=> ([],3)
=> ? = 1 - 2
[1,3,2,4,5] => ([(3,4)],5)
=> ([],4)
=> ([],4)
=> ? = 2 - 2
[1,3,2,5,4] => ([(1,4),(2,3)],5)
=> ([],3)
=> ([],3)
=> ? = 2 - 2
[1,3,4,2,5] => ([(2,4),(3,4)],5)
=> ([(2,4),(3,4)],5)
=> ([(2,4),(3,4)],5)
=> 0 = 2 - 2
[1,3,4,5,2] => ([(1,4),(2,4),(3,4)],5)
=> ([(1,4),(2,4),(3,4)],5)
=> ([(1,4),(2,4),(3,4)],5)
=> 0 = 2 - 2
[1,3,5,2,4] => ([(1,4),(2,3),(3,4)],5)
=> ([(1,5),(2,4),(3,4),(3,5)],6)
=> ([(1,2),(1,5),(2,4),(3,4),(3,5),(4,5)],6)
=> ? = 2 - 2
[1,3,5,4,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,3),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> 0 = 2 - 2
[1,4,2,3,5] => ([(2,4),(3,4)],5)
=> ([(2,4),(3,4)],5)
=> ([(2,4),(3,4)],5)
=> 0 = 2 - 2
[1,4,2,5,3] => ([(1,4),(2,3),(3,4)],5)
=> ([(1,5),(2,4),(3,4),(3,5)],6)
=> ([(1,2),(1,5),(2,4),(3,4),(3,5),(4,5)],6)
=> ? = 2 - 2
[1,4,3,2,5] => ([(2,3),(2,4),(3,4)],5)
=> ([],3)
=> ([],3)
=> ? = 2 - 2
[1,4,3,5,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,3),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> 0 = 2 - 2
[1,4,5,2,3] => ([(1,3),(1,4),(2,3),(2,4)],5)
=> ([],2)
=> ([],2)
=> ? = 2 - 2
[1,4,5,3,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([],2)
=> ([],2)
=> ? = 2 - 2
[1,5,2,3,4] => ([(1,4),(2,4),(3,4)],5)
=> ([(1,4),(2,4),(3,4)],5)
=> ([(1,4),(2,4),(3,4)],5)
=> 0 = 2 - 2
[1,5,2,4,3] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,3),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> 0 = 2 - 2
[1,5,3,2,4] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,3),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> 0 = 2 - 2
[1,5,3,4,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([],2)
=> ([],2)
=> ? = 2 - 2
[1,5,4,2,3] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([],2)
=> ([],2)
=> ? = 2 - 2
[1,5,4,3,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([],2)
=> ([],2)
=> ? = 1 - 2
[2,1,3,4,5] => ([(3,4)],5)
=> ([],4)
=> ([],4)
=> ? = 2 - 2
[2,1,3,5,4] => ([(1,4),(2,3)],5)
=> ([],3)
=> ([],3)
=> ? = 1 - 2
[2,1,4,3,5] => ([(1,4),(2,3)],5)
=> ([],3)
=> ([],3)
=> ? = 2 - 2
[2,1,4,5,3] => ([(0,1),(2,4),(3,4)],5)
=> ([(1,3),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> 0 = 2 - 2
[2,1,5,3,4] => ([(0,1),(2,4),(3,4)],5)
=> ([(1,3),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> 0 = 2 - 2
[2,1,5,4,3] => ([(0,1),(2,3),(2,4),(3,4)],5)
=> ([],2)
=> ([],2)
=> ? = 2 - 2
[2,3,1,4,5] => ([(2,4),(3,4)],5)
=> ([(2,4),(3,4)],5)
=> ([(2,4),(3,4)],5)
=> 0 = 2 - 2
[2,3,1,5,4] => ([(0,1),(2,4),(3,4)],5)
=> ([(1,3),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> 0 = 2 - 2
[2,3,4,1,5] => ([(1,4),(2,4),(3,4)],5)
=> ([(1,4),(2,4),(3,4)],5)
=> ([(1,4),(2,4),(3,4)],5)
=> 0 = 2 - 2
[2,3,5,1,4] => ([(0,4),(1,4),(2,3),(3,4)],5)
=> ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
=> ([(0,2),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6)
=> ? = 2 - 2
[2,4,1,3,5] => ([(1,4),(2,3),(3,4)],5)
=> ([(1,5),(2,4),(3,4),(3,5)],6)
=> ([(1,2),(1,5),(2,4),(3,4),(3,5),(4,5)],6)
=> ? = 2 - 2
[2,4,1,5,3] => ([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,6),(1,5),(2,3),(2,4),(3,5),(4,6)],7)
=> ([(0,1),(0,3),(0,4),(0,6),(1,2),(1,4),(1,5),(2,3),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 2 - 2
[2,4,3,1,5] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,3),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> 0 = 2 - 2
[2,4,5,1,3] => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> ([(0,2),(1,2)],3)
=> ([(0,2),(1,2)],3)
=> 0 = 2 - 2
[2,5,1,3,4] => ([(0,4),(1,4),(2,3),(3,4)],5)
=> ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
=> ([(0,2),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6)
=> ? = 2 - 2
[2,5,1,4,3] => ([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> ? = 2 - 2
[2,5,3,1,4] => ([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> ? = 2 - 2
[2,5,4,1,3] => ([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(0,2),(1,2)],3)
=> ([(0,2),(1,2)],3)
=> 0 = 2 - 2
[3,1,2,4,5] => ([(2,4),(3,4)],5)
=> ([(2,4),(3,4)],5)
=> ([(2,4),(3,4)],5)
=> 0 = 2 - 2
[3,1,2,5,4] => ([(0,1),(2,4),(3,4)],5)
=> ([(1,3),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> 0 = 2 - 2
[3,1,4,2,5] => ([(1,4),(2,3),(3,4)],5)
=> ([(1,5),(2,4),(3,4),(3,5)],6)
=> ([(1,2),(1,5),(2,4),(3,4),(3,5),(4,5)],6)
=> ? = 2 - 2
[3,1,4,5,2] => ([(0,4),(1,4),(2,3),(3,4)],5)
=> ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
=> ([(0,2),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6)
=> ? = 2 - 2
[3,1,5,2,4] => ([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,6),(1,5),(2,3),(2,4),(3,5),(4,6)],7)
=> ([(0,1),(0,3),(0,4),(0,6),(1,2),(1,4),(1,5),(2,3),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 2 - 2
[3,1,5,4,2] => ([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> ? = 2 - 2
[3,2,1,4,5] => ([(2,3),(2,4),(3,4)],5)
=> ([],3)
=> ([],3)
=> ? = 1 - 2
[3,2,1,5,4] => ([(0,1),(2,3),(2,4),(3,4)],5)
=> ([],2)
=> ([],2)
=> ? = 2 - 2
[3,2,4,1,5] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,3),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> 0 = 2 - 2
[3,2,5,1,4] => ([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> ? = 2 - 2
[3,4,1,2,5] => ([(1,3),(1,4),(2,3),(2,4)],5)
=> ([],2)
=> ([],2)
=> ? = 2 - 2
[3,4,1,5,2] => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> ([(0,2),(1,2)],3)
=> ([(0,2),(1,2)],3)
=> 0 = 2 - 2
[3,4,2,1,5] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([],2)
=> ([],2)
=> ? = 2 - 2
[3,5,1,2,4] => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> ([(0,2),(1,2)],3)
=> ([(0,2),(1,2)],3)
=> 0 = 2 - 2
[3,5,1,4,2] => ([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> ([],1)
=> ([],1)
=> ? = 2 - 2
[3,5,2,1,4] => ([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(0,2),(1,2)],3)
=> ([(0,2),(1,2)],3)
=> 0 = 2 - 2
[4,1,2,3,5] => ([(1,4),(2,4),(3,4)],5)
=> ([(1,4),(2,4),(3,4)],5)
=> ([(1,4),(2,4),(3,4)],5)
=> 0 = 2 - 2
[4,1,2,5,3] => ([(0,4),(1,4),(2,3),(3,4)],5)
=> ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
=> ([(0,2),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6)
=> ? = 2 - 2
[4,1,3,2,5] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,3),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> 0 = 2 - 2
[4,1,3,5,2] => ([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> ? = 2 - 2
[4,1,5,2,3] => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> ([(0,2),(1,2)],3)
=> ([(0,2),(1,2)],3)
=> 0 = 2 - 2
[4,1,5,3,2] => ([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(0,2),(1,2)],3)
=> ([(0,2),(1,2)],3)
=> 0 = 2 - 2
[4,2,1,3,5] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,3),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> 0 = 2 - 2
[4,3,1,5,2] => ([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(0,2),(1,2)],3)
=> ([(0,2),(1,2)],3)
=> 0 = 2 - 2
[1,2,3,5,6,4] => ([(3,5),(4,5)],6)
=> ([(3,5),(4,5)],6)
=> ([(3,5),(4,5)],6)
=> 0 = 2 - 2
[1,2,3,6,4,5] => ([(3,5),(4,5)],6)
=> ([(3,5),(4,5)],6)
=> ([(3,5),(4,5)],6)
=> 0 = 2 - 2
[1,2,4,5,3,6] => ([(3,5),(4,5)],6)
=> ([(3,5),(4,5)],6)
=> ([(3,5),(4,5)],6)
=> 0 = 2 - 2
[1,2,4,5,6,3] => ([(2,5),(3,5),(4,5)],6)
=> ([(2,5),(3,5),(4,5)],6)
=> ([(2,5),(3,5),(4,5)],6)
=> 0 = 2 - 2
[1,2,4,6,5,3] => ([(2,5),(3,4),(3,5),(4,5)],6)
=> ([(2,4),(3,4)],5)
=> ([(2,4),(3,4)],5)
=> 0 = 2 - 2
[1,2,5,3,4,6] => ([(3,5),(4,5)],6)
=> ([(3,5),(4,5)],6)
=> ([(3,5),(4,5)],6)
=> 0 = 2 - 2
[1,2,5,4,6,3] => ([(2,5),(3,4),(3,5),(4,5)],6)
=> ([(2,4),(3,4)],5)
=> ([(2,4),(3,4)],5)
=> 0 = 2 - 2
[1,2,6,3,4,5] => ([(2,5),(3,5),(4,5)],6)
=> ([(2,5),(3,5),(4,5)],6)
=> ([(2,5),(3,5),(4,5)],6)
=> 0 = 2 - 2
[1,2,6,3,5,4] => ([(2,5),(3,4),(3,5),(4,5)],6)
=> ([(2,4),(3,4)],5)
=> ([(2,4),(3,4)],5)
=> 0 = 2 - 2
[1,2,6,4,3,5] => ([(2,5),(3,4),(3,5),(4,5)],6)
=> ([(2,4),(3,4)],5)
=> ([(2,4),(3,4)],5)
=> 0 = 2 - 2
[1,3,2,5,6,4] => ([(1,2),(3,5),(4,5)],6)
=> ([(2,4),(3,4)],5)
=> ([(2,4),(3,4)],5)
=> 0 = 2 - 2
[1,3,2,6,4,5] => ([(1,2),(3,5),(4,5)],6)
=> ([(2,4),(3,4)],5)
=> ([(2,4),(3,4)],5)
=> 0 = 2 - 2
[1,3,4,2,5,6] => ([(3,5),(4,5)],6)
=> ([(3,5),(4,5)],6)
=> ([(3,5),(4,5)],6)
=> 0 = 2 - 2
[1,3,4,2,6,5] => ([(1,2),(3,5),(4,5)],6)
=> ([(2,4),(3,4)],5)
=> ([(2,4),(3,4)],5)
=> 0 = 2 - 2
[1,3,4,5,2,6] => ([(2,5),(3,5),(4,5)],6)
=> ([(2,5),(3,5),(4,5)],6)
=> ([(2,5),(3,5),(4,5)],6)
=> 0 = 2 - 2
[1,3,4,5,6,2] => ([(1,5),(2,5),(3,5),(4,5)],6)
=> ([(1,5),(2,5),(3,5),(4,5)],6)
=> ([(1,5),(2,5),(3,5),(4,5)],6)
=> 0 = 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.
Mp00160: Permutations graph of inversionsGraphs
Mp00274: Graphs block-cut treeGraphs
St000456: Graphs ⟶ ℤResult quality: 33% values known / values provided: 34%distinct values known / distinct values provided: 33%
Values
[1] => ([],1)
=> ([],1)
=> ? = 0 - 1
[1,2] => ([],2)
=> ([],2)
=> ? = 1 - 1
[1,2,3] => ([],3)
=> ([],3)
=> ? = 1 - 1
[1,3,2] => ([(1,2)],3)
=> ([],2)
=> ? = 1 - 1
[2,1,3] => ([(1,2)],3)
=> ([],2)
=> ? = 1 - 1
[1,2,3,4] => ([],4)
=> ([],4)
=> ? = 2 - 1
[1,2,4,3] => ([(2,3)],4)
=> ([],3)
=> ? = 1 - 1
[1,3,2,4] => ([(2,3)],4)
=> ([],3)
=> ? = 2 - 1
[1,3,4,2] => ([(1,3),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> ? = 2 - 1
[1,4,2,3] => ([(1,3),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> ? = 2 - 1
[1,4,3,2] => ([(1,2),(1,3),(2,3)],4)
=> ([],2)
=> ? = 1 - 1
[2,1,3,4] => ([(2,3)],4)
=> ([],3)
=> ? = 1 - 1
[2,1,4,3] => ([(0,3),(1,2)],4)
=> ([],2)
=> ? = 2 - 1
[2,3,1,4] => ([(1,3),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> ? = 2 - 1
[2,4,1,3] => ([(0,3),(1,2),(2,3)],4)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 1 = 2 - 1
[3,1,2,4] => ([(1,3),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> ? = 2 - 1
[3,1,4,2] => ([(0,3),(1,2),(2,3)],4)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 1 = 2 - 1
[3,2,1,4] => ([(1,2),(1,3),(2,3)],4)
=> ([],2)
=> ? = 1 - 1
[1,2,3,4,5] => ([],5)
=> ([],5)
=> ? = 2 - 1
[1,2,3,5,4] => ([(3,4)],5)
=> ([],4)
=> ? = 2 - 1
[1,2,4,3,5] => ([(3,4)],5)
=> ([],4)
=> ? = 2 - 1
[1,2,4,5,3] => ([(2,4),(3,4)],5)
=> ([(2,4),(3,4)],5)
=> ? = 2 - 1
[1,2,5,3,4] => ([(2,4),(3,4)],5)
=> ([(2,4),(3,4)],5)
=> ? = 2 - 1
[1,2,5,4,3] => ([(2,3),(2,4),(3,4)],5)
=> ([],3)
=> ? = 1 - 1
[1,3,2,4,5] => ([(3,4)],5)
=> ([],4)
=> ? = 2 - 1
[1,3,2,5,4] => ([(1,4),(2,3)],5)
=> ([],3)
=> ? = 2 - 1
[1,3,4,2,5] => ([(2,4),(3,4)],5)
=> ([(2,4),(3,4)],5)
=> ? = 2 - 1
[1,3,4,5,2] => ([(1,4),(2,4),(3,4)],5)
=> ([(1,4),(2,4),(3,4)],5)
=> ? = 2 - 1
[1,3,5,2,4] => ([(1,4),(2,3),(3,4)],5)
=> ([(1,5),(2,4),(3,4),(3,5)],6)
=> ? = 2 - 1
[1,3,5,4,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,3),(2,3)],4)
=> ? = 2 - 1
[1,4,2,3,5] => ([(2,4),(3,4)],5)
=> ([(2,4),(3,4)],5)
=> ? = 2 - 1
[1,4,2,5,3] => ([(1,4),(2,3),(3,4)],5)
=> ([(1,5),(2,4),(3,4),(3,5)],6)
=> ? = 2 - 1
[1,4,3,2,5] => ([(2,3),(2,4),(3,4)],5)
=> ([],3)
=> ? = 2 - 1
[1,4,3,5,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,3),(2,3)],4)
=> ? = 2 - 1
[1,4,5,2,3] => ([(1,3),(1,4),(2,3),(2,4)],5)
=> ([],2)
=> ? = 2 - 1
[1,4,5,3,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([],2)
=> ? = 2 - 1
[1,5,2,3,4] => ([(1,4),(2,4),(3,4)],5)
=> ([(1,4),(2,4),(3,4)],5)
=> ? = 2 - 1
[1,5,2,4,3] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,3),(2,3)],4)
=> ? = 2 - 1
[1,5,3,2,4] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,3),(2,3)],4)
=> ? = 2 - 1
[1,5,3,4,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([],2)
=> ? = 2 - 1
[1,5,4,2,3] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([],2)
=> ? = 2 - 1
[1,5,4,3,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([],2)
=> ? = 1 - 1
[2,1,3,4,5] => ([(3,4)],5)
=> ([],4)
=> ? = 2 - 1
[2,1,3,5,4] => ([(1,4),(2,3)],5)
=> ([],3)
=> ? = 1 - 1
[2,1,4,3,5] => ([(1,4),(2,3)],5)
=> ([],3)
=> ? = 2 - 1
[2,1,4,5,3] => ([(0,1),(2,4),(3,4)],5)
=> ([(1,3),(2,3)],4)
=> ? = 2 - 1
[2,1,5,3,4] => ([(0,1),(2,4),(3,4)],5)
=> ([(1,3),(2,3)],4)
=> ? = 2 - 1
[2,1,5,4,3] => ([(0,1),(2,3),(2,4),(3,4)],5)
=> ([],2)
=> ? = 2 - 1
[2,3,1,4,5] => ([(2,4),(3,4)],5)
=> ([(2,4),(3,4)],5)
=> ? = 2 - 1
[2,3,1,5,4] => ([(0,1),(2,4),(3,4)],5)
=> ([(1,3),(2,3)],4)
=> ? = 2 - 1
[2,3,4,1,5] => ([(1,4),(2,4),(3,4)],5)
=> ([(1,4),(2,4),(3,4)],5)
=> ? = 2 - 1
[2,3,5,1,4] => ([(0,4),(1,4),(2,3),(3,4)],5)
=> ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
=> 1 = 2 - 1
[2,4,1,3,5] => ([(1,4),(2,3),(3,4)],5)
=> ([(1,5),(2,4),(3,4),(3,5)],6)
=> ? = 2 - 1
[2,4,1,5,3] => ([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,6),(1,5),(2,3),(2,4),(3,5),(4,6)],7)
=> 1 = 2 - 1
[2,4,5,1,3] => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> ([(0,2),(1,2)],3)
=> 1 = 2 - 1
[2,5,1,3,4] => ([(0,4),(1,4),(2,3),(3,4)],5)
=> ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
=> 1 = 2 - 1
[2,5,1,4,3] => ([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 1 = 2 - 1
[2,5,3,1,4] => ([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 1 = 2 - 1
[2,5,4,1,3] => ([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(0,2),(1,2)],3)
=> 1 = 2 - 1
[3,1,4,5,2] => ([(0,4),(1,4),(2,3),(3,4)],5)
=> ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
=> 1 = 2 - 1
[3,1,5,2,4] => ([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,6),(1,5),(2,3),(2,4),(3,5),(4,6)],7)
=> 1 = 2 - 1
[3,1,5,4,2] => ([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 1 = 2 - 1
[3,2,5,1,4] => ([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 1 = 2 - 1
[3,4,1,5,2] => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> ([(0,2),(1,2)],3)
=> 1 = 2 - 1
[3,5,1,2,4] => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> ([(0,2),(1,2)],3)
=> 1 = 2 - 1
[3,5,2,1,4] => ([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(0,2),(1,2)],3)
=> 1 = 2 - 1
[4,1,2,5,3] => ([(0,4),(1,4),(2,3),(3,4)],5)
=> ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
=> 1 = 2 - 1
[4,1,3,5,2] => ([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 1 = 2 - 1
[4,1,5,2,3] => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> ([(0,2),(1,2)],3)
=> 1 = 2 - 1
[4,1,5,3,2] => ([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(0,2),(1,2)],3)
=> 1 = 2 - 1
[4,2,1,5,3] => ([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 1 = 2 - 1
[4,3,1,5,2] => ([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(0,2),(1,2)],3)
=> 1 = 2 - 1
[2,3,4,6,1,5] => ([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
=> ([(0,6),(1,6),(2,6),(3,4),(4,5),(5,6)],7)
=> 1 = 2 - 1
[2,3,5,6,1,4] => ([(0,5),(1,5),(2,3),(2,4),(3,5),(4,5)],6)
=> ([(0,3),(1,3),(2,3)],4)
=> 1 = 2 - 1
[2,3,6,1,4,5] => ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> ([(0,6),(1,6),(2,5),(3,5),(4,5),(4,6)],7)
=> 1 = 2 - 1
[2,3,6,1,5,4] => ([(0,4),(1,4),(2,3),(2,5),(3,5),(4,5)],6)
=> ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
=> 1 = 2 - 1
[2,3,6,4,1,5] => ([(0,5),(1,5),(2,4),(3,4),(3,5),(4,5)],6)
=> ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
=> 1 = 2 - 1
[2,3,6,5,1,4] => ([(0,5),(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6)
=> ([(0,3),(1,3),(2,3)],4)
=> 1 = 2 - 1
[2,4,1,6,5,3] => ([(0,4),(1,2),(1,5),(2,5),(3,4),(3,5)],6)
=> ([(0,6),(1,5),(2,3),(2,4),(3,5),(4,6)],7)
=> 1 = 2 - 1
[2,4,3,6,1,5] => ([(0,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)
=> ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
=> 1 = 2 - 1
[2,4,5,1,6,3] => ([(0,5),(1,4),(2,4),(2,5),(3,4),(3,5)],6)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 1 = 2 - 1
[2,4,5,6,1,3] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> ([(0,2),(1,2)],3)
=> 1 = 2 - 1
[2,4,6,1,3,5] => ([(0,5),(1,4),(2,3),(2,4),(3,5),(4,5)],6)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 1 = 2 - 1
[2,4,6,1,5,3] => ([(0,5),(1,3),(1,4),(2,4),(2,5),(3,4),(3,5)],6)
=> ([(0,2),(1,2)],3)
=> 1 = 2 - 1
[2,4,6,3,1,5] => ([(0,5),(1,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 1 = 2 - 1
[2,4,6,5,1,3] => ([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5)],6)
=> ([(0,2),(1,2)],3)
=> 1 = 2 - 1
[2,5,1,4,6,3] => ([(0,4),(1,2),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,6),(1,5),(2,3),(2,4),(3,5),(4,6)],7)
=> 1 = 2 - 1
[2,5,1,6,3,4] => ([(0,4),(1,2),(1,3),(2,5),(3,5),(4,5)],6)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 1 = 2 - 1
[2,5,1,6,4,3] => ([(0,2),(1,4),(1,5),(2,3),(3,4),(3,5),(4,5)],6)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 1 = 2 - 1
[2,5,3,1,6,4] => ([(0,4),(1,2),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,6),(1,5),(2,3),(2,4),(3,5),(4,6)],7)
=> 1 = 2 - 1
[2,5,3,6,1,4] => ([(0,5),(1,2),(1,5),(2,4),(3,4),(3,5),(4,5)],6)
=> ([(0,2),(1,2)],3)
=> 1 = 2 - 1
[2,5,4,1,6,3] => ([(0,5),(1,4),(2,3),(2,4),(2,5),(3,4),(3,5)],6)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 1 = 2 - 1
[2,5,4,6,1,3] => ([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5)],6)
=> ([(0,2),(1,2)],3)
=> 1 = 2 - 1
[2,5,6,1,3,4] => ([(0,5),(1,3),(1,4),(2,3),(2,4),(3,5),(4,5)],6)
=> ([(0,2),(1,2)],3)
=> 1 = 2 - 1
[2,5,6,1,4,3] => ([(0,5),(1,2),(1,3),(1,4),(2,3),(2,4),(3,5),(4,5)],6)
=> ([(0,2),(1,2)],3)
=> 1 = 2 - 1
[2,5,6,3,1,4] => ([(0,5),(1,3),(1,4),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
=> ([(0,2),(1,2)],3)
=> 1 = 2 - 1
[2,5,6,4,1,3] => ([(0,5),(1,2),(1,3),(1,4),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,2),(1,2)],3)
=> 1 = 2 - 1
[2,6,1,3,4,5] => ([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
=> ([(0,6),(1,6),(2,6),(3,4),(4,5),(5,6)],7)
=> 1 = 2 - 1
[2,6,1,3,5,4] => ([(0,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)
=> ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
=> 1 = 2 - 1
[2,6,1,4,3,5] => ([(0,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)
=> ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
=> 1 = 2 - 1
Description
The monochromatic index of a connected graph. This is the maximal number of colours such that there is a colouring of the edges where any two vertices can be joined by a monochromatic path. For example, a circle graph other than the triangle can be coloured with at most two colours: one edge blue, all the others red.
Matching statistic: St001603
Mp00223: Permutations runsortPermutations
Mp00060: Permutations Robinson-Schensted tableau shapeInteger partitions
Mp00202: Integer partitions first row removalInteger partitions
St001603: Integer partitions ⟶ ℤResult quality: 28% values known / values provided: 28%distinct values known / distinct values provided: 33%
Values
[1] => [1] => [1]
=> []
=> ? = 0 - 1
[1,2] => [1,2] => [2]
=> []
=> ? = 1 - 1
[1,2,3] => [1,2,3] => [3]
=> []
=> ? = 1 - 1
[1,3,2] => [1,3,2] => [2,1]
=> [1]
=> ? = 1 - 1
[2,1,3] => [1,3,2] => [2,1]
=> [1]
=> ? = 1 - 1
[1,2,3,4] => [1,2,3,4] => [4]
=> []
=> ? = 2 - 1
[1,2,4,3] => [1,2,4,3] => [3,1]
=> [1]
=> ? = 1 - 1
[1,3,2,4] => [1,3,2,4] => [3,1]
=> [1]
=> ? = 2 - 1
[1,3,4,2] => [1,3,4,2] => [3,1]
=> [1]
=> ? = 2 - 1
[1,4,2,3] => [1,4,2,3] => [3,1]
=> [1]
=> ? = 2 - 1
[1,4,3,2] => [1,4,2,3] => [3,1]
=> [1]
=> ? = 1 - 1
[2,1,3,4] => [1,3,4,2] => [3,1]
=> [1]
=> ? = 1 - 1
[2,1,4,3] => [1,4,2,3] => [3,1]
=> [1]
=> ? = 2 - 1
[2,3,1,4] => [1,4,2,3] => [3,1]
=> [1]
=> ? = 2 - 1
[2,4,1,3] => [1,3,2,4] => [3,1]
=> [1]
=> ? = 2 - 1
[3,1,2,4] => [1,2,4,3] => [3,1]
=> [1]
=> ? = 2 - 1
[3,1,4,2] => [1,4,2,3] => [3,1]
=> [1]
=> ? = 2 - 1
[3,2,1,4] => [1,4,2,3] => [3,1]
=> [1]
=> ? = 1 - 1
[1,2,3,4,5] => [1,2,3,4,5] => [5]
=> []
=> ? = 2 - 1
[1,2,3,5,4] => [1,2,3,5,4] => [4,1]
=> [1]
=> ? = 2 - 1
[1,2,4,3,5] => [1,2,4,3,5] => [4,1]
=> [1]
=> ? = 2 - 1
[1,2,4,5,3] => [1,2,4,5,3] => [4,1]
=> [1]
=> ? = 2 - 1
[1,2,5,3,4] => [1,2,5,3,4] => [4,1]
=> [1]
=> ? = 2 - 1
[1,2,5,4,3] => [1,2,5,3,4] => [4,1]
=> [1]
=> ? = 1 - 1
[1,3,2,4,5] => [1,3,2,4,5] => [4,1]
=> [1]
=> ? = 2 - 1
[1,3,2,5,4] => [1,3,2,5,4] => [3,2]
=> [2]
=> ? = 2 - 1
[1,3,4,2,5] => [1,3,4,2,5] => [4,1]
=> [1]
=> ? = 2 - 1
[1,3,4,5,2] => [1,3,4,5,2] => [4,1]
=> [1]
=> ? = 2 - 1
[1,3,5,2,4] => [1,3,5,2,4] => [3,2]
=> [2]
=> ? = 2 - 1
[1,3,5,4,2] => [1,3,5,2,4] => [3,2]
=> [2]
=> ? = 2 - 1
[1,4,2,3,5] => [1,4,2,3,5] => [4,1]
=> [1]
=> ? = 2 - 1
[1,4,2,5,3] => [1,4,2,5,3] => [3,2]
=> [2]
=> ? = 2 - 1
[1,4,3,2,5] => [1,4,2,5,3] => [3,2]
=> [2]
=> ? = 2 - 1
[1,4,3,5,2] => [1,4,2,3,5] => [4,1]
=> [1]
=> ? = 2 - 1
[1,4,5,2,3] => [1,4,5,2,3] => [3,2]
=> [2]
=> ? = 2 - 1
[1,4,5,3,2] => [1,4,5,2,3] => [3,2]
=> [2]
=> ? = 2 - 1
[1,5,2,3,4] => [1,5,2,3,4] => [4,1]
=> [1]
=> ? = 2 - 1
[1,5,2,4,3] => [1,5,2,4,3] => [3,1,1]
=> [1,1]
=> ? = 2 - 1
[1,5,3,2,4] => [1,5,2,4,3] => [3,1,1]
=> [1,1]
=> ? = 2 - 1
[1,5,3,4,2] => [1,5,2,3,4] => [4,1]
=> [1]
=> ? = 2 - 1
[1,5,4,2,3] => [1,5,2,3,4] => [4,1]
=> [1]
=> ? = 2 - 1
[1,5,4,3,2] => [1,5,2,3,4] => [4,1]
=> [1]
=> ? = 1 - 1
[2,1,3,4,5] => [1,3,4,5,2] => [4,1]
=> [1]
=> ? = 2 - 1
[2,1,3,5,4] => [1,3,5,2,4] => [3,2]
=> [2]
=> ? = 1 - 1
[2,1,4,3,5] => [1,4,2,3,5] => [4,1]
=> [1]
=> ? = 2 - 1
[2,1,4,5,3] => [1,4,5,2,3] => [3,2]
=> [2]
=> ? = 2 - 1
[2,1,5,3,4] => [1,5,2,3,4] => [4,1]
=> [1]
=> ? = 2 - 1
[2,1,5,4,3] => [1,5,2,3,4] => [4,1]
=> [1]
=> ? = 2 - 1
[2,3,1,4,5] => [1,4,5,2,3] => [3,2]
=> [2]
=> ? = 2 - 1
[2,3,1,5,4] => [1,5,2,3,4] => [4,1]
=> [1]
=> ? = 2 - 1
[1,3,6,2,5,4] => [1,3,6,2,5,4] => [3,2,1]
=> [2,1]
=> 1 = 2 - 1
[1,3,6,4,2,5] => [1,3,6,2,5,4] => [3,2,1]
=> [2,1]
=> 1 = 2 - 1
[1,4,6,2,5,3] => [1,4,6,2,5,3] => [3,2,1]
=> [2,1]
=> 1 = 2 - 1
[1,4,6,3,2,5] => [1,4,6,2,5,3] => [3,2,1]
=> [2,1]
=> 1 = 2 - 1
[1,5,6,2,4,3] => [1,5,6,2,4,3] => [3,2,1]
=> [2,1]
=> 1 = 2 - 1
[1,5,6,3,2,4] => [1,5,6,2,4,3] => [3,2,1]
=> [2,1]
=> 1 = 2 - 1
[2,4,1,5,6,3] => [1,5,6,2,4,3] => [3,2,1]
=> [2,1]
=> 1 = 2 - 1
[2,4,3,1,5,6] => [1,5,6,2,4,3] => [3,2,1]
=> [2,1]
=> 1 = 2 - 1
[2,5,1,3,6,4] => [1,3,6,2,5,4] => [3,2,1]
=> [2,1]
=> 1 = 2 - 1
[2,5,1,4,6,3] => [1,4,6,2,5,3] => [3,2,1]
=> [2,1]
=> 1 = 2 - 1
[2,5,3,1,4,6] => [1,4,6,2,5,3] => [3,2,1]
=> [2,1]
=> 1 = 2 - 1
[2,5,4,1,3,6] => [1,3,6,2,5,4] => [3,2,1]
=> [2,1]
=> 1 = 2 - 1
[3,1,4,6,2,5] => [1,4,6,2,5,3] => [3,2,1]
=> [2,1]
=> 1 = 2 - 1
[3,1,5,6,2,4] => [1,5,6,2,4,3] => [3,2,1]
=> [2,1]
=> 1 = 2 - 1
[3,2,4,1,5,6] => [1,5,6,2,4,3] => [3,2,1]
=> [2,1]
=> 1 = 2 - 1
[3,2,5,1,4,6] => [1,4,6,2,5,3] => [3,2,1]
=> [2,1]
=> 1 = 2 - 1
[4,1,3,6,2,5] => [1,3,6,2,5,4] => [3,2,1]
=> [2,1]
=> 1 = 2 - 1
[4,2,5,1,3,6] => [1,3,6,2,5,4] => [3,2,1]
=> [2,1]
=> 1 = 2 - 1
[1,2,4,7,3,6,5] => [1,2,4,7,3,6,5] => [4,2,1]
=> [2,1]
=> 1 = 2 - 1
[1,2,4,7,5,3,6] => [1,2,4,7,3,6,5] => [4,2,1]
=> [2,1]
=> 1 = 2 - 1
[1,2,5,7,3,6,4] => [1,2,5,7,3,6,4] => [4,2,1]
=> [2,1]
=> 1 = 2 - 1
[1,2,5,7,4,3,6] => [1,2,5,7,3,6,4] => [4,2,1]
=> [2,1]
=> 1 = 2 - 1
[1,2,6,7,3,5,4] => [1,2,6,7,3,5,4] => [4,2,1]
=> [2,1]
=> 1 = 2 - 1
[1,2,6,7,4,3,5] => [1,2,6,7,3,5,4] => [4,2,1]
=> [2,1]
=> 1 = 2 - 1
[1,3,2,5,4,7,6] => [1,3,2,5,4,7,6] => [4,3]
=> [3]
=> 1 = 2 - 1
[1,3,2,5,7,4,6] => [1,3,2,5,7,4,6] => [4,3]
=> [3]
=> 1 = 2 - 1
[1,3,2,5,7,6,4] => [1,3,2,5,7,4,6] => [4,3]
=> [3]
=> 1 = 2 - 1
[1,3,2,6,4,7,5] => [1,3,2,6,4,7,5] => [4,3]
=> [3]
=> 1 = 2 - 1
[1,3,2,6,5,4,7] => [1,3,2,6,4,7,5] => [4,3]
=> [3]
=> 1 = 2 - 1
[1,3,2,6,7,4,5] => [1,3,2,6,7,4,5] => [4,3]
=> [3]
=> 1 = 2 - 1
[1,3,2,6,7,5,4] => [1,3,2,6,7,4,5] => [4,3]
=> [3]
=> 1 = 2 - 1
[1,3,2,7,4,6,5] => [1,3,2,7,4,6,5] => [4,2,1]
=> [2,1]
=> 1 = 2 - 1
[1,3,2,7,5,4,6] => [1,3,2,7,4,6,5] => [4,2,1]
=> [2,1]
=> 1 = 2 - 1
[1,3,4,7,2,6,5] => [1,3,4,7,2,6,5] => [4,2,1]
=> [2,1]
=> 1 = 2 - 1
[1,3,4,7,5,2,6] => [1,3,4,7,2,6,5] => [4,2,1]
=> [2,1]
=> 1 = 2 - 1
[1,3,5,2,4,7,6] => [1,3,5,2,4,7,6] => [4,3]
=> [3]
=> 1 = 2 - 1
[1,3,5,2,7,4,6] => [1,3,5,2,7,4,6] => [4,3]
=> [3]
=> 1 = 2 - 1
[1,3,5,2,7,6,4] => [1,3,5,2,7,4,6] => [4,3]
=> [3]
=> 1 = 2 - 1
[1,3,5,4,2,7,6] => [1,3,5,2,7,4,6] => [4,3]
=> [3]
=> 1 = 2 - 1
[1,3,5,4,6,2,7] => [1,3,5,2,7,4,6] => [4,3]
=> [3]
=> 1 = 2 - 1
[1,3,5,4,7,6,2] => [1,3,5,2,4,7,6] => [4,3]
=> [3]
=> 1 = 2 - 1
[1,3,5,7,2,4,6] => [1,3,5,7,2,4,6] => [4,3]
=> [3]
=> 1 = 2 - 1
[1,3,5,7,2,6,4] => [1,3,5,7,2,6,4] => [4,2,1]
=> [2,1]
=> 1 = 2 - 1
[1,3,5,7,4,2,6] => [1,3,5,7,2,6,4] => [4,2,1]
=> [2,1]
=> 1 = 2 - 1
[1,3,5,7,4,6,2] => [1,3,5,7,2,4,6] => [4,3]
=> [3]
=> 1 = 2 - 1
[1,3,5,7,6,2,4] => [1,3,5,7,2,4,6] => [4,3]
=> [3]
=> 1 = 2 - 1
[1,3,5,7,6,4,2] => [1,3,5,7,2,4,6] => [4,3]
=> [3]
=> 1 = 2 - 1
[1,3,6,2,4,7,5] => [1,3,6,2,4,7,5] => [4,3]
=> [3]
=> 1 = 2 - 1
[1,3,6,2,5,4,7] => [1,3,6,2,5,4,7] => [4,2,1]
=> [2,1]
=> 1 = 2 - 1
[1,3,6,2,5,7,4] => [1,3,6,2,5,7,4] => [4,2,1]
=> [2,1]
=> 1 = 2 - 1
Description
The number of colourings of a polygon such that the multiplicities of a colour are given by a partition. Two colourings are considered equal, if they are obtained by an action of the dihedral group. This statistic is only defined for partitions of size at least 3, to avoid ambiguity.
Mp00064: Permutations reversePermutations
Mp00159: Permutations Demazure product with inversePermutations
Mp00127: Permutations left-to-right-maxima to Dyck pathDyck paths
St001498: Dyck paths ⟶ ℤResult quality: 26% values known / values provided: 26%distinct values known / distinct values provided: 33%
Values
[1] => [1] => [1] => [1,0]
=> ? = 0 - 2
[1,2] => [2,1] => [2,1] => [1,1,0,0]
=> ? = 1 - 2
[1,2,3] => [3,2,1] => [3,2,1] => [1,1,1,0,0,0]
=> ? = 1 - 2
[1,3,2] => [2,3,1] => [3,2,1] => [1,1,1,0,0,0]
=> ? = 1 - 2
[2,1,3] => [3,1,2] => [3,2,1] => [1,1,1,0,0,0]
=> ? = 1 - 2
[1,2,3,4] => [4,3,2,1] => [4,3,2,1] => [1,1,1,1,0,0,0,0]
=> ? = 2 - 2
[1,2,4,3] => [3,4,2,1] => [4,3,2,1] => [1,1,1,1,0,0,0,0]
=> ? = 1 - 2
[1,3,2,4] => [4,2,3,1] => [4,3,2,1] => [1,1,1,1,0,0,0,0]
=> ? = 2 - 2
[1,3,4,2] => [2,4,3,1] => [4,3,2,1] => [1,1,1,1,0,0,0,0]
=> ? = 2 - 2
[1,4,2,3] => [3,2,4,1] => [4,2,3,1] => [1,1,1,1,0,0,0,0]
=> ? = 2 - 2
[1,4,3,2] => [2,3,4,1] => [4,2,3,1] => [1,1,1,1,0,0,0,0]
=> ? = 1 - 2
[2,1,3,4] => [4,3,1,2] => [4,3,2,1] => [1,1,1,1,0,0,0,0]
=> ? = 1 - 2
[2,1,4,3] => [3,4,1,2] => [4,3,2,1] => [1,1,1,1,0,0,0,0]
=> ? = 2 - 2
[2,3,1,4] => [4,1,3,2] => [4,2,3,1] => [1,1,1,1,0,0,0,0]
=> ? = 2 - 2
[2,4,1,3] => [3,1,4,2] => [4,2,3,1] => [1,1,1,1,0,0,0,0]
=> ? = 2 - 2
[3,1,2,4] => [4,2,1,3] => [4,3,2,1] => [1,1,1,1,0,0,0,0]
=> ? = 2 - 2
[3,1,4,2] => [2,4,1,3] => [3,4,1,2] => [1,1,1,0,1,0,0,0]
=> 0 = 2 - 2
[3,2,1,4] => [4,1,2,3] => [4,2,3,1] => [1,1,1,1,0,0,0,0]
=> ? = 1 - 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 - 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 - 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 - 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 - 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 - 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]
=> ? = 1 - 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 - 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 - 2
[1,3,4,2,5] => [5,2,4,3,1] => [5,4,3,2,1] => [1,1,1,1,1,0,0,0,0,0]
=> ? = 2 - 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]
=> ? = 2 - 2
[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 - 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 - 2
[1,4,2,3,5] => [5,3,2,4,1] => [5,4,3,2,1] => [1,1,1,1,1,0,0,0,0,0]
=> ? = 2 - 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]
=> ? = 2 - 2
[1,4,3,2,5] => [5,2,3,4,1] => [5,4,3,2,1] => [1,1,1,1,1,0,0,0,0,0]
=> ? = 2 - 2
[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]
=> ? = 2 - 2
[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 - 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 - 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]
=> ? = 2 - 2
[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]
=> ? = 2 - 2
[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 - 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 - 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 - 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]
=> ? = 1 - 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]
=> ? = 2 - 2
[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]
=> ? = 1 - 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
[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
[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
[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
[2,3,1,4,5] => [5,4,1,3,2] => [5,4,3,2,1] => [1,1,1,1,1,0,0,0,0,0]
=> ? = 2 - 2
[2,3,1,5,4] => [4,5,1,3,2] => [5,4,3,2,1] => [1,1,1,1,1,0,0,0,0,0]
=> ? = 2 - 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
[3,1,4,5,2] => [2,5,4,1,3] => [4,5,3,1,2] => [1,1,1,1,0,1,0,0,0,0]
=> 0 = 2 - 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]
=> 0 = 2 - 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]
=> 0 = 2 - 2
[3,5,1,4,2] => [2,4,1,5,3] => [3,5,1,4,2] => [1,1,1,0,1,1,0,0,0,0]
=> 0 = 2 - 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]
=> 0 = 2 - 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]
=> 0 = 2 - 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]
=> 0 = 2 - 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]
=> 0 = 2 - 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]
=> 0 = 2 - 2
[4,2,5,1,3] => [3,1,5,2,4] => [4,2,5,1,3] => [1,1,1,1,0,0,1,0,0,0]
=> 0 = 2 - 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]
=> 0 = 2 - 2
[3,1,4,5,6,2] => [2,6,5,4,1,3] => [5,6,4,3,1,2] => [1,1,1,1,1,0,1,0,0,0,0,0]
=> 0 = 2 - 2
[3,1,4,6,5,2] => [2,5,6,4,1,3] => [5,6,4,3,1,2] => [1,1,1,1,1,0,1,0,0,0,0,0]
=> 0 = 2 - 2
[3,1,5,4,6,2] => [2,6,4,5,1,3] => [5,6,4,3,1,2] => [1,1,1,1,1,0,1,0,0,0,0,0]
=> 0 = 2 - 2
[3,1,5,6,4,2] => [2,4,6,5,1,3] => [5,6,4,3,1,2] => [1,1,1,1,1,0,1,0,0,0,0,0]
=> 0 = 2 - 2
[3,1,6,4,5,2] => [2,5,4,6,1,3] => [5,6,3,4,1,2] => [1,1,1,1,1,0,1,0,0,0,0,0]
=> 0 = 2 - 2
[3,1,6,5,4,2] => [2,4,5,6,1,3] => [5,6,3,4,1,2] => [1,1,1,1,1,0,1,0,0,0,0,0]
=> 0 = 2 - 2
[3,4,1,5,6,2] => [2,6,5,1,4,3] => [4,6,5,1,3,2] => [1,1,1,1,0,1,1,0,0,0,0,0]
=> 0 = 2 - 2
[3,4,1,6,5,2] => [2,5,6,1,4,3] => [4,6,5,1,3,2] => [1,1,1,1,0,1,1,0,0,0,0,0]
=> 0 = 2 - 2
[3,4,5,1,6,2] => [2,6,1,5,4,3] => [3,6,1,5,4,2] => [1,1,1,0,1,1,1,0,0,0,0,0]
=> 0 = 2 - 2
[3,4,6,1,5,2] => [2,5,1,6,4,3] => [3,6,1,5,4,2] => [1,1,1,0,1,1,1,0,0,0,0,0]
=> 0 = 2 - 2
[3,5,1,4,6,2] => [2,6,4,1,5,3] => [4,6,5,1,3,2] => [1,1,1,1,0,1,1,0,0,0,0,0]
=> 0 = 2 - 2
[3,5,1,6,4,2] => [2,4,6,1,5,3] => [4,6,5,1,3,2] => [1,1,1,1,0,1,1,0,0,0,0,0]
=> 0 = 2 - 2
[3,5,4,1,6,2] => [2,6,1,4,5,3] => [3,6,1,5,4,2] => [1,1,1,0,1,1,1,0,0,0,0,0]
=> 0 = 2 - 2
[3,5,6,1,4,2] => [2,4,1,6,5,3] => [3,6,1,5,4,2] => [1,1,1,0,1,1,1,0,0,0,0,0]
=> 0 = 2 - 2
[3,6,1,4,5,2] => [2,5,4,1,6,3] => [4,6,3,1,5,2] => [1,1,1,1,0,1,1,0,0,0,0,0]
=> 0 = 2 - 2
[3,6,1,5,4,2] => [2,4,5,1,6,3] => [4,6,3,1,5,2] => [1,1,1,1,0,1,1,0,0,0,0,0]
=> 0 = 2 - 2
[3,6,4,1,5,2] => [2,5,1,4,6,3] => [3,6,1,4,5,2] => [1,1,1,0,1,1,1,0,0,0,0,0]
=> 0 = 2 - 2
[3,6,5,1,4,2] => [2,4,1,5,6,3] => [3,6,1,4,5,2] => [1,1,1,0,1,1,1,0,0,0,0,0]
=> 0 = 2 - 2
[4,1,2,5,6,3] => [3,6,5,2,1,4] => [5,6,4,3,1,2] => [1,1,1,1,1,0,1,0,0,0,0,0]
=> 0 = 2 - 2
[4,1,2,6,5,3] => [3,5,6,2,1,4] => [5,6,4,3,1,2] => [1,1,1,1,1,0,1,0,0,0,0,0]
=> 0 = 2 - 2
[4,1,3,5,6,2] => [2,6,5,3,1,4] => [5,6,4,3,1,2] => [1,1,1,1,1,0,1,0,0,0,0,0]
=> 0 = 2 - 2
[4,1,3,6,5,2] => [2,5,6,3,1,4] => [5,6,4,3,1,2] => [1,1,1,1,1,0,1,0,0,0,0,0]
=> 0 = 2 - 2
[4,1,5,2,6,3] => [3,6,2,5,1,4] => [5,6,3,4,1,2] => [1,1,1,1,1,0,1,0,0,0,0,0]
=> 0 = 2 - 2
[4,1,5,3,6,2] => [2,6,3,5,1,4] => [5,6,3,4,1,2] => [1,1,1,1,1,0,1,0,0,0,0,0]
=> 0 = 2 - 2
[4,1,5,6,2,3] => [3,2,6,5,1,4] => [5,2,6,4,1,3] => [1,1,1,1,1,0,0,1,0,0,0,0]
=> 0 = 2 - 2
[4,1,5,6,3,2] => [2,3,6,5,1,4] => [5,2,6,4,1,3] => [1,1,1,1,1,0,0,1,0,0,0,0]
=> 0 = 2 - 2
[4,1,6,2,5,3] => [3,5,2,6,1,4] => [5,6,3,4,1,2] => [1,1,1,1,1,0,1,0,0,0,0,0]
=> 0 = 2 - 2
[4,1,6,3,5,2] => [2,5,3,6,1,4] => [5,6,3,4,1,2] => [1,1,1,1,1,0,1,0,0,0,0,0]
=> 0 = 2 - 2
[4,1,6,5,2,3] => [3,2,5,6,1,4] => [5,2,6,4,1,3] => [1,1,1,1,1,0,0,1,0,0,0,0]
=> 0 = 2 - 2
[4,1,6,5,3,2] => [2,3,5,6,1,4] => [5,2,6,4,1,3] => [1,1,1,1,1,0,0,1,0,0,0,0]
=> 0 = 2 - 2
[4,2,1,5,6,3] => [3,6,5,1,2,4] => [5,6,4,3,1,2] => [1,1,1,1,1,0,1,0,0,0,0,0]
=> 0 = 2 - 2
[4,2,1,6,5,3] => [3,5,6,1,2,4] => [5,6,4,3,1,2] => [1,1,1,1,1,0,1,0,0,0,0,0]
=> 0 = 2 - 2
[4,2,5,1,6,3] => [3,6,1,5,2,4] => [5,6,3,4,1,2] => [1,1,1,1,1,0,1,0,0,0,0,0]
=> 0 = 2 - 2
[4,2,5,6,1,3] => [3,1,6,5,2,4] => [5,2,6,4,1,3] => [1,1,1,1,1,0,0,1,0,0,0,0]
=> 0 = 2 - 2
[4,2,6,1,5,3] => [3,5,1,6,2,4] => [5,6,3,4,1,2] => [1,1,1,1,1,0,1,0,0,0,0,0]
=> 0 = 2 - 2
[4,2,6,5,1,3] => [3,1,5,6,2,4] => [5,2,6,4,1,3] => [1,1,1,1,1,0,0,1,0,0,0,0]
=> 0 = 2 - 2
[4,3,1,5,6,2] => [2,6,5,1,3,4] => [4,6,5,1,3,2] => [1,1,1,1,0,1,1,0,0,0,0,0]
=> 0 = 2 - 2
[4,3,1,6,5,2] => [2,5,6,1,3,4] => [4,6,5,1,3,2] => [1,1,1,1,0,1,1,0,0,0,0,0]
=> 0 = 2 - 2
Description
The normalised height of a Nakayama algebra with magnitude 1. We use the bijection (see code) suggested by Christian Stump, to have a bijection between such Nakayama algebras with magnitude 1 and Dyck paths. The normalised height is the height of the (periodic) Dyck path given by the top of the Auslander-Reiten quiver. Thus when having a CNakayama algebra it is the Loewy length minus the number of simple modules and for the LNakayama algebras it is the usual height.
Mp00223: Permutations runsortPermutations
Mp00061: Permutations to increasing treeBinary trees
Mp00011: Binary trees to graphGraphs
St000454: Graphs ⟶ ℤResult quality: 14% values known / values provided: 14%distinct values known / distinct values provided: 100%
Values
[1] => [1] => [.,.]
=> ([],1)
=> 0
[1,2] => [1,2] => [.,[.,.]]
=> ([(0,1)],2)
=> 1
[1,2,3] => [1,2,3] => [.,[.,[.,.]]]
=> ([(0,2),(1,2)],3)
=> ? = 1
[1,3,2] => [1,3,2] => [.,[[.,.],.]]
=> ([(0,2),(1,2)],3)
=> ? = 1
[2,1,3] => [1,3,2] => [.,[[.,.],.]]
=> ([(0,2),(1,2)],3)
=> ? = 1
[1,2,3,4] => [1,2,3,4] => [.,[.,[.,[.,.]]]]
=> ([(0,3),(1,2),(2,3)],4)
=> ? = 2
[1,2,4,3] => [1,2,4,3] => [.,[.,[[.,.],.]]]
=> ([(0,3),(1,2),(2,3)],4)
=> ? = 1
[1,3,2,4] => [1,3,2,4] => [.,[[.,.],[.,.]]]
=> ([(0,3),(1,3),(2,3)],4)
=> ? = 2
[1,3,4,2] => [1,3,4,2] => [.,[[.,[.,.]],.]]
=> ([(0,3),(1,2),(2,3)],4)
=> ? = 2
[1,4,2,3] => [1,4,2,3] => [.,[[.,.],[.,.]]]
=> ([(0,3),(1,3),(2,3)],4)
=> ? = 2
[1,4,3,2] => [1,4,2,3] => [.,[[.,.],[.,.]]]
=> ([(0,3),(1,3),(2,3)],4)
=> ? = 1
[2,1,3,4] => [1,3,4,2] => [.,[[.,[.,.]],.]]
=> ([(0,3),(1,2),(2,3)],4)
=> ? = 1
[2,1,4,3] => [1,4,2,3] => [.,[[.,.],[.,.]]]
=> ([(0,3),(1,3),(2,3)],4)
=> ? = 2
[2,3,1,4] => [1,4,2,3] => [.,[[.,.],[.,.]]]
=> ([(0,3),(1,3),(2,3)],4)
=> ? = 2
[2,4,1,3] => [1,3,2,4] => [.,[[.,.],[.,.]]]
=> ([(0,3),(1,3),(2,3)],4)
=> ? = 2
[3,1,2,4] => [1,2,4,3] => [.,[.,[[.,.],.]]]
=> ([(0,3),(1,2),(2,3)],4)
=> ? = 2
[3,1,4,2] => [1,4,2,3] => [.,[[.,.],[.,.]]]
=> ([(0,3),(1,3),(2,3)],4)
=> ? = 2
[3,2,1,4] => [1,4,2,3] => [.,[[.,.],[.,.]]]
=> ([(0,3),(1,3),(2,3)],4)
=> ? = 1
[1,2,3,4,5] => [1,2,3,4,5] => [.,[.,[.,[.,[.,.]]]]]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ? = 2
[1,2,3,5,4] => [1,2,3,5,4] => [.,[.,[.,[[.,.],.]]]]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ? = 2
[1,2,4,3,5] => [1,2,4,3,5] => [.,[.,[[.,.],[.,.]]]]
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> ? = 2
[1,2,4,5,3] => [1,2,4,5,3] => [.,[.,[[.,[.,.]],.]]]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ? = 2
[1,2,5,3,4] => [1,2,5,3,4] => [.,[.,[[.,.],[.,.]]]]
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> ? = 2
[1,2,5,4,3] => [1,2,5,3,4] => [.,[.,[[.,.],[.,.]]]]
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> ? = 1
[1,3,2,4,5] => [1,3,2,4,5] => [.,[[.,.],[.,[.,.]]]]
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> ? = 2
[1,3,2,5,4] => [1,3,2,5,4] => [.,[[.,.],[[.,.],.]]]
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> ? = 2
[1,3,4,2,5] => [1,3,4,2,5] => [.,[[.,[.,.]],[.,.]]]
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> ? = 2
[1,3,4,5,2] => [1,3,4,5,2] => [.,[[.,[.,[.,.]]],.]]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ? = 2
[1,3,5,2,4] => [1,3,5,2,4] => [.,[[.,[.,.]],[.,.]]]
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> ? = 2
[1,3,5,4,2] => [1,3,5,2,4] => [.,[[.,[.,.]],[.,.]]]
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> ? = 2
[1,4,2,3,5] => [1,4,2,3,5] => [.,[[.,.],[.,[.,.]]]]
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> ? = 2
[1,4,2,5,3] => [1,4,2,5,3] => [.,[[.,.],[[.,.],.]]]
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> ? = 2
[1,4,3,2,5] => [1,4,2,5,3] => [.,[[.,.],[[.,.],.]]]
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> ? = 2
[1,4,3,5,2] => [1,4,2,3,5] => [.,[[.,.],[.,[.,.]]]]
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> ? = 2
[1,4,5,2,3] => [1,4,5,2,3] => [.,[[.,[.,.]],[.,.]]]
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> ? = 2
[1,4,5,3,2] => [1,4,5,2,3] => [.,[[.,[.,.]],[.,.]]]
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> ? = 2
[1,5,2,3,4] => [1,5,2,3,4] => [.,[[.,.],[.,[.,.]]]]
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> ? = 2
[1,5,2,4,3] => [1,5,2,4,3] => [.,[[.,.],[[.,.],.]]]
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> ? = 2
[1,5,3,2,4] => [1,5,2,4,3] => [.,[[.,.],[[.,.],.]]]
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> ? = 2
[1,5,3,4,2] => [1,5,2,3,4] => [.,[[.,.],[.,[.,.]]]]
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> ? = 2
[1,5,4,2,3] => [1,5,2,3,4] => [.,[[.,.],[.,[.,.]]]]
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> ? = 2
[1,5,4,3,2] => [1,5,2,3,4] => [.,[[.,.],[.,[.,.]]]]
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> ? = 1
[2,1,3,4,5] => [1,3,4,5,2] => [.,[[.,[.,[.,.]]],.]]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ? = 2
[2,1,3,5,4] => [1,3,5,2,4] => [.,[[.,[.,.]],[.,.]]]
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> ? = 1
[2,1,4,3,5] => [1,4,2,3,5] => [.,[[.,.],[.,[.,.]]]]
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> ? = 2
[2,1,4,5,3] => [1,4,5,2,3] => [.,[[.,[.,.]],[.,.]]]
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> ? = 2
[2,1,5,3,4] => [1,5,2,3,4] => [.,[[.,.],[.,[.,.]]]]
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> ? = 2
[2,1,5,4,3] => [1,5,2,3,4] => [.,[[.,.],[.,[.,.]]]]
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> ? = 2
[2,3,1,4,5] => [1,4,5,2,3] => [.,[[.,[.,.]],[.,.]]]
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> ? = 2
[2,3,1,5,4] => [1,5,2,3,4] => [.,[[.,.],[.,[.,.]]]]
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> ? = 2
[2,3,4,1,5] => [1,5,2,3,4] => [.,[[.,.],[.,[.,.]]]]
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> ? = 2
[2,3,5,1,4] => [1,4,2,3,5] => [.,[[.,.],[.,[.,.]]]]
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> ? = 2
[1,3,2,5,4,6] => [1,3,2,5,4,6] => [.,[[.,.],[[.,.],[.,.]]]]
=> ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> 2
[1,3,2,6,4,5] => [1,3,2,6,4,5] => [.,[[.,.],[[.,.],[.,.]]]]
=> ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> 2
[1,3,2,6,5,4] => [1,3,2,6,4,5] => [.,[[.,.],[[.,.],[.,.]]]]
=> ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> 2
[1,4,2,5,3,6] => [1,4,2,5,3,6] => [.,[[.,.],[[.,.],[.,.]]]]
=> ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> 2
[1,4,2,6,3,5] => [1,4,2,6,3,5] => [.,[[.,.],[[.,.],[.,.]]]]
=> ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> 2
[1,4,2,6,5,3] => [1,4,2,6,3,5] => [.,[[.,.],[[.,.],[.,.]]]]
=> ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> 2
[1,4,3,2,6,5] => [1,4,2,6,3,5] => [.,[[.,.],[[.,.],[.,.]]]]
=> ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> 2
[1,4,3,5,2,6] => [1,4,2,6,3,5] => [.,[[.,.],[[.,.],[.,.]]]]
=> ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> 2
[1,4,3,6,2,5] => [1,4,2,5,3,6] => [.,[[.,.],[[.,.],[.,.]]]]
=> ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> 2
[1,5,2,4,3,6] => [1,5,2,4,3,6] => [.,[[.,.],[[.,.],[.,.]]]]
=> ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> 2
[1,5,2,6,3,4] => [1,5,2,6,3,4] => [.,[[.,.],[[.,.],[.,.]]]]
=> ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> 2
[1,5,2,6,4,3] => [1,5,2,6,3,4] => [.,[[.,.],[[.,.],[.,.]]]]
=> ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> 2
[1,5,3,2,6,4] => [1,5,2,6,3,4] => [.,[[.,.],[[.,.],[.,.]]]]
=> ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> 2
[1,5,3,4,2,6] => [1,5,2,6,3,4] => [.,[[.,.],[[.,.],[.,.]]]]
=> ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> 2
[1,5,3,6,2,4] => [1,5,2,4,3,6] => [.,[[.,.],[[.,.],[.,.]]]]
=> ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> 2
[1,5,4,2,6,3] => [1,5,2,6,3,4] => [.,[[.,.],[[.,.],[.,.]]]]
=> ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> 2
[1,5,4,3,2,6] => [1,5,2,6,3,4] => [.,[[.,.],[[.,.],[.,.]]]]
=> ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> 2
[1,6,2,4,3,5] => [1,6,2,4,3,5] => [.,[[.,.],[[.,.],[.,.]]]]
=> ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> 2
[1,6,2,5,3,4] => [1,6,2,5,3,4] => [.,[[.,.],[[.,.],[.,.]]]]
=> ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> 2
[1,6,2,5,4,3] => [1,6,2,5,3,4] => [.,[[.,.],[[.,.],[.,.]]]]
=> ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> 2
[1,6,3,2,5,4] => [1,6,2,5,3,4] => [.,[[.,.],[[.,.],[.,.]]]]
=> ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> 2
[1,6,3,4,2,5] => [1,6,2,5,3,4] => [.,[[.,.],[[.,.],[.,.]]]]
=> ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> 2
[1,6,3,5,2,4] => [1,6,2,4,3,5] => [.,[[.,.],[[.,.],[.,.]]]]
=> ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> 2
[1,6,4,2,5,3] => [1,6,2,5,3,4] => [.,[[.,.],[[.,.],[.,.]]]]
=> ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> 2
[1,6,4,3,2,5] => [1,6,2,5,3,4] => [.,[[.,.],[[.,.],[.,.]]]]
=> ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> 2
[1,6,5,2,4,3] => [1,6,2,4,3,5] => [.,[[.,.],[[.,.],[.,.]]]]
=> ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> 2
[1,6,5,3,2,4] => [1,6,2,4,3,5] => [.,[[.,.],[[.,.],[.,.]]]]
=> ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> 2
[2,4,1,5,3,6] => [1,5,2,4,3,6] => [.,[[.,.],[[.,.],[.,.]]]]
=> ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> 2
[2,4,1,6,3,5] => [1,6,2,4,3,5] => [.,[[.,.],[[.,.],[.,.]]]]
=> ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> 2
[2,4,1,6,5,3] => [1,6,2,4,3,5] => [.,[[.,.],[[.,.],[.,.]]]]
=> ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> 2
[2,4,3,1,6,5] => [1,6,2,4,3,5] => [.,[[.,.],[[.,.],[.,.]]]]
=> ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> 2
[2,4,3,5,1,6] => [1,6,2,4,3,5] => [.,[[.,.],[[.,.],[.,.]]]]
=> ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> 2
[2,4,3,6,1,5] => [1,5,2,4,3,6] => [.,[[.,.],[[.,.],[.,.]]]]
=> ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> 2
[2,5,1,4,3,6] => [1,4,2,5,3,6] => [.,[[.,.],[[.,.],[.,.]]]]
=> ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> 2
[2,5,1,6,3,4] => [1,6,2,5,3,4] => [.,[[.,.],[[.,.],[.,.]]]]
=> ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> 2
[2,5,1,6,4,3] => [1,6,2,5,3,4] => [.,[[.,.],[[.,.],[.,.]]]]
=> ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> 2
[2,5,3,1,6,4] => [1,6,2,5,3,4] => [.,[[.,.],[[.,.],[.,.]]]]
=> ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> 2
[2,5,3,4,1,6] => [1,6,2,5,3,4] => [.,[[.,.],[[.,.],[.,.]]]]
=> ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> 2
[2,5,3,6,1,4] => [1,4,2,5,3,6] => [.,[[.,.],[[.,.],[.,.]]]]
=> ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> 2
[2,5,4,1,6,3] => [1,6,2,5,3,4] => [.,[[.,.],[[.,.],[.,.]]]]
=> ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> 2
[2,5,4,3,1,6] => [1,6,2,5,3,4] => [.,[[.,.],[[.,.],[.,.]]]]
=> ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> 2
[2,5,4,6,1,3] => [1,3,2,5,4,6] => [.,[[.,.],[[.,.],[.,.]]]]
=> ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> 2
[2,6,1,4,3,5] => [1,4,2,6,3,5] => [.,[[.,.],[[.,.],[.,.]]]]
=> ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> 2
[2,6,1,5,3,4] => [1,5,2,6,3,4] => [.,[[.,.],[[.,.],[.,.]]]]
=> ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> 2
[2,6,1,5,4,3] => [1,5,2,6,3,4] => [.,[[.,.],[[.,.],[.,.]]]]
=> ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> 2
[2,6,3,1,5,4] => [1,5,2,6,3,4] => [.,[[.,.],[[.,.],[.,.]]]]
=> ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> 2
[2,6,3,4,1,5] => [1,5,2,6,3,4] => [.,[[.,.],[[.,.],[.,.]]]]
=> ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> 2
[2,6,3,5,1,4] => [1,4,2,6,3,5] => [.,[[.,.],[[.,.],[.,.]]]]
=> ([(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.
Mp00238: Permutations Clarke-Steingrimsson-ZengPermutations
Mp00160: Permutations graph of inversionsGraphs
Mp00154: Graphs coreGraphs
St001060: Graphs ⟶ ℤResult quality: 10% values known / values provided: 10%distinct values known / distinct values provided: 33%
Values
[1] => [1] => ([],1)
=> ([],1)
=> ? = 0 + 1
[1,2] => [1,2] => ([],2)
=> ([],1)
=> ? = 1 + 1
[1,2,3] => [1,2,3] => ([],3)
=> ([],1)
=> ? = 1 + 1
[1,3,2] => [1,3,2] => ([(1,2)],3)
=> ([(0,1)],2)
=> ? = 1 + 1
[2,1,3] => [2,1,3] => ([(1,2)],3)
=> ([(0,1)],2)
=> ? = 1 + 1
[1,2,3,4] => [1,2,3,4] => ([],4)
=> ([],1)
=> ? = 2 + 1
[1,2,4,3] => [1,2,4,3] => ([(2,3)],4)
=> ([(0,1)],2)
=> ? = 1 + 1
[1,3,2,4] => [1,3,2,4] => ([(2,3)],4)
=> ([(0,1)],2)
=> ? = 2 + 1
[1,3,4,2] => [1,4,3,2] => ([(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,2)],3)
=> 3 = 2 + 1
[1,4,2,3] => [1,4,2,3] => ([(1,3),(2,3)],4)
=> ([(0,1)],2)
=> ? = 2 + 1
[1,4,3,2] => [1,3,4,2] => ([(1,3),(2,3)],4)
=> ([(0,1)],2)
=> ? = 1 + 1
[2,1,3,4] => [2,1,3,4] => ([(2,3)],4)
=> ([(0,1)],2)
=> ? = 1 + 1
[2,1,4,3] => [2,1,4,3] => ([(0,3),(1,2)],4)
=> ([(0,1)],2)
=> ? = 2 + 1
[2,3,1,4] => [3,2,1,4] => ([(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,2)],3)
=> 3 = 2 + 1
[2,4,1,3] => [4,2,1,3] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,2)],3)
=> 3 = 2 + 1
[3,1,2,4] => [3,1,2,4] => ([(1,3),(2,3)],4)
=> ([(0,1)],2)
=> ? = 2 + 1
[3,1,4,2] => [4,3,1,2] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,2)],3)
=> 3 = 2 + 1
[3,2,1,4] => [2,3,1,4] => ([(1,3),(2,3)],4)
=> ([(0,1)],2)
=> ? = 1 + 1
[1,2,3,4,5] => [1,2,3,4,5] => ([],5)
=> ([],1)
=> ? = 2 + 1
[1,2,3,5,4] => [1,2,3,5,4] => ([(3,4)],5)
=> ([(0,1)],2)
=> ? = 2 + 1
[1,2,4,3,5] => [1,2,4,3,5] => ([(3,4)],5)
=> ([(0,1)],2)
=> ? = 2 + 1
[1,2,4,5,3] => [1,2,5,4,3] => ([(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 3 = 2 + 1
[1,2,5,3,4] => [1,2,5,3,4] => ([(2,4),(3,4)],5)
=> ([(0,1)],2)
=> ? = 2 + 1
[1,2,5,4,3] => [1,2,4,5,3] => ([(2,4),(3,4)],5)
=> ([(0,1)],2)
=> ? = 1 + 1
[1,3,2,4,5] => [1,3,2,4,5] => ([(3,4)],5)
=> ([(0,1)],2)
=> ? = 2 + 1
[1,3,2,5,4] => [1,3,2,5,4] => ([(1,4),(2,3)],5)
=> ([(0,1)],2)
=> ? = 2 + 1
[1,3,4,2,5] => [1,4,3,2,5] => ([(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 3 = 2 + 1
[1,3,4,5,2] => [1,5,3,4,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 3 = 2 + 1
[1,3,5,2,4] => [1,5,3,2,4] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 3 = 2 + 1
[1,3,5,4,2] => [1,4,3,5,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 3 = 2 + 1
[1,4,2,3,5] => [1,4,2,3,5] => ([(2,4),(3,4)],5)
=> ([(0,1)],2)
=> ? = 2 + 1
[1,4,2,5,3] => [1,5,4,2,3] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 3 = 2 + 1
[1,4,3,2,5] => [1,3,4,2,5] => ([(2,4),(3,4)],5)
=> ([(0,1)],2)
=> ? = 2 + 1
[1,4,3,5,2] => [1,5,4,3,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3 = 2 + 1
[1,4,5,2,3] => [1,5,2,4,3] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 3 = 2 + 1
[1,4,5,3,2] => [1,3,5,4,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 3 = 2 + 1
[1,5,2,3,4] => [1,5,2,3,4] => ([(1,4),(2,4),(3,4)],5)
=> ([(0,1)],2)
=> ? = 2 + 1
[1,5,2,4,3] => [1,4,5,2,3] => ([(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,1)],2)
=> ? = 2 + 1
[1,5,3,2,4] => [1,3,5,2,4] => ([(1,4),(2,3),(3,4)],5)
=> ([(0,1)],2)
=> ? = 2 + 1
[1,5,3,4,2] => [1,4,5,3,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 3 = 2 + 1
[1,5,4,2,3] => [1,4,2,5,3] => ([(1,4),(2,3),(3,4)],5)
=> ([(0,1)],2)
=> ? = 2 + 1
[1,5,4,3,2] => [1,3,4,5,2] => ([(1,4),(2,4),(3,4)],5)
=> ([(0,1)],2)
=> ? = 1 + 1
[2,1,3,4,5] => [2,1,3,4,5] => ([(3,4)],5)
=> ([(0,1)],2)
=> ? = 2 + 1
[2,1,3,5,4] => [2,1,3,5,4] => ([(1,4),(2,3)],5)
=> ([(0,1)],2)
=> ? = 1 + 1
[2,1,4,3,5] => [2,1,4,3,5] => ([(1,4),(2,3)],5)
=> ([(0,1)],2)
=> ? = 2 + 1
[2,1,4,5,3] => [2,1,5,4,3] => ([(0,1),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 3 = 2 + 1
[2,1,5,3,4] => [2,1,5,3,4] => ([(0,1),(2,4),(3,4)],5)
=> ([(0,1)],2)
=> ? = 2 + 1
[2,1,5,4,3] => [2,1,4,5,3] => ([(0,1),(2,4),(3,4)],5)
=> ([(0,1)],2)
=> ? = 2 + 1
[2,3,1,4,5] => [3,2,1,4,5] => ([(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 3 = 2 + 1
[2,3,1,5,4] => [3,2,1,5,4] => ([(0,1),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 3 = 2 + 1
[2,3,4,1,5] => [4,2,3,1,5] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 3 = 2 + 1
[2,3,5,1,4] => [5,2,3,1,4] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 3 = 2 + 1
[2,4,1,3,5] => [4,2,1,3,5] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 3 = 2 + 1
[2,4,1,5,3] => [5,2,4,1,3] => ([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 3 = 2 + 1
[2,4,3,1,5] => [3,2,4,1,5] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 3 = 2 + 1
[2,4,5,1,3] => [5,2,1,4,3] => ([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 3 = 2 + 1
[2,5,1,3,4] => [5,2,1,3,4] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 3 = 2 + 1
[2,5,1,4,3] => [4,2,5,1,3] => ([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 3 = 2 + 1
[2,5,3,1,4] => [3,2,5,1,4] => ([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 3 = 2 + 1
[2,5,4,1,3] => [4,2,1,5,3] => ([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 3 = 2 + 1
[3,1,2,4,5] => [3,1,2,4,5] => ([(2,4),(3,4)],5)
=> ([(0,1)],2)
=> ? = 2 + 1
[3,1,2,5,4] => [3,1,2,5,4] => ([(0,1),(2,4),(3,4)],5)
=> ([(0,1)],2)
=> ? = 2 + 1
[3,1,4,2,5] => [4,3,1,2,5] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 3 = 2 + 1
[3,1,4,5,2] => [5,3,1,4,2] => ([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 3 = 2 + 1
[3,1,5,2,4] => [5,3,1,2,4] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 3 = 2 + 1
[3,1,5,4,2] => [4,3,1,5,2] => ([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 3 = 2 + 1
[3,2,1,4,5] => [2,3,1,4,5] => ([(2,4),(3,4)],5)
=> ([(0,1)],2)
=> ? = 1 + 1
[3,2,1,5,4] => [2,3,1,5,4] => ([(0,1),(2,4),(3,4)],5)
=> ([(0,1)],2)
=> ? = 2 + 1
[3,2,4,1,5] => [4,3,2,1,5] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3 = 2 + 1
[3,2,5,1,4] => [5,3,2,1,4] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3 = 2 + 1
[3,4,1,2,5] => [4,1,3,2,5] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 3 = 2 + 1
[3,4,1,5,2] => [5,4,3,1,2] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3 = 2 + 1
[3,4,2,1,5] => [2,4,3,1,5] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 3 = 2 + 1
[3,5,1,2,4] => [5,1,3,2,4] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 3 = 2 + 1
[3,5,1,4,2] => [4,5,3,1,2] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 3 = 2 + 1
[3,5,2,1,4] => [2,5,3,1,4] => ([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 3 = 2 + 1
[4,1,2,3,5] => [4,1,2,3,5] => ([(1,4),(2,4),(3,4)],5)
=> ([(0,1)],2)
=> ? = 2 + 1
[4,1,2,5,3] => [5,1,4,2,3] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 3 = 2 + 1
[4,1,3,2,5] => [3,4,1,2,5] => ([(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,1)],2)
=> ? = 2 + 1
[4,1,3,5,2] => [5,4,1,3,2] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3 = 2 + 1
[4,1,5,2,3] => [5,4,1,2,3] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 3 = 2 + 1
[4,1,5,3,2] => [3,5,4,1,2] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 3 = 2 + 1
[4,2,1,3,5] => [2,4,1,3,5] => ([(1,4),(2,3),(3,4)],5)
=> ([(0,1)],2)
=> ? = 2 + 1
[4,2,1,5,3] => [2,5,4,1,3] => ([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 3 = 2 + 1
[4,2,3,1,5] => [3,4,2,1,5] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 3 = 2 + 1
[4,2,5,1,3] => [5,4,2,1,3] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3 = 2 + 1
[4,3,1,2,5] => [3,1,4,2,5] => ([(1,4),(2,3),(3,4)],5)
=> ([(0,1)],2)
=> ? = 2 + 1
[4,3,1,5,2] => [5,3,4,1,2] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 3 = 2 + 1
[4,3,2,1,5] => [2,3,4,1,5] => ([(1,4),(2,4),(3,4)],5)
=> ([(0,1)],2)
=> ? = 1 + 1
[1,2,3,4,5,6] => [1,2,3,4,5,6] => ([],6)
=> ([],1)
=> ? = 2 + 1
[1,2,3,4,6,5] => [1,2,3,4,6,5] => ([(4,5)],6)
=> ([(0,1)],2)
=> ? = 2 + 1
[1,2,3,5,4,6] => [1,2,3,5,4,6] => ([(4,5)],6)
=> ([(0,1)],2)
=> ? = 2 + 1
[1,2,3,5,6,4] => [1,2,3,6,5,4] => ([(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> 3 = 2 + 1
[1,2,3,6,4,5] => [1,2,3,6,4,5] => ([(3,5),(4,5)],6)
=> ([(0,1)],2)
=> ? = 2 + 1
[1,2,3,6,5,4] => [1,2,3,5,6,4] => ([(3,5),(4,5)],6)
=> ([(0,1)],2)
=> ? = 2 + 1
[1,2,4,3,5,6] => [1,2,4,3,5,6] => ([(4,5)],6)
=> ([(0,1)],2)
=> ? = 2 + 1
[1,2,4,3,6,5] => [1,2,4,3,6,5] => ([(2,5),(3,4)],6)
=> ([(0,1)],2)
=> ? = 2 + 1
[1,2,4,5,3,6] => [1,2,5,4,3,6] => ([(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> 3 = 2 + 1
[1,2,4,5,6,3] => [1,2,6,4,5,3] => ([(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> 3 = 2 + 1
[1,2,5,3,4,6] => [1,2,5,3,4,6] => ([(3,5),(4,5)],6)
=> ([(0,1)],2)
=> ? = 2 + 1
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.
The following 5 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St001570The minimal number of edges to add to make a graph Hamiltonian. St001632The number of indecomposable injective modules I with dimExt1(I,A)=1 for the incidence algebra A of a poset. St001880The number of 2-Gorenstein indecomposable injective modules in the incidence algebra of the lattice. St000699The toughness times the least common multiple of 1,. St001491The number of indecomposable projective-injective modules in the algebra corresponding to a subset.