Processing math: 100%

Your data matches 21 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)
Matching statistic: St000068
Mp00178: Binary words to compositionInteger compositions
Mp00180: Integer compositions to ribbonSkew partitions
Mp00185: Skew partitions cell posetPosets
St000068: Posets ⟶ ℤResult quality: 100% values known / values provided: 100%distinct values known / distinct values provided: 100%
Values
0 => [2] => [[2],[]]
=> ([(0,1)],2)
=> 1
1 => [1,1] => [[1,1],[]]
=> ([(0,1)],2)
=> 1
00 => [3] => [[3],[]]
=> ([(0,2),(2,1)],3)
=> 1
01 => [2,1] => [[2,2],[1]]
=> ([(0,2),(1,2)],3)
=> 2
10 => [1,2] => [[2,1],[]]
=> ([(0,1),(0,2)],3)
=> 1
11 => [1,1,1] => [[1,1,1],[]]
=> ([(0,2),(2,1)],3)
=> 1
000 => [4] => [[4],[]]
=> ([(0,3),(2,1),(3,2)],4)
=> 1
001 => [3,1] => [[3,3],[2]]
=> ([(0,3),(1,2),(2,3)],4)
=> 2
010 => [2,2] => [[3,2],[1]]
=> ([(0,3),(1,2),(1,3)],4)
=> 2
011 => [2,1,1] => [[2,2,2],[1,1]]
=> ([(0,3),(1,2),(2,3)],4)
=> 2
100 => [1,3] => [[3,1],[]]
=> ([(0,2),(0,3),(3,1)],4)
=> 1
101 => [1,2,1] => [[2,2,1],[1]]
=> ([(0,3),(1,2),(1,3)],4)
=> 2
110 => [1,1,2] => [[2,1,1],[]]
=> ([(0,2),(0,3),(3,1)],4)
=> 1
111 => [1,1,1,1] => [[1,1,1,1],[]]
=> ([(0,3),(2,1),(3,2)],4)
=> 1
0000 => [5] => [[5],[]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1
0001 => [4,1] => [[4,4],[3]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> 2
0010 => [3,2] => [[4,3],[2]]
=> ([(0,3),(1,2),(1,4),(3,4)],5)
=> 2
0011 => [3,1,1] => [[3,3,3],[2,2]]
=> ([(0,3),(1,2),(2,4),(3,4)],5)
=> 2
0100 => [2,3] => [[4,2],[1]]
=> ([(0,4),(1,2),(1,4),(2,3)],5)
=> 2
0101 => [2,2,1] => [[3,3,2],[2,1]]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 3
0110 => [2,1,2] => [[3,2,2],[1,1]]
=> ([(0,4),(1,2),(1,3),(3,4)],5)
=> 2
0111 => [2,1,1,1] => [[2,2,2,2],[1,1,1]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> 2
1000 => [1,4] => [[4,1],[]]
=> ([(0,2),(0,4),(3,1),(4,3)],5)
=> 1
1001 => [1,3,1] => [[3,3,1],[2]]
=> ([(0,4),(1,2),(1,3),(3,4)],5)
=> 2
1010 => [1,2,2] => [[3,2,1],[1]]
=> ([(0,3),(0,4),(1,2),(1,4)],5)
=> 2
1011 => [1,2,1,1] => [[2,2,2,1],[1,1]]
=> ([(0,3),(1,2),(1,4),(3,4)],5)
=> 2
1100 => [1,1,3] => [[3,1,1],[]]
=> ([(0,3),(0,4),(3,2),(4,1)],5)
=> 1
1101 => [1,1,2,1] => [[2,2,1,1],[1]]
=> ([(0,4),(1,2),(1,4),(2,3)],5)
=> 2
1110 => [1,1,1,2] => [[2,1,1,1],[]]
=> ([(0,2),(0,4),(3,1),(4,3)],5)
=> 1
1111 => [1,1,1,1,1] => [[1,1,1,1,1],[]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1
00000 => [6] => [[6],[]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 1
00001 => [5,1] => [[5,5],[4]]
=> ([(0,5),(1,4),(2,5),(3,2),(4,3)],6)
=> 2
00010 => [4,2] => [[5,4],[3]]
=> ([(0,4),(1,3),(1,5),(2,5),(4,2)],6)
=> 2
00011 => [4,1,1] => [[4,4,4],[3,3]]
=> ([(0,3),(1,4),(2,5),(3,5),(4,2)],6)
=> 2
00100 => [3,3] => [[5,3],[2]]
=> ([(0,3),(1,4),(1,5),(3,5),(4,2)],6)
=> 2
00101 => [3,2,1] => [[4,4,3],[3,2]]
=> ([(0,4),(1,4),(1,5),(2,3),(3,5)],6)
=> 3
00110 => [3,1,2] => [[4,3,3],[2,2]]
=> ([(0,4),(1,2),(1,3),(3,5),(4,5)],6)
=> 2
00111 => [3,1,1,1] => [[3,3,3,3],[2,2,2]]
=> ([(0,3),(1,4),(2,5),(3,5),(4,2)],6)
=> 2
01000 => [2,4] => [[5,2],[1]]
=> ([(0,5),(1,4),(1,5),(3,2),(4,3)],6)
=> 2
01001 => [2,3,1] => [[4,4,2],[3,1]]
=> ([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
=> 3
01010 => [2,2,2] => [[4,3,2],[2,1]]
=> ([(0,4),(1,4),(1,5),(2,3),(2,5)],6)
=> 3
01011 => [2,2,1,1] => [[3,3,3,2],[2,2,1]]
=> ([(0,4),(1,4),(1,5),(2,3),(3,5)],6)
=> 3
01100 => [2,1,3] => [[4,2,2],[1,1]]
=> ([(0,5),(1,3),(1,4),(3,5),(4,2)],6)
=> 2
01101 => [2,1,2,1] => [[3,3,2,2],[2,1,1]]
=> ([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
=> 3
01110 => [2,1,1,2] => [[3,2,2,2],[1,1,1]]
=> ([(0,5),(1,2),(1,4),(3,5),(4,3)],6)
=> 2
01111 => [2,1,1,1,1] => [[2,2,2,2,2],[1,1,1,1]]
=> ([(0,5),(1,4),(2,5),(3,2),(4,3)],6)
=> 2
10000 => [1,5] => [[5,1],[]]
=> ([(0,2),(0,5),(3,4),(4,1),(5,3)],6)
=> 1
10001 => [1,4,1] => [[4,4,1],[3]]
=> ([(0,5),(1,2),(1,4),(3,5),(4,3)],6)
=> 2
10010 => [1,3,2] => [[4,3,1],[2]]
=> ([(0,4),(0,5),(1,2),(1,3),(3,5)],6)
=> 2
10011 => [1,3,1,1] => [[3,3,3,1],[2,2]]
=> ([(0,4),(1,2),(1,3),(3,5),(4,5)],6)
=> 2
Description
The number of minimal elements in a poset.
Matching statistic: St000318
Mp00178: Binary words to compositionInteger compositions
Mp00180: Integer compositions to ribbonSkew partitions
Mp00183: Skew partitions inner shapeInteger partitions
St000318: Integer partitions ⟶ ℤResult quality: 100% values known / values provided: 100%distinct values known / distinct values provided: 100%
Values
0 => [2] => [[2],[]]
=> []
=> 1
1 => [1,1] => [[1,1],[]]
=> []
=> 1
00 => [3] => [[3],[]]
=> []
=> 1
01 => [2,1] => [[2,2],[1]]
=> [1]
=> 2
10 => [1,2] => [[2,1],[]]
=> []
=> 1
11 => [1,1,1] => [[1,1,1],[]]
=> []
=> 1
000 => [4] => [[4],[]]
=> []
=> 1
001 => [3,1] => [[3,3],[2]]
=> [2]
=> 2
010 => [2,2] => [[3,2],[1]]
=> [1]
=> 2
011 => [2,1,1] => [[2,2,2],[1,1]]
=> [1,1]
=> 2
100 => [1,3] => [[3,1],[]]
=> []
=> 1
101 => [1,2,1] => [[2,2,1],[1]]
=> [1]
=> 2
110 => [1,1,2] => [[2,1,1],[]]
=> []
=> 1
111 => [1,1,1,1] => [[1,1,1,1],[]]
=> []
=> 1
0000 => [5] => [[5],[]]
=> []
=> 1
0001 => [4,1] => [[4,4],[3]]
=> [3]
=> 2
0010 => [3,2] => [[4,3],[2]]
=> [2]
=> 2
0011 => [3,1,1] => [[3,3,3],[2,2]]
=> [2,2]
=> 2
0100 => [2,3] => [[4,2],[1]]
=> [1]
=> 2
0101 => [2,2,1] => [[3,3,2],[2,1]]
=> [2,1]
=> 3
0110 => [2,1,2] => [[3,2,2],[1,1]]
=> [1,1]
=> 2
0111 => [2,1,1,1] => [[2,2,2,2],[1,1,1]]
=> [1,1,1]
=> 2
1000 => [1,4] => [[4,1],[]]
=> []
=> 1
1001 => [1,3,1] => [[3,3,1],[2]]
=> [2]
=> 2
1010 => [1,2,2] => [[3,2,1],[1]]
=> [1]
=> 2
1011 => [1,2,1,1] => [[2,2,2,1],[1,1]]
=> [1,1]
=> 2
1100 => [1,1,3] => [[3,1,1],[]]
=> []
=> 1
1101 => [1,1,2,1] => [[2,2,1,1],[1]]
=> [1]
=> 2
1110 => [1,1,1,2] => [[2,1,1,1],[]]
=> []
=> 1
1111 => [1,1,1,1,1] => [[1,1,1,1,1],[]]
=> []
=> 1
00000 => [6] => [[6],[]]
=> []
=> 1
00001 => [5,1] => [[5,5],[4]]
=> [4]
=> 2
00010 => [4,2] => [[5,4],[3]]
=> [3]
=> 2
00011 => [4,1,1] => [[4,4,4],[3,3]]
=> [3,3]
=> 2
00100 => [3,3] => [[5,3],[2]]
=> [2]
=> 2
00101 => [3,2,1] => [[4,4,3],[3,2]]
=> [3,2]
=> 3
00110 => [3,1,2] => [[4,3,3],[2,2]]
=> [2,2]
=> 2
00111 => [3,1,1,1] => [[3,3,3,3],[2,2,2]]
=> [2,2,2]
=> 2
01000 => [2,4] => [[5,2],[1]]
=> [1]
=> 2
01001 => [2,3,1] => [[4,4,2],[3,1]]
=> [3,1]
=> 3
01010 => [2,2,2] => [[4,3,2],[2,1]]
=> [2,1]
=> 3
01011 => [2,2,1,1] => [[3,3,3,2],[2,2,1]]
=> [2,2,1]
=> 3
01100 => [2,1,3] => [[4,2,2],[1,1]]
=> [1,1]
=> 2
01101 => [2,1,2,1] => [[3,3,2,2],[2,1,1]]
=> [2,1,1]
=> 3
01110 => [2,1,1,2] => [[3,2,2,2],[1,1,1]]
=> [1,1,1]
=> 2
01111 => [2,1,1,1,1] => [[2,2,2,2,2],[1,1,1,1]]
=> [1,1,1,1]
=> 2
10000 => [1,5] => [[5,1],[]]
=> []
=> 1
10001 => [1,4,1] => [[4,4,1],[3]]
=> [3]
=> 2
10010 => [1,3,2] => [[4,3,1],[2]]
=> [2]
=> 2
10011 => [1,3,1,1] => [[3,3,3,1],[2,2]]
=> [2,2]
=> 2
Description
The number of addable cells of the Ferrers diagram of an integer partition.
Matching statistic: St000159
Mp00178: Binary words to compositionInteger compositions
Mp00180: Integer compositions to ribbonSkew partitions
Mp00183: Skew partitions inner shapeInteger partitions
St000159: Integer partitions ⟶ ℤResult quality: 100% values known / values provided: 100%distinct values known / distinct values provided: 100%
Values
0 => [2] => [[2],[]]
=> []
=> 0 = 1 - 1
1 => [1,1] => [[1,1],[]]
=> []
=> 0 = 1 - 1
00 => [3] => [[3],[]]
=> []
=> 0 = 1 - 1
01 => [2,1] => [[2,2],[1]]
=> [1]
=> 1 = 2 - 1
10 => [1,2] => [[2,1],[]]
=> []
=> 0 = 1 - 1
11 => [1,1,1] => [[1,1,1],[]]
=> []
=> 0 = 1 - 1
000 => [4] => [[4],[]]
=> []
=> 0 = 1 - 1
001 => [3,1] => [[3,3],[2]]
=> [2]
=> 1 = 2 - 1
010 => [2,2] => [[3,2],[1]]
=> [1]
=> 1 = 2 - 1
011 => [2,1,1] => [[2,2,2],[1,1]]
=> [1,1]
=> 1 = 2 - 1
100 => [1,3] => [[3,1],[]]
=> []
=> 0 = 1 - 1
101 => [1,2,1] => [[2,2,1],[1]]
=> [1]
=> 1 = 2 - 1
110 => [1,1,2] => [[2,1,1],[]]
=> []
=> 0 = 1 - 1
111 => [1,1,1,1] => [[1,1,1,1],[]]
=> []
=> 0 = 1 - 1
0000 => [5] => [[5],[]]
=> []
=> 0 = 1 - 1
0001 => [4,1] => [[4,4],[3]]
=> [3]
=> 1 = 2 - 1
0010 => [3,2] => [[4,3],[2]]
=> [2]
=> 1 = 2 - 1
0011 => [3,1,1] => [[3,3,3],[2,2]]
=> [2,2]
=> 1 = 2 - 1
0100 => [2,3] => [[4,2],[1]]
=> [1]
=> 1 = 2 - 1
0101 => [2,2,1] => [[3,3,2],[2,1]]
=> [2,1]
=> 2 = 3 - 1
0110 => [2,1,2] => [[3,2,2],[1,1]]
=> [1,1]
=> 1 = 2 - 1
0111 => [2,1,1,1] => [[2,2,2,2],[1,1,1]]
=> [1,1,1]
=> 1 = 2 - 1
1000 => [1,4] => [[4,1],[]]
=> []
=> 0 = 1 - 1
1001 => [1,3,1] => [[3,3,1],[2]]
=> [2]
=> 1 = 2 - 1
1010 => [1,2,2] => [[3,2,1],[1]]
=> [1]
=> 1 = 2 - 1
1011 => [1,2,1,1] => [[2,2,2,1],[1,1]]
=> [1,1]
=> 1 = 2 - 1
1100 => [1,1,3] => [[3,1,1],[]]
=> []
=> 0 = 1 - 1
1101 => [1,1,2,1] => [[2,2,1,1],[1]]
=> [1]
=> 1 = 2 - 1
1110 => [1,1,1,2] => [[2,1,1,1],[]]
=> []
=> 0 = 1 - 1
1111 => [1,1,1,1,1] => [[1,1,1,1,1],[]]
=> []
=> 0 = 1 - 1
00000 => [6] => [[6],[]]
=> []
=> 0 = 1 - 1
00001 => [5,1] => [[5,5],[4]]
=> [4]
=> 1 = 2 - 1
00010 => [4,2] => [[5,4],[3]]
=> [3]
=> 1 = 2 - 1
00011 => [4,1,1] => [[4,4,4],[3,3]]
=> [3,3]
=> 1 = 2 - 1
00100 => [3,3] => [[5,3],[2]]
=> [2]
=> 1 = 2 - 1
00101 => [3,2,1] => [[4,4,3],[3,2]]
=> [3,2]
=> 2 = 3 - 1
00110 => [3,1,2] => [[4,3,3],[2,2]]
=> [2,2]
=> 1 = 2 - 1
00111 => [3,1,1,1] => [[3,3,3,3],[2,2,2]]
=> [2,2,2]
=> 1 = 2 - 1
01000 => [2,4] => [[5,2],[1]]
=> [1]
=> 1 = 2 - 1
01001 => [2,3,1] => [[4,4,2],[3,1]]
=> [3,1]
=> 2 = 3 - 1
01010 => [2,2,2] => [[4,3,2],[2,1]]
=> [2,1]
=> 2 = 3 - 1
01011 => [2,2,1,1] => [[3,3,3,2],[2,2,1]]
=> [2,2,1]
=> 2 = 3 - 1
01100 => [2,1,3] => [[4,2,2],[1,1]]
=> [1,1]
=> 1 = 2 - 1
01101 => [2,1,2,1] => [[3,3,2,2],[2,1,1]]
=> [2,1,1]
=> 2 = 3 - 1
01110 => [2,1,1,2] => [[3,2,2,2],[1,1,1]]
=> [1,1,1]
=> 1 = 2 - 1
01111 => [2,1,1,1,1] => [[2,2,2,2,2],[1,1,1,1]]
=> [1,1,1,1]
=> 1 = 2 - 1
10000 => [1,5] => [[5,1],[]]
=> []
=> 0 = 1 - 1
10001 => [1,4,1] => [[4,4,1],[3]]
=> [3]
=> 1 = 2 - 1
10010 => [1,3,2] => [[4,3,1],[2]]
=> [2]
=> 1 = 2 - 1
10011 => [1,3,1,1] => [[3,3,3,1],[2,2]]
=> [2,2]
=> 1 = 2 - 1
Description
The number of distinct parts of the integer partition. This statistic is also the number of removeable cells of the partition, and the number of valleys of the Dyck path tracing the shape of the partition.
St000292: Binary words ⟶ ℤResult quality: 99% values known / values provided: 99%distinct values known / distinct values provided: 100%
Values
0 => 0 = 1 - 1
1 => 0 = 1 - 1
00 => 0 = 1 - 1
01 => 1 = 2 - 1
10 => 0 = 1 - 1
11 => 0 = 1 - 1
000 => 0 = 1 - 1
001 => 1 = 2 - 1
010 => 1 = 2 - 1
011 => 1 = 2 - 1
100 => 0 = 1 - 1
101 => 1 = 2 - 1
110 => 0 = 1 - 1
111 => 0 = 1 - 1
0000 => 0 = 1 - 1
0001 => 1 = 2 - 1
0010 => 1 = 2 - 1
0011 => 1 = 2 - 1
0100 => 1 = 2 - 1
0101 => 2 = 3 - 1
0110 => 1 = 2 - 1
0111 => 1 = 2 - 1
1000 => 0 = 1 - 1
1001 => 1 = 2 - 1
1010 => 1 = 2 - 1
1011 => 1 = 2 - 1
1100 => 0 = 1 - 1
1101 => 1 = 2 - 1
1110 => 0 = 1 - 1
1111 => 0 = 1 - 1
00000 => 0 = 1 - 1
00001 => 1 = 2 - 1
00010 => 1 = 2 - 1
00011 => 1 = 2 - 1
00100 => 1 = 2 - 1
00101 => 2 = 3 - 1
00110 => 1 = 2 - 1
00111 => 1 = 2 - 1
01000 => 1 = 2 - 1
01001 => 2 = 3 - 1
01010 => 2 = 3 - 1
01011 => 2 = 3 - 1
01100 => 1 = 2 - 1
01101 => 2 = 3 - 1
01110 => 1 = 2 - 1
01111 => 1 = 2 - 1
10000 => 0 = 1 - 1
10001 => 1 = 2 - 1
10010 => 1 = 2 - 1
10011 => 1 = 2 - 1
=> ? = 1 - 1
Description
The number of ascents of a binary word.
Mp00104: Binary words reverseBinary words
St000291: Binary words ⟶ ℤResult quality: 99% values known / values provided: 99%distinct values known / distinct values provided: 100%
Values
0 => 0 => 0 = 1 - 1
1 => 1 => 0 = 1 - 1
00 => 00 => 0 = 1 - 1
01 => 10 => 1 = 2 - 1
10 => 01 => 0 = 1 - 1
11 => 11 => 0 = 1 - 1
000 => 000 => 0 = 1 - 1
001 => 100 => 1 = 2 - 1
010 => 010 => 1 = 2 - 1
011 => 110 => 1 = 2 - 1
100 => 001 => 0 = 1 - 1
101 => 101 => 1 = 2 - 1
110 => 011 => 0 = 1 - 1
111 => 111 => 0 = 1 - 1
0000 => 0000 => 0 = 1 - 1
0001 => 1000 => 1 = 2 - 1
0010 => 0100 => 1 = 2 - 1
0011 => 1100 => 1 = 2 - 1
0100 => 0010 => 1 = 2 - 1
0101 => 1010 => 2 = 3 - 1
0110 => 0110 => 1 = 2 - 1
0111 => 1110 => 1 = 2 - 1
1000 => 0001 => 0 = 1 - 1
1001 => 1001 => 1 = 2 - 1
1010 => 0101 => 1 = 2 - 1
1011 => 1101 => 1 = 2 - 1
1100 => 0011 => 0 = 1 - 1
1101 => 1011 => 1 = 2 - 1
1110 => 0111 => 0 = 1 - 1
1111 => 1111 => 0 = 1 - 1
00000 => 00000 => 0 = 1 - 1
00001 => 10000 => 1 = 2 - 1
00010 => 01000 => 1 = 2 - 1
00011 => 11000 => 1 = 2 - 1
00100 => 00100 => 1 = 2 - 1
00101 => 10100 => 2 = 3 - 1
00110 => 01100 => 1 = 2 - 1
00111 => 11100 => 1 = 2 - 1
01000 => 00010 => 1 = 2 - 1
01001 => 10010 => 2 = 3 - 1
01010 => 01010 => 2 = 3 - 1
01011 => 11010 => 2 = 3 - 1
01100 => 00110 => 1 = 2 - 1
01101 => 10110 => 2 = 3 - 1
01110 => 01110 => 1 = 2 - 1
01111 => 11110 => 1 = 2 - 1
10000 => 00001 => 0 = 1 - 1
10001 => 10001 => 1 = 2 - 1
10010 => 01001 => 1 = 2 - 1
10011 => 11001 => 1 = 2 - 1
=> => ? = 1 - 1
Description
The number of descents of a binary word.
Mp00178: Binary words to compositionInteger compositions
Mp00094: Integer compositions to binary wordBinary words
Mp00104: Binary words reverseBinary words
St000390: Binary words ⟶ ℤResult quality: 99% values known / values provided: 99%distinct values known / distinct values provided: 100%
Values
0 => [2] => 10 => 01 => 1
1 => [1,1] => 11 => 11 => 1
00 => [3] => 100 => 001 => 1
01 => [2,1] => 101 => 101 => 2
10 => [1,2] => 110 => 011 => 1
11 => [1,1,1] => 111 => 111 => 1
000 => [4] => 1000 => 0001 => 1
001 => [3,1] => 1001 => 1001 => 2
010 => [2,2] => 1010 => 0101 => 2
011 => [2,1,1] => 1011 => 1101 => 2
100 => [1,3] => 1100 => 0011 => 1
101 => [1,2,1] => 1101 => 1011 => 2
110 => [1,1,2] => 1110 => 0111 => 1
111 => [1,1,1,1] => 1111 => 1111 => 1
0000 => [5] => 10000 => 00001 => 1
0001 => [4,1] => 10001 => 10001 => 2
0010 => [3,2] => 10010 => 01001 => 2
0011 => [3,1,1] => 10011 => 11001 => 2
0100 => [2,3] => 10100 => 00101 => 2
0101 => [2,2,1] => 10101 => 10101 => 3
0110 => [2,1,2] => 10110 => 01101 => 2
0111 => [2,1,1,1] => 10111 => 11101 => 2
1000 => [1,4] => 11000 => 00011 => 1
1001 => [1,3,1] => 11001 => 10011 => 2
1010 => [1,2,2] => 11010 => 01011 => 2
1011 => [1,2,1,1] => 11011 => 11011 => 2
1100 => [1,1,3] => 11100 => 00111 => 1
1101 => [1,1,2,1] => 11101 => 10111 => 2
1110 => [1,1,1,2] => 11110 => 01111 => 1
1111 => [1,1,1,1,1] => 11111 => 11111 => 1
00000 => [6] => 100000 => 000001 => 1
00001 => [5,1] => 100001 => 100001 => 2
00010 => [4,2] => 100010 => 010001 => 2
00011 => [4,1,1] => 100011 => 110001 => 2
00100 => [3,3] => 100100 => 001001 => 2
00101 => [3,2,1] => 100101 => 101001 => 3
00110 => [3,1,2] => 100110 => 011001 => 2
00111 => [3,1,1,1] => 100111 => 111001 => 2
01000 => [2,4] => 101000 => 000101 => 2
01001 => [2,3,1] => 101001 => 100101 => 3
01010 => [2,2,2] => 101010 => 010101 => 3
01011 => [2,2,1,1] => 101011 => 110101 => 3
01100 => [2,1,3] => 101100 => 001101 => 2
01101 => [2,1,2,1] => 101101 => 101101 => 3
01110 => [2,1,1,2] => 101110 => 011101 => 2
01111 => [2,1,1,1,1] => 101111 => 111101 => 2
10000 => [1,5] => 110000 => 000011 => 1
10001 => [1,4,1] => 110001 => 100011 => 2
10010 => [1,3,2] => 110010 => 010011 => 2
10011 => [1,3,1,1] => 110011 => 110011 => 2
111111100 => [1,1,1,1,1,1,1,3] => 1111111100 => 0011111111 => ? = 1
111111111 => [1,1,1,1,1,1,1,1,1,1] => 1111111111 => 1111111111 => ? = 1
Description
The number of runs of ones in a binary word.
Mp00178: Binary words to compositionInteger compositions
Mp00231: Integer compositions bounce pathDyck paths
St000386: Dyck paths ⟶ ℤResult quality: 93% values known / values provided: 93%distinct values known / distinct values provided: 100%
Values
0 => [2] => [1,1,0,0]
=> 0 = 1 - 1
1 => [1,1] => [1,0,1,0]
=> 0 = 1 - 1
00 => [3] => [1,1,1,0,0,0]
=> 0 = 1 - 1
01 => [2,1] => [1,1,0,0,1,0]
=> 1 = 2 - 1
10 => [1,2] => [1,0,1,1,0,0]
=> 0 = 1 - 1
11 => [1,1,1] => [1,0,1,0,1,0]
=> 0 = 1 - 1
000 => [4] => [1,1,1,1,0,0,0,0]
=> 0 = 1 - 1
001 => [3,1] => [1,1,1,0,0,0,1,0]
=> 1 = 2 - 1
010 => [2,2] => [1,1,0,0,1,1,0,0]
=> 1 = 2 - 1
011 => [2,1,1] => [1,1,0,0,1,0,1,0]
=> 1 = 2 - 1
100 => [1,3] => [1,0,1,1,1,0,0,0]
=> 0 = 1 - 1
101 => [1,2,1] => [1,0,1,1,0,0,1,0]
=> 1 = 2 - 1
110 => [1,1,2] => [1,0,1,0,1,1,0,0]
=> 0 = 1 - 1
111 => [1,1,1,1] => [1,0,1,0,1,0,1,0]
=> 0 = 1 - 1
0000 => [5] => [1,1,1,1,1,0,0,0,0,0]
=> 0 = 1 - 1
0001 => [4,1] => [1,1,1,1,0,0,0,0,1,0]
=> 1 = 2 - 1
0010 => [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> 1 = 2 - 1
0011 => [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> 1 = 2 - 1
0100 => [2,3] => [1,1,0,0,1,1,1,0,0,0]
=> 1 = 2 - 1
0101 => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> 2 = 3 - 1
0110 => [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
=> 1 = 2 - 1
0111 => [2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
=> 1 = 2 - 1
1000 => [1,4] => [1,0,1,1,1,1,0,0,0,0]
=> 0 = 1 - 1
1001 => [1,3,1] => [1,0,1,1,1,0,0,0,1,0]
=> 1 = 2 - 1
1010 => [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
=> 1 = 2 - 1
1011 => [1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
=> 1 = 2 - 1
1100 => [1,1,3] => [1,0,1,0,1,1,1,0,0,0]
=> 0 = 1 - 1
1101 => [1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
=> 1 = 2 - 1
1110 => [1,1,1,2] => [1,0,1,0,1,0,1,1,0,0]
=> 0 = 1 - 1
1111 => [1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
=> 0 = 1 - 1
00000 => [6] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> 0 = 1 - 1
00001 => [5,1] => [1,1,1,1,1,0,0,0,0,0,1,0]
=> 1 = 2 - 1
00010 => [4,2] => [1,1,1,1,0,0,0,0,1,1,0,0]
=> 1 = 2 - 1
00011 => [4,1,1] => [1,1,1,1,0,0,0,0,1,0,1,0]
=> 1 = 2 - 1
00100 => [3,3] => [1,1,1,0,0,0,1,1,1,0,0,0]
=> 1 = 2 - 1
00101 => [3,2,1] => [1,1,1,0,0,0,1,1,0,0,1,0]
=> 2 = 3 - 1
00110 => [3,1,2] => [1,1,1,0,0,0,1,0,1,1,0,0]
=> 1 = 2 - 1
00111 => [3,1,1,1] => [1,1,1,0,0,0,1,0,1,0,1,0]
=> 1 = 2 - 1
01000 => [2,4] => [1,1,0,0,1,1,1,1,0,0,0,0]
=> 1 = 2 - 1
01001 => [2,3,1] => [1,1,0,0,1,1,1,0,0,0,1,0]
=> 2 = 3 - 1
01010 => [2,2,2] => [1,1,0,0,1,1,0,0,1,1,0,0]
=> 2 = 3 - 1
01011 => [2,2,1,1] => [1,1,0,0,1,1,0,0,1,0,1,0]
=> 2 = 3 - 1
01100 => [2,1,3] => [1,1,0,0,1,0,1,1,1,0,0,0]
=> 1 = 2 - 1
01101 => [2,1,2,1] => [1,1,0,0,1,0,1,1,0,0,1,0]
=> 2 = 3 - 1
01110 => [2,1,1,2] => [1,1,0,0,1,0,1,0,1,1,0,0]
=> 1 = 2 - 1
01111 => [2,1,1,1,1] => [1,1,0,0,1,0,1,0,1,0,1,0]
=> 1 = 2 - 1
10000 => [1,5] => [1,0,1,1,1,1,1,0,0,0,0,0]
=> 0 = 1 - 1
10001 => [1,4,1] => [1,0,1,1,1,1,0,0,0,0,1,0]
=> 1 = 2 - 1
10010 => [1,3,2] => [1,0,1,1,1,0,0,0,1,1,0,0]
=> 1 = 2 - 1
10011 => [1,3,1,1] => [1,0,1,1,1,0,0,0,1,0,1,0]
=> 1 = 2 - 1
11100000 => [1,1,1,6] => [1,0,1,0,1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> ? = 1 - 1
11110000 => [1,1,1,1,5] => [1,0,1,0,1,0,1,0,1,1,1,1,1,0,0,0,0,0]
=> ? = 1 - 1
11111000 => [1,1,1,1,1,4] => [1,0,1,0,1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> ? = 1 - 1
11111100 => [1,1,1,1,1,1,3] => [1,0,1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> ? = 1 - 1
11111111 => [1,1,1,1,1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 1 - 1
111000000 => [1,1,1,7] => [1,0,1,0,1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> ? = 1 - 1
111100000 => [1,1,1,1,6] => [1,0,1,0,1,0,1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> ? = 1 - 1
111110000 => [1,1,1,1,1,5] => [1,0,1,0,1,0,1,0,1,0,1,1,1,1,1,0,0,0,0,0]
=> ? = 1 - 1
111111000 => [1,1,1,1,1,1,4] => [1,0,1,0,1,0,1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> ? = 1 - 1
111111100 => [1,1,1,1,1,1,1,3] => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> ? = 1 - 1
111111111 => [1,1,1,1,1,1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 1 - 1
Description
The number of factors DDU in a Dyck path.
Matching statistic: St000201
Mp00178: Binary words to compositionInteger compositions
Mp00231: Integer compositions bounce pathDyck paths
Mp00034: Dyck paths to binary tree: up step, left tree, down step, right treeBinary trees
St000201: Binary trees ⟶ ℤResult quality: 85% values known / values provided: 85%distinct values known / distinct values provided: 100%
Values
0 => [2] => [1,1,0,0]
=> [[.,.],.]
=> 1
1 => [1,1] => [1,0,1,0]
=> [.,[.,.]]
=> 1
00 => [3] => [1,1,1,0,0,0]
=> [[[.,.],.],.]
=> 1
01 => [2,1] => [1,1,0,0,1,0]
=> [[.,.],[.,.]]
=> 2
10 => [1,2] => [1,0,1,1,0,0]
=> [.,[[.,.],.]]
=> 1
11 => [1,1,1] => [1,0,1,0,1,0]
=> [.,[.,[.,.]]]
=> 1
000 => [4] => [1,1,1,1,0,0,0,0]
=> [[[[.,.],.],.],.]
=> 1
001 => [3,1] => [1,1,1,0,0,0,1,0]
=> [[[.,.],.],[.,.]]
=> 2
010 => [2,2] => [1,1,0,0,1,1,0,0]
=> [[.,.],[[.,.],.]]
=> 2
011 => [2,1,1] => [1,1,0,0,1,0,1,0]
=> [[.,.],[.,[.,.]]]
=> 2
100 => [1,3] => [1,0,1,1,1,0,0,0]
=> [.,[[[.,.],.],.]]
=> 1
101 => [1,2,1] => [1,0,1,1,0,0,1,0]
=> [.,[[.,.],[.,.]]]
=> 2
110 => [1,1,2] => [1,0,1,0,1,1,0,0]
=> [.,[.,[[.,.],.]]]
=> 1
111 => [1,1,1,1] => [1,0,1,0,1,0,1,0]
=> [.,[.,[.,[.,.]]]]
=> 1
0000 => [5] => [1,1,1,1,1,0,0,0,0,0]
=> [[[[[.,.],.],.],.],.]
=> 1
0001 => [4,1] => [1,1,1,1,0,0,0,0,1,0]
=> [[[[.,.],.],.],[.,.]]
=> 2
0010 => [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> [[[.,.],.],[[.,.],.]]
=> 2
0011 => [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> [[[.,.],.],[.,[.,.]]]
=> 2
0100 => [2,3] => [1,1,0,0,1,1,1,0,0,0]
=> [[.,.],[[[.,.],.],.]]
=> 2
0101 => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> [[.,.],[[.,.],[.,.]]]
=> 3
0110 => [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
=> [[.,.],[.,[[.,.],.]]]
=> 2
0111 => [2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
=> [[.,.],[.,[.,[.,.]]]]
=> 2
1000 => [1,4] => [1,0,1,1,1,1,0,0,0,0]
=> [.,[[[[.,.],.],.],.]]
=> 1
1001 => [1,3,1] => [1,0,1,1,1,0,0,0,1,0]
=> [.,[[[.,.],.],[.,.]]]
=> 2
1010 => [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
=> [.,[[.,.],[[.,.],.]]]
=> 2
1011 => [1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
=> [.,[[.,.],[.,[.,.]]]]
=> 2
1100 => [1,1,3] => [1,0,1,0,1,1,1,0,0,0]
=> [.,[.,[[[.,.],.],.]]]
=> 1
1101 => [1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
=> [.,[.,[[.,.],[.,.]]]]
=> 2
1110 => [1,1,1,2] => [1,0,1,0,1,0,1,1,0,0]
=> [.,[.,[.,[[.,.],.]]]]
=> 1
1111 => [1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
=> [.,[.,[.,[.,[.,.]]]]]
=> 1
00000 => [6] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> [[[[[[.,.],.],.],.],.],.]
=> 1
00001 => [5,1] => [1,1,1,1,1,0,0,0,0,0,1,0]
=> [[[[[.,.],.],.],.],[.,.]]
=> 2
00010 => [4,2] => [1,1,1,1,0,0,0,0,1,1,0,0]
=> [[[[.,.],.],.],[[.,.],.]]
=> 2
00011 => [4,1,1] => [1,1,1,1,0,0,0,0,1,0,1,0]
=> [[[[.,.],.],.],[.,[.,.]]]
=> 2
00100 => [3,3] => [1,1,1,0,0,0,1,1,1,0,0,0]
=> [[[.,.],.],[[[.,.],.],.]]
=> 2
00101 => [3,2,1] => [1,1,1,0,0,0,1,1,0,0,1,0]
=> [[[.,.],.],[[.,.],[.,.]]]
=> 3
00110 => [3,1,2] => [1,1,1,0,0,0,1,0,1,1,0,0]
=> [[[.,.],.],[.,[[.,.],.]]]
=> 2
00111 => [3,1,1,1] => [1,1,1,0,0,0,1,0,1,0,1,0]
=> [[[.,.],.],[.,[.,[.,.]]]]
=> 2
01000 => [2,4] => [1,1,0,0,1,1,1,1,0,0,0,0]
=> [[.,.],[[[[.,.],.],.],.]]
=> 2
01001 => [2,3,1] => [1,1,0,0,1,1,1,0,0,0,1,0]
=> [[.,.],[[[.,.],.],[.,.]]]
=> 3
01010 => [2,2,2] => [1,1,0,0,1,1,0,0,1,1,0,0]
=> [[.,.],[[.,.],[[.,.],.]]]
=> 3
01011 => [2,2,1,1] => [1,1,0,0,1,1,0,0,1,0,1,0]
=> [[.,.],[[.,.],[.,[.,.]]]]
=> 3
01100 => [2,1,3] => [1,1,0,0,1,0,1,1,1,0,0,0]
=> [[.,.],[.,[[[.,.],.],.]]]
=> 2
01101 => [2,1,2,1] => [1,1,0,0,1,0,1,1,0,0,1,0]
=> [[.,.],[.,[[.,.],[.,.]]]]
=> 3
01110 => [2,1,1,2] => [1,1,0,0,1,0,1,0,1,1,0,0]
=> [[.,.],[.,[.,[[.,.],.]]]]
=> 2
01111 => [2,1,1,1,1] => [1,1,0,0,1,0,1,0,1,0,1,0]
=> [[.,.],[.,[.,[.,[.,.]]]]]
=> 2
10000 => [1,5] => [1,0,1,1,1,1,1,0,0,0,0,0]
=> [.,[[[[[.,.],.],.],.],.]]
=> 1
10001 => [1,4,1] => [1,0,1,1,1,1,0,0,0,0,1,0]
=> [.,[[[[.,.],.],.],[.,.]]]
=> 2
10010 => [1,3,2] => [1,0,1,1,1,0,0,0,1,1,0,0]
=> [.,[[[.,.],.],[[.,.],.]]]
=> 2
10011 => [1,3,1,1] => [1,0,1,1,1,0,0,0,1,0,1,0]
=> [.,[[[.,.],.],[.,[.,.]]]]
=> 2
1100000 => [1,1,6] => [1,0,1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [.,[.,[[[[[[.,.],.],.],.],.],.]]]
=> ? = 1
1110000 => [1,1,1,5] => [1,0,1,0,1,0,1,1,1,1,1,0,0,0,0,0]
=> [.,[.,[.,[[[[[.,.],.],.],.],.]]]]
=> ? = 1
1111000 => [1,1,1,1,4] => [1,0,1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> [.,[.,[.,[.,[[[[.,.],.],.],.]]]]]
=> ? = 1
1111100 => [1,1,1,1,1,3] => [1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [.,[.,[.,[.,[.,[[[.,.],.],.]]]]]]
=> ? = 1
1111110 => [1,1,1,1,1,1,2] => [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [.,[.,[.,[.,[.,[.,[[.,.],.]]]]]]]
=> ? = 1
1111111 => [1,1,1,1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [.,[.,[.,[.,[.,[.,[.,[.,.]]]]]]]]
=> ? = 1
00000000 => [9] => [1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0]
=> [[[[[[[[[.,.],.],.],.],.],.],.],.],.]
=> ? = 1
11000000 => [1,1,7] => [1,0,1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [.,[.,[[[[[[[.,.],.],.],.],.],.],.]]]
=> ? = 1
11100000 => [1,1,1,6] => [1,0,1,0,1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [.,[.,[.,[[[[[[.,.],.],.],.],.],.]]]]
=> ? = 1
11110000 => [1,1,1,1,5] => [1,0,1,0,1,0,1,0,1,1,1,1,1,0,0,0,0,0]
=> [.,[.,[.,[.,[[[[[.,.],.],.],.],.]]]]]
=> ? = 1
11111000 => [1,1,1,1,1,4] => [1,0,1,0,1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> [.,[.,[.,[.,[.,[[[[.,.],.],.],.]]]]]]
=> ? = 1
11111100 => [1,1,1,1,1,1,3] => [1,0,1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [.,[.,[.,[.,[.,[.,[[[.,.],.],.]]]]]]]
=> ? = 1
11111110 => [1,1,1,1,1,1,1,2] => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [.,[.,[.,[.,[.,[.,[.,[[.,.],.]]]]]]]]
=> ? = 1
11111111 => [1,1,1,1,1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [.,[.,[.,[.,[.,[.,[.,[.,[.,.]]]]]]]]]
=> ? = 1
000000000 => [10] => [1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0]
=> [[[[[[[[[[.,.],.],.],.],.],.],.],.],.],.]
=> ? = 1
110000000 => [1,1,8] => [1,0,1,0,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [.,[.,[[[[[[[[.,.],.],.],.],.],.],.],.]]]
=> ? = 1
111000000 => [1,1,1,7] => [1,0,1,0,1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [.,[.,[.,[[[[[[[.,.],.],.],.],.],.],.]]]]
=> ? = 1
111100000 => [1,1,1,1,6] => [1,0,1,0,1,0,1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [.,[.,[.,[.,[[[[[[.,.],.],.],.],.],.]]]]]
=> ? = 1
111110000 => [1,1,1,1,1,5] => [1,0,1,0,1,0,1,0,1,0,1,1,1,1,1,0,0,0,0,0]
=> [.,[.,[.,[.,[.,[[[[[.,.],.],.],.],.]]]]]]
=> ? = 1
111111000 => [1,1,1,1,1,1,4] => [1,0,1,0,1,0,1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> [.,[.,[.,[.,[.,[.,[[[[.,.],.],.],.]]]]]]]
=> ? = 1
111111100 => [1,1,1,1,1,1,1,3] => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [.,[.,[.,[.,[.,[.,[.,[[[.,.],.],.]]]]]]]]
=> ? = 1
111111110 => [1,1,1,1,1,1,1,1,2] => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [.,[.,[.,[.,[.,[.,[.,[.,[[.,.],.]]]]]]]]]
=> ? = 1
111111111 => [1,1,1,1,1,1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [.,[.,[.,[.,[.,[.,[.,[.,[.,[.,.]]]]]]]]]]
=> ? = 1
Description
The number of leaf nodes in a binary tree. Equivalently, the number of cherries [1] in the complete binary tree. The number of binary trees of size n, at least 1, with exactly one leaf node for is 2n1, see [2]. The number of binary tree of size n, at least 3, with exactly two leaf nodes is n(n+1)2n2, see [3].
Mp00178: Binary words to compositionInteger compositions
Mp00231: Integer compositions bounce pathDyck paths
Mp00229: Dyck paths Delest-ViennotDyck paths
St001037: Dyck paths ⟶ ℤResult quality: 84% values known / values provided: 84%distinct values known / distinct values provided: 100%
Values
0 => [2] => [1,1,0,0]
=> [1,0,1,0]
=> 0 = 1 - 1
1 => [1,1] => [1,0,1,0]
=> [1,1,0,0]
=> 0 = 1 - 1
00 => [3] => [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> 0 = 1 - 1
01 => [2,1] => [1,1,0,0,1,0]
=> [1,0,1,1,0,0]
=> 1 = 2 - 1
10 => [1,2] => [1,0,1,1,0,0]
=> [1,1,0,0,1,0]
=> 0 = 1 - 1
11 => [1,1,1] => [1,0,1,0,1,0]
=> [1,1,0,1,0,0]
=> 0 = 1 - 1
000 => [4] => [1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> 0 = 1 - 1
001 => [3,1] => [1,1,1,0,0,0,1,0]
=> [1,0,1,0,1,1,0,0]
=> 1 = 2 - 1
010 => [2,2] => [1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,0,1,0]
=> 1 = 2 - 1
011 => [2,1,1] => [1,1,0,0,1,0,1,0]
=> [1,0,1,1,0,1,0,0]
=> 1 = 2 - 1
100 => [1,3] => [1,0,1,1,1,0,0,0]
=> [1,1,0,0,1,0,1,0]
=> 0 = 1 - 1
101 => [1,2,1] => [1,0,1,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0]
=> 1 = 2 - 1
110 => [1,1,2] => [1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> 0 = 1 - 1
111 => [1,1,1,1] => [1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,0]
=> 0 = 1 - 1
0000 => [5] => [1,1,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> 0 = 1 - 1
0001 => [4,1] => [1,1,1,1,0,0,0,0,1,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> 1 = 2 - 1
0010 => [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> 1 = 2 - 1
0011 => [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> [1,0,1,0,1,1,0,1,0,0]
=> 1 = 2 - 1
0100 => [2,3] => [1,1,0,0,1,1,1,0,0,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> 1 = 2 - 1
0101 => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> 2 = 3 - 1
0110 => [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
=> [1,0,1,1,0,1,0,0,1,0]
=> 1 = 2 - 1
0111 => [2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
=> [1,0,1,1,0,1,0,1,0,0]
=> 1 = 2 - 1
1000 => [1,4] => [1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> 0 = 1 - 1
1001 => [1,3,1] => [1,0,1,1,1,0,0,0,1,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> 1 = 2 - 1
1010 => [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> 1 = 2 - 1
1011 => [1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
=> [1,1,0,0,1,1,0,1,0,0]
=> 1 = 2 - 1
1100 => [1,1,3] => [1,0,1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,0,1,0,1,0]
=> 0 = 1 - 1
1101 => [1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,0,0,1,1,0,0]
=> 1 = 2 - 1
1110 => [1,1,1,2] => [1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,0,1,0]
=> 0 = 1 - 1
1111 => [1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> 0 = 1 - 1
00000 => [6] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> 0 = 1 - 1
00001 => [5,1] => [1,1,1,1,1,0,0,0,0,0,1,0]
=> [1,0,1,0,1,0,1,0,1,1,0,0]
=> 1 = 2 - 1
00010 => [4,2] => [1,1,1,1,0,0,0,0,1,1,0,0]
=> [1,0,1,0,1,0,1,1,0,0,1,0]
=> 1 = 2 - 1
00011 => [4,1,1] => [1,1,1,1,0,0,0,0,1,0,1,0]
=> [1,0,1,0,1,0,1,1,0,1,0,0]
=> 1 = 2 - 1
00100 => [3,3] => [1,1,1,0,0,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,0,0,1,0,1,0]
=> 1 = 2 - 1
00101 => [3,2,1] => [1,1,1,0,0,0,1,1,0,0,1,0]
=> [1,0,1,0,1,1,0,0,1,1,0,0]
=> 2 = 3 - 1
00110 => [3,1,2] => [1,1,1,0,0,0,1,0,1,1,0,0]
=> [1,0,1,0,1,1,0,1,0,0,1,0]
=> 1 = 2 - 1
00111 => [3,1,1,1] => [1,1,1,0,0,0,1,0,1,0,1,0]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> 1 = 2 - 1
01000 => [2,4] => [1,1,0,0,1,1,1,1,0,0,0,0]
=> [1,0,1,1,0,0,1,0,1,0,1,0]
=> 1 = 2 - 1
01001 => [2,3,1] => [1,1,0,0,1,1,1,0,0,0,1,0]
=> [1,0,1,1,0,0,1,0,1,1,0,0]
=> 2 = 3 - 1
01010 => [2,2,2] => [1,1,0,0,1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,0,1,1,0,0,1,0]
=> 2 = 3 - 1
01011 => [2,2,1,1] => [1,1,0,0,1,1,0,0,1,0,1,0]
=> [1,0,1,1,0,0,1,1,0,1,0,0]
=> 2 = 3 - 1
01100 => [2,1,3] => [1,1,0,0,1,0,1,1,1,0,0,0]
=> [1,0,1,1,0,1,0,0,1,0,1,0]
=> 1 = 2 - 1
01101 => [2,1,2,1] => [1,1,0,0,1,0,1,1,0,0,1,0]
=> [1,0,1,1,0,1,0,0,1,1,0,0]
=> 2 = 3 - 1
01110 => [2,1,1,2] => [1,1,0,0,1,0,1,0,1,1,0,0]
=> [1,0,1,1,0,1,0,1,0,0,1,0]
=> 1 = 2 - 1
01111 => [2,1,1,1,1] => [1,1,0,0,1,0,1,0,1,0,1,0]
=> [1,0,1,1,0,1,0,1,0,1,0,0]
=> 1 = 2 - 1
10000 => [1,5] => [1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,1,0,0,1,0,1,0,1,0,1,0]
=> 0 = 1 - 1
10001 => [1,4,1] => [1,0,1,1,1,1,0,0,0,0,1,0]
=> [1,1,0,0,1,0,1,0,1,1,0,0]
=> 1 = 2 - 1
10010 => [1,3,2] => [1,0,1,1,1,0,0,0,1,1,0,0]
=> [1,1,0,0,1,0,1,1,0,0,1,0]
=> 1 = 2 - 1
10011 => [1,3,1,1] => [1,0,1,1,1,0,0,0,1,0,1,0]
=> [1,1,0,0,1,0,1,1,0,1,0,0]
=> 1 = 2 - 1
1110000 => [1,1,1,5] => [1,0,1,0,1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,1,0,1,0,1,0,0,1,0,1,0,1,0,1,0]
=> ? = 1 - 1
1111000 => [1,1,1,1,4] => [1,0,1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,0,1,0,1,0,1,0]
=> ? = 1 - 1
1111100 => [1,1,1,1,1,3] => [1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,0,1,0,1,0,0,1,0,1,0]
=> ? = 1 - 1
1111110 => [1,1,1,1,1,1,2] => [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,0,1,0]
=> ? = 1 - 1
1111111 => [1,1,1,1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 1 - 1
00000000 => [9] => [1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 1 - 1
10000000 => [1,8] => [1,0,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 1 - 1
11000000 => [1,1,7] => [1,0,1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [1,1,0,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 1 - 1
11100000 => [1,1,1,6] => [1,0,1,0,1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,1,0,1,0,1,0,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 1 - 1
11110000 => [1,1,1,1,5] => [1,0,1,0,1,0,1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,0,1,0,1,0,1,0,1,0]
=> ? = 1 - 1
11111000 => [1,1,1,1,1,4] => [1,0,1,0,1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,1,0,0,1,0,1,0,1,0]
=> ? = 1 - 1
11111100 => [1,1,1,1,1,1,3] => [1,0,1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,0,1,0,1,0]
=> ? = 1 - 1
11111110 => [1,1,1,1,1,1,1,2] => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0,1,0]
=> ? = 1 - 1
11111111 => [1,1,1,1,1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 1 - 1
000000000 => [10] => [1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 1 - 1
100000000 => [1,9] => [1,0,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0]
=> [1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 1 - 1
110000000 => [1,1,8] => [1,0,1,0,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [1,1,0,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 1 - 1
111000000 => [1,1,1,7] => [1,0,1,0,1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [1,1,0,1,0,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 1 - 1
111100000 => [1,1,1,1,6] => [1,0,1,0,1,0,1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 1 - 1
111110000 => [1,1,1,1,1,5] => [1,0,1,0,1,0,1,0,1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,1,0,0,1,0,1,0,1,0,1,0]
=> ? = 1 - 1
111111000 => [1,1,1,1,1,1,4] => [1,0,1,0,1,0,1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,0,1,0,1,0,1,0]
=> ? = 1 - 1
111111100 => [1,1,1,1,1,1,1,3] => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0,1,0,1,0]
=> ? = 1 - 1
111111110 => [1,1,1,1,1,1,1,1,2] => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0,1,0]
=> ? = 1 - 1
111111111 => [1,1,1,1,1,1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 1 - 1
Description
The number of inner corners of the upper path of the parallelogram polyomino associated with the Dyck path.
Matching statistic: St000568
Mp00178: Binary words to compositionInteger compositions
Mp00231: Integer compositions bounce pathDyck paths
Mp00029: Dyck paths to binary tree: left tree, up step, right tree, down stepBinary trees
St000568: Binary trees ⟶ ℤResult quality: 83% values known / values provided: 83%distinct values known / distinct values provided: 100%
Values
0 => [2] => [1,1,0,0]
=> [.,[.,.]]
=> 1
1 => [1,1] => [1,0,1,0]
=> [[.,.],.]
=> 1
00 => [3] => [1,1,1,0,0,0]
=> [.,[.,[.,.]]]
=> 1
01 => [2,1] => [1,1,0,0,1,0]
=> [[.,[.,.]],.]
=> 2
10 => [1,2] => [1,0,1,1,0,0]
=> [[.,.],[.,.]]
=> 1
11 => [1,1,1] => [1,0,1,0,1,0]
=> [[[.,.],.],.]
=> 1
000 => [4] => [1,1,1,1,0,0,0,0]
=> [.,[.,[.,[.,.]]]]
=> 1
001 => [3,1] => [1,1,1,0,0,0,1,0]
=> [[.,[.,[.,.]]],.]
=> 2
010 => [2,2] => [1,1,0,0,1,1,0,0]
=> [[.,[.,.]],[.,.]]
=> 2
011 => [2,1,1] => [1,1,0,0,1,0,1,0]
=> [[[.,[.,.]],.],.]
=> 2
100 => [1,3] => [1,0,1,1,1,0,0,0]
=> [[.,.],[.,[.,.]]]
=> 1
101 => [1,2,1] => [1,0,1,1,0,0,1,0]
=> [[[.,.],[.,.]],.]
=> 2
110 => [1,1,2] => [1,0,1,0,1,1,0,0]
=> [[[.,.],.],[.,.]]
=> 1
111 => [1,1,1,1] => [1,0,1,0,1,0,1,0]
=> [[[[.,.],.],.],.]
=> 1
0000 => [5] => [1,1,1,1,1,0,0,0,0,0]
=> [.,[.,[.,[.,[.,.]]]]]
=> 1
0001 => [4,1] => [1,1,1,1,0,0,0,0,1,0]
=> [[.,[.,[.,[.,.]]]],.]
=> 2
0010 => [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> [[.,[.,[.,.]]],[.,.]]
=> 2
0011 => [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> [[[.,[.,[.,.]]],.],.]
=> 2
0100 => [2,3] => [1,1,0,0,1,1,1,0,0,0]
=> [[.,[.,.]],[.,[.,.]]]
=> 2
0101 => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> [[[.,[.,.]],[.,.]],.]
=> 3
0110 => [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
=> [[[.,[.,.]],.],[.,.]]
=> 2
0111 => [2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
=> [[[[.,[.,.]],.],.],.]
=> 2
1000 => [1,4] => [1,0,1,1,1,1,0,0,0,0]
=> [[.,.],[.,[.,[.,.]]]]
=> 1
1001 => [1,3,1] => [1,0,1,1,1,0,0,0,1,0]
=> [[[.,.],[.,[.,.]]],.]
=> 2
1010 => [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
=> [[[.,.],[.,.]],[.,.]]
=> 2
1011 => [1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
=> [[[[.,.],[.,.]],.],.]
=> 2
1100 => [1,1,3] => [1,0,1,0,1,1,1,0,0,0]
=> [[[.,.],.],[.,[.,.]]]
=> 1
1101 => [1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
=> [[[[.,.],.],[.,.]],.]
=> 2
1110 => [1,1,1,2] => [1,0,1,0,1,0,1,1,0,0]
=> [[[[.,.],.],.],[.,.]]
=> 1
1111 => [1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
=> [[[[[.,.],.],.],.],.]
=> 1
00000 => [6] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> [.,[.,[.,[.,[.,[.,.]]]]]]
=> 1
00001 => [5,1] => [1,1,1,1,1,0,0,0,0,0,1,0]
=> [[.,[.,[.,[.,[.,.]]]]],.]
=> 2
00010 => [4,2] => [1,1,1,1,0,0,0,0,1,1,0,0]
=> [[.,[.,[.,[.,.]]]],[.,.]]
=> 2
00011 => [4,1,1] => [1,1,1,1,0,0,0,0,1,0,1,0]
=> [[[.,[.,[.,[.,.]]]],.],.]
=> 2
00100 => [3,3] => [1,1,1,0,0,0,1,1,1,0,0,0]
=> [[.,[.,[.,.]]],[.,[.,.]]]
=> 2
00101 => [3,2,1] => [1,1,1,0,0,0,1,1,0,0,1,0]
=> [[[.,[.,[.,.]]],[.,.]],.]
=> 3
00110 => [3,1,2] => [1,1,1,0,0,0,1,0,1,1,0,0]
=> [[[.,[.,[.,.]]],.],[.,.]]
=> 2
00111 => [3,1,1,1] => [1,1,1,0,0,0,1,0,1,0,1,0]
=> [[[[.,[.,[.,.]]],.],.],.]
=> 2
01000 => [2,4] => [1,1,0,0,1,1,1,1,0,0,0,0]
=> [[.,[.,.]],[.,[.,[.,.]]]]
=> 2
01001 => [2,3,1] => [1,1,0,0,1,1,1,0,0,0,1,0]
=> [[[.,[.,.]],[.,[.,.]]],.]
=> 3
01010 => [2,2,2] => [1,1,0,0,1,1,0,0,1,1,0,0]
=> [[[.,[.,.]],[.,.]],[.,.]]
=> 3
01011 => [2,2,1,1] => [1,1,0,0,1,1,0,0,1,0,1,0]
=> [[[[.,[.,.]],[.,.]],.],.]
=> 3
01100 => [2,1,3] => [1,1,0,0,1,0,1,1,1,0,0,0]
=> [[[.,[.,.]],.],[.,[.,.]]]
=> 2
01101 => [2,1,2,1] => [1,1,0,0,1,0,1,1,0,0,1,0]
=> [[[[.,[.,.]],.],[.,.]],.]
=> 3
01110 => [2,1,1,2] => [1,1,0,0,1,0,1,0,1,1,0,0]
=> [[[[.,[.,.]],.],.],[.,.]]
=> 2
01111 => [2,1,1,1,1] => [1,1,0,0,1,0,1,0,1,0,1,0]
=> [[[[[.,[.,.]],.],.],.],.]
=> 2
10000 => [1,5] => [1,0,1,1,1,1,1,0,0,0,0,0]
=> [[.,.],[.,[.,[.,[.,.]]]]]
=> 1
10001 => [1,4,1] => [1,0,1,1,1,1,0,0,0,0,1,0]
=> [[[.,.],[.,[.,[.,.]]]],.]
=> 2
10010 => [1,3,2] => [1,0,1,1,1,0,0,0,1,1,0,0]
=> [[[.,.],[.,[.,.]]],[.,.]]
=> 2
10011 => [1,3,1,1] => [1,0,1,1,1,0,0,0,1,0,1,0]
=> [[[[.,.],[.,[.,.]]],.],.]
=> 2
1100000 => [1,1,6] => [1,0,1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [[[.,.],.],[.,[.,[.,[.,[.,.]]]]]]
=> ? = 1
1110000 => [1,1,1,5] => [1,0,1,0,1,0,1,1,1,1,1,0,0,0,0,0]
=> [[[[.,.],.],.],[.,[.,[.,[.,.]]]]]
=> ? = 1
1111000 => [1,1,1,1,4] => [1,0,1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> [[[[[.,.],.],.],.],[.,[.,[.,.]]]]
=> ? = 1
1111100 => [1,1,1,1,1,3] => [1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [[[[[[.,.],.],.],.],.],[.,[.,.]]]
=> ? = 1
1111110 => [1,1,1,1,1,1,2] => [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [[[[[[[.,.],.],.],.],.],.],[.,.]]
=> ? = 1
1111111 => [1,1,1,1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [[[[[[[[.,.],.],.],.],.],.],.],.]
=> ? = 1
00000000 => [9] => [1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0]
=> [.,[.,[.,[.,[.,[.,[.,[.,[.,.]]]]]]]]]
=> ? = 1
10000000 => [1,8] => [1,0,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [[.,.],[.,[.,[.,[.,[.,[.,[.,.]]]]]]]]
=> ? = 1
11000000 => [1,1,7] => [1,0,1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [[[.,.],.],[.,[.,[.,[.,[.,[.,.]]]]]]]
=> ? = 1
11100000 => [1,1,1,6] => [1,0,1,0,1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [[[[.,.],.],.],[.,[.,[.,[.,[.,.]]]]]]
=> ? = 1
11110000 => [1,1,1,1,5] => [1,0,1,0,1,0,1,0,1,1,1,1,1,0,0,0,0,0]
=> [[[[[.,.],.],.],.],[.,[.,[.,[.,.]]]]]
=> ? = 1
11111000 => [1,1,1,1,1,4] => [1,0,1,0,1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> [[[[[[.,.],.],.],.],.],[.,[.,[.,.]]]]
=> ? = 1
11111100 => [1,1,1,1,1,1,3] => [1,0,1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [[[[[[[.,.],.],.],.],.],.],[.,[.,.]]]
=> ? = 1
11111110 => [1,1,1,1,1,1,1,2] => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [[[[[[[[.,.],.],.],.],.],.],.],[.,.]]
=> ? = 1
11111111 => [1,1,1,1,1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [[[[[[[[[.,.],.],.],.],.],.],.],.],.]
=> ? = 1
000000000 => [10] => [1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0]
=> [.,[.,[.,[.,[.,[.,[.,[.,[.,[.,.]]]]]]]]]]
=> ? = 1
100000000 => [1,9] => [1,0,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0]
=> [[.,.],[.,[.,[.,[.,[.,[.,[.,[.,.]]]]]]]]]
=> ? = 1
110000000 => [1,1,8] => [1,0,1,0,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [[[.,.],.],[.,[.,[.,[.,[.,[.,[.,.]]]]]]]]
=> ? = 1
111000000 => [1,1,1,7] => [1,0,1,0,1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [[[[.,.],.],.],[.,[.,[.,[.,[.,[.,.]]]]]]]
=> ? = 1
111100000 => [1,1,1,1,6] => [1,0,1,0,1,0,1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [[[[[.,.],.],.],.],[.,[.,[.,[.,[.,.]]]]]]
=> ? = 1
111110000 => [1,1,1,1,1,5] => [1,0,1,0,1,0,1,0,1,0,1,1,1,1,1,0,0,0,0,0]
=> [[[[[[.,.],.],.],.],.],[.,[.,[.,[.,.]]]]]
=> ? = 1
111111000 => [1,1,1,1,1,1,4] => [1,0,1,0,1,0,1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> [[[[[[[.,.],.],.],.],.],.],[.,[.,[.,.]]]]
=> ? = 1
111111100 => [1,1,1,1,1,1,1,3] => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [[[[[[[[.,.],.],.],.],.],.],.],[.,[.,.]]]
=> ? = 1
111111110 => [1,1,1,1,1,1,1,1,2] => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [[[[[[[[[.,.],.],.],.],.],.],.],.],[.,.]]
=> ? = 1
111111111 => [1,1,1,1,1,1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [[[[[[[[[[.,.],.],.],.],.],.],.],.],.],.]
=> ? = 1
=> [1] => [1,0]
=> [.,.]
=> ? = 1
Description
The hook number of a binary tree. A hook of a binary tree is a vertex together with is left- and its right-most branch. Then there is a unique decomposition of the tree into hooks and the hook number is the number of hooks in this decomposition.
The following 11 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St001036The number of inner corners of the parallelogram polyomino associated with the Dyck path. St000196The number of occurrences of the contiguous pattern [[.,.],[.,. St000356The number of occurrences of the pattern 13-2. St001687The number of distinct positions of the pattern letter 2 in occurrences of 213 in a permutation. St000353The number of inner valleys of a permutation. St001124The multiplicity of the standard representation in the Kronecker square corresponding to a partition. St000092The number of outer peaks of a permutation. St000256The number of parts from which one can substract 2 and still get an integer partition. St001487The number of inner corners of a skew partition. St000354The number of recoils of a permutation. St001960The number of descents of a permutation minus one if its first entry is not one.