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Your data matches 138 different statistics following compositions of up to 3 maps.
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Matching statistic: St000259
(load all 19 compositions to match this statistic)
(load all 19 compositions to match this statistic)
Mp00178: Binary words —to composition⟶ 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%
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
1 => [1,1] => ([(0,1)],2)
=> 1
01 => [2,1] => ([(0,2),(1,2)],3)
=> 2
11 => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 1
001 => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 2
011 => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 2
101 => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 2
111 => [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 1
0001 => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 2
0011 => [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
0101 => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
0111 => [2,1,1,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
1001 => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
1011 => [1,2,1,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
1101 => [1,1,2,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
1111 => [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
00001 => [5,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> 2
00011 => [4,1,1] => ([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 2
00101 => [3,2,1] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 2
00111 => [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
01001 => [2,3,1] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 2
01011 => [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
01101 => [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
01111 => [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
10001 => [1,4,1] => ([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> 2
10011 => [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
10101 => [1,2,2,1] => ([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 2
10111 => [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
11001 => [1,1,3,1] => ([(0,5),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 2
11011 => [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
11101 => [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
11111 => [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
000001 => [6,1] => ([(0,6),(1,6),(2,6),(3,6),(4,6),(5,6)],7)
=> 2
000011 => [5,1,1] => ([(0,5),(0,6),(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 2
000101 => [4,2,1] => ([(0,6),(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 2
000111 => [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
001001 => [3,3,1] => ([(0,6),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 2
001011 => [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
001101 => [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
001111 => [3,1,1,1,1] => ([(0,3),(0,4),(0,5),(0,6),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 2
010001 => [2,4,1] => ([(0,6),(1,6),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 2
010011 => [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
010101 => [2,2,2,1] => ([(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
010111 => [2,2,1,1,1] => ([(0,4),(0,5),(0,6),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 2
011001 => [2,1,3,1] => ([(0,6),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 2
011011 => [2,1,2,1,1] => ([(0,5),(0,6),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 2
011101 => [2,1,1,2,1] => ([(0,6),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 2
011111 => [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
100001 => [1,5,1] => ([(0,6),(1,6),(2,6),(3,6),(4,5),(4,6),(5,6)],7)
=> 2
100011 => [1,4,1,1] => ([(0,5),(0,6),(1,5),(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 2
100101 => [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)
=> 2
Description
The diameter of a connected graph.
This is the greatest distance between any pair of vertices.
Matching statistic: St001093
(load all 7 compositions to match this statistic)
(load all 7 compositions to match this statistic)
Mp00224: Binary words —runsort⟶ Binary words
Mp00178: Binary words —to composition⟶ 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%
Mp00178: Binary words —to composition⟶ 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
1 => 1 => [1,1] => ([(0,1)],2)
=> 2 = 1 + 1
01 => 01 => [2,1] => ([(0,2),(1,2)],3)
=> 3 = 2 + 1
11 => 11 => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 2 = 1 + 1
001 => 001 => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 3 = 2 + 1
011 => 011 => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3 = 2 + 1
101 => 011 => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3 = 2 + 1
111 => 111 => [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 2 = 1 + 1
0001 => 0001 => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 3 = 2 + 1
0011 => 0011 => [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = 2 + 1
0101 => 0101 => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = 2 + 1
0111 => 0111 => [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
1001 => 0011 => [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = 2 + 1
1011 => 0111 => [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
1101 => 0111 => [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
1111 => 1111 => [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 = 1 + 1
00001 => 00001 => [5,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> 3 = 2 + 1
00011 => 00011 => [4,1,1] => ([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 3 = 2 + 1
00101 => 00101 => [3,2,1] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 3 = 2 + 1
00111 => 00111 => [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
01001 => 00101 => [3,2,1] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 3 = 2 + 1
01011 => 01011 => [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
01101 => 01011 => [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
01111 => 01111 => [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)
=> 3 = 2 + 1
10001 => 00011 => [4,1,1] => ([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 3 = 2 + 1
10011 => 00111 => [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
10101 => 01011 => [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
10111 => 01111 => [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)
=> 3 = 2 + 1
11001 => 00111 => [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
11011 => 01111 => [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)
=> 3 = 2 + 1
11101 => 01111 => [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)
=> 3 = 2 + 1
11111 => 11111 => [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
000001 => 000001 => [6,1] => ([(0,6),(1,6),(2,6),(3,6),(4,6),(5,6)],7)
=> 3 = 2 + 1
000011 => 000011 => [5,1,1] => ([(0,5),(0,6),(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 3 = 2 + 1
000101 => 000101 => [4,2,1] => ([(0,6),(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 3 = 2 + 1
000111 => 000111 => [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
001001 => 001001 => [3,3,1] => ([(0,6),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 3 = 2 + 1
001011 => 001011 => [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
001101 => 001101 => [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
001111 => 001111 => [3,1,1,1,1] => ([(0,3),(0,4),(0,5),(0,6),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 3 = 2 + 1
010001 => 000101 => [4,2,1] => ([(0,6),(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 3 = 2 + 1
010011 => 001101 => [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
010101 => 010101 => [2,2,2,1] => ([(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
010111 => 010111 => [2,2,1,1,1] => ([(0,4),(0,5),(0,6),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 3 = 2 + 1
011001 => 001011 => [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
011011 => 011011 => [2,1,2,1,1] => ([(0,5),(0,6),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 3 = 2 + 1
011101 => 010111 => [2,2,1,1,1] => ([(0,4),(0,5),(0,6),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 3 = 2 + 1
011111 => 011111 => [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
100001 => 000011 => [5,1,1] => ([(0,5),(0,6),(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 3 = 2 + 1
100011 => 000111 => [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
100101 => 001011 => [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
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: St001486
Mp00178: Binary words —to composition⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
Mp00324: Graphs —chromatic difference sequence⟶ Integer compositions
St001486: Integer compositions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00184: Integer compositions —to threshold graph⟶ Graphs
Mp00324: Graphs —chromatic difference sequence⟶ Integer compositions
St001486: Integer compositions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
1 => [1,1] => ([(0,1)],2)
=> [1,1] => 2 = 1 + 1
01 => [2,1] => ([(0,2),(1,2)],3)
=> [2,1] => 3 = 2 + 1
11 => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> [1,1,1] => 2 = 1 + 1
001 => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> [3,1] => 3 = 2 + 1
011 => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [2,1,1] => 3 = 2 + 1
101 => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> [2,1,1] => 3 = 2 + 1
111 => [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [1,1,1,1] => 2 = 1 + 1
0001 => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> [4,1] => 3 = 2 + 1
0011 => [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [3,1,1] => 3 = 2 + 1
0101 => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [3,1,1] => 3 = 2 + 1
0111 => [2,1,1,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [2,1,1,1] => 3 = 2 + 1
1001 => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> [3,1,1] => 3 = 2 + 1
1011 => [1,2,1,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [2,1,1,1] => 3 = 2 + 1
1101 => [1,1,2,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [2,1,1,1] => 3 = 2 + 1
1111 => [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)
=> [1,1,1,1,1] => 2 = 1 + 1
00001 => [5,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> [5,1] => 3 = 2 + 1
00011 => [4,1,1] => ([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [4,1,1] => 3 = 2 + 1
00101 => [3,2,1] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [4,1,1] => 3 = 2 + 1
00111 => [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)
=> [3,1,1,1] => 3 = 2 + 1
01001 => [2,3,1] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [4,1,1] => 3 = 2 + 1
01011 => [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)
=> [3,1,1,1] => 3 = 2 + 1
01101 => [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)
=> [3,1,1,1] => 3 = 2 + 1
01111 => [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)
=> [2,1,1,1,1] => 3 = 2 + 1
10001 => [1,4,1] => ([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> [4,1,1] => 3 = 2 + 1
10011 => [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)
=> [3,1,1,1] => 3 = 2 + 1
10101 => [1,2,2,1] => ([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [3,1,1,1] => 3 = 2 + 1
10111 => [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)
=> [2,1,1,1,1] => 3 = 2 + 1
11001 => [1,1,3,1] => ([(0,5),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [3,1,1,1] => 3 = 2 + 1
11011 => [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)
=> [2,1,1,1,1] => 3 = 2 + 1
11101 => [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)
=> [2,1,1,1,1] => 3 = 2 + 1
11111 => [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)
=> [1,1,1,1,1,1] => 2 = 1 + 1
000001 => [6,1] => ([(0,6),(1,6),(2,6),(3,6),(4,6),(5,6)],7)
=> [6,1] => 3 = 2 + 1
000011 => [5,1,1] => ([(0,5),(0,6),(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [5,1,1] => 3 = 2 + 1
000101 => [4,2,1] => ([(0,6),(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [5,1,1] => 3 = 2 + 1
000111 => [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)
=> [4,1,1,1] => 3 = 2 + 1
001001 => [3,3,1] => ([(0,6),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [5,1,1] => 3 = 2 + 1
001011 => [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)
=> [4,1,1,1] => 3 = 2 + 1
001101 => [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)
=> [4,1,1,1] => 3 = 2 + 1
001111 => [3,1,1,1,1] => ([(0,3),(0,4),(0,5),(0,6),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [3,1,1,1,1] => 3 = 2 + 1
010001 => [2,4,1] => ([(0,6),(1,6),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [5,1,1] => 3 = 2 + 1
010011 => [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)
=> [4,1,1,1] => 3 = 2 + 1
010101 => [2,2,2,1] => ([(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)
=> [4,1,1,1] => 3 = 2 + 1
010111 => [2,2,1,1,1] => ([(0,4),(0,5),(0,6),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [3,1,1,1,1] => 3 = 2 + 1
011001 => [2,1,3,1] => ([(0,6),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [4,1,1,1] => 3 = 2 + 1
011011 => [2,1,2,1,1] => ([(0,5),(0,6),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [3,1,1,1,1] => 3 = 2 + 1
011101 => [2,1,1,2,1] => ([(0,6),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [3,1,1,1,1] => 3 = 2 + 1
011111 => [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)
=> [2,1,1,1,1,1] => 3 = 2 + 1
100001 => [1,5,1] => ([(0,6),(1,6),(2,6),(3,6),(4,5),(4,6),(5,6)],7)
=> [5,1,1] => 3 = 2 + 1
100011 => [1,4,1,1] => ([(0,5),(0,6),(1,5),(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [4,1,1,1] => 3 = 2 + 1
100101 => [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)
=> [4,1,1,1] => 3 = 2 + 1
Description
The number of corners of the ribbon associated with an integer composition.
We associate a ribbon shape to a composition $c=(c_1,\dots,c_n)$ with $c_i$ cells in the $i$-th row from bottom to top, such that the cells in two rows overlap in precisely one cell.
This statistic records the total number of corners of the ribbon shape.
Matching statistic: St000326
(load all 5 compositions to match this statistic)
(load all 5 compositions to match this statistic)
Mp00261: Binary words —Burrows-Wheeler⟶ Binary words
Mp00136: Binary words —rotate back-to-front⟶ Binary words
St000326: Binary words ⟶ ℤResult quality: 67% ●values known / values provided: 98%●distinct values known / distinct values provided: 67%
Mp00136: Binary words —rotate back-to-front⟶ Binary words
St000326: Binary words ⟶ ℤResult quality: 67% ●values known / values provided: 98%●distinct values known / distinct values provided: 67%
Values
1 => 1 => 1 => 1
01 => 10 => 01 => 2
11 => 11 => 11 => 1
001 => 100 => 010 => 2
011 => 110 => 011 => 2
101 => 110 => 011 => 2
111 => 111 => 111 => 1
0001 => 1000 => 0100 => 2
0011 => 1010 => 0101 => 2
0101 => 1100 => 0110 => 2
0111 => 1110 => 0111 => 2
1001 => 1010 => 0101 => 2
1011 => 1110 => 0111 => 2
1101 => 1110 => 0111 => 2
1111 => 1111 => 1111 => 1
00001 => 10000 => 01000 => 2
00011 => 10010 => 01001 => 2
00101 => 11000 => 01100 => 2
00111 => 10110 => 01011 => 2
01001 => 11000 => 01100 => 2
01011 => 11100 => 01110 => 2
01101 => 11100 => 01110 => 2
01111 => 11110 => 01111 => 2
10001 => 10010 => 01001 => 2
10011 => 10110 => 01011 => 2
10101 => 11100 => 01110 => 2
10111 => 11110 => 01111 => 2
11001 => 10110 => 01011 => 2
11011 => 11110 => 01111 => 2
11101 => 11110 => 01111 => 2
11111 => 11111 => 11111 => 1
000001 => 100000 => 010000 => 2
000011 => 100010 => 010001 => 2
000101 => 101000 => 010100 => 2
000111 => 100110 => 010011 => 2
001001 => 110000 => 011000 => 2
001011 => 101100 => 010110 => 2
001101 => 110010 => 011001 => 2
001111 => 101110 => 010111 => 2
010001 => 101000 => 010100 => 2
010011 => 110010 => 011001 => 2
010101 => 111000 => 011100 => 2
010111 => 111010 => 011101 => 2
011001 => 101100 => 010110 => 2
011011 => 111100 => 011110 => 2
011101 => 111010 => 011101 => 2
011111 => 111110 => 011111 => 2
100001 => 100010 => 010001 => 2
100011 => 100110 => 010011 => 2
100101 => 101100 => 010110 => 2
=> ? => ? => ? = 0
Description
The position of the first one in a binary word after appending a 1 at the end.
Regarding the binary word as a subset of $\{1,\dots,n,n+1\}$ that contains $n+1$, this is the minimal element of the set.
Matching statistic: St000630
(load all 4 compositions to match this statistic)
(load all 4 compositions to match this statistic)
Mp00261: Binary words —Burrows-Wheeler⟶ Binary words
Mp00224: Binary words —runsort⟶ Binary words
St000630: Binary words ⟶ ℤResult quality: 67% ●values known / values provided: 98%●distinct values known / distinct values provided: 67%
Mp00224: Binary words —runsort⟶ Binary words
St000630: Binary words ⟶ ℤResult quality: 67% ●values known / values provided: 98%●distinct values known / distinct values provided: 67%
Values
1 => 1 => 1 => 1
01 => 10 => 01 => 2
11 => 11 => 11 => 1
001 => 100 => 001 => 2
011 => 110 => 011 => 2
101 => 110 => 011 => 2
111 => 111 => 111 => 1
0001 => 1000 => 0001 => 2
0011 => 1010 => 0011 => 2
0101 => 1100 => 0011 => 2
0111 => 1110 => 0111 => 2
1001 => 1010 => 0011 => 2
1011 => 1110 => 0111 => 2
1101 => 1110 => 0111 => 2
1111 => 1111 => 1111 => 1
00001 => 10000 => 00001 => 2
00011 => 10010 => 00011 => 2
00101 => 11000 => 00011 => 2
00111 => 10110 => 00111 => 2
01001 => 11000 => 00011 => 2
01011 => 11100 => 00111 => 2
01101 => 11100 => 00111 => 2
01111 => 11110 => 01111 => 2
10001 => 10010 => 00011 => 2
10011 => 10110 => 00111 => 2
10101 => 11100 => 00111 => 2
10111 => 11110 => 01111 => 2
11001 => 10110 => 00111 => 2
11011 => 11110 => 01111 => 2
11101 => 11110 => 01111 => 2
11111 => 11111 => 11111 => 1
000001 => 100000 => 000001 => 2
000011 => 100010 => 000011 => 2
000101 => 101000 => 000011 => 2
000111 => 100110 => 000111 => 2
001001 => 110000 => 000011 => 2
001011 => 101100 => 000111 => 2
001101 => 110010 => 000111 => 2
001111 => 101110 => 001111 => 2
010001 => 101000 => 000011 => 2
010011 => 110010 => 000111 => 2
010101 => 111000 => 000111 => 2
010111 => 111010 => 001111 => 2
011001 => 101100 => 000111 => 2
011011 => 111100 => 001111 => 2
011101 => 111010 => 001111 => 2
011111 => 111110 => 011111 => 2
100001 => 100010 => 000011 => 2
100011 => 100110 => 000111 => 2
100101 => 101100 => 000111 => 2
=> ? => ? => ? = 0
Description
The length of the shortest palindromic decomposition of a binary word.
A palindromic decomposition (paldec for short) of a word $w=a_1,\dots,a_n$ is any list of factors $p_1,\dots,p_k$ such that $w=p_1\dots p_k$ and each $p_i$ is a palindrome, i.e. coincides with itself read backwards.
Matching statistic: St000758
(load all 10 compositions to match this statistic)
(load all 10 compositions to match this statistic)
Mp00234: Binary words —valleys-to-peaks⟶ Binary words
Mp00178: Binary words —to composition⟶ Integer compositions
St000758: Integer compositions ⟶ ℤResult quality: 67% ●values known / values provided: 98%●distinct values known / distinct values provided: 67%
Mp00178: Binary words —to composition⟶ Integer compositions
St000758: Integer compositions ⟶ ℤResult quality: 67% ●values known / values provided: 98%●distinct values known / distinct values provided: 67%
Values
1 => 1 => [1,1] => 1
01 => 10 => [1,2] => 2
11 => 11 => [1,1,1] => 1
001 => 010 => [2,2] => 2
011 => 101 => [1,2,1] => 2
101 => 110 => [1,1,2] => 2
111 => 111 => [1,1,1,1] => 1
0001 => 0010 => [3,2] => 2
0011 => 0101 => [2,2,1] => 2
0101 => 1010 => [1,2,2] => 2
0111 => 1011 => [1,2,1,1] => 2
1001 => 1010 => [1,2,2] => 2
1011 => 1101 => [1,1,2,1] => 2
1101 => 1110 => [1,1,1,2] => 2
1111 => 1111 => [1,1,1,1,1] => 1
00001 => 00010 => [4,2] => 2
00011 => 00101 => [3,2,1] => 2
00101 => 01010 => [2,2,2] => 2
00111 => 01011 => [2,2,1,1] => 2
01001 => 10010 => [1,3,2] => 2
01011 => 10101 => [1,2,2,1] => 2
01101 => 10110 => [1,2,1,2] => 2
01111 => 10111 => [1,2,1,1,1] => 2
10001 => 10010 => [1,3,2] => 2
10011 => 10101 => [1,2,2,1] => 2
10101 => 11010 => [1,1,2,2] => 2
10111 => 11011 => [1,1,2,1,1] => 2
11001 => 11010 => [1,1,2,2] => 2
11011 => 11101 => [1,1,1,2,1] => 2
11101 => 11110 => [1,1,1,1,2] => 2
11111 => 11111 => [1,1,1,1,1,1] => 1
000001 => 000010 => [5,2] => 2
000011 => 000101 => [4,2,1] => 2
000101 => 001010 => [3,2,2] => 2
000111 => 001011 => [3,2,1,1] => 2
001001 => 010010 => [2,3,2] => 2
001011 => 010101 => [2,2,2,1] => 2
001101 => 010110 => [2,2,1,2] => 2
001111 => 010111 => [2,2,1,1,1] => 2
010001 => 100010 => [1,4,2] => 2
010011 => 100101 => [1,3,2,1] => 2
010101 => 101010 => [1,2,2,2] => 2
010111 => 101011 => [1,2,2,1,1] => 2
011001 => 101010 => [1,2,2,2] => 2
011011 => 101101 => [1,2,1,2,1] => 2
011101 => 101110 => [1,2,1,1,2] => 2
011111 => 101111 => [1,2,1,1,1,1] => 2
100001 => 100010 => [1,4,2] => 2
100011 => 100101 => [1,3,2,1] => 2
100101 => 101010 => [1,2,2,2] => 2
=> ? => ? => ? = 0
Description
The length of the longest staircase fitting into an integer composition.
For a given composition $c_1,\dots,c_n$, this is the maximal number $\ell$ such that there are indices $i_1 < \dots < i_\ell$ with $c_{i_k} \geq k$, see [def.3.1, 1]
Matching statistic: St000983
(load all 8 compositions to match this statistic)
(load all 8 compositions to match this statistic)
Mp00261: Binary words —Burrows-Wheeler⟶ Binary words
Mp00224: Binary words —runsort⟶ Binary words
St000983: Binary words ⟶ ℤResult quality: 67% ●values known / values provided: 98%●distinct values known / distinct values provided: 67%
Mp00224: Binary words —runsort⟶ Binary words
St000983: Binary words ⟶ ℤResult quality: 67% ●values known / values provided: 98%●distinct values known / distinct values provided: 67%
Values
1 => 1 => 1 => 1
01 => 10 => 01 => 2
11 => 11 => 11 => 1
001 => 100 => 001 => 2
011 => 110 => 011 => 2
101 => 110 => 011 => 2
111 => 111 => 111 => 1
0001 => 1000 => 0001 => 2
0011 => 1010 => 0011 => 2
0101 => 1100 => 0011 => 2
0111 => 1110 => 0111 => 2
1001 => 1010 => 0011 => 2
1011 => 1110 => 0111 => 2
1101 => 1110 => 0111 => 2
1111 => 1111 => 1111 => 1
00001 => 10000 => 00001 => 2
00011 => 10010 => 00011 => 2
00101 => 11000 => 00011 => 2
00111 => 10110 => 00111 => 2
01001 => 11000 => 00011 => 2
01011 => 11100 => 00111 => 2
01101 => 11100 => 00111 => 2
01111 => 11110 => 01111 => 2
10001 => 10010 => 00011 => 2
10011 => 10110 => 00111 => 2
10101 => 11100 => 00111 => 2
10111 => 11110 => 01111 => 2
11001 => 10110 => 00111 => 2
11011 => 11110 => 01111 => 2
11101 => 11110 => 01111 => 2
11111 => 11111 => 11111 => 1
000001 => 100000 => 000001 => 2
000011 => 100010 => 000011 => 2
000101 => 101000 => 000011 => 2
000111 => 100110 => 000111 => 2
001001 => 110000 => 000011 => 2
001011 => 101100 => 000111 => 2
001101 => 110010 => 000111 => 2
001111 => 101110 => 001111 => 2
010001 => 101000 => 000011 => 2
010011 => 110010 => 000111 => 2
010101 => 111000 => 000111 => 2
010111 => 111010 => 001111 => 2
011001 => 101100 => 000111 => 2
011011 => 111100 => 001111 => 2
011101 => 111010 => 001111 => 2
011111 => 111110 => 011111 => 2
100001 => 100010 => 000011 => 2
100011 => 100110 => 000111 => 2
100101 => 101100 => 000111 => 2
=> ? => ? => ? = 0
Description
The length of the longest alternating subword.
This is the length of the longest consecutive subword of the form $010...$ or of the form $101...$.
Matching statistic: St001239
(load all 10 compositions to match this statistic)
(load all 10 compositions to match this statistic)
Mp00097: Binary words —delta morphism⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St001239: Dyck paths ⟶ ℤResult quality: 67% ●values known / values provided: 98%●distinct values known / distinct values provided: 67%
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St001239: Dyck paths ⟶ ℤResult quality: 67% ●values known / values provided: 98%●distinct values known / distinct values provided: 67%
Values
1 => [1] => [1,0]
=> 1
01 => [1,1] => [1,0,1,0]
=> 2
11 => [2] => [1,1,0,0]
=> 1
001 => [2,1] => [1,1,0,0,1,0]
=> 2
011 => [1,2] => [1,0,1,1,0,0]
=> 2
101 => [1,1,1] => [1,0,1,0,1,0]
=> 2
111 => [3] => [1,1,1,0,0,0]
=> 1
0001 => [3,1] => [1,1,1,0,0,0,1,0]
=> 2
0011 => [2,2] => [1,1,0,0,1,1,0,0]
=> 2
0101 => [1,1,1,1] => [1,0,1,0,1,0,1,0]
=> 2
0111 => [1,3] => [1,0,1,1,1,0,0,0]
=> 2
1001 => [1,2,1] => [1,0,1,1,0,0,1,0]
=> 2
1011 => [1,1,2] => [1,0,1,0,1,1,0,0]
=> 2
1101 => [2,1,1] => [1,1,0,0,1,0,1,0]
=> 2
1111 => [4] => [1,1,1,1,0,0,0,0]
=> 1
00001 => [4,1] => [1,1,1,1,0,0,0,0,1,0]
=> 2
00011 => [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> 2
00101 => [2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
=> 2
00111 => [2,3] => [1,1,0,0,1,1,1,0,0,0]
=> 2
01001 => [1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
=> 2
01011 => [1,1,1,2] => [1,0,1,0,1,0,1,1,0,0]
=> 2
01101 => [1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
=> 2
01111 => [1,4] => [1,0,1,1,1,1,0,0,0,0]
=> 2
10001 => [1,3,1] => [1,0,1,1,1,0,0,0,1,0]
=> 2
10011 => [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
=> 2
10101 => [1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
=> 2
10111 => [1,1,3] => [1,0,1,0,1,1,1,0,0,0]
=> 2
11001 => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> 2
11011 => [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
=> 2
11101 => [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> 2
11111 => [5] => [1,1,1,1,1,0,0,0,0,0]
=> 1
000001 => [5,1] => [1,1,1,1,1,0,0,0,0,0,1,0]
=> 2
000011 => [4,2] => [1,1,1,1,0,0,0,0,1,1,0,0]
=> 2
000101 => [3,1,1,1] => [1,1,1,0,0,0,1,0,1,0,1,0]
=> 2
000111 => [3,3] => [1,1,1,0,0,0,1,1,1,0,0,0]
=> 2
001001 => [2,1,2,1] => [1,1,0,0,1,0,1,1,0,0,1,0]
=> 2
001011 => [2,1,1,2] => [1,1,0,0,1,0,1,0,1,1,0,0]
=> 2
001101 => [2,2,1,1] => [1,1,0,0,1,1,0,0,1,0,1,0]
=> 2
001111 => [2,4] => [1,1,0,0,1,1,1,1,0,0,0,0]
=> 2
010001 => [1,1,3,1] => [1,0,1,0,1,1,1,0,0,0,1,0]
=> 2
010011 => [1,1,2,2] => [1,0,1,0,1,1,0,0,1,1,0,0]
=> 2
010101 => [1,1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0,1,0]
=> 2
010111 => [1,1,1,3] => [1,0,1,0,1,0,1,1,1,0,0,0]
=> 2
011001 => [1,2,2,1] => [1,0,1,1,0,0,1,1,0,0,1,0]
=> 2
011011 => [1,2,1,2] => [1,0,1,1,0,0,1,0,1,1,0,0]
=> 2
011101 => [1,3,1,1] => [1,0,1,1,1,0,0,0,1,0,1,0]
=> 2
011111 => [1,5] => [1,0,1,1,1,1,1,0,0,0,0,0]
=> 2
100001 => [1,4,1] => [1,0,1,1,1,1,0,0,0,0,1,0]
=> 2
100011 => [1,3,2] => [1,0,1,1,1,0,0,0,1,1,0,0]
=> 2
100101 => [1,2,1,1,1] => [1,0,1,1,0,0,1,0,1,0,1,0]
=> 2
=> [] => ?
=> ? = 0
Description
The largest vector space dimension of the double dual of a simple module in the corresponding Nakayama algebra.
Matching statistic: St001261
(load all 18 compositions to match this statistic)
(load all 18 compositions to match this statistic)
Mp00097: Binary words —delta morphism⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St001261: Graphs ⟶ ℤResult quality: 67% ●values known / values provided: 98%●distinct values known / distinct values provided: 67%
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St001261: Graphs ⟶ ℤResult quality: 67% ●values known / values provided: 98%●distinct values known / distinct values provided: 67%
Values
1 => [1] => ([],1)
=> 1
01 => [1,1] => ([(0,1)],2)
=> 2
11 => [2] => ([],2)
=> 1
001 => [2,1] => ([(0,2),(1,2)],3)
=> 2
011 => [1,2] => ([(1,2)],3)
=> 2
101 => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 2
111 => [3] => ([],3)
=> 1
0001 => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 2
0011 => [2,2] => ([(1,3),(2,3)],4)
=> 2
0101 => [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 2
0111 => [1,3] => ([(2,3)],4)
=> 2
1001 => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 2
1011 => [1,1,2] => ([(1,2),(1,3),(2,3)],4)
=> 2
1101 => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 2
1111 => [4] => ([],4)
=> 1
00001 => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 2
00011 => [3,2] => ([(1,4),(2,4),(3,4)],5)
=> 2
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
00111 => [2,3] => ([(2,4),(3,4)],5)
=> 2
01001 => [1,1,2,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
01011 => [1,1,1,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
01101 => [1,2,1,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
01111 => [1,4] => ([(3,4)],5)
=> 2
10001 => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
10011 => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> 2
10101 => [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
10111 => [1,1,3] => ([(2,3),(2,4),(3,4)],5)
=> 2
11001 => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
11011 => [2,1,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
11101 => [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
11111 => [5] => ([],5)
=> 1
000001 => [5,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> 2
000011 => [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6)
=> 2
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
000111 => [3,3] => ([(2,5),(3,5),(4,5)],6)
=> 2
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
001011 => [2,1,1,2] => ([(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 2
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
001111 => [2,4] => ([(3,5),(4,5)],6)
=> 2
010001 => [1,1,3,1] => ([(0,5),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 2
010011 => [1,1,2,2] => ([(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 2
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
010111 => [1,1,1,3] => ([(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 2
011001 => [1,2,2,1] => ([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 2
011011 => [1,2,1,2] => ([(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 2
011101 => [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
011111 => [1,5] => ([(4,5)],6)
=> 2
100001 => [1,4,1] => ([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> 2
100011 => [1,3,2] => ([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> 2
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)
=> 2
=> [] => ?
=> ? = 0
Description
The Castelnuovo-Mumford regularity of a graph.
Matching statistic: St000292
(load all 7 compositions to match this statistic)
(load all 7 compositions to match this statistic)
Mp00261: Binary words —Burrows-Wheeler⟶ Binary words
Mp00224: Binary words —runsort⟶ Binary words
St000292: Binary words ⟶ ℤResult quality: 67% ●values known / values provided: 98%●distinct values known / distinct values provided: 67%
Mp00224: Binary words —runsort⟶ Binary words
St000292: Binary words ⟶ ℤResult quality: 67% ●values known / values provided: 98%●distinct values known / distinct values provided: 67%
Values
1 => 1 => 1 => 0 = 1 - 1
01 => 10 => 01 => 1 = 2 - 1
11 => 11 => 11 => 0 = 1 - 1
001 => 100 => 001 => 1 = 2 - 1
011 => 110 => 011 => 1 = 2 - 1
101 => 110 => 011 => 1 = 2 - 1
111 => 111 => 111 => 0 = 1 - 1
0001 => 1000 => 0001 => 1 = 2 - 1
0011 => 1010 => 0011 => 1 = 2 - 1
0101 => 1100 => 0011 => 1 = 2 - 1
0111 => 1110 => 0111 => 1 = 2 - 1
1001 => 1010 => 0011 => 1 = 2 - 1
1011 => 1110 => 0111 => 1 = 2 - 1
1101 => 1110 => 0111 => 1 = 2 - 1
1111 => 1111 => 1111 => 0 = 1 - 1
00001 => 10000 => 00001 => 1 = 2 - 1
00011 => 10010 => 00011 => 1 = 2 - 1
00101 => 11000 => 00011 => 1 = 2 - 1
00111 => 10110 => 00111 => 1 = 2 - 1
01001 => 11000 => 00011 => 1 = 2 - 1
01011 => 11100 => 00111 => 1 = 2 - 1
01101 => 11100 => 00111 => 1 = 2 - 1
01111 => 11110 => 01111 => 1 = 2 - 1
10001 => 10010 => 00011 => 1 = 2 - 1
10011 => 10110 => 00111 => 1 = 2 - 1
10101 => 11100 => 00111 => 1 = 2 - 1
10111 => 11110 => 01111 => 1 = 2 - 1
11001 => 10110 => 00111 => 1 = 2 - 1
11011 => 11110 => 01111 => 1 = 2 - 1
11101 => 11110 => 01111 => 1 = 2 - 1
11111 => 11111 => 11111 => 0 = 1 - 1
000001 => 100000 => 000001 => 1 = 2 - 1
000011 => 100010 => 000011 => 1 = 2 - 1
000101 => 101000 => 000011 => 1 = 2 - 1
000111 => 100110 => 000111 => 1 = 2 - 1
001001 => 110000 => 000011 => 1 = 2 - 1
001011 => 101100 => 000111 => 1 = 2 - 1
001101 => 110010 => 000111 => 1 = 2 - 1
001111 => 101110 => 001111 => 1 = 2 - 1
010001 => 101000 => 000011 => 1 = 2 - 1
010011 => 110010 => 000111 => 1 = 2 - 1
010101 => 111000 => 000111 => 1 = 2 - 1
010111 => 111010 => 001111 => 1 = 2 - 1
011001 => 101100 => 000111 => 1 = 2 - 1
011011 => 111100 => 001111 => 1 = 2 - 1
011101 => 111010 => 001111 => 1 = 2 - 1
011111 => 111110 => 011111 => 1 = 2 - 1
100001 => 100010 => 000011 => 1 = 2 - 1
100011 => 100110 => 000111 => 1 = 2 - 1
100101 => 101100 => 000111 => 1 = 2 - 1
=> ? => ? => ? = 0 - 1
Description
The number of ascents of a binary word.
The following 128 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000535The rank-width of a graph. St000691The number of changes of a binary word. St001192The maximal dimension of $Ext_A^2(S,A)$ for a simple module $S$ over the corresponding Nakayama algebra $A$. St001333The cardinality of a minimal edge-isolating set of a graph. St001393The induced matching number of a graph. St001418Half of the global dimension of the stable Auslander algebra of the Nakayama algebra corresponding to the Dyck path. St000013The height of a Dyck path. St000258The burning number of a graph. St000299The number of nonisomorphic vertex-induced subtrees. St000383The last part of an integer composition. St000397The Strahler number of a rooted tree. St000451The length of the longest pattern of the form k 1 2. St000452The number of distinct eigenvalues of a graph. St000453The number of distinct Laplacian eigenvalues of a graph. St000542The number of left-to-right-minima of a permutation. St000760The length of the longest strictly decreasing subsequence of parts of an integer composition. St000764The number of strong records in an integer composition. St000777The number of distinct eigenvalues of the distance Laplacian of a connected graph. St000918The 2-limited packing number of a graph. St001315The dissociation number of a graph. 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. St001390The number of bumps occurring when Schensted-inserting the letter 1 of a permutation. St001674The number of vertices of the largest induced star graph in the graph. St000141The maximum drop size of a permutation. St000253The crossing number of a set partition. St000254The nesting number of a set partition. St000260The radius of a connected graph. St000291The number of descents of a binary word. St000297The number of leading ones in a binary word. St000389The number of runs of ones of odd length in a binary word. St000392The length of the longest run of ones in a binary word. St000480The number of lower covers of a partition in dominance order. St000659The number of rises of length at least 2 of a Dyck path. St000662The staircase size of the code of a permutation. St000730The maximal arc length of a set partition. St000742The number of big ascents of a permutation after prepending zero. St000864The number of circled entries of the shifted recording tableau of a permutation. St000875The semilength of the longest Dyck word in the Catalan factorisation of a binary word. St000985The number of positive eigenvalues of the adjacency matrix of the graph. St001011Number of simple modules of projective dimension 2 in the Nakayama algebra corresponding to the Dyck path. St001035The convexity degree of the parallelogram polyomino associated with the Dyck path. St001194The injective dimension of $A/AfA$ in the corresponding Nakayama algebra $A$ when $Af$ is the minimal faithful projective-injective left $A$-module St001212The number of simple modules in the corresponding Nakayama algebra that have non-zero second Ext-group with the regular module. St001271The competition number of a graph. St001280The number of parts of an integer partition that are at least two. St001335The cardinality of a minimal cycle-isolating set of a graph. St001340The cardinality of a minimal non-edge isolating set of a graph. St001349The number of different graphs obtained from the given graph by removing an edge. St001354The number of series nodes in the modular decomposition of a graph. St001421Half the length of a longest factor which is its own reverse-complement and begins with a one of a binary word. St001512The minimum rank of a graph. St001737The number of descents of type 2 in a permutation. St001761The maximal multiplicity of a letter in a reduced word of a permutation. St000444The length of the maximal rise of a Dyck path. St000442The maximal area to the right of an up step of a Dyck path. St000541The number of indices greater than or equal to 2 of a permutation such that all smaller indices appear to its right. St000919The number of maximal left branches of a binary tree. St001031The height of the bicoloured Motzkin path associated with the Dyck path. St001859The number of factors of the Stanley symmetric function associated with a permutation. St001198The 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 $eA$. St001206The 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 $eA$. St001704The size of the largest multi-subset-intersection of the deck of a graph with the deck of another graph. St000264The girth of a graph, which is not a tree. St000781The number of proper colouring schemes of a Ferrers diagram. St001901The largest multiplicity of an irreducible representation contained in the higher Lie character for an integer partition. St000205Number of non-integral Gelfand-Tsetlin polytopes with prescribed top row and partition weight. St000206Number of non-integral Gelfand-Tsetlin polytopes with prescribed top row and integer composition weight. St000980The number of boxes weakly below the path and above the diagonal that lie below at least two peaks. St001498The normalised height of a Nakayama algebra with magnitude 1. St000455The second largest eigenvalue of a graph if it is integral. St000485The length of the longest cycle of a permutation. St001195The global dimension of the algebra $A/AfA$ of the corresponding Nakayama algebra $A$ with minimal left faithful projective-injective module $Af$. St001570The minimal number of edges to add to make a graph Hamiltonian. St000846The maximal number of elements covering an element of a poset. St000862The number of parts of the shifted shape of a permutation. St001092The number of distinct even parts of a partition. St001734The lettericity of a graph. St001174The Gorenstein dimension of the algebra $A/I$ when $I$ is the tilting module corresponding to the permutation in the Auslander algebra of $K[x]/(x^n)$. St001431Half of the Loewy length minus one of a modified stable Auslander algebra of the Nakayama algebra corresponding to the Dyck path. St001488The number of corners of a skew partition. St001804The minimal height of the rectangular inner shape in a cylindrical tableau associated to a tableau. St000093The cardinality of a maximal independent set of vertices of a graph. St000097The order of the largest clique of the graph. St000208Number of integral Gelfand-Tsetlin polytopes with prescribed top row and integer partition weight. St000098The chromatic number of a graph. St001491The number of indecomposable projective-injective modules in the algebra corresponding to a subset. St000298The order dimension or Dushnik-Miller dimension of a poset. St000845The maximal number of elements covered by an element in a poset. St001632The number of indecomposable injective modules $I$ with $dim Ext^1(I,A)=1$ for the incidence algebra A of a poset. St000307The number of rowmotion orbits of a poset. St000544The cop number of a graph. St000785The number of distinct colouring schemes of a graph. St001029The size of the core of a graph. St001277The degeneracy of a graph. St001358The largest degree of a regular subgraph of a graph. St001494The Alon-Tarsi number of a graph. St001580The acyclic chromatic number of a graph. St001792The arboricity of a graph. St001883The mutual visibility number of a graph. St001951The number of factors in the disjoint direct product decomposition of the automorphism group of a graph. St000272The treewidth of a graph. St000536The pathwidth of a graph. St000537The cutwidth of a graph. St000632The jump number of the poset. St001270The bandwidth of a graph. St001592The maximal number of simple paths between any two different vertices of a graph. St001644The dimension of a graph. St001743The discrepancy of a graph. St001746The coalition number of a graph. St001826The maximal number of leaves on a vertex of a graph. St001962The proper pathwidth of a graph. St000771The largest multiplicity of a distance Laplacian eigenvalue in a connected graph. St000786The maximal number of occurrences of a colour in a proper colouring of a graph. St001057The Grundy value of the game of creating an independent set in a graph. St000640The rank of the largest boolean interval in a poset. St000261The edge connectivity of a graph. St000262The vertex connectivity of a graph. St000310The minimal degree of a vertex of a graph. St000822The Hadwiger number of the graph. St001330The hat guessing number of a graph. St001642The Prague dimension of a graph. St001337The upper domination number of a graph. St001338The upper irredundance number of a graph. St001323The independence gap of a graph. St000454The largest eigenvalue of a graph if it is integral. St001624The breadth of a lattice. St001877Number of indecomposable injective modules with projective dimension 2.
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