Your data matches 4 different statistics following compositions of up to 3 maps.
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Matching statistic: St000922
(load all 5 compositions to match this statistic)
Mapping
St000922: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
0 => 1
1 => 1
00 => 2
01 => 1
10 => 1
11 => 2
000 => 3
001 => 2
010 => 2
011 => 2
100 => 2
101 => 2
110 => 2
111 => 3
0000 => 4
0001 => 3
0010 => 2
0011 => 2
0100 => 2
0101 => 3
0110 => 2
0111 => 3
1000 => 3
1001 => 2
1010 => 3
1011 => 2
1100 => 2
1101 => 2
1110 => 3
1111 => 4
00000 => 5
00001 => 4
00010 => 3
00011 => 3
00100 => 3
00101 => 3
00110 => 2
00111 => 3
01000 => 3
01001 => 3
01010 => 4
01011 => 3
01100 => 2
01101 => 3
01110 => 3
01111 => 4
10000 => 4
10001 => 3
10010 => 3
10011 => 2
Description
The minimal number such that all substrings of this length are unique.
Matching statistic: St000923
(load all 2 compositions to match this statistic)
Mapping
Mp00178: Binary words —to composition⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
Mp00023: Dyck paths —to non-crossing permutation⟶ Permutations
St000923: Permutations ⟶ ℤResult quality: 8% ●values known / values provided: 8%●distinct values known / distinct values provided: 60%
Values
0 => [2] => [1,1,0,0]
 => [2,1] => 2 = 1 + 1
1 => [1,1] => [1,0,1,0]
 => [1,2] => 2 = 1 + 1
00 => [3] => [1,1,1,0,0,0]
 => [3,2,1] => 3 = 2 + 1
01 => [2,1] => [1,1,0,0,1,0]
 => [2,1,3] => 2 = 1 + 1
10 => [1,2] => [1,0,1,1,0,0]
 => [1,3,2] => 2 = 1 + 1
11 => [1,1,1] => [1,0,1,0,1,0]
 => [1,2,3] => 3 = 2 + 1
000 => [4] => [1,1,1,1,0,0,0,0]
 => [4,3,2,1] => 4 = 3 + 1
001 => [3,1] => [1,1,1,0,0,0,1,0]
 => [3,2,1,4] => 3 = 2 + 1
010 => [2,2] => [1,1,0,0,1,1,0,0]
 => [2,1,4,3] => 3 = 2 + 1
011 => [2,1,1] => [1,1,0,0,1,0,1,0]
 => [2,1,3,4] => 3 = 2 + 1
100 => [1,3] => [1,0,1,1,1,0,0,0]
 => [1,4,3,2] => 3 = 2 + 1
101 => [1,2,1] => [1,0,1,1,0,0,1,0]
 => [1,3,2,4] => 3 = 2 + 1
110 => [1,1,2] => [1,0,1,0,1,1,0,0]
 => [1,2,4,3] => 3 = 2 + 1
111 => [1,1,1,1] => [1,0,1,0,1,0,1,0]
 => [1,2,3,4] => 4 = 3 + 1
0000 => [5] => [1,1,1,1,1,0,0,0,0,0]
 => [5,4,3,2,1] => 5 = 4 + 1
0001 => [4,1] => [1,1,1,1,0,0,0,0,1,0]
 => [4,3,2,1,5] => 4 = 3 + 1
0010 => [3,2] => [1,1,1,0,0,0,1,1,0,0]
 => [3,2,1,5,4] => 3 = 2 + 1
0011 => [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
 => [3,2,1,4,5] => 3 = 2 + 1
0100 => [2,3] => [1,1,0,0,1,1,1,0,0,0]
 => [2,1,5,4,3] => 3 = 2 + 1
0101 => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
 => [2,1,4,3,5] => 4 = 3 + 1
0110 => [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
 => [2,1,3,5,4] => 3 = 2 + 1
0111 => [2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
 => [2,1,3,4,5] => 4 = 3 + 1
1000 => [1,4] => [1,0,1,1,1,1,0,0,0,0]
 => [1,5,4,3,2] => 4 = 3 + 1
1001 => [1,3,1] => [1,0,1,1,1,0,0,0,1,0]
 => [1,4,3,2,5] => 3 = 2 + 1
1010 => [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
 => [1,3,2,5,4] => 4 = 3 + 1
1011 => [1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
 => [1,3,2,4,5] => 3 = 2 + 1
1100 => [1,1,3] => [1,0,1,0,1,1,1,0,0,0]
 => [1,2,5,4,3] => 3 = 2 + 1
1101 => [1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
 => [1,2,4,3,5] => 3 = 2 + 1
1110 => [1,1,1,2] => [1,0,1,0,1,0,1,1,0,0]
 => [1,2,3,5,4] => 4 = 3 + 1
1111 => [1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
 => [1,2,3,4,5] => 5 = 4 + 1
00000 => [6] => [1,1,1,1,1,1,0,0,0,0,0,0]
 => [6,5,4,3,2,1] => 6 = 5 + 1
00001 => [5,1] => [1,1,1,1,1,0,0,0,0,0,1,0]
 => [5,4,3,2,1,6] => 5 = 4 + 1
00010 => [4,2] => [1,1,1,1,0,0,0,0,1,1,0,0]
 => [4,3,2,1,6,5] => 4 = 3 + 1
00011 => [4,1,1] => [1,1,1,1,0,0,0,0,1,0,1,0]
 => [4,3,2,1,5,6] => 4 = 3 + 1
00100 => [3,3] => [1,1,1,0,0,0,1,1,1,0,0,0]
 => [3,2,1,6,5,4] => 4 = 3 + 1
00101 => [3,2,1] => [1,1,1,0,0,0,1,1,0,0,1,0]
 => [3,2,1,5,4,6] => 4 = 3 + 1
00110 => [3,1,2] => [1,1,1,0,0,0,1,0,1,1,0,0]
 => [3,2,1,4,6,5] => 3 = 2 + 1
00111 => [3,1,1,1] => [1,1,1,0,0,0,1,0,1,0,1,0]
 => [3,2,1,4,5,6] => 4 = 3 + 1
01000 => [2,4] => [1,1,0,0,1,1,1,1,0,0,0,0]
 => [2,1,6,5,4,3] => 4 = 3 + 1
01001 => [2,3,1] => [1,1,0,0,1,1,1,0,0,0,1,0]
 => [2,1,5,4,3,6] => 4 = 3 + 1
01010 => [2,2,2] => [1,1,0,0,1,1,0,0,1,1,0,0]
 => [2,1,4,3,6,5] => 5 = 4 + 1
01011 => [2,2,1,1] => [1,1,0,0,1,1,0,0,1,0,1,0]
 => [2,1,4,3,5,6] => 4 = 3 + 1
01100 => [2,1,3] => [1,1,0,0,1,0,1,1,1,0,0,0]
 => [2,1,3,6,5,4] => 3 = 2 + 1
01101 => [2,1,2,1] => [1,1,0,0,1,0,1,1,0,0,1,0]
 => [2,1,3,5,4,6] => 4 = 3 + 1
01110 => [2,1,1,2] => [1,1,0,0,1,0,1,0,1,1,0,0]
 => [2,1,3,4,6,5] => 4 = 3 + 1
01111 => [2,1,1,1,1] => [1,1,0,0,1,0,1,0,1,0,1,0]
 => [2,1,3,4,5,6] => 5 = 4 + 1
10000 => [1,5] => [1,0,1,1,1,1,1,0,0,0,0,0]
 => [1,6,5,4,3,2] => 5 = 4 + 1
10001 => [1,4,1] => [1,0,1,1,1,1,0,0,0,0,1,0]
 => [1,5,4,3,2,6] => 4 = 3 + 1
10010 => [1,3,2] => [1,0,1,1,1,0,0,0,1,1,0,0]
 => [1,4,3,2,6,5] => 4 = 3 + 1
10011 => [1,3,1,1] => [1,0,1,1,1,0,0,0,1,0,1,0]
 => [1,4,3,2,5,6] => 3 = 2 + 1
000000 => [7] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
 => [7,6,5,4,3,2,1] => ? = 6 + 1
000001 => [6,1] => [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
 => [6,5,4,3,2,1,7] => ? = 5 + 1
000010 => [5,2] => [1,1,1,1,1,0,0,0,0,0,1,1,0,0]
 => [5,4,3,2,1,7,6] => ? = 4 + 1
000011 => [5,1,1] => [1,1,1,1,1,0,0,0,0,0,1,0,1,0]
 => [5,4,3,2,1,6,7] => ? = 4 + 1
000100 => [4,3] => [1,1,1,1,0,0,0,0,1,1,1,0,0,0]
 => [4,3,2,1,7,6,5] => ? = 3 + 1
000101 => [4,2,1] => [1,1,1,1,0,0,0,0,1,1,0,0,1,0]
 => [4,3,2,1,6,5,7] => ? = 3 + 1
000110 => [4,1,2] => [1,1,1,1,0,0,0,0,1,0,1,1,0,0]
 => [4,3,2,1,5,7,6] => ? = 3 + 1
000111 => [4,1,1,1] => [1,1,1,1,0,0,0,0,1,0,1,0,1,0]
 => [4,3,2,1,5,6,7] => ? = 3 + 1
001000 => [3,4] => [1,1,1,0,0,0,1,1,1,1,0,0,0,0]
 => [3,2,1,7,6,5,4] => ? = 3 + 1
001001 => [3,3,1] => [1,1,1,0,0,0,1,1,1,0,0,0,1,0]
 => [3,2,1,6,5,4,7] => ? = 4 + 1
001010 => [3,2,2] => [1,1,1,0,0,0,1,1,0,0,1,1,0,0]
 => [3,2,1,5,4,7,6] => ? = 4 + 1
001011 => [3,2,1,1] => [1,1,1,0,0,0,1,1,0,0,1,0,1,0]
 => [3,2,1,5,4,6,7] => ? = 3 + 1
001100 => [3,1,3] => [1,1,1,0,0,0,1,0,1,1,1,0,0,0]
 => [3,2,1,4,7,6,5] => ? = 3 + 1
001101 => [3,1,2,1] => [1,1,1,0,0,0,1,0,1,1,0,0,1,0]
 => [3,2,1,4,6,5,7] => ? = 3 + 1
001110 => [3,1,1,2] => [1,1,1,0,0,0,1,0,1,0,1,1,0,0]
 => [3,2,1,4,5,7,6] => ? = 3 + 1
001111 => [3,1,1,1,1] => [1,1,1,0,0,0,1,0,1,0,1,0,1,0]
 => [3,2,1,4,5,6,7] => ? = 4 + 1
010000 => [2,5] => [1,1,0,0,1,1,1,1,1,0,0,0,0,0]
 => [2,1,7,6,5,4,3] => ? = 4 + 1
010001 => [2,4,1] => [1,1,0,0,1,1,1,1,0,0,0,0,1,0]
 => [2,1,6,5,4,3,7] => ? = 3 + 1
010010 => [2,3,2] => [1,1,0,0,1,1,1,0,0,0,1,1,0,0]
 => [2,1,5,4,3,7,6] => ? = 4 + 1
010011 => [2,3,1,1] => [1,1,0,0,1,1,1,0,0,0,1,0,1,0]
 => [2,1,5,4,3,6,7] => ? = 3 + 1
010100 => [2,2,3] => [1,1,0,0,1,1,0,0,1,1,1,0,0,0]
 => [2,1,4,3,7,6,5] => ? = 4 + 1
010101 => [2,2,2,1] => [1,1,0,0,1,1,0,0,1,1,0,0,1,0]
 => [2,1,4,3,6,5,7] => ? = 5 + 1
010110 => [2,2,1,2] => [1,1,0,0,1,1,0,0,1,0,1,1,0,0]
 => [2,1,4,3,5,7,6] => ? = 3 + 1
010111 => [2,2,1,1,1] => [1,1,0,0,1,1,0,0,1,0,1,0,1,0]
 => [2,1,4,3,5,6,7] => ? = 3 + 1
011000 => [2,1,4] => [1,1,0,0,1,0,1,1,1,1,0,0,0,0]
 => [2,1,3,7,6,5,4] => ? = 3 + 1
011001 => [2,1,3,1] => [1,1,0,0,1,0,1,1,1,0,0,0,1,0]
 => [2,1,3,6,5,4,7] => ? = 3 + 1
011010 => [2,1,2,2] => [1,1,0,0,1,0,1,1,0,0,1,1,0,0]
 => [2,1,3,5,4,7,6] => ? = 3 + 1
011011 => [2,1,2,1,1] => [1,1,0,0,1,0,1,1,0,0,1,0,1,0]
 => [2,1,3,5,4,6,7] => ? = 4 + 1
011100 => [2,1,1,3] => [1,1,0,0,1,0,1,0,1,1,1,0,0,0]
 => [2,1,3,4,7,6,5] => ? = 3 + 1
011101 => [2,1,1,2,1] => [1,1,0,0,1,0,1,0,1,1,0,0,1,0]
 => [2,1,3,4,6,5,7] => ? = 3 + 1
011110 => [2,1,1,1,2] => [1,1,0,0,1,0,1,0,1,0,1,1,0,0]
 => [2,1,3,4,5,7,6] => ? = 4 + 1
011111 => [2,1,1,1,1,1] => [1,1,0,0,1,0,1,0,1,0,1,0,1,0]
 => [2,1,3,4,5,6,7] => ? = 5 + 1
100000 => [1,6] => [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
 => [1,7,6,5,4,3,2] => ? = 5 + 1
100001 => [1,5,1] => [1,0,1,1,1,1,1,0,0,0,0,0,1,0]
 => [1,6,5,4,3,2,7] => ? = 4 + 1
100010 => [1,4,2] => [1,0,1,1,1,1,0,0,0,0,1,1,0,0]
 => [1,5,4,3,2,7,6] => ? = 3 + 1
100011 => [1,4,1,1] => [1,0,1,1,1,1,0,0,0,0,1,0,1,0]
 => [1,5,4,3,2,6,7] => ? = 3 + 1
0000000 => [8] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
 => [8,7,6,5,4,3,2,1] => ? = 7 + 1
0000001 => [7,1] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
 => [7,6,5,4,3,2,1,8] => ? = 6 + 1
0000010 => [6,2] => [1,1,1,1,1,1,0,0,0,0,0,0,1,1,0,0]
 => [6,5,4,3,2,1,8,7] => ? = 5 + 1
0000011 => [6,1,1] => [1,1,1,1,1,1,0,0,0,0,0,0,1,0,1,0]
 => [6,5,4,3,2,1,7,8] => ? = 5 + 1
0000100 => [5,3] => [1,1,1,1,1,0,0,0,0,0,1,1,1,0,0,0]
 => [5,4,3,2,1,8,7,6] => ? = 4 + 1
0000101 => [5,2,1] => [1,1,1,1,1,0,0,0,0,0,1,1,0,0,1,0]
 => [5,4,3,2,1,7,6,8] => ? = 4 + 1
0000110 => [5,1,2] => [1,1,1,1,1,0,0,0,0,0,1,0,1,1,0,0]
 => [5,4,3,2,1,6,8,7] => ? = 4 + 1
0000111 => [5,1,1,1] => [1,1,1,1,1,0,0,0,0,0,1,0,1,0,1,0]
 => [5,4,3,2,1,6,7,8] => ? = 4 + 1
0001000 => [4,4] => [1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0]
 => [4,3,2,1,8,7,6,5] => ? = 4 + 1
0001001 => [4,3,1] => [1,1,1,1,0,0,0,0,1,1,1,0,0,0,1,0]
 => [4,3,2,1,7,6,5,8] => ? = 4 + 1
0001010 => [4,2,2] => [1,1,1,1,0,0,0,0,1,1,0,0,1,1,0,0]
 => [4,3,2,1,6,5,8,7] => ? = 4 + 1
0001011 => [4,2,1,1] => [1,1,1,1,0,0,0,0,1,1,0,0,1,0,1,0]
 => [4,3,2,1,6,5,7,8] => ? = 3 + 1
0001100 => [4,1,3] => [1,1,1,1,0,0,0,0,1,0,1,1,1,0,0,0]
 => [4,3,2,1,5,8,7,6] => ? = 3 + 1
0001101 => [4,1,2,1] => [1,1,1,1,0,0,0,0,1,0,1,1,0,0,1,0]
 => [4,3,2,1,5,7,6,8] => ? = 3 + 1
Description
The minimal number with no two order isomorphic substrings of this length in a permutation. For example, the length $3$ substrings of the permutation $12435$ are $124$, $243$ and $435$, whereas its length $2$ substrings are $12$, $24$, $43$ and $35$. No two sequences among $124$, $243$ and $435$ are order isomorphic, but $12$ and $24$ are, so the statistic on $12435$ is $3$. This is inspired by [[St000922]].
Matching statistic: St001091
Mapping
Mp00262: Binary words —poset of factors⟶ Posets
Mp00110: Posets —Greene-Kleitman invariant⟶ Integer partitions
Mp00044: Integer partitions —conjugate⟶ Integer partitions
St001091: Integer partitions ⟶ ℤResult quality: 6% ●values known / values provided: 6%●distinct values known / distinct values provided: 90%
Values
0 => ([(0,1)],2)
 => [2]
 => [1,1]
 => 1
1 => ([(0,1)],2)
 => [2]
 => [1,1]
 => 1
00 => ([(0,2),(2,1)],3)
 => [3]
 => [1,1,1]
 => 2
01 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => [3,1]
 => [2,1,1]
 => 1
10 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => [3,1]
 => [2,1,1]
 => 1
11 => ([(0,2),(2,1)],3)
 => [3]
 => [1,1,1]
 => 2
000 => ([(0,3),(2,1),(3,2)],4)
 => [4]
 => [1,1,1,1]
 => 3
001 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => [4,2]
 => [2,2,1,1]
 => 2
010 => ([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)
 => [4,2]
 => [2,2,1,1]
 => 2
011 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => [4,2]
 => [2,2,1,1]
 => 2
100 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => [4,2]
 => [2,2,1,1]
 => 2
101 => ([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)
 => [4,2]
 => [2,2,1,1]
 => 2
110 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => [4,2]
 => [2,2,1,1]
 => 2
111 => ([(0,3),(2,1),(3,2)],4)
 => [4]
 => [1,1,1,1]
 => 3
0000 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => [5]
 => [1,1,1,1,1]
 => 4
0001 => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
 => [5,3]
 => [2,2,2,1,1]
 => 3
0010 => ([(0,2),(0,3),(1,6),(2,7),(2,8),(3,1),(3,7),(3,8),(5,4),(6,4),(7,5),(8,5),(8,6)],9)
 => [5,3,1]
 => [3,2,2,1,1]
 => 2
0011 => ([(0,3),(0,4),(1,7),(2,6),(3,2),(3,5),(4,1),(4,5),(5,6),(5,7),(6,8),(7,8)],9)
 => [5,3,1]
 => [3,2,2,1,1]
 => 2
0100 => ([(0,2),(0,3),(1,6),(2,7),(2,8),(3,1),(3,7),(3,8),(5,4),(6,4),(7,5),(8,5),(8,6)],9)
 => [5,3,1]
 => [3,2,2,1,1]
 => 2
0101 => ([(0,1),(0,2),(1,6),(1,7),(2,6),(2,7),(4,3),(5,3),(6,4),(6,5),(7,4),(7,5)],8)
 => [5,3]
 => [2,2,2,1,1]
 => 3
0110 => ([(0,2),(0,3),(1,5),(1,6),(2,7),(2,8),(3,1),(3,7),(3,8),(5,4),(6,4),(7,6),(8,5)],9)
 => [5,3,1]
 => [3,2,2,1,1]
 => 2
0111 => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
 => [5,3]
 => [2,2,2,1,1]
 => 3
1000 => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
 => [5,3]
 => [2,2,2,1,1]
 => 3
1001 => ([(0,2),(0,3),(1,5),(1,6),(2,7),(2,8),(3,1),(3,7),(3,8),(5,4),(6,4),(7,6),(8,5)],9)
 => [5,3,1]
 => [3,2,2,1,1]
 => 2
1010 => ([(0,1),(0,2),(1,6),(1,7),(2,6),(2,7),(4,3),(5,3),(6,4),(6,5),(7,4),(7,5)],8)
 => [5,3]
 => [2,2,2,1,1]
 => 3
1011 => ([(0,2),(0,3),(1,6),(2,7),(2,8),(3,1),(3,7),(3,8),(5,4),(6,4),(7,5),(8,5),(8,6)],9)
 => [5,3,1]
 => [3,2,2,1,1]
 => 2
1100 => ([(0,3),(0,4),(1,7),(2,6),(3,2),(3,5),(4,1),(4,5),(5,6),(5,7),(6,8),(7,8)],9)
 => [5,3,1]
 => [3,2,2,1,1]
 => 2
1101 => ([(0,2),(0,3),(1,6),(2,7),(2,8),(3,1),(3,7),(3,8),(5,4),(6,4),(7,5),(8,5),(8,6)],9)
 => [5,3,1]
 => [3,2,2,1,1]
 => 2
1110 => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
 => [5,3]
 => [2,2,2,1,1]
 => 3
1111 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => [5]
 => [1,1,1,1,1]
 => 4
00000 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => [6]
 => [1,1,1,1,1,1]
 => 5
00001 => ([(0,2),(0,5),(1,7),(2,6),(3,4),(3,9),(4,1),(4,8),(5,3),(5,6),(6,9),(8,7),(9,8)],10)
 => [6,4]
 => [2,2,2,2,1,1]
 => 4
00010 => ([(0,3),(0,4),(1,2),(1,11),(2,8),(3,9),(3,10),(4,1),(4,9),(4,10),(6,7),(7,5),(8,5),(9,6),(10,6),(10,11),(11,7),(11,8)],12)
 => [6,4,2]
 => [3,3,2,2,1,1]
 => 3
00011 => ([(0,4),(0,5),(1,9),(2,3),(2,11),(3,8),(4,1),(4,10),(5,2),(5,10),(7,6),(8,6),(9,7),(10,9),(10,11),(11,7),(11,8)],12)
 => [6,4,2]
 => [3,3,2,2,1,1]
 => 3
00100 => ([(0,2),(0,3),(1,5),(1,6),(2,10),(2,11),(3,1),(3,10),(3,11),(5,8),(6,7),(7,4),(8,4),(9,7),(9,8),(10,6),(10,9),(11,5),(11,9)],12)
 => [6,4,2]
 => [3,3,2,2,1,1]
 => 3
00101 => ([(0,2),(0,3),(1,8),(2,10),(2,11),(3,1),(3,10),(3,11),(5,6),(6,4),(7,4),(8,7),(9,6),(9,7),(10,5),(10,9),(11,5),(11,8),(11,9)],12)
 => [6,4,2]
 => [3,3,2,2,1,1]
 => 3
00110 => ([(0,3),(0,4),(1,9),(2,6),(2,11),(3,2),(3,10),(3,12),(4,1),(4,10),(4,12),(6,7),(7,5),(8,5),(9,8),(10,6),(11,7),(11,8),(12,9),(12,11)],13)
 => [6,4,2,1]
 => [4,3,2,2,1,1]
 => ? = 2
00111 => ([(0,4),(0,5),(1,9),(2,3),(2,11),(3,8),(4,1),(4,10),(5,2),(5,10),(7,6),(8,6),(9,7),(10,9),(10,11),(11,7),(11,8)],12)
 => [6,4,2]
 => [3,3,2,2,1,1]
 => 3
01000 => ([(0,3),(0,4),(1,2),(1,11),(2,8),(3,9),(3,10),(4,1),(4,9),(4,10),(6,7),(7,5),(8,5),(9,6),(10,6),(10,11),(11,7),(11,8)],12)
 => [6,4,2]
 => [3,3,2,2,1,1]
 => 3
01001 => ([(0,2),(0,3),(1,5),(1,9),(2,10),(2,11),(3,1),(3,10),(3,11),(5,7),(6,8),(7,4),(8,4),(9,7),(9,8),(10,5),(10,6),(11,6),(11,9)],12)
 => [6,4,2]
 => [3,3,2,2,1,1]
 => 3
01010 => ([(0,1),(0,2),(1,8),(1,9),(2,8),(2,9),(4,3),(5,3),(6,4),(6,5),(7,4),(7,5),(8,6),(8,7),(9,6),(9,7)],10)
 => [6,4]
 => [2,2,2,2,1,1]
 => 4
01011 => ([(0,2),(0,3),(1,8),(2,10),(2,11),(3,1),(3,10),(3,11),(5,6),(6,4),(7,4),(8,7),(9,6),(9,7),(10,5),(10,9),(11,5),(11,8),(11,9)],12)
 => [6,4,2]
 => [3,3,2,2,1,1]
 => 3
01100 => ([(0,3),(0,4),(1,9),(2,6),(2,11),(3,2),(3,10),(3,12),(4,1),(4,10),(4,12),(6,7),(7,5),(8,5),(9,8),(10,6),(11,7),(11,8),(12,9),(12,11)],13)
 => [6,4,2,1]
 => [4,3,2,2,1,1]
 => ? = 2
01101 => ([(0,2),(0,3),(1,5),(1,9),(2,10),(2,11),(3,1),(3,10),(3,11),(5,7),(6,8),(7,4),(8,4),(9,7),(9,8),(10,5),(10,6),(11,6),(11,9)],12)
 => [6,4,2]
 => [3,3,2,2,1,1]
 => 3
01110 => ([(0,3),(0,4),(1,2),(1,10),(1,11),(2,8),(2,9),(3,6),(3,7),(4,1),(4,6),(4,7),(6,11),(7,10),(8,5),(9,5),(10,8),(11,9)],12)
 => [6,4,2]
 => [3,3,2,2,1,1]
 => 3
01111 => ([(0,2),(0,5),(1,7),(2,6),(3,4),(3,9),(4,1),(4,8),(5,3),(5,6),(6,9),(8,7),(9,8)],10)
 => [6,4]
 => [2,2,2,2,1,1]
 => 4
10000 => ([(0,2),(0,5),(1,7),(2,6),(3,4),(3,9),(4,1),(4,8),(5,3),(5,6),(6,9),(8,7),(9,8)],10)
 => [6,4]
 => [2,2,2,2,1,1]
 => 4
10001 => ([(0,3),(0,4),(1,2),(1,10),(1,11),(2,8),(2,9),(3,6),(3,7),(4,1),(4,6),(4,7),(6,11),(7,10),(8,5),(9,5),(10,8),(11,9)],12)
 => [6,4,2]
 => [3,3,2,2,1,1]
 => 3
10010 => ([(0,2),(0,3),(1,5),(1,9),(2,10),(2,11),(3,1),(3,10),(3,11),(5,7),(6,8),(7,4),(8,4),(9,7),(9,8),(10,5),(10,6),(11,6),(11,9)],12)
 => [6,4,2]
 => [3,3,2,2,1,1]
 => 3
10011 => ([(0,3),(0,4),(1,9),(2,6),(2,11),(3,2),(3,10),(3,12),(4,1),(4,10),(4,12),(6,7),(7,5),(8,5),(9,8),(10,6),(11,7),(11,8),(12,9),(12,11)],13)
 => [6,4,2,1]
 => [4,3,2,2,1,1]
 => ? = 2
10100 => ([(0,2),(0,3),(1,8),(2,10),(2,11),(3,1),(3,10),(3,11),(5,6),(6,4),(7,4),(8,7),(9,6),(9,7),(10,5),(10,9),(11,5),(11,8),(11,9)],12)
 => [6,4,2]
 => [3,3,2,2,1,1]
 => 3
10101 => ([(0,1),(0,2),(1,8),(1,9),(2,8),(2,9),(4,3),(5,3),(6,4),(6,5),(7,4),(7,5),(8,6),(8,7),(9,6),(9,7)],10)
 => [6,4]
 => [2,2,2,2,1,1]
 => 4
10110 => ([(0,2),(0,3),(1,5),(1,9),(2,10),(2,11),(3,1),(3,10),(3,11),(5,7),(6,8),(7,4),(8,4),(9,7),(9,8),(10,5),(10,6),(11,6),(11,9)],12)
 => [6,4,2]
 => [3,3,2,2,1,1]
 => 3
11001 => ([(0,3),(0,4),(1,9),(2,6),(2,11),(3,2),(3,10),(3,12),(4,1),(4,10),(4,12),(6,7),(7,5),(8,5),(9,8),(10,6),(11,7),(11,8),(12,9),(12,11)],13)
 => [6,4,2,1]
 => [4,3,2,2,1,1]
 => ? = 2
000010 => ([(0,4),(0,5),(1,3),(1,12),(2,11),(3,2),(3,14),(4,10),(4,13),(5,1),(5,10),(5,13),(7,8),(8,9),(9,6),(10,7),(11,6),(12,8),(12,14),(13,7),(13,12),(14,9),(14,11)],15)
 => [7,5,3]
 => [3,3,3,2,2,1,1]
 => ? = 4
000011 => ([(0,5),(0,6),(1,4),(1,14),(2,11),(3,10),(4,3),(4,12),(5,1),(5,13),(6,2),(6,13),(8,9),(9,7),(10,7),(11,8),(12,9),(12,10),(13,11),(13,14),(14,8),(14,12)],15)
 => [7,5,3]
 => [3,3,3,2,2,1,1]
 => ? = 4
000100 => ([(0,3),(0,4),(1,10),(2,1),(2,6),(2,12),(3,14),(3,15),(4,2),(4,14),(4,15),(6,7),(7,8),(8,5),(9,5),(10,9),(11,7),(11,13),(12,10),(12,13),(13,8),(13,9),(14,6),(14,11),(15,11),(15,12)],16)
 => [7,5,3,1]
 => [4,3,3,2,2,1,1]
 => ? = 3
000101 => ([(0,3),(0,4),(1,2),(1,14),(2,6),(3,13),(3,15),(4,1),(4,13),(4,15),(6,9),(7,8),(8,10),(9,5),(10,5),(11,8),(11,12),(12,9),(12,10),(13,7),(13,11),(14,6),(14,12),(15,7),(15,11),(15,14)],16)
 => [7,5,3,1]
 => [4,3,3,2,2,1,1]
 => ? = 3
000110 => ([(0,4),(0,5),(1,11),(2,1),(2,13),(3,7),(3,14),(4,2),(4,12),(4,16),(5,3),(5,12),(5,16),(7,8),(8,9),(9,6),(10,6),(11,10),(12,7),(13,11),(13,15),(14,8),(14,15),(15,9),(15,10),(16,13),(16,14)],17)
 => [7,5,3,2]
 => [4,4,3,2,2,1,1]
 => ? = 3
000111 => ([(0,5),(0,6),(1,4),(1,15),(2,3),(2,14),(3,8),(4,9),(5,2),(5,13),(6,1),(6,13),(8,10),(9,11),(10,7),(11,7),(12,10),(12,11),(13,14),(13,15),(14,8),(14,12),(15,9),(15,12)],16)
 => [7,5,3,1]
 => [4,3,3,2,2,1,1]
 => ? = 3
001000 => ([(0,3),(0,4),(1,10),(2,1),(2,6),(2,12),(3,14),(3,15),(4,2),(4,14),(4,15),(6,7),(7,8),(8,5),(9,5),(10,9),(11,7),(11,13),(12,10),(12,13),(13,8),(13,9),(14,6),(14,11),(15,11),(15,12)],16)
 => [7,5,3,1]
 => [4,3,3,2,2,1,1]
 => ? = 3
001001 => ([(0,2),(0,3),(1,11),(1,12),(2,13),(2,14),(3,1),(3,13),(3,14),(5,7),(6,8),(7,4),(8,4),(9,7),(9,8),(10,5),(10,9),(11,6),(11,9),(12,5),(12,6),(13,10),(13,11),(14,10),(14,12)],15)
 => [7,5,3]
 => [3,3,3,2,2,1,1]
 => ? = 4
001010 => ([(0,2),(0,3),(1,9),(2,12),(2,14),(3,1),(3,12),(3,14),(5,7),(6,8),(7,4),(8,4),(9,5),(10,6),(10,11),(11,7),(11,8),(12,10),(12,13),(13,5),(13,6),(13,11),(14,9),(14,10),(14,13)],15)
 => [7,5,3]
 => [3,3,3,2,2,1,1]
 => ? = 4
001011 => ([(0,3),(0,4),(1,11),(2,10),(3,2),(3,15),(3,16),(4,1),(4,15),(4,16),(6,8),(7,9),(8,5),(9,5),(10,6),(11,7),(12,6),(12,14),(13,7),(13,14),(14,8),(14,9),(15,12),(15,13),(16,10),(16,11),(16,12),(16,13)],17)
 => [7,5,3,2]
 => [4,4,3,2,2,1,1]
 => ? = 3
001100 => ([(0,3),(0,4),(1,15),(1,16),(2,10),(2,11),(3,1),(3,13),(3,14),(4,2),(4,13),(4,14),(6,9),(7,8),(8,5),(9,5),(10,7),(11,6),(12,8),(12,9),(13,10),(13,15),(14,11),(14,16),(15,7),(15,12),(16,6),(16,12)],17)
 => [7,5,3,2]
 => [4,4,3,2,2,1,1]
 => ? = 3
001101 => ([(0,3),(0,4),(1,11),(2,12),(2,13),(3,2),(3,15),(3,16),(4,1),(4,15),(4,16),(6,7),(7,9),(8,10),(9,5),(10,5),(11,8),(12,7),(12,14),(13,8),(13,14),(14,9),(14,10),(15,6),(15,12),(16,6),(16,11),(16,13)],17)
 => [7,5,3,2]
 => [4,4,3,2,2,1,1]
 => ? = 3
001110 => ([(0,4),(0,5),(1,12),(2,3),(2,13),(2,16),(3,8),(3,14),(4,1),(4,9),(4,15),(5,2),(5,9),(5,15),(7,10),(8,11),(9,13),(10,6),(11,6),(12,7),(13,8),(14,10),(14,11),(15,12),(15,16),(16,7),(16,14)],17)
 => [7,5,3,2]
 => [4,4,3,2,2,1,1]
 => ? = 3
001111 => ([(0,5),(0,6),(1,4),(1,14),(2,11),(3,10),(4,3),(4,12),(5,1),(5,13),(6,2),(6,13),(8,9),(9,7),(10,7),(11,8),(12,9),(12,10),(13,11),(13,14),(14,8),(14,12)],15)
 => [7,5,3]
 => [3,3,3,2,2,1,1]
 => ? = 4
010000 => ([(0,4),(0,5),(1,3),(1,12),(2,11),(3,2),(3,14),(4,10),(4,13),(5,1),(5,10),(5,13),(7,8),(8,9),(9,6),(10,7),(11,6),(12,8),(12,14),(13,7),(13,12),(14,9),(14,11)],15)
 => [7,5,3]
 => [3,3,3,2,2,1,1]
 => ? = 4
010001 => ([(0,3),(0,4),(1,2),(1,11),(1,15),(2,7),(2,12),(3,13),(3,14),(4,1),(4,13),(4,14),(6,9),(7,10),(8,6),(9,5),(10,5),(11,7),(12,9),(12,10),(13,8),(13,15),(14,8),(14,11),(15,6),(15,12)],16)
 => [7,5,3,1]
 => [4,3,3,2,2,1,1]
 => ? = 3
010010 => ([(0,2),(0,3),(1,10),(1,11),(2,13),(2,14),(3,1),(3,13),(3,14),(5,8),(6,7),(7,4),(8,4),(9,7),(9,8),(10,6),(10,9),(11,5),(11,9),(12,5),(12,6),(13,10),(13,12),(14,11),(14,12)],15)
 => [7,5,3]
 => [3,3,3,2,2,1,1]
 => ? = 4
010011 => ([(0,3),(0,4),(1,11),(2,12),(2,13),(3,2),(3,15),(3,16),(4,1),(4,15),(4,16),(6,7),(7,9),(8,10),(9,5),(10,5),(11,8),(12,7),(12,14),(13,8),(13,14),(14,9),(14,10),(15,6),(15,12),(16,6),(16,11),(16,13)],17)
 => [7,5,3,2]
 => [4,4,3,2,2,1,1]
 => ? = 3
010100 => ([(0,2),(0,3),(1,9),(2,12),(2,14),(3,1),(3,12),(3,14),(5,7),(6,8),(7,4),(8,4),(9,5),(10,6),(10,11),(11,7),(11,8),(12,10),(12,13),(13,5),(13,6),(13,11),(14,9),(14,10),(14,13)],15)
 => [7,5,3]
 => [3,3,3,2,2,1,1]
 => ? = 4
010110 => ([(0,2),(0,3),(1,6),(1,11),(2,14),(2,15),(3,1),(3,14),(3,15),(5,8),(6,7),(7,9),(8,10),(9,4),(10,4),(11,7),(11,13),(12,8),(12,13),(13,9),(13,10),(14,5),(14,6),(14,12),(15,5),(15,11),(15,12)],16)
 => [7,5,3,1]
 => [4,3,3,2,2,1,1]
 => ? = 3
010111 => ([(0,3),(0,4),(1,2),(1,14),(2,6),(3,13),(3,15),(4,1),(4,13),(4,15),(6,9),(7,8),(8,10),(9,5),(10,5),(11,8),(11,12),(12,9),(12,10),(13,7),(13,11),(14,6),(14,12),(15,7),(15,11),(15,14)],16)
 => [7,5,3,1]
 => [4,3,3,2,2,1,1]
 => ? = 3
011000 => ([(0,4),(0,5),(1,11),(2,1),(2,13),(3,7),(3,14),(4,2),(4,12),(4,16),(5,3),(5,12),(5,16),(7,8),(8,9),(9,6),(10,6),(11,10),(12,7),(13,11),(13,15),(14,8),(14,15),(15,9),(15,10),(16,13),(16,14)],17)
 => [7,5,3,2]
 => [4,4,3,2,2,1,1]
 => ? = 3
011001 => ([(0,3),(0,4),(1,11),(1,16),(2,10),(2,15),(3,2),(3,13),(3,14),(4,1),(4,13),(4,14),(6,8),(7,9),(8,5),(9,5),(10,6),(11,7),(12,8),(12,9),(13,15),(13,16),(14,10),(14,11),(15,6),(15,12),(16,7),(16,12)],17)
 => [7,5,3,2]
 => [4,4,3,2,2,1,1]
 => ? = 3
011010 => ([(0,2),(0,3),(1,6),(1,11),(2,14),(2,15),(3,1),(3,14),(3,15),(5,8),(6,7),(7,9),(8,10),(9,4),(10,4),(11,7),(11,13),(12,8),(12,13),(13,9),(13,10),(14,5),(14,6),(14,12),(15,5),(15,11),(15,12)],16)
 => [7,5,3,1]
 => [4,3,3,2,2,1,1]
 => ? = 3
011011 => ([(0,2),(0,3),(1,11),(1,12),(2,13),(2,14),(3,1),(3,13),(3,14),(5,7),(6,8),(7,4),(8,4),(9,7),(9,8),(10,5),(10,9),(11,6),(11,9),(12,5),(12,6),(13,10),(13,11),(14,10),(14,12)],15)
 => [7,5,3]
 => [3,3,3,2,2,1,1]
 => ? = 4
011100 => ([(0,4),(0,5),(1,12),(2,3),(2,13),(2,16),(3,8),(3,14),(4,1),(4,9),(4,15),(5,2),(5,9),(5,15),(7,10),(8,11),(9,13),(10,6),(11,6),(12,7),(13,8),(14,10),(14,11),(15,12),(15,16),(16,7),(16,14)],17)
 => [7,5,3,2]
 => [4,4,3,2,2,1,1]
 => ? = 3
011101 => ([(0,3),(0,4),(1,2),(1,11),(1,15),(2,7),(2,12),(3,13),(3,14),(4,1),(4,13),(4,14),(6,9),(7,10),(8,6),(9,5),(10,5),(11,7),(12,9),(12,10),(13,8),(13,15),(14,8),(14,11),(15,6),(15,12)],16)
 => [7,5,3,1]
 => [4,3,3,2,2,1,1]
 => ? = 3
011110 => ([(0,4),(0,5),(1,3),(1,7),(1,8),(2,13),(2,14),(3,2),(3,11),(3,12),(4,9),(4,10),(5,1),(5,9),(5,10),(7,12),(8,11),(9,8),(10,7),(11,13),(12,14),(13,6),(14,6)],15)
 => [7,5,3]
 => [3,3,3,2,2,1,1]
 => ? = 4
100001 => ([(0,4),(0,5),(1,3),(1,7),(1,8),(2,13),(2,14),(3,2),(3,11),(3,12),(4,9),(4,10),(5,1),(5,9),(5,10),(7,12),(8,11),(9,8),(10,7),(11,13),(12,14),(13,6),(14,6)],15)
 => [7,5,3]
 => [3,3,3,2,2,1,1]
 => ? = 4
100010 => ([(0,3),(0,4),(1,2),(1,11),(1,15),(2,7),(2,12),(3,13),(3,14),(4,1),(4,13),(4,14),(6,9),(7,10),(8,6),(9,5),(10,5),(11,7),(12,9),(12,10),(13,8),(13,15),(14,8),(14,11),(15,6),(15,12)],16)
 => [7,5,3,1]
 => [4,3,3,2,2,1,1]
 => ? = 3
100011 => ([(0,4),(0,5),(1,12),(2,3),(2,13),(2,16),(3,8),(3,14),(4,1),(4,9),(4,15),(5,2),(5,9),(5,15),(7,10),(8,11),(9,13),(10,6),(11,6),(12,7),(13,8),(14,10),(14,11),(15,12),(15,16),(16,7),(16,14)],17)
 => [7,5,3,2]
 => [4,4,3,2,2,1,1]
 => ? = 3
100100 => ([(0,2),(0,3),(1,11),(1,12),(2,13),(2,14),(3,1),(3,13),(3,14),(5,7),(6,8),(7,4),(8,4),(9,7),(9,8),(10,5),(10,9),(11,6),(11,9),(12,5),(12,6),(13,10),(13,11),(14,10),(14,12)],15)
 => [7,5,3]
 => [3,3,3,2,2,1,1]
 => ? = 4
100101 => ([(0,2),(0,3),(1,6),(1,11),(2,14),(2,15),(3,1),(3,14),(3,15),(5,8),(6,7),(7,9),(8,10),(9,4),(10,4),(11,7),(11,13),(12,8),(12,13),(13,9),(13,10),(14,5),(14,6),(14,12),(15,5),(15,11),(15,12)],16)
 => [7,5,3,1]
 => [4,3,3,2,2,1,1]
 => ? = 3
100110 => ([(0,3),(0,4),(1,11),(1,16),(2,10),(2,15),(3,2),(3,13),(3,14),(4,1),(4,13),(4,14),(6,8),(7,9),(8,5),(9,5),(10,6),(11,7),(12,8),(12,9),(13,15),(13,16),(14,10),(14,11),(15,6),(15,12),(16,7),(16,12)],17)
 => [7,5,3,2]
 => [4,4,3,2,2,1,1]
 => ? = 3
100111 => ([(0,4),(0,5),(1,11),(2,1),(2,13),(3,7),(3,14),(4,2),(4,12),(4,16),(5,3),(5,12),(5,16),(7,8),(8,9),(9,6),(10,6),(11,10),(12,7),(13,11),(13,15),(14,8),(14,15),(15,9),(15,10),(16,13),(16,14)],17)
 => [7,5,3,2]
 => [4,4,3,2,2,1,1]
 => ? = 3
101000 => ([(0,3),(0,4),(1,2),(1,14),(2,6),(3,13),(3,15),(4,1),(4,13),(4,15),(6,9),(7,8),(8,10),(9,5),(10,5),(11,8),(11,12),(12,9),(12,10),(13,7),(13,11),(14,6),(14,12),(15,7),(15,11),(15,14)],16)
 => [7,5,3,1]
 => [4,3,3,2,2,1,1]
 => ? = 3
101001 => ([(0,2),(0,3),(1,6),(1,11),(2,14),(2,15),(3,1),(3,14),(3,15),(5,8),(6,7),(7,9),(8,10),(9,4),(10,4),(11,7),(11,13),(12,8),(12,13),(13,9),(13,10),(14,5),(14,6),(14,12),(15,5),(15,11),(15,12)],16)
 => [7,5,3,1]
 => [4,3,3,2,2,1,1]
 => ? = 3
101011 => ([(0,2),(0,3),(1,9),(2,12),(2,14),(3,1),(3,12),(3,14),(5,7),(6,8),(7,4),(8,4),(9,5),(10,6),(10,11),(11,7),(11,8),(12,10),(12,13),(13,5),(13,6),(13,11),(14,9),(14,10),(14,13)],15)
 => [7,5,3]
 => [3,3,3,2,2,1,1]
 => ? = 4
101100 => ([(0,3),(0,4),(1,11),(2,12),(2,13),(3,2),(3,15),(3,16),(4,1),(4,15),(4,16),(6,7),(7,9),(8,10),(9,5),(10,5),(11,8),(12,7),(12,14),(13,8),(13,14),(14,9),(14,10),(15,6),(15,12),(16,6),(16,11),(16,13)],17)
 => [7,5,3,2]
 => [4,4,3,2,2,1,1]
 => ? = 3
101101 => ([(0,2),(0,3),(1,10),(1,11),(2,13),(2,14),(3,1),(3,13),(3,14),(5,8),(6,7),(7,4),(8,4),(9,7),(9,8),(10,6),(10,9),(11,5),(11,9),(12,5),(12,6),(13,10),(13,12),(14,11),(14,12)],15)
 => [7,5,3]
 => [3,3,3,2,2,1,1]
 => ? = 4
101110 => ([(0,3),(0,4),(1,2),(1,11),(1,15),(2,7),(2,12),(3,13),(3,14),(4,1),(4,13),(4,14),(6,9),(7,10),(8,6),(9,5),(10,5),(11,7),(12,9),(12,10),(13,8),(13,15),(14,8),(14,11),(15,6),(15,12)],16)
 => [7,5,3,1]
 => [4,3,3,2,2,1,1]
 => ? = 3
101111 => ([(0,4),(0,5),(1,3),(1,12),(2,11),(3,2),(3,14),(4,10),(4,13),(5,1),(5,10),(5,13),(7,8),(8,9),(9,6),(10,7),(11,6),(12,8),(12,14),(13,7),(13,12),(14,9),(14,11)],15)
 => [7,5,3]
 => [3,3,3,2,2,1,1]
 => ? = 4
110000 => ([(0,5),(0,6),(1,4),(1,14),(2,11),(3,10),(4,3),(4,12),(5,1),(5,13),(6,2),(6,13),(8,9),(9,7),(10,7),(11,8),(12,9),(12,10),(13,11),(13,14),(14,8),(14,12)],15)
 => [7,5,3]
 => [3,3,3,2,2,1,1]
 => ? = 4
110001 => ([(0,4),(0,5),(1,12),(2,3),(2,13),(2,16),(3,8),(3,14),(4,1),(4,9),(4,15),(5,2),(5,9),(5,15),(7,10),(8,11),(9,13),(10,6),(11,6),(12,7),(13,8),(14,10),(14,11),(15,12),(15,16),(16,7),(16,14)],17)
 => [7,5,3,2]
 => [4,4,3,2,2,1,1]
 => ? = 3
110010 => ([(0,3),(0,4),(1,11),(2,12),(2,13),(3,2),(3,15),(3,16),(4,1),(4,15),(4,16),(6,7),(7,9),(8,10),(9,5),(10,5),(11,8),(12,7),(12,14),(13,8),(13,14),(14,9),(14,10),(15,6),(15,12),(16,6),(16,11),(16,13)],17)
 => [7,5,3,2]
 => [4,4,3,2,2,1,1]
 => ? = 3
110011 => ([(0,3),(0,4),(1,15),(1,16),(2,10),(2,11),(3,1),(3,13),(3,14),(4,2),(4,13),(4,14),(6,9),(7,8),(8,5),(9,5),(10,7),(11,6),(12,8),(12,9),(13,10),(13,15),(14,11),(14,16),(15,7),(15,12),(16,6),(16,12)],17)
 => [7,5,3,2]
 => [4,4,3,2,2,1,1]
 => ? = 3
Description
The number of parts in an integer partition whose next smaller part has the same size. In other words, this is the number of distinct parts subtracted from the number of all parts.
Matching statistic: St000145
Mapping
Mp00262: Binary words —poset of factors⟶ Posets
Mp00110: Posets —Greene-Kleitman invariant⟶ Integer partitions
St000145: Integer partitions ⟶ ℤResult quality: 4% ●values known / values provided: 4%●distinct values known / distinct values provided: 90%
Values
0 => ([(0,1)],2)
 => [2]
 => 1
1 => ([(0,1)],2)
 => [2]
 => 1
00 => ([(0,2),(2,1)],3)
 => [3]
 => 2
01 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => [3,1]
 => 1
10 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => [3,1]
 => 1
11 => ([(0,2),(2,1)],3)
 => [3]
 => 2
000 => ([(0,3),(2,1),(3,2)],4)
 => [4]
 => 3
001 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => [4,2]
 => 2
010 => ([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)
 => [4,2]
 => 2
011 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => [4,2]
 => 2
100 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => [4,2]
 => 2
101 => ([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)
 => [4,2]
 => 2
110 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => [4,2]
 => 2
111 => ([(0,3),(2,1),(3,2)],4)
 => [4]
 => 3
0000 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => [5]
 => 4
0001 => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
 => [5,3]
 => 3
0010 => ([(0,2),(0,3),(1,6),(2,7),(2,8),(3,1),(3,7),(3,8),(5,4),(6,4),(7,5),(8,5),(8,6)],9)
 => [5,3,1]
 => 2
0011 => ([(0,3),(0,4),(1,7),(2,6),(3,2),(3,5),(4,1),(4,5),(5,6),(5,7),(6,8),(7,8)],9)
 => [5,3,1]
 => 2
0100 => ([(0,2),(0,3),(1,6),(2,7),(2,8),(3,1),(3,7),(3,8),(5,4),(6,4),(7,5),(8,5),(8,6)],9)
 => [5,3,1]
 => 2
0101 => ([(0,1),(0,2),(1,6),(1,7),(2,6),(2,7),(4,3),(5,3),(6,4),(6,5),(7,4),(7,5)],8)
 => [5,3]
 => 3
0110 => ([(0,2),(0,3),(1,5),(1,6),(2,7),(2,8),(3,1),(3,7),(3,8),(5,4),(6,4),(7,6),(8,5)],9)
 => [5,3,1]
 => 2
0111 => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
 => [5,3]
 => 3
1000 => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
 => [5,3]
 => 3
1001 => ([(0,2),(0,3),(1,5),(1,6),(2,7),(2,8),(3,1),(3,7),(3,8),(5,4),(6,4),(7,6),(8,5)],9)
 => [5,3,1]
 => 2
1010 => ([(0,1),(0,2),(1,6),(1,7),(2,6),(2,7),(4,3),(5,3),(6,4),(6,5),(7,4),(7,5)],8)
 => [5,3]
 => 3
1011 => ([(0,2),(0,3),(1,6),(2,7),(2,8),(3,1),(3,7),(3,8),(5,4),(6,4),(7,5),(8,5),(8,6)],9)
 => [5,3,1]
 => 2
1100 => ([(0,3),(0,4),(1,7),(2,6),(3,2),(3,5),(4,1),(4,5),(5,6),(5,7),(6,8),(7,8)],9)
 => [5,3,1]
 => 2
1101 => ([(0,2),(0,3),(1,6),(2,7),(2,8),(3,1),(3,7),(3,8),(5,4),(6,4),(7,5),(8,5),(8,6)],9)
 => [5,3,1]
 => 2
1110 => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
 => [5,3]
 => 3
1111 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => [5]
 => 4
00000 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => [6]
 => 5
00001 => ([(0,2),(0,5),(1,7),(2,6),(3,4),(3,9),(4,1),(4,8),(5,3),(5,6),(6,9),(8,7),(9,8)],10)
 => [6,4]
 => 4
00010 => ([(0,3),(0,4),(1,2),(1,11),(2,8),(3,9),(3,10),(4,1),(4,9),(4,10),(6,7),(7,5),(8,5),(9,6),(10,6),(10,11),(11,7),(11,8)],12)
 => [6,4,2]
 => ? = 3
00011 => ([(0,4),(0,5),(1,9),(2,3),(2,11),(3,8),(4,1),(4,10),(5,2),(5,10),(7,6),(8,6),(9,7),(10,9),(10,11),(11,7),(11,8)],12)
 => [6,4,2]
 => ? = 3
00100 => ([(0,2),(0,3),(1,5),(1,6),(2,10),(2,11),(3,1),(3,10),(3,11),(5,8),(6,7),(7,4),(8,4),(9,7),(9,8),(10,6),(10,9),(11,5),(11,9)],12)
 => [6,4,2]
 => ? = 3
00101 => ([(0,2),(0,3),(1,8),(2,10),(2,11),(3,1),(3,10),(3,11),(5,6),(6,4),(7,4),(8,7),(9,6),(9,7),(10,5),(10,9),(11,5),(11,8),(11,9)],12)
 => [6,4,2]
 => ? = 3
00110 => ([(0,3),(0,4),(1,9),(2,6),(2,11),(3,2),(3,10),(3,12),(4,1),(4,10),(4,12),(6,7),(7,5),(8,5),(9,8),(10,6),(11,7),(11,8),(12,9),(12,11)],13)
 => [6,4,2,1]
 => ? = 2
00111 => ([(0,4),(0,5),(1,9),(2,3),(2,11),(3,8),(4,1),(4,10),(5,2),(5,10),(7,6),(8,6),(9,7),(10,9),(10,11),(11,7),(11,8)],12)
 => [6,4,2]
 => ? = 3
01000 => ([(0,3),(0,4),(1,2),(1,11),(2,8),(3,9),(3,10),(4,1),(4,9),(4,10),(6,7),(7,5),(8,5),(9,6),(10,6),(10,11),(11,7),(11,8)],12)
 => [6,4,2]
 => ? = 3
01001 => ([(0,2),(0,3),(1,5),(1,9),(2,10),(2,11),(3,1),(3,10),(3,11),(5,7),(6,8),(7,4),(8,4),(9,7),(9,8),(10,5),(10,6),(11,6),(11,9)],12)
 => [6,4,2]
 => ? = 3
01010 => ([(0,1),(0,2),(1,8),(1,9),(2,8),(2,9),(4,3),(5,3),(6,4),(6,5),(7,4),(7,5),(8,6),(8,7),(9,6),(9,7)],10)
 => [6,4]
 => 4
01011 => ([(0,2),(0,3),(1,8),(2,10),(2,11),(3,1),(3,10),(3,11),(5,6),(6,4),(7,4),(8,7),(9,6),(9,7),(10,5),(10,9),(11,5),(11,8),(11,9)],12)
 => [6,4,2]
 => ? = 3
01100 => ([(0,3),(0,4),(1,9),(2,6),(2,11),(3,2),(3,10),(3,12),(4,1),(4,10),(4,12),(6,7),(7,5),(8,5),(9,8),(10,6),(11,7),(11,8),(12,9),(12,11)],13)
 => [6,4,2,1]
 => ? = 2
01101 => ([(0,2),(0,3),(1,5),(1,9),(2,10),(2,11),(3,1),(3,10),(3,11),(5,7),(6,8),(7,4),(8,4),(9,7),(9,8),(10,5),(10,6),(11,6),(11,9)],12)
 => [6,4,2]
 => ? = 3
01110 => ([(0,3),(0,4),(1,2),(1,10),(1,11),(2,8),(2,9),(3,6),(3,7),(4,1),(4,6),(4,7),(6,11),(7,10),(8,5),(9,5),(10,8),(11,9)],12)
 => [6,4,2]
 => ? = 3
01111 => ([(0,2),(0,5),(1,7),(2,6),(3,4),(3,9),(4,1),(4,8),(5,3),(5,6),(6,9),(8,7),(9,8)],10)
 => [6,4]
 => 4
10000 => ([(0,2),(0,5),(1,7),(2,6),(3,4),(3,9),(4,1),(4,8),(5,3),(5,6),(6,9),(8,7),(9,8)],10)
 => [6,4]
 => 4
10001 => ([(0,3),(0,4),(1,2),(1,10),(1,11),(2,8),(2,9),(3,6),(3,7),(4,1),(4,6),(4,7),(6,11),(7,10),(8,5),(9,5),(10,8),(11,9)],12)
 => [6,4,2]
 => ? = 3
10010 => ([(0,2),(0,3),(1,5),(1,9),(2,10),(2,11),(3,1),(3,10),(3,11),(5,7),(6,8),(7,4),(8,4),(9,7),(9,8),(10,5),(10,6),(11,6),(11,9)],12)
 => [6,4,2]
 => ? = 3
10011 => ([(0,3),(0,4),(1,9),(2,6),(2,11),(3,2),(3,10),(3,12),(4,1),(4,10),(4,12),(6,7),(7,5),(8,5),(9,8),(10,6),(11,7),(11,8),(12,9),(12,11)],13)
 => [6,4,2,1]
 => ? = 2
10100 => ([(0,2),(0,3),(1,8),(2,10),(2,11),(3,1),(3,10),(3,11),(5,6),(6,4),(7,4),(8,7),(9,6),(9,7),(10,5),(10,9),(11,5),(11,8),(11,9)],12)
 => [6,4,2]
 => ? = 3
10101 => ([(0,1),(0,2),(1,8),(1,9),(2,8),(2,9),(4,3),(5,3),(6,4),(6,5),(7,4),(7,5),(8,6),(8,7),(9,6),(9,7)],10)
 => [6,4]
 => 4
10110 => ([(0,2),(0,3),(1,5),(1,9),(2,10),(2,11),(3,1),(3,10),(3,11),(5,7),(6,8),(7,4),(8,4),(9,7),(9,8),(10,5),(10,6),(11,6),(11,9)],12)
 => [6,4,2]
 => ? = 3
10111 => ([(0,3),(0,4),(1,2),(1,11),(2,8),(3,9),(3,10),(4,1),(4,9),(4,10),(6,7),(7,5),(8,5),(9,6),(10,6),(10,11),(11,7),(11,8)],12)
 => [6,4,2]
 => ? = 3
11000 => ([(0,4),(0,5),(1,9),(2,3),(2,11),(3,8),(4,1),(4,10),(5,2),(5,10),(7,6),(8,6),(9,7),(10,9),(10,11),(11,7),(11,8)],12)
 => [6,4,2]
 => ? = 3
11001 => ([(0,3),(0,4),(1,9),(2,6),(2,11),(3,2),(3,10),(3,12),(4,1),(4,10),(4,12),(6,7),(7,5),(8,5),(9,8),(10,6),(11,7),(11,8),(12,9),(12,11)],13)
 => [6,4,2,1]
 => ? = 2
11010 => ([(0,2),(0,3),(1,8),(2,10),(2,11),(3,1),(3,10),(3,11),(5,6),(6,4),(7,4),(8,7),(9,6),(9,7),(10,5),(10,9),(11,5),(11,8),(11,9)],12)
 => [6,4,2]
 => ? = 3
11011 => ([(0,2),(0,3),(1,5),(1,6),(2,10),(2,11),(3,1),(3,10),(3,11),(5,8),(6,7),(7,4),(8,4),(9,7),(9,8),(10,6),(10,9),(11,5),(11,9)],12)
 => [6,4,2]
 => ? = 3
11100 => ([(0,4),(0,5),(1,9),(2,3),(2,11),(3,8),(4,1),(4,10),(5,2),(5,10),(7,6),(8,6),(9,7),(10,9),(10,11),(11,7),(11,8)],12)
 => [6,4,2]
 => ? = 3
11101 => ([(0,3),(0,4),(1,2),(1,11),(2,8),(3,9),(3,10),(4,1),(4,9),(4,10),(6,7),(7,5),(8,5),(9,6),(10,6),(10,11),(11,7),(11,8)],12)
 => [6,4,2]
 => ? = 3
11110 => ([(0,2),(0,5),(1,7),(2,6),(3,4),(3,9),(4,1),(4,8),(5,3),(5,6),(6,9),(8,7),(9,8)],10)
 => [6,4]
 => 4
11111 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => [6]
 => 5
000000 => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
 => [7]
 => 6
000001 => ([(0,2),(0,6),(1,8),(2,7),(3,5),(3,9),(4,3),(4,11),(5,1),(5,10),(6,4),(6,7),(7,11),(9,10),(10,8),(11,9)],12)
 => [7,5]
 => ? = 5
000010 => ([(0,4),(0,5),(1,3),(1,12),(2,11),(3,2),(3,14),(4,10),(4,13),(5,1),(5,10),(5,13),(7,8),(8,9),(9,6),(10,7),(11,6),(12,8),(12,14),(13,7),(13,12),(14,9),(14,11)],15)
 => [7,5,3]
 => ? = 4
000011 => ([(0,5),(0,6),(1,4),(1,14),(2,11),(3,10),(4,3),(4,12),(5,1),(5,13),(6,2),(6,13),(8,9),(9,7),(10,7),(11,8),(12,9),(12,10),(13,11),(13,14),(14,8),(14,12)],15)
 => [7,5,3]
 => ? = 4
000100 => ([(0,3),(0,4),(1,10),(2,1),(2,6),(2,12),(3,14),(3,15),(4,2),(4,14),(4,15),(6,7),(7,8),(8,5),(9,5),(10,9),(11,7),(11,13),(12,10),(12,13),(13,8),(13,9),(14,6),(14,11),(15,11),(15,12)],16)
 => [7,5,3,1]
 => ? = 3
000101 => ([(0,3),(0,4),(1,2),(1,14),(2,6),(3,13),(3,15),(4,1),(4,13),(4,15),(6,9),(7,8),(8,10),(9,5),(10,5),(11,8),(11,12),(12,9),(12,10),(13,7),(13,11),(14,6),(14,12),(15,7),(15,11),(15,14)],16)
 => [7,5,3,1]
 => ? = 3
000110 => ([(0,4),(0,5),(1,11),(2,1),(2,13),(3,7),(3,14),(4,2),(4,12),(4,16),(5,3),(5,12),(5,16),(7,8),(8,9),(9,6),(10,6),(11,10),(12,7),(13,11),(13,15),(14,8),(14,15),(15,9),(15,10),(16,13),(16,14)],17)
 => [7,5,3,2]
 => ? = 3
000111 => ([(0,5),(0,6),(1,4),(1,15),(2,3),(2,14),(3,8),(4,9),(5,2),(5,13),(6,1),(6,13),(8,10),(9,11),(10,7),(11,7),(12,10),(12,11),(13,14),(13,15),(14,8),(14,12),(15,9),(15,12)],16)
 => [7,5,3,1]
 => ? = 3
001000 => ([(0,3),(0,4),(1,10),(2,1),(2,6),(2,12),(3,14),(3,15),(4,2),(4,14),(4,15),(6,7),(7,8),(8,5),(9,5),(10,9),(11,7),(11,13),(12,10),(12,13),(13,8),(13,9),(14,6),(14,11),(15,11),(15,12)],16)
 => [7,5,3,1]
 => ? = 3
001001 => ([(0,2),(0,3),(1,11),(1,12),(2,13),(2,14),(3,1),(3,13),(3,14),(5,7),(6,8),(7,4),(8,4),(9,7),(9,8),(10,5),(10,9),(11,6),(11,9),(12,5),(12,6),(13,10),(13,11),(14,10),(14,12)],15)
 => [7,5,3]
 => ? = 4
001010 => ([(0,2),(0,3),(1,9),(2,12),(2,14),(3,1),(3,12),(3,14),(5,7),(6,8),(7,4),(8,4),(9,5),(10,6),(10,11),(11,7),(11,8),(12,10),(12,13),(13,5),(13,6),(13,11),(14,9),(14,10),(14,13)],15)
 => [7,5,3]
 => ? = 4
001011 => ([(0,3),(0,4),(1,11),(2,10),(3,2),(3,15),(3,16),(4,1),(4,15),(4,16),(6,8),(7,9),(8,5),(9,5),(10,6),(11,7),(12,6),(12,14),(13,7),(13,14),(14,8),(14,9),(15,12),(15,13),(16,10),(16,11),(16,12),(16,13)],17)
 => [7,5,3,2]
 => ? = 3
001100 => ([(0,3),(0,4),(1,15),(1,16),(2,10),(2,11),(3,1),(3,13),(3,14),(4,2),(4,13),(4,14),(6,9),(7,8),(8,5),(9,5),(10,7),(11,6),(12,8),(12,9),(13,10),(13,15),(14,11),(14,16),(15,7),(15,12),(16,6),(16,12)],17)
 => [7,5,3,2]
 => ? = 3
001101 => ([(0,3),(0,4),(1,11),(2,12),(2,13),(3,2),(3,15),(3,16),(4,1),(4,15),(4,16),(6,7),(7,9),(8,10),(9,5),(10,5),(11,8),(12,7),(12,14),(13,8),(13,14),(14,9),(14,10),(15,6),(15,12),(16,6),(16,11),(16,13)],17)
 => [7,5,3,2]
 => ? = 3
001110 => ([(0,4),(0,5),(1,12),(2,3),(2,13),(2,16),(3,8),(3,14),(4,1),(4,9),(4,15),(5,2),(5,9),(5,15),(7,10),(8,11),(9,13),(10,6),(11,6),(12,7),(13,8),(14,10),(14,11),(15,12),(15,16),(16,7),(16,14)],17)
 => [7,5,3,2]
 => ? = 3
001111 => ([(0,5),(0,6),(1,4),(1,14),(2,11),(3,10),(4,3),(4,12),(5,1),(5,13),(6,2),(6,13),(8,9),(9,7),(10,7),(11,8),(12,9),(12,10),(13,11),(13,14),(14,8),(14,12)],15)
 => [7,5,3]
 => ? = 4
010000 => ([(0,4),(0,5),(1,3),(1,12),(2,11),(3,2),(3,14),(4,10),(4,13),(5,1),(5,10),(5,13),(7,8),(8,9),(9,6),(10,7),(11,6),(12,8),(12,14),(13,7),(13,12),(14,9),(14,11)],15)
 => [7,5,3]
 => ? = 4
010001 => ([(0,3),(0,4),(1,2),(1,11),(1,15),(2,7),(2,12),(3,13),(3,14),(4,1),(4,13),(4,14),(6,9),(7,10),(8,6),(9,5),(10,5),(11,7),(12,9),(12,10),(13,8),(13,15),(14,8),(14,11),(15,6),(15,12)],16)
 => [7,5,3,1]
 => ? = 3
010010 => ([(0,2),(0,3),(1,10),(1,11),(2,13),(2,14),(3,1),(3,13),(3,14),(5,8),(6,7),(7,4),(8,4),(9,7),(9,8),(10,6),(10,9),(11,5),(11,9),(12,5),(12,6),(13,10),(13,12),(14,11),(14,12)],15)
 => [7,5,3]
 => ? = 4
010011 => ([(0,3),(0,4),(1,11),(2,12),(2,13),(3,2),(3,15),(3,16),(4,1),(4,15),(4,16),(6,7),(7,9),(8,10),(9,5),(10,5),(11,8),(12,7),(12,14),(13,8),(13,14),(14,9),(14,10),(15,6),(15,12),(16,6),(16,11),(16,13)],17)
 => [7,5,3,2]
 => ? = 3
010100 => ([(0,2),(0,3),(1,9),(2,12),(2,14),(3,1),(3,12),(3,14),(5,7),(6,8),(7,4),(8,4),(9,5),(10,6),(10,11),(11,7),(11,8),(12,10),(12,13),(13,5),(13,6),(13,11),(14,9),(14,10),(14,13)],15)
 => [7,5,3]
 => ? = 4
010101 => ([(0,1),(0,2),(1,10),(1,11),(2,10),(2,11),(4,3),(5,3),(6,8),(6,9),(7,8),(7,9),(8,4),(8,5),(9,4),(9,5),(10,6),(10,7),(11,6),(11,7)],12)
 => [7,5]
 => ? = 5
010110 => ([(0,2),(0,3),(1,6),(1,11),(2,14),(2,15),(3,1),(3,14),(3,15),(5,8),(6,7),(7,9),(8,10),(9,4),(10,4),(11,7),(11,13),(12,8),(12,13),(13,9),(13,10),(14,5),(14,6),(14,12),(15,5),(15,11),(15,12)],16)
 => [7,5,3,1]
 => ? = 3
010111 => ([(0,3),(0,4),(1,2),(1,14),(2,6),(3,13),(3,15),(4,1),(4,13),(4,15),(6,9),(7,8),(8,10),(9,5),(10,5),(11,8),(11,12),(12,9),(12,10),(13,7),(13,11),(14,6),(14,12),(15,7),(15,11),(15,14)],16)
 => [7,5,3,1]
 => ? = 3
011000 => ([(0,4),(0,5),(1,11),(2,1),(2,13),(3,7),(3,14),(4,2),(4,12),(4,16),(5,3),(5,12),(5,16),(7,8),(8,9),(9,6),(10,6),(11,10),(12,7),(13,11),(13,15),(14,8),(14,15),(15,9),(15,10),(16,13),(16,14)],17)
 => [7,5,3,2]
 => ? = 3
011001 => ([(0,3),(0,4),(1,11),(1,16),(2,10),(2,15),(3,2),(3,13),(3,14),(4,1),(4,13),(4,14),(6,8),(7,9),(8,5),(9,5),(10,6),(11,7),(12,8),(12,9),(13,15),(13,16),(14,10),(14,11),(15,6),(15,12),(16,7),(16,12)],17)
 => [7,5,3,2]
 => ? = 3
011010 => ([(0,2),(0,3),(1,6),(1,11),(2,14),(2,15),(3,1),(3,14),(3,15),(5,8),(6,7),(7,9),(8,10),(9,4),(10,4),(11,7),(11,13),(12,8),(12,13),(13,9),(13,10),(14,5),(14,6),(14,12),(15,5),(15,11),(15,12)],16)
 => [7,5,3,1]
 => ? = 3
111111 => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
 => [7]
 => 6
0000000 => ([(0,7),(2,4),(3,2),(4,6),(5,3),(6,1),(7,5)],8)
 => [8]
 => 7
1111111 => ([(0,7),(2,4),(3,2),(4,6),(5,3),(6,1),(7,5)],8)
 => [8]
 => 7
00000000 => ([(0,8),(2,3),(3,5),(4,2),(5,7),(6,4),(7,1),(8,6)],9)
 => [9]
 => 8
11111111 => ([(0,8),(2,3),(3,5),(4,2),(5,7),(6,4),(7,1),(8,6)],9)
 => [9]
 => 8
000000000 => ([(0,9),(2,4),(3,2),(4,6),(5,3),(6,8),(7,5),(8,1),(9,7)],10)
 => [10]
 => 9
111111111 => ([(0,9),(2,4),(3,2),(4,6),(5,3),(6,8),(7,5),(8,1),(9,7)],10)
 => [10]
 => 9
Description
The Dyson rank of a partition. This rank is defined as the largest part minus the number of parts. It was introduced by Dyson [1] in connection to Ramanujan's partition congruences $$p(5n+4) \equiv 0 \pmod 5$$ and $$p(7n+6) \equiv 0 \pmod 7.$$