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Your data matches 25 different statistics following compositions of up to 3 maps.
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Matching statistic: St000326
Mp00104: Binary words —reverse⟶ Binary words
Mp00269: Binary words —flag zeros to zeros⟶ Binary words
Mp00096: Binary words —Foata bijection⟶ Binary words
St000326: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00269: Binary words —flag zeros to zeros⟶ Binary words
Mp00096: Binary words —Foata bijection⟶ Binary words
St000326: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
0001 => 1000 => 0110 => 1010 => 1
0010 => 0100 => 0100 => 0100 => 2
1101 => 1011 => 1100 => 0110 => 2
1110 => 0111 => 1110 => 1110 => 1
00001 => 10000 => 01110 => 10110 => 1
00010 => 01000 => 01100 => 01010 => 2
00011 => 11000 => 01101 => 10101 => 1
00100 => 00100 => 01001 => 01001 => 2
00101 => 10100 => 01000 => 00100 => 3
00110 => 01100 => 01010 => 11000 => 1
01101 => 10110 => 00100 => 01000 => 2
01110 => 01110 => 00110 => 10010 => 1
10001 => 10001 => 10110 => 11010 => 1
10010 => 01001 => 10100 => 01100 => 2
11001 => 10011 => 11010 => 11100 => 1
11010 => 01011 => 11000 => 00110 => 3
11011 => 11011 => 11001 => 01101 => 2
11100 => 00111 => 11101 => 11101 => 1
11101 => 10111 => 11100 => 01110 => 2
11110 => 01111 => 11110 => 11110 => 1
000001 => 100000 => 011110 => 101110 => 1
000010 => 010000 => 011100 => 010110 => 2
000011 => 110000 => 011101 => 101101 => 1
000100 => 001000 => 011001 => 010101 => 2
000101 => 101000 => 011000 => 001010 => 3
000110 => 011000 => 011010 => 110100 => 1
000111 => 111000 => 011011 => 101011 => 1
001000 => 000100 => 010011 => 010011 => 2
001001 => 100100 => 010010 => 101000 => 1
001010 => 010100 => 010000 => 000100 => 4
001011 => 110100 => 010001 => 001001 => 3
001100 => 001100 => 010101 => 110001 => 1
001101 => 101100 => 010100 => 011000 => 2
001110 => 011100 => 010110 => 110010 => 1
010001 => 100010 => 000110 => 100010 => 1
010010 => 010010 => 000100 => 010000 => 2
011001 => 100110 => 001010 => 110000 => 1
011010 => 010110 => 001000 => 001000 => 3
011011 => 110110 => 001001 => 010001 => 2
011100 => 001110 => 001101 => 100101 => 1
011101 => 101110 => 001100 => 010010 => 2
011110 => 011110 => 001110 => 100110 => 1
100001 => 100001 => 101110 => 110110 => 1
100010 => 010001 => 101100 => 011010 => 2
100011 => 110001 => 101101 => 110101 => 1
100100 => 001001 => 101001 => 011001 => 2
100101 => 101001 => 101000 => 001100 => 3
100110 => 011001 => 101010 => 111000 => 1
101101 => 101101 => 100100 => 010100 => 2
101110 => 011101 => 100110 => 101010 => 1
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,…,n,n+1} that contains n+1, this is the minimal element of the set.
Matching statistic: St000382
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00097: Binary words —delta morphism⟶ Integer compositions
Mp00172: Integer compositions —rotate back to front⟶ Integer compositions
Mp00041: Integer compositions —conjugate⟶ Integer compositions
St000382: Integer compositions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00172: Integer compositions —rotate back to front⟶ Integer compositions
Mp00041: Integer compositions —conjugate⟶ Integer compositions
St000382: Integer compositions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
0001 => [3,1] => [1,3] => [1,1,2] => 1
0010 => [2,1,1] => [1,2,1] => [2,2] => 2
1101 => [2,1,1] => [1,2,1] => [2,2] => 2
1110 => [3,1] => [1,3] => [1,1,2] => 1
00001 => [4,1] => [1,4] => [1,1,1,2] => 1
00010 => [3,1,1] => [1,3,1] => [2,1,2] => 2
00011 => [3,2] => [2,3] => [1,1,2,1] => 1
00100 => [2,1,2] => [2,2,1] => [2,2,1] => 2
00101 => [2,1,1,1] => [1,2,1,1] => [3,2] => 3
00110 => [2,2,1] => [1,2,2] => [1,2,2] => 1
01101 => [1,2,1,1] => [1,1,2,1] => [2,3] => 2
01110 => [1,3,1] => [1,1,3] => [1,1,3] => 1
10001 => [1,3,1] => [1,1,3] => [1,1,3] => 1
10010 => [1,2,1,1] => [1,1,2,1] => [2,3] => 2
11001 => [2,2,1] => [1,2,2] => [1,2,2] => 1
11010 => [2,1,1,1] => [1,2,1,1] => [3,2] => 3
11011 => [2,1,2] => [2,2,1] => [2,2,1] => 2
11100 => [3,2] => [2,3] => [1,1,2,1] => 1
11101 => [3,1,1] => [1,3,1] => [2,1,2] => 2
11110 => [4,1] => [1,4] => [1,1,1,2] => 1
000001 => [5,1] => [1,5] => [1,1,1,1,2] => 1
000010 => [4,1,1] => [1,4,1] => [2,1,1,2] => 2
000011 => [4,2] => [2,4] => [1,1,1,2,1] => 1
000100 => [3,1,2] => [2,3,1] => [2,1,2,1] => 2
000101 => [3,1,1,1] => [1,3,1,1] => [3,1,2] => 3
000110 => [3,2,1] => [1,3,2] => [1,2,1,2] => 1
000111 => [3,3] => [3,3] => [1,1,2,1,1] => 1
001000 => [2,1,3] => [3,2,1] => [2,2,1,1] => 2
001001 => [2,1,2,1] => [1,2,1,2] => [1,3,2] => 1
001010 => [2,1,1,1,1] => [1,2,1,1,1] => [4,2] => 4
001011 => [2,1,1,2] => [2,2,1,1] => [3,2,1] => 3
001100 => [2,2,2] => [2,2,2] => [1,2,2,1] => 1
001101 => [2,2,1,1] => [1,2,2,1] => [2,2,2] => 2
001110 => [2,3,1] => [1,2,3] => [1,1,2,2] => 1
010001 => [1,1,3,1] => [1,1,1,3] => [1,1,4] => 1
010010 => [1,1,2,1,1] => [1,1,1,2,1] => [2,4] => 2
011001 => [1,2,2,1] => [1,1,2,2] => [1,2,3] => 1
011010 => [1,2,1,1,1] => [1,1,2,1,1] => [3,3] => 3
011011 => [1,2,1,2] => [2,1,2,1] => [2,3,1] => 2
011100 => [1,3,2] => [2,1,3] => [1,1,3,1] => 1
011101 => [1,3,1,1] => [1,1,3,1] => [2,1,3] => 2
011110 => [1,4,1] => [1,1,4] => [1,1,1,3] => 1
100001 => [1,4,1] => [1,1,4] => [1,1,1,3] => 1
100010 => [1,3,1,1] => [1,1,3,1] => [2,1,3] => 2
100011 => [1,3,2] => [2,1,3] => [1,1,3,1] => 1
100100 => [1,2,1,2] => [2,1,2,1] => [2,3,1] => 2
100101 => [1,2,1,1,1] => [1,1,2,1,1] => [3,3] => 3
100110 => [1,2,2,1] => [1,1,2,2] => [1,2,3] => 1
101101 => [1,1,2,1,1] => [1,1,1,2,1] => [2,4] => 2
101110 => [1,1,3,1] => [1,1,1,3] => [1,1,4] => 1
Description
The first part of an integer composition.
Matching statistic: St000993
Mp00097: Binary words —delta morphism⟶ Integer compositions
Mp00180: Integer compositions —to ribbon⟶ Skew partitions
Mp00183: Skew partitions —inner shape⟶ Integer partitions
St000993: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00180: Integer compositions —to ribbon⟶ Skew partitions
Mp00183: Skew partitions —inner shape⟶ Integer partitions
St000993: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
0001 => [3,1] => [[3,3],[2]]
=> [2]
=> 1
0010 => [2,1,1] => [[2,2,2],[1,1]]
=> [1,1]
=> 2
1101 => [2,1,1] => [[2,2,2],[1,1]]
=> [1,1]
=> 2
1110 => [3,1] => [[3,3],[2]]
=> [2]
=> 1
00001 => [4,1] => [[4,4],[3]]
=> [3]
=> 1
00010 => [3,1,1] => [[3,3,3],[2,2]]
=> [2,2]
=> 2
00011 => [3,2] => [[4,3],[2]]
=> [2]
=> 1
00100 => [2,1,2] => [[3,2,2],[1,1]]
=> [1,1]
=> 2
00101 => [2,1,1,1] => [[2,2,2,2],[1,1,1]]
=> [1,1,1]
=> 3
00110 => [2,2,1] => [[3,3,2],[2,1]]
=> [2,1]
=> 1
01101 => [1,2,1,1] => [[2,2,2,1],[1,1]]
=> [1,1]
=> 2
01110 => [1,3,1] => [[3,3,1],[2]]
=> [2]
=> 1
10001 => [1,3,1] => [[3,3,1],[2]]
=> [2]
=> 1
10010 => [1,2,1,1] => [[2,2,2,1],[1,1]]
=> [1,1]
=> 2
11001 => [2,2,1] => [[3,3,2],[2,1]]
=> [2,1]
=> 1
11010 => [2,1,1,1] => [[2,2,2,2],[1,1,1]]
=> [1,1,1]
=> 3
11011 => [2,1,2] => [[3,2,2],[1,1]]
=> [1,1]
=> 2
11100 => [3,2] => [[4,3],[2]]
=> [2]
=> 1
11101 => [3,1,1] => [[3,3,3],[2,2]]
=> [2,2]
=> 2
11110 => [4,1] => [[4,4],[3]]
=> [3]
=> 1
000001 => [5,1] => [[5,5],[4]]
=> [4]
=> 1
000010 => [4,1,1] => [[4,4,4],[3,3]]
=> [3,3]
=> 2
000011 => [4,2] => [[5,4],[3]]
=> [3]
=> 1
000100 => [3,1,2] => [[4,3,3],[2,2]]
=> [2,2]
=> 2
000101 => [3,1,1,1] => [[3,3,3,3],[2,2,2]]
=> [2,2,2]
=> 3
000110 => [3,2,1] => [[4,4,3],[3,2]]
=> [3,2]
=> 1
000111 => [3,3] => [[5,3],[2]]
=> [2]
=> 1
001000 => [2,1,3] => [[4,2,2],[1,1]]
=> [1,1]
=> 2
001001 => [2,1,2,1] => [[3,3,2,2],[2,1,1]]
=> [2,1,1]
=> 1
001010 => [2,1,1,1,1] => [[2,2,2,2,2],[1,1,1,1]]
=> [1,1,1,1]
=> 4
001011 => [2,1,1,2] => [[3,2,2,2],[1,1,1]]
=> [1,1,1]
=> 3
001100 => [2,2,2] => [[4,3,2],[2,1]]
=> [2,1]
=> 1
001101 => [2,2,1,1] => [[3,3,3,2],[2,2,1]]
=> [2,2,1]
=> 2
001110 => [2,3,1] => [[4,4,2],[3,1]]
=> [3,1]
=> 1
010001 => [1,1,3,1] => [[3,3,1,1],[2]]
=> [2]
=> 1
010010 => [1,1,2,1,1] => [[2,2,2,1,1],[1,1]]
=> [1,1]
=> 2
011001 => [1,2,2,1] => [[3,3,2,1],[2,1]]
=> [2,1]
=> 1
011010 => [1,2,1,1,1] => [[2,2,2,2,1],[1,1,1]]
=> [1,1,1]
=> 3
011011 => [1,2,1,2] => [[3,2,2,1],[1,1]]
=> [1,1]
=> 2
011100 => [1,3,2] => [[4,3,1],[2]]
=> [2]
=> 1
011101 => [1,3,1,1] => [[3,3,3,1],[2,2]]
=> [2,2]
=> 2
011110 => [1,4,1] => [[4,4,1],[3]]
=> [3]
=> 1
100001 => [1,4,1] => [[4,4,1],[3]]
=> [3]
=> 1
100010 => [1,3,1,1] => [[3,3,3,1],[2,2]]
=> [2,2]
=> 2
100011 => [1,3,2] => [[4,3,1],[2]]
=> [2]
=> 1
100100 => [1,2,1,2] => [[3,2,2,1],[1,1]]
=> [1,1]
=> 2
100101 => [1,2,1,1,1] => [[2,2,2,2,1],[1,1,1]]
=> [1,1,1]
=> 3
100110 => [1,2,2,1] => [[3,3,2,1],[2,1]]
=> [2,1]
=> 1
101101 => [1,1,2,1,1] => [[2,2,2,1,1],[1,1]]
=> [1,1]
=> 2
101110 => [1,1,3,1] => [[3,3,1,1],[2]]
=> [2]
=> 1
Description
The multiplicity of the largest part of an integer partition.
Matching statistic: St000678
Mp00097: Binary words —delta morphism⟶ Integer compositions
Mp00172: Integer compositions —rotate back to front⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St000678: Dyck paths ⟶ ℤResult quality: 75% ●values known / values provided: 90%●distinct values known / distinct values provided: 75%
Mp00172: Integer compositions —rotate back to front⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St000678: Dyck paths ⟶ ℤResult quality: 75% ●values known / values provided: 90%●distinct values known / distinct values provided: 75%
Values
0001 => [3,1] => [1,3] => [1,0,1,1,1,0,0,0]
=> 1
0010 => [2,1,1] => [1,2,1] => [1,0,1,1,0,0,1,0]
=> 2
1101 => [2,1,1] => [1,2,1] => [1,0,1,1,0,0,1,0]
=> 2
1110 => [3,1] => [1,3] => [1,0,1,1,1,0,0,0]
=> 1
00001 => [4,1] => [1,4] => [1,0,1,1,1,1,0,0,0,0]
=> 1
00010 => [3,1,1] => [1,3,1] => [1,0,1,1,1,0,0,0,1,0]
=> 2
00011 => [3,2] => [2,3] => [1,1,0,0,1,1,1,0,0,0]
=> 1
00100 => [2,1,2] => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> 2
00101 => [2,1,1,1] => [1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
=> 3
00110 => [2,2,1] => [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
=> 1
01101 => [1,2,1,1] => [1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
=> 2
01110 => [1,3,1] => [1,1,3] => [1,0,1,0,1,1,1,0,0,0]
=> 1
10001 => [1,3,1] => [1,1,3] => [1,0,1,0,1,1,1,0,0,0]
=> 1
10010 => [1,2,1,1] => [1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
=> 2
11001 => [2,2,1] => [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
=> 1
11010 => [2,1,1,1] => [1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
=> 3
11011 => [2,1,2] => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> 2
11100 => [3,2] => [2,3] => [1,1,0,0,1,1,1,0,0,0]
=> 1
11101 => [3,1,1] => [1,3,1] => [1,0,1,1,1,0,0,0,1,0]
=> 2
11110 => [4,1] => [1,4] => [1,0,1,1,1,1,0,0,0,0]
=> 1
000001 => [5,1] => [1,5] => [1,0,1,1,1,1,1,0,0,0,0,0]
=> 1
000010 => [4,1,1] => [1,4,1] => [1,0,1,1,1,1,0,0,0,0,1,0]
=> 2
000011 => [4,2] => [2,4] => [1,1,0,0,1,1,1,1,0,0,0,0]
=> 1
000100 => [3,1,2] => [2,3,1] => [1,1,0,0,1,1,1,0,0,0,1,0]
=> 2
000101 => [3,1,1,1] => [1,3,1,1] => [1,0,1,1,1,0,0,0,1,0,1,0]
=> 3
000110 => [3,2,1] => [1,3,2] => [1,0,1,1,1,0,0,0,1,1,0,0]
=> 1
000111 => [3,3] => [3,3] => [1,1,1,0,0,0,1,1,1,0,0,0]
=> 1
001000 => [2,1,3] => [3,2,1] => [1,1,1,0,0,0,1,1,0,0,1,0]
=> 2
001001 => [2,1,2,1] => [1,2,1,2] => [1,0,1,1,0,0,1,0,1,1,0,0]
=> 1
001010 => [2,1,1,1,1] => [1,2,1,1,1] => [1,0,1,1,0,0,1,0,1,0,1,0]
=> 4
001011 => [2,1,1,2] => [2,2,1,1] => [1,1,0,0,1,1,0,0,1,0,1,0]
=> 3
001100 => [2,2,2] => [2,2,2] => [1,1,0,0,1,1,0,0,1,1,0,0]
=> 1
001101 => [2,2,1,1] => [1,2,2,1] => [1,0,1,1,0,0,1,1,0,0,1,0]
=> 2
001110 => [2,3,1] => [1,2,3] => [1,0,1,1,0,0,1,1,1,0,0,0]
=> 1
010001 => [1,1,3,1] => [1,1,1,3] => [1,0,1,0,1,0,1,1,1,0,0,0]
=> 1
010010 => [1,1,2,1,1] => [1,1,1,2,1] => [1,0,1,0,1,0,1,1,0,0,1,0]
=> 2
011001 => [1,2,2,1] => [1,1,2,2] => [1,0,1,0,1,1,0,0,1,1,0,0]
=> 1
011010 => [1,2,1,1,1] => [1,1,2,1,1] => [1,0,1,0,1,1,0,0,1,0,1,0]
=> 3
011011 => [1,2,1,2] => [2,1,2,1] => [1,1,0,0,1,0,1,1,0,0,1,0]
=> 2
011100 => [1,3,2] => [2,1,3] => [1,1,0,0,1,0,1,1,1,0,0,0]
=> 1
011101 => [1,3,1,1] => [1,1,3,1] => [1,0,1,0,1,1,1,0,0,0,1,0]
=> 2
011110 => [1,4,1] => [1,1,4] => [1,0,1,0,1,1,1,1,0,0,0,0]
=> 1
100001 => [1,4,1] => [1,1,4] => [1,0,1,0,1,1,1,1,0,0,0,0]
=> 1
100010 => [1,3,1,1] => [1,1,3,1] => [1,0,1,0,1,1,1,0,0,0,1,0]
=> 2
100011 => [1,3,2] => [2,1,3] => [1,1,0,0,1,0,1,1,1,0,0,0]
=> 1
100100 => [1,2,1,2] => [2,1,2,1] => [1,1,0,0,1,0,1,1,0,0,1,0]
=> 2
100101 => [1,2,1,1,1] => [1,1,2,1,1] => [1,0,1,0,1,1,0,0,1,0,1,0]
=> 3
100110 => [1,2,2,1] => [1,1,2,2] => [1,0,1,0,1,1,0,0,1,1,0,0]
=> 1
101101 => [1,1,2,1,1] => [1,1,1,2,1] => [1,0,1,0,1,0,1,1,0,0,1,0]
=> 2
101110 => [1,1,3,1] => [1,1,1,3] => [1,0,1,0,1,0,1,1,1,0,0,0]
=> 1
00001000 => [4,1,3] => [3,4,1] => [1,1,1,0,0,0,1,1,1,1,0,0,0,0,1,0]
=> ? = 2
00001111 => [4,4] => [4,4] => [1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0]
=> ? = 1
00010111 => [3,1,1,3] => [3,3,1,1] => [1,1,1,0,0,0,1,1,1,0,0,0,1,0,1,0]
=> ? = 3
00011000 => [3,2,3] => [3,3,2] => [1,1,1,0,0,0,1,1,1,0,0,0,1,1,0,0]
=> ? = 1
00100111 => [2,1,2,3] => [3,2,1,2] => [1,1,1,0,0,0,1,1,0,0,1,0,1,1,0,0]
=> ? = 1
00110111 => [2,2,1,3] => [3,2,2,1] => [1,1,1,0,0,0,1,1,0,0,1,1,0,0,1,0]
=> ? = 2
00111000 => [2,3,3] => [3,2,3] => [1,1,1,0,0,0,1,1,0,0,1,1,1,0,0,0]
=> ? = 1
01000111 => [1,1,3,3] => [3,1,1,3] => [1,1,1,0,0,0,1,0,1,0,1,1,1,0,0,0]
=> ? = 1
01100111 => [1,2,2,3] => [3,1,2,2] => [1,1,1,0,0,0,1,0,1,1,0,0,1,1,0,0]
=> ? = 1
01110111 => [1,3,1,3] => [3,1,3,1] => [1,1,1,0,0,0,1,0,1,1,1,0,0,0,1,0]
=> ? = 2
01111000 => [1,4,3] => [3,1,4] => [1,1,1,0,0,0,1,0,1,1,1,1,0,0,0,0]
=> ? = 1
10000111 => [1,4,3] => [3,1,4] => [1,1,1,0,0,0,1,0,1,1,1,1,0,0,0,0]
=> ? = 1
10001000 => [1,3,1,3] => [3,1,3,1] => [1,1,1,0,0,0,1,0,1,1,1,0,0,0,1,0]
=> ? = 2
10011000 => [1,2,2,3] => [3,1,2,2] => [1,1,1,0,0,0,1,0,1,1,0,0,1,1,0,0]
=> ? = 1
10111000 => [1,1,3,3] => [3,1,1,3] => [1,1,1,0,0,0,1,0,1,0,1,1,1,0,0,0]
=> ? = 1
11000111 => [2,3,3] => [3,2,3] => [1,1,1,0,0,0,1,1,0,0,1,1,1,0,0,0]
=> ? = 1
11001000 => [2,2,1,3] => [3,2,2,1] => [1,1,1,0,0,0,1,1,0,0,1,1,0,0,1,0]
=> ? = 2
11011000 => [2,1,2,3] => [3,2,1,2] => [1,1,1,0,0,0,1,1,0,0,1,0,1,1,0,0]
=> ? = 1
11100111 => [3,2,3] => [3,3,2] => [1,1,1,0,0,0,1,1,1,0,0,0,1,1,0,0]
=> ? = 1
11101000 => [3,1,1,3] => [3,3,1,1] => [1,1,1,0,0,0,1,1,1,0,0,0,1,0,1,0]
=> ? = 3
11110000 => [4,4] => [4,4] => [1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0]
=> ? = 1
11110111 => [4,1,3] => [3,4,1] => [1,1,1,0,0,0,1,1,1,1,0,0,0,0,1,0]
=> ? = 2
000000001 => [8,1] => [1,8] => [1,0,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 1
000000010 => [7,1,1] => [1,7,1] => [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> ? = 2
000101010 => [3,1,1,1,1,1,1] => [1,3,1,1,1,1,1] => [1,0,1,1,1,0,0,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 6
001010101 => [2,1,1,1,1,1,1,1] => [1,2,1,1,1,1,1,1] => [1,0,1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 7
110101010 => [2,1,1,1,1,1,1,1] => [1,2,1,1,1,1,1,1] => [1,0,1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 7
111010101 => [3,1,1,1,1,1,1] => [1,3,1,1,1,1,1] => [1,0,1,1,1,0,0,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 6
111111101 => [7,1,1] => [1,7,1] => [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> ? = 2
111111110 => [8,1] => [1,8] => [1,0,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 1
1111111110 => [9,1] => [1,9] => [1,0,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0]
=> ? = 1
1101010101 => [2,1,1,1,1,1,1,1,1] => [1,2,1,1,1,1,1,1,1] => [1,0,1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 8
0000000001 => [9,1] => [1,9] => [1,0,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0]
=> ? = 1
0010101010 => [2,1,1,1,1,1,1,1,1] => [1,2,1,1,1,1,1,1,1] => [1,0,1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 8
Description
The number of up steps after the last double rise of a Dyck path.
Matching statistic: St000383
Mp00097: Binary words —delta morphism⟶ Integer compositions
Mp00172: Integer compositions —rotate back to front⟶ Integer compositions
Mp00039: Integer compositions —complement⟶ Integer compositions
St000383: Integer compositions ⟶ ℤResult quality: 62% ●values known / values provided: 85%●distinct values known / distinct values provided: 62%
Mp00172: Integer compositions —rotate back to front⟶ Integer compositions
Mp00039: Integer compositions —complement⟶ Integer compositions
St000383: Integer compositions ⟶ ℤResult quality: 62% ●values known / values provided: 85%●distinct values known / distinct values provided: 62%
Values
0001 => [3,1] => [1,3] => [2,1,1] => 1
0010 => [2,1,1] => [1,2,1] => [2,2] => 2
1101 => [2,1,1] => [1,2,1] => [2,2] => 2
1110 => [3,1] => [1,3] => [2,1,1] => 1
00001 => [4,1] => [1,4] => [2,1,1,1] => 1
00010 => [3,1,1] => [1,3,1] => [2,1,2] => 2
00011 => [3,2] => [2,3] => [1,2,1,1] => 1
00100 => [2,1,2] => [2,2,1] => [1,2,2] => 2
00101 => [2,1,1,1] => [1,2,1,1] => [2,3] => 3
00110 => [2,2,1] => [1,2,2] => [2,2,1] => 1
01101 => [1,2,1,1] => [1,1,2,1] => [3,2] => 2
01110 => [1,3,1] => [1,1,3] => [3,1,1] => 1
10001 => [1,3,1] => [1,1,3] => [3,1,1] => 1
10010 => [1,2,1,1] => [1,1,2,1] => [3,2] => 2
11001 => [2,2,1] => [1,2,2] => [2,2,1] => 1
11010 => [2,1,1,1] => [1,2,1,1] => [2,3] => 3
11011 => [2,1,2] => [2,2,1] => [1,2,2] => 2
11100 => [3,2] => [2,3] => [1,2,1,1] => 1
11101 => [3,1,1] => [1,3,1] => [2,1,2] => 2
11110 => [4,1] => [1,4] => [2,1,1,1] => 1
000001 => [5,1] => [1,5] => [2,1,1,1,1] => 1
000010 => [4,1,1] => [1,4,1] => [2,1,1,2] => 2
000011 => [4,2] => [2,4] => [1,2,1,1,1] => 1
000100 => [3,1,2] => [2,3,1] => [1,2,1,2] => 2
000101 => [3,1,1,1] => [1,3,1,1] => [2,1,3] => 3
000110 => [3,2,1] => [1,3,2] => [2,1,2,1] => 1
000111 => [3,3] => [3,3] => [1,1,2,1,1] => 1
001000 => [2,1,3] => [3,2,1] => [1,1,2,2] => 2
001001 => [2,1,2,1] => [1,2,1,2] => [2,3,1] => 1
001010 => [2,1,1,1,1] => [1,2,1,1,1] => [2,4] => 4
001011 => [2,1,1,2] => [2,2,1,1] => [1,2,3] => 3
001100 => [2,2,2] => [2,2,2] => [1,2,2,1] => 1
001101 => [2,2,1,1] => [1,2,2,1] => [2,2,2] => 2
001110 => [2,3,1] => [1,2,3] => [2,2,1,1] => 1
010001 => [1,1,3,1] => [1,1,1,3] => [4,1,1] => 1
010010 => [1,1,2,1,1] => [1,1,1,2,1] => [4,2] => 2
011001 => [1,2,2,1] => [1,1,2,2] => [3,2,1] => 1
011010 => [1,2,1,1,1] => [1,1,2,1,1] => [3,3] => 3
011011 => [1,2,1,2] => [2,1,2,1] => [1,3,2] => 2
011100 => [1,3,2] => [2,1,3] => [1,3,1,1] => 1
011101 => [1,3,1,1] => [1,1,3,1] => [3,1,2] => 2
011110 => [1,4,1] => [1,1,4] => [3,1,1,1] => 1
100001 => [1,4,1] => [1,1,4] => [3,1,1,1] => 1
100010 => [1,3,1,1] => [1,1,3,1] => [3,1,2] => 2
100011 => [1,3,2] => [2,1,3] => [1,3,1,1] => 1
100100 => [1,2,1,2] => [2,1,2,1] => [1,3,2] => 2
100101 => [1,2,1,1,1] => [1,1,2,1,1] => [3,3] => 3
100110 => [1,2,2,1] => [1,1,2,2] => [3,2,1] => 1
101101 => [1,1,2,1,1] => [1,1,1,2,1] => [4,2] => 2
101110 => [1,1,3,1] => [1,1,1,3] => [4,1,1] => 1
00000001 => [7,1] => [1,7] => [2,1,1,1,1,1,1] => ? = 1
00000101 => [5,1,1,1] => [1,5,1,1] => [2,1,1,1,3] => ? = 3
00001010 => [4,1,1,1,1] => [1,4,1,1,1] => [2,1,1,4] => ? = 4
00001011 => [4,1,1,2] => [2,4,1,1] => [1,2,1,1,3] => ? = 3
00010101 => [3,1,1,1,1,1] => [1,3,1,1,1,1] => [2,1,5] => ? = 5
00010111 => [3,1,1,3] => [3,3,1,1] => [1,1,2,1,3] => ? = 3
00101001 => [2,1,1,1,2,1] => [1,2,1,1,1,2] => [2,5,1] => ? = 1
00101010 => [2,1,1,1,1,1,1] => [1,2,1,1,1,1,1] => [2,6] => ? = 6
00101011 => [2,1,1,1,1,2] => [2,2,1,1,1,1] => [1,2,5] => ? = 5
00101110 => [2,1,1,3,1] => [1,2,1,1,3] => [2,4,1,1] => ? = 1
00110101 => [2,2,1,1,1,1] => [1,2,2,1,1,1] => [2,2,4] => ? = 4
01000111 => [1,1,3,3] => [3,1,1,3] => [1,1,4,1,1] => ? = 1
01001101 => [1,1,2,2,1,1] => [1,1,1,2,2,1] => [4,2,2] => ? = 2
01001110 => [1,1,2,3,1] => [1,1,1,2,3] => [4,2,1,1] => ? = 1
01010010 => [1,1,1,1,2,1,1] => [1,1,1,1,1,2,1] => [6,2] => ? = 2
01011010 => [1,1,1,2,1,1,1] => [1,1,1,1,2,1,1] => [5,3] => ? = 3
01101010 => [1,2,1,1,1,1,1] => [1,1,2,1,1,1,1] => [3,5] => ? = 5
01101011 => [1,2,1,1,1,2] => [2,1,2,1,1,1] => [1,3,4] => ? = 4
01110001 => [1,3,3,1] => [1,1,3,3] => [3,1,2,1,1] => ? = 1
01111001 => [1,4,2,1] => [1,1,4,2] => [3,1,1,2,1] => ? = 1
10000110 => [1,4,2,1] => [1,1,4,2] => [3,1,1,2,1] => ? = 1
10001110 => [1,3,3,1] => [1,1,3,3] => [3,1,2,1,1] => ? = 1
10010100 => [1,2,1,1,1,2] => [2,1,2,1,1,1] => [1,3,4] => ? = 4
10010101 => [1,2,1,1,1,1,1] => [1,1,2,1,1,1,1] => [3,5] => ? = 5
10100101 => [1,1,1,2,1,1,1] => [1,1,1,1,2,1,1] => [5,3] => ? = 3
10101101 => [1,1,1,1,2,1,1] => [1,1,1,1,1,2,1] => [6,2] => ? = 2
10110001 => [1,1,2,3,1] => [1,1,1,2,3] => [4,2,1,1] => ? = 1
10110010 => [1,1,2,2,1,1] => [1,1,1,2,2,1] => [4,2,2] => ? = 2
10111000 => [1,1,3,3] => [3,1,1,3] => [1,1,4,1,1] => ? = 1
11001010 => [2,2,1,1,1,1] => [1,2,2,1,1,1] => [2,2,4] => ? = 4
11010001 => [2,1,1,3,1] => [1,2,1,1,3] => [2,4,1,1] => ? = 1
11010100 => [2,1,1,1,1,2] => [2,2,1,1,1,1] => [1,2,5] => ? = 5
11010101 => [2,1,1,1,1,1,1] => [1,2,1,1,1,1,1] => [2,6] => ? = 6
11010110 => [2,1,1,1,2,1] => [1,2,1,1,1,2] => [2,5,1] => ? = 1
11101000 => [3,1,1,3] => [3,3,1,1] => [1,1,2,1,3] => ? = 3
11101010 => [3,1,1,1,1,1] => [1,3,1,1,1,1] => [2,1,5] => ? = 5
11110100 => [4,1,1,2] => [2,4,1,1] => [1,2,1,1,3] => ? = 3
11110101 => [4,1,1,1,1] => [1,4,1,1,1] => [2,1,1,4] => ? = 4
11111010 => [5,1,1,1] => [1,5,1,1] => [2,1,1,1,3] => ? = 3
11111110 => [7,1] => [1,7] => [2,1,1,1,1,1,1] => ? = 1
000000001 => [8,1] => [1,8] => [2,1,1,1,1,1,1,1] => ? = 1
000000010 => [7,1,1] => [1,7,1] => [2,1,1,1,1,1,2] => ? = 2
000101010 => [3,1,1,1,1,1,1] => [1,3,1,1,1,1,1] => [2,1,6] => ? = 6
001010101 => [2,1,1,1,1,1,1,1] => [1,2,1,1,1,1,1,1] => [2,7] => ? = 7
110101010 => [2,1,1,1,1,1,1,1] => [1,2,1,1,1,1,1,1] => [2,7] => ? = 7
111010101 => [3,1,1,1,1,1,1] => [1,3,1,1,1,1,1] => [2,1,6] => ? = 6
111111101 => [7,1,1] => [1,7,1] => [2,1,1,1,1,1,2] => ? = 2
111111110 => [8,1] => [1,8] => [2,1,1,1,1,1,1,1] => ? = 1
1111111110 => [9,1] => [1,9] => [2,1,1,1,1,1,1,1,1] => ? = 1
1101010101 => [2,1,1,1,1,1,1,1,1] => [1,2,1,1,1,1,1,1,1] => [2,8] => ? = 8
Description
The last part of an integer composition.
Matching statistic: St000773
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00097: Binary words —delta morphism⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
Mp00247: Graphs —de-duplicate⟶ Graphs
St000773: Graphs ⟶ ℤResult quality: 49% ●values known / values provided: 49%●distinct values known / distinct values provided: 62%
Mp00184: Integer compositions —to threshold graph⟶ Graphs
Mp00247: Graphs —de-duplicate⟶ Graphs
St000773: Graphs ⟶ ℤResult quality: 49% ●values known / values provided: 49%●distinct values known / distinct values provided: 62%
Values
0001 => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> ([(0,1)],2)
=> 1
0010 => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,2)],3)
=> 2
1101 => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,2)],3)
=> 2
1110 => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> ([(0,1)],2)
=> 1
00001 => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> ([(0,1)],2)
=> 1
00010 => [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 2
00011 => [3,2] => ([(1,4),(2,4),(3,4)],5)
=> ([(1,2)],3)
=> 1
00100 => [2,1,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(2,3)],4)
=> 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)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3
00110 => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> 1
01101 => [1,2,1,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
01110 => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> 1
10001 => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> 1
10010 => [1,2,1,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
11001 => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> 1
11010 => [2,1,1,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3
11011 => [2,1,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(2,3)],4)
=> 2
11100 => [3,2] => ([(1,4),(2,4),(3,4)],5)
=> ([(1,2)],3)
=> 1
11101 => [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 2
11110 => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> ([(0,1)],2)
=> 1
000001 => [5,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> ([(0,1)],2)
=> 1
000010 => [4,1,1] => ([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> 2
000011 => [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6)
=> ([(1,2)],3)
=> 1
000100 => [3,1,2] => ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(1,2),(1,3),(2,3)],4)
=> 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)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3
000110 => [3,2,1] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> 1
000111 => [3,3] => ([(2,5),(3,5),(4,5)],6)
=> ([(1,2)],3)
=> 1
001000 => [2,1,3] => ([(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(1,2),(1,3),(2,3)],4)
=> 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)
=> ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
001010 => [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)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 4
001011 => [2,1,1,2] => ([(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3
001100 => [2,2,2] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(1,4),(2,3),(2,4),(3,4)],5)
=> 1
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)
=> ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
001110 => [2,3,1] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> 1
010001 => [1,1,3,1] => ([(0,5),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
010010 => [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)
=> ([(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
011001 => [1,2,2,1] => ([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 1
011010 => [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)
=> ([(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
011011 => [1,2,1,2] => ([(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 2
011100 => [1,3,2] => ([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(1,4),(2,3),(2,4),(3,4)],5)
=> 1
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)
=> ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
011110 => [1,4,1] => ([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> 1
100001 => [1,4,1] => ([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> 1
100010 => [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)
=> ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
100011 => [1,3,2] => ([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(1,4),(2,3),(2,4),(3,4)],5)
=> 1
100100 => [1,2,1,2] => ([(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(1,4),(1,5),(2,3),(2,4),(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)
=> ([(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
100110 => [1,2,2,1] => ([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 1
101101 => [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)
=> ([(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
101110 => [1,1,3,1] => ([(0,5),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
00000001 => [7,1] => ([(0,7),(1,7),(2,7),(3,7),(4,7),(5,7),(6,7)],8)
=> ?
=> ? = 1
00000010 => [6,1,1] => ([(0,6),(0,7),(1,6),(1,7),(2,6),(2,7),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ?
=> ? = 2
00000101 => [5,1,1,1] => ([(0,5),(0,6),(0,7),(1,5),(1,6),(1,7),(2,5),(2,6),(2,7),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ?
=> ? = 3
00001000 => [4,1,3] => ([(2,6),(2,7),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ?
=> ? = 2
00001001 => [4,1,2,1] => ([(0,7),(1,5),(1,6),(1,7),(2,5),(2,6),(2,7),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ?
=> ? = 1
00001011 => [4,1,1,2] => ([(1,5),(1,6),(1,7),(2,5),(2,6),(2,7),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ?
=> ? = 3
00001100 => [4,2,2] => ([(1,7),(2,6),(2,7),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ?
=> ? = 1
00001101 => [4,2,1,1] => ([(0,6),(0,7),(1,5),(1,6),(1,7),(2,5),(2,6),(2,7),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ?
=> ? = 2
00001110 => [4,3,1] => ([(0,7),(1,7),(2,6),(2,7),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ?
=> ? = 1
00001111 => [4,4] => ([(3,7),(4,7),(5,7),(6,7)],8)
=> ?
=> ? = 1
00010001 => [3,1,3,1] => ([(0,7),(1,7),(2,5),(2,6),(2,7),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ?
=> ? = 1
00010010 => [3,1,2,1,1] => ([(0,6),(0,7),(1,4),(1,5),(1,6),(1,7),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ?
=> ? = 2
00010011 => [3,1,2,2] => ([(1,7),(2,5),(2,6),(2,7),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ?
=> ? = 1
00010101 => [3,1,1,1,1,1] => ([(0,3),(0,4),(0,5),(0,6),(0,7),(1,3),(1,4),(1,5),(1,6),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ?
=> ? = 5
00010110 => [3,1,1,2,1] => ([(0,7),(1,4),(1,5),(1,6),(1,7),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ?
=> ? = 1
00010111 => [3,1,1,3] => ([(2,5),(2,6),(2,7),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ?
=> ? = 3
00011000 => [3,2,3] => ([(2,7),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ?
=> ? = 1
00011001 => [3,2,2,1] => ([(0,7),(1,6),(1,7),(2,5),(2,6),(2,7),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ?
=> ? = 1
00011010 => [3,2,1,1,1] => ([(0,5),(0,6),(0,7),(1,4),(1,5),(1,6),(1,7),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ?
=> ? = 3
00011011 => [3,2,1,2] => ([(1,6),(1,7),(2,5),(2,6),(2,7),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ?
=> ? = 2
00011100 => [3,3,2] => ([(1,7),(2,7),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ?
=> ? = 1
00011101 => [3,3,1,1] => ([(0,6),(0,7),(1,6),(1,7),(2,5),(2,6),(2,7),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ?
=> ? = 2
00011110 => [3,4,1] => ([(0,7),(1,7),(2,7),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ?
=> ? = 1
00100010 => [2,1,3,1,1] => ([(0,6),(0,7),(1,6),(1,7),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ?
=> ? = 2
00100011 => [2,1,3,2] => ([(1,7),(2,7),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ?
=> ? = 1
00100100 => [2,1,2,1,2] => ([(1,6),(1,7),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ?
=> ? = 2
00100101 => [2,1,2,1,1,1] => ([(0,5),(0,6),(0,7),(1,3),(1,4),(1,5),(1,6),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ?
=> ? = 3
00100110 => [2,1,2,2,1] => ([(0,7),(1,6),(1,7),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ?
=> ? = 1
00100111 => [2,1,2,3] => ([(2,7),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ?
=> ? = 1
00101001 => [2,1,1,1,2,1] => ([(0,7),(1,3),(1,4),(1,5),(1,6),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ?
=> ? = 1
00101010 => [2,1,1,1,1,1,1] => ([(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(1,2),(1,3),(1,4),(1,5),(1,6),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ?
=> ? = 6
00101011 => [2,1,1,1,1,2] => ([(1,3),(1,4),(1,5),(1,6),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ?
=> ? = 5
00101100 => [2,1,1,2,2] => ([(1,7),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ?
=> ? = 1
00101101 => [2,1,1,2,1,1] => ([(0,6),(0,7),(1,3),(1,4),(1,5),(1,6),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ?
=> ? = 2
00101110 => [2,1,1,3,1] => ([(0,7),(1,7),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ?
=> ? = 1
00110001 => [2,2,3,1] => ([(0,7),(1,7),(2,6),(2,7),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ?
=> ? = 1
00110010 => [2,2,2,1,1] => ([(0,6),(0,7),(1,5),(1,6),(1,7),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ?
=> ? = 2
00110011 => [2,2,2,2] => ([(1,7),(2,6),(2,7),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ?
=> ? = 1
00110100 => [2,2,1,1,2] => ([(1,5),(1,6),(1,7),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ?
=> ? = 3
00110101 => [2,2,1,1,1,1] => ([(0,4),(0,5),(0,6),(0,7),(1,3),(1,4),(1,5),(1,6),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ?
=> ? = 4
00110110 => [2,2,1,2,1] => ([(0,7),(1,5),(1,6),(1,7),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ?
=> ? = 1
00110111 => [2,2,1,3] => ([(2,6),(2,7),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ?
=> ? = 2
00111000 => [2,3,3] => ([(2,7),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ?
=> ? = 1
00111001 => [2,3,2,1] => ([(0,7),(1,6),(1,7),(2,6),(2,7),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ?
=> ? = 1
00111010 => [2,3,1,1,1] => ([(0,5),(0,6),(0,7),(1,5),(1,6),(1,7),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ?
=> ? = 3
00111011 => [2,3,1,2] => ([(1,6),(1,7),(2,6),(2,7),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ?
=> ? = 2
00111100 => [2,4,2] => ([(1,7),(2,7),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ?
=> ? = 1
00111101 => [2,4,1,1] => ([(0,6),(0,7),(1,6),(1,7),(2,6),(2,7),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ?
=> ? = 2
01000100 => [1,1,3,1,2] => ([(1,6),(1,7),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ?
=> ? = 2
01000110 => [1,1,3,2,1] => ([(0,7),(1,6),(1,7),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ?
=> ? = 1
Description
The multiplicity of the largest Laplacian eigenvalue in a graph.
Matching statistic: St001135
Mp00097: Binary words —delta morphism⟶ Integer compositions
Mp00172: Integer compositions —rotate back to front⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St001135: Dyck paths ⟶ ℤResult quality: 48% ●values known / values provided: 48%●distinct values known / distinct values provided: 62%
Mp00172: Integer compositions —rotate back to front⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St001135: Dyck paths ⟶ ℤResult quality: 48% ●values known / values provided: 48%●distinct values known / distinct values provided: 62%
Values
0001 => [3,1] => [1,3] => [1,0,1,1,1,0,0,0]
=> 1
0010 => [2,1,1] => [1,2,1] => [1,0,1,1,0,0,1,0]
=> 2
1101 => [2,1,1] => [1,2,1] => [1,0,1,1,0,0,1,0]
=> 2
1110 => [3,1] => [1,3] => [1,0,1,1,1,0,0,0]
=> 1
00001 => [4,1] => [1,4] => [1,0,1,1,1,1,0,0,0,0]
=> 1
00010 => [3,1,1] => [1,3,1] => [1,0,1,1,1,0,0,0,1,0]
=> 2
00011 => [3,2] => [2,3] => [1,1,0,0,1,1,1,0,0,0]
=> 1
00100 => [2,1,2] => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> 2
00101 => [2,1,1,1] => [1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
=> 3
00110 => [2,2,1] => [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
=> 1
01101 => [1,2,1,1] => [1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
=> 2
01110 => [1,3,1] => [1,1,3] => [1,0,1,0,1,1,1,0,0,0]
=> 1
10001 => [1,3,1] => [1,1,3] => [1,0,1,0,1,1,1,0,0,0]
=> 1
10010 => [1,2,1,1] => [1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
=> 2
11001 => [2,2,1] => [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
=> 1
11010 => [2,1,1,1] => [1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
=> 3
11011 => [2,1,2] => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> 2
11100 => [3,2] => [2,3] => [1,1,0,0,1,1,1,0,0,0]
=> 1
11101 => [3,1,1] => [1,3,1] => [1,0,1,1,1,0,0,0,1,0]
=> 2
11110 => [4,1] => [1,4] => [1,0,1,1,1,1,0,0,0,0]
=> 1
000001 => [5,1] => [1,5] => [1,0,1,1,1,1,1,0,0,0,0,0]
=> 1
000010 => [4,1,1] => [1,4,1] => [1,0,1,1,1,1,0,0,0,0,1,0]
=> 2
000011 => [4,2] => [2,4] => [1,1,0,0,1,1,1,1,0,0,0,0]
=> 1
000100 => [3,1,2] => [2,3,1] => [1,1,0,0,1,1,1,0,0,0,1,0]
=> 2
000101 => [3,1,1,1] => [1,3,1,1] => [1,0,1,1,1,0,0,0,1,0,1,0]
=> 3
000110 => [3,2,1] => [1,3,2] => [1,0,1,1,1,0,0,0,1,1,0,0]
=> 1
000111 => [3,3] => [3,3] => [1,1,1,0,0,0,1,1,1,0,0,0]
=> 1
001000 => [2,1,3] => [3,2,1] => [1,1,1,0,0,0,1,1,0,0,1,0]
=> 2
001001 => [2,1,2,1] => [1,2,1,2] => [1,0,1,1,0,0,1,0,1,1,0,0]
=> 1
001010 => [2,1,1,1,1] => [1,2,1,1,1] => [1,0,1,1,0,0,1,0,1,0,1,0]
=> 4
001011 => [2,1,1,2] => [2,2,1,1] => [1,1,0,0,1,1,0,0,1,0,1,0]
=> 3
001100 => [2,2,2] => [2,2,2] => [1,1,0,0,1,1,0,0,1,1,0,0]
=> 1
001101 => [2,2,1,1] => [1,2,2,1] => [1,0,1,1,0,0,1,1,0,0,1,0]
=> 2
001110 => [2,3,1] => [1,2,3] => [1,0,1,1,0,0,1,1,1,0,0,0]
=> 1
010001 => [1,1,3,1] => [1,1,1,3] => [1,0,1,0,1,0,1,1,1,0,0,0]
=> 1
010010 => [1,1,2,1,1] => [1,1,1,2,1] => [1,0,1,0,1,0,1,1,0,0,1,0]
=> 2
011001 => [1,2,2,1] => [1,1,2,2] => [1,0,1,0,1,1,0,0,1,1,0,0]
=> 1
011010 => [1,2,1,1,1] => [1,1,2,1,1] => [1,0,1,0,1,1,0,0,1,0,1,0]
=> 3
011011 => [1,2,1,2] => [2,1,2,1] => [1,1,0,0,1,0,1,1,0,0,1,0]
=> 2
011100 => [1,3,2] => [2,1,3] => [1,1,0,0,1,0,1,1,1,0,0,0]
=> 1
011101 => [1,3,1,1] => [1,1,3,1] => [1,0,1,0,1,1,1,0,0,0,1,0]
=> 2
011110 => [1,4,1] => [1,1,4] => [1,0,1,0,1,1,1,1,0,0,0,0]
=> 1
100001 => [1,4,1] => [1,1,4] => [1,0,1,0,1,1,1,1,0,0,0,0]
=> 1
100010 => [1,3,1,1] => [1,1,3,1] => [1,0,1,0,1,1,1,0,0,0,1,0]
=> 2
100011 => [1,3,2] => [2,1,3] => [1,1,0,0,1,0,1,1,1,0,0,0]
=> 1
100100 => [1,2,1,2] => [2,1,2,1] => [1,1,0,0,1,0,1,1,0,0,1,0]
=> 2
100101 => [1,2,1,1,1] => [1,1,2,1,1] => [1,0,1,0,1,1,0,0,1,0,1,0]
=> 3
100110 => [1,2,2,1] => [1,1,2,2] => [1,0,1,0,1,1,0,0,1,1,0,0]
=> 1
101101 => [1,1,2,1,1] => [1,1,1,2,1] => [1,0,1,0,1,0,1,1,0,0,1,0]
=> 2
101110 => [1,1,3,1] => [1,1,1,3] => [1,0,1,0,1,0,1,1,1,0,0,0]
=> 1
00000001 => [7,1] => [1,7] => [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> ? = 1
00000010 => [6,1,1] => [1,6,1] => [1,0,1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> ? = 2
00000101 => [5,1,1,1] => [1,5,1,1] => [1,0,1,1,1,1,1,0,0,0,0,0,1,0,1,0]
=> ? = 3
00001000 => [4,1,3] => [3,4,1] => [1,1,1,0,0,0,1,1,1,1,0,0,0,0,1,0]
=> ? = 2
00001001 => [4,1,2,1] => [1,4,1,2] => [1,0,1,1,1,1,0,0,0,0,1,0,1,1,0,0]
=> ? = 1
00001010 => [4,1,1,1,1] => [1,4,1,1,1] => [1,0,1,1,1,1,0,0,0,0,1,0,1,0,1,0]
=> ? = 4
00001011 => [4,1,1,2] => [2,4,1,1] => [1,1,0,0,1,1,1,1,0,0,0,0,1,0,1,0]
=> ? = 3
00001100 => [4,2,2] => [2,4,2] => [1,1,0,0,1,1,1,1,0,0,0,0,1,1,0,0]
=> ? = 1
00001101 => [4,2,1,1] => [1,4,2,1] => [1,0,1,1,1,1,0,0,0,0,1,1,0,0,1,0]
=> ? = 2
00001110 => [4,3,1] => [1,4,3] => [1,0,1,1,1,1,0,0,0,0,1,1,1,0,0,0]
=> ? = 1
00001111 => [4,4] => [4,4] => [1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0]
=> ? = 1
00010001 => [3,1,3,1] => [1,3,1,3] => [1,0,1,1,1,0,0,0,1,0,1,1,1,0,0,0]
=> ? = 1
00010010 => [3,1,2,1,1] => [1,3,1,2,1] => [1,0,1,1,1,0,0,0,1,0,1,1,0,0,1,0]
=> ? = 2
00010011 => [3,1,2,2] => [2,3,1,2] => [1,1,0,0,1,1,1,0,0,0,1,0,1,1,0,0]
=> ? = 1
00010101 => [3,1,1,1,1,1] => [1,3,1,1,1,1] => [1,0,1,1,1,0,0,0,1,0,1,0,1,0,1,0]
=> ? = 5
00010110 => [3,1,1,2,1] => [1,3,1,1,2] => [1,0,1,1,1,0,0,0,1,0,1,0,1,1,0,0]
=> ? = 1
00010111 => [3,1,1,3] => [3,3,1,1] => [1,1,1,0,0,0,1,1,1,0,0,0,1,0,1,0]
=> ? = 3
00011000 => [3,2,3] => [3,3,2] => [1,1,1,0,0,0,1,1,1,0,0,0,1,1,0,0]
=> ? = 1
00011001 => [3,2,2,1] => [1,3,2,2] => [1,0,1,1,1,0,0,0,1,1,0,0,1,1,0,0]
=> ? = 1
00011010 => [3,2,1,1,1] => [1,3,2,1,1] => [1,0,1,1,1,0,0,0,1,1,0,0,1,0,1,0]
=> ? = 3
00011011 => [3,2,1,2] => [2,3,2,1] => [1,1,0,0,1,1,1,0,0,0,1,1,0,0,1,0]
=> ? = 2
00011100 => [3,3,2] => [2,3,3] => [1,1,0,0,1,1,1,0,0,0,1,1,1,0,0,0]
=> ? = 1
00011101 => [3,3,1,1] => [1,3,3,1] => [1,0,1,1,1,0,0,0,1,1,1,0,0,0,1,0]
=> ? = 2
00011110 => [3,4,1] => [1,3,4] => [1,0,1,1,1,0,0,0,1,1,1,1,0,0,0,0]
=> ? = 1
00100010 => [2,1,3,1,1] => [1,2,1,3,1] => [1,0,1,1,0,0,1,0,1,1,1,0,0,0,1,0]
=> ? = 2
00100011 => [2,1,3,2] => [2,2,1,3] => [1,1,0,0,1,1,0,0,1,0,1,1,1,0,0,0]
=> ? = 1
00100100 => [2,1,2,1,2] => [2,2,1,2,1] => [1,1,0,0,1,1,0,0,1,0,1,1,0,0,1,0]
=> ? = 2
00100101 => [2,1,2,1,1,1] => [1,2,1,2,1,1] => [1,0,1,1,0,0,1,0,1,1,0,0,1,0,1,0]
=> ? = 3
00100110 => [2,1,2,2,1] => [1,2,1,2,2] => [1,0,1,1,0,0,1,0,1,1,0,0,1,1,0,0]
=> ? = 1
00100111 => [2,1,2,3] => [3,2,1,2] => [1,1,1,0,0,0,1,1,0,0,1,0,1,1,0,0]
=> ? = 1
00101001 => [2,1,1,1,2,1] => [1,2,1,1,1,2] => [1,0,1,1,0,0,1,0,1,0,1,0,1,1,0,0]
=> ? = 1
00101010 => [2,1,1,1,1,1,1] => [1,2,1,1,1,1,1] => [1,0,1,1,0,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 6
00101011 => [2,1,1,1,1,2] => [2,2,1,1,1,1] => [1,1,0,0,1,1,0,0,1,0,1,0,1,0,1,0]
=> ? = 5
00101100 => [2,1,1,2,2] => [2,2,1,1,2] => [1,1,0,0,1,1,0,0,1,0,1,0,1,1,0,0]
=> ? = 1
00101101 => [2,1,1,2,1,1] => [1,2,1,1,2,1] => [1,0,1,1,0,0,1,0,1,0,1,1,0,0,1,0]
=> ? = 2
00101110 => [2,1,1,3,1] => [1,2,1,1,3] => [1,0,1,1,0,0,1,0,1,0,1,1,1,0,0,0]
=> ? = 1
00110001 => [2,2,3,1] => [1,2,2,3] => [1,0,1,1,0,0,1,1,0,0,1,1,1,0,0,0]
=> ? = 1
00110010 => [2,2,2,1,1] => [1,2,2,2,1] => [1,0,1,1,0,0,1,1,0,0,1,1,0,0,1,0]
=> ? = 2
00110011 => [2,2,2,2] => [2,2,2,2] => [1,1,0,0,1,1,0,0,1,1,0,0,1,1,0,0]
=> ? = 1
00110100 => [2,2,1,1,2] => [2,2,2,1,1] => [1,1,0,0,1,1,0,0,1,1,0,0,1,0,1,0]
=> ? = 3
00110101 => [2,2,1,1,1,1] => [1,2,2,1,1,1] => [1,0,1,1,0,0,1,1,0,0,1,0,1,0,1,0]
=> ? = 4
00110110 => [2,2,1,2,1] => [1,2,2,1,2] => [1,0,1,1,0,0,1,1,0,0,1,0,1,1,0,0]
=> ? = 1
00110111 => [2,2,1,3] => [3,2,2,1] => [1,1,1,0,0,0,1,1,0,0,1,1,0,0,1,0]
=> ? = 2
00111000 => [2,3,3] => [3,2,3] => [1,1,1,0,0,0,1,1,0,0,1,1,1,0,0,0]
=> ? = 1
00111001 => [2,3,2,1] => [1,2,3,2] => [1,0,1,1,0,0,1,1,1,0,0,0,1,1,0,0]
=> ? = 1
00111010 => [2,3,1,1,1] => [1,2,3,1,1] => [1,0,1,1,0,0,1,1,1,0,0,0,1,0,1,0]
=> ? = 3
00111011 => [2,3,1,2] => [2,2,3,1] => [1,1,0,0,1,1,0,0,1,1,1,0,0,0,1,0]
=> ? = 2
00111100 => [2,4,2] => [2,2,4] => [1,1,0,0,1,1,0,0,1,1,1,1,0,0,0,0]
=> ? = 1
00111101 => [2,4,1,1] => [1,2,4,1] => [1,0,1,1,0,0,1,1,1,1,0,0,0,0,1,0]
=> ? = 2
01000100 => [1,1,3,1,2] => [2,1,1,3,1] => [1,1,0,0,1,0,1,0,1,1,1,0,0,0,1,0]
=> ? = 2
Description
The projective dimension of the first simple module in the Nakayama algebra corresponding to the Dyck path.
Matching statistic: St001368
Mp00097: Binary words —delta morphism⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St001368: Graphs ⟶ ℤResult quality: 46% ●values known / values provided: 46%●distinct values known / distinct values provided: 50%
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St001368: Graphs ⟶ ℤResult quality: 46% ●values known / values provided: 46%●distinct values known / distinct values provided: 50%
Values
0001 => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 1
0010 => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 2
1101 => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 2
1110 => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 1
00001 => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 1
00010 => [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
00011 => [3,2] => ([(1,4),(2,4),(3,4)],5)
=> 1
00100 => [2,1,2] => ([(1,3),(1,4),(2,3),(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)
=> 3
00110 => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
01101 => [1,2,1,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
01110 => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
10001 => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
10010 => [1,2,1,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(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)
=> 1
11010 => [2,1,1,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3
11011 => [2,1,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
11100 => [3,2] => ([(1,4),(2,4),(3,4)],5)
=> 1
11101 => [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
11110 => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 1
000001 => [5,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> 1
000010 => [4,1,1] => ([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 2
000011 => [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6)
=> 1
000100 => [3,1,2] => ([(1,4),(1,5),(2,4),(2,5),(3,4),(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)
=> 3
000110 => [3,2,1] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 1
000111 => [3,3] => ([(2,5),(3,5),(4,5)],6)
=> 1
001000 => [2,1,3] => ([(2,4),(2,5),(3,4),(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)
=> 1
001010 => [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)
=> 4
001011 => [2,1,1,2] => ([(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 3
001100 => [2,2,2] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 1
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
001110 => [2,3,1] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 1
010001 => [1,1,3,1] => ([(0,5),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 1
010010 => [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
011001 => [1,2,2,1] => ([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 1
011010 => [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
011011 => [1,2,1,2] => ([(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 2
011100 => [1,3,2] => ([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> 1
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
011110 => [1,4,1] => ([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> 1
100001 => [1,4,1] => ([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> 1
100010 => [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
100011 => [1,3,2] => ([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> 1
100100 => [1,2,1,2] => ([(1,4),(1,5),(2,3),(2,4),(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)
=> 3
100110 => [1,2,2,1] => ([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 1
101101 => [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
101110 => [1,1,3,1] => ([(0,5),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 1
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)
=> ? = 5
0100101 => [1,1,2,1,1,1] => ([(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
0101101 => [1,1,1,2,1,1] => ([(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
0110101 => [1,2,1,1,1,1] => ([(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)
=> ? = 4
1001010 => [1,2,1,1,1,1] => ([(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)
=> ? = 4
1010010 => [1,1,1,2,1,1] => ([(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
1011010 => [1,1,2,1,1,1] => ([(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
1101010 => [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)
=> ? = 5
00000001 => [7,1] => ([(0,7),(1,7),(2,7),(3,7),(4,7),(5,7),(6,7)],8)
=> ? = 1
00000010 => [6,1,1] => ([(0,6),(0,7),(1,6),(1,7),(2,6),(2,7),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 2
00000101 => [5,1,1,1] => ([(0,5),(0,6),(0,7),(1,5),(1,6),(1,7),(2,5),(2,6),(2,7),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 3
00001000 => [4,1,3] => ([(2,6),(2,7),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 2
00001001 => [4,1,2,1] => ([(0,7),(1,5),(1,6),(1,7),(2,5),(2,6),(2,7),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 1
00001010 => [4,1,1,1,1] => ([(0,4),(0,5),(0,6),(0,7),(1,4),(1,5),(1,6),(1,7),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 4
00001011 => [4,1,1,2] => ([(1,5),(1,6),(1,7),(2,5),(2,6),(2,7),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 3
00001100 => [4,2,2] => ([(1,7),(2,6),(2,7),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 1
00001101 => [4,2,1,1] => ([(0,6),(0,7),(1,5),(1,6),(1,7),(2,5),(2,6),(2,7),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 2
00001110 => [4,3,1] => ([(0,7),(1,7),(2,6),(2,7),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 1
00001111 => [4,4] => ([(3,7),(4,7),(5,7),(6,7)],8)
=> ? = 1
00010001 => [3,1,3,1] => ([(0,7),(1,7),(2,5),(2,6),(2,7),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 1
00010010 => [3,1,2,1,1] => ([(0,6),(0,7),(1,4),(1,5),(1,6),(1,7),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 2
00010011 => [3,1,2,2] => ([(1,7),(2,5),(2,6),(2,7),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 1
00010101 => [3,1,1,1,1,1] => ([(0,3),(0,4),(0,5),(0,6),(0,7),(1,3),(1,4),(1,5),(1,6),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 5
00010110 => [3,1,1,2,1] => ([(0,7),(1,4),(1,5),(1,6),(1,7),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 1
00010111 => [3,1,1,3] => ([(2,5),(2,6),(2,7),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 3
00011000 => [3,2,3] => ([(2,7),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 1
00011001 => [3,2,2,1] => ([(0,7),(1,6),(1,7),(2,5),(2,6),(2,7),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 1
00011010 => [3,2,1,1,1] => ([(0,5),(0,6),(0,7),(1,4),(1,5),(1,6),(1,7),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 3
00011011 => [3,2,1,2] => ([(1,6),(1,7),(2,5),(2,6),(2,7),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 2
00011100 => [3,3,2] => ([(1,7),(2,7),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 1
00011101 => [3,3,1,1] => ([(0,6),(0,7),(1,6),(1,7),(2,5),(2,6),(2,7),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 2
00011110 => [3,4,1] => ([(0,7),(1,7),(2,7),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 1
00100010 => [2,1,3,1,1] => ([(0,6),(0,7),(1,6),(1,7),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 2
00100011 => [2,1,3,2] => ([(1,7),(2,7),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 1
00100100 => [2,1,2,1,2] => ([(1,6),(1,7),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 2
00100101 => [2,1,2,1,1,1] => ([(0,5),(0,6),(0,7),(1,3),(1,4),(1,5),(1,6),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 3
00100110 => [2,1,2,2,1] => ([(0,7),(1,6),(1,7),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 1
00100111 => [2,1,2,3] => ([(2,7),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 1
00101001 => [2,1,1,1,2,1] => ([(0,7),(1,3),(1,4),(1,5),(1,6),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 1
00101010 => [2,1,1,1,1,1,1] => ([(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(1,2),(1,3),(1,4),(1,5),(1,6),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 6
00101011 => [2,1,1,1,1,2] => ([(1,3),(1,4),(1,5),(1,6),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 5
00101100 => [2,1,1,2,2] => ([(1,7),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 1
00101101 => [2,1,1,2,1,1] => ([(0,6),(0,7),(1,3),(1,4),(1,5),(1,6),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 2
00101110 => [2,1,1,3,1] => ([(0,7),(1,7),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 1
00110001 => [2,2,3,1] => ([(0,7),(1,7),(2,6),(2,7),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 1
00110010 => [2,2,2,1,1] => ([(0,6),(0,7),(1,5),(1,6),(1,7),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 2
00110011 => [2,2,2,2] => ([(1,7),(2,6),(2,7),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 1
00110100 => [2,2,1,1,2] => ([(1,5),(1,6),(1,7),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 3
00110101 => [2,2,1,1,1,1] => ([(0,4),(0,5),(0,6),(0,7),(1,3),(1,4),(1,5),(1,6),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 4
00110110 => [2,2,1,2,1] => ([(0,7),(1,5),(1,6),(1,7),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 1
Description
The number of vertices of maximal degree in a graph.
Matching statistic: St001316
Mp00097: Binary words —delta morphism⟶ Integer compositions
Mp00172: Integer compositions —rotate back to front⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St001316: Graphs ⟶ ℤResult quality: 46% ●values known / values provided: 46%●distinct values known / distinct values provided: 50%
Mp00172: Integer compositions —rotate back to front⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St001316: Graphs ⟶ ℤResult quality: 46% ●values known / values provided: 46%●distinct values known / distinct values provided: 50%
Values
0001 => [3,1] => [1,3] => ([(2,3)],4)
=> 1
0010 => [2,1,1] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 2
1101 => [2,1,1] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 2
1110 => [3,1] => [1,3] => ([(2,3)],4)
=> 1
00001 => [4,1] => [1,4] => ([(3,4)],5)
=> 1
00010 => [3,1,1] => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
00011 => [3,2] => [2,3] => ([(2,4),(3,4)],5)
=> 1
00100 => [2,1,2] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
00101 => [2,1,1,1] => [1,2,1,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3
00110 => [2,2,1] => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> 1
01101 => [1,2,1,1] => [1,1,2,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
01110 => [1,3,1] => [1,1,3] => ([(2,3),(2,4),(3,4)],5)
=> 1
10001 => [1,3,1] => [1,1,3] => ([(2,3),(2,4),(3,4)],5)
=> 1
10010 => [1,2,1,1] => [1,1,2,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
11001 => [2,2,1] => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> 1
11010 => [2,1,1,1] => [1,2,1,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3
11011 => [2,1,2] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
11100 => [3,2] => [2,3] => ([(2,4),(3,4)],5)
=> 1
11101 => [3,1,1] => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
11110 => [4,1] => [1,4] => ([(3,4)],5)
=> 1
000001 => [5,1] => [1,5] => ([(4,5)],6)
=> 1
000010 => [4,1,1] => [1,4,1] => ([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> 2
000011 => [4,2] => [2,4] => ([(3,5),(4,5)],6)
=> 1
000100 => [3,1,2] => [2,3,1] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 2
000101 => [3,1,1,1] => [1,3,1,1] => ([(0,4),(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 3
000110 => [3,2,1] => [1,3,2] => ([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> 1
000111 => [3,3] => [3,3] => ([(2,5),(3,5),(4,5)],6)
=> 1
001000 => [2,1,3] => [3,2,1] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 2
001001 => [2,1,2,1] => [1,2,1,2] => ([(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 1
001010 => [2,1,1,1,1] => [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)
=> 4
001011 => [2,1,1,2] => [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
001100 => [2,2,2] => [2,2,2] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 1
001101 => [2,2,1,1] => [1,2,2,1] => ([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 2
001110 => [2,3,1] => [1,2,3] => ([(2,5),(3,4),(3,5),(4,5)],6)
=> 1
010001 => [1,1,3,1] => [1,1,1,3] => ([(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 1
010010 => [1,1,2,1,1] => [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
011001 => [1,2,2,1] => [1,1,2,2] => ([(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 1
011010 => [1,2,1,1,1] => [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)
=> 3
011011 => [1,2,1,2] => [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
011100 => [1,3,2] => [2,1,3] => ([(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 1
011101 => [1,3,1,1] => [1,1,3,1] => ([(0,5),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 2
011110 => [1,4,1] => [1,1,4] => ([(3,4),(3,5),(4,5)],6)
=> 1
100001 => [1,4,1] => [1,1,4] => ([(3,4),(3,5),(4,5)],6)
=> 1
100010 => [1,3,1,1] => [1,1,3,1] => ([(0,5),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 2
100011 => [1,3,2] => [2,1,3] => ([(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 1
100100 => [1,2,1,2] => [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
100101 => [1,2,1,1,1] => [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)
=> 3
100110 => [1,2,2,1] => [1,1,2,2] => ([(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 1
101101 => [1,1,2,1,1] => [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
101110 => [1,1,3,1] => [1,1,1,3] => ([(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 1
0010101 => [2,1,1,1,1,1] => [1,2,1,1,1,1] => ([(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)
=> ? = 5
0100101 => [1,1,2,1,1,1] => [1,1,1,2,1,1] => ([(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
0101101 => [1,1,1,2,1,1] => [1,1,1,1,2,1] => ([(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
0110101 => [1,2,1,1,1,1] => [1,1,2,1,1,1] => ([(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)
=> ? = 4
1001010 => [1,2,1,1,1,1] => [1,1,2,1,1,1] => ([(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)
=> ? = 4
1010010 => [1,1,1,2,1,1] => [1,1,1,1,2,1] => ([(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
1011010 => [1,1,2,1,1,1] => [1,1,1,2,1,1] => ([(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
1101010 => [2,1,1,1,1,1] => [1,2,1,1,1,1] => ([(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)
=> ? = 5
00000001 => [7,1] => [1,7] => ([(6,7)],8)
=> ? = 1
00000010 => [6,1,1] => [1,6,1] => ([(0,7),(1,7),(2,7),(3,7),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 2
00000101 => [5,1,1,1] => [1,5,1,1] => ([(0,6),(0,7),(1,6),(1,7),(2,6),(2,7),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 3
00001000 => [4,1,3] => [3,4,1] => ([(0,7),(1,7),(2,7),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 2
00001001 => [4,1,2,1] => [1,4,1,2] => ([(1,6),(1,7),(2,6),(2,7),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 1
00001010 => [4,1,1,1,1] => [1,4,1,1,1] => ([(0,5),(0,6),(0,7),(1,5),(1,6),(1,7),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 4
00001011 => [4,1,1,2] => [2,4,1,1] => ([(0,6),(0,7),(1,6),(1,7),(2,6),(2,7),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 3
00001100 => [4,2,2] => [2,4,2] => ([(1,7),(2,7),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 1
00001101 => [4,2,1,1] => [1,4,2,1] => ([(0,7),(1,6),(1,7),(2,6),(2,7),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 2
00001110 => [4,3,1] => [1,4,3] => ([(2,7),(3,7),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 1
00001111 => [4,4] => [4,4] => ([(3,7),(4,7),(5,7),(6,7)],8)
=> ? = 1
00010001 => [3,1,3,1] => [1,3,1,3] => ([(2,6),(2,7),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 1
00010010 => [3,1,2,1,1] => [1,3,1,2,1] => ([(0,7),(1,5),(1,6),(1,7),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 2
00010011 => [3,1,2,2] => [2,3,1,2] => ([(1,6),(1,7),(2,6),(2,7),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 1
00010101 => [3,1,1,1,1,1] => [1,3,1,1,1,1] => ([(0,4),(0,5),(0,6),(0,7),(1,4),(1,5),(1,6),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 5
00010110 => [3,1,1,2,1] => [1,3,1,1,2] => ([(1,5),(1,6),(1,7),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 1
00010111 => [3,1,1,3] => [3,3,1,1] => ([(0,6),(0,7),(1,6),(1,7),(2,5),(2,6),(2,7),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 3
00011000 => [3,2,3] => [3,3,2] => ([(1,7),(2,7),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 1
00011001 => [3,2,2,1] => [1,3,2,2] => ([(1,7),(2,6),(2,7),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 1
00011010 => [3,2,1,1,1] => [1,3,2,1,1] => ([(0,6),(0,7),(1,5),(1,6),(1,7),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 3
00011011 => [3,2,1,2] => [2,3,2,1] => ([(0,7),(1,6),(1,7),(2,6),(2,7),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 2
00011100 => [3,3,2] => [2,3,3] => ([(2,7),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 1
00011101 => [3,3,1,1] => [1,3,3,1] => ([(0,7),(1,7),(2,6),(2,7),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 2
00011110 => [3,4,1] => [1,3,4] => ([(3,7),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 1
00100010 => [2,1,3,1,1] => [1,2,1,3,1] => ([(0,7),(1,7),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 2
00100011 => [2,1,3,2] => [2,2,1,3] => ([(2,6),(2,7),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 1
00100100 => [2,1,2,1,2] => [2,2,1,2,1] => ([(0,7),(1,5),(1,6),(1,7),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 2
00100101 => [2,1,2,1,1,1] => [1,2,1,2,1,1] => ([(0,6),(0,7),(1,4),(1,5),(1,6),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 3
00100110 => [2,1,2,2,1] => [1,2,1,2,2] => ([(1,7),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 1
00100111 => [2,1,2,3] => [3,2,1,2] => ([(1,6),(1,7),(2,5),(2,6),(2,7),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 1
00101001 => [2,1,1,1,2,1] => [1,2,1,1,1,2] => ([(1,4),(1,5),(1,6),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 1
00101010 => [2,1,1,1,1,1,1] => [1,2,1,1,1,1,1] => ([(0,3),(0,4),(0,5),(0,6),(0,7),(1,2),(1,3),(1,4),(1,5),(1,6),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 6
00101011 => [2,1,1,1,1,2] => [2,2,1,1,1,1] => ([(0,4),(0,5),(0,6),(0,7),(1,3),(1,4),(1,5),(1,6),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 5
00101100 => [2,1,1,2,2] => [2,2,1,1,2] => ([(1,5),(1,6),(1,7),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 1
00101101 => [2,1,1,2,1,1] => [1,2,1,1,2,1] => ([(0,7),(1,4),(1,5),(1,6),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 2
00101110 => [2,1,1,3,1] => [1,2,1,1,3] => ([(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 1
00110001 => [2,2,3,1] => [1,2,2,3] => ([(2,7),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 1
00110010 => [2,2,2,1,1] => [1,2,2,2,1] => ([(0,7),(1,6),(1,7),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 2
00110011 => [2,2,2,2] => [2,2,2,2] => ([(1,7),(2,6),(2,7),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 1
00110100 => [2,2,1,1,2] => [2,2,2,1,1] => ([(0,6),(0,7),(1,5),(1,6),(1,7),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 3
00110101 => [2,2,1,1,1,1] => [1,2,2,1,1,1] => ([(0,5),(0,6),(0,7),(1,4),(1,5),(1,6),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 4
00110110 => [2,2,1,2,1] => [1,2,2,1,2] => ([(1,6),(1,7),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 1
Description
The domatic number of a graph.
This is the maximal size of a partition of the vertices into dominating sets.
Matching statistic: St000286
Mp00097: Binary words —delta morphism⟶ Integer compositions
Mp00172: Integer compositions —rotate back to front⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St000286: Graphs ⟶ ℤResult quality: 18% ●values known / values provided: 18%●distinct values known / distinct values provided: 50%
Mp00172: Integer compositions —rotate back to front⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St000286: Graphs ⟶ ℤResult quality: 18% ●values known / values provided: 18%●distinct values known / distinct values provided: 50%
Values
0001 => [3,1] => [1,3] => ([(2,3)],4)
=> 1
0010 => [2,1,1] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 2
1101 => [2,1,1] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 2
1110 => [3,1] => [1,3] => ([(2,3)],4)
=> 1
00001 => [4,1] => [1,4] => ([(3,4)],5)
=> 1
00010 => [3,1,1] => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
00011 => [3,2] => [2,3] => ([(2,4),(3,4)],5)
=> 1
00100 => [2,1,2] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
00101 => [2,1,1,1] => [1,2,1,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3
00110 => [2,2,1] => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> 1
01101 => [1,2,1,1] => [1,1,2,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
01110 => [1,3,1] => [1,1,3] => ([(2,3),(2,4),(3,4)],5)
=> 1
10001 => [1,3,1] => [1,1,3] => ([(2,3),(2,4),(3,4)],5)
=> 1
10010 => [1,2,1,1] => [1,1,2,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
11001 => [2,2,1] => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> 1
11010 => [2,1,1,1] => [1,2,1,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3
11011 => [2,1,2] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
11100 => [3,2] => [2,3] => ([(2,4),(3,4)],5)
=> 1
11101 => [3,1,1] => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
11110 => [4,1] => [1,4] => ([(3,4)],5)
=> 1
000001 => [5,1] => [1,5] => ([(4,5)],6)
=> 1
000010 => [4,1,1] => [1,4,1] => ([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> 2
000011 => [4,2] => [2,4] => ([(3,5),(4,5)],6)
=> 1
000100 => [3,1,2] => [2,3,1] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 2
000101 => [3,1,1,1] => [1,3,1,1] => ([(0,4),(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 3
000110 => [3,2,1] => [1,3,2] => ([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> 1
000111 => [3,3] => [3,3] => ([(2,5),(3,5),(4,5)],6)
=> 1
001000 => [2,1,3] => [3,2,1] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 2
001001 => [2,1,2,1] => [1,2,1,2] => ([(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 1
001010 => [2,1,1,1,1] => [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)
=> 4
001011 => [2,1,1,2] => [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
001100 => [2,2,2] => [2,2,2] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 1
001101 => [2,2,1,1] => [1,2,2,1] => ([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 2
001110 => [2,3,1] => [1,2,3] => ([(2,5),(3,4),(3,5),(4,5)],6)
=> 1
010001 => [1,1,3,1] => [1,1,1,3] => ([(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 1
010010 => [1,1,2,1,1] => [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
011001 => [1,2,2,1] => [1,1,2,2] => ([(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 1
011010 => [1,2,1,1,1] => [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)
=> 3
011011 => [1,2,1,2] => [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
011100 => [1,3,2] => [2,1,3] => ([(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 1
011101 => [1,3,1,1] => [1,1,3,1] => ([(0,5),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 2
011110 => [1,4,1] => [1,1,4] => ([(3,4),(3,5),(4,5)],6)
=> 1
100001 => [1,4,1] => [1,1,4] => ([(3,4),(3,5),(4,5)],6)
=> 1
100010 => [1,3,1,1] => [1,1,3,1] => ([(0,5),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 2
100011 => [1,3,2] => [2,1,3] => ([(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 1
100100 => [1,2,1,2] => [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
100101 => [1,2,1,1,1] => [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)
=> 3
100110 => [1,2,2,1] => [1,1,2,2] => ([(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 1
101101 => [1,1,2,1,1] => [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
101110 => [1,1,3,1] => [1,1,1,3] => ([(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 1
0000001 => [6,1] => [1,6] => ([(5,6)],7)
=> ? = 1
0000010 => [5,1,1] => [1,5,1] => ([(0,6),(1,6),(2,6),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 2
0000011 => [5,2] => [2,5] => ([(4,6),(5,6)],7)
=> ? = 1
0000100 => [4,1,2] => [2,4,1] => ([(0,6),(1,6),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 2
0000101 => [4,1,1,1] => [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)
=> ? = 3
0000110 => [4,2,1] => [1,4,2] => ([(1,6),(2,6),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 1
0000111 => [4,3] => [3,4] => ([(3,6),(4,6),(5,6)],7)
=> ? = 1
0001000 => [3,1,3] => [3,3,1] => ([(0,6),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 2
0001001 => [3,1,2,1] => [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)
=> ? = 1
0001010 => [3,1,1,1,1] => [1,3,1,1,1] => ([(0,4),(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)
=> ? = 4
0001011 => [3,1,1,2] => [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
0001100 => [3,2,2] => [2,3,2] => ([(1,6),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 1
0001101 => [3,2,1,1] => [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
0001110 => [3,3,1] => [1,3,3] => ([(2,6),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 1
0001111 => [3,4] => [4,3] => ([(2,6),(3,6),(4,6),(5,6)],7)
=> ? = 1
0010000 => [2,1,4] => [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
0010001 => [2,1,3,1] => [1,2,1,3] => ([(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 1
0010010 => [2,1,2,1,1] => [1,2,1,2,1] => ([(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)
=> ? = 2
0010011 => [2,1,2,2] => [2,2,1,2] => ([(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 1
0010100 => [2,1,1,1,2] => [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)
=> ? = 4
0010101 => [2,1,1,1,1,1] => [1,2,1,1,1,1] => ([(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)
=> ? = 5
0010110 => [2,1,1,2,1] => [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)
=> ? = 1
0010111 => [2,1,1,3] => [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
0011000 => [2,2,3] => [3,2,2] => ([(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 1
0011001 => [2,2,2,1] => [1,2,2,2] => ([(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 1
0011010 => [2,2,1,1,1] => [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
0011011 => [2,2,1,2] => [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
0011100 => [2,3,2] => [2,2,3] => ([(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 1
0011101 => [2,3,1,1] => [1,2,3,1] => ([(0,6),(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 2
0011110 => [2,4,1] => [1,2,4] => ([(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 1
0100001 => [1,1,4,1] => [1,1,1,4] => ([(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 1
0100010 => [1,1,3,1,1] => [1,1,1,3,1] => ([(0,6),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 2
0100011 => [1,1,3,2] => [2,1,1,3] => ([(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 1
0100100 => [1,1,2,1,2] => [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
0100101 => [1,1,2,1,1,1] => [1,1,1,2,1,1] => ([(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
0100110 => [1,1,2,2,1] => [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)
=> ? = 1
0101101 => [1,1,1,2,1,1] => [1,1,1,1,2,1] => ([(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
0101110 => [1,1,1,3,1] => [1,1,1,1,3] => ([(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 1
0110001 => [1,2,3,1] => [1,1,2,3] => ([(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 1
0110010 => [1,2,2,1,1] => [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)
=> ? = 2
0110011 => [1,2,2,2] => [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)
=> ? = 1
0110100 => [1,2,1,1,2] => [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
0110101 => [1,2,1,1,1,1] => [1,1,2,1,1,1] => ([(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)
=> ? = 4
0110110 => [1,2,1,2,1] => [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)
=> ? = 1
0110111 => [1,2,1,3] => [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
0111000 => [1,3,3] => [3,1,3] => ([(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 1
0111001 => [1,3,2,1] => [1,1,3,2] => ([(1,6),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 1
0111010 => [1,3,1,1,1] => [1,1,3,1,1] => ([(0,5),(0,6),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 3
0111011 => [1,3,1,2] => [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
0111100 => [1,4,2] => [2,1,4] => ([(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 1
Description
The number of connected components of the complement of a graph.
The complement of a graph is the graph on the same vertex set with complementary edges.
The following 15 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000772The multiplicity of the largest distance Laplacian eigenvalue in a connected graph. St001201The grade of the simple module S0 in the special CNakayama algebra corresponding to the Dyck path. St000261The edge connectivity of a graph. St000262The vertex connectivity of a graph. St000310The minimal degree of a vertex of a graph. St000454The largest eigenvalue of a graph if it is integral. St001330The hat guessing number of a graph. St000141The maximum drop size of a permutation. St000455The second largest eigenvalue of a graph if it is integral. St000451The length of the longest pattern of the form k 1 2. St001238The number of simple modules S such that the Auslander-Reiten translate of S is isomorphic to the Nakayama functor applied to the second syzygy of S. St001431Half of the Loewy length minus one of a modified stable Auslander algebra of the Nakayama algebra corresponding to the Dyck path. St001186Number of simple modules with grade at least 3 in the corresponding Nakayama algebra. St001314The number of tilting modules of arbitrary projective dimension that have no simple modules as a direct summand in the corresponding Nakayama algebra. St001526The Loewy length of the Auslander-Reiten translate of the regular module as a bimodule of the Nakayama algebra corresponding to the Dyck path.
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