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Your data matches 22 different statistics following compositions of up to 3 maps.
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Matching statistic: St000288
Mp00152: Graphs —Laplacian multiplicities⟶ Integer compositions
Mp00094: Integer compositions —to binary word⟶ Binary words
Mp00234: Binary words —valleys-to-peaks⟶ Binary words
St000288: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00094: Integer compositions —to binary word⟶ Binary words
Mp00234: Binary words —valleys-to-peaks⟶ Binary words
St000288: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
([],1)
=> [1] => 1 => 1 => 1
([],2)
=> [2] => 10 => 11 => 2
([(0,1)],2)
=> [1,1] => 11 => 11 => 2
([],3)
=> [3] => 100 => 101 => 2
([(1,2)],3)
=> [1,2] => 110 => 111 => 3
([(0,2),(1,2)],3)
=> [1,1,1] => 111 => 111 => 3
([(0,1),(0,2),(1,2)],3)
=> [2,1] => 101 => 110 => 2
([],4)
=> [4] => 1000 => 1001 => 2
([(2,3)],4)
=> [1,3] => 1100 => 1101 => 3
([(1,3),(2,3)],4)
=> [1,1,2] => 1110 => 1111 => 4
([(0,3),(1,3),(2,3)],4)
=> [1,2,1] => 1101 => 1110 => 3
([(0,3),(1,2)],4)
=> [2,2] => 1010 => 1101 => 3
([(0,3),(1,2),(2,3)],4)
=> [1,1,1,1] => 1111 => 1111 => 4
([(1,2),(1,3),(2,3)],4)
=> [2,2] => 1010 => 1101 => 3
([(0,3),(1,2),(1,3),(2,3)],4)
=> [1,1,1,1] => 1111 => 1111 => 4
([(0,2),(0,3),(1,2),(1,3)],4)
=> [1,2,1] => 1101 => 1110 => 3
([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [2,1,1] => 1011 => 1101 => 3
([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [3,1] => 1001 => 1010 => 2
([],5)
=> [5] => 10000 => 10001 => 2
([(3,4)],5)
=> [1,4] => 11000 => 11001 => 3
([(2,4),(3,4)],5)
=> [1,1,3] => 11100 => 11101 => 4
([(1,4),(2,4),(3,4)],5)
=> [1,2,2] => 11010 => 11101 => 4
([(0,4),(1,4),(2,4),(3,4)],5)
=> [1,3,1] => 11001 => 11010 => 3
([(1,4),(2,3)],5)
=> [2,3] => 10100 => 11001 => 3
([(1,4),(2,3),(3,4)],5)
=> [1,1,1,2] => 11110 => 11111 => 5
([(0,1),(2,4),(3,4)],5)
=> [1,1,1,2] => 11110 => 11111 => 5
([(2,3),(2,4),(3,4)],5)
=> [2,3] => 10100 => 11001 => 3
([(0,4),(1,4),(2,3),(3,4)],5)
=> [1,1,1,1,1] => 11111 => 11111 => 5
([(1,4),(2,3),(2,4),(3,4)],5)
=> [1,1,1,2] => 11110 => 11111 => 5
([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> [1,1,2,1] => 11101 => 11110 => 4
([(1,3),(1,4),(2,3),(2,4)],5)
=> [1,2,2] => 11010 => 11101 => 4
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> [1,1,1,1,1] => 11111 => 11111 => 5
([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [2,1,2] => 10110 => 11011 => 4
([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> [1,1,1,1,1] => 11111 => 11111 => 5
([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [1,1,1,1,1] => 11111 => 11111 => 5
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> [1,1,2,1] => 11101 => 11110 => 4
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [2,2,1] => 10101 => 11010 => 3
([(0,4),(1,3),(2,3),(2,4)],5)
=> [1,1,1,1,1] => 11111 => 11111 => 5
([(0,1),(2,3),(2,4),(3,4)],5)
=> [2,1,2] => 10110 => 11011 => 4
([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> [1,1,1,1,1] => 11111 => 11111 => 5
([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
=> [1,2,1,1] => 11011 => 11101 => 4
([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
=> [2,2,1] => 10101 => 11010 => 3
([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> [1,1,1,1,1] => 11111 => 11111 => 5
([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
=> [1,1,1,1,1] => 11111 => 11111 => 5
([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
=> [1,1,1,1,1] => 11111 => 11111 => 5
([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [3,2] => 10010 => 10101 => 3
([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [1,2,1,1] => 11011 => 11101 => 4
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [2,1,1,1] => 10111 => 11011 => 4
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
=> [1,1,1,1,1] => 11111 => 11111 => 5
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> [2,2,1] => 10101 => 11010 => 3
Description
The number of ones in a binary word.
This is also known as the Hamming weight of the word.
Matching statistic: St001315
Mp00152: Graphs —Laplacian multiplicities⟶ Integer compositions
Mp00041: Integer compositions —conjugate⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St001315: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00041: Integer compositions —conjugate⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St001315: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
([],1)
=> [1] => [1] => ([],1)
=> 1
([],2)
=> [2] => [1,1] => ([(0,1)],2)
=> 2
([(0,1)],2)
=> [1,1] => [2] => ([],2)
=> 2
([],3)
=> [3] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 2
([(1,2)],3)
=> [1,2] => [1,2] => ([(1,2)],3)
=> 3
([(0,2),(1,2)],3)
=> [1,1,1] => [3] => ([],3)
=> 3
([(0,1),(0,2),(1,2)],3)
=> [2,1] => [2,1] => ([(0,2),(1,2)],3)
=> 2
([],4)
=> [4] => [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 2
([(2,3)],4)
=> [1,3] => [1,1,2] => ([(1,2),(1,3),(2,3)],4)
=> 3
([(1,3),(2,3)],4)
=> [1,1,2] => [1,3] => ([(2,3)],4)
=> 4
([(0,3),(1,3),(2,3)],4)
=> [1,2,1] => [2,2] => ([(1,3),(2,3)],4)
=> 3
([(0,3),(1,2)],4)
=> [2,2] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 3
([(0,3),(1,2),(2,3)],4)
=> [1,1,1,1] => [4] => ([],4)
=> 4
([(1,2),(1,3),(2,3)],4)
=> [2,2] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 3
([(0,3),(1,2),(1,3),(2,3)],4)
=> [1,1,1,1] => [4] => ([],4)
=> 4
([(0,2),(0,3),(1,2),(1,3)],4)
=> [1,2,1] => [2,2] => ([(1,3),(2,3)],4)
=> 3
([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [2,1,1] => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 3
([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [3,1] => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 2
([],5)
=> [5] => [1,1,1,1,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
([(3,4)],5)
=> [1,4] => [1,1,1,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3
([(2,4),(3,4)],5)
=> [1,1,3] => [1,1,3] => ([(2,3),(2,4),(3,4)],5)
=> 4
([(1,4),(2,4),(3,4)],5)
=> [1,2,2] => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> 4
([(0,4),(1,4),(2,4),(3,4)],5)
=> [1,3,1] => [2,1,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3
([(1,4),(2,3)],5)
=> [2,3] => [1,1,2,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3
([(1,4),(2,3),(3,4)],5)
=> [1,1,1,2] => [1,4] => ([(3,4)],5)
=> 5
([(0,1),(2,4),(3,4)],5)
=> [1,1,1,2] => [1,4] => ([(3,4)],5)
=> 5
([(2,3),(2,4),(3,4)],5)
=> [2,3] => [1,1,2,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3
([(0,4),(1,4),(2,3),(3,4)],5)
=> [1,1,1,1,1] => [5] => ([],5)
=> 5
([(1,4),(2,3),(2,4),(3,4)],5)
=> [1,1,1,2] => [1,4] => ([(3,4)],5)
=> 5
([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> [1,1,2,1] => [2,3] => ([(2,4),(3,4)],5)
=> 4
([(1,3),(1,4),(2,3),(2,4)],5)
=> [1,2,2] => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> 4
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> [1,1,1,1,1] => [5] => ([],5)
=> 5
([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [2,1,2] => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> 4
([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> [1,1,1,1,1] => [5] => ([],5)
=> 5
([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [1,1,1,1,1] => [5] => ([],5)
=> 5
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> [1,1,2,1] => [2,3] => ([(2,4),(3,4)],5)
=> 4
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [2,2,1] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3
([(0,4),(1,3),(2,3),(2,4)],5)
=> [1,1,1,1,1] => [5] => ([],5)
=> 5
([(0,1),(2,3),(2,4),(3,4)],5)
=> [2,1,2] => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> 4
([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> [1,1,1,1,1] => [5] => ([],5)
=> 5
([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
=> [1,2,1,1] => [3,2] => ([(1,4),(2,4),(3,4)],5)
=> 4
([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
=> [2,2,1] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3
([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> [1,1,1,1,1] => [5] => ([],5)
=> 5
([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
=> [1,1,1,1,1] => [5] => ([],5)
=> 5
([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
=> [1,1,1,1,1] => [5] => ([],5)
=> 5
([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [3,2] => [1,2,1,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3
([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [1,2,1,1] => [3,2] => ([(1,4),(2,4),(3,4)],5)
=> 4
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [2,1,1,1] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 4
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
=> [1,1,1,1,1] => [5] => ([],5)
=> 5
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> [2,2,1] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3
Description
The dissociation number of a graph.
Matching statistic: St001526
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00152: Graphs —Laplacian multiplicities⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
Mp00101: Dyck paths —decomposition reverse⟶ Dyck paths
St001526: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00231: Integer compositions —bounce path⟶ Dyck paths
Mp00101: Dyck paths —decomposition reverse⟶ Dyck paths
St001526: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
([],1)
=> [1] => [1,0]
=> [1,0]
=> 1
([],2)
=> [2] => [1,1,0,0]
=> [1,0,1,0]
=> 2
([(0,1)],2)
=> [1,1] => [1,0,1,0]
=> [1,1,0,0]
=> 2
([],3)
=> [3] => [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> 2
([(1,2)],3)
=> [1,2] => [1,0,1,1,0,0]
=> [1,1,0,1,0,0]
=> 3
([(0,2),(1,2)],3)
=> [1,1,1] => [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> 3
([(0,1),(0,2),(1,2)],3)
=> [2,1] => [1,1,0,0,1,0]
=> [1,1,0,0,1,0]
=> 2
([],4)
=> [4] => [1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> 2
([(2,3)],4)
=> [1,3] => [1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,0,0]
=> 3
([(1,3),(2,3)],4)
=> [1,1,2] => [1,0,1,0,1,1,0,0]
=> [1,1,1,0,1,0,0,0]
=> 4
([(0,3),(1,3),(2,3)],4)
=> [1,2,1] => [1,0,1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> 3
([(0,3),(1,2)],4)
=> [2,2] => [1,1,0,0,1,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> 3
([(0,3),(1,2),(2,3)],4)
=> [1,1,1,1] => [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> 4
([(1,2),(1,3),(2,3)],4)
=> [2,2] => [1,1,0,0,1,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> 3
([(0,3),(1,2),(1,3),(2,3)],4)
=> [1,1,1,1] => [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> 4
([(0,2),(0,3),(1,2),(1,3)],4)
=> [1,2,1] => [1,0,1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> 3
([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [2,1,1] => [1,1,0,0,1,0,1,0]
=> [1,1,1,0,0,0,1,0]
=> 3
([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [3,1] => [1,1,1,0,0,0,1,0]
=> [1,1,0,0,1,0,1,0]
=> 2
([],5)
=> [5] => [1,1,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> 2
([(3,4)],5)
=> [1,4] => [1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> 3
([(2,4),(3,4)],5)
=> [1,1,3] => [1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> 4
([(1,4),(2,4),(3,4)],5)
=> [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> 4
([(0,4),(1,4),(2,4),(3,4)],5)
=> [1,3,1] => [1,0,1,1,1,0,0,0,1,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> 3
([(1,4),(2,3)],5)
=> [2,3] => [1,1,0,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,0,0,1,0]
=> 3
([(1,4),(2,3),(3,4)],5)
=> [1,1,1,2] => [1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> 5
([(0,1),(2,4),(3,4)],5)
=> [1,1,1,2] => [1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> 5
([(2,3),(2,4),(3,4)],5)
=> [2,3] => [1,1,0,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,0,0,1,0]
=> 3
([(0,4),(1,4),(2,3),(3,4)],5)
=> [1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> 5
([(1,4),(2,3),(2,4),(3,4)],5)
=> [1,1,1,2] => [1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> 5
([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> [1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> 4
([(1,3),(1,4),(2,3),(2,4)],5)
=> [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> 4
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> [1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> 5
([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
=> [1,1,1,0,1,0,0,0,1,0]
=> 4
([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> [1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> 5
([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> 5
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> [1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> 4
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0,1,0]
=> 3
([(0,4),(1,3),(2,3),(2,4)],5)
=> [1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> 5
([(0,1),(2,3),(2,4),(3,4)],5)
=> [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
=> [1,1,1,0,1,0,0,0,1,0]
=> 4
([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> [1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> 5
([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
=> [1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> 4
([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
=> [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0,1,0]
=> 3
([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> [1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> 5
([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
=> [1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> 5
([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
=> [1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> 5
([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> [1,1,0,1,0,0,1,0,1,0]
=> 3
([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> 4
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> 4
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
=> [1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> 5
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0,1,0]
=> 3
Description
The Loewy length of the Auslander-Reiten translate of the regular module as a bimodule of the Nakayama algebra corresponding to the Dyck path.
Matching statistic: St001499
Mp00152: Graphs —Laplacian multiplicities⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
Mp00132: Dyck paths —switch returns and last double rise⟶ Dyck paths
St001499: Dyck paths ⟶ ℤResult quality: 80% ●values known / values provided: 98%●distinct values known / distinct values provided: 80%
Mp00231: Integer compositions —bounce path⟶ Dyck paths
Mp00132: Dyck paths —switch returns and last double rise⟶ Dyck paths
St001499: Dyck paths ⟶ ℤResult quality: 80% ●values known / values provided: 98%●distinct values known / distinct values provided: 80%
Values
([],1)
=> [1] => [1,0]
=> [1,0]
=> ? = 1 - 1
([],2)
=> [2] => [1,1,0,0]
=> [1,1,0,0]
=> 1 = 2 - 1
([(0,1)],2)
=> [1,1] => [1,0,1,0]
=> [1,0,1,0]
=> 1 = 2 - 1
([],3)
=> [3] => [1,1,1,0,0,0]
=> [1,1,1,0,0,0]
=> 1 = 2 - 1
([(1,2)],3)
=> [1,2] => [1,0,1,1,0,0]
=> [1,1,0,1,0,0]
=> 2 = 3 - 1
([(0,2),(1,2)],3)
=> [1,1,1] => [1,0,1,0,1,0]
=> [1,0,1,0,1,0]
=> 2 = 3 - 1
([(0,1),(0,2),(1,2)],3)
=> [2,1] => [1,1,0,0,1,0]
=> [1,1,0,0,1,0]
=> 1 = 2 - 1
([],4)
=> [4] => [1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> 1 = 2 - 1
([(2,3)],4)
=> [1,3] => [1,0,1,1,1,0,0,0]
=> [1,1,1,0,1,0,0,0]
=> 2 = 3 - 1
([(1,3),(2,3)],4)
=> [1,1,2] => [1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,0]
=> 3 = 4 - 1
([(0,3),(1,3),(2,3)],4)
=> [1,2,1] => [1,0,1,1,0,0,1,0]
=> [1,1,0,1,0,0,1,0]
=> 2 = 3 - 1
([(0,3),(1,2)],4)
=> [2,2] => [1,1,0,0,1,1,0,0]
=> [1,1,1,0,0,1,0,0]
=> 2 = 3 - 1
([(0,3),(1,2),(2,3)],4)
=> [1,1,1,1] => [1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0]
=> 3 = 4 - 1
([(1,2),(1,3),(2,3)],4)
=> [2,2] => [1,1,0,0,1,1,0,0]
=> [1,1,1,0,0,1,0,0]
=> 2 = 3 - 1
([(0,3),(1,2),(1,3),(2,3)],4)
=> [1,1,1,1] => [1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0]
=> 3 = 4 - 1
([(0,2),(0,3),(1,2),(1,3)],4)
=> [1,2,1] => [1,0,1,1,0,0,1,0]
=> [1,1,0,1,0,0,1,0]
=> 2 = 3 - 1
([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [2,1,1] => [1,1,0,0,1,0,1,0]
=> [1,1,0,0,1,0,1,0]
=> 2 = 3 - 1
([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [3,1] => [1,1,1,0,0,0,1,0]
=> [1,1,1,0,0,0,1,0]
=> 1 = 2 - 1
([],5)
=> [5] => [1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> 1 = 2 - 1
([(3,4)],5)
=> [1,4] => [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> 2 = 3 - 1
([(2,4),(3,4)],5)
=> [1,1,3] => [1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> 3 = 4 - 1
([(1,4),(2,4),(3,4)],5)
=> [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> 3 = 4 - 1
([(0,4),(1,4),(2,4),(3,4)],5)
=> [1,3,1] => [1,0,1,1,1,0,0,0,1,0]
=> [1,1,1,0,1,0,0,0,1,0]
=> 2 = 3 - 1
([(1,4),(2,3)],5)
=> [2,3] => [1,1,0,0,1,1,1,0,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> 2 = 3 - 1
([(1,4),(2,3),(3,4)],5)
=> [1,1,1,2] => [1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> 4 = 5 - 1
([(0,1),(2,4),(3,4)],5)
=> [1,1,1,2] => [1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> 4 = 5 - 1
([(2,3),(2,4),(3,4)],5)
=> [2,3] => [1,1,0,0,1,1,1,0,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> 2 = 3 - 1
([(0,4),(1,4),(2,3),(3,4)],5)
=> [1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> 4 = 5 - 1
([(1,4),(2,3),(2,4),(3,4)],5)
=> [1,1,1,2] => [1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> 4 = 5 - 1
([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> [1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,0,1,0,0,1,0]
=> 3 = 4 - 1
([(1,3),(1,4),(2,3),(2,4)],5)
=> [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> 3 = 4 - 1
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> [1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> 4 = 5 - 1
([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> 3 = 4 - 1
([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> [1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> 4 = 5 - 1
([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> 4 = 5 - 1
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> [1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,0,1,0,0,1,0]
=> 3 = 4 - 1
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0,1,0]
=> 2 = 3 - 1
([(0,4),(1,3),(2,3),(2,4)],5)
=> [1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> 4 = 5 - 1
([(0,1),(2,3),(2,4),(3,4)],5)
=> [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> 3 = 4 - 1
([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> [1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> 4 = 5 - 1
([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
=> [1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
=> [1,1,0,1,0,0,1,0,1,0]
=> 3 = 4 - 1
([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
=> [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0,1,0]
=> 2 = 3 - 1
([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> [1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> 4 = 5 - 1
([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
=> [1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> 4 = 5 - 1
([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
=> [1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> 4 = 5 - 1
([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> 2 = 3 - 1
([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
=> [1,1,0,1,0,0,1,0,1,0]
=> 3 = 4 - 1
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> 3 = 4 - 1
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
=> [1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> 4 = 5 - 1
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0,1,0]
=> 2 = 3 - 1
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> 2 = 3 - 1
Description
The number of indecomposable projective-injective modules of a magnitude 1 Nakayama algebra.
We use the bijection in the code by Christian Stump to have a bijection to Dyck paths.
Matching statistic: St001240
Mp00276: Graphs —to edge-partition of biconnected components⟶ Integer partitions
Mp00044: Integer partitions —conjugate⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
St001240: Dyck paths ⟶ ℤResult quality: 75% ●values known / values provided: 75%●distinct values known / distinct values provided: 80%
Mp00044: Integer partitions —conjugate⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
St001240: Dyck paths ⟶ ℤResult quality: 75% ●values known / values provided: 75%●distinct values known / distinct values provided: 80%
Values
([],1)
=> []
=> []
=> []
=> ? = 1
([],2)
=> []
=> []
=> []
=> ? = 2
([(0,1)],2)
=> [1]
=> [1]
=> [1,0]
=> 2
([],3)
=> []
=> []
=> []
=> ? = 2
([(1,2)],3)
=> [1]
=> [1]
=> [1,0]
=> 2
([(0,2),(1,2)],3)
=> [1,1]
=> [2]
=> [1,0,1,0]
=> 3
([(0,1),(0,2),(1,2)],3)
=> [3]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> 3
([],4)
=> []
=> []
=> []
=> ? = 2
([(2,3)],4)
=> [1]
=> [1]
=> [1,0]
=> 2
([(1,3),(2,3)],4)
=> [1,1]
=> [2]
=> [1,0,1,0]
=> 3
([(0,3),(1,3),(2,3)],4)
=> [1,1,1]
=> [3]
=> [1,0,1,0,1,0]
=> 4
([(0,3),(1,2)],4)
=> [1,1]
=> [2]
=> [1,0,1,0]
=> 3
([(0,3),(1,2),(2,3)],4)
=> [1,1,1]
=> [3]
=> [1,0,1,0,1,0]
=> 4
([(1,2),(1,3),(2,3)],4)
=> [3]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> 3
([(0,3),(1,2),(1,3),(2,3)],4)
=> [3,1]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 4
([(0,2),(0,3),(1,2),(1,3)],4)
=> [4]
=> [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> 3
([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [5]
=> [1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,0]
=> 3
([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [6]
=> [1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,0]
=> 3
([],5)
=> []
=> []
=> []
=> ? ∊ {2,4,4,5,5,5,5,5,5}
([(3,4)],5)
=> [1]
=> [1]
=> [1,0]
=> 2
([(2,4),(3,4)],5)
=> [1,1]
=> [2]
=> [1,0,1,0]
=> 3
([(1,4),(2,4),(3,4)],5)
=> [1,1,1]
=> [3]
=> [1,0,1,0,1,0]
=> 4
([(0,4),(1,4),(2,4),(3,4)],5)
=> [1,1,1,1]
=> [4]
=> [1,0,1,0,1,0,1,0]
=> 5
([(1,4),(2,3)],5)
=> [1,1]
=> [2]
=> [1,0,1,0]
=> 3
([(1,4),(2,3),(3,4)],5)
=> [1,1,1]
=> [3]
=> [1,0,1,0,1,0]
=> 4
([(0,1),(2,4),(3,4)],5)
=> [1,1,1]
=> [3]
=> [1,0,1,0,1,0]
=> 4
([(2,3),(2,4),(3,4)],5)
=> [3]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> 3
([(0,4),(1,4),(2,3),(3,4)],5)
=> [1,1,1,1]
=> [4]
=> [1,0,1,0,1,0,1,0]
=> 5
([(1,4),(2,3),(2,4),(3,4)],5)
=> [3,1]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 4
([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> [3,1,1]
=> [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> 5
([(1,3),(1,4),(2,3),(2,4)],5)
=> [4]
=> [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> 3
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> [4,1]
=> [2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> 4
([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> [1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,0]
=> 3
([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> [3,1,1]
=> [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> 5
([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [5,1]
=> [2,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,0]
=> 4
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> [6]
=> [1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,0]
=> 3
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [7]
=> [1,1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? ∊ {2,4,4,5,5,5,5,5,5}
([(0,4),(1,3),(2,3),(2,4)],5)
=> [1,1,1,1]
=> [4]
=> [1,0,1,0,1,0,1,0]
=> 5
([(0,1),(2,3),(2,4),(3,4)],5)
=> [3,1]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 4
([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> [3,1,1]
=> [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> 5
([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
=> [3,3]
=> [2,2,2]
=> [1,1,1,1,0,0,0,0]
=> 5
([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
=> [5]
=> [1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,0]
=> 3
([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> [6]
=> [1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,0]
=> 3
([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
=> [7]
=> [1,1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? ∊ {2,4,4,5,5,5,5,5,5}
([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
=> [5,1]
=> [2,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,0]
=> 4
([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [6]
=> [1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,0]
=> 3
([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [6,1]
=> [2,1,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,1,0,0]
=> ? ∊ {2,4,4,5,5,5,5,5,5}
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [8]
=> [1,1,1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? ∊ {2,4,4,5,5,5,5,5,5}
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
=> [7]
=> [1,1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? ∊ {2,4,4,5,5,5,5,5,5}
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> [8]
=> [1,1,1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? ∊ {2,4,4,5,5,5,5,5,5}
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [9]
=> [1,1,1,1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? ∊ {2,4,4,5,5,5,5,5,5}
([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [10]
=> [1,1,1,1,1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? ∊ {2,4,4,5,5,5,5,5,5}
Description
The number of indecomposable modules e_i J^2 that have injective dimension at most one in the corresponding Nakayama algebra
Matching statistic: St000777
(load all 16 compositions to match this statistic)
(load all 16 compositions to match this statistic)
Values
([],1)
=> ([],1)
=> ([],1)
=> 1
([],2)
=> ([],2)
=> ([(0,1)],2)
=> 2
([(0,1)],2)
=> ([(0,1)],2)
=> ([],2)
=> ? = 2
([],3)
=> ([],3)
=> ([(0,1),(0,2),(1,2)],3)
=> 2
([(1,2)],3)
=> ([(1,2)],3)
=> ([(0,2),(1,2)],3)
=> 3
([(0,2),(1,2)],3)
=> ([(0,2),(1,2)],3)
=> ([(1,2)],3)
=> ? ∊ {2,3}
([(0,1),(0,2),(1,2)],3)
=> ([(0,1),(0,2),(1,2)],3)
=> ([],3)
=> ? ∊ {2,3}
([],4)
=> ([],4)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 2
([(2,3)],4)
=> ([(2,3)],4)
=> ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3
([(1,3),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> 4
([(0,3),(1,3),(2,3)],4)
=> ([(0,3),(1,3),(2,3)],4)
=> ([(1,2),(1,3),(2,3)],4)
=> ? ∊ {2,3,3,3,4}
([(0,3),(1,2)],4)
=> ([(0,3),(1,2)],4)
=> ([(0,2),(0,3),(1,2),(1,3)],4)
=> 3
([(0,3),(1,2),(2,3)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> 4
([(1,2),(1,3),(2,3)],4)
=> ([(1,2),(1,3),(2,3)],4)
=> ([(0,3),(1,3),(2,3)],4)
=> 3
([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> ? ∊ {2,3,3,3,4}
([(0,2),(0,3),(1,2),(1,3)],4)
=> ([(0,2),(0,3),(1,2),(1,3)],4)
=> ([(0,3),(1,2)],4)
=> ? ∊ {2,3,3,3,4}
([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(2,3)],4)
=> ? ∊ {2,3,3,3,4}
([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([],4)
=> ? ∊ {2,3,3,3,4}
([],5)
=> ([],5)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
([(3,4)],5)
=> ([(3,4)],5)
=> ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3
([(2,4),(3,4)],5)
=> ([(2,4),(3,4)],5)
=> ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 4
([(1,4),(2,4),(3,4)],5)
=> ([(1,4),(2,4),(3,4)],5)
=> ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 4
([(0,4),(1,4),(2,4),(3,4)],5)
=> ([(0,4),(1,4),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {2,3,3,3,3,4,4,4,4,5,5,5}
([(1,4),(2,3)],5)
=> ([(1,4),(2,3)],5)
=> ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> 3
([(1,4),(2,3),(3,4)],5)
=> ([(1,4),(2,3),(3,4)],5)
=> ([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
=> 5
([(0,1),(2,4),(3,4)],5)
=> ([(0,1),(2,4),(3,4)],5)
=> ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
=> 5
([(2,3),(2,4),(3,4)],5)
=> ([(2,3),(2,4),(3,4)],5)
=> ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3
([(0,4),(1,4),(2,3),(3,4)],5)
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> ([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
=> 5
([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 5
([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {2,3,3,3,3,4,4,4,4,5,5,5}
([(1,3),(1,4),(2,3),(2,4)],5)
=> ([(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
=> 4
([(0,4),(1,2),(1,3),(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)
=> 5
([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> 4
([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> 5
([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {2,3,3,3,3,4,4,4,4,5,5,5}
([(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)
=> ([(0,1),(2,3),(2,4),(3,4)],5)
=> ? ∊ {2,3,3,3,3,4,4,4,4,5,5,5}
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(2,3),(2,4),(3,4)],5)
=> ? ∊ {2,3,3,3,3,4,4,4,4,5,5,5}
([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> 5
([(0,1),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> 4
([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> ([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> 5
([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
=> ([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
=> ([(1,3),(1,4),(2,3),(2,4)],5)
=> ? ∊ {2,3,3,3,3,4,4,4,4,5,5,5}
([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
=> ([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
=> ([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
=> 3
([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 5
([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,4),(2,3),(3,4)],5)
=> ? ∊ {2,3,3,3,3,4,4,4,4,5,5,5}
([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> 5
([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(1,4),(2,4),(3,4)],5)
=> 3
([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,4),(2,4),(3,4)],5)
=> ? ∊ {2,3,3,3,3,4,4,4,4,5,5,5}
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(2,4),(3,4)],5)
=> ? ∊ {2,3,3,3,3,4,4,4,4,5,5,5}
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,1),(2,4),(3,4)],5)
=> ? ∊ {2,3,3,3,3,4,4,4,4,5,5,5}
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> ([(1,4),(2,3)],5)
=> ? ∊ {2,3,3,3,3,4,4,4,4,5,5,5}
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(0,4),(0,5),(1,2),(1,4),(1,5),(2,3),(2,5),(3,4)],6)
=> ([(0,4),(0,5),(1,2),(1,3),(2,5),(3,4)],6)
=> 4
([(0,1),(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),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([],5)
=> ? ∊ {2,3,3,3,3,4,4,4,4,5,5,5}
Description
The number of distinct eigenvalues of the distance Laplacian of a connected graph.
Matching statistic: St000454
(load all 9 compositions to match this statistic)
(load all 9 compositions to match this statistic)
Values
([],1)
=> ([(0,1)],2)
=> [1,1] => ([(0,1)],2)
=> 1
([],2)
=> ([(0,2),(1,2)],3)
=> [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 2
([(0,1)],2)
=> ([(0,1),(0,2),(1,2)],3)
=> [2,1] => ([(0,2),(1,2)],3)
=> ? = 2
([],3)
=> ([(0,3),(1,3),(2,3)],4)
=> [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> ? ∊ {2,2,3}
([(1,2)],3)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3
([(0,2),(1,2)],3)
=> ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ? ∊ {2,2,3}
([(0,1),(0,2),(1,2)],3)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [3,1] => ([(0,3),(1,3),(2,3)],4)
=> ? ∊ {2,2,3}
([],4)
=> ([(0,4),(1,4),(2,4),(3,4)],5)
=> [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {2,3,3,3,3,3,4}
([(2,3)],4)
=> ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> [1,1,2,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {2,3,3,3,3,3,4}
([(1,3),(2,3)],4)
=> ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [1,1,1,1,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 4
([(0,3),(1,3),(2,3)],4)
=> ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {2,3,3,3,3,3,4}
([(0,3),(1,2)],4)
=> ([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
=> [1,2,1,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {2,3,3,3,3,3,4}
([(0,3),(1,2),(2,3)],4)
=> ([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
=> [1,1,1,1,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 4
([(1,2),(1,3),(2,3)],4)
=> ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [1,2,1,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {2,3,3,3,3,3,4}
([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [2,1,1,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {2,3,3,3,3,3,4}
([(0,2),(0,3),(1,2),(1,3)],4)
=> ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {2,3,3,3,3,3,4}
([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3
([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 2
([],5)
=> ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> [1,4,1] => ([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> ? ∊ {2,2,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,5,5,5}
([(3,4)],5)
=> ([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> [1,1,3,1] => ([(0,5),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? ∊ {2,2,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,5,5,5}
([(2,4),(3,4)],5)
=> ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [1,1,1,2,1] => ([(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? ∊ {2,2,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,5,5,5}
([(1,4),(2,4),(3,4)],5)
=> ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [1,1,2,1,1] => ([(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? ∊ {2,2,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,5,5,5}
([(0,4),(1,4),(2,4),(3,4)],5)
=> ([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [2,3,1] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? ∊ {2,2,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,5,5,5}
([(1,4),(2,3)],5)
=> ([(0,5),(1,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)
=> [1,2,2,1] => ([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? ∊ {2,2,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,5,5,5}
([(1,4),(2,3),(3,4)],5)
=> ([(0,5),(1,4),(1,5),(2,3),(2,5),(3,4),(3,5),(4,5)],6)
=> [1,1,1,1,1,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 5
([(0,1),(2,4),(3,4)],5)
=> ([(0,1),(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [1,1,1,1,1,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 5
([(2,3),(2,4),(3,4)],5)
=> ([(0,5),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [1,2,2,1] => ([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? ∊ {2,2,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,5,5,5}
([(0,4),(1,4),(2,3),(3,4)],5)
=> ([(0,4),(0,5),(1,4),(1,5),(2,3),(2,5),(3,4),(3,5),(4,5)],6)
=> [1,1,1,1,1,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 5
([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [1,1,1,1,1,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 5
([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [2,1,2,1] => ([(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? ∊ {2,2,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,5,5,5}
([(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
=> [1,1,2,1,1] => ([(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? ∊ {2,2,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,5,5,5}
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> ([(0,4),(0,5),(1,2),(1,3),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [1,1,1,1,1,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 5
([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [1,2,1,1,1] => ([(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? ∊ {2,2,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,5,5,5}
([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(0,5),(1,3),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [1,1,1,1,1,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 5
([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [2,1,1,1,1] => ([(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? ∊ {2,2,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,5,5,5}
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
=> [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)
=> ? ∊ {2,2,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,5,5,5}
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(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)
=> [3,2,1] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? ∊ {2,2,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,5,5,5}
([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,4),(0,5),(1,3),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
=> [1,1,1,1,1,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 5
([(0,1),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,5),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [1,2,1,1,1] => ([(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? ∊ {2,2,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,5,5,5}
([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> ([(0,3),(0,5),(1,2),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [1,1,1,1,1,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 5
([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
=> ([(0,3),(0,4),(0,5),(1,2),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [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)
=> ? ∊ {2,2,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,5,5,5}
([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
=> ([(0,3),(0,4),(0,5),(1,2),(1,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)
=> [1,2,2,1] => ([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? ∊ {2,2,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,5,5,5}
([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,4),(0,5),(1,3),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [1,1,1,1,1,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 5
([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(0,4),(0,5),(1,2),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [2,1,1,1,1] => ([(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? ∊ {2,2,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,5,5,5}
([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(0,5),(1,2),(1,3),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [1,1,1,1,1,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 5
([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [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)
=> ? ∊ {2,2,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,5,5,5}
([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [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)
=> ? ∊ {2,2,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,5,5,5}
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [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)
=> ? ∊ {2,2,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,5,5,5}
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
=> [2,1,1,1,1] => ([(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? ∊ {2,2,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,5,5,5}
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> ([(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [3,2,1] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? ∊ {2,2,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,5,5,5}
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [4,1,1] => ([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? ∊ {2,2,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,5,5,5}
([(0,1),(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),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [5,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> ? ∊ {2,2,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,5,5,5}
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.
Matching statistic: St001060
(load all 7 compositions to match this statistic)
(load all 7 compositions to match this statistic)
Values
([],1)
=> ([],1)
=> ([],1)
=> ([],1)
=> ? = 1
([],2)
=> ([],2)
=> ([],2)
=> ([(0,1)],2)
=> ? ∊ {2,2}
([(0,1)],2)
=> ([(0,1)],2)
=> ([],1)
=> ([],1)
=> ? ∊ {2,2}
([],3)
=> ([],3)
=> ([],3)
=> ([(0,1),(0,2),(1,2)],3)
=> 3
([(1,2)],3)
=> ([(1,2)],3)
=> ([],2)
=> ([(0,1)],2)
=> ? ∊ {2,2,3}
([(0,2),(1,2)],3)
=> ([(0,2),(1,2)],3)
=> ([],1)
=> ([],1)
=> ? ∊ {2,2,3}
([(0,1),(0,2),(1,2)],3)
=> ([(0,1),(0,2),(1,2)],3)
=> ([(0,1),(0,2),(1,2)],3)
=> ([],3)
=> ? ∊ {2,2,3}
([],4)
=> ([],4)
=> ([],4)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3
([(2,3)],4)
=> ([(2,3)],4)
=> ([],3)
=> ([(0,1),(0,2),(1,2)],3)
=> 3
([(1,3),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> ([],2)
=> ([(0,1)],2)
=> ? ∊ {2,2,3,3,3,4,4,4}
([(0,3),(1,3),(2,3)],4)
=> ([(0,3),(1,3),(2,3)],4)
=> ([],1)
=> ([],1)
=> ? ∊ {2,2,3,3,3,4,4,4}
([(0,3),(1,2)],4)
=> ([(0,3),(1,2)],4)
=> ([],2)
=> ([(0,1)],2)
=> ? ∊ {2,2,3,3,3,4,4,4}
([(0,3),(1,2),(2,3)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> ([],1)
=> ([],1)
=> ? ∊ {2,2,3,3,3,4,4,4}
([(1,2),(1,3),(2,3)],4)
=> ([(1,2),(1,3),(2,3)],4)
=> ([(1,2),(1,3),(2,3)],4)
=> ([(0,3),(1,3),(2,3)],4)
=> 3
([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,2)],3)
=> ([],3)
=> ? ∊ {2,2,3,3,3,4,4,4}
([(0,2),(0,3),(1,2),(1,3)],4)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([],4)
=> ? ∊ {2,2,3,3,3,4,4,4}
([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([],4)
=> ? ∊ {2,2,3,3,3,4,4,4}
([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([],4)
=> ? ∊ {2,2,3,3,3,4,4,4}
([],5)
=> ([],5)
=> ([],5)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3
([(3,4)],5)
=> ([(3,4)],5)
=> ([],4)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3
([(2,4),(3,4)],5)
=> ([(2,4),(3,4)],5)
=> ([],3)
=> ([(0,1),(0,2),(1,2)],3)
=> 3
([(1,4),(2,4),(3,4)],5)
=> ([(1,4),(2,4),(3,4)],5)
=> ([],2)
=> ([(0,1)],2)
=> ? ∊ {2,3,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5}
([(0,4),(1,4),(2,4),(3,4)],5)
=> ([(0,4),(1,4),(2,4),(3,4)],5)
=> ([],1)
=> ([],1)
=> ? ∊ {2,3,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5}
([(1,4),(2,3)],5)
=> ([(1,4),(2,3)],5)
=> ([],3)
=> ([(0,1),(0,2),(1,2)],3)
=> 3
([(1,4),(2,3),(3,4)],5)
=> ([(1,4),(2,3),(3,4)],5)
=> ([],2)
=> ([(0,1)],2)
=> ? ∊ {2,3,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5}
([(0,1),(2,4),(3,4)],5)
=> ([(0,1),(2,4),(3,4)],5)
=> ([],2)
=> ([(0,1)],2)
=> ? ∊ {2,3,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5}
([(2,3),(2,4),(3,4)],5)
=> ([(2,3),(2,4),(3,4)],5)
=> ([(2,3),(2,4),(3,4)],5)
=> ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
([(0,4),(1,4),(2,3),(3,4)],5)
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> ([],1)
=> ([],1)
=> ? ∊ {2,3,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5}
([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(2,3)],4)
=> ([(0,3),(1,3),(2,3)],4)
=> 3
([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> ([],3)
=> ? ∊ {2,3,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5}
([(1,3),(1,4),(2,3),(2,4)],5)
=> ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(1,4),(2,4),(3,4)],5)
=> 4
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(2,3)],4)
=> ? ∊ {2,3,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5}
([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(1,4),(2,4),(3,4)],5)
=> 4
([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> ([],3)
=> ? ∊ {2,3,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5}
([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(2,3)],4)
=> ? ∊ {2,3,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5}
([(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),(3,4)],5)
=> ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(2,3),(2,4),(3,4)],5)
=> ? ∊ {2,3,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5}
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(2,3),(2,4),(3,4)],5)
=> ? ∊ {2,3,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5}
([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ([],1)
=> ([],1)
=> ? ∊ {2,3,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5}
([(0,1),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(2,3),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(2,3)],4)
=> ([(0,3),(1,3),(2,3)],4)
=> 3
([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> ([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> ([],3)
=> ? ∊ {2,3,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5}
([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
=> ([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
=> ([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
=> ([(1,3),(1,4),(2,3),(2,4)],5)
=> 3
([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
=> ([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
=> ([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
=> ([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
=> 3
([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(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),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([],5)
=> ? ∊ {2,3,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5}
([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(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),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([],5)
=> ? ∊ {2,3,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5}
([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(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)
=> ([],4)
=> ? ∊ {2,3,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5}
([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(1,4),(2,4),(3,4)],5)
=> 4
([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(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)
=> ([],4)
=> ? ∊ {2,3,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5}
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(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),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([],5)
=> ? ∊ {2,3,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5}
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,1),(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),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([],5)
=> ? ∊ {2,3,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5}
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> ([(0,1),(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),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([],5)
=> ? ∊ {2,3,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5}
([(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),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([],5)
=> ? ∊ {2,3,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5}
([(0,1),(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),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([],5)
=> ? ∊ {2,3,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5}
Description
The distinguishing index of a graph.
This is the smallest number of colours such that there is a colouring of the edges which is not preserved by any automorphism.
If the graph has a connected component which is a single edge, or at least two isolated vertices, this statistic is undefined.
Matching statistic: St001880
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00152: Graphs —Laplacian multiplicities⟶ Integer compositions
Mp00180: Integer compositions —to ribbon⟶ Skew partitions
Mp00185: Skew partitions —cell poset⟶ Posets
St001880: Posets ⟶ ℤResult quality: 31% ●values known / values provided: 31%●distinct values known / distinct values provided: 60%
Mp00180: Integer compositions —to ribbon⟶ Skew partitions
Mp00185: Skew partitions —cell poset⟶ Posets
St001880: Posets ⟶ ℤResult quality: 31% ●values known / values provided: 31%●distinct values known / distinct values provided: 60%
Values
([],1)
=> [1] => [[1],[]]
=> ([],1)
=> ? = 1
([],2)
=> [2] => [[2],[]]
=> ([(0,1)],2)
=> ? ∊ {2,2}
([(0,1)],2)
=> [1,1] => [[1,1],[]]
=> ([(0,1)],2)
=> ? ∊ {2,2}
([],3)
=> [3] => [[3],[]]
=> ([(0,2),(2,1)],3)
=> 3
([(1,2)],3)
=> [1,2] => [[2,1],[]]
=> ([(0,1),(0,2)],3)
=> ? ∊ {2,2}
([(0,2),(1,2)],3)
=> [1,1,1] => [[1,1,1],[]]
=> ([(0,2),(2,1)],3)
=> 3
([(0,1),(0,2),(1,2)],3)
=> [2,1] => [[2,2],[1]]
=> ([(0,2),(1,2)],3)
=> ? ∊ {2,2}
([],4)
=> [4] => [[4],[]]
=> ([(0,3),(2,1),(3,2)],4)
=> 4
([(2,3)],4)
=> [1,3] => [[3,1],[]]
=> ([(0,2),(0,3),(3,1)],4)
=> ? ∊ {2,2,3,3,3,3,3,3}
([(1,3),(2,3)],4)
=> [1,1,2] => [[2,1,1],[]]
=> ([(0,2),(0,3),(3,1)],4)
=> ? ∊ {2,2,3,3,3,3,3,3}
([(0,3),(1,3),(2,3)],4)
=> [1,2,1] => [[2,2,1],[1]]
=> ([(0,3),(1,2),(1,3)],4)
=> ? ∊ {2,2,3,3,3,3,3,3}
([(0,3),(1,2)],4)
=> [2,2] => [[3,2],[1]]
=> ([(0,3),(1,2),(1,3)],4)
=> ? ∊ {2,2,3,3,3,3,3,3}
([(0,3),(1,2),(2,3)],4)
=> [1,1,1,1] => [[1,1,1,1],[]]
=> ([(0,3),(2,1),(3,2)],4)
=> 4
([(1,2),(1,3),(2,3)],4)
=> [2,2] => [[3,2],[1]]
=> ([(0,3),(1,2),(1,3)],4)
=> ? ∊ {2,2,3,3,3,3,3,3}
([(0,3),(1,2),(1,3),(2,3)],4)
=> [1,1,1,1] => [[1,1,1,1],[]]
=> ([(0,3),(2,1),(3,2)],4)
=> 4
([(0,2),(0,3),(1,2),(1,3)],4)
=> [1,2,1] => [[2,2,1],[1]]
=> ([(0,3),(1,2),(1,3)],4)
=> ? ∊ {2,2,3,3,3,3,3,3}
([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [2,1,1] => [[2,2,2],[1,1]]
=> ([(0,3),(1,2),(2,3)],4)
=> ? ∊ {2,2,3,3,3,3,3,3}
([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [3,1] => [[3,3],[2]]
=> ([(0,3),(1,2),(2,3)],4)
=> ? ∊ {2,2,3,3,3,3,3,3}
([],5)
=> [5] => [[5],[]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 5
([(3,4)],5)
=> [1,4] => [[4,1],[]]
=> ([(0,2),(0,4),(3,1),(4,3)],5)
=> ? ∊ {2,2,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,5,5}
([(2,4),(3,4)],5)
=> [1,1,3] => [[3,1,1],[]]
=> ([(0,3),(0,4),(3,2),(4,1)],5)
=> ? ∊ {2,2,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,5,5}
([(1,4),(2,4),(3,4)],5)
=> [1,2,2] => [[3,2,1],[1]]
=> ([(0,3),(0,4),(1,2),(1,4)],5)
=> ? ∊ {2,2,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,5,5}
([(0,4),(1,4),(2,4),(3,4)],5)
=> [1,3,1] => [[3,3,1],[2]]
=> ([(0,4),(1,2),(1,3),(3,4)],5)
=> ? ∊ {2,2,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,5,5}
([(1,4),(2,3)],5)
=> [2,3] => [[4,2],[1]]
=> ([(0,4),(1,2),(1,4),(2,3)],5)
=> ? ∊ {2,2,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,5,5}
([(1,4),(2,3),(3,4)],5)
=> [1,1,1,2] => [[2,1,1,1],[]]
=> ([(0,2),(0,4),(3,1),(4,3)],5)
=> ? ∊ {2,2,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,5,5}
([(0,1),(2,4),(3,4)],5)
=> [1,1,1,2] => [[2,1,1,1],[]]
=> ([(0,2),(0,4),(3,1),(4,3)],5)
=> ? ∊ {2,2,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,5,5}
([(2,3),(2,4),(3,4)],5)
=> [2,3] => [[4,2],[1]]
=> ([(0,4),(1,2),(1,4),(2,3)],5)
=> ? ∊ {2,2,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,5,5}
([(0,4),(1,4),(2,3),(3,4)],5)
=> [1,1,1,1,1] => [[1,1,1,1,1],[]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 5
([(1,4),(2,3),(2,4),(3,4)],5)
=> [1,1,1,2] => [[2,1,1,1],[]]
=> ([(0,2),(0,4),(3,1),(4,3)],5)
=> ? ∊ {2,2,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,5,5}
([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> [1,1,2,1] => [[2,2,1,1],[1]]
=> ([(0,4),(1,2),(1,4),(2,3)],5)
=> ? ∊ {2,2,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,5,5}
([(1,3),(1,4),(2,3),(2,4)],5)
=> [1,2,2] => [[3,2,1],[1]]
=> ([(0,3),(0,4),(1,2),(1,4)],5)
=> ? ∊ {2,2,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,5,5}
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> [1,1,1,1,1] => [[1,1,1,1,1],[]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 5
([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [2,1,2] => [[3,2,2],[1,1]]
=> ([(0,4),(1,2),(1,3),(3,4)],5)
=> ? ∊ {2,2,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,5,5}
([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> [1,1,1,1,1] => [[1,1,1,1,1],[]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 5
([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [1,1,1,1,1] => [[1,1,1,1,1],[]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 5
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> [1,1,2,1] => [[2,2,1,1],[1]]
=> ([(0,4),(1,2),(1,4),(2,3)],5)
=> ? ∊ {2,2,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,5,5}
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [2,2,1] => [[3,3,2],[2,1]]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ? ∊ {2,2,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,5,5}
([(0,4),(1,3),(2,3),(2,4)],5)
=> [1,1,1,1,1] => [[1,1,1,1,1],[]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 5
([(0,1),(2,3),(2,4),(3,4)],5)
=> [2,1,2] => [[3,2,2],[1,1]]
=> ([(0,4),(1,2),(1,3),(3,4)],5)
=> ? ∊ {2,2,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,5,5}
([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> [1,1,1,1,1] => [[1,1,1,1,1],[]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 5
([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
=> [1,2,1,1] => [[2,2,2,1],[1,1]]
=> ([(0,3),(1,2),(1,4),(3,4)],5)
=> ? ∊ {2,2,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,5,5}
([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
=> [2,2,1] => [[3,3,2],[2,1]]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ? ∊ {2,2,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,5,5}
([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> [1,1,1,1,1] => [[1,1,1,1,1],[]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 5
([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
=> [1,1,1,1,1] => [[1,1,1,1,1],[]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 5
([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
=> [1,1,1,1,1] => [[1,1,1,1,1],[]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 5
([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [3,2] => [[4,3],[2]]
=> ([(0,3),(1,2),(1,4),(3,4)],5)
=> ? ∊ {2,2,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,5,5}
([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [1,2,1,1] => [[2,2,2,1],[1,1]]
=> ([(0,3),(1,2),(1,4),(3,4)],5)
=> ? ∊ {2,2,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,5,5}
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [2,1,1,1] => [[2,2,2,2],[1,1,1]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> ? ∊ {2,2,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,5,5}
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
=> [1,1,1,1,1] => [[1,1,1,1,1],[]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 5
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> [2,2,1] => [[3,3,2],[2,1]]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ? ∊ {2,2,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,5,5}
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [3,1,1] => [[3,3,3],[2,2]]
=> ([(0,3),(1,2),(2,4),(3,4)],5)
=> ? ∊ {2,2,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,5,5}
([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [4,1] => [[4,4],[3]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> ? ∊ {2,2,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,5,5}
Description
The number of 2-Gorenstein indecomposable injective modules in the incidence algebra of the lattice.
Matching statistic: St000259
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Mp00324: Graphs —chromatic difference sequence⟶ Integer compositions
Mp00039: Integer compositions —complement⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St000259: Graphs ⟶ ℤResult quality: 31% ●values known / values provided: 31%●distinct values known / distinct values provided: 60%
Mp00039: Integer compositions —complement⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St000259: Graphs ⟶ ℤResult quality: 31% ●values known / values provided: 31%●distinct values known / distinct values provided: 60%
Values
([],1)
=> [1] => [1] => ([],1)
=> 0 = 1 - 1
([],2)
=> [2] => [1,1] => ([(0,1)],2)
=> 1 = 2 - 1
([(0,1)],2)
=> [1,1] => [2] => ([],2)
=> ? = 2 - 1
([],3)
=> [3] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 1 = 2 - 1
([(1,2)],3)
=> [2,1] => [1,2] => ([(1,2)],3)
=> ? ∊ {2,3,3} - 1
([(0,2),(1,2)],3)
=> [2,1] => [1,2] => ([(1,2)],3)
=> ? ∊ {2,3,3} - 1
([(0,1),(0,2),(1,2)],3)
=> [1,1,1] => [3] => ([],3)
=> ? ∊ {2,3,3} - 1
([],4)
=> [4] => [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 1 = 2 - 1
([(2,3)],4)
=> [3,1] => [1,1,2] => ([(1,2),(1,3),(2,3)],4)
=> ? ∊ {2,3,3,3,4,4,4} - 1
([(1,3),(2,3)],4)
=> [3,1] => [1,1,2] => ([(1,2),(1,3),(2,3)],4)
=> ? ∊ {2,3,3,3,4,4,4} - 1
([(0,3),(1,3),(2,3)],4)
=> [3,1] => [1,1,2] => ([(1,2),(1,3),(2,3)],4)
=> ? ∊ {2,3,3,3,4,4,4} - 1
([(0,3),(1,2)],4)
=> [2,2] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 2 = 3 - 1
([(0,3),(1,2),(2,3)],4)
=> [2,2] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 2 = 3 - 1
([(1,2),(1,3),(2,3)],4)
=> [2,1,1] => [1,3] => ([(2,3)],4)
=> ? ∊ {2,3,3,3,4,4,4} - 1
([(0,3),(1,2),(1,3),(2,3)],4)
=> [2,1,1] => [1,3] => ([(2,3)],4)
=> ? ∊ {2,3,3,3,4,4,4} - 1
([(0,2),(0,3),(1,2),(1,3)],4)
=> [2,2] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 2 = 3 - 1
([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [2,1,1] => [1,3] => ([(2,3)],4)
=> ? ∊ {2,3,3,3,4,4,4} - 1
([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [1,1,1,1] => [4] => ([],4)
=> ? ∊ {2,3,3,3,4,4,4} - 1
([],5)
=> [5] => [1,1,1,1,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1 = 2 - 1
([(3,4)],5)
=> [4,1] => [1,1,1,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {2,3,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5} - 1
([(2,4),(3,4)],5)
=> [4,1] => [1,1,1,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {2,3,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5} - 1
([(1,4),(2,4),(3,4)],5)
=> [4,1] => [1,1,1,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {2,3,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5} - 1
([(0,4),(1,4),(2,4),(3,4)],5)
=> [4,1] => [1,1,1,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {2,3,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5} - 1
([(1,4),(2,3)],5)
=> [3,2] => [1,1,2,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2 = 3 - 1
([(1,4),(2,3),(3,4)],5)
=> [3,2] => [1,1,2,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2 = 3 - 1
([(0,1),(2,4),(3,4)],5)
=> [3,2] => [1,1,2,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2 = 3 - 1
([(2,3),(2,4),(3,4)],5)
=> [3,1,1] => [1,1,3] => ([(2,3),(2,4),(3,4)],5)
=> ? ∊ {2,3,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5} - 1
([(0,4),(1,4),(2,3),(3,4)],5)
=> [3,2] => [1,1,2,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2 = 3 - 1
([(1,4),(2,3),(2,4),(3,4)],5)
=> [3,1,1] => [1,1,3] => ([(2,3),(2,4),(3,4)],5)
=> ? ∊ {2,3,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5} - 1
([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> [3,1,1] => [1,1,3] => ([(2,3),(2,4),(3,4)],5)
=> ? ∊ {2,3,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5} - 1
([(1,3),(1,4),(2,3),(2,4)],5)
=> [3,2] => [1,1,2,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2 = 3 - 1
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> [3,2] => [1,1,2,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2 = 3 - 1
([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [3,1,1] => [1,1,3] => ([(2,3),(2,4),(3,4)],5)
=> ? ∊ {2,3,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5} - 1
([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> [3,1,1] => [1,1,3] => ([(2,3),(2,4),(3,4)],5)
=> ? ∊ {2,3,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5} - 1
([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [3,1,1] => [1,1,3] => ([(2,3),(2,4),(3,4)],5)
=> ? ∊ {2,3,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5} - 1
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> [3,2] => [1,1,2,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2 = 3 - 1
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [3,1,1] => [1,1,3] => ([(2,3),(2,4),(3,4)],5)
=> ? ∊ {2,3,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5} - 1
([(0,4),(1,3),(2,3),(2,4)],5)
=> [3,2] => [1,1,2,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2 = 3 - 1
([(0,1),(2,3),(2,4),(3,4)],5)
=> [2,2,1] => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {2,3,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5} - 1
([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> [2,2,1] => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {2,3,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5} - 1
([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
=> [2,2,1] => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {2,3,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5} - 1
([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
=> [2,2,1] => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {2,3,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5} - 1
([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> [2,2,1] => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {2,3,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5} - 1
([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
=> [2,2,1] => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {2,3,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5} - 1
([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
=> [2,2,1] => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {2,3,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5} - 1
([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [2,1,1,1] => [1,4] => ([(3,4)],5)
=> ? ∊ {2,3,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5} - 1
([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [2,1,1,1] => [1,4] => ([(3,4)],5)
=> ? ∊ {2,3,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5} - 1
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [2,1,1,1] => [1,4] => ([(3,4)],5)
=> ? ∊ {2,3,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5} - 1
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
=> [2,2,1] => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {2,3,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5} - 1
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> [2,2,1] => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {2,3,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5} - 1
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [2,1,1,1] => [1,4] => ([(3,4)],5)
=> ? ∊ {2,3,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5} - 1
([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [1,1,1,1,1] => [5] => ([],5)
=> ? ∊ {2,3,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5} - 1
Description
The diameter of a connected graph.
This is the greatest distance between any pair of vertices.
The following 12 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000264The girth of a graph, which is not a tree. St000680The Grundy value for Hackendot on posets. St000717The number of ordinal summands of a poset. St000906The length of the shortest maximal chain in a poset. St001875The number of simple modules with projective dimension at most 1. St001207The Lowey length of the algebra $A/T$ when $T$ is the 1-tilting module corresponding to the permutation in the Auslander algebra of $K[x]/(x^n)$. St001630The global dimension of the incidence algebra of the lattice over the rational numbers. St001878The projective dimension of the simple modules corresponding to the minimum of L in the incidence algebra of the lattice L. St001879The number of indecomposable summands of the top of the first syzygy of the dual of the regular module in the incidence algebra of the lattice. St001330The hat guessing number of a graph. St001645The pebbling number of a connected graph. St000260The radius of a connected graph.
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