Your data matches 2 different statistics following compositions of up to 3 maps.
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Matching statistic: St001657
Mapping
Mp00097: Binary words —delta morphism⟶ Integer compositions
Mp00180: Integer compositions —to ribbon⟶ Skew partitions
Mp00183: Skew partitions —inner shape⟶ Integer partitions
St001657: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
001 => [2,1] => [[2,2],[1]]
 => [1]
 => 0
110 => [2,1] => [[2,2],[1]]
 => [1]
 => 0
0001 => [3,1] => [[3,3],[2]]
 => [2]
 => 1
0010 => [2,1,1] => [[2,2,2],[1,1]]
 => [1,1]
 => 0
0011 => [2,2] => [[3,2],[1]]
 => [1]
 => 0
0110 => [1,2,1] => [[2,2,1],[1]]
 => [1]
 => 0
1001 => [1,2,1] => [[2,2,1],[1]]
 => [1]
 => 0
1100 => [2,2] => [[3,2],[1]]
 => [1]
 => 0
1101 => [2,1,1] => [[2,2,2],[1,1]]
 => [1,1]
 => 0
1110 => [3,1] => [[3,3],[2]]
 => [2]
 => 1
00001 => [4,1] => [[4,4],[3]]
 => [3]
 => 0
00010 => [3,1,1] => [[3,3,3],[2,2]]
 => [2,2]
 => 2
00011 => [3,2] => [[4,3],[2]]
 => [2]
 => 1
00100 => [2,1,2] => [[3,2,2],[1,1]]
 => [1,1]
 => 0
00101 => [2,1,1,1] => [[2,2,2,2],[1,1,1]]
 => [1,1,1]
 => 0
00110 => [2,2,1] => [[3,3,2],[2,1]]
 => [2,1]
 => 1
00111 => [2,3] => [[4,2],[1]]
 => [1]
 => 0
01001 => [1,1,2,1] => [[2,2,1,1],[1]]
 => [1]
 => 0
01100 => [1,2,2] => [[3,2,1],[1]]
 => [1]
 => 0
01101 => [1,2,1,1] => [[2,2,2,1],[1,1]]
 => [1,1]
 => 0
01110 => [1,3,1] => [[3,3,1],[2]]
 => [2]
 => 1
10001 => [1,3,1] => [[3,3,1],[2]]
 => [2]
 => 1
10010 => [1,2,1,1] => [[2,2,2,1],[1,1]]
 => [1,1]
 => 0
10011 => [1,2,2] => [[3,2,1],[1]]
 => [1]
 => 0
10110 => [1,1,2,1] => [[2,2,1,1],[1]]
 => [1]
 => 0
11000 => [2,3] => [[4,2],[1]]
 => [1]
 => 0
11001 => [2,2,1] => [[3,3,2],[2,1]]
 => [2,1]
 => 1
11010 => [2,1,1,1] => [[2,2,2,2],[1,1,1]]
 => [1,1,1]
 => 0
11011 => [2,1,2] => [[3,2,2],[1,1]]
 => [1,1]
 => 0
11100 => [3,2] => [[4,3],[2]]
 => [2]
 => 1
11101 => [3,1,1] => [[3,3,3],[2,2]]
 => [2,2]
 => 2
11110 => [4,1] => [[4,4],[3]]
 => [3]
 => 0
000001 => [5,1] => [[5,5],[4]]
 => [4]
 => 0
000010 => [4,1,1] => [[4,4,4],[3,3]]
 => [3,3]
 => 0
000011 => [4,2] => [[5,4],[3]]
 => [3]
 => 0
000100 => [3,1,2] => [[4,3,3],[2,2]]
 => [2,2]
 => 2
000101 => [3,1,1,1] => [[3,3,3,3],[2,2,2]]
 => [2,2,2]
 => 3
000110 => [3,2,1] => [[4,4,3],[3,2]]
 => [3,2]
 => 1
000111 => [3,3] => [[5,3],[2]]
 => [2]
 => 1
001000 => [2,1,3] => [[4,2,2],[1,1]]
 => [1,1]
 => 0
001001 => [2,1,2,1] => [[3,3,2,2],[2,1,1]]
 => [2,1,1]
 => 1
001010 => [2,1,1,1,1] => [[2,2,2,2,2],[1,1,1,1]]
 => [1,1,1,1]
 => 0
001011 => [2,1,1,2] => [[3,2,2,2],[1,1,1]]
 => [1,1,1]
 => 0
001100 => [2,2,2] => [[4,3,2],[2,1]]
 => [2,1]
 => 1
001101 => [2,2,1,1] => [[3,3,3,2],[2,2,1]]
 => [2,2,1]
 => 2
001110 => [2,3,1] => [[4,4,2],[3,1]]
 => [3,1]
 => 0
001111 => [2,4] => [[5,2],[1]]
 => [1]
 => 0
010001 => [1,1,3,1] => [[3,3,1,1],[2]]
 => [2]
 => 1
010010 => [1,1,2,1,1] => [[2,2,2,1,1],[1,1]]
 => [1,1]
 => 0
010011 => [1,1,2,2] => [[3,2,1,1],[1]]
 => [1]
 => 0
Description
The number of twos in an integer partition. The total number of twos in all partitions of $n$ is equal to the total number of singletons [[St001484]] in all partitions of $n-1$, see [1].
Matching statistic: St001330
Mapping
Mp00097: Binary words —delta morphism⟶ Integer compositions
Mp00041: Integer compositions —conjugate⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St001330: Graphs ⟶ ℤResult quality: 6% ●values known / values provided: 6%●distinct values known / distinct values provided: 12%
Values
001 => [2,1] => [2,1] => ([(0,2),(1,2)],3)
 => 2 = 0 + 2
110 => [2,1] => [2,1] => ([(0,2),(1,2)],3)
 => 2 = 0 + 2
0001 => [3,1] => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => ? = 1 + 2
0010 => [2,1,1] => [3,1] => ([(0,3),(1,3),(2,3)],4)
 => 2 = 0 + 2
0011 => [2,2] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => ? = 0 + 2
0110 => [1,2,1] => [2,2] => ([(1,3),(2,3)],4)
 => 2 = 0 + 2
1001 => [1,2,1] => [2,2] => ([(1,3),(2,3)],4)
 => 2 = 0 + 2
1100 => [2,2] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => ? = 0 + 2
1101 => [2,1,1] => [3,1] => ([(0,3),(1,3),(2,3)],4)
 => 2 = 0 + 2
1110 => [3,1] => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => ? = 1 + 2
00001 => [4,1] => [2,1,1,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 0 + 2
00010 => [3,1,1] => [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 2 + 2
00011 => [3,2] => [1,2,1,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 1 + 2
00100 => [2,1,2] => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 0 + 2
00101 => [2,1,1,1] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 2 = 0 + 2
00110 => [2,2,1] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 1 + 2
00111 => [2,3] => [1,1,2,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 0 + 2
01001 => [1,1,2,1] => [2,3] => ([(2,4),(3,4)],5)
 => 2 = 0 + 2
01100 => [1,2,2] => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 0 + 2
01101 => [1,2,1,1] => [3,2] => ([(1,4),(2,4),(3,4)],5)
 => 2 = 0 + 2
01110 => [1,3,1] => [2,1,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 1 + 2
10001 => [1,3,1] => [2,1,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 1 + 2
10010 => [1,2,1,1] => [3,2] => ([(1,4),(2,4),(3,4)],5)
 => 2 = 0 + 2
10011 => [1,2,2] => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 0 + 2
10110 => [1,1,2,1] => [2,3] => ([(2,4),(3,4)],5)
 => 2 = 0 + 2
11000 => [2,3] => [1,1,2,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 0 + 2
11001 => [2,2,1] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 1 + 2
11010 => [2,1,1,1] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 2 = 0 + 2
11011 => [2,1,2] => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 0 + 2
11100 => [3,2] => [1,2,1,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 1 + 2
11101 => [3,1,1] => [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 2 + 2
11110 => [4,1] => [2,1,1,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 0 + 2
000001 => [5,1] => [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 + 2
000010 => [4,1,1] => [3,1,1,1] => ([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 0 + 2
000011 => [4,2] => [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 + 2
000100 => [3,1,2] => [1,3,1,1] => ([(0,4),(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 2 + 2
000101 => [3,1,1,1] => [4,1,1] => ([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 3 + 2
000110 => [3,2,1] => [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)
 => ? = 1 + 2
000111 => [3,3] => [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)
 => ? = 1 + 2
001000 => [2,1,3] => [1,1,3,1] => ([(0,5),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 0 + 2
001001 => [2,1,2,1] => [2,3,1] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 1 + 2
001010 => [2,1,1,1,1] => [5,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
 => 2 = 0 + 2
001011 => [2,1,1,2] => [1,4,1] => ([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 0 + 2
001100 => [2,2,2] => [1,2,2,1] => ([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 1 + 2
001101 => [2,2,1,1] => [3,2,1] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 2 + 2
001110 => [2,3,1] => [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 + 2
001111 => [2,4] => [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 + 2
010001 => [1,1,3,1] => [2,1,3] => ([(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 1 + 2
010010 => [1,1,2,1,1] => [3,3] => ([(2,5),(3,5),(4,5)],6)
 => 2 = 0 + 2
010011 => [1,1,2,2] => [1,2,3] => ([(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 0 + 2
010110 => [1,1,1,2,1] => [2,4] => ([(3,5),(4,5)],6)
 => 2 = 0 + 2
011000 => [1,2,3] => [1,1,2,2] => ([(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 0 + 2
011001 => [1,2,2,1] => [2,2,2] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 1 + 2
011010 => [1,2,1,1,1] => [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6)
 => 2 = 0 + 2
011011 => [1,2,1,2] => [1,3,2] => ([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 0 + 2
011100 => [1,3,2] => [1,2,1,2] => ([(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 1 + 2
011101 => [1,3,1,1] => [3,1,2] => ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 2 + 2
011110 => [1,4,1] => [2,1,1,2] => ([(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 0 + 2
100001 => [1,4,1] => [2,1,1,2] => ([(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 0 + 2
100010 => [1,3,1,1] => [3,1,2] => ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 2 + 2
100011 => [1,3,2] => [1,2,1,2] => ([(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 1 + 2
100100 => [1,2,1,2] => [1,3,2] => ([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 0 + 2
100101 => [1,2,1,1,1] => [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6)
 => 2 = 0 + 2
100110 => [1,2,2,1] => [2,2,2] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 1 + 2
100111 => [1,2,3] => [1,1,2,2] => ([(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 0 + 2
101001 => [1,1,1,2,1] => [2,4] => ([(3,5),(4,5)],6)
 => 2 = 0 + 2
101100 => [1,1,2,2] => [1,2,3] => ([(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 0 + 2
101101 => [1,1,2,1,1] => [3,3] => ([(2,5),(3,5),(4,5)],6)
 => 2 = 0 + 2
101110 => [1,1,3,1] => [2,1,3] => ([(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 1 + 2
110101 => [2,1,1,1,1] => [5,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
 => 2 = 0 + 2
0010101 => [2,1,1,1,1,1] => [6,1] => ([(0,6),(1,6),(2,6),(3,6),(4,6),(5,6)],7)
 => 2 = 0 + 2
1101010 => [2,1,1,1,1,1] => [6,1] => ([(0,6),(1,6),(2,6),(3,6),(4,6),(5,6)],7)
 => 2 = 0 + 2
00101010 => [2,1,1,1,1,1,1] => [7,1] => ([(0,7),(1,7),(2,7),(3,7),(4,7),(5,7),(6,7)],8)
 => 2 = 0 + 2
11010101 => [2,1,1,1,1,1,1] => [7,1] => ([(0,7),(1,7),(2,7),(3,7),(4,7),(5,7),(6,7)],8)
 => 2 = 0 + 2
Description
The hat guessing number of a graph. Suppose that each vertex of a graph corresponds to a player, wearing a hat whose color is arbitrarily chosen from a set of $q$ possible colors. Each player can see the hat colors of his neighbors, but not his own hat color. All of the players are asked to guess their own hat colors simultaneously, according to a predetermined guessing strategy and the hat colors they see, where no communication between them is allowed. The hat guessing number $HG(G)$ of a graph $G$ is the largest integer $q$ such that there exists a guessing strategy guaranteeing at least one correct guess for any hat assignment of $q$ possible colors. Because it suffices that a single player guesses correctly, the hat guessing number of a graph is the maximum of the hat guessing numbers of its connected components.