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Your data matches 83 different statistics following compositions of up to 3 maps.
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Matching statistic: St000259
(load all 6 compositions to match this statistic)
(load all 6 compositions to match this statistic)
Mp00224: Binary words —runsort⟶ Binary words
Mp00097: Binary words —delta morphism⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St000259: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00097: Binary words —delta morphism⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St000259: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
0 => 0 => [1] => ([],1)
=> 0
1 => 1 => [1] => ([],1)
=> 0
01 => 01 => [1,1] => ([(0,1)],2)
=> 1
10 => 01 => [1,1] => ([(0,1)],2)
=> 1
001 => 001 => [2,1] => ([(0,2),(1,2)],3)
=> 2
010 => 001 => [2,1] => ([(0,2),(1,2)],3)
=> 2
100 => 001 => [2,1] => ([(0,2),(1,2)],3)
=> 2
0001 => 0001 => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 2
0010 => 0001 => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 2
0100 => 0001 => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 2
0101 => 0101 => [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 1
1000 => 0001 => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 2
00001 => 00001 => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 2
00010 => 00001 => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 2
00100 => 00001 => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 2
00101 => 00101 => [2,1,1,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
01000 => 00001 => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 2
01001 => 00101 => [2,1,1,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
01010 => 00101 => [2,1,1,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
10000 => 00001 => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 2
000001 => 000001 => [5,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> 2
000010 => 000001 => [5,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> 2
000100 => 000001 => [5,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> 2
000101 => 000101 => [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
001000 => 000001 => [5,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> 2
001001 => 001001 => [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
001010 => 000101 => [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
001101 => 001101 => [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
010000 => 000001 => [5,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> 2
010001 => 000101 => [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
010010 => 000101 => [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
010011 => 001101 => [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
010100 => 000101 => [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
010101 => 010101 => [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)
=> 1
100000 => 000001 => [5,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> 2
0000001 => 0000001 => [6,1] => ([(0,6),(1,6),(2,6),(3,6),(4,6),(5,6)],7)
=> 2
0000010 => 0000001 => [6,1] => ([(0,6),(1,6),(2,6),(3,6),(4,6),(5,6)],7)
=> 2
0000100 => 0000001 => [6,1] => ([(0,6),(1,6),(2,6),(3,6),(4,6),(5,6)],7)
=> 2
0000101 => 0000101 => [4,1,1,1] => ([(0,4),(0,5),(0,6),(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 2
0001000 => 0000001 => [6,1] => ([(0,6),(1,6),(2,6),(3,6),(4,6),(5,6)],7)
=> 2
0001001 => 0001001 => [3,1,2,1] => ([(0,6),(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 2
0001010 => 0000101 => [4,1,1,1] => ([(0,4),(0,5),(0,6),(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 2
0001101 => 0001101 => [3,2,1,1] => ([(0,5),(0,6),(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 2
0010000 => 0000001 => [6,1] => ([(0,6),(1,6),(2,6),(3,6),(4,6),(5,6)],7)
=> 2
0010001 => 0001001 => [3,1,2,1] => ([(0,6),(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 2
0010010 => 0001001 => [3,1,2,1] => ([(0,6),(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 2
0010100 => 0000101 => [4,1,1,1] => ([(0,4),(0,5),(0,6),(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 2
0010101 => 0010101 => [2,1,1,1,1,1] => ([(0,2),(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 2
0011010 => 0001101 => [3,2,1,1] => ([(0,5),(0,6),(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 2
0011101 => 0011101 => [2,3,1,1] => ([(0,5),(0,6),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 2
Description
The diameter of a connected graph.
This is the greatest distance between any pair of vertices.
Matching statistic: St001093
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00224: Binary words —runsort⟶ Binary words
Mp00097: Binary words —delta morphism⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St001093: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00097: Binary words —delta morphism⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St001093: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
0 => 0 => [1] => ([],1)
=> 1 = 0 + 1
1 => 1 => [1] => ([],1)
=> 1 = 0 + 1
01 => 01 => [1,1] => ([(0,1)],2)
=> 2 = 1 + 1
10 => 01 => [1,1] => ([(0,1)],2)
=> 2 = 1 + 1
001 => 001 => [2,1] => ([(0,2),(1,2)],3)
=> 3 = 2 + 1
010 => 001 => [2,1] => ([(0,2),(1,2)],3)
=> 3 = 2 + 1
100 => 001 => [2,1] => ([(0,2),(1,2)],3)
=> 3 = 2 + 1
0001 => 0001 => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 3 = 2 + 1
0010 => 0001 => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 3 = 2 + 1
0100 => 0001 => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 3 = 2 + 1
0101 => 0101 => [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 2 = 1 + 1
1000 => 0001 => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 3 = 2 + 1
00001 => 00001 => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 3 = 2 + 1
00010 => 00001 => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 3 = 2 + 1
00100 => 00001 => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 3 = 2 + 1
00101 => 00101 => [2,1,1,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = 2 + 1
01000 => 00001 => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 3 = 2 + 1
01001 => 00101 => [2,1,1,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = 2 + 1
01010 => 00101 => [2,1,1,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = 2 + 1
10000 => 00001 => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 3 = 2 + 1
000001 => 000001 => [5,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> 3 = 2 + 1
000010 => 000001 => [5,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> 3 = 2 + 1
000100 => 000001 => [5,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> 3 = 2 + 1
000101 => 000101 => [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)
=> 3 = 2 + 1
001000 => 000001 => [5,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> 3 = 2 + 1
001001 => 001001 => [2,1,2,1] => ([(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 3 = 2 + 1
001010 => 000101 => [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)
=> 3 = 2 + 1
001101 => 001101 => [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)
=> 3 = 2 + 1
010000 => 000001 => [5,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> 3 = 2 + 1
010001 => 000101 => [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)
=> 3 = 2 + 1
010010 => 000101 => [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)
=> 3 = 2 + 1
010011 => 001101 => [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)
=> 3 = 2 + 1
010100 => 000101 => [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)
=> 3 = 2 + 1
010101 => 010101 => [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)
=> 2 = 1 + 1
100000 => 000001 => [5,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> 3 = 2 + 1
0000001 => 0000001 => [6,1] => ([(0,6),(1,6),(2,6),(3,6),(4,6),(5,6)],7)
=> 3 = 2 + 1
0000010 => 0000001 => [6,1] => ([(0,6),(1,6),(2,6),(3,6),(4,6),(5,6)],7)
=> 3 = 2 + 1
0000100 => 0000001 => [6,1] => ([(0,6),(1,6),(2,6),(3,6),(4,6),(5,6)],7)
=> 3 = 2 + 1
0000101 => 0000101 => [4,1,1,1] => ([(0,4),(0,5),(0,6),(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 3 = 2 + 1
0001000 => 0000001 => [6,1] => ([(0,6),(1,6),(2,6),(3,6),(4,6),(5,6)],7)
=> 3 = 2 + 1
0001001 => 0001001 => [3,1,2,1] => ([(0,6),(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 3 = 2 + 1
0001010 => 0000101 => [4,1,1,1] => ([(0,4),(0,5),(0,6),(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 3 = 2 + 1
0001101 => 0001101 => [3,2,1,1] => ([(0,5),(0,6),(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 3 = 2 + 1
0010000 => 0000001 => [6,1] => ([(0,6),(1,6),(2,6),(3,6),(4,6),(5,6)],7)
=> 3 = 2 + 1
0010001 => 0001001 => [3,1,2,1] => ([(0,6),(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 3 = 2 + 1
0010010 => 0001001 => [3,1,2,1] => ([(0,6),(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 3 = 2 + 1
0010100 => 0000101 => [4,1,1,1] => ([(0,4),(0,5),(0,6),(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 3 = 2 + 1
0010101 => 0010101 => [2,1,1,1,1,1] => ([(0,2),(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 3 = 2 + 1
0011010 => 0001101 => [3,2,1,1] => ([(0,5),(0,6),(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 3 = 2 + 1
0011101 => 0011101 => [2,3,1,1] => ([(0,5),(0,6),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 3 = 2 + 1
Description
The detour number of a graph.
This is the number of vertices in a longest induced path in a graph.
Note that [1] defines the detour number as the number of edges in a longest induced path, which is unsuitable for the empty graph.
Matching statistic: St001198
(load all 4 compositions to match this statistic)
(load all 4 compositions to match this statistic)
Mp00097: Binary words —delta morphism⟶ Integer compositions
Mp00133: Integer compositions —delta morphism⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St001198: Dyck paths ⟶ ℤResult quality: 33% ●values known / values provided: 85%●distinct values known / distinct values provided: 33%
Mp00133: Integer compositions —delta morphism⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St001198: Dyck paths ⟶ ℤResult quality: 33% ●values known / values provided: 85%●distinct values known / distinct values provided: 33%
Values
0 => [1] => [1] => [1,0]
=> ? = 0
1 => [1] => [1] => [1,0]
=> ? = 0
01 => [1,1] => [2] => [1,1,0,0]
=> ? = 1
10 => [1,1] => [2] => [1,1,0,0]
=> ? = 1
001 => [2,1] => [1,1] => [1,0,1,0]
=> 2
010 => [1,1,1] => [3] => [1,1,1,0,0,0]
=> ? = 2
100 => [1,2] => [1,1] => [1,0,1,0]
=> 2
0001 => [3,1] => [1,1] => [1,0,1,0]
=> 2
0010 => [2,1,1] => [1,2] => [1,0,1,1,0,0]
=> 2
0100 => [1,1,2] => [2,1] => [1,1,0,0,1,0]
=> 2
0101 => [1,1,1,1] => [4] => [1,1,1,1,0,0,0,0]
=> ? = 1
1000 => [1,3] => [1,1] => [1,0,1,0]
=> 2
00001 => [4,1] => [1,1] => [1,0,1,0]
=> 2
00010 => [3,1,1] => [1,2] => [1,0,1,1,0,0]
=> 2
00100 => [2,1,2] => [1,1,1] => [1,0,1,0,1,0]
=> 2
00101 => [2,1,1,1] => [1,3] => [1,0,1,1,1,0,0,0]
=> 2
01000 => [1,1,3] => [2,1] => [1,1,0,0,1,0]
=> 2
01001 => [1,1,2,1] => [2,1,1] => [1,1,0,0,1,0,1,0]
=> 2
01010 => [1,1,1,1,1] => [5] => [1,1,1,1,1,0,0,0,0,0]
=> ? = 2
10000 => [1,4] => [1,1] => [1,0,1,0]
=> 2
000001 => [5,1] => [1,1] => [1,0,1,0]
=> 2
000010 => [4,1,1] => [1,2] => [1,0,1,1,0,0]
=> 2
000100 => [3,1,2] => [1,1,1] => [1,0,1,0,1,0]
=> 2
000101 => [3,1,1,1] => [1,3] => [1,0,1,1,1,0,0,0]
=> 2
001000 => [2,1,3] => [1,1,1] => [1,0,1,0,1,0]
=> 2
001001 => [2,1,2,1] => [1,1,1,1] => [1,0,1,0,1,0,1,0]
=> 2
001010 => [2,1,1,1,1] => [1,4] => [1,0,1,1,1,1,0,0,0,0]
=> 2
001101 => [2,2,1,1] => [2,2] => [1,1,0,0,1,1,0,0]
=> 2
010000 => [1,1,4] => [2,1] => [1,1,0,0,1,0]
=> 2
010001 => [1,1,3,1] => [2,1,1] => [1,1,0,0,1,0,1,0]
=> 2
010010 => [1,1,2,1,1] => [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
=> 2
010011 => [1,1,2,2] => [2,2] => [1,1,0,0,1,1,0,0]
=> 2
010100 => [1,1,1,1,2] => [4,1] => [1,1,1,1,0,0,0,0,1,0]
=> 2
010101 => [1,1,1,1,1,1] => [6] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> ? = 1
100000 => [1,5] => [1,1] => [1,0,1,0]
=> 2
0000001 => [6,1] => [1,1] => [1,0,1,0]
=> 2
0000010 => [5,1,1] => [1,2] => [1,0,1,1,0,0]
=> 2
0000100 => [4,1,2] => [1,1,1] => [1,0,1,0,1,0]
=> 2
0000101 => [4,1,1,1] => [1,3] => [1,0,1,1,1,0,0,0]
=> 2
0001000 => [3,1,3] => [1,1,1] => [1,0,1,0,1,0]
=> 2
0001001 => [3,1,2,1] => [1,1,1,1] => [1,0,1,0,1,0,1,0]
=> 2
0001010 => [3,1,1,1,1] => [1,4] => [1,0,1,1,1,1,0,0,0,0]
=> 2
0001101 => [3,2,1,1] => [1,1,2] => [1,0,1,0,1,1,0,0]
=> 2
0010000 => [2,1,4] => [1,1,1] => [1,0,1,0,1,0]
=> 2
0010001 => [2,1,3,1] => [1,1,1,1] => [1,0,1,0,1,0,1,0]
=> 2
0010010 => [2,1,2,1,1] => [1,1,1,2] => [1,0,1,0,1,0,1,1,0,0]
=> 2
0010100 => [2,1,1,1,2] => [1,3,1] => [1,0,1,1,1,0,0,0,1,0]
=> 2
0010101 => [2,1,1,1,1,1] => [1,5] => [1,0,1,1,1,1,1,0,0,0,0,0]
=> 2
0011010 => [2,2,1,1,1] => [2,3] => [1,1,0,0,1,1,1,0,0,0]
=> 2
0011101 => [2,3,1,1] => [1,1,2] => [1,0,1,0,1,1,0,0]
=> 2
0100000 => [1,1,5] => [2,1] => [1,1,0,0,1,0]
=> 2
0100001 => [1,1,4,1] => [2,1,1] => [1,1,0,0,1,0,1,0]
=> 2
0100010 => [1,1,3,1,1] => [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
=> 2
0100011 => [1,1,3,2] => [2,1,1] => [1,1,0,0,1,0,1,0]
=> 2
0100100 => [1,1,2,1,2] => [2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
=> 2
0100101 => [1,1,2,1,1,1] => [2,1,3] => [1,1,0,0,1,0,1,1,1,0,0,0]
=> 2
0100110 => [1,1,2,2,1] => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> 2
0100111 => [1,1,2,3] => [2,1,1] => [1,1,0,0,1,0,1,0]
=> 2
0101010 => [1,1,1,1,1,1,1] => [7] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> ? = 2
Description
The number of simple modules in the algebra $eAe$ with projective dimension at most 1 in the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$.
Matching statistic: St001206
(load all 4 compositions to match this statistic)
(load all 4 compositions to match this statistic)
Mp00097: Binary words —delta morphism⟶ Integer compositions
Mp00133: Integer compositions —delta morphism⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St001206: Dyck paths ⟶ ℤResult quality: 33% ●values known / values provided: 85%●distinct values known / distinct values provided: 33%
Mp00133: Integer compositions —delta morphism⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St001206: Dyck paths ⟶ ℤResult quality: 33% ●values known / values provided: 85%●distinct values known / distinct values provided: 33%
Values
0 => [1] => [1] => [1,0]
=> ? = 0
1 => [1] => [1] => [1,0]
=> ? = 0
01 => [1,1] => [2] => [1,1,0,0]
=> ? = 1
10 => [1,1] => [2] => [1,1,0,0]
=> ? = 1
001 => [2,1] => [1,1] => [1,0,1,0]
=> 2
010 => [1,1,1] => [3] => [1,1,1,0,0,0]
=> ? = 2
100 => [1,2] => [1,1] => [1,0,1,0]
=> 2
0001 => [3,1] => [1,1] => [1,0,1,0]
=> 2
0010 => [2,1,1] => [1,2] => [1,0,1,1,0,0]
=> 2
0100 => [1,1,2] => [2,1] => [1,1,0,0,1,0]
=> 2
0101 => [1,1,1,1] => [4] => [1,1,1,1,0,0,0,0]
=> ? = 1
1000 => [1,3] => [1,1] => [1,0,1,0]
=> 2
00001 => [4,1] => [1,1] => [1,0,1,0]
=> 2
00010 => [3,1,1] => [1,2] => [1,0,1,1,0,0]
=> 2
00100 => [2,1,2] => [1,1,1] => [1,0,1,0,1,0]
=> 2
00101 => [2,1,1,1] => [1,3] => [1,0,1,1,1,0,0,0]
=> 2
01000 => [1,1,3] => [2,1] => [1,1,0,0,1,0]
=> 2
01001 => [1,1,2,1] => [2,1,1] => [1,1,0,0,1,0,1,0]
=> 2
01010 => [1,1,1,1,1] => [5] => [1,1,1,1,1,0,0,0,0,0]
=> ? = 2
10000 => [1,4] => [1,1] => [1,0,1,0]
=> 2
000001 => [5,1] => [1,1] => [1,0,1,0]
=> 2
000010 => [4,1,1] => [1,2] => [1,0,1,1,0,0]
=> 2
000100 => [3,1,2] => [1,1,1] => [1,0,1,0,1,0]
=> 2
000101 => [3,1,1,1] => [1,3] => [1,0,1,1,1,0,0,0]
=> 2
001000 => [2,1,3] => [1,1,1] => [1,0,1,0,1,0]
=> 2
001001 => [2,1,2,1] => [1,1,1,1] => [1,0,1,0,1,0,1,0]
=> 2
001010 => [2,1,1,1,1] => [1,4] => [1,0,1,1,1,1,0,0,0,0]
=> 2
001101 => [2,2,1,1] => [2,2] => [1,1,0,0,1,1,0,0]
=> 2
010000 => [1,1,4] => [2,1] => [1,1,0,0,1,0]
=> 2
010001 => [1,1,3,1] => [2,1,1] => [1,1,0,0,1,0,1,0]
=> 2
010010 => [1,1,2,1,1] => [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
=> 2
010011 => [1,1,2,2] => [2,2] => [1,1,0,0,1,1,0,0]
=> 2
010100 => [1,1,1,1,2] => [4,1] => [1,1,1,1,0,0,0,0,1,0]
=> 2
010101 => [1,1,1,1,1,1] => [6] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> ? = 1
100000 => [1,5] => [1,1] => [1,0,1,0]
=> 2
0000001 => [6,1] => [1,1] => [1,0,1,0]
=> 2
0000010 => [5,1,1] => [1,2] => [1,0,1,1,0,0]
=> 2
0000100 => [4,1,2] => [1,1,1] => [1,0,1,0,1,0]
=> 2
0000101 => [4,1,1,1] => [1,3] => [1,0,1,1,1,0,0,0]
=> 2
0001000 => [3,1,3] => [1,1,1] => [1,0,1,0,1,0]
=> 2
0001001 => [3,1,2,1] => [1,1,1,1] => [1,0,1,0,1,0,1,0]
=> 2
0001010 => [3,1,1,1,1] => [1,4] => [1,0,1,1,1,1,0,0,0,0]
=> 2
0001101 => [3,2,1,1] => [1,1,2] => [1,0,1,0,1,1,0,0]
=> 2
0010000 => [2,1,4] => [1,1,1] => [1,0,1,0,1,0]
=> 2
0010001 => [2,1,3,1] => [1,1,1,1] => [1,0,1,0,1,0,1,0]
=> 2
0010010 => [2,1,2,1,1] => [1,1,1,2] => [1,0,1,0,1,0,1,1,0,0]
=> 2
0010100 => [2,1,1,1,2] => [1,3,1] => [1,0,1,1,1,0,0,0,1,0]
=> 2
0010101 => [2,1,1,1,1,1] => [1,5] => [1,0,1,1,1,1,1,0,0,0,0,0]
=> 2
0011010 => [2,2,1,1,1] => [2,3] => [1,1,0,0,1,1,1,0,0,0]
=> 2
0011101 => [2,3,1,1] => [1,1,2] => [1,0,1,0,1,1,0,0]
=> 2
0100000 => [1,1,5] => [2,1] => [1,1,0,0,1,0]
=> 2
0100001 => [1,1,4,1] => [2,1,1] => [1,1,0,0,1,0,1,0]
=> 2
0100010 => [1,1,3,1,1] => [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
=> 2
0100011 => [1,1,3,2] => [2,1,1] => [1,1,0,0,1,0,1,0]
=> 2
0100100 => [1,1,2,1,2] => [2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
=> 2
0100101 => [1,1,2,1,1,1] => [2,1,3] => [1,1,0,0,1,0,1,1,1,0,0,0]
=> 2
0100110 => [1,1,2,2,1] => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> 2
0100111 => [1,1,2,3] => [2,1,1] => [1,1,0,0,1,0,1,0]
=> 2
0101010 => [1,1,1,1,1,1,1] => [7] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> ? = 2
Description
The maximal dimension of an indecomposable projective $eAe$-module (that is the height of the corresponding Dyck path) of the corresponding Nakayama algebra with minimal faithful projective-injective module $eA$.
Matching statistic: St000455
(load all 5 compositions to match this statistic)
(load all 5 compositions to match this statistic)
Mp00224: Binary words —runsort⟶ Binary words
Mp00097: Binary words —delta morphism⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St000455: Graphs ⟶ ℤResult quality: 67% ●values known / values provided: 77%●distinct values known / distinct values provided: 67%
Mp00097: Binary words —delta morphism⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St000455: Graphs ⟶ ℤResult quality: 67% ●values known / values provided: 77%●distinct values known / distinct values provided: 67%
Values
0 => 0 => [1] => ([],1)
=> ? = 0 - 2
1 => 1 => [1] => ([],1)
=> ? = 0 - 2
01 => 01 => [1,1] => ([(0,1)],2)
=> -1 = 1 - 2
10 => 01 => [1,1] => ([(0,1)],2)
=> -1 = 1 - 2
001 => 001 => [2,1] => ([(0,2),(1,2)],3)
=> 0 = 2 - 2
010 => 001 => [2,1] => ([(0,2),(1,2)],3)
=> 0 = 2 - 2
100 => 001 => [2,1] => ([(0,2),(1,2)],3)
=> 0 = 2 - 2
0001 => 0001 => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 0 = 2 - 2
0010 => 0001 => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 0 = 2 - 2
0100 => 0001 => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 0 = 2 - 2
0101 => 0101 => [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> -1 = 1 - 2
1000 => 0001 => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 0 = 2 - 2
00001 => 00001 => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 0 = 2 - 2
00010 => 00001 => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 0 = 2 - 2
00100 => 00001 => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 0 = 2 - 2
00101 => 00101 => [2,1,1,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 0 = 2 - 2
01000 => 00001 => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 0 = 2 - 2
01001 => 00101 => [2,1,1,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 0 = 2 - 2
01010 => 00101 => [2,1,1,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 0 = 2 - 2
10000 => 00001 => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 0 = 2 - 2
000001 => 000001 => [5,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> 0 = 2 - 2
000010 => 000001 => [5,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> 0 = 2 - 2
000100 => 000001 => [5,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> 0 = 2 - 2
000101 => 000101 => [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)
=> 0 = 2 - 2
001000 => 000001 => [5,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> 0 = 2 - 2
001001 => 001001 => [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
001010 => 000101 => [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)
=> 0 = 2 - 2
001101 => 001101 => [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
010000 => 000001 => [5,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> 0 = 2 - 2
010001 => 000101 => [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)
=> 0 = 2 - 2
010010 => 000101 => [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)
=> 0 = 2 - 2
010011 => 001101 => [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
010100 => 000101 => [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)
=> 0 = 2 - 2
010101 => 010101 => [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)
=> -1 = 1 - 2
100000 => 000001 => [5,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> 0 = 2 - 2
0000001 => 0000001 => [6,1] => ([(0,6),(1,6),(2,6),(3,6),(4,6),(5,6)],7)
=> 0 = 2 - 2
0000010 => 0000001 => [6,1] => ([(0,6),(1,6),(2,6),(3,6),(4,6),(5,6)],7)
=> 0 = 2 - 2
0000100 => 0000001 => [6,1] => ([(0,6),(1,6),(2,6),(3,6),(4,6),(5,6)],7)
=> 0 = 2 - 2
0000101 => 0000101 => [4,1,1,1] => ([(0,4),(0,5),(0,6),(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 0 = 2 - 2
0001000 => 0000001 => [6,1] => ([(0,6),(1,6),(2,6),(3,6),(4,6),(5,6)],7)
=> 0 = 2 - 2
0001001 => 0001001 => [3,1,2,1] => ([(0,6),(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 2 - 2
0001010 => 0000101 => [4,1,1,1] => ([(0,4),(0,5),(0,6),(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 0 = 2 - 2
0001101 => 0001101 => [3,2,1,1] => ([(0,5),(0,6),(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 2 - 2
0010000 => 0000001 => [6,1] => ([(0,6),(1,6),(2,6),(3,6),(4,6),(5,6)],7)
=> 0 = 2 - 2
0010001 => 0001001 => [3,1,2,1] => ([(0,6),(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 2 - 2
0010010 => 0001001 => [3,1,2,1] => ([(0,6),(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 2 - 2
0010100 => 0000101 => [4,1,1,1] => ([(0,4),(0,5),(0,6),(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 0 = 2 - 2
0010101 => 0010101 => [2,1,1,1,1,1] => ([(0,2),(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 0 = 2 - 2
0011010 => 0001101 => [3,2,1,1] => ([(0,5),(0,6),(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 2 - 2
0011101 => 0011101 => [2,3,1,1] => ([(0,5),(0,6),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 2 - 2
0100000 => 0000001 => [6,1] => ([(0,6),(1,6),(2,6),(3,6),(4,6),(5,6)],7)
=> 0 = 2 - 2
0100001 => 0000101 => [4,1,1,1] => ([(0,4),(0,5),(0,6),(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 0 = 2 - 2
0100010 => 0000101 => [4,1,1,1] => ([(0,4),(0,5),(0,6),(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 0 = 2 - 2
0100011 => 0001101 => [3,2,1,1] => ([(0,5),(0,6),(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 2 - 2
0100100 => 0000101 => [4,1,1,1] => ([(0,4),(0,5),(0,6),(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 0 = 2 - 2
0100101 => 0010101 => [2,1,1,1,1,1] => ([(0,2),(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 0 = 2 - 2
0100110 => 0001101 => [3,2,1,1] => ([(0,5),(0,6),(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 2 - 2
0100111 => 0011101 => [2,3,1,1] => ([(0,5),(0,6),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 2 - 2
0101000 => 0000101 => [4,1,1,1] => ([(0,4),(0,5),(0,6),(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 0 = 2 - 2
0101001 => 0010101 => [2,1,1,1,1,1] => ([(0,2),(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 0 = 2 - 2
0101010 => 0010101 => [2,1,1,1,1,1] => ([(0,2),(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 0 = 2 - 2
1000000 => 0000001 => [6,1] => ([(0,6),(1,6),(2,6),(3,6),(4,6),(5,6)],7)
=> 0 = 2 - 2
Description
The second largest eigenvalue of a graph if it is integral.
This statistic is undefined if the second largest eigenvalue of the graph is not integral.
Chapter 4 of [1] provides lots of context.
Matching statistic: St000264
(load all 6 compositions to match this statistic)
(load all 6 compositions to match this statistic)
Mp00269: Binary words —flag zeros to zeros⟶ Binary words
Mp00097: Binary words —delta morphism⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St000264: Graphs ⟶ ℤResult quality: 33% ●values known / values provided: 76%●distinct values known / distinct values provided: 33%
Mp00097: Binary words —delta morphism⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St000264: Graphs ⟶ ℤResult quality: 33% ●values known / values provided: 76%●distinct values known / distinct values provided: 33%
Values
0 => 0 => [1] => ([],1)
=> ? = 0 + 1
1 => 1 => [1] => ([],1)
=> ? = 0 + 1
01 => 10 => [1,1] => ([(0,1)],2)
=> ? = 1 + 1
10 => 00 => [2] => ([],2)
=> ? = 1 + 1
001 => 101 => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 3 = 2 + 1
010 => 000 => [3] => ([],3)
=> ? = 2 + 1
100 => 010 => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 3 = 2 + 1
0001 => 1011 => [1,1,2] => ([(1,2),(1,3),(2,3)],4)
=> 3 = 2 + 1
0010 => 0001 => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> ? = 2 + 1
0100 => 0100 => [1,1,2] => ([(1,2),(1,3),(2,3)],4)
=> 3 = 2 + 1
0101 => 1000 => [1,3] => ([(2,3)],4)
=> ? = 1 + 1
1000 => 0110 => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 3 = 2 + 1
00001 => 10111 => [1,1,3] => ([(2,3),(2,4),(3,4)],5)
=> 3 = 2 + 1
00010 => 00011 => [3,2] => ([(1,4),(2,4),(3,4)],5)
=> ? = 2 + 1
00100 => 01001 => [1,1,2,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = 2 + 1
00101 => 10001 => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = 2 + 1
01000 => 01100 => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = 2 + 1
01001 => 10100 => [1,1,1,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = 2 + 1
01010 => 00000 => [5] => ([],5)
=> ? = 2 + 1
10000 => 01110 => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = 2 + 1
000001 => 101111 => [1,1,4] => ([(3,4),(3,5),(4,5)],6)
=> 3 = 2 + 1
000010 => 000111 => [3,3] => ([(2,5),(3,5),(4,5)],6)
=> ? = 2 + 1
000100 => 010011 => [1,1,2,2] => ([(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 3 = 2 + 1
000101 => 100011 => [1,3,2] => ([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> 3 = 2 + 1
001000 => 011001 => [1,2,2,1] => ([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 3 = 2 + 1
001001 => 101001 => [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)
=> 3 = 2 + 1
001010 => 000001 => [5,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> ? = 2 + 1
001101 => 100101 => [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)
=> 3 = 2 + 1
010000 => 011100 => [1,3,2] => ([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> 3 = 2 + 1
010001 => 101100 => [1,1,2,2] => ([(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 3 = 2 + 1
010010 => 000100 => [3,1,2] => ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 3 = 2 + 1
010011 => 110100 => [2,1,1,2] => ([(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 3 = 2 + 1
010100 => 010000 => [1,1,4] => ([(3,4),(3,5),(4,5)],6)
=> 3 = 2 + 1
010101 => 100000 => [1,5] => ([(4,5)],6)
=> ? = 1 + 1
100000 => 011110 => [1,4,1] => ([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> 3 = 2 + 1
0000001 => 1011111 => [1,1,5] => ([(4,5),(4,6),(5,6)],7)
=> 3 = 2 + 1
0000010 => 0001111 => [3,4] => ([(3,6),(4,6),(5,6)],7)
=> ? = 2 + 1
0000100 => 0100111 => [1,1,2,3] => ([(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 3 = 2 + 1
0000101 => 1000111 => [1,3,3] => ([(2,6),(3,6),(4,5),(4,6),(5,6)],7)
=> 3 = 2 + 1
0001000 => 0110011 => [1,2,2,2] => ([(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 3 = 2 + 1
0001001 => 1010011 => [1,1,1,2,2] => ([(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 3 = 2 + 1
0001010 => 0000011 => [5,2] => ([(1,6),(2,6),(3,6),(4,6),(5,6)],7)
=> ? = 2 + 1
0001101 => 1001011 => [1,2,1,1,2] => ([(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 3 = 2 + 1
0010000 => 0111001 => [1,3,2,1] => ([(0,6),(1,5),(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 3 = 2 + 1
0010001 => 1011001 => [1,1,2,2,1] => ([(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)
=> 3 = 2 + 1
0010010 => 0001001 => [3,1,2,1] => ([(0,6),(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 3 = 2 + 1
0010100 => 0100001 => [1,1,4,1] => ([(0,6),(1,6),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 3 = 2 + 1
0010101 => 1000001 => [1,5,1] => ([(0,6),(1,6),(2,6),(3,6),(4,5),(4,6),(5,6)],7)
=> 3 = 2 + 1
0011010 => 0000101 => [4,1,1,1] => ([(0,4),(0,5),(0,6),(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 3 = 2 + 1
0011101 => 1001101 => [1,2,2,1,1] => ([(0,5),(0,6),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 3 = 2 + 1
0100000 => 0111100 => [1,4,2] => ([(1,6),(2,6),(3,6),(4,5),(4,6),(5,6)],7)
=> 3 = 2 + 1
0100001 => 1011100 => [1,1,3,2] => ([(1,6),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 3 = 2 + 1
0100010 => 0001100 => [3,2,2] => ([(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 3 = 2 + 1
0100011 => 1101100 => [2,1,2,2] => ([(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 3 = 2 + 1
0100100 => 0100100 => [1,1,2,1,2] => ([(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)
=> 3 = 2 + 1
0100101 => 1000100 => [1,3,1,2] => ([(1,5),(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 3 = 2 + 1
0100110 => 0010100 => [2,1,1,1,2] => ([(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 3 = 2 + 1
0100111 => 1110100 => [3,1,1,2] => ([(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 3 = 2 + 1
0101000 => 0110000 => [1,2,4] => ([(3,6),(4,5),(4,6),(5,6)],7)
=> 3 = 2 + 1
0101001 => 1010000 => [1,1,1,4] => ([(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 3 = 2 + 1
0101010 => 0000000 => [7] => ([],7)
=> ? = 2 + 1
1000000 => 0111110 => [1,5,1] => ([(0,6),(1,6),(2,6),(3,6),(4,5),(4,6),(5,6)],7)
=> 3 = 2 + 1
Description
The girth of a graph, which is not a tree.
This is the length of the shortest cycle in the graph.
Matching statistic: St000781
Mp00097: Binary words —delta morphism⟶ Integer compositions
Mp00180: Integer compositions —to ribbon⟶ Skew partitions
Mp00183: Skew partitions —inner shape⟶ Integer partitions
St000781: Integer partitions ⟶ ℤResult quality: 33% ●values known / values provided: 68%●distinct values known / distinct values provided: 33%
Mp00180: Integer compositions —to ribbon⟶ Skew partitions
Mp00183: Skew partitions —inner shape⟶ Integer partitions
St000781: Integer partitions ⟶ ℤResult quality: 33% ●values known / values provided: 68%●distinct values known / distinct values provided: 33%
Values
0 => [1] => [[1],[]]
=> []
=> ? = 0 - 1
1 => [1] => [[1],[]]
=> []
=> ? = 0 - 1
01 => [1,1] => [[1,1],[]]
=> []
=> ? = 1 - 1
10 => [1,1] => [[1,1],[]]
=> []
=> ? = 1 - 1
001 => [2,1] => [[2,2],[1]]
=> [1]
=> 1 = 2 - 1
010 => [1,1,1] => [[1,1,1],[]]
=> []
=> ? = 2 - 1
100 => [1,2] => [[2,1],[]]
=> []
=> ? = 2 - 1
0001 => [3,1] => [[3,3],[2]]
=> [2]
=> 1 = 2 - 1
0010 => [2,1,1] => [[2,2,2],[1,1]]
=> [1,1]
=> 1 = 2 - 1
0100 => [1,1,2] => [[2,1,1],[]]
=> []
=> ? = 2 - 1
0101 => [1,1,1,1] => [[1,1,1,1],[]]
=> []
=> ? = 1 - 1
1000 => [1,3] => [[3,1],[]]
=> []
=> ? = 2 - 1
00001 => [4,1] => [[4,4],[3]]
=> [3]
=> 1 = 2 - 1
00010 => [3,1,1] => [[3,3,3],[2,2]]
=> [2,2]
=> 1 = 2 - 1
00100 => [2,1,2] => [[3,2,2],[1,1]]
=> [1,1]
=> 1 = 2 - 1
00101 => [2,1,1,1] => [[2,2,2,2],[1,1,1]]
=> [1,1,1]
=> 1 = 2 - 1
01000 => [1,1,3] => [[3,1,1],[]]
=> []
=> ? = 2 - 1
01001 => [1,1,2,1] => [[2,2,1,1],[1]]
=> [1]
=> 1 = 2 - 1
01010 => [1,1,1,1,1] => [[1,1,1,1,1],[]]
=> []
=> ? = 2 - 1
10000 => [1,4] => [[4,1],[]]
=> []
=> ? = 2 - 1
000001 => [5,1] => [[5,5],[4]]
=> [4]
=> 1 = 2 - 1
000010 => [4,1,1] => [[4,4,4],[3,3]]
=> [3,3]
=> 1 = 2 - 1
000100 => [3,1,2] => [[4,3,3],[2,2]]
=> [2,2]
=> 1 = 2 - 1
000101 => [3,1,1,1] => [[3,3,3,3],[2,2,2]]
=> [2,2,2]
=> 1 = 2 - 1
001000 => [2,1,3] => [[4,2,2],[1,1]]
=> [1,1]
=> 1 = 2 - 1
001001 => [2,1,2,1] => [[3,3,2,2],[2,1,1]]
=> [2,1,1]
=> 1 = 2 - 1
001010 => [2,1,1,1,1] => [[2,2,2,2,2],[1,1,1,1]]
=> [1,1,1,1]
=> 1 = 2 - 1
001101 => [2,2,1,1] => [[3,3,3,2],[2,2,1]]
=> [2,2,1]
=> 1 = 2 - 1
010000 => [1,1,4] => [[4,1,1],[]]
=> []
=> ? = 2 - 1
010001 => [1,1,3,1] => [[3,3,1,1],[2]]
=> [2]
=> 1 = 2 - 1
010010 => [1,1,2,1,1] => [[2,2,2,1,1],[1,1]]
=> [1,1]
=> 1 = 2 - 1
010011 => [1,1,2,2] => [[3,2,1,1],[1]]
=> [1]
=> 1 = 2 - 1
010100 => [1,1,1,1,2] => [[2,1,1,1,1],[]]
=> []
=> ? = 2 - 1
010101 => [1,1,1,1,1,1] => [[1,1,1,1,1,1],[]]
=> []
=> ? = 1 - 1
100000 => [1,5] => [[5,1],[]]
=> []
=> ? = 2 - 1
0000001 => [6,1] => [[6,6],[5]]
=> [5]
=> 1 = 2 - 1
0000010 => [5,1,1] => [[5,5,5],[4,4]]
=> [4,4]
=> 1 = 2 - 1
0000100 => [4,1,2] => [[5,4,4],[3,3]]
=> [3,3]
=> 1 = 2 - 1
0000101 => [4,1,1,1] => [[4,4,4,4],[3,3,3]]
=> [3,3,3]
=> 1 = 2 - 1
0001000 => [3,1,3] => [[5,3,3],[2,2]]
=> [2,2]
=> 1 = 2 - 1
0001001 => [3,1,2,1] => [[4,4,3,3],[3,2,2]]
=> [3,2,2]
=> 1 = 2 - 1
0001010 => [3,1,1,1,1] => [[3,3,3,3,3],[2,2,2,2]]
=> [2,2,2,2]
=> 1 = 2 - 1
0001101 => [3,2,1,1] => [[4,4,4,3],[3,3,2]]
=> [3,3,2]
=> 1 = 2 - 1
0010000 => [2,1,4] => [[5,2,2],[1,1]]
=> [1,1]
=> 1 = 2 - 1
0010001 => [2,1,3,1] => [[4,4,2,2],[3,1,1]]
=> [3,1,1]
=> 1 = 2 - 1
0010010 => [2,1,2,1,1] => [[3,3,3,2,2],[2,2,1,1]]
=> [2,2,1,1]
=> 1 = 2 - 1
0010100 => [2,1,1,1,2] => [[3,2,2,2,2],[1,1,1,1]]
=> [1,1,1,1]
=> 1 = 2 - 1
0010101 => [2,1,1,1,1,1] => [[2,2,2,2,2,2],[1,1,1,1,1]]
=> [1,1,1,1,1]
=> 1 = 2 - 1
0011010 => [2,2,1,1,1] => [[3,3,3,3,2],[2,2,2,1]]
=> [2,2,2,1]
=> 1 = 2 - 1
0011101 => [2,3,1,1] => [[4,4,4,2],[3,3,1]]
=> [3,3,1]
=> 1 = 2 - 1
0100000 => [1,1,5] => [[5,1,1],[]]
=> []
=> ? = 2 - 1
0100001 => [1,1,4,1] => [[4,4,1,1],[3]]
=> [3]
=> 1 = 2 - 1
0100010 => [1,1,3,1,1] => [[3,3,3,1,1],[2,2]]
=> [2,2]
=> 1 = 2 - 1
0100011 => [1,1,3,2] => [[4,3,1,1],[2]]
=> [2]
=> 1 = 2 - 1
0100100 => [1,1,2,1,2] => [[3,2,2,1,1],[1,1]]
=> [1,1]
=> 1 = 2 - 1
0100101 => [1,1,2,1,1,1] => [[2,2,2,2,1,1],[1,1,1]]
=> [1,1,1]
=> 1 = 2 - 1
0100110 => [1,1,2,2,1] => [[3,3,2,1,1],[2,1]]
=> [2,1]
=> 1 = 2 - 1
0100111 => [1,1,2,3] => [[4,2,1,1],[1]]
=> [1]
=> 1 = 2 - 1
0101000 => [1,1,1,1,3] => [[3,1,1,1,1],[]]
=> []
=> ? = 2 - 1
0101001 => [1,1,1,1,2,1] => [[2,2,1,1,1,1],[1]]
=> [1]
=> 1 = 2 - 1
0101010 => [1,1,1,1,1,1,1] => [[1,1,1,1,1,1,1],[]]
=> []
=> ? = 2 - 1
1000000 => [1,6] => [[6,1],[]]
=> []
=> ? = 2 - 1
Description
The number of proper colouring schemes of a Ferrers diagram.
A colouring of a Ferrers diagram is proper if no two cells in a row or in a column have the same colour. The minimal number of colours needed is the maximum of the length and the first part of the partition, because we can restrict a latin square to the shape. We can associate to each colouring the integer partition recording how often each colour is used, see [1].
This statistic is the number of distinct such integer partitions that occur.
Matching statistic: St001488
Mp00224: Binary words —runsort⟶ Binary words
Mp00097: Binary words —delta morphism⟶ Integer compositions
Mp00180: Integer compositions —to ribbon⟶ Skew partitions
St001488: Skew partitions ⟶ ℤResult quality: 32% ●values known / values provided: 32%●distinct values known / distinct values provided: 100%
Mp00097: Binary words —delta morphism⟶ Integer compositions
Mp00180: Integer compositions —to ribbon⟶ Skew partitions
St001488: Skew partitions ⟶ ℤResult quality: 32% ●values known / values provided: 32%●distinct values known / distinct values provided: 100%
Values
0 => 0 => [1] => [[1],[]]
=> 1 = 0 + 1
1 => 1 => [1] => [[1],[]]
=> 1 = 0 + 1
01 => 01 => [1,1] => [[1,1],[]]
=> 2 = 1 + 1
10 => 01 => [1,1] => [[1,1],[]]
=> 2 = 1 + 1
001 => 001 => [2,1] => [[2,2],[1]]
=> 3 = 2 + 1
010 => 001 => [2,1] => [[2,2],[1]]
=> 3 = 2 + 1
100 => 001 => [2,1] => [[2,2],[1]]
=> 3 = 2 + 1
0001 => 0001 => [3,1] => [[3,3],[2]]
=> 3 = 2 + 1
0010 => 0001 => [3,1] => [[3,3],[2]]
=> 3 = 2 + 1
0100 => 0001 => [3,1] => [[3,3],[2]]
=> 3 = 2 + 1
0101 => 0101 => [1,1,1,1] => [[1,1,1,1],[]]
=> 2 = 1 + 1
1000 => 0001 => [3,1] => [[3,3],[2]]
=> 3 = 2 + 1
00001 => 00001 => [4,1] => [[4,4],[3]]
=> 3 = 2 + 1
00010 => 00001 => [4,1] => [[4,4],[3]]
=> 3 = 2 + 1
00100 => 00001 => [4,1] => [[4,4],[3]]
=> 3 = 2 + 1
00101 => 00101 => [2,1,1,1] => [[2,2,2,2],[1,1,1]]
=> 3 = 2 + 1
01000 => 00001 => [4,1] => [[4,4],[3]]
=> 3 = 2 + 1
01001 => 00101 => [2,1,1,1] => [[2,2,2,2],[1,1,1]]
=> 3 = 2 + 1
01010 => 00101 => [2,1,1,1] => [[2,2,2,2],[1,1,1]]
=> 3 = 2 + 1
10000 => 00001 => [4,1] => [[4,4],[3]]
=> 3 = 2 + 1
000001 => 000001 => [5,1] => [[5,5],[4]]
=> ? = 2 + 1
000010 => 000001 => [5,1] => [[5,5],[4]]
=> ? = 2 + 1
000100 => 000001 => [5,1] => [[5,5],[4]]
=> ? = 2 + 1
000101 => 000101 => [3,1,1,1] => [[3,3,3,3],[2,2,2]]
=> ? = 2 + 1
001000 => 000001 => [5,1] => [[5,5],[4]]
=> ? = 2 + 1
001001 => 001001 => [2,1,2,1] => [[3,3,2,2],[2,1,1]]
=> ? = 2 + 1
001010 => 000101 => [3,1,1,1] => [[3,3,3,3],[2,2,2]]
=> ? = 2 + 1
001101 => 001101 => [2,2,1,1] => [[3,3,3,2],[2,2,1]]
=> ? = 2 + 1
010000 => 000001 => [5,1] => [[5,5],[4]]
=> ? = 2 + 1
010001 => 000101 => [3,1,1,1] => [[3,3,3,3],[2,2,2]]
=> ? = 2 + 1
010010 => 000101 => [3,1,1,1] => [[3,3,3,3],[2,2,2]]
=> ? = 2 + 1
010011 => 001101 => [2,2,1,1] => [[3,3,3,2],[2,2,1]]
=> ? = 2 + 1
010100 => 000101 => [3,1,1,1] => [[3,3,3,3],[2,2,2]]
=> ? = 2 + 1
010101 => 010101 => [1,1,1,1,1,1] => [[1,1,1,1,1,1],[]]
=> ? = 1 + 1
100000 => 000001 => [5,1] => [[5,5],[4]]
=> ? = 2 + 1
0000001 => 0000001 => [6,1] => [[6,6],[5]]
=> ? = 2 + 1
0000010 => 0000001 => [6,1] => [[6,6],[5]]
=> ? = 2 + 1
0000100 => 0000001 => [6,1] => [[6,6],[5]]
=> ? = 2 + 1
0000101 => 0000101 => [4,1,1,1] => [[4,4,4,4],[3,3,3]]
=> ? = 2 + 1
0001000 => 0000001 => [6,1] => [[6,6],[5]]
=> ? = 2 + 1
0001001 => 0001001 => [3,1,2,1] => [[4,4,3,3],[3,2,2]]
=> ? = 2 + 1
0001010 => 0000101 => [4,1,1,1] => [[4,4,4,4],[3,3,3]]
=> ? = 2 + 1
0001101 => 0001101 => [3,2,1,1] => [[4,4,4,3],[3,3,2]]
=> ? = 2 + 1
0010000 => 0000001 => [6,1] => [[6,6],[5]]
=> ? = 2 + 1
0010001 => 0001001 => [3,1,2,1] => [[4,4,3,3],[3,2,2]]
=> ? = 2 + 1
0010010 => 0001001 => [3,1,2,1] => [[4,4,3,3],[3,2,2]]
=> ? = 2 + 1
0010100 => 0000101 => [4,1,1,1] => [[4,4,4,4],[3,3,3]]
=> ? = 2 + 1
0010101 => 0010101 => [2,1,1,1,1,1] => [[2,2,2,2,2,2],[1,1,1,1,1]]
=> ? = 2 + 1
0011010 => 0001101 => [3,2,1,1] => [[4,4,4,3],[3,3,2]]
=> ? = 2 + 1
0011101 => 0011101 => [2,3,1,1] => [[4,4,4,2],[3,3,1]]
=> ? = 2 + 1
0100000 => 0000001 => [6,1] => [[6,6],[5]]
=> ? = 2 + 1
0100001 => 0000101 => [4,1,1,1] => [[4,4,4,4],[3,3,3]]
=> ? = 2 + 1
0100010 => 0000101 => [4,1,1,1] => [[4,4,4,4],[3,3,3]]
=> ? = 2 + 1
0100011 => 0001101 => [3,2,1,1] => [[4,4,4,3],[3,3,2]]
=> ? = 2 + 1
0100100 => 0000101 => [4,1,1,1] => [[4,4,4,4],[3,3,3]]
=> ? = 2 + 1
0100101 => 0010101 => [2,1,1,1,1,1] => [[2,2,2,2,2,2],[1,1,1,1,1]]
=> ? = 2 + 1
0100110 => 0001101 => [3,2,1,1] => [[4,4,4,3],[3,3,2]]
=> ? = 2 + 1
0100111 => 0011101 => [2,3,1,1] => [[4,4,4,2],[3,3,1]]
=> ? = 2 + 1
0101000 => 0000101 => [4,1,1,1] => [[4,4,4,4],[3,3,3]]
=> ? = 2 + 1
0101001 => 0010101 => [2,1,1,1,1,1] => [[2,2,2,2,2,2],[1,1,1,1,1]]
=> ? = 2 + 1
0101010 => 0010101 => [2,1,1,1,1,1] => [[2,2,2,2,2,2],[1,1,1,1,1]]
=> ? = 2 + 1
1000000 => 0000001 => [6,1] => [[6,6],[5]]
=> ? = 2 + 1
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: St001878
Mp00316: Binary words —inverse Foata bijection⟶ Binary words
Mp00262: Binary words —poset of factors⟶ Posets
Mp00206: Posets —antichains of maximal size⟶ Lattices
St001878: Lattices ⟶ ℤResult quality: 19% ●values known / values provided: 19%●distinct values known / distinct values provided: 33%
Mp00262: Binary words —poset of factors⟶ Posets
Mp00206: Posets —antichains of maximal size⟶ Lattices
St001878: Lattices ⟶ ℤResult quality: 19% ●values known / values provided: 19%●distinct values known / distinct values provided: 33%
Values
0 => 0 => ([(0,1)],2)
=> ([(0,1)],2)
=> ? = 0 - 1
1 => 1 => ([(0,1)],2)
=> ([(0,1)],2)
=> ? = 0 - 1
01 => 01 => ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([],1)
=> ? = 1 - 1
10 => 10 => ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([],1)
=> ? = 1 - 1
001 => 001 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ([(0,2),(2,1)],3)
=> 1 = 2 - 1
010 => 100 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ([(0,2),(2,1)],3)
=> 1 = 2 - 1
100 => 010 => ([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)
=> ([(0,1)],2)
=> ? = 2 - 1
0001 => 0001 => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> 1 = 2 - 1
0010 => 1000 => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> 1 = 2 - 1
0100 => 0100 => ([(0,2),(0,3),(1,6),(2,7),(2,8),(3,1),(3,7),(3,8),(5,4),(6,4),(7,5),(8,5),(8,6)],9)
=> ([],1)
=> ? = 2 - 1
0101 => 1001 => ([(0,2),(0,3),(1,5),(1,6),(2,7),(2,8),(3,1),(3,7),(3,8),(5,4),(6,4),(7,6),(8,5)],9)
=> ([],1)
=> ? = 1 - 1
1000 => 0010 => ([(0,2),(0,3),(1,6),(2,7),(2,8),(3,1),(3,7),(3,8),(5,4),(6,4),(7,5),(8,5),(8,6)],9)
=> ([],1)
=> ? = 2 - 1
00001 => 00001 => ([(0,2),(0,5),(1,7),(2,6),(3,4),(3,9),(4,1),(4,8),(5,3),(5,6),(6,9),(8,7),(9,8)],10)
=> ([(0,6),(1,7),(2,9),(4,8),(5,1),(5,9),(6,2),(6,5),(7,8),(8,3),(9,4),(9,7)],10)
=> ? = 2 - 1
00010 => 10000 => ([(0,2),(0,5),(1,7),(2,6),(3,4),(3,9),(4,1),(4,8),(5,3),(5,6),(6,9),(8,7),(9,8)],10)
=> ([(0,6),(1,7),(2,9),(4,8),(5,1),(5,9),(6,2),(6,5),(7,8),(8,3),(9,4),(9,7)],10)
=> ? = 2 - 1
00100 => 01000 => ([(0,3),(0,4),(1,2),(1,11),(2,8),(3,9),(3,10),(4,1),(4,9),(4,10),(6,7),(7,5),(8,5),(9,6),(10,6),(10,11),(11,7),(11,8)],12)
=> ([(0,3),(2,1),(3,2)],4)
=> 1 = 2 - 1
00101 => 10001 => ([(0,3),(0,4),(1,2),(1,10),(1,11),(2,8),(2,9),(3,6),(3,7),(4,1),(4,6),(4,7),(6,11),(7,10),(8,5),(9,5),(10,8),(11,9)],12)
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> 1 = 2 - 1
01000 => 00100 => ([(0,2),(0,3),(1,5),(1,6),(2,10),(2,11),(3,1),(3,10),(3,11),(5,8),(6,7),(7,4),(8,4),(9,7),(9,8),(10,6),(10,9),(11,5),(11,9)],12)
=> ([(0,1)],2)
=> ? = 2 - 1
01001 => 01001 => ([(0,2),(0,3),(1,5),(1,9),(2,10),(2,11),(3,1),(3,10),(3,11),(5,7),(6,8),(7,4),(8,4),(9,7),(9,8),(10,5),(10,6),(11,6),(11,9)],12)
=> ([(0,1)],2)
=> ? = 2 - 1
01010 => 01100 => ([(0,3),(0,4),(1,9),(2,6),(2,11),(3,2),(3,10),(3,12),(4,1),(4,10),(4,12),(6,7),(7,5),(8,5),(9,8),(10,6),(11,7),(11,8),(12,9),(12,11)],13)
=> ([],1)
=> ? = 2 - 1
10000 => 00010 => ([(0,3),(0,4),(1,2),(1,11),(2,8),(3,9),(3,10),(4,1),(4,9),(4,10),(6,7),(7,5),(8,5),(9,6),(10,6),(10,11),(11,7),(11,8)],12)
=> ([(0,3),(2,1),(3,2)],4)
=> 1 = 2 - 1
000001 => 000001 => ([(0,2),(0,6),(1,8),(2,7),(3,5),(3,9),(4,3),(4,11),(5,1),(5,10),(6,4),(6,7),(7,11),(9,10),(10,8),(11,9)],12)
=> ([(0,8),(1,14),(3,13),(4,12),(5,11),(6,7),(6,12),(7,5),(7,9),(8,4),(8,6),(9,11),(9,13),(10,14),(11,10),(12,3),(12,9),(13,1),(13,10),(14,2)],15)
=> ? = 2 - 1
000010 => 100000 => ([(0,2),(0,6),(1,8),(2,7),(3,5),(3,9),(4,3),(4,11),(5,1),(5,10),(6,4),(6,7),(7,11),(9,10),(10,8),(11,9)],12)
=> ([(0,8),(1,14),(3,13),(4,12),(5,11),(6,7),(6,12),(7,5),(7,9),(8,4),(8,6),(9,11),(9,13),(10,14),(11,10),(12,3),(12,9),(13,1),(13,10),(14,2)],15)
=> ? = 2 - 1
000100 => 010000 => ([(0,4),(0,5),(1,3),(1,12),(2,11),(3,2),(3,14),(4,10),(4,13),(5,1),(5,10),(5,13),(7,8),(8,9),(9,6),(10,7),(11,6),(12,8),(12,14),(13,7),(13,12),(14,9),(14,11)],15)
=> ([(0,6),(1,7),(2,9),(4,8),(5,1),(5,9),(6,2),(6,5),(7,8),(8,3),(9,4),(9,7)],10)
=> ? = 2 - 1
000101 => 100001 => ([(0,4),(0,5),(1,3),(1,7),(1,8),(2,13),(2,14),(3,2),(3,11),(3,12),(4,9),(4,10),(5,1),(5,9),(5,10),(7,12),(8,11),(9,8),(10,7),(11,13),(12,14),(13,6),(14,6)],15)
=> ([(0,6),(1,11),(1,12),(2,9),(3,8),(4,7),(4,11),(5,7),(5,12),(6,1),(6,4),(6,5),(7,13),(8,10),(9,10),(11,3),(11,13),(12,2),(12,13),(13,8),(13,9)],14)
=> ? = 2 - 1
001000 => 001000 => ([(0,3),(0,4),(1,10),(2,1),(2,6),(2,12),(3,14),(3,15),(4,2),(4,14),(4,15),(6,7),(7,8),(8,5),(9,5),(10,9),(11,7),(11,13),(12,10),(12,13),(13,8),(13,9),(14,6),(14,11),(15,11),(15,12)],16)
=> ([],1)
=> ? = 2 - 1
001001 => 010001 => ([(0,3),(0,4),(1,2),(1,11),(1,15),(2,7),(2,12),(3,13),(3,14),(4,1),(4,13),(4,14),(6,9),(7,10),(8,6),(9,5),(10,5),(11,7),(12,9),(12,10),(13,8),(13,15),(14,8),(14,11),(15,6),(15,12)],16)
=> ([],1)
=> ? = 2 - 1
001010 => 011000 => ([(0,4),(0,5),(1,11),(2,1),(2,13),(3,7),(3,14),(4,2),(4,12),(4,16),(5,3),(5,12),(5,16),(7,8),(8,9),(9,6),(10,6),(11,10),(12,7),(13,11),(13,15),(14,8),(14,15),(15,9),(15,10),(16,13),(16,14)],17)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1 = 2 - 1
001101 => 110001 => ([(0,4),(0,5),(1,12),(2,3),(2,13),(2,16),(3,8),(3,14),(4,1),(4,9),(4,15),(5,2),(5,9),(5,15),(7,10),(8,11),(9,13),(10,6),(11,6),(12,7),(13,8),(14,10),(14,11),(15,12),(15,16),(16,7),(16,14)],17)
=> ([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
=> 1 = 2 - 1
010000 => 000100 => ([(0,3),(0,4),(1,10),(2,1),(2,6),(2,12),(3,14),(3,15),(4,2),(4,14),(4,15),(6,7),(7,8),(8,5),(9,5),(10,9),(11,7),(11,13),(12,10),(12,13),(13,8),(13,9),(14,6),(14,11),(15,11),(15,12)],16)
=> ([],1)
=> ? = 2 - 1
010001 => 001001 => ([(0,2),(0,3),(1,11),(1,12),(2,13),(2,14),(3,1),(3,13),(3,14),(5,7),(6,8),(7,4),(8,4),(9,7),(9,8),(10,5),(10,9),(11,6),(11,9),(12,5),(12,6),(13,10),(13,11),(14,10),(14,12)],15)
=> ([(0,2),(2,1)],3)
=> 1 = 2 - 1
010010 => 001100 => ([(0,3),(0,4),(1,15),(1,16),(2,10),(2,11),(3,1),(3,13),(3,14),(4,2),(4,13),(4,14),(6,9),(7,8),(8,5),(9,5),(10,7),(11,6),(12,8),(12,9),(13,10),(13,15),(14,11),(14,16),(15,7),(15,12),(16,6),(16,12)],17)
=> ([(0,1)],2)
=> ? = 2 - 1
010011 => 010011 => ([(0,3),(0,4),(1,11),(2,12),(2,13),(3,2),(3,15),(3,16),(4,1),(4,15),(4,16),(6,7),(7,9),(8,10),(9,5),(10,5),(11,8),(12,7),(12,14),(13,8),(13,14),(14,9),(14,10),(15,6),(15,12),(16,6),(16,11),(16,13)],17)
=> ([(0,2),(2,1)],3)
=> 1 = 2 - 1
010100 => 100100 => ([(0,2),(0,3),(1,11),(1,12),(2,13),(2,14),(3,1),(3,13),(3,14),(5,7),(6,8),(7,4),(8,4),(9,7),(9,8),(10,5),(10,9),(11,6),(11,9),(12,5),(12,6),(13,10),(13,11),(14,10),(14,12)],15)
=> ([(0,2),(2,1)],3)
=> 1 = 2 - 1
010101 => 011001 => ([(0,3),(0,4),(1,11),(1,16),(2,10),(2,15),(3,2),(3,13),(3,14),(4,1),(4,13),(4,14),(6,8),(7,9),(8,5),(9,5),(10,6),(11,7),(12,8),(12,9),(13,15),(13,16),(14,10),(14,11),(15,6),(15,12),(16,7),(16,12)],17)
=> ([(0,1)],2)
=> ? = 1 - 1
100000 => 000010 => ([(0,4),(0,5),(1,3),(1,12),(2,11),(3,2),(3,14),(4,10),(4,13),(5,1),(5,10),(5,13),(7,8),(8,9),(9,6),(10,7),(11,6),(12,8),(12,14),(13,7),(13,12),(14,9),(14,11)],15)
=> ([(0,6),(1,7),(2,9),(4,8),(5,1),(5,9),(6,2),(6,5),(7,8),(8,3),(9,4),(9,7)],10)
=> ? = 2 - 1
0000001 => 0000001 => ([(0,2),(0,7),(1,9),(2,8),(3,4),(3,11),(4,6),(4,10),(5,3),(5,13),(6,1),(6,12),(7,5),(7,8),(8,13),(10,12),(11,10),(12,9),(13,11)],14)
=> ?
=> ? = 2 - 1
0000010 => 1000000 => ([(0,2),(0,7),(1,9),(2,8),(3,4),(3,11),(4,6),(4,10),(5,3),(5,13),(6,1),(6,12),(7,5),(7,8),(8,13),(10,12),(11,10),(12,9),(13,11)],14)
=> ?
=> ? = 2 - 1
0000100 => 0100000 => ([(0,5),(0,6),(1,4),(1,14),(2,13),(3,2),(3,16),(4,3),(4,17),(5,12),(5,15),(6,1),(6,12),(6,15),(8,11),(9,10),(10,8),(11,7),(12,9),(13,7),(14,10),(14,17),(15,9),(15,14),(16,11),(16,13),(17,8),(17,16)],18)
=> ?
=> ? = 2 - 1
0000101 => 1000001 => ([(0,5),(0,6),(1,4),(1,8),(1,9),(2,16),(2,17),(3,2),(3,12),(3,13),(4,3),(4,14),(4,15),(5,10),(5,11),(6,1),(6,10),(6,11),(8,15),(9,14),(10,9),(11,8),(12,16),(13,17),(14,12),(15,13),(16,7),(17,7)],18)
=> ?
=> ? = 2 - 1
0001000 => 0010000 => ([(0,4),(0,5),(1,2),(1,17),(2,7),(3,1),(3,8),(3,16),(4,18),(4,19),(5,3),(5,18),(5,19),(7,12),(8,9),(9,10),(10,11),(11,6),(12,6),(13,10),(13,14),(14,11),(14,12),(15,9),(15,13),(16,13),(16,17),(17,7),(17,14),(18,15),(18,16),(19,8),(19,15)],20)
=> ?
=> ? = 2 - 1
0001001 => 0100001 => ([(0,4),(0,5),(1,3),(1,9),(1,17),(2,14),(2,19),(3,2),(3,13),(3,18),(4,15),(4,16),(5,1),(5,15),(5,16),(7,11),(8,7),(9,13),(10,8),(11,6),(12,6),(13,14),(14,12),(15,9),(15,10),(16,10),(16,17),(17,8),(17,18),(18,7),(18,19),(19,11),(19,12)],20)
=> ?
=> ? = 2 - 1
0001010 => 0110000 => ([(0,5),(0,6),(1,3),(1,15),(2,14),(3,2),(3,20),(4,8),(4,16),(5,1),(5,13),(5,19),(6,4),(6,13),(6,19),(8,9),(9,10),(10,11),(11,7),(12,7),(13,8),(14,12),(15,18),(15,20),(16,9),(16,18),(17,11),(17,12),(18,10),(18,17),(19,15),(19,16),(20,14),(20,17)],21)
=> ?
=> ? = 2 - 1
0001101 => 1100001 => ([(0,5),(0,6),(1,14),(2,4),(2,9),(2,18),(3,16),(3,20),(4,3),(4,15),(4,19),(5,2),(5,10),(5,17),(6,1),(6,10),(6,17),(8,12),(9,15),(10,9),(11,8),(12,7),(13,7),(14,11),(15,16),(16,13),(17,14),(17,18),(18,11),(18,19),(19,8),(19,20),(20,12),(20,13)],21)
=> ?
=> ? = 2 - 1
0010000 => 0001000 => ([(0,3),(0,4),(1,2),(1,18),(1,19),(2,6),(2,7),(3,16),(3,17),(4,1),(4,16),(4,17),(6,9),(7,8),(8,10),(9,11),(10,5),(11,5),(12,14),(12,15),(13,10),(13,11),(14,8),(14,13),(15,9),(15,13),(16,12),(16,18),(17,12),(17,19),(18,7),(18,14),(19,6),(19,15)],20)
=> ?
=> ? = 2 - 1
0010001 => 0010001 => ([(0,3),(0,4),(1,2),(1,18),(1,19),(2,7),(2,14),(3,15),(3,16),(4,1),(4,15),(4,16),(6,8),(7,9),(8,10),(9,11),(10,5),(11,5),(12,10),(12,11),(13,8),(13,12),(14,9),(14,12),(15,17),(15,19),(16,17),(16,18),(17,6),(17,13),(18,13),(18,14),(19,6),(19,7)],20)
=> ?
=> ? = 2 - 1
0010010 => 0011000 => ([(0,4),(0,5),(1,12),(2,19),(2,21),(3,1),(3,13),(3,20),(4,2),(4,16),(4,17),(5,3),(5,16),(5,17),(7,11),(8,9),(9,10),(10,6),(11,6),(12,7),(13,8),(14,10),(14,11),(15,9),(15,14),(16,19),(16,20),(17,13),(17,21),(18,7),(18,14),(19,15),(19,18),(20,12),(20,18),(21,8),(21,15)],22)
=> ?
=> ? = 2 - 1
0010100 => 1001000 => ([(0,3),(0,4),(1,11),(2,1),(2,15),(2,19),(3,17),(3,18),(4,2),(4,17),(4,18),(6,10),(7,8),(8,9),(9,5),(10,5),(11,6),(12,8),(12,13),(13,9),(13,10),(14,12),(14,16),(15,7),(15,12),(16,6),(16,13),(17,14),(17,15),(18,14),(18,19),(19,7),(19,11),(19,16)],20)
=> ?
=> ? = 2 - 1
0010101 => 0110001 => ([(0,4),(0,5),(1,13),(1,20),(2,3),(2,14),(2,21),(3,8),(3,16),(4,1),(4,17),(4,18),(5,2),(5,17),(5,18),(7,9),(8,10),(9,11),(10,12),(11,6),(12,6),(13,7),(14,8),(15,9),(15,19),(16,10),(16,19),(17,20),(17,21),(18,13),(18,14),(19,11),(19,12),(20,7),(20,15),(21,15),(21,16)],22)
=> ?
=> ? = 2 - 1
0011010 => 1011000 => ([(0,4),(0,5),(1,13),(2,1),(2,15),(3,14),(3,16),(4,2),(4,20),(4,21),(5,3),(5,20),(5,21),(7,8),(8,9),(9,11),(10,12),(11,6),(12,6),(13,10),(14,8),(14,18),(15,13),(15,17),(16,17),(16,18),(17,10),(17,19),(18,9),(18,19),(19,11),(19,12),(20,7),(20,14),(21,7),(21,15),(21,16)],22)
=> ?
=> ? = 2 - 1
0011101 => 1110001 => ([(0,5),(0,6),(1,2),(1,20),(2,8),(3,4),(3,15),(3,21),(4,9),(4,17),(5,1),(5,10),(5,18),(6,3),(6,10),(6,18),(8,11),(9,12),(10,15),(11,13),(12,14),(13,7),(14,7),(15,9),(16,11),(16,19),(17,12),(17,19),(18,20),(18,21),(19,13),(19,14),(20,8),(20,16),(21,16),(21,17)],22)
=> ?
=> ? = 2 - 1
0100000 => 0000100 => ([(0,4),(0,5),(1,2),(1,17),(2,7),(3,1),(3,8),(3,16),(4,18),(4,19),(5,3),(5,18),(5,19),(7,12),(8,9),(9,10),(10,11),(11,6),(12,6),(13,10),(13,14),(14,11),(14,12),(15,9),(15,13),(16,13),(16,17),(17,7),(17,14),(18,15),(18,16),(19,8),(19,15)],20)
=> ?
=> ? = 2 - 1
0100001 => 0001001 => ([(0,3),(0,4),(1,11),(2,1),(2,15),(2,19),(3,17),(3,18),(4,2),(4,17),(4,18),(6,10),(7,8),(8,9),(9,5),(10,5),(11,6),(12,8),(12,13),(13,9),(13,10),(14,12),(14,16),(15,7),(15,12),(16,6),(16,13),(17,14),(17,15),(18,14),(18,19),(19,7),(19,11),(19,16)],20)
=> ?
=> ? = 2 - 1
0100010 => 0001100 => ([(0,4),(0,5),(1,12),(2,19),(2,21),(3,1),(3,13),(3,20),(4,2),(4,16),(4,17),(5,3),(5,16),(5,17),(7,11),(8,9),(9,10),(10,6),(11,6),(12,7),(13,8),(14,10),(14,11),(15,9),(15,14),(16,19),(16,20),(17,13),(17,21),(18,7),(18,14),(19,15),(19,18),(20,12),(20,18),(21,8),(21,15)],22)
=> ?
=> ? = 2 - 1
0100011 => 0010011 => ([(0,3),(0,4),(1,12),(2,14),(2,19),(3,2),(3,18),(3,20),(4,1),(4,18),(4,20),(6,8),(7,10),(8,11),(9,7),(10,5),(11,5),(12,6),(13,9),(13,16),(14,15),(14,16),(15,8),(15,17),(16,7),(16,17),(17,10),(17,11),(18,13),(18,14),(19,6),(19,9),(19,15),(20,12),(20,13),(20,19)],21)
=> ?
=> ? = 2 - 1
0100100 => 1000100 => ([(0,3),(0,4),(1,2),(1,18),(1,19),(2,7),(2,14),(3,15),(3,16),(4,1),(4,15),(4,16),(6,8),(7,9),(8,10),(9,11),(10,5),(11,5),(12,10),(12,11),(13,8),(13,12),(14,9),(14,12),(15,17),(15,19),(16,17),(16,18),(17,6),(17,13),(18,13),(18,14),(19,6),(19,7)],20)
=> ?
=> ? = 2 - 1
0100101 => 0011001 => ([(0,3),(0,4),(1,18),(1,20),(2,17),(2,19),(3,1),(3,15),(3,16),(4,2),(4,15),(4,16),(6,8),(7,9),(8,10),(9,11),(10,5),(11,5),(12,10),(12,11),(13,8),(13,12),(14,9),(14,12),(15,19),(15,20),(16,17),(16,18),(17,13),(17,14),(18,6),(18,13),(19,7),(19,14),(20,6),(20,7)],21)
=> ?
=> ? = 2 - 1
0100110 => 0011100 => ([(0,4),(0,5),(1,13),(1,14),(2,3),(2,20),(2,21),(3,15),(3,16),(4,1),(4,17),(4,18),(5,2),(5,17),(5,18),(7,10),(8,9),(9,11),(10,12),(11,6),(12,6),(13,8),(14,7),(15,9),(15,19),(16,10),(16,19),(17,13),(17,20),(18,14),(18,21),(19,11),(19,12),(20,8),(20,15),(21,7),(21,16)],22)
=> ?
=> ? = 2 - 1
0100111 => 0100111 => ([(0,4),(0,5),(1,13),(2,1),(2,15),(3,14),(3,16),(4,2),(4,20),(4,21),(5,3),(5,20),(5,21),(7,8),(8,9),(9,11),(10,12),(11,6),(12,6),(13,10),(14,8),(14,18),(15,13),(15,17),(16,17),(16,18),(17,10),(17,19),(18,9),(18,19),(19,11),(19,12),(20,7),(20,14),(21,7),(21,15),(21,16)],22)
=> ?
=> ? = 2 - 1
0101000 => 0100100 => ([(0,2),(0,3),(1,11),(1,12),(2,16),(2,17),(3,1),(3,16),(3,17),(5,7),(6,8),(7,4),(8,4),(9,7),(9,8),(10,13),(10,15),(11,14),(11,15),(12,13),(12,14),(13,5),(13,9),(14,6),(14,9),(15,5),(15,6),(16,10),(16,12),(17,10),(17,11)],18)
=> ?
=> ? = 2 - 1
0101001 => 1001001 => ([(0,2),(0,3),(1,11),(1,12),(2,16),(2,17),(3,1),(3,16),(3,17),(5,8),(6,7),(7,4),(8,4),(9,7),(9,8),(10,13),(10,14),(11,13),(11,15),(12,14),(12,15),(13,6),(13,9),(14,5),(14,9),(15,5),(15,6),(16,10),(16,11),(17,10),(17,12)],18)
=> ?
=> ? = 2 - 1
0101010 => 1001100 => ([(0,3),(0,4),(1,18),(1,20),(2,17),(2,19),(3,1),(3,15),(3,16),(4,2),(4,15),(4,16),(6,8),(7,9),(8,10),(9,11),(10,5),(11,5),(12,10),(12,11),(13,8),(13,12),(14,9),(14,12),(15,19),(15,20),(16,17),(16,18),(17,13),(17,14),(18,6),(18,13),(19,7),(19,14),(20,6),(20,7)],21)
=> ?
=> ? = 2 - 1
1000000 => 0000010 => ([(0,5),(0,6),(1,4),(1,14),(2,13),(3,2),(3,16),(4,3),(4,17),(5,12),(5,15),(6,1),(6,12),(6,15),(8,11),(9,10),(10,8),(11,7),(12,9),(13,7),(14,10),(14,17),(15,9),(15,14),(16,11),(16,13),(17,8),(17,16)],18)
=> ?
=> ? = 2 - 1
Description
The projective dimension of the simple modules corresponding to the minimum of L in the incidence algebra of the lattice L.
Matching statistic: St000271
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Values
0 => ([(0,1)],2)
=> ([(0,1)],2)
=> 1 = 0 + 1
1 => ([(0,1)],2)
=> ([(0,1)],2)
=> 1 = 0 + 1
01 => ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(0,2),(0,3),(1,2),(1,3)],4)
=> 2 = 1 + 1
10 => ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(0,2),(0,3),(1,2),(1,3)],4)
=> 2 = 1 + 1
001 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ([(0,3),(0,5),(1,2),(1,5),(2,4),(3,4),(4,5)],6)
=> 3 = 2 + 1
010 => ([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)
=> ([(0,4),(0,5),(1,2),(1,3),(2,4),(2,5),(3,4),(3,5)],6)
=> 3 = 2 + 1
100 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ([(0,3),(0,5),(1,2),(1,5),(2,4),(3,4),(4,5)],6)
=> 3 = 2 + 1
0001 => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ([(0,3),(0,7),(1,2),(1,4),(2,5),(3,6),(4,5),(4,6),(5,7),(6,7)],8)
=> 3 = 2 + 1
0010 => ([(0,2),(0,3),(1,6),(2,7),(2,8),(3,1),(3,7),(3,8),(5,4),(6,4),(7,5),(8,5),(8,6)],9)
=> ([(0,5),(0,8),(1,4),(1,6),(2,6),(2,8),(3,4),(3,5),(3,8),(4,7),(5,7),(6,7),(7,8)],9)
=> ? = 2 + 1
0100 => ([(0,2),(0,3),(1,6),(2,7),(2,8),(3,1),(3,7),(3,8),(5,4),(6,4),(7,5),(8,5),(8,6)],9)
=> ([(0,5),(0,8),(1,4),(1,6),(2,6),(2,8),(3,4),(3,5),(3,8),(4,7),(5,7),(6,7),(7,8)],9)
=> ? = 2 + 1
0101 => ([(0,1),(0,2),(1,6),(1,7),(2,6),(2,7),(4,3),(5,3),(6,4),(6,5),(7,4),(7,5)],8)
=> ([(0,4),(0,5),(1,2),(1,3),(2,6),(2,7),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7)],8)
=> ? = 1 + 1
1000 => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ([(0,3),(0,7),(1,2),(1,4),(2,5),(3,6),(4,5),(4,6),(5,7),(6,7)],8)
=> 3 = 2 + 1
00001 => ([(0,2),(0,5),(1,7),(2,6),(3,4),(3,9),(4,1),(4,8),(5,3),(5,6),(6,9),(8,7),(9,8)],10)
=> ([(0,3),(0,7),(1,2),(1,6),(2,8),(3,9),(4,5),(4,8),(4,9),(5,6),(5,7),(6,8),(7,9)],10)
=> ? = 2 + 1
00010 => ([(0,3),(0,4),(1,2),(1,11),(2,8),(3,9),(3,10),(4,1),(4,9),(4,10),(6,7),(7,5),(8,5),(9,6),(10,6),(10,11),(11,7),(11,8)],12)
=> ([(0,7),(0,11),(1,5),(1,8),(2,6),(2,8),(3,4),(3,7),(3,11),(4,5),(4,9),(5,10),(6,10),(6,11),(7,9),(8,10),(9,10),(9,11)],12)
=> ? = 2 + 1
00100 => ([(0,2),(0,3),(1,5),(1,6),(2,10),(2,11),(3,1),(3,10),(3,11),(5,8),(6,7),(7,4),(8,4),(9,7),(9,8),(10,6),(10,9),(11,5),(11,9)],12)
=> ([(0,7),(0,11),(1,5),(1,6),(2,3),(2,4),(2,11),(3,6),(3,9),(4,5),(4,8),(5,10),(6,10),(7,8),(7,9),(8,10),(8,11),(9,10),(9,11)],12)
=> ? = 2 + 1
00101 => ([(0,2),(0,3),(1,8),(2,10),(2,11),(3,1),(3,10),(3,11),(5,6),(6,4),(7,4),(8,7),(9,6),(9,7),(10,5),(10,9),(11,5),(11,8),(11,9)],12)
=> ([(0,5),(0,6),(1,7),(1,10),(2,4),(2,10),(3,6),(3,8),(3,11),(4,5),(4,11),(5,9),(6,9),(7,8),(7,11),(8,9),(8,10),(9,11),(10,11)],12)
=> ? = 2 + 1
01000 => ([(0,3),(0,4),(1,2),(1,11),(2,8),(3,9),(3,10),(4,1),(4,9),(4,10),(6,7),(7,5),(8,5),(9,6),(10,6),(10,11),(11,7),(11,8)],12)
=> ([(0,7),(0,11),(1,5),(1,8),(2,6),(2,8),(3,4),(3,7),(3,11),(4,5),(4,9),(5,10),(6,10),(6,11),(7,9),(8,10),(9,10),(9,11)],12)
=> ? = 2 + 1
01001 => ([(0,2),(0,3),(1,5),(1,9),(2,10),(2,11),(3,1),(3,10),(3,11),(5,7),(6,8),(7,4),(8,4),(9,7),(9,8),(10,5),(10,6),(11,6),(11,9)],12)
=> ([(0,7),(0,11),(1,5),(1,6),(2,3),(2,10),(2,11),(3,5),(3,8),(4,6),(4,8),(4,9),(5,10),(6,10),(7,8),(7,9),(8,11),(9,10),(9,11)],12)
=> ? = 2 + 1
01010 => ([(0,1),(0,2),(1,8),(1,9),(2,8),(2,9),(4,3),(5,3),(6,4),(6,5),(7,4),(7,5),(8,6),(8,7),(9,6),(9,7)],10)
=> ([(0,4),(0,5),(1,2),(1,3),(2,6),(2,7),(3,6),(3,7),(4,8),(4,9),(5,8),(5,9),(6,8),(6,9),(7,8),(7,9)],10)
=> ? = 2 + 1
10000 => ([(0,2),(0,5),(1,7),(2,6),(3,4),(3,9),(4,1),(4,8),(5,3),(5,6),(6,9),(8,7),(9,8)],10)
=> ([(0,3),(0,7),(1,2),(1,6),(2,8),(3,9),(4,5),(4,8),(4,9),(5,6),(5,7),(6,8),(7,9)],10)
=> ? = 2 + 1
000001 => ([(0,2),(0,6),(1,8),(2,7),(3,5),(3,9),(4,3),(4,11),(5,1),(5,10),(6,4),(6,7),(7,11),(9,10),(10,8),(11,9)],12)
=> ([(0,3),(0,11),(1,2),(1,8),(2,9),(3,10),(4,5),(4,6),(4,8),(5,7),(5,9),(6,7),(6,10),(7,11),(8,9),(10,11)],12)
=> ? = 2 + 1
000010 => ([(0,4),(0,5),(1,3),(1,12),(2,11),(3,2),(3,14),(4,10),(4,13),(5,1),(5,10),(5,13),(7,8),(8,9),(9,6),(10,7),(11,6),(12,8),(12,14),(13,7),(13,12),(14,9),(14,11)],15)
=> ([(0,9),(0,14),(1,7),(1,10),(2,8),(2,10),(3,4),(3,9),(3,14),(4,5),(4,11),(5,7),(5,13),(6,8),(6,13),(6,14),(7,12),(8,12),(9,11),(10,12),(11,13),(11,14),(12,13)],15)
=> ? = 2 + 1
000100 => ([(0,3),(0,4),(1,10),(2,1),(2,6),(2,12),(3,14),(3,15),(4,2),(4,14),(4,15),(6,7),(7,8),(8,5),(9,5),(10,9),(11,7),(11,13),(12,10),(12,13),(13,8),(13,9),(14,6),(14,11),(15,11),(15,12)],16)
=> ([(0,7),(0,15),(1,5),(1,14),(2,6),(2,8),(3,4),(3,13),(3,15),(4,6),(4,10),(5,12),(5,13),(6,11),(7,8),(7,9),(8,11),(9,11),(9,12),(9,15),(10,11),(10,12),(10,13),(12,14),(13,14),(14,15)],16)
=> ? = 2 + 1
000101 => ([(0,3),(0,4),(1,2),(1,14),(2,6),(3,13),(3,15),(4,1),(4,13),(4,15),(6,9),(7,8),(8,10),(9,5),(10,5),(11,8),(11,12),(12,9),(12,10),(13,7),(13,11),(14,6),(14,12),(15,7),(15,11),(15,14)],16)
=> ([(0,5),(0,7),(1,6),(1,8),(2,9),(2,14),(3,4),(3,8),(3,13),(4,10),(4,15),(5,6),(5,11),(6,12),(7,11),(7,14),(8,12),(9,10),(9,15),(10,13),(10,14),(11,12),(11,15),(12,13),(13,15),(14,15)],16)
=> ? = 2 + 1
001000 => ([(0,3),(0,4),(1,10),(2,1),(2,6),(2,12),(3,14),(3,15),(4,2),(4,14),(4,15),(6,7),(7,8),(8,5),(9,5),(10,9),(11,7),(11,13),(12,10),(12,13),(13,8),(13,9),(14,6),(14,11),(15,11),(15,12)],16)
=> ([(0,7),(0,15),(1,5),(1,14),(2,6),(2,8),(3,4),(3,13),(3,15),(4,6),(4,10),(5,12),(5,13),(6,11),(7,8),(7,9),(8,11),(9,11),(9,12),(9,15),(10,11),(10,12),(10,13),(12,14),(13,14),(14,15)],16)
=> ? = 2 + 1
001001 => ([(0,2),(0,3),(1,11),(1,12),(2,13),(2,14),(3,1),(3,13),(3,14),(5,7),(6,8),(7,4),(8,4),(9,7),(9,8),(10,5),(10,9),(11,6),(11,9),(12,5),(12,6),(13,10),(13,11),(14,10),(14,12)],15)
=> ([(0,5),(0,14),(1,6),(1,7),(2,12),(2,13),(2,14),(3,6),(3,12),(3,13),(4,7),(4,11),(4,13),(5,8),(5,9),(6,10),(7,10),(8,11),(8,13),(8,14),(9,11),(9,12),(9,14),(10,11),(10,12)],15)
=> ? = 2 + 1
001010 => ([(0,2),(0,3),(1,9),(2,12),(2,14),(3,1),(3,12),(3,14),(5,7),(6,8),(7,4),(8,4),(9,5),(10,6),(10,11),(11,7),(11,8),(12,10),(12,13),(13,5),(13,6),(13,11),(14,9),(14,10),(14,13)],15)
=> ([(0,6),(0,7),(1,8),(1,12),(2,5),(2,12),(3,5),(3,7),(3,14),(4,6),(4,11),(4,14),(5,13),(6,9),(7,9),(8,10),(8,13),(9,11),(9,14),(10,11),(10,12),(10,14),(11,13),(12,13),(13,14)],15)
=> ? = 2 + 1
001101 => ([(0,3),(0,4),(1,11),(2,12),(2,13),(3,2),(3,15),(3,16),(4,1),(4,15),(4,16),(6,7),(7,9),(8,10),(9,5),(10,5),(11,8),(12,7),(12,14),(13,8),(13,14),(14,9),(14,10),(15,6),(15,12),(16,6),(16,11),(16,13)],17)
=> ([(0,7),(0,15),(1,14),(1,15),(2,8),(2,9),(3,5),(3,8),(3,12),(4,7),(4,9),(4,11),(5,10),(5,16),(6,11),(6,12),(6,14),(7,16),(8,13),(9,13),(10,12),(10,14),(10,15),(11,13),(11,16),(12,13),(14,16),(15,16)],17)
=> ? = 2 + 1
010000 => ([(0,4),(0,5),(1,3),(1,12),(2,11),(3,2),(3,14),(4,10),(4,13),(5,1),(5,10),(5,13),(7,8),(8,9),(9,6),(10,7),(11,6),(12,8),(12,14),(13,7),(13,12),(14,9),(14,11)],15)
=> ([(0,9),(0,14),(1,7),(1,10),(2,8),(2,10),(3,4),(3,9),(3,14),(4,5),(4,11),(5,7),(5,13),(6,8),(6,13),(6,14),(7,12),(8,12),(9,11),(10,12),(11,13),(11,14),(12,13)],15)
=> ? = 2 + 1
010001 => ([(0,3),(0,4),(1,2),(1,11),(1,15),(2,7),(2,12),(3,13),(3,14),(4,1),(4,13),(4,14),(6,9),(7,10),(8,6),(9,5),(10,5),(11,7),(12,9),(12,10),(13,8),(13,15),(14,8),(14,11),(15,6),(15,12)],16)
=> ([(0,9),(0,15),(1,7),(1,8),(2,5),(2,6),(2,8),(3,4),(3,7),(3,14),(4,10),(4,12),(5,11),(5,13),(6,10),(6,11),(7,13),(8,13),(9,10),(9,12),(10,15),(11,14),(11,15),(12,14),(12,15),(13,14)],16)
=> ? = 2 + 1
010010 => ([(0,2),(0,3),(1,10),(1,11),(2,13),(2,14),(3,1),(3,13),(3,14),(5,8),(6,7),(7,4),(8,4),(9,7),(9,8),(10,6),(10,9),(11,5),(11,9),(12,5),(12,6),(13,10),(13,12),(14,11),(14,12)],15)
=> ([(0,5),(0,14),(1,6),(1,7),(2,12),(2,13),(2,14),(3,7),(3,11),(3,13),(4,6),(4,11),(4,12),(5,8),(5,9),(6,10),(7,10),(8,11),(8,12),(8,14),(9,11),(9,13),(9,14),(10,12),(10,13)],15)
=> ? = 2 + 1
010011 => ([(0,3),(0,4),(1,11),(2,12),(2,13),(3,2),(3,15),(3,16),(4,1),(4,15),(4,16),(6,7),(7,9),(8,10),(9,5),(10,5),(11,8),(12,7),(12,14),(13,8),(13,14),(14,9),(14,10),(15,6),(15,12),(16,6),(16,11),(16,13)],17)
=> ([(0,7),(0,15),(1,14),(1,15),(2,8),(2,9),(3,5),(3,8),(3,12),(4,7),(4,9),(4,11),(5,10),(5,16),(6,11),(6,12),(6,14),(7,16),(8,13),(9,13),(10,12),(10,14),(10,15),(11,13),(11,16),(12,13),(14,16),(15,16)],17)
=> ? = 2 + 1
010100 => ([(0,2),(0,3),(1,9),(2,12),(2,14),(3,1),(3,12),(3,14),(5,7),(6,8),(7,4),(8,4),(9,5),(10,6),(10,11),(11,7),(11,8),(12,10),(12,13),(13,5),(13,6),(13,11),(14,9),(14,10),(14,13)],15)
=> ([(0,6),(0,7),(1,8),(1,12),(2,5),(2,12),(3,5),(3,7),(3,14),(4,6),(4,11),(4,14),(5,13),(6,9),(7,9),(8,10),(8,13),(9,11),(9,14),(10,11),(10,12),(10,14),(11,13),(12,13),(13,14)],15)
=> ? = 2 + 1
010101 => ([(0,1),(0,2),(1,10),(1,11),(2,10),(2,11),(4,3),(5,3),(6,8),(6,9),(7,8),(7,9),(8,4),(8,5),(9,4),(9,5),(10,6),(10,7),(11,6),(11,7)],12)
=> ([(0,4),(0,5),(1,2),(1,3),(2,8),(2,9),(3,8),(3,9),(4,10),(4,11),(5,10),(5,11),(6,8),(6,9),(6,10),(6,11),(7,8),(7,9),(7,10),(7,11)],12)
=> ? = 1 + 1
100000 => ([(0,2),(0,6),(1,8),(2,7),(3,5),(3,9),(4,3),(4,11),(5,1),(5,10),(6,4),(6,7),(7,11),(9,10),(10,8),(11,9)],12)
=> ([(0,3),(0,11),(1,2),(1,8),(2,9),(3,10),(4,5),(4,6),(4,8),(5,7),(5,9),(6,7),(6,10),(7,11),(8,9),(10,11)],12)
=> ? = 2 + 1
0000001 => ([(0,2),(0,7),(1,9),(2,8),(3,4),(3,11),(4,6),(4,10),(5,3),(5,13),(6,1),(6,12),(7,5),(7,8),(8,13),(10,12),(11,10),(12,9),(13,11)],14)
=> ?
=> ? = 2 + 1
0000010 => ([(0,5),(0,6),(1,4),(1,14),(2,13),(3,2),(3,16),(4,3),(4,17),(5,12),(5,15),(6,1),(6,12),(6,15),(8,11),(9,10),(10,8),(11,7),(12,9),(13,7),(14,10),(14,17),(15,9),(15,14),(16,11),(16,13),(17,8),(17,16)],18)
=> ?
=> ? = 2 + 1
0000100 => ([(0,4),(0,5),(1,2),(1,17),(2,7),(3,1),(3,8),(3,16),(4,18),(4,19),(5,3),(5,18),(5,19),(7,12),(8,9),(9,10),(10,11),(11,6),(12,6),(13,10),(13,14),(14,11),(14,12),(15,9),(15,13),(16,13),(16,17),(17,7),(17,14),(18,15),(18,16),(19,8),(19,15)],20)
=> ?
=> ? = 2 + 1
0000101 => ([(0,4),(0,5),(1,3),(1,14),(2,12),(3,2),(3,18),(4,17),(4,19),(5,1),(5,17),(5,19),(7,8),(8,9),(9,10),(10,6),(11,6),(12,11),(13,8),(13,16),(14,16),(14,18),(15,10),(15,11),(16,9),(16,15),(17,7),(17,13),(18,12),(18,15),(19,7),(19,13),(19,14)],20)
=> ?
=> ? = 2 + 1
0001000 => ([(0,3),(0,4),(1,2),(1,18),(1,19),(2,6),(2,7),(3,16),(3,17),(4,1),(4,16),(4,17),(6,9),(7,8),(8,10),(9,11),(10,5),(11,5),(12,14),(12,15),(13,10),(13,11),(14,8),(14,13),(15,9),(15,13),(16,12),(16,18),(17,12),(17,19),(18,7),(18,14),(19,6),(19,15)],20)
=> ?
=> ? = 2 + 1
0001001 => ([(0,3),(0,4),(1,11),(2,1),(2,15),(2,19),(3,17),(3,18),(4,2),(4,17),(4,18),(6,10),(7,8),(8,9),(9,5),(10,5),(11,6),(12,8),(12,13),(13,9),(13,10),(14,12),(14,16),(15,7),(15,12),(16,6),(16,13),(17,14),(17,15),(18,14),(18,19),(19,7),(19,11),(19,16)],20)
=> ?
=> ? = 2 + 1
0001010 => ([(0,3),(0,4),(1,2),(1,17),(2,6),(3,16),(3,19),(4,1),(4,16),(4,19),(6,8),(7,10),(8,11),(9,7),(10,5),(11,5),(12,9),(12,14),(13,8),(13,15),(14,7),(14,15),(15,10),(15,11),(16,12),(16,18),(17,6),(17,13),(18,9),(18,13),(18,14),(19,12),(19,17),(19,18)],20)
=> ?
=> ? = 2 + 1
0001101 => ([(0,4),(0,5),(1,13),(2,1),(2,15),(3,14),(3,16),(4,2),(4,20),(4,21),(5,3),(5,20),(5,21),(7,8),(8,9),(9,11),(10,12),(11,6),(12,6),(13,10),(14,8),(14,18),(15,13),(15,17),(16,17),(16,18),(17,10),(17,19),(18,9),(18,19),(19,11),(19,12),(20,7),(20,14),(21,7),(21,15),(21,16)],22)
=> ?
=> ? = 2 + 1
0010000 => ([(0,4),(0,5),(1,2),(1,17),(2,7),(3,1),(3,8),(3,16),(4,18),(4,19),(5,3),(5,18),(5,19),(7,12),(8,9),(9,10),(10,11),(11,6),(12,6),(13,10),(13,14),(14,11),(14,12),(15,9),(15,13),(16,13),(16,17),(17,7),(17,14),(18,15),(18,16),(19,8),(19,15)],20)
=> ?
=> ? = 2 + 1
0010001 => ([(0,3),(0,4),(1,2),(1,18),(1,19),(2,7),(2,14),(3,15),(3,16),(4,1),(4,15),(4,16),(6,8),(7,9),(8,10),(9,11),(10,5),(11,5),(12,10),(12,11),(13,8),(13,12),(14,9),(14,12),(15,17),(15,19),(16,17),(16,18),(17,6),(17,13),(18,13),(18,14),(19,6),(19,7)],20)
=> ?
=> ? = 2 + 1
0010010 => ([(0,2),(0,3),(1,11),(1,12),(2,16),(2,17),(3,1),(3,16),(3,17),(5,7),(6,8),(7,4),(8,4),(9,7),(9,8),(10,13),(10,15),(11,14),(11,15),(12,13),(12,14),(13,5),(13,9),(14,6),(14,9),(15,5),(15,6),(16,10),(16,12),(17,10),(17,11)],18)
=> ?
=> ? = 2 + 1
0010100 => ([(0,2),(0,3),(1,7),(1,8),(2,17),(2,18),(3,1),(3,17),(3,18),(5,10),(6,9),(7,6),(8,5),(9,11),(10,12),(11,4),(12,4),(13,14),(13,15),(14,9),(14,16),(15,10),(15,16),(16,11),(16,12),(17,7),(17,13),(17,19),(18,8),(18,13),(18,19),(19,5),(19,6),(19,14),(19,15)],20)
=> ?
=> ? = 2 + 1
0010101 => ([(0,2),(0,3),(1,10),(2,14),(2,17),(3,1),(3,14),(3,17),(5,8),(6,5),(7,9),(8,4),(9,4),(10,6),(11,13),(11,16),(12,8),(12,9),(13,7),(13,12),(14,11),(14,15),(15,6),(15,13),(15,16),(16,5),(16,7),(16,12),(17,10),(17,11),(17,15)],18)
=> ?
=> ? = 2 + 1
0011010 => ([(0,3),(0,4),(1,13),(2,14),(2,15),(3,2),(3,20),(3,21),(4,1),(4,20),(4,21),(6,7),(7,8),(8,11),(9,12),(10,9),(11,5),(12,5),(13,10),(14,10),(14,18),(15,17),(15,18),(16,7),(16,17),(17,8),(17,19),(18,9),(18,19),(19,11),(19,12),(20,6),(20,15),(20,16),(21,6),(21,13),(21,14),(21,16)],22)
=> ?
=> ? = 2 + 1
0011101 => ([(0,4),(0,5),(1,14),(2,3),(2,19),(2,20),(3,15),(3,16),(4,1),(4,17),(4,21),(5,2),(5,17),(5,21),(7,10),(8,9),(9,12),(10,13),(11,7),(12,6),(13,6),(14,8),(15,10),(15,18),(16,9),(16,18),(17,11),(17,20),(18,12),(18,13),(19,8),(19,16),(20,7),(20,15),(21,11),(21,14),(21,19)],22)
=> ?
=> ? = 2 + 1
0100000 => ([(0,5),(0,6),(1,4),(1,14),(2,13),(3,2),(3,16),(4,3),(4,17),(5,12),(5,15),(6,1),(6,12),(6,15),(8,11),(9,10),(10,8),(11,7),(12,9),(13,7),(14,10),(14,17),(15,9),(15,14),(16,11),(16,13),(17,8),(17,16)],18)
=> ?
=> ? = 2 + 1
0100001 => ([(0,4),(0,5),(1,3),(1,9),(1,17),(2,14),(2,19),(3,2),(3,13),(3,18),(4,15),(4,16),(5,1),(5,15),(5,16),(7,11),(8,7),(9,13),(10,8),(11,6),(12,6),(13,14),(14,12),(15,9),(15,10),(16,10),(16,17),(17,8),(17,18),(18,7),(18,19),(19,11),(19,12)],20)
=> ?
=> ? = 2 + 1
0100010 => ([(0,3),(0,4),(1,2),(1,18),(1,19),(2,13),(2,14),(3,15),(3,16),(4,1),(4,15),(4,16),(6,9),(7,8),(8,10),(9,11),(10,5),(11,5),(12,10),(12,11),(13,8),(13,12),(14,9),(14,12),(15,17),(15,18),(16,17),(16,19),(17,6),(17,7),(18,7),(18,13),(19,6),(19,14)],20)
=> ?
=> ? = 2 + 1
0100011 => ([(0,4),(0,5),(1,14),(2,3),(2,19),(2,20),(3,15),(3,16),(4,1),(4,17),(4,21),(5,2),(5,17),(5,21),(7,10),(8,9),(9,12),(10,13),(11,7),(12,6),(13,6),(14,8),(15,10),(15,18),(16,9),(16,18),(17,11),(17,20),(18,12),(18,13),(19,8),(19,16),(20,7),(20,15),(21,11),(21,14),(21,19)],22)
=> ?
=> ? = 2 + 1
0100100 => ([(0,2),(0,3),(1,11),(1,12),(2,16),(2,17),(3,1),(3,16),(3,17),(5,7),(6,8),(7,4),(8,4),(9,7),(9,8),(10,13),(10,15),(11,14),(11,15),(12,13),(12,14),(13,5),(13,9),(14,6),(14,9),(15,5),(15,6),(16,10),(16,12),(17,10),(17,11)],18)
=> ?
=> ? = 2 + 1
0100101 => ([(0,2),(0,3),(1,12),(1,13),(2,18),(2,19),(3,1),(3,18),(3,19),(5,8),(6,5),(7,10),(8,11),(9,7),(10,4),(11,4),(12,9),(12,15),(13,14),(13,15),(14,8),(14,16),(15,7),(15,16),(16,10),(16,11),(17,5),(17,9),(17,14),(18,6),(18,12),(18,17),(19,6),(19,13),(19,17)],20)
=> ?
=> ? = 2 + 1
0100110 => ([(0,3),(0,4),(1,17),(1,19),(2,13),(2,18),(3,1),(3,20),(3,21),(4,2),(4,20),(4,21),(6,9),(7,10),(8,6),(9,11),(10,12),(11,5),(12,5),(13,7),(14,9),(14,16),(15,10),(15,16),(16,11),(16,12),(17,6),(17,14),(18,7),(18,15),(19,14),(19,15),(20,8),(20,18),(20,19),(21,8),(21,13),(21,17)],22)
=> ?
=> ? = 2 + 1
0100111 => ([(0,4),(0,5),(1,13),(2,1),(2,15),(3,14),(3,16),(4,2),(4,20),(4,21),(5,3),(5,20),(5,21),(7,8),(8,9),(9,11),(10,12),(11,6),(12,6),(13,10),(14,8),(14,18),(15,13),(15,17),(16,17),(16,18),(17,10),(17,19),(18,9),(18,19),(19,11),(19,12),(20,7),(20,14),(21,7),(21,15),(21,16)],22)
=> ?
=> ? = 2 + 1
0101000 => ([(0,3),(0,4),(1,2),(1,17),(2,6),(3,16),(3,19),(4,1),(4,16),(4,19),(6,8),(7,10),(8,11),(9,7),(10,5),(11,5),(12,9),(12,14),(13,8),(13,15),(14,7),(14,15),(15,10),(15,11),(16,12),(16,18),(17,6),(17,13),(18,9),(18,13),(18,14),(19,12),(19,17),(19,18)],20)
=> ?
=> ? = 2 + 1
Description
The chromatic index of a graph.
This is the minimal number of colours needed such that no two adjacent edges have the same colour.
The following 73 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000848The balance constant multiplied with the number of linear extensions of a poset. St000849The number of 1/3-balanced pairs in a poset. St001633The number of simple modules with projective dimension two in the incidence algebra of the poset. St000080The rank of the poset. St000528The height of a poset. St000906The length of the shortest maximal chain in a poset. St000643The size of the largest orbit of antichains under Panyushev complementation. St000362The size of a minimal vertex cover of a graph. St000387The matching number of a graph. St000985The number of positive eigenvalues of the adjacency matrix of the graph. St001723The differential of a graph. St001724The 2-packing differential of a graph. St001812The biclique partition number of a graph. St000171The degree of the graph. St000786The maximal number of occurrences of a colour in a proper colouring of a graph. St000910The number of maximal chains of minimal length in a poset. St001105The number of greedy linear extensions of a poset. St001106The number of supergreedy linear extensions of a poset. St001112The 3-weak dynamic number of a graph. St001286The annihilation number of a graph. St001337The upper domination number of a graph. St001338The upper irredundance number of a graph. St001340The cardinality of a minimal non-edge isolating set of a graph. St001510The number of self-evacuating linear extensions of a finite poset. St001725The harmonious chromatic number of a graph. St001829The common independence number of a graph. St000273The domination number of a graph. St000680The Grundy value for Hackendot on posets. St000717The number of ordinal summands of a poset. St000912The number of maximal antichains in a poset. St000916The packing number of a graph. St001322The size of a minimal independent dominating set in a graph. St001339The irredundance number of a graph. St001343The dimension of the reduced incidence algebra of a poset. St001636The number of indecomposable injective modules with projective dimension at most one in the incidence algebra of the poset. St001670The connected partition number of a graph. St001674The number of vertices of the largest induced star graph in the graph. St001704The size of the largest multi-subset-intersection of the deck of a graph with the deck of another graph. St001883The mutual visibility number of a graph. St001963The tree-depth of a graph. St001782The order of rowmotion on the set of order ideals of a poset. St000272The treewidth of a graph. St000310The minimal degree of a vertex of a graph. St000536The pathwidth of a graph. St000671The maximin edge-connectivity for choosing a subgraph. St001270The bandwidth of a graph. St001277The degeneracy of a graph. St001331The size of the minimal feedback vertex set. St001336The minimal number of vertices in a graph whose complement is triangle-free. St001357The maximal degree of a regular spanning subgraph of a graph. St001358The largest degree of a regular subgraph of a graph. St001792The arboricity of a graph. St001962The proper pathwidth of a graph. St000778The metric dimension of a graph. St000822The Hadwiger number of the graph. St001029The size of the core of a graph. St001108The 2-dynamic chromatic number of a graph. St001316The domatic number of a graph. St001494The Alon-Tarsi number of a graph. St001580The acyclic chromatic number of a graph. St001642The Prague dimension of a graph. St001716The 1-improper chromatic number of a graph. St000093The cardinality of a maximal independent set of vertices of a graph. St000299The number of nonisomorphic vertex-induced subtrees. St000744The length of the path to the largest entry in a standard Young tableau. St001318The number of vertices of the largest induced subforest with the same number of connected components of a graph. St001321The number of vertices of the largest induced subforest of a graph. St001655The general position number of a graph. St001656The monophonic position number of a graph. St000044The number of vertices of the unicellular map given by a perfect matching. St000642The size of the smallest orbit of antichains under Panyushev complementation. St001638The book thickness of a graph. St000741The Colin de Verdière graph invariant.
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