Your data matches 9 different statistics following compositions of up to 3 maps.
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Matching statistic: St001486
(load all 5 compositions to match this statistic)
Mapping
Mp00248: Permutations —DEX composition⟶ Integer compositions
St001486: Integer compositions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1] => 1
[3,2,1] => [2,1] => 3
[1,4,3,2] => [1,2,1] => 4
[2,4,3,1] => [3,1] => 3
[3,4,2,1] => [3,1] => 3
[4,1,3,2] => [3,1] => 3
[4,2,3,1] => [3,1] => 3
[4,3,2,1] => [1,2,1] => 4
[1,2,5,4,3] => [2,2,1] => 5
[1,3,5,4,2] => [1,3,1] => 4
[1,4,5,3,2] => [1,3,1] => 4
[1,5,2,4,3] => [1,3,1] => 4
[1,5,3,4,2] => [1,3,1] => 4
[1,5,4,3,2] => [1,1,2,1] => 4
[2,1,5,4,3] => [2,2,1] => 5
[2,3,5,4,1] => [4,1] => 3
[2,4,5,3,1] => [4,1] => 3
[2,5,1,4,3] => [4,1] => 3
[2,5,3,4,1] => [4,1] => 3
[2,5,4,3,1] => [2,2,1] => 5
[3,1,5,4,2] => [2,2,1] => 5
[3,2,5,4,1] => [2,2,1] => 5
[3,4,5,2,1] => [4,1] => 3
[3,5,1,4,2] => [4,1] => 3
[3,5,2,4,1] => [4,1] => 3
[3,5,4,2,1] => [2,2,1] => 5
[4,1,5,3,2] => [2,2,1] => 5
[4,2,5,3,1] => [2,2,1] => 5
[4,3,5,2,1] => [1,3,1] => 4
[4,5,1,3,2] => [4,1] => 3
[4,5,2,3,1] => [4,1] => 3
[4,5,3,2,1] => [3,1,1] => 3
[5,1,2,4,3] => [4,1] => 3
[5,1,3,4,2] => [4,1] => 3
[5,1,4,3,2] => [2,2,1] => 5
[5,2,1,4,3] => [2,2,1] => 5
[5,2,3,4,1] => [4,1] => 3
[5,2,4,3,1] => [2,2,1] => 5
[5,3,1,4,2] => [1,3,1] => 4
[5,3,2,4,1] => [1,3,1] => 4
[5,3,4,2,1] => [1,3,1] => 4
[5,4,1,3,2] => [1,3,1] => 4
[5,4,2,3,1] => [1,3,1] => 4
[5,4,3,2,1] => [1,2,1,1] => 4
[1,2,3,6,5,4] => [3,2,1] => 5
[1,2,4,6,5,3] => [2,3,1] => 5
[1,2,5,6,4,3] => [2,3,1] => 5
[1,2,6,3,5,4] => [2,3,1] => 5
[1,2,6,4,5,3] => [2,3,1] => 5
[1,2,6,5,4,3] => [2,1,2,1] => 5
Description
The number of corners of the ribbon associated with an integer composition. We associate a ribbon shape to a composition $c=(c_1,\dots,c_n)$ with $c_i$ cells in the $i$-th row from bottom to top, such that the cells in two rows overlap in precisely one cell. This statistic records the total number of corners of the ribbon shape.
Matching statistic: St000777
(load all 3 compositions to match this statistic)
Mapping
Mp00248: Permutations —DEX composition⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St000777: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1] => ([],1)
 => 1
[3,2,1] => [2,1] => ([(0,2),(1,2)],3)
 => 3
[1,4,3,2] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => 4
[2,4,3,1] => [3,1] => ([(0,3),(1,3),(2,3)],4)
 => 3
[3,4,2,1] => [3,1] => ([(0,3),(1,3),(2,3)],4)
 => 3
[4,1,3,2] => [3,1] => ([(0,3),(1,3),(2,3)],4)
 => 3
[4,2,3,1] => [3,1] => ([(0,3),(1,3),(2,3)],4)
 => 3
[4,3,2,1] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => 4
[1,2,5,4,3] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 5
[1,3,5,4,2] => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => 4
[1,4,5,3,2] => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => 4
[1,5,2,4,3] => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => 4
[1,5,3,4,2] => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => 4
[1,5,4,3,2] => [1,1,2,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 4
[2,1,5,4,3] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 5
[2,3,5,4,1] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 3
[2,4,5,3,1] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 3
[2,5,1,4,3] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 3
[2,5,3,4,1] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 3
[2,5,4,3,1] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 5
[3,1,5,4,2] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 5
[3,2,5,4,1] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 5
[3,4,5,2,1] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 3
[3,5,1,4,2] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 3
[3,5,2,4,1] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 3
[3,5,4,2,1] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 5
[4,1,5,3,2] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 5
[4,2,5,3,1] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 5
[4,3,5,2,1] => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => 4
[4,5,1,3,2] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 3
[4,5,2,3,1] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 3
[4,5,3,2,1] => [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 3
[5,1,2,4,3] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 3
[5,1,3,4,2] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 3
[5,1,4,3,2] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 5
[5,2,1,4,3] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 5
[5,2,3,4,1] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 3
[5,2,4,3,1] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 5
[5,3,1,4,2] => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => 4
[5,3,2,4,1] => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => 4
[5,3,4,2,1] => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => 4
[5,4,1,3,2] => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => 4
[5,4,2,3,1] => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => 4
[5,4,3,2,1] => [1,2,1,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 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)
 => 5
[1,2,4,6,5,3] => [2,3,1] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 5
[1,2,5,6,4,3] => [2,3,1] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 5
[1,2,6,3,5,4] => [2,3,1] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 5
[1,2,6,4,5,3] => [2,3,1] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 5
[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)
 => 5
Description
The number of distinct eigenvalues of the distance Laplacian of a connected graph.
Matching statistic: St000340
Mapping
Mp00248: Permutations —DEX composition⟶ Integer compositions
Mp00041: Integer compositions —conjugate⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St000340: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1] => [1] => [1,0]
 => 0 = 1 - 1
[3,2,1] => [2,1] => [2,1] => [1,1,0,0,1,0]
 => 2 = 3 - 1
[1,4,3,2] => [1,2,1] => [2,2] => [1,1,0,0,1,1,0,0]
 => 3 = 4 - 1
[2,4,3,1] => [3,1] => [2,1,1] => [1,1,0,0,1,0,1,0]
 => 2 = 3 - 1
[3,4,2,1] => [3,1] => [2,1,1] => [1,1,0,0,1,0,1,0]
 => 2 = 3 - 1
[4,1,3,2] => [3,1] => [2,1,1] => [1,1,0,0,1,0,1,0]
 => 2 = 3 - 1
[4,2,3,1] => [3,1] => [2,1,1] => [1,1,0,0,1,0,1,0]
 => 2 = 3 - 1
[4,3,2,1] => [1,2,1] => [2,2] => [1,1,0,0,1,1,0,0]
 => 3 = 4 - 1
[1,2,5,4,3] => [2,2,1] => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
 => 4 = 5 - 1
[1,3,5,4,2] => [1,3,1] => [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
 => 3 = 4 - 1
[1,4,5,3,2] => [1,3,1] => [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
 => 3 = 4 - 1
[1,5,2,4,3] => [1,3,1] => [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
 => 3 = 4 - 1
[1,5,3,4,2] => [1,3,1] => [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
 => 3 = 4 - 1
[1,5,4,3,2] => [1,1,2,1] => [2,3] => [1,1,0,0,1,1,1,0,0,0]
 => 3 = 4 - 1
[2,1,5,4,3] => [2,2,1] => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
 => 4 = 5 - 1
[2,3,5,4,1] => [4,1] => [2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
 => 2 = 3 - 1
[2,4,5,3,1] => [4,1] => [2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
 => 2 = 3 - 1
[2,5,1,4,3] => [4,1] => [2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
 => 2 = 3 - 1
[2,5,3,4,1] => [4,1] => [2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
 => 2 = 3 - 1
[2,5,4,3,1] => [2,2,1] => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
 => 4 = 5 - 1
[3,1,5,4,2] => [2,2,1] => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
 => 4 = 5 - 1
[3,2,5,4,1] => [2,2,1] => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
 => 4 = 5 - 1
[3,4,5,2,1] => [4,1] => [2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
 => 2 = 3 - 1
[3,5,1,4,2] => [4,1] => [2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
 => 2 = 3 - 1
[3,5,2,4,1] => [4,1] => [2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
 => 2 = 3 - 1
[3,5,4,2,1] => [2,2,1] => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
 => 4 = 5 - 1
[4,1,5,3,2] => [2,2,1] => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
 => 4 = 5 - 1
[4,2,5,3,1] => [2,2,1] => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
 => 4 = 5 - 1
[4,3,5,2,1] => [1,3,1] => [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
 => 3 = 4 - 1
[4,5,1,3,2] => [4,1] => [2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
 => 2 = 3 - 1
[4,5,2,3,1] => [4,1] => [2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
 => 2 = 3 - 1
[4,5,3,2,1] => [3,1,1] => [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
 => 2 = 3 - 1
[5,1,2,4,3] => [4,1] => [2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
 => 2 = 3 - 1
[5,1,3,4,2] => [4,1] => [2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
 => 2 = 3 - 1
[5,1,4,3,2] => [2,2,1] => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
 => 4 = 5 - 1
[5,2,1,4,3] => [2,2,1] => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
 => 4 = 5 - 1
[5,2,3,4,1] => [4,1] => [2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
 => 2 = 3 - 1
[5,2,4,3,1] => [2,2,1] => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
 => 4 = 5 - 1
[5,3,1,4,2] => [1,3,1] => [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
 => 3 = 4 - 1
[5,3,2,4,1] => [1,3,1] => [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
 => 3 = 4 - 1
[5,3,4,2,1] => [1,3,1] => [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
 => 3 = 4 - 1
[5,4,1,3,2] => [1,3,1] => [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
 => 3 = 4 - 1
[5,4,2,3,1] => [1,3,1] => [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
 => 3 = 4 - 1
[5,4,3,2,1] => [1,2,1,1] => [3,2] => [1,1,1,0,0,0,1,1,0,0]
 => 3 = 4 - 1
[1,2,3,6,5,4] => [3,2,1] => [2,2,1,1] => [1,1,0,0,1,1,0,0,1,0,1,0]
 => 4 = 5 - 1
[1,2,4,6,5,3] => [2,3,1] => [2,1,2,1] => [1,1,0,0,1,0,1,1,0,0,1,0]
 => 4 = 5 - 1
[1,2,5,6,4,3] => [2,3,1] => [2,1,2,1] => [1,1,0,0,1,0,1,1,0,0,1,0]
 => 4 = 5 - 1
[1,2,6,3,5,4] => [2,3,1] => [2,1,2,1] => [1,1,0,0,1,0,1,1,0,0,1,0]
 => 4 = 5 - 1
[1,2,6,4,5,3] => [2,3,1] => [2,1,2,1] => [1,1,0,0,1,0,1,1,0,0,1,0]
 => 4 = 5 - 1
[1,2,6,5,4,3] => [2,1,2,1] => [2,3,1] => [1,1,0,0,1,1,1,0,0,0,1,0]
 => 4 = 5 - 1
Description
The number of non-final maximal constant sub-paths of length greater than one. This is the total number of occurrences of the patterns $110$ and $001$.
Matching statistic: St000691
Mapping
Mp00248: Permutations —DEX composition⟶ Integer compositions
Mp00041: Integer compositions —conjugate⟶ Integer compositions
Mp00094: Integer compositions —to binary word⟶ Binary words
St000691: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1] => [1] => 1 => 0 = 1 - 1
[3,2,1] => [2,1] => [2,1] => 101 => 2 = 3 - 1
[1,4,3,2] => [1,2,1] => [2,2] => 1010 => 3 = 4 - 1
[2,4,3,1] => [3,1] => [2,1,1] => 1011 => 2 = 3 - 1
[3,4,2,1] => [3,1] => [2,1,1] => 1011 => 2 = 3 - 1
[4,1,3,2] => [3,1] => [2,1,1] => 1011 => 2 = 3 - 1
[4,2,3,1] => [3,1] => [2,1,1] => 1011 => 2 = 3 - 1
[4,3,2,1] => [1,2,1] => [2,2] => 1010 => 3 = 4 - 1
[1,2,5,4,3] => [2,2,1] => [2,2,1] => 10101 => 4 = 5 - 1
[1,3,5,4,2] => [1,3,1] => [2,1,2] => 10110 => 3 = 4 - 1
[1,4,5,3,2] => [1,3,1] => [2,1,2] => 10110 => 3 = 4 - 1
[1,5,2,4,3] => [1,3,1] => [2,1,2] => 10110 => 3 = 4 - 1
[1,5,3,4,2] => [1,3,1] => [2,1,2] => 10110 => 3 = 4 - 1
[1,5,4,3,2] => [1,1,2,1] => [2,3] => 10100 => 3 = 4 - 1
[2,1,5,4,3] => [2,2,1] => [2,2,1] => 10101 => 4 = 5 - 1
[2,3,5,4,1] => [4,1] => [2,1,1,1] => 10111 => 2 = 3 - 1
[2,4,5,3,1] => [4,1] => [2,1,1,1] => 10111 => 2 = 3 - 1
[2,5,1,4,3] => [4,1] => [2,1,1,1] => 10111 => 2 = 3 - 1
[2,5,3,4,1] => [4,1] => [2,1,1,1] => 10111 => 2 = 3 - 1
[2,5,4,3,1] => [2,2,1] => [2,2,1] => 10101 => 4 = 5 - 1
[3,1,5,4,2] => [2,2,1] => [2,2,1] => 10101 => 4 = 5 - 1
[3,2,5,4,1] => [2,2,1] => [2,2,1] => 10101 => 4 = 5 - 1
[3,4,5,2,1] => [4,1] => [2,1,1,1] => 10111 => 2 = 3 - 1
[3,5,1,4,2] => [4,1] => [2,1,1,1] => 10111 => 2 = 3 - 1
[3,5,2,4,1] => [4,1] => [2,1,1,1] => 10111 => 2 = 3 - 1
[3,5,4,2,1] => [2,2,1] => [2,2,1] => 10101 => 4 = 5 - 1
[4,1,5,3,2] => [2,2,1] => [2,2,1] => 10101 => 4 = 5 - 1
[4,2,5,3,1] => [2,2,1] => [2,2,1] => 10101 => 4 = 5 - 1
[4,3,5,2,1] => [1,3,1] => [2,1,2] => 10110 => 3 = 4 - 1
[4,5,1,3,2] => [4,1] => [2,1,1,1] => 10111 => 2 = 3 - 1
[4,5,2,3,1] => [4,1] => [2,1,1,1] => 10111 => 2 = 3 - 1
[4,5,3,2,1] => [3,1,1] => [3,1,1] => 10011 => 2 = 3 - 1
[5,1,2,4,3] => [4,1] => [2,1,1,1] => 10111 => 2 = 3 - 1
[5,1,3,4,2] => [4,1] => [2,1,1,1] => 10111 => 2 = 3 - 1
[5,1,4,3,2] => [2,2,1] => [2,2,1] => 10101 => 4 = 5 - 1
[5,2,1,4,3] => [2,2,1] => [2,2,1] => 10101 => 4 = 5 - 1
[5,2,3,4,1] => [4,1] => [2,1,1,1] => 10111 => 2 = 3 - 1
[5,2,4,3,1] => [2,2,1] => [2,2,1] => 10101 => 4 = 5 - 1
[5,3,1,4,2] => [1,3,1] => [2,1,2] => 10110 => 3 = 4 - 1
[5,3,2,4,1] => [1,3,1] => [2,1,2] => 10110 => 3 = 4 - 1
[5,3,4,2,1] => [1,3,1] => [2,1,2] => 10110 => 3 = 4 - 1
[5,4,1,3,2] => [1,3,1] => [2,1,2] => 10110 => 3 = 4 - 1
[5,4,2,3,1] => [1,3,1] => [2,1,2] => 10110 => 3 = 4 - 1
[5,4,3,2,1] => [1,2,1,1] => [3,2] => 10010 => 3 = 4 - 1
[1,2,3,6,5,4] => [3,2,1] => [2,2,1,1] => 101011 => 4 = 5 - 1
[1,2,4,6,5,3] => [2,3,1] => [2,1,2,1] => 101101 => 4 = 5 - 1
[1,2,5,6,4,3] => [2,3,1] => [2,1,2,1] => 101101 => 4 = 5 - 1
[1,2,6,3,5,4] => [2,3,1] => [2,1,2,1] => 101101 => 4 = 5 - 1
[1,2,6,4,5,3] => [2,3,1] => [2,1,2,1] => 101101 => 4 = 5 - 1
[1,2,6,5,4,3] => [2,1,2,1] => [2,3,1] => 101001 => 4 = 5 - 1
Description
The number of changes of a binary word. This is the number of indices $i$ such that $w_i \neq w_{i+1}$.
Matching statistic: St001035
(load all 4 compositions to match this statistic)
Mapping
Mp00248: Permutations —DEX composition⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St001035: Dyck paths ⟶ ℤResult quality: 83% ●values known / values provided: 100%●distinct values known / distinct values provided: 83%
Values
[1] => [1] => [1,0]
 => ? = 1 - 2
[3,2,1] => [2,1] => [1,1,0,0,1,0]
 => 1 = 3 - 2
[1,4,3,2] => [1,2,1] => [1,0,1,1,0,0,1,0]
 => 2 = 4 - 2
[2,4,3,1] => [3,1] => [1,1,1,0,0,0,1,0]
 => 1 = 3 - 2
[3,4,2,1] => [3,1] => [1,1,1,0,0,0,1,0]
 => 1 = 3 - 2
[4,1,3,2] => [3,1] => [1,1,1,0,0,0,1,0]
 => 1 = 3 - 2
[4,2,3,1] => [3,1] => [1,1,1,0,0,0,1,0]
 => 1 = 3 - 2
[4,3,2,1] => [1,2,1] => [1,0,1,1,0,0,1,0]
 => 2 = 4 - 2
[1,2,5,4,3] => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
 => 3 = 5 - 2
[1,3,5,4,2] => [1,3,1] => [1,0,1,1,1,0,0,0,1,0]
 => 2 = 4 - 2
[1,4,5,3,2] => [1,3,1] => [1,0,1,1,1,0,0,0,1,0]
 => 2 = 4 - 2
[1,5,2,4,3] => [1,3,1] => [1,0,1,1,1,0,0,0,1,0]
 => 2 = 4 - 2
[1,5,3,4,2] => [1,3,1] => [1,0,1,1,1,0,0,0,1,0]
 => 2 = 4 - 2
[1,5,4,3,2] => [1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
 => 2 = 4 - 2
[2,1,5,4,3] => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
 => 3 = 5 - 2
[2,3,5,4,1] => [4,1] => [1,1,1,1,0,0,0,0,1,0]
 => 1 = 3 - 2
[2,4,5,3,1] => [4,1] => [1,1,1,1,0,0,0,0,1,0]
 => 1 = 3 - 2
[2,5,1,4,3] => [4,1] => [1,1,1,1,0,0,0,0,1,0]
 => 1 = 3 - 2
[2,5,3,4,1] => [4,1] => [1,1,1,1,0,0,0,0,1,0]
 => 1 = 3 - 2
[2,5,4,3,1] => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
 => 3 = 5 - 2
[3,1,5,4,2] => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
 => 3 = 5 - 2
[3,2,5,4,1] => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
 => 3 = 5 - 2
[3,4,5,2,1] => [4,1] => [1,1,1,1,0,0,0,0,1,0]
 => 1 = 3 - 2
[3,5,1,4,2] => [4,1] => [1,1,1,1,0,0,0,0,1,0]
 => 1 = 3 - 2
[3,5,2,4,1] => [4,1] => [1,1,1,1,0,0,0,0,1,0]
 => 1 = 3 - 2
[3,5,4,2,1] => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
 => 3 = 5 - 2
[4,1,5,3,2] => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
 => 3 = 5 - 2
[4,2,5,3,1] => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
 => 3 = 5 - 2
[4,3,5,2,1] => [1,3,1] => [1,0,1,1,1,0,0,0,1,0]
 => 2 = 4 - 2
[4,5,1,3,2] => [4,1] => [1,1,1,1,0,0,0,0,1,0]
 => 1 = 3 - 2
[4,5,2,3,1] => [4,1] => [1,1,1,1,0,0,0,0,1,0]
 => 1 = 3 - 2
[4,5,3,2,1] => [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
 => 1 = 3 - 2
[5,1,2,4,3] => [4,1] => [1,1,1,1,0,0,0,0,1,0]
 => 1 = 3 - 2
[5,1,3,4,2] => [4,1] => [1,1,1,1,0,0,0,0,1,0]
 => 1 = 3 - 2
[5,1,4,3,2] => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
 => 3 = 5 - 2
[5,2,1,4,3] => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
 => 3 = 5 - 2
[5,2,3,4,1] => [4,1] => [1,1,1,1,0,0,0,0,1,0]
 => 1 = 3 - 2
[5,2,4,3,1] => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
 => 3 = 5 - 2
[5,3,1,4,2] => [1,3,1] => [1,0,1,1,1,0,0,0,1,0]
 => 2 = 4 - 2
[5,3,2,4,1] => [1,3,1] => [1,0,1,1,1,0,0,0,1,0]
 => 2 = 4 - 2
[5,3,4,2,1] => [1,3,1] => [1,0,1,1,1,0,0,0,1,0]
 => 2 = 4 - 2
[5,4,1,3,2] => [1,3,1] => [1,0,1,1,1,0,0,0,1,0]
 => 2 = 4 - 2
[5,4,2,3,1] => [1,3,1] => [1,0,1,1,1,0,0,0,1,0]
 => 2 = 4 - 2
[5,4,3,2,1] => [1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
 => 2 = 4 - 2
[1,2,3,6,5,4] => [3,2,1] => [1,1,1,0,0,0,1,1,0,0,1,0]
 => 3 = 5 - 2
[1,2,4,6,5,3] => [2,3,1] => [1,1,0,0,1,1,1,0,0,0,1,0]
 => 3 = 5 - 2
[1,2,5,6,4,3] => [2,3,1] => [1,1,0,0,1,1,1,0,0,0,1,0]
 => 3 = 5 - 2
[1,2,6,3,5,4] => [2,3,1] => [1,1,0,0,1,1,1,0,0,0,1,0]
 => 3 = 5 - 2
[1,2,6,4,5,3] => [2,3,1] => [1,1,0,0,1,1,1,0,0,0,1,0]
 => 3 = 5 - 2
[1,2,6,5,4,3] => [2,1,2,1] => [1,1,0,0,1,0,1,1,0,0,1,0]
 => 3 = 5 - 2
[1,3,2,6,5,4] => [1,2,2,1] => [1,0,1,1,0,0,1,1,0,0,1,0]
 => 4 = 6 - 2
Description
The convexity degree of the parallelogram polyomino associated with the Dyck path. A parallelogram polyomino is $k$-convex if $k$ is the maximal number of turns an axis-parallel path must take to connect two cells of the polyomino. For example, any rotation of a Ferrers shape has convexity degree at most one. The (bivariate) generating function is given in Theorem 2 of [1].
Matching statistic: St000453
Mapping
Mp00248: Permutations —DEX composition⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St000453: Graphs ⟶ ℤResult quality: 99% ●values known / values provided: 99%●distinct values known / distinct values provided: 100%
Values
[1] => [1] => ([],1)
 => 1
[3,2,1] => [2,1] => ([(0,2),(1,2)],3)
 => 3
[1,4,3,2] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => 4
[2,4,3,1] => [3,1] => ([(0,3),(1,3),(2,3)],4)
 => 3
[3,4,2,1] => [3,1] => ([(0,3),(1,3),(2,3)],4)
 => 3
[4,1,3,2] => [3,1] => ([(0,3),(1,3),(2,3)],4)
 => 3
[4,2,3,1] => [3,1] => ([(0,3),(1,3),(2,3)],4)
 => 3
[4,3,2,1] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => 4
[1,2,5,4,3] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 5
[1,3,5,4,2] => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => 4
[1,4,5,3,2] => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => 4
[1,5,2,4,3] => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => 4
[1,5,3,4,2] => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => 4
[1,5,4,3,2] => [1,1,2,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 4
[2,1,5,4,3] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 5
[2,3,5,4,1] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 3
[2,4,5,3,1] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 3
[2,5,1,4,3] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 3
[2,5,3,4,1] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 3
[2,5,4,3,1] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 5
[3,1,5,4,2] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 5
[3,2,5,4,1] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 5
[3,4,5,2,1] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 3
[3,5,1,4,2] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 3
[3,5,2,4,1] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 3
[3,5,4,2,1] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 5
[4,1,5,3,2] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 5
[4,2,5,3,1] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 5
[4,3,5,2,1] => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => 4
[4,5,1,3,2] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 3
[4,5,2,3,1] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 3
[4,5,3,2,1] => [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 3
[5,1,2,4,3] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 3
[5,1,3,4,2] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 3
[5,1,4,3,2] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 5
[5,2,1,4,3] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 5
[5,2,3,4,1] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 3
[5,2,4,3,1] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 5
[5,3,1,4,2] => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => 4
[5,3,2,4,1] => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => 4
[5,3,4,2,1] => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => 4
[5,4,1,3,2] => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => 4
[5,4,2,3,1] => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => 4
[5,4,3,2,1] => [1,2,1,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 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)
 => 5
[1,2,4,6,5,3] => [2,3,1] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 5
[1,2,5,6,4,3] => [2,3,1] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 5
[1,2,6,3,5,4] => [2,3,1] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 5
[1,2,6,4,5,3] => [2,3,1] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 5
[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)
 => 5
[1,6,4,7,5,3,2] => [1,1,3,1,1] => ([(0,5),(0,6),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 4
[1,6,5,7,4,3,2] => [1,1,3,1,1] => ([(0,5),(0,6),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 4
[1,7,4,6,5,3,2] => [1,1,3,1,1] => ([(0,5),(0,6),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 4
[1,7,5,6,4,3,2] => [1,1,3,1,1] => ([(0,5),(0,6),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 4
[1,7,6,2,5,4,3] => [1,1,3,1,1] => ([(0,5),(0,6),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 4
[1,7,6,3,5,4,2] => [1,1,3,1,1] => ([(0,5),(0,6),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 4
[1,7,6,4,5,3,2] => [1,1,3,1,1] => ([(0,5),(0,6),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 4
[6,5,4,7,3,2,1] => [1,1,3,1,1] => ([(0,5),(0,6),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 4
[7,5,4,6,3,2,1] => [1,1,3,1,1] => ([(0,5),(0,6),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 4
[7,6,4,1,5,3,2] => [1,1,3,1,1] => ([(0,5),(0,6),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 4
[7,6,4,2,5,3,1] => [1,1,3,1,1] => ([(0,5),(0,6),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 4
[7,6,4,3,5,2,1] => [1,1,3,1,1] => ([(0,5),(0,6),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 4
[7,6,4,5,3,2,1] => [1,1,3,1,1] => ([(0,5),(0,6),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 4
[7,6,5,1,4,3,2] => [1,1,3,1,1] => ([(0,5),(0,6),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 4
[7,6,5,2,4,3,1] => [1,1,3,1,1] => ([(0,5),(0,6),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 4
[7,6,5,3,4,2,1] => [1,1,3,1,1] => ([(0,5),(0,6),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 4
Description
The number of distinct Laplacian eigenvalues of a graph.
Matching statistic: St000455
(load all 2 compositions to match this statistic)
Mapping
Mp00248: Permutations —DEX composition⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St000455: Graphs ⟶ ℤResult quality: 8% ●values known / values provided: 8%●distinct values known / distinct values provided: 17%
Values
[1] => [1] => ([],1)
 => ? = 1 - 3
[3,2,1] => [2,1] => ([(0,2),(1,2)],3)
 => 0 = 3 - 3
[1,4,3,2] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => ? = 4 - 3
[2,4,3,1] => [3,1] => ([(0,3),(1,3),(2,3)],4)
 => 0 = 3 - 3
[3,4,2,1] => [3,1] => ([(0,3),(1,3),(2,3)],4)
 => 0 = 3 - 3
[4,1,3,2] => [3,1] => ([(0,3),(1,3),(2,3)],4)
 => 0 = 3 - 3
[4,2,3,1] => [3,1] => ([(0,3),(1,3),(2,3)],4)
 => 0 = 3 - 3
[4,3,2,1] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => ? = 4 - 3
[1,2,5,4,3] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 5 - 3
[1,3,5,4,2] => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 4 - 3
[1,4,5,3,2] => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 4 - 3
[1,5,2,4,3] => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 4 - 3
[1,5,3,4,2] => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 4 - 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)
 => ? = 4 - 3
[2,1,5,4,3] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 5 - 3
[2,3,5,4,1] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 0 = 3 - 3
[2,4,5,3,1] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 0 = 3 - 3
[2,5,1,4,3] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 0 = 3 - 3
[2,5,3,4,1] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 0 = 3 - 3
[2,5,4,3,1] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 5 - 3
[3,1,5,4,2] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 5 - 3
[3,2,5,4,1] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 5 - 3
[3,4,5,2,1] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 0 = 3 - 3
[3,5,1,4,2] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 0 = 3 - 3
[3,5,2,4,1] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 0 = 3 - 3
[3,5,4,2,1] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 5 - 3
[4,1,5,3,2] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 5 - 3
[4,2,5,3,1] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 5 - 3
[4,3,5,2,1] => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 4 - 3
[4,5,1,3,2] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 0 = 3 - 3
[4,5,2,3,1] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 0 = 3 - 3
[4,5,3,2,1] => [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 0 = 3 - 3
[5,1,2,4,3] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 0 = 3 - 3
[5,1,3,4,2] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 0 = 3 - 3
[5,1,4,3,2] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 5 - 3
[5,2,1,4,3] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 5 - 3
[5,2,3,4,1] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 0 = 3 - 3
[5,2,4,3,1] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 5 - 3
[5,3,1,4,2] => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 4 - 3
[5,3,2,4,1] => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 4 - 3
[5,3,4,2,1] => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 4 - 3
[5,4,1,3,2] => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 4 - 3
[5,4,2,3,1] => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 4 - 3
[5,4,3,2,1] => [1,2,1,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 4 - 3
[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)
 => ? = 5 - 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)
 => ? = 5 - 3
[1,2,5,6,4,3] => [2,3,1] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 5 - 3
[1,2,6,3,5,4] => [2,3,1] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 5 - 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)
 => ? = 5 - 3
[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)
 => ? = 5 - 3
[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)
 => ? = 6 - 3
[1,3,4,6,5,2] => [1,4,1] => ([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 4 - 3
[1,3,5,6,4,2] => [1,4,1] => ([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 4 - 3
[1,3,6,2,5,4] => [1,4,1] => ([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 4 - 3
[1,3,6,4,5,2] => [1,4,1] => ([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 4 - 3
[1,3,6,5,4,2] => [1,2,2,1] => ([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 6 - 3
[1,4,2,6,5,3] => [1,2,2,1] => ([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 6 - 3
[1,4,3,6,5,2] => [1,2,2,1] => ([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 6 - 3
[1,4,5,6,3,2] => [1,4,1] => ([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 4 - 3
[1,4,6,2,5,3] => [1,4,1] => ([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 4 - 3
[1,4,6,3,5,2] => [1,4,1] => ([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 4 - 3
[1,4,6,5,3,2] => [1,2,2,1] => ([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 6 - 3
[1,5,2,6,4,3] => [1,2,2,1] => ([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 6 - 3
[1,5,3,6,4,2] => [1,2,2,1] => ([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 6 - 3
[1,5,4,6,3,2] => [1,1,3,1] => ([(0,5),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 4 - 3
[1,5,6,2,4,3] => [1,4,1] => ([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 4 - 3
[1,5,6,3,4,2] => [1,4,1] => ([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 4 - 3
[1,5,6,4,3,2] => [1,3,1,1] => ([(0,4),(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 4 - 3
[2,3,4,6,5,1] => [5,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
 => 0 = 3 - 3
[2,3,5,6,4,1] => [5,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
 => 0 = 3 - 3
[2,3,6,1,5,4] => [5,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
 => 0 = 3 - 3
[2,3,6,4,5,1] => [5,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
 => 0 = 3 - 3
[2,4,5,6,3,1] => [5,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
 => 0 = 3 - 3
[2,4,6,1,5,3] => [5,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
 => 0 = 3 - 3
[2,4,6,3,5,1] => [5,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
 => 0 = 3 - 3
[2,5,6,1,4,3] => [5,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
 => 0 = 3 - 3
[2,5,6,3,4,1] => [5,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
 => 0 = 3 - 3
[2,5,6,4,3,1] => [4,1,1] => ([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 0 = 3 - 3
[2,6,1,3,5,4] => [5,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
 => 0 = 3 - 3
[2,6,1,4,5,3] => [5,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
 => 0 = 3 - 3
[2,6,3,4,5,1] => [5,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
 => 0 = 3 - 3
[3,4,5,6,2,1] => [5,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
 => 0 = 3 - 3
[3,4,6,1,5,2] => [5,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
 => 0 = 3 - 3
[3,4,6,2,5,1] => [5,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
 => 0 = 3 - 3
[3,5,6,1,4,2] => [5,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
 => 0 = 3 - 3
[3,5,6,2,4,1] => [5,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
 => 0 = 3 - 3
[3,5,6,4,2,1] => [4,1,1] => ([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 0 = 3 - 3
[3,6,1,2,5,4] => [5,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
 => 0 = 3 - 3
[3,6,1,4,5,2] => [5,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
 => 0 = 3 - 3
[3,6,2,4,5,1] => [5,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
 => 0 = 3 - 3
[4,5,6,1,3,2] => [5,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
 => 0 = 3 - 3
[4,5,6,2,3,1] => [5,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
 => 0 = 3 - 3
[4,5,6,3,2,1] => [4,1,1] => ([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 0 = 3 - 3
[4,6,1,2,5,3] => [5,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
 => 0 = 3 - 3
[4,6,1,3,5,2] => [5,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
 => 0 = 3 - 3
[4,6,2,3,5,1] => [5,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
 => 0 = 3 - 3
[5,6,1,2,4,3] => [5,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
 => 0 = 3 - 3
[5,6,1,3,4,2] => [5,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
 => 0 = 3 - 3
[5,6,1,4,3,2] => [4,1,1] => ([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 0 = 3 - 3
[5,6,2,3,4,1] => [5,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
 => 0 = 3 - 3
Description
The second largest eigenvalue of a graph if it is integral. This statistic is undefined if the second largest eigenvalue of the graph is not integral. Chapter 4 of [1] provides lots of context.
Matching statistic: St001488
(load all 4 compositions to match this statistic)
Mapping
Mp00248: Permutations —DEX composition⟶ Integer compositions
Mp00180: Integer compositions —to ribbon⟶ Skew partitions
St001488: Skew partitions ⟶ ℤResult quality: 2% ●values known / values provided: 2%●distinct values known / distinct values provided: 67%
Values
[1] => [1] => [[1],[]]
 => 1
[3,2,1] => [2,1] => [[2,2],[1]]
 => 3
[1,4,3,2] => [1,2,1] => [[2,2,1],[1]]
 => 4
[2,4,3,1] => [3,1] => [[3,3],[2]]
 => 3
[3,4,2,1] => [3,1] => [[3,3],[2]]
 => 3
[4,1,3,2] => [3,1] => [[3,3],[2]]
 => 3
[4,2,3,1] => [3,1] => [[3,3],[2]]
 => 3
[4,3,2,1] => [1,2,1] => [[2,2,1],[1]]
 => 4
[1,2,5,4,3] => [2,2,1] => [[3,3,2],[2,1]]
 => 5
[1,3,5,4,2] => [1,3,1] => [[3,3,1],[2]]
 => 4
[1,4,5,3,2] => [1,3,1] => [[3,3,1],[2]]
 => 4
[1,5,2,4,3] => [1,3,1] => [[3,3,1],[2]]
 => 4
[1,5,3,4,2] => [1,3,1] => [[3,3,1],[2]]
 => 4
[1,5,4,3,2] => [1,1,2,1] => [[2,2,1,1],[1]]
 => 4
[2,1,5,4,3] => [2,2,1] => [[3,3,2],[2,1]]
 => 5
[2,3,5,4,1] => [4,1] => [[4,4],[3]]
 => 3
[2,4,5,3,1] => [4,1] => [[4,4],[3]]
 => 3
[2,5,1,4,3] => [4,1] => [[4,4],[3]]
 => 3
[2,5,3,4,1] => [4,1] => [[4,4],[3]]
 => 3
[2,5,4,3,1] => [2,2,1] => [[3,3,2],[2,1]]
 => 5
[3,1,5,4,2] => [2,2,1] => [[3,3,2],[2,1]]
 => 5
[3,2,5,4,1] => [2,2,1] => [[3,3,2],[2,1]]
 => 5
[3,4,5,2,1] => [4,1] => [[4,4],[3]]
 => 3
[3,5,1,4,2] => [4,1] => [[4,4],[3]]
 => 3
[3,5,2,4,1] => [4,1] => [[4,4],[3]]
 => 3
[3,5,4,2,1] => [2,2,1] => [[3,3,2],[2,1]]
 => 5
[4,1,5,3,2] => [2,2,1] => [[3,3,2],[2,1]]
 => 5
[4,2,5,3,1] => [2,2,1] => [[3,3,2],[2,1]]
 => 5
[4,3,5,2,1] => [1,3,1] => [[3,3,1],[2]]
 => 4
[4,5,1,3,2] => [4,1] => [[4,4],[3]]
 => 3
[4,5,2,3,1] => [4,1] => [[4,4],[3]]
 => 3
[4,5,3,2,1] => [3,1,1] => [[3,3,3],[2,2]]
 => 3
[5,1,2,4,3] => [4,1] => [[4,4],[3]]
 => 3
[5,1,3,4,2] => [4,1] => [[4,4],[3]]
 => 3
[5,1,4,3,2] => [2,2,1] => [[3,3,2],[2,1]]
 => 5
[5,2,1,4,3] => [2,2,1] => [[3,3,2],[2,1]]
 => 5
[5,2,3,4,1] => [4,1] => [[4,4],[3]]
 => 3
[5,2,4,3,1] => [2,2,1] => [[3,3,2],[2,1]]
 => 5
[5,3,1,4,2] => [1,3,1] => [[3,3,1],[2]]
 => 4
[5,3,2,4,1] => [1,3,1] => [[3,3,1],[2]]
 => 4
[5,3,4,2,1] => [1,3,1] => [[3,3,1],[2]]
 => 4
[5,4,1,3,2] => [1,3,1] => [[3,3,1],[2]]
 => 4
[5,4,2,3,1] => [1,3,1] => [[3,3,1],[2]]
 => 4
[5,4,3,2,1] => [1,2,1,1] => [[2,2,2,1],[1,1]]
 => 4
[1,2,3,6,5,4] => [3,2,1] => [[4,4,3],[3,2]]
 => ? = 5
[1,2,4,6,5,3] => [2,3,1] => [[4,4,2],[3,1]]
 => ? = 5
[1,2,5,6,4,3] => [2,3,1] => [[4,4,2],[3,1]]
 => ? = 5
[1,2,6,3,5,4] => [2,3,1] => [[4,4,2],[3,1]]
 => ? = 5
[1,2,6,4,5,3] => [2,3,1] => [[4,4,2],[3,1]]
 => ? = 5
[1,2,6,5,4,3] => [2,1,2,1] => [[3,3,2,2],[2,1,1]]
 => ? = 5
[1,3,2,6,5,4] => [1,2,2,1] => [[3,3,2,1],[2,1]]
 => ? = 6
[1,3,4,6,5,2] => [1,4,1] => [[4,4,1],[3]]
 => ? = 4
[1,3,5,6,4,2] => [1,4,1] => [[4,4,1],[3]]
 => ? = 4
[1,3,6,2,5,4] => [1,4,1] => [[4,4,1],[3]]
 => ? = 4
[1,3,6,4,5,2] => [1,4,1] => [[4,4,1],[3]]
 => ? = 4
[1,3,6,5,4,2] => [1,2,2,1] => [[3,3,2,1],[2,1]]
 => ? = 6
[1,4,2,6,5,3] => [1,2,2,1] => [[3,3,2,1],[2,1]]
 => ? = 6
[1,4,3,6,5,2] => [1,2,2,1] => [[3,3,2,1],[2,1]]
 => ? = 6
[1,4,5,6,3,2] => [1,4,1] => [[4,4,1],[3]]
 => ? = 4
[1,4,6,2,5,3] => [1,4,1] => [[4,4,1],[3]]
 => ? = 4
[1,4,6,3,5,2] => [1,4,1] => [[4,4,1],[3]]
 => ? = 4
[1,4,6,5,3,2] => [1,2,2,1] => [[3,3,2,1],[2,1]]
 => ? = 6
[1,5,2,6,4,3] => [1,2,2,1] => [[3,3,2,1],[2,1]]
 => ? = 6
[1,5,3,6,4,2] => [1,2,2,1] => [[3,3,2,1],[2,1]]
 => ? = 6
[1,5,4,6,3,2] => [1,1,3,1] => [[3,3,1,1],[2]]
 => ? = 4
[1,5,6,2,4,3] => [1,4,1] => [[4,4,1],[3]]
 => ? = 4
[1,5,6,3,4,2] => [1,4,1] => [[4,4,1],[3]]
 => ? = 4
[1,5,6,4,3,2] => [1,3,1,1] => [[3,3,3,1],[2,2]]
 => ? = 4
[1,6,2,3,5,4] => [1,4,1] => [[4,4,1],[3]]
 => ? = 4
[1,6,2,4,5,3] => [1,4,1] => [[4,4,1],[3]]
 => ? = 4
[1,6,2,5,4,3] => [1,2,2,1] => [[3,3,2,1],[2,1]]
 => ? = 6
[1,6,3,2,5,4] => [1,2,2,1] => [[3,3,2,1],[2,1]]
 => ? = 6
[1,6,3,4,5,2] => [1,4,1] => [[4,4,1],[3]]
 => ? = 4
[1,6,3,5,4,2] => [1,2,2,1] => [[3,3,2,1],[2,1]]
 => ? = 6
[1,6,4,2,5,3] => [1,1,3,1] => [[3,3,1,1],[2]]
 => ? = 4
[1,6,4,3,5,2] => [1,1,3,1] => [[3,3,1,1],[2]]
 => ? = 4
[1,6,4,5,3,2] => [1,1,3,1] => [[3,3,1,1],[2]]
 => ? = 4
[1,6,5,2,4,3] => [1,1,3,1] => [[3,3,1,1],[2]]
 => ? = 4
[1,6,5,3,4,2] => [1,1,3,1] => [[3,3,1,1],[2]]
 => ? = 4
[1,6,5,4,3,2] => [1,1,2,1,1] => [[2,2,2,1,1],[1,1]]
 => ? = 4
[2,1,3,6,5,4] => [3,2,1] => [[4,4,3],[3,2]]
 => ? = 5
[2,1,4,6,5,3] => [2,3,1] => [[4,4,2],[3,1]]
 => ? = 5
[2,1,5,6,4,3] => [2,3,1] => [[4,4,2],[3,1]]
 => ? = 5
[2,1,6,3,5,4] => [2,3,1] => [[4,4,2],[3,1]]
 => ? = 5
[2,1,6,4,5,3] => [2,3,1] => [[4,4,2],[3,1]]
 => ? = 5
[2,1,6,5,4,3] => [2,1,2,1] => [[3,3,2,2],[2,1,1]]
 => ? = 5
[2,3,1,6,5,4] => [3,2,1] => [[4,4,3],[3,2]]
 => ? = 5
[2,3,4,6,5,1] => [5,1] => [[5,5],[4]]
 => ? = 3
[2,3,5,6,4,1] => [5,1] => [[5,5],[4]]
 => ? = 3
[2,3,6,1,5,4] => [5,1] => [[5,5],[4]]
 => ? = 3
[2,3,6,4,5,1] => [5,1] => [[5,5],[4]]
 => ? = 3
[2,3,6,5,4,1] => [3,2,1] => [[4,4,3],[3,2]]
 => ? = 5
[2,4,1,6,5,3] => [3,2,1] => [[4,4,3],[3,2]]
 => ? = 5
[2,4,3,6,5,1] => [3,2,1] => [[4,4,3],[3,2]]
 => ? = 5
Description
The number of corners of a skew partition. This is also known as the number of removable cells of the skew partition.
Matching statistic: St000454
(load all 2 compositions to match this statistic)
Mapping
Mp00248: Permutations —DEX composition⟶ Integer compositions
Mp00314: Integer compositions —Foata bijection⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St000454: Graphs ⟶ ℤResult quality: 1% ●values known / values provided: 1%●distinct values known / distinct values provided: 50%
Values
[1] => [1] => [1] => ([],1)
 => 0 = 1 - 1
[3,2,1] => [2,1] => [2,1] => ([(0,2),(1,2)],3)
 => ? = 3 - 1
[1,4,3,2] => [1,2,1] => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => ? = 4 - 1
[2,4,3,1] => [3,1] => [3,1] => ([(0,3),(1,3),(2,3)],4)
 => ? = 3 - 1
[3,4,2,1] => [3,1] => [3,1] => ([(0,3),(1,3),(2,3)],4)
 => ? = 3 - 1
[4,1,3,2] => [3,1] => [3,1] => ([(0,3),(1,3),(2,3)],4)
 => ? = 3 - 1
[4,2,3,1] => [3,1] => [3,1] => ([(0,3),(1,3),(2,3)],4)
 => ? = 3 - 1
[4,3,2,1] => [1,2,1] => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => ? = 4 - 1
[1,2,5,4,3] => [2,2,1] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 5 - 1
[1,3,5,4,2] => [1,3,1] => [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 3 = 4 - 1
[1,4,5,3,2] => [1,3,1] => [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 3 = 4 - 1
[1,5,2,4,3] => [1,3,1] => [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 3 = 4 - 1
[1,5,3,4,2] => [1,3,1] => [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 3 = 4 - 1
[1,5,4,3,2] => [1,1,2,1] => [2,1,1,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 4 - 1
[2,1,5,4,3] => [2,2,1] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 5 - 1
[2,3,5,4,1] => [4,1] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 2 = 3 - 1
[2,4,5,3,1] => [4,1] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 2 = 3 - 1
[2,5,1,4,3] => [4,1] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 2 = 3 - 1
[2,5,3,4,1] => [4,1] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 2 = 3 - 1
[2,5,4,3,1] => [2,2,1] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 5 - 1
[3,1,5,4,2] => [2,2,1] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 5 - 1
[3,2,5,4,1] => [2,2,1] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 5 - 1
[3,4,5,2,1] => [4,1] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 2 = 3 - 1
[3,5,1,4,2] => [4,1] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 2 = 3 - 1
[3,5,2,4,1] => [4,1] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 2 = 3 - 1
[3,5,4,2,1] => [2,2,1] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 5 - 1
[4,1,5,3,2] => [2,2,1] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 5 - 1
[4,2,5,3,1] => [2,2,1] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 5 - 1
[4,3,5,2,1] => [1,3,1] => [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 3 = 4 - 1
[4,5,1,3,2] => [4,1] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 2 = 3 - 1
[4,5,2,3,1] => [4,1] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 2 = 3 - 1
[4,5,3,2,1] => [3,1,1] => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 3 - 1
[5,1,2,4,3] => [4,1] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 2 = 3 - 1
[5,1,3,4,2] => [4,1] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 2 = 3 - 1
[5,1,4,3,2] => [2,2,1] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 5 - 1
[5,2,1,4,3] => [2,2,1] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 5 - 1
[5,2,3,4,1] => [4,1] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 2 = 3 - 1
[5,2,4,3,1] => [2,2,1] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 5 - 1
[5,3,1,4,2] => [1,3,1] => [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 3 = 4 - 1
[5,3,2,4,1] => [1,3,1] => [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 3 = 4 - 1
[5,3,4,2,1] => [1,3,1] => [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 3 = 4 - 1
[5,4,1,3,2] => [1,3,1] => [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 3 = 4 - 1
[5,4,2,3,1] => [1,3,1] => [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 3 = 4 - 1
[5,4,3,2,1] => [1,2,1,1] => [1,2,1,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 4 - 1
[1,2,3,6,5,4] => [3,2,1] => [3,2,1] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 5 - 1
[1,2,4,6,5,3] => [2,3,1] => [2,3,1] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 5 - 1
[1,2,5,6,4,3] => [2,3,1] => [2,3,1] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 5 - 1
[1,2,6,3,5,4] => [2,3,1] => [2,3,1] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 5 - 1
[1,2,6,4,5,3] => [2,3,1] => [2,3,1] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 5 - 1
[1,2,6,5,4,3] => [2,1,2,1] => [2,2,1,1] => ([(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 5 - 1
[1,3,2,6,5,4] => [1,2,2,1] => [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)
 => ? = 6 - 1
[1,3,4,6,5,2] => [1,4,1] => [4,1,1] => ([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 4 - 1
[1,3,5,6,4,2] => [1,4,1] => [4,1,1] => ([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 4 - 1
[1,3,6,2,5,4] => [1,4,1] => [4,1,1] => ([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 4 - 1
[1,3,6,4,5,2] => [1,4,1] => [4,1,1] => ([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 4 - 1
[1,3,6,5,4,2] => [1,2,2,1] => [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)
 => ? = 6 - 1
[1,4,2,6,5,3] => [1,2,2,1] => [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)
 => ? = 6 - 1
[1,4,3,6,5,2] => [1,2,2,1] => [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)
 => ? = 6 - 1
[1,4,5,6,3,2] => [1,4,1] => [4,1,1] => ([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 4 - 1
[1,4,6,2,5,3] => [1,4,1] => [4,1,1] => ([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 4 - 1
[1,4,6,3,5,2] => [1,4,1] => [4,1,1] => ([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 4 - 1
[1,4,6,5,3,2] => [1,2,2,1] => [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)
 => ? = 6 - 1
[1,5,2,6,4,3] => [1,2,2,1] => [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)
 => ? = 6 - 1
[1,5,3,6,4,2] => [1,2,2,1] => [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)
 => ? = 6 - 1
[1,5,4,6,3,2] => [1,1,3,1] => [3,1,1,1] => ([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 4 - 1
[1,5,6,2,4,3] => [1,4,1] => [4,1,1] => ([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 4 - 1
[1,5,6,3,4,2] => [1,4,1] => [4,1,1] => ([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 4 - 1
[1,5,6,4,3,2] => [1,3,1,1] => [1,3,1,1] => ([(0,4),(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 4 - 1
[1,6,2,3,5,4] => [1,4,1] => [4,1,1] => ([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 4 - 1
[1,6,2,4,5,3] => [1,4,1] => [4,1,1] => ([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 4 - 1
[1,6,2,5,4,3] => [1,2,2,1] => [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)
 => ? = 6 - 1
[1,6,3,2,5,4] => [1,2,2,1] => [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)
 => ? = 6 - 1
[1,6,3,4,5,2] => [1,4,1] => [4,1,1] => ([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 4 - 1
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.