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Your data matches 99 different statistics following compositions of up to 3 maps.
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Matching statistic: St000293
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Mp00224: Binary words —runsort⟶ Binary words
St000293: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
St000293: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
0 => 0 => 0
1 => 1 => 0
00 => 00 => 0
01 => 01 => 0
10 => 01 => 0
11 => 11 => 0
000 => 000 => 0
001 => 001 => 0
010 => 001 => 0
011 => 011 => 0
100 => 001 => 0
101 => 011 => 0
110 => 011 => 0
111 => 111 => 0
0000 => 0000 => 0
0001 => 0001 => 0
0010 => 0001 => 0
0011 => 0011 => 0
0100 => 0001 => 0
0101 => 0101 => 1
0110 => 0011 => 0
0111 => 0111 => 0
1000 => 0001 => 0
1001 => 0011 => 0
1010 => 0011 => 0
1011 => 0111 => 0
1100 => 0011 => 0
1101 => 0111 => 0
1110 => 0111 => 0
1111 => 1111 => 0
00000 => 00000 => 0
00001 => 00001 => 0
00010 => 00001 => 0
00011 => 00011 => 0
00100 => 00001 => 0
00101 => 00101 => 1
00110 => 00011 => 0
00111 => 00111 => 0
01000 => 00001 => 0
01001 => 00101 => 1
01010 => 00101 => 1
01011 => 01011 => 1
01100 => 00011 => 0
01101 => 01011 => 1
01110 => 00111 => 0
01111 => 01111 => 0
10000 => 00001 => 0
10001 => 00011 => 0
10010 => 00011 => 0
10011 => 00111 => 0
Description
The number of inversions of a binary word.
Matching statistic: St000290
Mp00224: Binary words —runsort⟶ Binary words
Mp00316: Binary words —inverse Foata bijection⟶ Binary words
St000290: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00316: Binary words —inverse Foata bijection⟶ Binary words
St000290: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
0 => 0 => 0 => 0
1 => 1 => 1 => 0
00 => 00 => 00 => 0
01 => 01 => 01 => 0
10 => 01 => 01 => 0
11 => 11 => 11 => 0
000 => 000 => 000 => 0
001 => 001 => 001 => 0
010 => 001 => 001 => 0
011 => 011 => 011 => 0
100 => 001 => 001 => 0
101 => 011 => 011 => 0
110 => 011 => 011 => 0
111 => 111 => 111 => 0
0000 => 0000 => 0000 => 0
0001 => 0001 => 0001 => 0
0010 => 0001 => 0001 => 0
0011 => 0011 => 0011 => 0
0100 => 0001 => 0001 => 0
0101 => 0101 => 1001 => 1
0110 => 0011 => 0011 => 0
0111 => 0111 => 0111 => 0
1000 => 0001 => 0001 => 0
1001 => 0011 => 0011 => 0
1010 => 0011 => 0011 => 0
1011 => 0111 => 0111 => 0
1100 => 0011 => 0011 => 0
1101 => 0111 => 0111 => 0
1110 => 0111 => 0111 => 0
1111 => 1111 => 1111 => 0
00000 => 00000 => 00000 => 0
00001 => 00001 => 00001 => 0
00010 => 00001 => 00001 => 0
00011 => 00011 => 00011 => 0
00100 => 00001 => 00001 => 0
00101 => 00101 => 10001 => 1
00110 => 00011 => 00011 => 0
00111 => 00111 => 00111 => 0
01000 => 00001 => 00001 => 0
01001 => 00101 => 10001 => 1
01010 => 00101 => 10001 => 1
01011 => 01011 => 10011 => 1
01100 => 00011 => 00011 => 0
01101 => 01011 => 10011 => 1
01110 => 00111 => 00111 => 0
01111 => 01111 => 01111 => 0
10000 => 00001 => 00001 => 0
10001 => 00011 => 00011 => 0
10010 => 00011 => 00011 => 0
10011 => 00111 => 00111 => 0
Description
The major index of a binary word.
This is the sum of the positions of descents, i.e., a one followed by a zero.
For words of length $n$ with $a$ zeros, the generating function for the major index is the $q$-binomial coefficient $\binom{n}{a}_q$.
Matching statistic: St001485
Mp00224: Binary words —runsort⟶ Binary words
Mp00316: Binary words —inverse Foata bijection⟶ Binary words
St001485: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00316: Binary words —inverse Foata bijection⟶ Binary words
St001485: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
0 => 0 => 0 => 0
1 => 1 => 1 => 0
00 => 00 => 00 => 0
01 => 01 => 01 => 0
10 => 01 => 01 => 0
11 => 11 => 11 => 0
000 => 000 => 000 => 0
001 => 001 => 001 => 0
010 => 001 => 001 => 0
011 => 011 => 011 => 0
100 => 001 => 001 => 0
101 => 011 => 011 => 0
110 => 011 => 011 => 0
111 => 111 => 111 => 0
0000 => 0000 => 0000 => 0
0001 => 0001 => 0001 => 0
0010 => 0001 => 0001 => 0
0011 => 0011 => 0011 => 0
0100 => 0001 => 0001 => 0
0101 => 0101 => 1001 => 1
0110 => 0011 => 0011 => 0
0111 => 0111 => 0111 => 0
1000 => 0001 => 0001 => 0
1001 => 0011 => 0011 => 0
1010 => 0011 => 0011 => 0
1011 => 0111 => 0111 => 0
1100 => 0011 => 0011 => 0
1101 => 0111 => 0111 => 0
1110 => 0111 => 0111 => 0
1111 => 1111 => 1111 => 0
00000 => 00000 => 00000 => 0
00001 => 00001 => 00001 => 0
00010 => 00001 => 00001 => 0
00011 => 00011 => 00011 => 0
00100 => 00001 => 00001 => 0
00101 => 00101 => 10001 => 1
00110 => 00011 => 00011 => 0
00111 => 00111 => 00111 => 0
01000 => 00001 => 00001 => 0
01001 => 00101 => 10001 => 1
01010 => 00101 => 10001 => 1
01011 => 01011 => 10011 => 1
01100 => 00011 => 00011 => 0
01101 => 01011 => 10011 => 1
01110 => 00111 => 00111 => 0
01111 => 01111 => 01111 => 0
10000 => 00001 => 00001 => 0
10001 => 00011 => 00011 => 0
10010 => 00011 => 00011 => 0
10011 => 00111 => 00111 => 0
Description
The modular major index of a binary word.
This is [[St000290]] modulo the length of the word.
Matching statistic: St001877
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Mp00097: Binary words —delta morphism⟶ Integer compositions
Mp00180: Integer compositions —to ribbon⟶ Skew partitions
Mp00192: Skew partitions —dominating sublattice⟶ Lattices
St001877: Lattices ⟶ ℤResult quality: 19% ●values known / values provided: 19%●distinct values known / distinct values provided: 50%
Mp00180: Integer compositions —to ribbon⟶ Skew partitions
Mp00192: Skew partitions —dominating sublattice⟶ Lattices
St001877: Lattices ⟶ ℤResult quality: 19% ●values known / values provided: 19%●distinct values known / distinct values provided: 50%
Values
0 => [1] => [[1],[]]
=> ([],1)
=> ? ∊ {0,0}
1 => [1] => [[1],[]]
=> ([],1)
=> ? ∊ {0,0}
00 => [2] => [[2],[]]
=> ([],1)
=> ? ∊ {0,0,0,0}
01 => [1,1] => [[1,1],[]]
=> ([],1)
=> ? ∊ {0,0,0,0}
10 => [1,1] => [[1,1],[]]
=> ([],1)
=> ? ∊ {0,0,0,0}
11 => [2] => [[2],[]]
=> ([],1)
=> ? ∊ {0,0,0,0}
000 => [3] => [[3],[]]
=> ([],1)
=> ? ∊ {0,0,0,0,0,0,0,0}
001 => [2,1] => [[2,2],[1]]
=> ([],1)
=> ? ∊ {0,0,0,0,0,0,0,0}
010 => [1,1,1] => [[1,1,1],[]]
=> ([],1)
=> ? ∊ {0,0,0,0,0,0,0,0}
011 => [1,2] => [[2,1],[]]
=> ([],1)
=> ? ∊ {0,0,0,0,0,0,0,0}
100 => [1,2] => [[2,1],[]]
=> ([],1)
=> ? ∊ {0,0,0,0,0,0,0,0}
101 => [1,1,1] => [[1,1,1],[]]
=> ([],1)
=> ? ∊ {0,0,0,0,0,0,0,0}
110 => [2,1] => [[2,2],[1]]
=> ([],1)
=> ? ∊ {0,0,0,0,0,0,0,0}
111 => [3] => [[3],[]]
=> ([],1)
=> ? ∊ {0,0,0,0,0,0,0,0}
0000 => [4] => [[4],[]]
=> ([],1)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1}
0001 => [3,1] => [[3,3],[2]]
=> ([],1)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1}
0010 => [2,1,1] => [[2,2,2],[1,1]]
=> ([],1)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1}
0011 => [2,2] => [[3,2],[1]]
=> ([(0,1)],2)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1}
0100 => [1,1,2] => [[2,1,1],[]]
=> ([],1)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1}
0101 => [1,1,1,1] => [[1,1,1,1],[]]
=> ([],1)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1}
0110 => [1,2,1] => [[2,2,1],[1]]
=> ([(0,1)],2)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1}
0111 => [1,3] => [[3,1],[]]
=> ([],1)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1}
1000 => [1,3] => [[3,1],[]]
=> ([],1)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1}
1001 => [1,2,1] => [[2,2,1],[1]]
=> ([(0,1)],2)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1}
1010 => [1,1,1,1] => [[1,1,1,1],[]]
=> ([],1)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1}
1011 => [1,1,2] => [[2,1,1],[]]
=> ([],1)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1}
1100 => [2,2] => [[3,2],[1]]
=> ([(0,1)],2)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1}
1101 => [2,1,1] => [[2,2,2],[1,1]]
=> ([],1)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1}
1110 => [3,1] => [[3,3],[2]]
=> ([],1)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1}
1111 => [4] => [[4],[]]
=> ([],1)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1}
00000 => [5] => [[5],[]]
=> ([],1)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1}
00001 => [4,1] => [[4,4],[3]]
=> ([],1)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1}
00010 => [3,1,1] => [[3,3,3],[2,2]]
=> ([],1)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1}
00011 => [3,2] => [[4,3],[2]]
=> ([(0,1)],2)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1}
00100 => [2,1,2] => [[3,2,2],[1,1]]
=> ([(0,1)],2)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1}
00101 => [2,1,1,1] => [[2,2,2,2],[1,1,1]]
=> ([],1)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1}
00110 => [2,2,1] => [[3,3,2],[2,1]]
=> ([(0,2),(2,1)],3)
=> 0
00111 => [2,3] => [[4,2],[1]]
=> ([(0,1)],2)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1}
01000 => [1,1,3] => [[3,1,1],[]]
=> ([],1)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1}
01001 => [1,1,2,1] => [[2,2,1,1],[1]]
=> ([(0,1)],2)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1}
01010 => [1,1,1,1,1] => [[1,1,1,1,1],[]]
=> ([],1)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1}
01011 => [1,1,1,2] => [[2,1,1,1],[]]
=> ([],1)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1}
01100 => [1,2,2] => [[3,2,1],[1]]
=> ([(0,2),(2,1)],3)
=> 0
01101 => [1,2,1,1] => [[2,2,2,1],[1,1]]
=> ([(0,1)],2)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1}
01110 => [1,3,1] => [[3,3,1],[2]]
=> ([(0,1)],2)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1}
01111 => [1,4] => [[4,1],[]]
=> ([],1)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1}
10000 => [1,4] => [[4,1],[]]
=> ([],1)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1}
10001 => [1,3,1] => [[3,3,1],[2]]
=> ([(0,1)],2)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1}
10010 => [1,2,1,1] => [[2,2,2,1],[1,1]]
=> ([(0,1)],2)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1}
10011 => [1,2,2] => [[3,2,1],[1]]
=> ([(0,2),(2,1)],3)
=> 0
10100 => [1,1,1,2] => [[2,1,1,1],[]]
=> ([],1)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1}
10101 => [1,1,1,1,1] => [[1,1,1,1,1],[]]
=> ([],1)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1}
10110 => [1,1,2,1] => [[2,2,1,1],[1]]
=> ([(0,1)],2)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1}
11001 => [2,2,1] => [[3,3,2],[2,1]]
=> ([(0,2),(2,1)],3)
=> 0
000110 => [3,2,1] => [[4,4,3],[3,2]]
=> ([(0,2),(2,1)],3)
=> 0
000111 => [3,3] => [[5,3],[2]]
=> ([(0,2),(2,1)],3)
=> 0
001001 => [2,1,2,1] => [[3,3,2,2],[2,1,1]]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1
001100 => [2,2,2] => [[4,3,2],[2,1]]
=> ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> 1
001101 => [2,2,1,1] => [[3,3,3,2],[2,2,1]]
=> ([(0,2),(2,1)],3)
=> 0
001110 => [2,3,1] => [[4,4,2],[3,1]]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1
010010 => [1,1,2,1,1] => [[2,2,2,1,1],[1,1]]
=> ([(0,2),(2,1)],3)
=> 0
010011 => [1,1,2,2] => [[3,2,1,1],[1]]
=> ([(0,2),(2,1)],3)
=> 0
011000 => [1,2,3] => [[4,2,1],[1]]
=> ([(0,2),(2,1)],3)
=> 0
011001 => [1,2,2,1] => [[3,3,2,1],[2,1]]
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> 1
011011 => [1,2,1,2] => [[3,2,2,1],[1,1]]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1
011100 => [1,3,2] => [[4,3,1],[2]]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1
100011 => [1,3,2] => [[4,3,1],[2]]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1
100100 => [1,2,1,2] => [[3,2,2,1],[1,1]]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1
100110 => [1,2,2,1] => [[3,3,2,1],[2,1]]
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> 1
100111 => [1,2,3] => [[4,2,1],[1]]
=> ([(0,2),(2,1)],3)
=> 0
101100 => [1,1,2,2] => [[3,2,1,1],[1]]
=> ([(0,2),(2,1)],3)
=> 0
101101 => [1,1,2,1,1] => [[2,2,2,1,1],[1,1]]
=> ([(0,2),(2,1)],3)
=> 0
110001 => [2,3,1] => [[4,4,2],[3,1]]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1
110010 => [2,2,1,1] => [[3,3,3,2],[2,2,1]]
=> ([(0,2),(2,1)],3)
=> 0
110011 => [2,2,2] => [[4,3,2],[2,1]]
=> ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> 1
110110 => [2,1,2,1] => [[3,3,2,2],[2,1,1]]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1
111000 => [3,3] => [[5,3],[2]]
=> ([(0,2),(2,1)],3)
=> 0
111001 => [3,2,1] => [[4,4,3],[3,2]]
=> ([(0,2),(2,1)],3)
=> 0
0000110 => [4,2,1] => [[5,5,4],[4,3]]
=> ([(0,2),(2,1)],3)
=> 0
0000111 => [4,3] => [[6,4],[3]]
=> ([(0,2),(2,1)],3)
=> 0
0001000 => [3,1,3] => [[5,3,3],[2,2]]
=> ([(0,2),(2,1)],3)
=> 0
0001001 => [3,1,2,1] => [[4,4,3,3],[3,2,2]]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1
0001100 => [3,2,2] => [[5,4,3],[3,2]]
=> ([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6)
=> 1
0001101 => [3,2,1,1] => [[4,4,4,3],[3,3,2]]
=> ([(0,2),(2,1)],3)
=> 0
0001110 => [3,3,1] => [[5,5,3],[4,2]]
=> ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> 1
0001111 => [3,4] => [[6,3],[2]]
=> ([(0,2),(2,1)],3)
=> 0
0010001 => [2,1,3,1] => [[4,4,2,2],[3,1,1]]
=> ([(0,2),(0,3),(1,4),(2,4),(3,1)],5)
=> 2
0010010 => [2,1,2,1,1] => [[3,3,3,2,2],[2,2,1,1]]
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> 1
0010011 => [2,1,2,2] => [[4,3,2,2],[2,1,1]]
=> ([(0,3),(0,4),(1,5),(3,5),(4,1),(5,2)],6)
=> 2
0010110 => [2,1,1,2,1] => [[3,3,2,2,2],[2,1,1,1]]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1
0011000 => [2,2,3] => [[5,3,2],[2,1]]
=> ([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6)
=> 1
0011001 => [2,2,2,1] => [[4,4,3,2],[3,2,1]]
=> ([(0,5),(1,6),(2,6),(4,2),(5,1),(5,4),(6,3)],7)
=> 2
0011010 => [2,2,1,1,1] => [[3,3,3,3,2],[2,2,2,1]]
=> ([(0,2),(2,1)],3)
=> 0
0011011 => [2,2,1,2] => [[4,3,3,2],[2,2,1]]
=> ([(0,3),(0,4),(1,5),(3,5),(4,1),(5,2)],6)
=> 2
0011100 => [2,3,2] => [[5,4,2],[3,1]]
=> ([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6)
=> 1
0011101 => [2,3,1,1] => [[4,4,4,2],[3,3,1]]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1
0011110 => [2,4,1] => [[5,5,2],[4,1]]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1
0100010 => [1,1,3,1,1] => [[3,3,3,1,1],[2,2]]
=> ([(0,2),(2,1)],3)
=> 0
0100011 => [1,1,3,2] => [[4,3,1,1],[2]]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1
0100100 => [1,1,2,1,2] => [[3,2,2,1,1],[1,1]]
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> 1
Description
Number of indecomposable injective modules with projective dimension 2.
Matching statistic: St001876
Mp00097: Binary words —delta morphism⟶ Integer compositions
Mp00180: Integer compositions —to ribbon⟶ Skew partitions
Mp00192: Skew partitions —dominating sublattice⟶ Lattices
St001876: Lattices ⟶ ℤResult quality: 16% ●values known / values provided: 16%●distinct values known / distinct values provided: 38%
Mp00180: Integer compositions —to ribbon⟶ Skew partitions
Mp00192: Skew partitions —dominating sublattice⟶ Lattices
St001876: Lattices ⟶ ℤResult quality: 16% ●values known / values provided: 16%●distinct values known / distinct values provided: 38%
Values
0 => [1] => [[1],[]]
=> ([],1)
=> ? ∊ {0,0}
1 => [1] => [[1],[]]
=> ([],1)
=> ? ∊ {0,0}
00 => [2] => [[2],[]]
=> ([],1)
=> ? ∊ {0,0,0,0}
01 => [1,1] => [[1,1],[]]
=> ([],1)
=> ? ∊ {0,0,0,0}
10 => [1,1] => [[1,1],[]]
=> ([],1)
=> ? ∊ {0,0,0,0}
11 => [2] => [[2],[]]
=> ([],1)
=> ? ∊ {0,0,0,0}
000 => [3] => [[3],[]]
=> ([],1)
=> ? ∊ {0,0,0,0,0,0,0,0}
001 => [2,1] => [[2,2],[1]]
=> ([],1)
=> ? ∊ {0,0,0,0,0,0,0,0}
010 => [1,1,1] => [[1,1,1],[]]
=> ([],1)
=> ? ∊ {0,0,0,0,0,0,0,0}
011 => [1,2] => [[2,1],[]]
=> ([],1)
=> ? ∊ {0,0,0,0,0,0,0,0}
100 => [1,2] => [[2,1],[]]
=> ([],1)
=> ? ∊ {0,0,0,0,0,0,0,0}
101 => [1,1,1] => [[1,1,1],[]]
=> ([],1)
=> ? ∊ {0,0,0,0,0,0,0,0}
110 => [2,1] => [[2,2],[1]]
=> ([],1)
=> ? ∊ {0,0,0,0,0,0,0,0}
111 => [3] => [[3],[]]
=> ([],1)
=> ? ∊ {0,0,0,0,0,0,0,0}
0000 => [4] => [[4],[]]
=> ([],1)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1}
0001 => [3,1] => [[3,3],[2]]
=> ([],1)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1}
0010 => [2,1,1] => [[2,2,2],[1,1]]
=> ([],1)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1}
0011 => [2,2] => [[3,2],[1]]
=> ([(0,1)],2)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1}
0100 => [1,1,2] => [[2,1,1],[]]
=> ([],1)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1}
0101 => [1,1,1,1] => [[1,1,1,1],[]]
=> ([],1)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1}
0110 => [1,2,1] => [[2,2,1],[1]]
=> ([(0,1)],2)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1}
0111 => [1,3] => [[3,1],[]]
=> ([],1)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1}
1000 => [1,3] => [[3,1],[]]
=> ([],1)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1}
1001 => [1,2,1] => [[2,2,1],[1]]
=> ([(0,1)],2)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1}
1010 => [1,1,1,1] => [[1,1,1,1],[]]
=> ([],1)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1}
1011 => [1,1,2] => [[2,1,1],[]]
=> ([],1)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1}
1100 => [2,2] => [[3,2],[1]]
=> ([(0,1)],2)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1}
1101 => [2,1,1] => [[2,2,2],[1,1]]
=> ([],1)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1}
1110 => [3,1] => [[3,3],[2]]
=> ([],1)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1}
1111 => [4] => [[4],[]]
=> ([],1)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1}
00000 => [5] => [[5],[]]
=> ([],1)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1}
00001 => [4,1] => [[4,4],[3]]
=> ([],1)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1}
00010 => [3,1,1] => [[3,3,3],[2,2]]
=> ([],1)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1}
00011 => [3,2] => [[4,3],[2]]
=> ([(0,1)],2)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1}
00100 => [2,1,2] => [[3,2,2],[1,1]]
=> ([(0,1)],2)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1}
00101 => [2,1,1,1] => [[2,2,2,2],[1,1,1]]
=> ([],1)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1}
00110 => [2,2,1] => [[3,3,2],[2,1]]
=> ([(0,2),(2,1)],3)
=> 0
00111 => [2,3] => [[4,2],[1]]
=> ([(0,1)],2)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1}
01000 => [1,1,3] => [[3,1,1],[]]
=> ([],1)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1}
01001 => [1,1,2,1] => [[2,2,1,1],[1]]
=> ([(0,1)],2)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1}
01010 => [1,1,1,1,1] => [[1,1,1,1,1],[]]
=> ([],1)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1}
01011 => [1,1,1,2] => [[2,1,1,1],[]]
=> ([],1)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1}
01100 => [1,2,2] => [[3,2,1],[1]]
=> ([(0,2),(2,1)],3)
=> 0
01101 => [1,2,1,1] => [[2,2,2,1],[1,1]]
=> ([(0,1)],2)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1}
01110 => [1,3,1] => [[3,3,1],[2]]
=> ([(0,1)],2)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1}
01111 => [1,4] => [[4,1],[]]
=> ([],1)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1}
10000 => [1,4] => [[4,1],[]]
=> ([],1)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1}
10001 => [1,3,1] => [[3,3,1],[2]]
=> ([(0,1)],2)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1}
10010 => [1,2,1,1] => [[2,2,2,1],[1,1]]
=> ([(0,1)],2)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1}
10011 => [1,2,2] => [[3,2,1],[1]]
=> ([(0,2),(2,1)],3)
=> 0
10100 => [1,1,1,2] => [[2,1,1,1],[]]
=> ([],1)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1}
10101 => [1,1,1,1,1] => [[1,1,1,1,1],[]]
=> ([],1)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1}
10110 => [1,1,2,1] => [[2,2,1,1],[1]]
=> ([(0,1)],2)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1}
11001 => [2,2,1] => [[3,3,2],[2,1]]
=> ([(0,2),(2,1)],3)
=> 0
000110 => [3,2,1] => [[4,4,3],[3,2]]
=> ([(0,2),(2,1)],3)
=> 0
000111 => [3,3] => [[5,3],[2]]
=> ([(0,2),(2,1)],3)
=> 0
001001 => [2,1,2,1] => [[3,3,2,2],[2,1,1]]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1
001100 => [2,2,2] => [[4,3,2],[2,1]]
=> ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> 1
001101 => [2,2,1,1] => [[3,3,3,2],[2,2,1]]
=> ([(0,2),(2,1)],3)
=> 0
001110 => [2,3,1] => [[4,4,2],[3,1]]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1
010010 => [1,1,2,1,1] => [[2,2,2,1,1],[1,1]]
=> ([(0,2),(2,1)],3)
=> 0
010011 => [1,1,2,2] => [[3,2,1,1],[1]]
=> ([(0,2),(2,1)],3)
=> 0
011000 => [1,2,3] => [[4,2,1],[1]]
=> ([(0,2),(2,1)],3)
=> 0
011001 => [1,2,2,1] => [[3,3,2,1],[2,1]]
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> 1
011011 => [1,2,1,2] => [[3,2,2,1],[1,1]]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1
011100 => [1,3,2] => [[4,3,1],[2]]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1
100011 => [1,3,2] => [[4,3,1],[2]]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1
100100 => [1,2,1,2] => [[3,2,2,1],[1,1]]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1
100110 => [1,2,2,1] => [[3,3,2,1],[2,1]]
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> 1
100111 => [1,2,3] => [[4,2,1],[1]]
=> ([(0,2),(2,1)],3)
=> 0
101100 => [1,1,2,2] => [[3,2,1,1],[1]]
=> ([(0,2),(2,1)],3)
=> 0
101101 => [1,1,2,1,1] => [[2,2,2,1,1],[1,1]]
=> ([(0,2),(2,1)],3)
=> 0
110001 => [2,3,1] => [[4,4,2],[3,1]]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1
110010 => [2,2,1,1] => [[3,3,3,2],[2,2,1]]
=> ([(0,2),(2,1)],3)
=> 0
110011 => [2,2,2] => [[4,3,2],[2,1]]
=> ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> 1
110110 => [2,1,2,1] => [[3,3,2,2],[2,1,1]]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1
111000 => [3,3] => [[5,3],[2]]
=> ([(0,2),(2,1)],3)
=> 0
111001 => [3,2,1] => [[4,4,3],[3,2]]
=> ([(0,2),(2,1)],3)
=> 0
0000110 => [4,2,1] => [[5,5,4],[4,3]]
=> ([(0,2),(2,1)],3)
=> 0
0000111 => [4,3] => [[6,4],[3]]
=> ([(0,2),(2,1)],3)
=> 0
0001000 => [3,1,3] => [[5,3,3],[2,2]]
=> ([(0,2),(2,1)],3)
=> 0
0001001 => [3,1,2,1] => [[4,4,3,3],[3,2,2]]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1
0001100 => [3,2,2] => [[5,4,3],[3,2]]
=> ([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6)
=> 1
0001101 => [3,2,1,1] => [[4,4,4,3],[3,3,2]]
=> ([(0,2),(2,1)],3)
=> 0
0001110 => [3,3,1] => [[5,5,3],[4,2]]
=> ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> 1
0001111 => [3,4] => [[6,3],[2]]
=> ([(0,2),(2,1)],3)
=> 0
0010010 => [2,1,2,1,1] => [[3,3,3,2,2],[2,2,1,1]]
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> 1
0010110 => [2,1,1,2,1] => [[3,3,2,2,2],[2,1,1,1]]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1
0011000 => [2,2,3] => [[5,3,2],[2,1]]
=> ([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6)
=> 1
0011010 => [2,2,1,1,1] => [[3,3,3,3,2],[2,2,2,1]]
=> ([(0,2),(2,1)],3)
=> 0
0011100 => [2,3,2] => [[5,4,2],[3,1]]
=> ([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6)
=> 1
0011101 => [2,3,1,1] => [[4,4,4,2],[3,3,1]]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1
0011110 => [2,4,1] => [[5,5,2],[4,1]]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1
0100010 => [1,1,3,1,1] => [[3,3,3,1,1],[2,2]]
=> ([(0,2),(2,1)],3)
=> 0
0100011 => [1,1,3,2] => [[4,3,1,1],[2]]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1
0100100 => [1,1,2,1,2] => [[3,2,2,1,1],[1,1]]
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> 1
0100101 => [1,1,2,1,1,1] => [[2,2,2,2,1,1],[1,1,1]]
=> ([(0,2),(2,1)],3)
=> 0
0100110 => [1,1,2,2,1] => [[3,3,2,1,1],[2,1]]
=> ([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6)
=> 1
0100111 => [1,1,2,3] => [[4,2,1,1],[1]]
=> ([(0,2),(2,1)],3)
=> 0
0101100 => [1,1,1,2,2] => [[3,2,1,1,1],[1]]
=> ([(0,2),(2,1)],3)
=> 0
Description
The number of 2-regular simple modules in the incidence algebra of the lattice.
Matching statistic: St000068
(load all 4 compositions to match this statistic)
(load all 4 compositions to match this statistic)
Mp00224: Binary words —runsort⟶ Binary words
Mp00262: Binary words —poset of factors⟶ Posets
St000068: Posets ⟶ ℤResult quality: 6% ●values known / values provided: 6%●distinct values known / distinct values provided: 12%
Mp00262: Binary words —poset of factors⟶ Posets
St000068: Posets ⟶ ℤResult quality: 6% ●values known / values provided: 6%●distinct values known / distinct values provided: 12%
Values
0 => 0 => ([(0,1)],2)
=> 1 = 0 + 1
1 => 1 => ([(0,1)],2)
=> 1 = 0 + 1
00 => 00 => ([(0,2),(2,1)],3)
=> 1 = 0 + 1
01 => 01 => ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1 = 0 + 1
10 => 01 => ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1 = 0 + 1
11 => 11 => ([(0,2),(2,1)],3)
=> 1 = 0 + 1
000 => 000 => ([(0,3),(2,1),(3,2)],4)
=> 1 = 0 + 1
001 => 001 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 1 = 0 + 1
010 => 001 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 1 = 0 + 1
011 => 011 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 1 = 0 + 1
100 => 001 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 1 = 0 + 1
101 => 011 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 1 = 0 + 1
110 => 011 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 1 = 0 + 1
111 => 111 => ([(0,3),(2,1),(3,2)],4)
=> 1 = 0 + 1
0000 => 0000 => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1 = 0 + 1
0001 => 0001 => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> 1 = 0 + 1
0010 => 0001 => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> 1 = 0 + 1
0011 => 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)
=> 1 = 0 + 1
0100 => 0001 => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> 1 = 0 + 1
0101 => 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)
=> ? = 1 + 1
0110 => 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)
=> 1 = 0 + 1
0111 => 0111 => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> 1 = 0 + 1
1000 => 0001 => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> 1 = 0 + 1
1001 => 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)
=> 1 = 0 + 1
1010 => 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)
=> 1 = 0 + 1
1011 => 0111 => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> 1 = 0 + 1
1100 => 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)
=> 1 = 0 + 1
1101 => 0111 => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> 1 = 0 + 1
1110 => 0111 => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> 1 = 0 + 1
1111 => 1111 => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1 = 0 + 1
00000 => 00000 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 1 = 0 + 1
00001 => 00001 => ([(0,2),(0,5),(1,7),(2,6),(3,4),(3,9),(4,1),(4,8),(5,3),(5,6),(6,9),(8,7),(9,8)],10)
=> 1 = 0 + 1
00010 => 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)
=> 1 = 0 + 1
00011 => 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)
=> 1 = 0 + 1
00100 => 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)
=> 1 = 0 + 1
00101 => 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)
=> ? ∊ {1,1,1,1,1,1} + 1
00110 => 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)
=> 1 = 0 + 1
00111 => 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)
=> 1 = 0 + 1
01000 => 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)
=> 1 = 0 + 1
01001 => 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)
=> ? ∊ {1,1,1,1,1,1} + 1
01010 => 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)
=> ? ∊ {1,1,1,1,1,1} + 1
01011 => 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)
=> ? ∊ {1,1,1,1,1,1} + 1
01100 => 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)
=> 1 = 0 + 1
01101 => 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)
=> ? ∊ {1,1,1,1,1,1} + 1
01110 => 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)
=> 1 = 0 + 1
01111 => 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)
=> 1 = 0 + 1
10000 => 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)
=> 1 = 0 + 1
10001 => 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)
=> 1 = 0 + 1
10010 => 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)
=> 1 = 0 + 1
10011 => 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)
=> 1 = 0 + 1
10100 => 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)
=> 1 = 0 + 1
10101 => 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)
=> ? ∊ {1,1,1,1,1,1} + 1
10110 => 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)
=> 1 = 0 + 1
10111 => 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)
=> 1 = 0 + 1
11000 => 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)
=> 1 = 0 + 1
11001 => 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)
=> 1 = 0 + 1
11010 => 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)
=> 1 = 0 + 1
000001 => 000001 => ([(0,2),(0,6),(1,8),(2,7),(3,5),(3,9),(4,3),(4,11),(5,1),(5,10),(6,4),(6,7),(7,11),(9,10),(10,8),(11,9)],12)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,3} + 1
000010 => 000001 => ([(0,2),(0,6),(1,8),(2,7),(3,5),(3,9),(4,3),(4,11),(5,1),(5,10),(6,4),(6,7),(7,11),(9,10),(10,8),(11,9)],12)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,3} + 1
000011 => 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)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,3} + 1
000100 => 000001 => ([(0,2),(0,6),(1,8),(2,7),(3,5),(3,9),(4,3),(4,11),(5,1),(5,10),(6,4),(6,7),(7,11),(9,10),(10,8),(11,9)],12)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,3} + 1
000101 => 000101 => ([(0,3),(0,4),(1,2),(1,14),(2,6),(3,13),(3,15),(4,1),(4,13),(4,15),(6,9),(7,8),(8,10),(9,5),(10,5),(11,8),(11,12),(12,9),(12,10),(13,7),(13,11),(14,6),(14,12),(15,7),(15,11),(15,14)],16)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,3} + 1
000110 => 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)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,3} + 1
000111 => 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)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,3} + 1
001000 => 000001 => ([(0,2),(0,6),(1,8),(2,7),(3,5),(3,9),(4,3),(4,11),(5,1),(5,10),(6,4),(6,7),(7,11),(9,10),(10,8),(11,9)],12)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,3} + 1
001001 => 001001 => ([(0,2),(0,3),(1,11),(1,12),(2,13),(2,14),(3,1),(3,13),(3,14),(5,7),(6,8),(7,4),(8,4),(9,7),(9,8),(10,5),(10,9),(11,6),(11,9),(12,5),(12,6),(13,10),(13,11),(14,10),(14,12)],15)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,3} + 1
001010 => 000101 => ([(0,3),(0,4),(1,2),(1,14),(2,6),(3,13),(3,15),(4,1),(4,13),(4,15),(6,9),(7,8),(8,10),(9,5),(10,5),(11,8),(11,12),(12,9),(12,10),(13,7),(13,11),(14,6),(14,12),(15,7),(15,11),(15,14)],16)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,3} + 1
001011 => 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)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,3} + 1
001100 => 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)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,3} + 1
001101 => 001101 => ([(0,3),(0,4),(1,11),(2,12),(2,13),(3,2),(3,15),(3,16),(4,1),(4,15),(4,16),(6,7),(7,9),(8,10),(9,5),(10,5),(11,8),(12,7),(12,14),(13,8),(13,14),(14,9),(14,10),(15,6),(15,12),(16,6),(16,11),(16,13)],17)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,3} + 1
001110 => 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)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,3} + 1
001111 => 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)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,3} + 1
010000 => 000001 => ([(0,2),(0,6),(1,8),(2,7),(3,5),(3,9),(4,3),(4,11),(5,1),(5,10),(6,4),(6,7),(7,11),(9,10),(10,8),(11,9)],12)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,3} + 1
010001 => 000101 => ([(0,3),(0,4),(1,2),(1,14),(2,6),(3,13),(3,15),(4,1),(4,13),(4,15),(6,9),(7,8),(8,10),(9,5),(10,5),(11,8),(11,12),(12,9),(12,10),(13,7),(13,11),(14,6),(14,12),(15,7),(15,11),(15,14)],16)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,3} + 1
010010 => 000101 => ([(0,3),(0,4),(1,2),(1,14),(2,6),(3,13),(3,15),(4,1),(4,13),(4,15),(6,9),(7,8),(8,10),(9,5),(10,5),(11,8),(11,12),(12,9),(12,10),(13,7),(13,11),(14,6),(14,12),(15,7),(15,11),(15,14)],16)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,3} + 1
010011 => 001101 => ([(0,3),(0,4),(1,11),(2,12),(2,13),(3,2),(3,15),(3,16),(4,1),(4,15),(4,16),(6,7),(7,9),(8,10),(9,5),(10,5),(11,8),(12,7),(12,14),(13,8),(13,14),(14,9),(14,10),(15,6),(15,12),(16,6),(16,11),(16,13)],17)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,3} + 1
010100 => 000101 => ([(0,3),(0,4),(1,2),(1,14),(2,6),(3,13),(3,15),(4,1),(4,13),(4,15),(6,9),(7,8),(8,10),(9,5),(10,5),(11,8),(11,12),(12,9),(12,10),(13,7),(13,11),(14,6),(14,12),(15,7),(15,11),(15,14)],16)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,3} + 1
010101 => 010101 => ([(0,1),(0,2),(1,10),(1,11),(2,10),(2,11),(4,3),(5,3),(6,8),(6,9),(7,8),(7,9),(8,4),(8,5),(9,4),(9,5),(10,6),(10,7),(11,6),(11,7)],12)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,3} + 1
010110 => 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)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,3} + 1
010111 => 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)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,3} + 1
011000 => 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)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,3} + 1
011001 => 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)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,3} + 1
011010 => 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)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,3} + 1
011011 => 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)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,3} + 1
011100 => 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)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,3} + 1
011101 => 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)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,3} + 1
011110 => 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)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,3} + 1
011111 => 011111 => ([(0,2),(0,6),(1,8),(2,7),(3,5),(3,9),(4,3),(4,11),(5,1),(5,10),(6,4),(6,7),(7,11),(9,10),(10,8),(11,9)],12)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,3} + 1
100000 => 000001 => ([(0,2),(0,6),(1,8),(2,7),(3,5),(3,9),(4,3),(4,11),(5,1),(5,10),(6,4),(6,7),(7,11),(9,10),(10,8),(11,9)],12)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,3} + 1
100001 => 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)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,3} + 1
100010 => 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)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,3} + 1
100011 => 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)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,3} + 1
100100 => 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)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,3} + 1
100101 => 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)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,3} + 1
100110 => 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)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,3} + 1
100111 => 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)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,3} + 1
101000 => 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)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,3} + 1
101001 => 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)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,3} + 1
101010 => 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)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,3} + 1
101011 => 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)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,3} + 1
Description
The number of minimal elements in a poset.
Matching statistic: St000370
(load all 6 compositions to match this statistic)
(load all 6 compositions to match this statistic)
Mp00178: Binary words —to composition⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
Mp00154: Graphs —core⟶ Graphs
St000370: Graphs ⟶ ℤResult quality: 6% ●values known / values provided: 6%●distinct values known / distinct values provided: 25%
Mp00184: Integer compositions —to threshold graph⟶ Graphs
Mp00154: Graphs —core⟶ Graphs
St000370: Graphs ⟶ ℤResult quality: 6% ●values known / values provided: 6%●distinct values known / distinct values provided: 25%
Values
0 => [2] => ([],2)
=> ([],1)
=> 0
1 => [1,1] => ([(0,1)],2)
=> ([(0,1)],2)
=> 0
00 => [3] => ([],3)
=> ([],1)
=> 0
01 => [2,1] => ([(0,2),(1,2)],3)
=> ([(0,1)],2)
=> 0
10 => [1,2] => ([(1,2)],3)
=> ([(0,1)],2)
=> 0
11 => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> ([(0,1),(0,2),(1,2)],3)
=> 0
000 => [4] => ([],4)
=> ([],1)
=> 0
001 => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> ([(0,1)],2)
=> 0
010 => [2,2] => ([(1,3),(2,3)],4)
=> ([(0,1)],2)
=> 0
011 => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,2)],3)
=> 0
100 => [1,3] => ([(2,3)],4)
=> ([(0,1)],2)
=> 0
101 => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,2)],3)
=> 0
110 => [1,1,2] => ([(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,2)],3)
=> 0
111 => [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 0
0000 => [5] => ([],5)
=> ([],1)
=> 0
0001 => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> ([(0,1)],2)
=> 0
0010 => [3,2] => ([(1,4),(2,4),(3,4)],5)
=> ([(0,1)],2)
=> 0
0011 => [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)
=> 0
0100 => [2,3] => ([(2,4),(3,4)],5)
=> ([(0,1)],2)
=> 0
0101 => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 0
0110 => [2,1,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 0
0111 => [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)
=> 0
1000 => [1,4] => ([(3,4)],5)
=> ([(0,1)],2)
=> 0
1001 => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 0
1010 => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 0
1011 => [1,2,1,1] => ([(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)
=> 0
1100 => [1,1,3] => ([(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 0
1101 => [1,1,2,1] => ([(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)
=> 0
1110 => [1,1,1,2] => ([(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)
=> 0
1111 => [1,1,1,1,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
00000 => [6] => ([],6)
=> ([],1)
=> 0
00001 => [5,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> ([(0,1)],2)
=> 0
00010 => [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6)
=> ([(0,1)],2)
=> 0
00011 => [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)
=> 0
00100 => [3,3] => ([(2,5),(3,5),(4,5)],6)
=> ([(0,1)],2)
=> 0
00101 => [3,2,1] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> 0
00110 => [3,1,2] => ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> 0
00111 => [3,1,1,1] => ([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 0
01000 => [2,4] => ([(3,5),(4,5)],6)
=> ([(0,1)],2)
=> 0
01001 => [2,3,1] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> 0
01010 => [2,2,2] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> 0
01011 => [2,2,1,1] => ([(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 0
01100 => [2,1,3] => ([(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> 0
01101 => [2,1,2,1] => ([(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 0
01110 => [2,1,1,2] => ([(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),(1,2),(1,3),(2,3)],4)
=> 0
01111 => [2,1,1,1,1] => ([(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
10000 => [1,5] => ([(4,5)],6)
=> ([(0,1)],2)
=> 0
10001 => [1,4,1] => ([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> 0
10010 => [1,3,2] => ([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> 0
10011 => [1,3,1,1] => ([(0,4),(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 0
000000 => [7] => ([],7)
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,3}
000001 => [6,1] => ([(0,6),(1,6),(2,6),(3,6),(4,6),(5,6)],7)
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,3}
000010 => [5,2] => ([(1,6),(2,6),(3,6),(4,6),(5,6)],7)
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,3}
000011 => [5,1,1] => ([(0,5),(0,6),(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,3}
000100 => [4,3] => ([(2,6),(3,6),(4,6),(5,6)],7)
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,3}
000101 => [4,2,1] => ([(0,6),(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,3}
000110 => [4,1,2] => ([(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,3}
000111 => [4,1,1,1] => ([(0,4),(0,5),(0,6),(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,3}
001000 => [3,4] => ([(3,6),(4,6),(5,6)],7)
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,3}
001001 => [3,3,1] => ([(0,6),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,3}
001010 => [3,2,2] => ([(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,3}
001011 => [3,2,1,1] => ([(0,5),(0,6),(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,3}
001100 => [3,1,3] => ([(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,3}
001101 => [3,1,2,1] => ([(0,6),(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,3}
001110 => [3,1,1,2] => ([(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,3}
001111 => [3,1,1,1,1] => ([(0,3),(0,4),(0,5),(0,6),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,3}
010000 => [2,5] => ([(4,6),(5,6)],7)
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,3}
010001 => [2,4,1] => ([(0,6),(1,6),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,3}
010010 => [2,3,2] => ([(1,6),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,3}
010011 => [2,3,1,1] => ([(0,5),(0,6),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,3}
010100 => [2,2,3] => ([(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,3}
010101 => [2,2,2,1] => ([(0,6),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,3}
010110 => [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)
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,3}
010111 => [2,2,1,1,1] => ([(0,4),(0,5),(0,6),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,3}
011000 => [2,1,4] => ([(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,3}
011001 => [2,1,3,1] => ([(0,6),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,3}
011010 => [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)
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,3}
011011 => [2,1,2,1,1] => ([(0,5),(0,6),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,3}
011100 => [2,1,1,3] => ([(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,3}
011101 => [2,1,1,2,1] => ([(0,6),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,3}
011110 => [2,1,1,1,2] => ([(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,3}
011111 => [2,1,1,1,1,1] => ([(0,2),(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,3}
100000 => [1,6] => ([(5,6)],7)
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,3}
100001 => [1,5,1] => ([(0,6),(1,6),(2,6),(3,6),(4,5),(4,6),(5,6)],7)
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,3}
100010 => [1,4,2] => ([(1,6),(2,6),(3,6),(4,5),(4,6),(5,6)],7)
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,3}
100011 => [1,4,1,1] => ([(0,5),(0,6),(1,5),(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,3}
100100 => [1,3,3] => ([(2,6),(3,6),(4,5),(4,6),(5,6)],7)
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,3}
100101 => [1,3,2,1] => ([(0,6),(1,5),(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,3}
100110 => [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)
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,3}
100111 => [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)
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,3}
101000 => [1,2,4] => ([(3,6),(4,5),(4,6),(5,6)],7)
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,3}
101001 => [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)
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,3}
101010 => [1,2,2,2] => ([(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,3}
101011 => [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)
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,3}
101100 => [1,2,1,3] => ([(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,3}
101101 => [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)
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,3}
101110 => [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)
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,3}
101111 => [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)
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,3}
110000 => [1,1,5] => ([(4,5),(4,6),(5,6)],7)
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,3}
110001 => [1,1,4,1] => ([(0,6),(1,6),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,3}
Description
The genus of a graph.
This is the smallest genus of an oriented surface on which the graph can be embedded without crossings. One can indeed compute the genus as the sum of the genuses for the connected components.
Matching statistic: St001570
Mp00178: Binary words —to composition⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
Mp00154: Graphs —core⟶ Graphs
St001570: Graphs ⟶ ℤResult quality: 4% ●values known / values provided: 4%●distinct values known / distinct values provided: 12%
Mp00184: Integer compositions —to threshold graph⟶ Graphs
Mp00154: Graphs —core⟶ Graphs
St001570: Graphs ⟶ ℤResult quality: 4% ●values known / values provided: 4%●distinct values known / distinct values provided: 12%
Values
0 => [2] => ([],2)
=> ([],1)
=> ? ∊ {0,0}
1 => [1,1] => ([(0,1)],2)
=> ([(0,1)],2)
=> ? ∊ {0,0}
00 => [3] => ([],3)
=> ([],1)
=> ? ∊ {0,0,0}
01 => [2,1] => ([(0,2),(1,2)],3)
=> ([(0,1)],2)
=> ? ∊ {0,0,0}
10 => [1,2] => ([(1,2)],3)
=> ([(0,1)],2)
=> ? ∊ {0,0,0}
11 => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> ([(0,1),(0,2),(1,2)],3)
=> 0
000 => [4] => ([],4)
=> ([],1)
=> ? ∊ {0,0,0,0}
001 => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> ([(0,1)],2)
=> ? ∊ {0,0,0,0}
010 => [2,2] => ([(1,3),(2,3)],4)
=> ([(0,1)],2)
=> ? ∊ {0,0,0,0}
011 => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,2)],3)
=> 0
100 => [1,3] => ([(2,3)],4)
=> ([(0,1)],2)
=> ? ∊ {0,0,0,0}
101 => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,2)],3)
=> 0
110 => [1,1,2] => ([(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,2)],3)
=> 0
111 => [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 0
0000 => [5] => ([],5)
=> ([],1)
=> ? ∊ {0,0,0,0,1}
0001 => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> ([(0,1)],2)
=> ? ∊ {0,0,0,0,1}
0010 => [3,2] => ([(1,4),(2,4),(3,4)],5)
=> ([(0,1)],2)
=> ? ∊ {0,0,0,0,1}
0011 => [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)
=> 0
0100 => [2,3] => ([(2,4),(3,4)],5)
=> ([(0,1)],2)
=> ? ∊ {0,0,0,0,1}
0101 => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 0
0110 => [2,1,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 0
0111 => [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)
=> 0
1000 => [1,4] => ([(3,4)],5)
=> ([(0,1)],2)
=> ? ∊ {0,0,0,0,1}
1001 => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 0
1010 => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 0
1011 => [1,2,1,1] => ([(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)
=> 0
1100 => [1,1,3] => ([(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 0
1101 => [1,1,2,1] => ([(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)
=> 0
1110 => [1,1,1,2] => ([(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)
=> 0
1111 => [1,1,1,1,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 0
00000 => [6] => ([],6)
=> ([],1)
=> ? ∊ {1,1,1,1,1,1}
00001 => [5,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> ([(0,1)],2)
=> ? ∊ {1,1,1,1,1,1}
00010 => [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6)
=> ([(0,1)],2)
=> ? ∊ {1,1,1,1,1,1}
00011 => [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)
=> 0
00100 => [3,3] => ([(2,5),(3,5),(4,5)],6)
=> ([(0,1)],2)
=> ? ∊ {1,1,1,1,1,1}
00101 => [3,2,1] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> 0
00110 => [3,1,2] => ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> 0
00111 => [3,1,1,1] => ([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 0
01000 => [2,4] => ([(3,5),(4,5)],6)
=> ([(0,1)],2)
=> ? ∊ {1,1,1,1,1,1}
01001 => [2,3,1] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> 0
01010 => [2,2,2] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> 0
01011 => [2,2,1,1] => ([(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 0
01100 => [2,1,3] => ([(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> 0
01101 => [2,1,2,1] => ([(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 0
01110 => [2,1,1,2] => ([(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),(1,2),(1,3),(2,3)],4)
=> 0
01111 => [2,1,1,1,1] => ([(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 0
10000 => [1,5] => ([(4,5)],6)
=> ([(0,1)],2)
=> ? ∊ {1,1,1,1,1,1}
10001 => [1,4,1] => ([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> 0
10010 => [1,3,2] => ([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> 0
10011 => [1,3,1,1] => ([(0,4),(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 0
10100 => [1,2,3] => ([(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> 0
10101 => [1,2,2,1] => ([(0,5),(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)
=> 0
10110 => [1,2,1,2] => ([(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)
=> 0
10111 => [1,2,1,1,1] => ([(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 0
11000 => [1,1,4] => ([(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> 0
11001 => [1,1,3,1] => ([(0,5),(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)
=> 0
11010 => [1,1,2,2] => ([(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)
=> 0
11011 => [1,1,2,1,1] => ([(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 0
11100 => [1,1,1,3] => ([(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)
=> 0
11101 => [1,1,1,2,1] => ([(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 0
11110 => [1,1,1,1,2] => ([(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)
=> 0
11111 => [1,1,1,1,1,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 0
000000 => [7] => ([],7)
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,3}
000001 => [6,1] => ([(0,6),(1,6),(2,6),(3,6),(4,6),(5,6)],7)
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,3}
000010 => [5,2] => ([(1,6),(2,6),(3,6),(4,6),(5,6)],7)
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,3}
000011 => [5,1,1] => ([(0,5),(0,6),(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,3}
000100 => [4,3] => ([(2,6),(3,6),(4,6),(5,6)],7)
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,3}
000101 => [4,2,1] => ([(0,6),(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,3}
000110 => [4,1,2] => ([(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,3}
000111 => [4,1,1,1] => ([(0,4),(0,5),(0,6),(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,3}
001000 => [3,4] => ([(3,6),(4,6),(5,6)],7)
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,3}
001001 => [3,3,1] => ([(0,6),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,3}
001010 => [3,2,2] => ([(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,3}
001011 => [3,2,1,1] => ([(0,5),(0,6),(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,3}
001100 => [3,1,3] => ([(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,3}
001101 => [3,1,2,1] => ([(0,6),(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,3}
001110 => [3,1,1,2] => ([(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,3}
001111 => [3,1,1,1,1] => ([(0,3),(0,4),(0,5),(0,6),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,3}
010000 => [2,5] => ([(4,6),(5,6)],7)
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,3}
010001 => [2,4,1] => ([(0,6),(1,6),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,3}
010010 => [2,3,2] => ([(1,6),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,3}
010011 => [2,3,1,1] => ([(0,5),(0,6),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,3}
010100 => [2,2,3] => ([(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,3}
010101 => [2,2,2,1] => ([(0,6),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,3}
010110 => [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)
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,3}
010111 => [2,2,1,1,1] => ([(0,4),(0,5),(0,6),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,3}
011000 => [2,1,4] => ([(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,3}
011001 => [2,1,3,1] => ([(0,6),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,3}
011010 => [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)
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,3}
011011 => [2,1,2,1,1] => ([(0,5),(0,6),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,3}
011100 => [2,1,1,3] => ([(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,3}
011101 => [2,1,1,2,1] => ([(0,6),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,3}
Description
The minimal number of edges to add to make a graph Hamiltonian.
A graph is Hamiltonian if it contains a cycle as a subgraph, which contains all vertices.
Matching statistic: St000666
Mp00178: Binary words —to composition⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
Mp00201: Dyck paths —Ringel⟶ Permutations
St000666: Permutations ⟶ ℤResult quality: 3% ●values known / values provided: 3%●distinct values known / distinct values provided: 25%
Mp00231: Integer compositions —bounce path⟶ Dyck paths
Mp00201: Dyck paths —Ringel⟶ Permutations
St000666: Permutations ⟶ ℤResult quality: 3% ●values known / values provided: 3%●distinct values known / distinct values provided: 25%
Values
0 => [2] => [1,1,0,0]
=> [2,3,1] => 0
1 => [1,1] => [1,0,1,0]
=> [3,1,2] => 0
00 => [3] => [1,1,1,0,0,0]
=> [2,3,4,1] => 0
01 => [2,1] => [1,1,0,0,1,0]
=> [2,4,1,3] => 0
10 => [1,2] => [1,0,1,1,0,0]
=> [3,1,4,2] => 0
11 => [1,1,1] => [1,0,1,0,1,0]
=> [4,1,2,3] => 0
000 => [4] => [1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => 0
001 => [3,1] => [1,1,1,0,0,0,1,0]
=> [2,3,5,1,4] => 0
010 => [2,2] => [1,1,0,0,1,1,0,0]
=> [2,4,1,5,3] => 0
011 => [2,1,1] => [1,1,0,0,1,0,1,0]
=> [2,5,1,3,4] => 0
100 => [1,3] => [1,0,1,1,1,0,0,0]
=> [3,1,4,5,2] => 0
101 => [1,2,1] => [1,0,1,1,0,0,1,0]
=> [3,1,5,2,4] => 0
110 => [1,1,2] => [1,0,1,0,1,1,0,0]
=> [4,1,2,5,3] => 0
111 => [1,1,1,1] => [1,0,1,0,1,0,1,0]
=> [5,1,2,3,4] => 0
0000 => [5] => [1,1,1,1,1,0,0,0,0,0]
=> [2,3,4,5,6,1] => 0
0001 => [4,1] => [1,1,1,1,0,0,0,0,1,0]
=> [2,3,4,6,1,5] => 0
0010 => [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> [2,3,5,1,6,4] => 0
0011 => [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> [2,3,6,1,4,5] => 0
0100 => [2,3] => [1,1,0,0,1,1,1,0,0,0]
=> [2,4,1,5,6,3] => 0
0101 => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> [2,4,1,6,3,5] => 0
0110 => [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
=> [2,5,1,3,6,4] => 0
0111 => [2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
=> [2,6,1,3,4,5] => 0
1000 => [1,4] => [1,0,1,1,1,1,0,0,0,0]
=> [3,1,4,5,6,2] => 0
1001 => [1,3,1] => [1,0,1,1,1,0,0,0,1,0]
=> [3,1,4,6,2,5] => 0
1010 => [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
=> [3,1,5,2,6,4] => 0
1011 => [1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
=> [3,1,6,2,4,5] => 0
1100 => [1,1,3] => [1,0,1,0,1,1,1,0,0,0]
=> [4,1,2,5,6,3] => 1
1101 => [1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
=> [4,1,2,6,3,5] => 0
1110 => [1,1,1,2] => [1,0,1,0,1,0,1,1,0,0]
=> [5,1,2,3,6,4] => 0
1111 => [1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
=> [6,1,2,3,4,5] => 0
00000 => [6] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> [2,3,4,5,6,7,1] => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1}
00001 => [5,1] => [1,1,1,1,1,0,0,0,0,0,1,0]
=> [2,3,4,5,7,1,6] => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1}
00010 => [4,2] => [1,1,1,1,0,0,0,0,1,1,0,0]
=> [2,3,4,6,1,7,5] => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1}
00011 => [4,1,1] => [1,1,1,1,0,0,0,0,1,0,1,0]
=> [2,3,4,7,1,5,6] => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1}
00100 => [3,3] => [1,1,1,0,0,0,1,1,1,0,0,0]
=> [2,3,5,1,6,7,4] => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1}
00101 => [3,2,1] => [1,1,1,0,0,0,1,1,0,0,1,0]
=> [2,3,5,1,7,4,6] => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1}
00110 => [3,1,2] => [1,1,1,0,0,0,1,0,1,1,0,0]
=> [2,3,6,1,4,7,5] => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1}
00111 => [3,1,1,1] => [1,1,1,0,0,0,1,0,1,0,1,0]
=> [2,3,7,1,4,5,6] => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1}
01000 => [2,4] => [1,1,0,0,1,1,1,1,0,0,0,0]
=> [2,4,1,5,6,7,3] => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1}
01001 => [2,3,1] => [1,1,0,0,1,1,1,0,0,0,1,0]
=> [2,4,1,5,7,3,6] => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1}
01010 => [2,2,2] => [1,1,0,0,1,1,0,0,1,1,0,0]
=> [2,4,1,6,3,7,5] => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1}
01011 => [2,2,1,1] => [1,1,0,0,1,1,0,0,1,0,1,0]
=> [2,4,1,7,3,5,6] => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1}
01100 => [2,1,3] => [1,1,0,0,1,0,1,1,1,0,0,0]
=> [2,5,1,3,6,7,4] => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1}
01101 => [2,1,2,1] => [1,1,0,0,1,0,1,1,0,0,1,0]
=> [2,5,1,3,7,4,6] => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1}
01110 => [2,1,1,2] => [1,1,0,0,1,0,1,0,1,1,0,0]
=> [2,6,1,3,4,7,5] => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1}
01111 => [2,1,1,1,1] => [1,1,0,0,1,0,1,0,1,0,1,0]
=> [2,7,1,3,4,5,6] => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1}
10000 => [1,5] => [1,0,1,1,1,1,1,0,0,0,0,0]
=> [3,1,4,5,6,7,2] => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1}
10001 => [1,4,1] => [1,0,1,1,1,1,0,0,0,0,1,0]
=> [3,1,4,5,7,2,6] => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1}
10010 => [1,3,2] => [1,0,1,1,1,0,0,0,1,1,0,0]
=> [3,1,4,6,2,7,5] => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1}
10011 => [1,3,1,1] => [1,0,1,1,1,0,0,0,1,0,1,0]
=> [3,1,4,7,2,5,6] => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1}
10100 => [1,2,3] => [1,0,1,1,0,0,1,1,1,0,0,0]
=> [3,1,5,2,6,7,4] => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1}
10101 => [1,2,2,1] => [1,0,1,1,0,0,1,1,0,0,1,0]
=> [3,1,5,2,7,4,6] => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1}
10110 => [1,2,1,2] => [1,0,1,1,0,0,1,0,1,1,0,0]
=> [3,1,6,2,4,7,5] => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1}
10111 => [1,2,1,1,1] => [1,0,1,1,0,0,1,0,1,0,1,0]
=> [3,1,7,2,4,5,6] => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1}
11000 => [1,1,4] => [1,0,1,0,1,1,1,1,0,0,0,0]
=> [4,1,2,5,6,7,3] => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1}
11001 => [1,1,3,1] => [1,0,1,0,1,1,1,0,0,0,1,0]
=> [4,1,2,5,7,3,6] => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1}
11010 => [1,1,2,2] => [1,0,1,0,1,1,0,0,1,1,0,0]
=> [4,1,2,6,3,7,5] => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1}
11011 => [1,1,2,1,1] => [1,0,1,0,1,1,0,0,1,0,1,0]
=> [4,1,2,7,3,5,6] => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1}
11100 => [1,1,1,3] => [1,0,1,0,1,0,1,1,1,0,0,0]
=> [5,1,2,3,6,7,4] => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1}
11101 => [1,1,1,2,1] => [1,0,1,0,1,0,1,1,0,0,1,0]
=> [5,1,2,3,7,4,6] => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1}
11110 => [1,1,1,1,2] => [1,0,1,0,1,0,1,0,1,1,0,0]
=> [6,1,2,3,4,7,5] => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1}
11111 => [1,1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0,1,0]
=> [7,1,2,3,4,5,6] => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1}
000000 => [7] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [2,3,4,5,6,7,8,1] => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,3}
000001 => [6,1] => [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> [2,3,4,5,6,8,1,7] => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,3}
000010 => [5,2] => [1,1,1,1,1,0,0,0,0,0,1,1,0,0]
=> [2,3,4,5,7,1,8,6] => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,3}
000011 => [5,1,1] => [1,1,1,1,1,0,0,0,0,0,1,0,1,0]
=> [2,3,4,5,8,1,6,7] => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,3}
000100 => [4,3] => [1,1,1,1,0,0,0,0,1,1,1,0,0,0]
=> [2,3,4,6,1,7,8,5] => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,3}
000101 => [4,2,1] => [1,1,1,1,0,0,0,0,1,1,0,0,1,0]
=> [2,3,4,6,1,8,5,7] => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,3}
000110 => [4,1,2] => [1,1,1,1,0,0,0,0,1,0,1,1,0,0]
=> [2,3,4,7,1,5,8,6] => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,3}
000111 => [4,1,1,1] => [1,1,1,1,0,0,0,0,1,0,1,0,1,0]
=> [2,3,4,8,1,5,6,7] => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,3}
001000 => [3,4] => [1,1,1,0,0,0,1,1,1,1,0,0,0,0]
=> [2,3,5,1,6,7,8,4] => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,3}
001001 => [3,3,1] => [1,1,1,0,0,0,1,1,1,0,0,0,1,0]
=> [2,3,5,1,6,8,4,7] => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,3}
001010 => [3,2,2] => [1,1,1,0,0,0,1,1,0,0,1,1,0,0]
=> [2,3,5,1,7,4,8,6] => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,3}
001011 => [3,2,1,1] => [1,1,1,0,0,0,1,1,0,0,1,0,1,0]
=> [2,3,5,1,8,4,6,7] => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,3}
001100 => [3,1,3] => [1,1,1,0,0,0,1,0,1,1,1,0,0,0]
=> [2,3,6,1,4,7,8,5] => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,3}
001101 => [3,1,2,1] => [1,1,1,0,0,0,1,0,1,1,0,0,1,0]
=> [2,3,6,1,4,8,5,7] => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,3}
001110 => [3,1,1,2] => [1,1,1,0,0,0,1,0,1,0,1,1,0,0]
=> [2,3,7,1,4,5,8,6] => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,3}
001111 => [3,1,1,1,1] => [1,1,1,0,0,0,1,0,1,0,1,0,1,0]
=> [2,3,8,1,4,5,6,7] => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,3}
010000 => [2,5] => [1,1,0,0,1,1,1,1,1,0,0,0,0,0]
=> [2,4,1,5,6,7,8,3] => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,3}
010001 => [2,4,1] => [1,1,0,0,1,1,1,1,0,0,0,0,1,0]
=> [2,4,1,5,6,8,3,7] => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,3}
Description
The number of right tethers of a permutation.
Let $\pi$ be a permutation of length $n$. A raft of $\pi$ is a non-empty maximal sequence of consecutive small ascents, [[St000441]], and a right tether is a large ascent between two consecutive rafts of $\pi$.
See Definition 3.10 and Example 3.11 in [1].
Matching statistic: St001493
Mp00178: Binary words —to composition⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
Mp00199: Dyck paths —prime Dyck path⟶ Dyck paths
St001493: Dyck paths ⟶ ℤResult quality: 3% ●values known / values provided: 3%●distinct values known / distinct values provided: 25%
Mp00231: Integer compositions —bounce path⟶ Dyck paths
Mp00199: Dyck paths —prime Dyck path⟶ Dyck paths
St001493: Dyck paths ⟶ ℤResult quality: 3% ●values known / values provided: 3%●distinct values known / distinct values provided: 25%
Values
0 => [2] => [1,1,0,0]
=> [1,1,1,0,0,0]
=> 1 = 0 + 1
1 => [1,1] => [1,0,1,0]
=> [1,1,0,1,0,0]
=> 1 = 0 + 1
00 => [3] => [1,1,1,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> 1 = 0 + 1
01 => [2,1] => [1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> 1 = 0 + 1
10 => [1,2] => [1,0,1,1,0,0]
=> [1,1,0,1,1,0,0,0]
=> 1 = 0 + 1
11 => [1,1,1] => [1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,0]
=> 1 = 0 + 1
000 => [4] => [1,1,1,1,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> 1 = 0 + 1
001 => [3,1] => [1,1,1,0,0,0,1,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> 1 = 0 + 1
010 => [2,2] => [1,1,0,0,1,1,0,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> 1 = 0 + 1
011 => [2,1,1] => [1,1,0,0,1,0,1,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> 1 = 0 + 1
100 => [1,3] => [1,0,1,1,1,0,0,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> 1 = 0 + 1
101 => [1,2,1] => [1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> 1 = 0 + 1
110 => [1,1,2] => [1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> 1 = 0 + 1
111 => [1,1,1,1] => [1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> 1 = 0 + 1
0000 => [5] => [1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0]
=> 1 = 0 + 1
0001 => [4,1] => [1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,1,1,0,0,0,0,1,0,0]
=> 1 = 0 + 1
0010 => [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> [1,1,1,1,0,0,0,1,1,0,0,0]
=> 1 = 0 + 1
0011 => [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> [1,1,1,1,0,0,0,1,0,1,0,0]
=> 1 = 0 + 1
0100 => [2,3] => [1,1,0,0,1,1,1,0,0,0]
=> [1,1,1,0,0,1,1,1,0,0,0,0]
=> 1 = 0 + 1
0101 => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> [1,1,1,0,0,1,1,0,0,1,0,0]
=> 1 = 0 + 1
0110 => [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
=> [1,1,1,0,0,1,0,1,1,0,0,0]
=> 1 = 0 + 1
0111 => [2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
=> [1,1,1,0,0,1,0,1,0,1,0,0]
=> 1 = 0 + 1
1000 => [1,4] => [1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,1,1,1,0,0,0,0,0]
=> 1 = 0 + 1
1001 => [1,3,1] => [1,0,1,1,1,0,0,0,1,0]
=> [1,1,0,1,1,1,0,0,0,1,0,0]
=> 2 = 1 + 1
1010 => [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,1,1,0,0,1,1,0,0,0]
=> 1 = 0 + 1
1011 => [1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
=> [1,1,0,1,1,0,0,1,0,1,0,0]
=> 1 = 0 + 1
1100 => [1,1,3] => [1,0,1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,1,1,0,0,0,0]
=> 1 = 0 + 1
1101 => [1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,0,1,1,0,0,1,0,0]
=> 1 = 0 + 1
1110 => [1,1,1,2] => [1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,1,1,0,0,0]
=> 1 = 0 + 1
1111 => [1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,1,0,0]
=> 1 = 0 + 1
00000 => [6] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1} + 1
00001 => [5,1] => [1,1,1,1,1,0,0,0,0,0,1,0]
=> [1,1,1,1,1,1,0,0,0,0,0,1,0,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1} + 1
00010 => [4,2] => [1,1,1,1,0,0,0,0,1,1,0,0]
=> [1,1,1,1,1,0,0,0,0,1,1,0,0,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1} + 1
00011 => [4,1,1] => [1,1,1,1,0,0,0,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,1,0,1,0,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1} + 1
00100 => [3,3] => [1,1,1,0,0,0,1,1,1,0,0,0]
=> [1,1,1,1,0,0,0,1,1,1,0,0,0,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1} + 1
00101 => [3,2,1] => [1,1,1,0,0,0,1,1,0,0,1,0]
=> [1,1,1,1,0,0,0,1,1,0,0,1,0,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1} + 1
00110 => [3,1,2] => [1,1,1,0,0,0,1,0,1,1,0,0]
=> [1,1,1,1,0,0,0,1,0,1,1,0,0,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1} + 1
00111 => [3,1,1,1] => [1,1,1,0,0,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,1,0,1,0,1,0,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1} + 1
01000 => [2,4] => [1,1,0,0,1,1,1,1,0,0,0,0]
=> [1,1,1,0,0,1,1,1,1,0,0,0,0,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1} + 1
01001 => [2,3,1] => [1,1,0,0,1,1,1,0,0,0,1,0]
=> [1,1,1,0,0,1,1,1,0,0,0,1,0,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1} + 1
01010 => [2,2,2] => [1,1,0,0,1,1,0,0,1,1,0,0]
=> [1,1,1,0,0,1,1,0,0,1,1,0,0,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1} + 1
01011 => [2,2,1,1] => [1,1,0,0,1,1,0,0,1,0,1,0]
=> [1,1,1,0,0,1,1,0,0,1,0,1,0,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1} + 1
01100 => [2,1,3] => [1,1,0,0,1,0,1,1,1,0,0,0]
=> [1,1,1,0,0,1,0,1,1,1,0,0,0,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1} + 1
01101 => [2,1,2,1] => [1,1,0,0,1,0,1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,1,1,0,0,1,0,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1} + 1
01110 => [2,1,1,2] => [1,1,0,0,1,0,1,0,1,1,0,0]
=> [1,1,1,0,0,1,0,1,0,1,1,0,0,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1} + 1
01111 => [2,1,1,1,1] => [1,1,0,0,1,0,1,0,1,0,1,0]
=> [1,1,1,0,0,1,0,1,0,1,0,1,0,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1} + 1
10000 => [1,5] => [1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,1,0,1,1,1,1,1,0,0,0,0,0,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1} + 1
10001 => [1,4,1] => [1,0,1,1,1,1,0,0,0,0,1,0]
=> [1,1,0,1,1,1,1,0,0,0,0,1,0,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1} + 1
10010 => [1,3,2] => [1,0,1,1,1,0,0,0,1,1,0,0]
=> [1,1,0,1,1,1,0,0,0,1,1,0,0,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1} + 1
10011 => [1,3,1,1] => [1,0,1,1,1,0,0,0,1,0,1,0]
=> [1,1,0,1,1,1,0,0,0,1,0,1,0,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1} + 1
10100 => [1,2,3] => [1,0,1,1,0,0,1,1,1,0,0,0]
=> [1,1,0,1,1,0,0,1,1,1,0,0,0,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1} + 1
10101 => [1,2,2,1] => [1,0,1,1,0,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,1,1,0,0,1,0,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1} + 1
10110 => [1,2,1,2] => [1,0,1,1,0,0,1,0,1,1,0,0]
=> [1,1,0,1,1,0,0,1,0,1,1,0,0,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1} + 1
10111 => [1,2,1,1,1] => [1,0,1,1,0,0,1,0,1,0,1,0]
=> [1,1,0,1,1,0,0,1,0,1,0,1,0,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1} + 1
11000 => [1,1,4] => [1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,0,1,1,1,1,0,0,0,0,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1} + 1
11001 => [1,1,3,1] => [1,0,1,0,1,1,1,0,0,0,1,0]
=> [1,1,0,1,0,1,1,1,0,0,0,1,0,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1} + 1
11010 => [1,1,2,2] => [1,0,1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,1,0,1,1,0,0,1,1,0,0,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1} + 1
11011 => [1,1,2,1,1] => [1,0,1,0,1,1,0,0,1,0,1,0]
=> [1,1,0,1,0,1,1,0,0,1,0,1,0,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1} + 1
11100 => [1,1,1,3] => [1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,0,1,1,1,0,0,0,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1} + 1
11101 => [1,1,1,2,1] => [1,0,1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,0,1,0,1,1,0,0,1,0,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1} + 1
11110 => [1,1,1,1,2] => [1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,1,0,1,1,0,0,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1} + 1
11111 => [1,1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1} + 1
000000 => [7] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,3} + 1
000001 => [6,1] => [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,1,0,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,3} + 1
000010 => [5,2] => [1,1,1,1,1,0,0,0,0,0,1,1,0,0]
=> [1,1,1,1,1,1,0,0,0,0,0,1,1,0,0,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,3} + 1
000011 => [5,1,1] => [1,1,1,1,1,0,0,0,0,0,1,0,1,0]
=> [1,1,1,1,1,1,0,0,0,0,0,1,0,1,0,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,3} + 1
000100 => [4,3] => [1,1,1,1,0,0,0,0,1,1,1,0,0,0]
=> [1,1,1,1,1,0,0,0,0,1,1,1,0,0,0,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,3} + 1
000101 => [4,2,1] => [1,1,1,1,0,0,0,0,1,1,0,0,1,0]
=> [1,1,1,1,1,0,0,0,0,1,1,0,0,1,0,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,3} + 1
000110 => [4,1,2] => [1,1,1,1,0,0,0,0,1,0,1,1,0,0]
=> [1,1,1,1,1,0,0,0,0,1,0,1,1,0,0,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,3} + 1
000111 => [4,1,1,1] => [1,1,1,1,0,0,0,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,1,0,1,0,1,0,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,3} + 1
001000 => [3,4] => [1,1,1,0,0,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,1,1,1,1,0,0,0,0,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,3} + 1
001001 => [3,3,1] => [1,1,1,0,0,0,1,1,1,0,0,0,1,0]
=> [1,1,1,1,0,0,0,1,1,1,0,0,0,1,0,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,3} + 1
001010 => [3,2,2] => [1,1,1,0,0,0,1,1,0,0,1,1,0,0]
=> [1,1,1,1,0,0,0,1,1,0,0,1,1,0,0,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,3} + 1
001011 => [3,2,1,1] => [1,1,1,0,0,0,1,1,0,0,1,0,1,0]
=> [1,1,1,1,0,0,0,1,1,0,0,1,0,1,0,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,3} + 1
001100 => [3,1,3] => [1,1,1,0,0,0,1,0,1,1,1,0,0,0]
=> [1,1,1,1,0,0,0,1,0,1,1,1,0,0,0,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,3} + 1
001101 => [3,1,2,1] => [1,1,1,0,0,0,1,0,1,1,0,0,1,0]
=> [1,1,1,1,0,0,0,1,0,1,1,0,0,1,0,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,3} + 1
001110 => [3,1,1,2] => [1,1,1,0,0,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,0,0,0,1,0,1,0,1,1,0,0,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,3} + 1
001111 => [3,1,1,1,1] => [1,1,1,0,0,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,1,0,1,0,1,0,1,0,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,3} + 1
010000 => [2,5] => [1,1,0,0,1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,0,0,1,1,1,1,1,0,0,0,0,0,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,3} + 1
010001 => [2,4,1] => [1,1,0,0,1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,0,0,1,1,1,1,0,0,0,0,1,0,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,3} + 1
Description
The number of simple modules with maximal even projective dimension in the corresponding Nakayama algebra.
The following 89 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000879The number of long braid edges in the graph of braid moves of a permutation. St000882The number of connected components of short braid edges in the graph of braid moves of a permutation. St001621The number of atoms of a lattice. St001624The breadth of a lattice. St000093The cardinality of a maximal independent set of vertices of a graph. St001878The projective dimension of the simple modules corresponding to the minimum of L in the incidence algebra of the lattice L. St000098The chromatic number of a graph. St000283The size of the preimage of the map 'to graph' from Binary trees to Graphs. St000322The skewness of a graph. St000323The minimal crossing number of a graph. St000351The determinant of the adjacency matrix of a graph. St000368The Altshuler-Steinberg determinant of a graph. St000379The number of Hamiltonian cycles in a graph. St000403The Szeged index minus the Wiener index of a graph. St000637The length of the longest cycle in a graph. St000671The maximin edge-connectivity for choosing a subgraph. St000699The toughness times the least common multiple of 1,. St000948The chromatic discriminant of a graph. St001069The coefficient of the monomial xy of the Tutte polynomial of the graph. St001119The length of a shortest maximal path in a graph. St001271The competition number of a graph. St001281The normalized isoperimetric number of a graph. St001305The number of induced cycles on four vertices in a graph. St001307The number of induced stars on four vertices in a graph. St001309The number of four-cliques in a graph. St001310The number of induced diamond graphs in a graph. St001311The cyclomatic number of a graph. St001317The minimal number of occurrences of the forest-pattern in a linear ordering of the vertices of the graph. St001320The minimal number of occurrences of the path-pattern in a linear ordering of the vertices of the graph. St001323The independence gap of a graph. St001324The minimal number of occurrences of the chordal-pattern in a linear ordering of the vertices of the graph. St001325The minimal number of occurrences of the comparability-pattern in a linear ordering of the vertices of the graph. St001326The minimal number of occurrences of the interval-pattern in a linear ordering of the vertices of the graph. St001328The minimal number of occurrences of the bipartite-pattern in a linear ordering of the vertices of the graph. St001329The minimal number of occurrences of the outerplanar pattern in a linear ordering of the vertices of the graph. St001331The size of the minimal feedback vertex set. St001334The minimal number of occurrences of the 3-colorable pattern in a linear ordering of the vertices of the graph. St001335The cardinality of a minimal cycle-isolating set of a graph. St001336The minimal number of vertices in a graph whose complement is triangle-free. St001357The maximal degree of a regular spanning subgraph of a graph. St001367The smallest number which does not occur as degree of a vertex in a graph. St001395The number of strictly unfriendly partitions of a graph. St001638The book thickness of a graph. St001689The number of celebrities in a graph. St001702The absolute value of the determinant of the adjacency matrix of a graph. St001736The total number of cycles in a graph. St001793The difference between the clique number and the chromatic number of a graph. St001794Half the number of sets of vertices in a graph which are dominating and non-blocking. St001795The binary logarithm of the evaluation of the Tutte polynomial of the graph at (x,y) equal to (-1,-1). St001796The absolute value of the quotient of the Tutte polynomial of the graph at (1,1) and (-1,-1). St001797The number of overfull subgraphs of a graph. St001970The signature of a graph. St000266The number of spanning subgraphs of a graph with the same connected components. St000267The number of maximal spanning forests contained in a graph. St000773The multiplicity of the largest Laplacian eigenvalue in a graph. St000775The multiplicity of the largest eigenvalue in a graph. St000785The number of distinct colouring schemes of a graph. St001272The number of graphs with the same degree sequence. St001316The domatic number of a graph. St001333The cardinality of a minimal edge-isolating set of a graph. St001475The evaluation of the Tutte polynomial of the graph at (x,y) equal to (1,0). St001476The evaluation of the Tutte polynomial of the graph at (x,y) equal to (1,-1). St001496The number of graphs with the same Laplacian spectrum as the given graph. St001546The number of monomials in the Tutte polynomial of a graph. St000636The hull number of a graph. St001029The size of the core of a graph. St001109The number of proper colourings of a graph with as few colours as possible. St001654The monophonic hull number of a graph. St000447The number of pairs of vertices of a graph with distance 3. St000449The number of pairs of vertices of a graph with distance 4. St000552The number of cut vertices of a graph. St001057The Grundy value of the game of creating an independent set in a graph. St001691The number of kings in a graph. St000260The radius of a connected graph. St000273The domination number of a graph. St000544The cop number of a graph. St000553The number of blocks of a graph. St000916The packing number of a graph. St001739The number of graphs with the same edge polytope as the given graph. St001740The number of graphs with the same symmetric edge polytope as the given graph. St001776The degree of the minimal polynomial of the largest Laplacian eigenvalue of a graph. St001829The common independence number of a graph. St001934The number of monotone factorisations of genus zero of a permutation of given cycle type. St000258The burning number of a graph. St000786The maximal number of occurrences of a colour in a proper colouring of a graph. St000918The 2-limited packing number of a graph. St001322The size of a minimal independent dominating set in a graph. St001337The upper domination number of a graph. St001338The upper irredundance number of a graph.
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