Your data matches 42 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)
Matching statistic: St000292
(load all 13 compositions to match this statistic)
Mapping
Mp00104: Binary words —reverse⟶ Binary words
Mp00178: Binary words —to composition⟶ Integer compositions
Mp00094: Integer compositions —to binary word⟶ Binary words
St000292: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
0 => 0 => [2] => 10 => 0
1 => 1 => [1,1] => 11 => 0
00 => 00 => [3] => 100 => 0
01 => 10 => [1,2] => 110 => 0
10 => 01 => [2,1] => 101 => 1
11 => 11 => [1,1,1] => 111 => 0
000 => 000 => [4] => 1000 => 0
001 => 100 => [1,3] => 1100 => 0
010 => 010 => [2,2] => 1010 => 1
011 => 110 => [1,1,2] => 1110 => 0
100 => 001 => [3,1] => 1001 => 1
101 => 101 => [1,2,1] => 1101 => 1
110 => 011 => [2,1,1] => 1011 => 1
111 => 111 => [1,1,1,1] => 1111 => 0
0000 => 0000 => [5] => 10000 => 0
0001 => 1000 => [1,4] => 11000 => 0
0010 => 0100 => [2,3] => 10100 => 1
0011 => 1100 => [1,1,3] => 11100 => 0
0100 => 0010 => [3,2] => 10010 => 1
0101 => 1010 => [1,2,2] => 11010 => 1
0110 => 0110 => [2,1,2] => 10110 => 1
0111 => 1110 => [1,1,1,2] => 11110 => 0
1000 => 0001 => [4,1] => 10001 => 1
1001 => 1001 => [1,3,1] => 11001 => 1
1010 => 0101 => [2,2,1] => 10101 => 2
1011 => 1101 => [1,1,2,1] => 11101 => 1
1100 => 0011 => [3,1,1] => 10011 => 1
1101 => 1011 => [1,2,1,1] => 11011 => 1
1110 => 0111 => [2,1,1,1] => 10111 => 1
1111 => 1111 => [1,1,1,1,1] => 11111 => 0
00000 => 00000 => [6] => 100000 => 0
00001 => 10000 => [1,5] => 110000 => 0
00010 => 01000 => [2,4] => 101000 => 1
00011 => 11000 => [1,1,4] => 111000 => 0
00100 => 00100 => [3,3] => 100100 => 1
00101 => 10100 => [1,2,3] => 110100 => 1
00110 => 01100 => [2,1,3] => 101100 => 1
00111 => 11100 => [1,1,1,3] => 111100 => 0
01000 => 00010 => [4,2] => 100010 => 1
01001 => 10010 => [1,3,2] => 110010 => 1
01010 => 01010 => [2,2,2] => 101010 => 2
01011 => 11010 => [1,1,2,2] => 111010 => 1
01100 => 00110 => [3,1,2] => 100110 => 1
01101 => 10110 => [1,2,1,2] => 110110 => 1
01110 => 01110 => [2,1,1,2] => 101110 => 1
01111 => 11110 => [1,1,1,1,2] => 111110 => 0
10000 => 00001 => [5,1] => 100001 => 1
10001 => 10001 => [1,4,1] => 110001 => 1
10010 => 01001 => [2,3,1] => 101001 => 2
10011 => 11001 => [1,1,3,1] => 111001 => 1
Description
The number of ascents of a binary word.
Matching statistic: St001036
Mapping
Mp00178: Binary words —to composition⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
Mp00032: Dyck paths —inverse zeta map⟶ Dyck paths
St001036: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
0 => [2] => [1,1,0,0]
 => [1,0,1,0]
 => 0
1 => [1,1] => [1,0,1,0]
 => [1,1,0,0]
 => 0
00 => [3] => [1,1,1,0,0,0]
 => [1,0,1,0,1,0]
 => 0
01 => [2,1] => [1,1,0,0,1,0]
 => [1,1,0,1,0,0]
 => 0
10 => [1,2] => [1,0,1,1,0,0]
 => [1,0,1,1,0,0]
 => 1
11 => [1,1,1] => [1,0,1,0,1,0]
 => [1,1,1,0,0,0]
 => 0
000 => [4] => [1,1,1,1,0,0,0,0]
 => [1,0,1,0,1,0,1,0]
 => 0
001 => [3,1] => [1,1,1,0,0,0,1,0]
 => [1,1,0,1,0,1,0,0]
 => 0
010 => [2,2] => [1,1,0,0,1,1,0,0]
 => [1,0,1,1,0,1,0,0]
 => 1
011 => [2,1,1] => [1,1,0,0,1,0,1,0]
 => [1,1,1,0,1,0,0,0]
 => 0
100 => [1,3] => [1,0,1,1,1,0,0,0]
 => [1,0,1,0,1,1,0,0]
 => 1
101 => [1,2,1] => [1,0,1,1,0,0,1,0]
 => [1,1,0,1,1,0,0,0]
 => 1
110 => [1,1,2] => [1,0,1,0,1,1,0,0]
 => [1,0,1,1,1,0,0,0]
 => 1
111 => [1,1,1,1] => [1,0,1,0,1,0,1,0]
 => [1,1,1,1,0,0,0,0]
 => 0
0000 => [5] => [1,1,1,1,1,0,0,0,0,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => 0
0001 => [4,1] => [1,1,1,1,0,0,0,0,1,0]
 => [1,1,0,1,0,1,0,1,0,0]
 => 0
0010 => [3,2] => [1,1,1,0,0,0,1,1,0,0]
 => [1,0,1,1,0,1,0,1,0,0]
 => 1
0011 => [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
 => [1,1,1,0,1,0,1,0,0,0]
 => 0
0100 => [2,3] => [1,1,0,0,1,1,1,0,0,0]
 => [1,0,1,0,1,1,0,1,0,0]
 => 1
0101 => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
 => [1,1,0,1,1,0,1,0,0,0]
 => 1
0110 => [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
 => [1,0,1,1,1,0,1,0,0,0]
 => 1
0111 => [2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
 => [1,1,1,1,0,1,0,0,0,0]
 => 0
1000 => [1,4] => [1,0,1,1,1,1,0,0,0,0]
 => [1,0,1,0,1,0,1,1,0,0]
 => 1
1001 => [1,3,1] => [1,0,1,1,1,0,0,0,1,0]
 => [1,1,0,1,0,1,1,0,0,0]
 => 1
1010 => [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
 => [1,0,1,1,0,1,1,0,0,0]
 => 2
1011 => [1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
 => [1,1,1,0,1,1,0,0,0,0]
 => 1
1100 => [1,1,3] => [1,0,1,0,1,1,1,0,0,0]
 => [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]
 => [1,1,0,1,1,1,0,0,0,0]
 => 1
1110 => [1,1,1,2] => [1,0,1,0,1,0,1,1,0,0]
 => [1,0,1,1,1,1,0,0,0,0]
 => 1
1111 => [1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => 0
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
00001 => [5,1] => [1,1,1,1,1,0,0,0,0,0,1,0]
 => [1,1,0,1,0,1,0,1,0,1,0,0]
 => 0
00010 => [4,2] => [1,1,1,1,0,0,0,0,1,1,0,0]
 => [1,0,1,1,0,1,0,1,0,1,0,0]
 => 1
00011 => [4,1,1] => [1,1,1,1,0,0,0,0,1,0,1,0]
 => [1,1,1,0,1,0,1,0,1,0,0,0]
 => 0
00100 => [3,3] => [1,1,1,0,0,0,1,1,1,0,0,0]
 => [1,0,1,0,1,1,0,1,0,1,0,0]
 => 1
00101 => [3,2,1] => [1,1,1,0,0,0,1,1,0,0,1,0]
 => [1,1,0,1,1,0,1,0,1,0,0,0]
 => 1
00110 => [3,1,2] => [1,1,1,0,0,0,1,0,1,1,0,0]
 => [1,0,1,1,1,0,1,0,1,0,0,0]
 => 1
00111 => [3,1,1,1] => [1,1,1,0,0,0,1,0,1,0,1,0]
 => [1,1,1,1,0,1,0,1,0,0,0,0]
 => 0
01000 => [2,4] => [1,1,0,0,1,1,1,1,0,0,0,0]
 => [1,0,1,0,1,0,1,1,0,1,0,0]
 => 1
01001 => [2,3,1] => [1,1,0,0,1,1,1,0,0,0,1,0]
 => [1,1,0,1,0,1,1,0,1,0,0,0]
 => 1
01010 => [2,2,2] => [1,1,0,0,1,1,0,0,1,1,0,0]
 => [1,0,1,1,0,1,1,0,1,0,0,0]
 => 2
01011 => [2,2,1,1] => [1,1,0,0,1,1,0,0,1,0,1,0]
 => [1,1,1,0,1,1,0,1,0,0,0,0]
 => 1
01100 => [2,1,3] => [1,1,0,0,1,0,1,1,1,0,0,0]
 => [1,0,1,0,1,1,1,0,1,0,0,0]
 => 1
01101 => [2,1,2,1] => [1,1,0,0,1,0,1,1,0,0,1,0]
 => [1,1,0,1,1,1,0,1,0,0,0,0]
 => 1
01110 => [2,1,1,2] => [1,1,0,0,1,0,1,0,1,1,0,0]
 => [1,0,1,1,1,1,0,1,0,0,0,0]
 => 1
01111 => [2,1,1,1,1] => [1,1,0,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,0,1,0,0,0,0,0]
 => 0
10000 => [1,5] => [1,0,1,1,1,1,1,0,0,0,0,0]
 => [1,0,1,0,1,0,1,0,1,1,0,0]
 => 1
10001 => [1,4,1] => [1,0,1,1,1,1,0,0,0,0,1,0]
 => [1,1,0,1,0,1,0,1,1,0,0,0]
 => 1
10010 => [1,3,2] => [1,0,1,1,1,0,0,0,1,1,0,0]
 => [1,0,1,1,0,1,0,1,1,0,0,0]
 => 2
10011 => [1,3,1,1] => [1,0,1,1,1,0,0,0,1,0,1,0]
 => [1,1,1,0,1,0,1,1,0,0,0,0]
 => 1
Description
The number of inner corners of the parallelogram polyomino associated with the Dyck path.
Matching statistic: St001037
(load all 2 compositions to match this statistic)
Mapping
Mp00178: Binary words —to composition⟶ Integer compositions
Mp00041: Integer compositions —conjugate⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St001037: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
0 => [2] => [1,1] => [1,0,1,0]
 => 0
1 => [1,1] => [2] => [1,1,0,0]
 => 0
00 => [3] => [1,1,1] => [1,0,1,0,1,0]
 => 0
01 => [2,1] => [2,1] => [1,1,0,0,1,0]
 => 0
10 => [1,2] => [1,2] => [1,0,1,1,0,0]
 => 1
11 => [1,1,1] => [3] => [1,1,1,0,0,0]
 => 0
000 => [4] => [1,1,1,1] => [1,0,1,0,1,0,1,0]
 => 0
001 => [3,1] => [2,1,1] => [1,1,0,0,1,0,1,0]
 => 0
010 => [2,2] => [1,2,1] => [1,0,1,1,0,0,1,0]
 => 1
011 => [2,1,1] => [3,1] => [1,1,1,0,0,0,1,0]
 => 0
100 => [1,3] => [1,1,2] => [1,0,1,0,1,1,0,0]
 => 1
101 => [1,2,1] => [2,2] => [1,1,0,0,1,1,0,0]
 => 1
110 => [1,1,2] => [1,3] => [1,0,1,1,1,0,0,0]
 => 1
111 => [1,1,1,1] => [4] => [1,1,1,1,0,0,0,0]
 => 0
0000 => [5] => [1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
 => 0
0001 => [4,1] => [2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
 => 0
0010 => [3,2] => [1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
 => 1
0011 => [3,1,1] => [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
 => 0
0100 => [2,3] => [1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
 => 1
0101 => [2,2,1] => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
 => 1
0110 => [2,1,2] => [1,3,1] => [1,0,1,1,1,0,0,0,1,0]
 => 1
0111 => [2,1,1,1] => [4,1] => [1,1,1,1,0,0,0,0,1,0]
 => 0
1000 => [1,4] => [1,1,1,2] => [1,0,1,0,1,0,1,1,0,0]
 => 1
1001 => [1,3,1] => [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
 => 1
1010 => [1,2,2] => [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
 => 2
1011 => [1,2,1,1] => [3,2] => [1,1,1,0,0,0,1,1,0,0]
 => 1
1100 => [1,1,3] => [1,1,3] => [1,0,1,0,1,1,1,0,0,0]
 => 1
1101 => [1,1,2,1] => [2,3] => [1,1,0,0,1,1,1,0,0,0]
 => 1
1110 => [1,1,1,2] => [1,4] => [1,0,1,1,1,1,0,0,0,0]
 => 1
1111 => [1,1,1,1,1] => [5] => [1,1,1,1,1,0,0,0,0,0]
 => 0
00000 => [6] => [1,1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0,1,0]
 => 0
00001 => [5,1] => [2,1,1,1,1] => [1,1,0,0,1,0,1,0,1,0,1,0]
 => 0
00010 => [4,2] => [1,2,1,1,1] => [1,0,1,1,0,0,1,0,1,0,1,0]
 => 1
00011 => [4,1,1] => [3,1,1,1] => [1,1,1,0,0,0,1,0,1,0,1,0]
 => 0
00100 => [3,3] => [1,1,2,1,1] => [1,0,1,0,1,1,0,0,1,0,1,0]
 => 1
00101 => [3,2,1] => [2,2,1,1] => [1,1,0,0,1,1,0,0,1,0,1,0]
 => 1
00110 => [3,1,2] => [1,3,1,1] => [1,0,1,1,1,0,0,0,1,0,1,0]
 => 1
00111 => [3,1,1,1] => [4,1,1] => [1,1,1,1,0,0,0,0,1,0,1,0]
 => 0
01000 => [2,4] => [1,1,1,2,1] => [1,0,1,0,1,0,1,1,0,0,1,0]
 => 1
01001 => [2,3,1] => [2,1,2,1] => [1,1,0,0,1,0,1,1,0,0,1,0]
 => 1
01010 => [2,2,2] => [1,2,2,1] => [1,0,1,1,0,0,1,1,0,0,1,0]
 => 2
01011 => [2,2,1,1] => [3,2,1] => [1,1,1,0,0,0,1,1,0,0,1,0]
 => 1
01100 => [2,1,3] => [1,1,3,1] => [1,0,1,0,1,1,1,0,0,0,1,0]
 => 1
01101 => [2,1,2,1] => [2,3,1] => [1,1,0,0,1,1,1,0,0,0,1,0]
 => 1
01110 => [2,1,1,2] => [1,4,1] => [1,0,1,1,1,1,0,0,0,0,1,0]
 => 1
01111 => [2,1,1,1,1] => [5,1] => [1,1,1,1,1,0,0,0,0,0,1,0]
 => 0
10000 => [1,5] => [1,1,1,1,2] => [1,0,1,0,1,0,1,0,1,1,0,0]
 => 1
10001 => [1,4,1] => [2,1,1,2] => [1,1,0,0,1,0,1,0,1,1,0,0]
 => 1
10010 => [1,3,2] => [1,2,1,2] => [1,0,1,1,0,0,1,0,1,1,0,0]
 => 2
10011 => [1,3,1,1] => [3,1,2] => [1,1,1,0,0,0,1,0,1,1,0,0]
 => 1
Description
The number of inner corners of the upper path of the parallelogram polyomino associated with the Dyck path.
Matching statistic: St000390
(load all 3 compositions to match this statistic)
Mapping
Mp00104: Binary words —reverse⟶ Binary words
Mp00178: Binary words —to composition⟶ Integer compositions
Mp00094: Integer compositions —to binary word⟶ Binary words
St000390: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
0 => 0 => [2] => 10 => 1 = 0 + 1
1 => 1 => [1,1] => 11 => 1 = 0 + 1
00 => 00 => [3] => 100 => 1 = 0 + 1
01 => 10 => [1,2] => 110 => 1 = 0 + 1
10 => 01 => [2,1] => 101 => 2 = 1 + 1
11 => 11 => [1,1,1] => 111 => 1 = 0 + 1
000 => 000 => [4] => 1000 => 1 = 0 + 1
001 => 100 => [1,3] => 1100 => 1 = 0 + 1
010 => 010 => [2,2] => 1010 => 2 = 1 + 1
011 => 110 => [1,1,2] => 1110 => 1 = 0 + 1
100 => 001 => [3,1] => 1001 => 2 = 1 + 1
101 => 101 => [1,2,1] => 1101 => 2 = 1 + 1
110 => 011 => [2,1,1] => 1011 => 2 = 1 + 1
111 => 111 => [1,1,1,1] => 1111 => 1 = 0 + 1
0000 => 0000 => [5] => 10000 => 1 = 0 + 1
0001 => 1000 => [1,4] => 11000 => 1 = 0 + 1
0010 => 0100 => [2,3] => 10100 => 2 = 1 + 1
0011 => 1100 => [1,1,3] => 11100 => 1 = 0 + 1
0100 => 0010 => [3,2] => 10010 => 2 = 1 + 1
0101 => 1010 => [1,2,2] => 11010 => 2 = 1 + 1
0110 => 0110 => [2,1,2] => 10110 => 2 = 1 + 1
0111 => 1110 => [1,1,1,2] => 11110 => 1 = 0 + 1
1000 => 0001 => [4,1] => 10001 => 2 = 1 + 1
1001 => 1001 => [1,3,1] => 11001 => 2 = 1 + 1
1010 => 0101 => [2,2,1] => 10101 => 3 = 2 + 1
1011 => 1101 => [1,1,2,1] => 11101 => 2 = 1 + 1
1100 => 0011 => [3,1,1] => 10011 => 2 = 1 + 1
1101 => 1011 => [1,2,1,1] => 11011 => 2 = 1 + 1
1110 => 0111 => [2,1,1,1] => 10111 => 2 = 1 + 1
1111 => 1111 => [1,1,1,1,1] => 11111 => 1 = 0 + 1
00000 => 00000 => [6] => 100000 => 1 = 0 + 1
00001 => 10000 => [1,5] => 110000 => 1 = 0 + 1
00010 => 01000 => [2,4] => 101000 => 2 = 1 + 1
00011 => 11000 => [1,1,4] => 111000 => 1 = 0 + 1
00100 => 00100 => [3,3] => 100100 => 2 = 1 + 1
00101 => 10100 => [1,2,3] => 110100 => 2 = 1 + 1
00110 => 01100 => [2,1,3] => 101100 => 2 = 1 + 1
00111 => 11100 => [1,1,1,3] => 111100 => 1 = 0 + 1
01000 => 00010 => [4,2] => 100010 => 2 = 1 + 1
01001 => 10010 => [1,3,2] => 110010 => 2 = 1 + 1
01010 => 01010 => [2,2,2] => 101010 => 3 = 2 + 1
01011 => 11010 => [1,1,2,2] => 111010 => 2 = 1 + 1
01100 => 00110 => [3,1,2] => 100110 => 2 = 1 + 1
01101 => 10110 => [1,2,1,2] => 110110 => 2 = 1 + 1
01110 => 01110 => [2,1,1,2] => 101110 => 2 = 1 + 1
01111 => 11110 => [1,1,1,1,2] => 111110 => 1 = 0 + 1
10000 => 00001 => [5,1] => 100001 => 2 = 1 + 1
10001 => 10001 => [1,4,1] => 110001 => 2 = 1 + 1
10010 => 01001 => [2,3,1] => 101001 => 3 = 2 + 1
10011 => 11001 => [1,1,3,1] => 111001 => 2 = 1 + 1
Description
The number of runs of ones in a binary word.
Matching statistic: St001732
Mapping
Mp00178: Binary words —to composition⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
Mp00032: Dyck paths —inverse zeta map⟶ Dyck paths
St001732: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
0 => [2] => [1,1,0,0]
 => [1,0,1,0]
 => 1 = 0 + 1
1 => [1,1] => [1,0,1,0]
 => [1,1,0,0]
 => 1 = 0 + 1
00 => [3] => [1,1,1,0,0,0]
 => [1,0,1,0,1,0]
 => 1 = 0 + 1
01 => [2,1] => [1,1,0,0,1,0]
 => [1,1,0,1,0,0]
 => 1 = 0 + 1
10 => [1,2] => [1,0,1,1,0,0]
 => [1,0,1,1,0,0]
 => 2 = 1 + 1
11 => [1,1,1] => [1,0,1,0,1,0]
 => [1,1,1,0,0,0]
 => 1 = 0 + 1
000 => [4] => [1,1,1,1,0,0,0,0]
 => [1,0,1,0,1,0,1,0]
 => 1 = 0 + 1
001 => [3,1] => [1,1,1,0,0,0,1,0]
 => [1,1,0,1,0,1,0,0]
 => 1 = 0 + 1
010 => [2,2] => [1,1,0,0,1,1,0,0]
 => [1,0,1,1,0,1,0,0]
 => 2 = 1 + 1
011 => [2,1,1] => [1,1,0,0,1,0,1,0]
 => [1,1,1,0,1,0,0,0]
 => 1 = 0 + 1
100 => [1,3] => [1,0,1,1,1,0,0,0]
 => [1,0,1,0,1,1,0,0]
 => 2 = 1 + 1
101 => [1,2,1] => [1,0,1,1,0,0,1,0]
 => [1,1,0,1,1,0,0,0]
 => 2 = 1 + 1
110 => [1,1,2] => [1,0,1,0,1,1,0,0]
 => [1,0,1,1,1,0,0,0]
 => 2 = 1 + 1
111 => [1,1,1,1] => [1,0,1,0,1,0,1,0]
 => [1,1,1,1,0,0,0,0]
 => 1 = 0 + 1
0000 => [5] => [1,1,1,1,1,0,0,0,0,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => 1 = 0 + 1
0001 => [4,1] => [1,1,1,1,0,0,0,0,1,0]
 => [1,1,0,1,0,1,0,1,0,0]
 => 1 = 0 + 1
0010 => [3,2] => [1,1,1,0,0,0,1,1,0,0]
 => [1,0,1,1,0,1,0,1,0,0]
 => 2 = 1 + 1
0011 => [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
 => [1,1,1,0,1,0,1,0,0,0]
 => 1 = 0 + 1
0100 => [2,3] => [1,1,0,0,1,1,1,0,0,0]
 => [1,0,1,0,1,1,0,1,0,0]
 => 2 = 1 + 1
0101 => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
 => [1,1,0,1,1,0,1,0,0,0]
 => 2 = 1 + 1
0110 => [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
 => [1,0,1,1,1,0,1,0,0,0]
 => 2 = 1 + 1
0111 => [2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
 => [1,1,1,1,0,1,0,0,0,0]
 => 1 = 0 + 1
1000 => [1,4] => [1,0,1,1,1,1,0,0,0,0]
 => [1,0,1,0,1,0,1,1,0,0]
 => 2 = 1 + 1
1001 => [1,3,1] => [1,0,1,1,1,0,0,0,1,0]
 => [1,1,0,1,0,1,1,0,0,0]
 => 2 = 1 + 1
1010 => [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
 => [1,0,1,1,0,1,1,0,0,0]
 => 3 = 2 + 1
1011 => [1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
 => [1,1,1,0,1,1,0,0,0,0]
 => 2 = 1 + 1
1100 => [1,1,3] => [1,0,1,0,1,1,1,0,0,0]
 => [1,0,1,0,1,1,1,0,0,0]
 => 2 = 1 + 1
1101 => [1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
 => [1,1,0,1,1,1,0,0,0,0]
 => 2 = 1 + 1
1110 => [1,1,1,2] => [1,0,1,0,1,0,1,1,0,0]
 => [1,0,1,1,1,1,0,0,0,0]
 => 2 = 1 + 1
1111 => [1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => 1 = 0 + 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]
 => 1 = 0 + 1
00001 => [5,1] => [1,1,1,1,1,0,0,0,0,0,1,0]
 => [1,1,0,1,0,1,0,1,0,1,0,0]
 => 1 = 0 + 1
00010 => [4,2] => [1,1,1,1,0,0,0,0,1,1,0,0]
 => [1,0,1,1,0,1,0,1,0,1,0,0]
 => 2 = 1 + 1
00011 => [4,1,1] => [1,1,1,1,0,0,0,0,1,0,1,0]
 => [1,1,1,0,1,0,1,0,1,0,0,0]
 => 1 = 0 + 1
00100 => [3,3] => [1,1,1,0,0,0,1,1,1,0,0,0]
 => [1,0,1,0,1,1,0,1,0,1,0,0]
 => 2 = 1 + 1
00101 => [3,2,1] => [1,1,1,0,0,0,1,1,0,0,1,0]
 => [1,1,0,1,1,0,1,0,1,0,0,0]
 => 2 = 1 + 1
00110 => [3,1,2] => [1,1,1,0,0,0,1,0,1,1,0,0]
 => [1,0,1,1,1,0,1,0,1,0,0,0]
 => 2 = 1 + 1
00111 => [3,1,1,1] => [1,1,1,0,0,0,1,0,1,0,1,0]
 => [1,1,1,1,0,1,0,1,0,0,0,0]
 => 1 = 0 + 1
01000 => [2,4] => [1,1,0,0,1,1,1,1,0,0,0,0]
 => [1,0,1,0,1,0,1,1,0,1,0,0]
 => 2 = 1 + 1
01001 => [2,3,1] => [1,1,0,0,1,1,1,0,0,0,1,0]
 => [1,1,0,1,0,1,1,0,1,0,0,0]
 => 2 = 1 + 1
01010 => [2,2,2] => [1,1,0,0,1,1,0,0,1,1,0,0]
 => [1,0,1,1,0,1,1,0,1,0,0,0]
 => 3 = 2 + 1
01011 => [2,2,1,1] => [1,1,0,0,1,1,0,0,1,0,1,0]
 => [1,1,1,0,1,1,0,1,0,0,0,0]
 => 2 = 1 + 1
01100 => [2,1,3] => [1,1,0,0,1,0,1,1,1,0,0,0]
 => [1,0,1,0,1,1,1,0,1,0,0,0]
 => 2 = 1 + 1
01101 => [2,1,2,1] => [1,1,0,0,1,0,1,1,0,0,1,0]
 => [1,1,0,1,1,1,0,1,0,0,0,0]
 => 2 = 1 + 1
01110 => [2,1,1,2] => [1,1,0,0,1,0,1,0,1,1,0,0]
 => [1,0,1,1,1,1,0,1,0,0,0,0]
 => 2 = 1 + 1
01111 => [2,1,1,1,1] => [1,1,0,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,0,1,0,0,0,0,0]
 => 1 = 0 + 1
10000 => [1,5] => [1,0,1,1,1,1,1,0,0,0,0,0]
 => [1,0,1,0,1,0,1,0,1,1,0,0]
 => 2 = 1 + 1
10001 => [1,4,1] => [1,0,1,1,1,1,0,0,0,0,1,0]
 => [1,1,0,1,0,1,0,1,1,0,0,0]
 => 2 = 1 + 1
10010 => [1,3,2] => [1,0,1,1,1,0,0,0,1,1,0,0]
 => [1,0,1,1,0,1,0,1,1,0,0,0]
 => 3 = 2 + 1
10011 => [1,3,1,1] => [1,0,1,1,1,0,0,0,1,0,1,0]
 => [1,1,1,0,1,0,1,1,0,0,0,0]
 => 2 = 1 + 1
Description
The number of peaks visible from the left. This is, the number of left-to-right maxima of the heights of the peaks of a Dyck path.
Matching statistic: St000291
(load all 11 compositions to match this statistic)
Mapping
St000291: Binary words ⟶ ℤResult quality: 99% ●values known / values provided: 99%●distinct values known / distinct values provided: 100%
Values
0 => 0
1 => 0
00 => 0
01 => 0
10 => 1
11 => 0
000 => 0
001 => 0
010 => 1
011 => 0
100 => 1
101 => 1
110 => 1
111 => 0
0000 => 0
0001 => 0
0010 => 1
0011 => 0
0100 => 1
0101 => 1
0110 => 1
0111 => 0
1000 => 1
1001 => 1
1010 => 2
1011 => 1
1100 => 1
1101 => 1
1110 => 1
1111 => 0
00000 => 0
00001 => 0
00010 => 1
00011 => 0
00100 => 1
00101 => 1
00110 => 1
00111 => 0
01000 => 1
01001 => 1
01010 => 2
01011 => 1
01100 => 1
01101 => 1
01110 => 1
01111 => 0
10000 => 1
10001 => 1
10010 => 2
10011 => 1
=> ? = 0
Description
The number of descents of a binary word.
Matching statistic: St000386
(load all 6 compositions to match this statistic)
Mapping
Mp00178: Binary words —to composition⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
Mp00229: Dyck paths —Delest-Viennot⟶ Dyck paths
St000386: Dyck paths ⟶ ℤResult quality: 96% ●values known / values provided: 96%●distinct values known / distinct values provided: 100%
Values
0 => [2] => [1,1,0,0]
 => [1,0,1,0]
 => 0
1 => [1,1] => [1,0,1,0]
 => [1,1,0,0]
 => 0
00 => [3] => [1,1,1,0,0,0]
 => [1,0,1,0,1,0]
 => 0
01 => [2,1] => [1,1,0,0,1,0]
 => [1,0,1,1,0,0]
 => 0
10 => [1,2] => [1,0,1,1,0,0]
 => [1,1,0,0,1,0]
 => 1
11 => [1,1,1] => [1,0,1,0,1,0]
 => [1,1,0,1,0,0]
 => 0
000 => [4] => [1,1,1,1,0,0,0,0]
 => [1,0,1,0,1,0,1,0]
 => 0
001 => [3,1] => [1,1,1,0,0,0,1,0]
 => [1,0,1,0,1,1,0,0]
 => 0
010 => [2,2] => [1,1,0,0,1,1,0,0]
 => [1,0,1,1,0,0,1,0]
 => 1
011 => [2,1,1] => [1,1,0,0,1,0,1,0]
 => [1,0,1,1,0,1,0,0]
 => 0
100 => [1,3] => [1,0,1,1,1,0,0,0]
 => [1,1,0,0,1,0,1,0]
 => 1
101 => [1,2,1] => [1,0,1,1,0,0,1,0]
 => [1,1,0,0,1,1,0,0]
 => 1
110 => [1,1,2] => [1,0,1,0,1,1,0,0]
 => [1,1,0,1,0,0,1,0]
 => 1
111 => [1,1,1,1] => [1,0,1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,0]
 => 0
0000 => [5] => [1,1,1,1,1,0,0,0,0,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => 0
0001 => [4,1] => [1,1,1,1,0,0,0,0,1,0]
 => [1,0,1,0,1,0,1,1,0,0]
 => 0
0010 => [3,2] => [1,1,1,0,0,0,1,1,0,0]
 => [1,0,1,0,1,1,0,0,1,0]
 => 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]
 => 0
0100 => [2,3] => [1,1,0,0,1,1,1,0,0,0]
 => [1,0,1,1,0,0,1,0,1,0]
 => 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]
 => 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
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]
 => 0
1000 => [1,4] => [1,0,1,1,1,1,0,0,0,0]
 => [1,1,0,0,1,0,1,0,1,0]
 => 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
1010 => [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
 => [1,1,0,0,1,1,0,0,1,0]
 => 2
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
1100 => [1,1,3] => [1,0,1,0,1,1,1,0,0,0]
 => [1,1,0,1,0,0,1,0,1,0]
 => 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
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]
 => 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
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
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]
 => 0
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
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]
 => 0
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
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]
 => 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
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]
 => 0
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
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]
 => 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
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]
 => 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
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]
 => 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
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]
 => 0
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]
 => 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
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]
 => 2
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
1101010 => [1,1,2,2,2] => [1,0,1,0,1,1,0,0,1,1,0,0,1,1,0,0]
 => [1,1,0,1,0,0,1,1,0,0,1,1,0,0,1,0]
 => ? = 3
1101100 => [1,1,2,1,3] => [1,0,1,0,1,1,0,0,1,0,1,1,1,0,0,0]
 => [1,1,0,1,0,0,1,1,0,1,0,0,1,0,1,0]
 => ? = 2
1101110 => [1,1,2,1,1,2] => [1,0,1,0,1,1,0,0,1,0,1,0,1,1,0,0]
 => [1,1,0,1,0,0,1,1,0,1,0,1,0,0,1,0]
 => ? = 2
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
1110010 => [1,1,1,3,2] => [1,0,1,0,1,0,1,1,1,0,0,0,1,1,0,0]
 => [1,1,0,1,0,1,0,0,1,0,1,1,0,0,1,0]
 => ? = 2
1110100 => [1,1,1,2,3] => [1,0,1,0,1,0,1,1,0,0,1,1,1,0,0,0]
 => [1,1,0,1,0,1,0,0,1,1,0,0,1,0,1,0]
 => ? = 2
1110110 => [1,1,1,2,1,2] => [1,0,1,0,1,0,1,1,0,0,1,0,1,1,0,0]
 => [1,1,0,1,0,1,0,0,1,1,0,1,0,0,1,0]
 => ? = 2
1111010 => [1,1,1,1,2,2] => [1,0,1,0,1,0,1,0,1,1,0,0,1,1,0,0]
 => [1,1,0,1,0,1,0,1,0,0,1,1,0,0,1,0]
 => ? = 2
Description
The number of factors DDU in a Dyck path.
Matching statistic: St000568
Mapping
Mp00178: Binary words —to composition⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
Mp00034: Dyck paths —to binary tree: up step, left tree, down step, right tree⟶ Binary trees
St000568: Binary trees ⟶ ℤResult quality: 91% ●values known / values provided: 91%●distinct values known / distinct values provided: 100%
Values
0 => [2] => [1,1,0,0]
 => [[.,.],.]
 => 1 = 0 + 1
1 => [1,1] => [1,0,1,0]
 => [.,[.,.]]
 => 1 = 0 + 1
00 => [3] => [1,1,1,0,0,0]
 => [[[.,.],.],.]
 => 1 = 0 + 1
01 => [2,1] => [1,1,0,0,1,0]
 => [[.,.],[.,.]]
 => 1 = 0 + 1
10 => [1,2] => [1,0,1,1,0,0]
 => [.,[[.,.],.]]
 => 2 = 1 + 1
11 => [1,1,1] => [1,0,1,0,1,0]
 => [.,[.,[.,.]]]
 => 1 = 0 + 1
000 => [4] => [1,1,1,1,0,0,0,0]
 => [[[[.,.],.],.],.]
 => 1 = 0 + 1
001 => [3,1] => [1,1,1,0,0,0,1,0]
 => [[[.,.],.],[.,.]]
 => 1 = 0 + 1
010 => [2,2] => [1,1,0,0,1,1,0,0]
 => [[.,.],[[.,.],.]]
 => 2 = 1 + 1
011 => [2,1,1] => [1,1,0,0,1,0,1,0]
 => [[.,.],[.,[.,.]]]
 => 1 = 0 + 1
100 => [1,3] => [1,0,1,1,1,0,0,0]
 => [.,[[[.,.],.],.]]
 => 2 = 1 + 1
101 => [1,2,1] => [1,0,1,1,0,0,1,0]
 => [.,[[.,.],[.,.]]]
 => 2 = 1 + 1
110 => [1,1,2] => [1,0,1,0,1,1,0,0]
 => [.,[.,[[.,.],.]]]
 => 2 = 1 + 1
111 => [1,1,1,1] => [1,0,1,0,1,0,1,0]
 => [.,[.,[.,[.,.]]]]
 => 1 = 0 + 1
0000 => [5] => [1,1,1,1,1,0,0,0,0,0]
 => [[[[[.,.],.],.],.],.]
 => 1 = 0 + 1
0001 => [4,1] => [1,1,1,1,0,0,0,0,1,0]
 => [[[[.,.],.],.],[.,.]]
 => 1 = 0 + 1
0010 => [3,2] => [1,1,1,0,0,0,1,1,0,0]
 => [[[.,.],.],[[.,.],.]]
 => 2 = 1 + 1
0011 => [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
 => [[[.,.],.],[.,[.,.]]]
 => 1 = 0 + 1
0100 => [2,3] => [1,1,0,0,1,1,1,0,0,0]
 => [[.,.],[[[.,.],.],.]]
 => 2 = 1 + 1
0101 => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
 => [[.,.],[[.,.],[.,.]]]
 => 2 = 1 + 1
0110 => [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
 => [[.,.],[.,[[.,.],.]]]
 => 2 = 1 + 1
0111 => [2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
 => [[.,.],[.,[.,[.,.]]]]
 => 1 = 0 + 1
1000 => [1,4] => [1,0,1,1,1,1,0,0,0,0]
 => [.,[[[[.,.],.],.],.]]
 => 2 = 1 + 1
1001 => [1,3,1] => [1,0,1,1,1,0,0,0,1,0]
 => [.,[[[.,.],.],[.,.]]]
 => 2 = 1 + 1
1010 => [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
 => [.,[[.,.],[[.,.],.]]]
 => 3 = 2 + 1
1011 => [1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
 => [.,[[.,.],[.,[.,.]]]]
 => 2 = 1 + 1
1100 => [1,1,3] => [1,0,1,0,1,1,1,0,0,0]
 => [.,[.,[[[.,.],.],.]]]
 => 2 = 1 + 1
1101 => [1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
 => [.,[.,[[.,.],[.,.]]]]
 => 2 = 1 + 1
1110 => [1,1,1,2] => [1,0,1,0,1,0,1,1,0,0]
 => [.,[.,[.,[[.,.],.]]]]
 => 2 = 1 + 1
1111 => [1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
 => [.,[.,[.,[.,[.,.]]]]]
 => 1 = 0 + 1
00000 => [6] => [1,1,1,1,1,1,0,0,0,0,0,0]
 => [[[[[[.,.],.],.],.],.],.]
 => 1 = 0 + 1
00001 => [5,1] => [1,1,1,1,1,0,0,0,0,0,1,0]
 => [[[[[.,.],.],.],.],[.,.]]
 => 1 = 0 + 1
00010 => [4,2] => [1,1,1,1,0,0,0,0,1,1,0,0]
 => [[[[.,.],.],.],[[.,.],.]]
 => 2 = 1 + 1
00011 => [4,1,1] => [1,1,1,1,0,0,0,0,1,0,1,0]
 => [[[[.,.],.],.],[.,[.,.]]]
 => 1 = 0 + 1
00100 => [3,3] => [1,1,1,0,0,0,1,1,1,0,0,0]
 => [[[.,.],.],[[[.,.],.],.]]
 => 2 = 1 + 1
00101 => [3,2,1] => [1,1,1,0,0,0,1,1,0,0,1,0]
 => [[[.,.],.],[[.,.],[.,.]]]
 => 2 = 1 + 1
00110 => [3,1,2] => [1,1,1,0,0,0,1,0,1,1,0,0]
 => [[[.,.],.],[.,[[.,.],.]]]
 => 2 = 1 + 1
00111 => [3,1,1,1] => [1,1,1,0,0,0,1,0,1,0,1,0]
 => [[[.,.],.],[.,[.,[.,.]]]]
 => 1 = 0 + 1
01000 => [2,4] => [1,1,0,0,1,1,1,1,0,0,0,0]
 => [[.,.],[[[[.,.],.],.],.]]
 => 2 = 1 + 1
01001 => [2,3,1] => [1,1,0,0,1,1,1,0,0,0,1,0]
 => [[.,.],[[[.,.],.],[.,.]]]
 => 2 = 1 + 1
01010 => [2,2,2] => [1,1,0,0,1,1,0,0,1,1,0,0]
 => [[.,.],[[.,.],[[.,.],.]]]
 => 3 = 2 + 1
01011 => [2,2,1,1] => [1,1,0,0,1,1,0,0,1,0,1,0]
 => [[.,.],[[.,.],[.,[.,.]]]]
 => 2 = 1 + 1
01100 => [2,1,3] => [1,1,0,0,1,0,1,1,1,0,0,0]
 => [[.,.],[.,[[[.,.],.],.]]]
 => 2 = 1 + 1
01101 => [2,1,2,1] => [1,1,0,0,1,0,1,1,0,0,1,0]
 => [[.,.],[.,[[.,.],[.,.]]]]
 => 2 = 1 + 1
01110 => [2,1,1,2] => [1,1,0,0,1,0,1,0,1,1,0,0]
 => [[.,.],[.,[.,[[.,.],.]]]]
 => 2 = 1 + 1
01111 => [2,1,1,1,1] => [1,1,0,0,1,0,1,0,1,0,1,0]
 => [[.,.],[.,[.,[.,[.,.]]]]]
 => 1 = 0 + 1
10000 => [1,5] => [1,0,1,1,1,1,1,0,0,0,0,0]
 => [.,[[[[[.,.],.],.],.],.]]
 => 2 = 1 + 1
10001 => [1,4,1] => [1,0,1,1,1,1,0,0,0,0,1,0]
 => [.,[[[[.,.],.],.],[.,.]]]
 => 2 = 1 + 1
10010 => [1,3,2] => [1,0,1,1,1,0,0,0,1,1,0,0]
 => [.,[[[.,.],.],[[.,.],.]]]
 => 3 = 2 + 1
10011 => [1,3,1,1] => [1,0,1,1,1,0,0,0,1,0,1,0]
 => [.,[[[.,.],.],[.,[.,.]]]]
 => 2 = 1 + 1
0000000 => [8] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
 => [[[[[[[[.,.],.],.],.],.],.],.],.]
 => ? = 0 + 1
0000010 => [6,2] => [1,1,1,1,1,1,0,0,0,0,0,0,1,1,0,0]
 => [[[[[[.,.],.],.],.],.],[[.,.],.]]
 => ? = 1 + 1
0000100 => [5,3] => [1,1,1,1,1,0,0,0,0,0,1,1,1,0,0,0]
 => [[[[[.,.],.],.],.],[[[.,.],.],.]]
 => ? = 1 + 1
0000110 => [5,1,2] => [1,1,1,1,1,0,0,0,0,0,1,0,1,1,0,0]
 => [[[[[.,.],.],.],.],[.,[[.,.],.]]]
 => ? = 1 + 1
0001000 => [4,4] => [1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0]
 => [[[[.,.],.],.],[[[[.,.],.],.],.]]
 => ? = 1 + 1
0001010 => [4,2,2] => [1,1,1,1,0,0,0,0,1,1,0,0,1,1,0,0]
 => [[[[.,.],.],.],[[.,.],[[.,.],.]]]
 => ? = 2 + 1
0001100 => [4,1,3] => [1,1,1,1,0,0,0,0,1,0,1,1,1,0,0,0]
 => [[[[.,.],.],.],[.,[[[.,.],.],.]]]
 => ? = 1 + 1
0001110 => [4,1,1,2] => [1,1,1,1,0,0,0,0,1,0,1,0,1,1,0,0]
 => [[[[.,.],.],.],[.,[.,[[.,.],.]]]]
 => ? = 1 + 1
0010000 => [3,5] => [1,1,1,0,0,0,1,1,1,1,1,0,0,0,0,0]
 => [[[.,.],.],[[[[[.,.],.],.],.],.]]
 => ? = 1 + 1
0010010 => [3,3,2] => [1,1,1,0,0,0,1,1,1,0,0,0,1,1,0,0]
 => [[[.,.],.],[[[.,.],.],[[.,.],.]]]
 => ? = 2 + 1
0010100 => [3,2,3] => [1,1,1,0,0,0,1,1,0,0,1,1,1,0,0,0]
 => [[[.,.],.],[[.,.],[[[.,.],.],.]]]
 => ? = 2 + 1
0010110 => [3,2,1,2] => [1,1,1,0,0,0,1,1,0,0,1,0,1,1,0,0]
 => [[[.,.],.],[[.,.],[.,[[.,.],.]]]]
 => ? = 2 + 1
0011000 => [3,1,4] => [1,1,1,0,0,0,1,0,1,1,1,1,0,0,0,0]
 => [[[.,.],.],[.,[[[[.,.],.],.],.]]]
 => ? = 1 + 1
0011010 => [3,1,2,2] => [1,1,1,0,0,0,1,0,1,1,0,0,1,1,0,0]
 => [[[.,.],.],[.,[[.,.],[[.,.],.]]]]
 => ? = 2 + 1
0011100 => [3,1,1,3] => [1,1,1,0,0,0,1,0,1,0,1,1,1,0,0,0]
 => [[[.,.],.],[.,[.,[[[.,.],.],.]]]]
 => ? = 1 + 1
0011110 => [3,1,1,1,2] => [1,1,1,0,0,0,1,0,1,0,1,0,1,1,0,0]
 => [[[.,.],.],[.,[.,[.,[[.,.],.]]]]]
 => ? = 1 + 1
=> [1] => [1,0]
 => [.,.]
 => ? = 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.
Matching statistic: St000201
Mapping
Mp00178: Binary words —to composition⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
Mp00029: Dyck paths —to binary tree: left tree, up step, right tree, down step⟶ Binary trees
St000201: Binary trees ⟶ ℤResult quality: 76% ●values known / values provided: 76%●distinct values known / distinct values provided: 100%
Values
0 => [2] => [1,1,0,0]
 => [.,[.,.]]
 => 1 = 0 + 1
1 => [1,1] => [1,0,1,0]
 => [[.,.],.]
 => 1 = 0 + 1
00 => [3] => [1,1,1,0,0,0]
 => [.,[.,[.,.]]]
 => 1 = 0 + 1
01 => [2,1] => [1,1,0,0,1,0]
 => [[.,[.,.]],.]
 => 1 = 0 + 1
10 => [1,2] => [1,0,1,1,0,0]
 => [[.,.],[.,.]]
 => 2 = 1 + 1
11 => [1,1,1] => [1,0,1,0,1,0]
 => [[[.,.],.],.]
 => 1 = 0 + 1
000 => [4] => [1,1,1,1,0,0,0,0]
 => [.,[.,[.,[.,.]]]]
 => 1 = 0 + 1
001 => [3,1] => [1,1,1,0,0,0,1,0]
 => [[.,[.,[.,.]]],.]
 => 1 = 0 + 1
010 => [2,2] => [1,1,0,0,1,1,0,0]
 => [[.,[.,.]],[.,.]]
 => 2 = 1 + 1
011 => [2,1,1] => [1,1,0,0,1,0,1,0]
 => [[[.,[.,.]],.],.]
 => 1 = 0 + 1
100 => [1,3] => [1,0,1,1,1,0,0,0]
 => [[.,.],[.,[.,.]]]
 => 2 = 1 + 1
101 => [1,2,1] => [1,0,1,1,0,0,1,0]
 => [[[.,.],[.,.]],.]
 => 2 = 1 + 1
110 => [1,1,2] => [1,0,1,0,1,1,0,0]
 => [[[.,.],.],[.,.]]
 => 2 = 1 + 1
111 => [1,1,1,1] => [1,0,1,0,1,0,1,0]
 => [[[[.,.],.],.],.]
 => 1 = 0 + 1
0000 => [5] => [1,1,1,1,1,0,0,0,0,0]
 => [.,[.,[.,[.,[.,.]]]]]
 => 1 = 0 + 1
0001 => [4,1] => [1,1,1,1,0,0,0,0,1,0]
 => [[.,[.,[.,[.,.]]]],.]
 => 1 = 0 + 1
0010 => [3,2] => [1,1,1,0,0,0,1,1,0,0]
 => [[.,[.,[.,.]]],[.,.]]
 => 2 = 1 + 1
0011 => [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
 => [[[.,[.,[.,.]]],.],.]
 => 1 = 0 + 1
0100 => [2,3] => [1,1,0,0,1,1,1,0,0,0]
 => [[.,[.,.]],[.,[.,.]]]
 => 2 = 1 + 1
0101 => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
 => [[[.,[.,.]],[.,.]],.]
 => 2 = 1 + 1
0110 => [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
 => [[[.,[.,.]],.],[.,.]]
 => 2 = 1 + 1
0111 => [2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
 => [[[[.,[.,.]],.],.],.]
 => 1 = 0 + 1
1000 => [1,4] => [1,0,1,1,1,1,0,0,0,0]
 => [[.,.],[.,[.,[.,.]]]]
 => 2 = 1 + 1
1001 => [1,3,1] => [1,0,1,1,1,0,0,0,1,0]
 => [[[.,.],[.,[.,.]]],.]
 => 2 = 1 + 1
1010 => [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
 => [[[.,.],[.,.]],[.,.]]
 => 3 = 2 + 1
1011 => [1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
 => [[[[.,.],[.,.]],.],.]
 => 2 = 1 + 1
1100 => [1,1,3] => [1,0,1,0,1,1,1,0,0,0]
 => [[[.,.],.],[.,[.,.]]]
 => 2 = 1 + 1
1101 => [1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
 => [[[[.,.],.],[.,.]],.]
 => 2 = 1 + 1
1110 => [1,1,1,2] => [1,0,1,0,1,0,1,1,0,0]
 => [[[[.,.],.],.],[.,.]]
 => 2 = 1 + 1
1111 => [1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
 => [[[[[.,.],.],.],.],.]
 => 1 = 0 + 1
00000 => [6] => [1,1,1,1,1,1,0,0,0,0,0,0]
 => [.,[.,[.,[.,[.,[.,.]]]]]]
 => 1 = 0 + 1
00001 => [5,1] => [1,1,1,1,1,0,0,0,0,0,1,0]
 => [[.,[.,[.,[.,[.,.]]]]],.]
 => 1 = 0 + 1
00010 => [4,2] => [1,1,1,1,0,0,0,0,1,1,0,0]
 => [[.,[.,[.,[.,.]]]],[.,.]]
 => 2 = 1 + 1
00011 => [4,1,1] => [1,1,1,1,0,0,0,0,1,0,1,0]
 => [[[.,[.,[.,[.,.]]]],.],.]
 => 1 = 0 + 1
00100 => [3,3] => [1,1,1,0,0,0,1,1,1,0,0,0]
 => [[.,[.,[.,.]]],[.,[.,.]]]
 => 2 = 1 + 1
00101 => [3,2,1] => [1,1,1,0,0,0,1,1,0,0,1,0]
 => [[[.,[.,[.,.]]],[.,.]],.]
 => 2 = 1 + 1
00110 => [3,1,2] => [1,1,1,0,0,0,1,0,1,1,0,0]
 => [[[.,[.,[.,.]]],.],[.,.]]
 => 2 = 1 + 1
00111 => [3,1,1,1] => [1,1,1,0,0,0,1,0,1,0,1,0]
 => [[[[.,[.,[.,.]]],.],.],.]
 => 1 = 0 + 1
01000 => [2,4] => [1,1,0,0,1,1,1,1,0,0,0,0]
 => [[.,[.,.]],[.,[.,[.,.]]]]
 => 2 = 1 + 1
01001 => [2,3,1] => [1,1,0,0,1,1,1,0,0,0,1,0]
 => [[[.,[.,.]],[.,[.,.]]],.]
 => 2 = 1 + 1
01010 => [2,2,2] => [1,1,0,0,1,1,0,0,1,1,0,0]
 => [[[.,[.,.]],[.,.]],[.,.]]
 => 3 = 2 + 1
01011 => [2,2,1,1] => [1,1,0,0,1,1,0,0,1,0,1,0]
 => [[[[.,[.,.]],[.,.]],.],.]
 => 2 = 1 + 1
01100 => [2,1,3] => [1,1,0,0,1,0,1,1,1,0,0,0]
 => [[[.,[.,.]],.],[.,[.,.]]]
 => 2 = 1 + 1
01101 => [2,1,2,1] => [1,1,0,0,1,0,1,1,0,0,1,0]
 => [[[[.,[.,.]],.],[.,.]],.]
 => 2 = 1 + 1
01110 => [2,1,1,2] => [1,1,0,0,1,0,1,0,1,1,0,0]
 => [[[[.,[.,.]],.],.],[.,.]]
 => 2 = 1 + 1
01111 => [2,1,1,1,1] => [1,1,0,0,1,0,1,0,1,0,1,0]
 => [[[[[.,[.,.]],.],.],.],.]
 => 1 = 0 + 1
10000 => [1,5] => [1,0,1,1,1,1,1,0,0,0,0,0]
 => [[.,.],[.,[.,[.,[.,.]]]]]
 => 2 = 1 + 1
10001 => [1,4,1] => [1,0,1,1,1,1,0,0,0,0,1,0]
 => [[[.,.],[.,[.,[.,.]]]],.]
 => 2 = 1 + 1
10010 => [1,3,2] => [1,0,1,1,1,0,0,0,1,1,0,0]
 => [[[.,.],[.,[.,.]]],[.,.]]
 => 3 = 2 + 1
10011 => [1,3,1,1] => [1,0,1,1,1,0,0,0,1,0,1,0]
 => [[[[.,.],[.,[.,.]]],.],.]
 => 2 = 1 + 1
0000000 => [8] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
 => [.,[.,[.,[.,[.,[.,[.,[.,.]]]]]]]]
 => ? = 0 + 1
0000010 => [6,2] => [1,1,1,1,1,1,0,0,0,0,0,0,1,1,0,0]
 => [[.,[.,[.,[.,[.,[.,.]]]]]],[.,.]]
 => ? = 1 + 1
0000100 => [5,3] => [1,1,1,1,1,0,0,0,0,0,1,1,1,0,0,0]
 => [[.,[.,[.,[.,[.,.]]]]],[.,[.,.]]]
 => ? = 1 + 1
0000110 => [5,1,2] => [1,1,1,1,1,0,0,0,0,0,1,0,1,1,0,0]
 => [[[.,[.,[.,[.,[.,.]]]]],.],[.,.]]
 => ? = 1 + 1
0001010 => [4,2,2] => [1,1,1,1,0,0,0,0,1,1,0,0,1,1,0,0]
 => [[[.,[.,[.,[.,.]]]],[.,.]],[.,.]]
 => ? = 2 + 1
0001100 => [4,1,3] => [1,1,1,1,0,0,0,0,1,0,1,1,1,0,0,0]
 => [[[.,[.,[.,[.,.]]]],.],[.,[.,.]]]
 => ? = 1 + 1
0001110 => [4,1,1,2] => [1,1,1,1,0,0,0,0,1,0,1,0,1,1,0,0]
 => [[[[.,[.,[.,[.,.]]]],.],.],[.,.]]
 => ? = 1 + 1
0010000 => [3,5] => [1,1,1,0,0,0,1,1,1,1,1,0,0,0,0,0]
 => [[.,[.,[.,.]]],[.,[.,[.,[.,.]]]]]
 => ? = 1 + 1
0010010 => [3,3,2] => [1,1,1,0,0,0,1,1,1,0,0,0,1,1,0,0]
 => [[[.,[.,[.,.]]],[.,[.,.]]],[.,.]]
 => ? = 2 + 1
0010100 => [3,2,3] => [1,1,1,0,0,0,1,1,0,0,1,1,1,0,0,0]
 => [[[.,[.,[.,.]]],[.,.]],[.,[.,.]]]
 => ? = 2 + 1
0010110 => [3,2,1,2] => [1,1,1,0,0,0,1,1,0,0,1,0,1,1,0,0]
 => [[[[.,[.,[.,.]]],[.,.]],.],[.,.]]
 => ? = 2 + 1
0011000 => [3,1,4] => [1,1,1,0,0,0,1,0,1,1,1,1,0,0,0,0]
 => [[[.,[.,[.,.]]],.],[.,[.,[.,.]]]]
 => ? = 1 + 1
0011010 => [3,1,2,2] => [1,1,1,0,0,0,1,0,1,1,0,0,1,1,0,0]
 => [[[[.,[.,[.,.]]],.],[.,.]],[.,.]]
 => ? = 2 + 1
0100000 => [2,6] => [1,1,0,0,1,1,1,1,1,1,0,0,0,0,0,0]
 => [[.,[.,.]],[.,[.,[.,[.,[.,.]]]]]]
 => ? = 1 + 1
0100010 => [2,4,2] => [1,1,0,0,1,1,1,1,0,0,0,0,1,1,0,0]
 => [[[.,[.,.]],[.,[.,[.,.]]]],[.,.]]
 => ? = 2 + 1
0100100 => [2,3,3] => [1,1,0,0,1,1,1,0,0,0,1,1,1,0,0,0]
 => [[[.,[.,.]],[.,[.,.]]],[.,[.,.]]]
 => ? = 2 + 1
0100110 => [2,3,1,2] => [1,1,0,0,1,1,1,0,0,0,1,0,1,1,0,0]
 => [[[[.,[.,.]],[.,[.,.]]],.],[.,.]]
 => ? = 2 + 1
0101000 => [2,2,4] => [1,1,0,0,1,1,0,0,1,1,1,1,0,0,0,0]
 => [[[.,[.,.]],[.,.]],[.,[.,[.,.]]]]
 => ? = 2 + 1
0101110 => [2,2,1,1,2] => [1,1,0,0,1,1,0,0,1,0,1,0,1,1,0,0]
 => [[[[[.,[.,.]],[.,.]],.],.],[.,.]]
 => ? = 2 + 1
0110000 => [2,1,5] => [1,1,0,0,1,0,1,1,1,1,1,0,0,0,0,0]
 => [[[.,[.,.]],.],[.,[.,[.,[.,.]]]]]
 => ? = 1 + 1
0110010 => [2,1,3,2] => [1,1,0,0,1,0,1,1,1,0,0,0,1,1,0,0]
 => [[[[.,[.,.]],.],[.,[.,.]]],[.,.]]
 => ? = 2 + 1
0110110 => [2,1,2,1,2] => [1,1,0,0,1,0,1,1,0,0,1,0,1,1,0,0]
 => [[[[[.,[.,.]],.],[.,.]],.],[.,.]]
 => ? = 2 + 1
0111000 => [2,1,1,4] => [1,1,0,0,1,0,1,0,1,1,1,1,0,0,0,0]
 => [[[[.,[.,.]],.],.],[.,[.,[.,.]]]]
 => ? = 1 + 1
0111010 => [2,1,1,2,2] => [1,1,0,0,1,0,1,0,1,1,0,0,1,1,0,0]
 => [[[[[.,[.,.]],.],.],[.,.]],[.,.]]
 => ? = 2 + 1
0111100 => [2,1,1,1,3] => [1,1,0,0,1,0,1,0,1,0,1,1,1,0,0,0]
 => [[[[[.,[.,.]],.],.],.],[.,[.,.]]]
 => ? = 1 + 1
1000010 => [1,5,2] => [1,0,1,1,1,1,1,0,0,0,0,0,1,1,0,0]
 => [[[.,.],[.,[.,[.,[.,.]]]]],[.,.]]
 => ? = 2 + 1
1000100 => [1,4,3] => [1,0,1,1,1,1,0,0,0,0,1,1,1,0,0,0]
 => [[[.,.],[.,[.,[.,.]]]],[.,[.,.]]]
 => ? = 2 + 1
1000110 => [1,4,1,2] => [1,0,1,1,1,1,0,0,0,0,1,0,1,1,0,0]
 => [[[[.,.],[.,[.,[.,.]]]],.],[.,.]]
 => ? = 2 + 1
1001110 => [1,3,1,1,2] => [1,0,1,1,1,0,0,0,1,0,1,0,1,1,0,0]
 => [[[[[.,.],[.,[.,.]]],.],.],[.,.]]
 => ? = 2 + 1
1010000 => [1,2,5] => [1,0,1,1,0,0,1,1,1,1,1,0,0,0,0,0]
 => [[[.,.],[.,.]],[.,[.,[.,[.,.]]]]]
 => ? = 2 + 1
1010110 => [1,2,2,1,2] => [1,0,1,1,0,0,1,1,0,0,1,0,1,1,0,0]
 => [[[[[.,.],[.,.]],[.,.]],.],[.,.]]
 => ? = 3 + 1
1011000 => [1,2,1,4] => [1,0,1,1,0,0,1,0,1,1,1,1,0,0,0,0]
 => [[[[.,.],[.,.]],.],[.,[.,[.,.]]]]
 => ? = 2 + 1
1011010 => [1,2,1,2,2] => [1,0,1,1,0,0,1,0,1,1,0,0,1,1,0,0]
 => [[[[[.,.],[.,.]],.],[.,.]],[.,.]]
 => ? = 3 + 1
1011100 => [1,2,1,1,3] => [1,0,1,1,0,0,1,0,1,0,1,1,1,0,0,0]
 => [[[[[.,.],[.,.]],.],.],[.,[.,.]]]
 => ? = 2 + 1
1100000 => [1,1,6] => [1,0,1,0,1,1,1,1,1,1,0,0,0,0,0,0]
 => [[[.,.],.],[.,[.,[.,[.,[.,.]]]]]]
 => ? = 1 + 1
1100010 => [1,1,4,2] => [1,0,1,0,1,1,1,1,0,0,0,0,1,1,0,0]
 => [[[[.,.],.],[.,[.,[.,.]]]],[.,.]]
 => ? = 2 + 1
1100110 => [1,1,3,1,2] => [1,0,1,0,1,1,1,0,0,0,1,0,1,1,0,0]
 => [[[[[.,.],.],[.,[.,.]]],.],[.,.]]
 => ? = 2 + 1
1101000 => [1,1,2,4] => [1,0,1,0,1,1,0,0,1,1,1,1,0,0,0,0]
 => [[[[.,.],.],[.,.]],[.,[.,[.,.]]]]
 => ? = 2 + 1
1101010 => [1,1,2,2,2] => [1,0,1,0,1,1,0,0,1,1,0,0,1,1,0,0]
 => [[[[[.,.],.],[.,.]],[.,.]],[.,.]]
 => ? = 3 + 1
1101100 => [1,1,2,1,3] => [1,0,1,0,1,1,0,0,1,0,1,1,1,0,0,0]
 => [[[[[.,.],.],[.,.]],.],[.,[.,.]]]
 => ? = 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
1110010 => [1,1,1,3,2] => [1,0,1,0,1,0,1,1,1,0,0,0,1,1,0,0]
 => [[[[[.,.],.],.],[.,[.,.]]],[.,.]]
 => ? = 2 + 1
1110100 => [1,1,1,2,3] => [1,0,1,0,1,0,1,1,0,0,1,1,1,0,0,0]
 => [[[[[.,.],.],.],[.,.]],[.,[.,.]]]
 => ? = 2 + 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
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
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 $2^{n-1}$, see [2]. The number of binary tree of size $n$, at least $3$, with exactly two leaf nodes is $n(n+1)2^{n-2}$, see [3].
Matching statistic: St000257
Mapping
Mp00178: Binary words —to composition⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
Mp00027: Dyck paths —to partition⟶ Integer partitions
St000257: Integer partitions ⟶ ℤResult quality: 74% ●values known / values provided: 74%●distinct values known / distinct values provided: 100%
Values
0 => [2] => [1,1,0,0]
 => []
 => 0
1 => [1,1] => [1,0,1,0]
 => [1]
 => 0
00 => [3] => [1,1,1,0,0,0]
 => []
 => 0
01 => [2,1] => [1,1,0,0,1,0]
 => [2]
 => 0
10 => [1,2] => [1,0,1,1,0,0]
 => [1,1]
 => 1
11 => [1,1,1] => [1,0,1,0,1,0]
 => [2,1]
 => 0
000 => [4] => [1,1,1,1,0,0,0,0]
 => []
 => 0
001 => [3,1] => [1,1,1,0,0,0,1,0]
 => [3]
 => 0
010 => [2,2] => [1,1,0,0,1,1,0,0]
 => [2,2]
 => 1
011 => [2,1,1] => [1,1,0,0,1,0,1,0]
 => [3,2]
 => 0
100 => [1,3] => [1,0,1,1,1,0,0,0]
 => [1,1,1]
 => 1
101 => [1,2,1] => [1,0,1,1,0,0,1,0]
 => [3,1,1]
 => 1
110 => [1,1,2] => [1,0,1,0,1,1,0,0]
 => [2,2,1]
 => 1
111 => [1,1,1,1] => [1,0,1,0,1,0,1,0]
 => [3,2,1]
 => 0
0000 => [5] => [1,1,1,1,1,0,0,0,0,0]
 => []
 => 0
0001 => [4,1] => [1,1,1,1,0,0,0,0,1,0]
 => [4]
 => 0
0010 => [3,2] => [1,1,1,0,0,0,1,1,0,0]
 => [3,3]
 => 1
0011 => [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
 => [4,3]
 => 0
0100 => [2,3] => [1,1,0,0,1,1,1,0,0,0]
 => [2,2,2]
 => 1
0101 => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
 => [4,2,2]
 => 1
0110 => [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
 => [3,3,2]
 => 1
0111 => [2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
 => [4,3,2]
 => 0
1000 => [1,4] => [1,0,1,1,1,1,0,0,0,0]
 => [1,1,1,1]
 => 1
1001 => [1,3,1] => [1,0,1,1,1,0,0,0,1,0]
 => [4,1,1,1]
 => 1
1010 => [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
 => [3,3,1,1]
 => 2
1011 => [1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
 => [4,3,1,1]
 => 1
1100 => [1,1,3] => [1,0,1,0,1,1,1,0,0,0]
 => [2,2,2,1]
 => 1
1101 => [1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
 => [4,2,2,1]
 => 1
1110 => [1,1,1,2] => [1,0,1,0,1,0,1,1,0,0]
 => [3,3,2,1]
 => 1
1111 => [1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
 => [4,3,2,1]
 => 0
00000 => [6] => [1,1,1,1,1,1,0,0,0,0,0,0]
 => []
 => 0
00001 => [5,1] => [1,1,1,1,1,0,0,0,0,0,1,0]
 => [5]
 => 0
00010 => [4,2] => [1,1,1,1,0,0,0,0,1,1,0,0]
 => [4,4]
 => 1
00011 => [4,1,1] => [1,1,1,1,0,0,0,0,1,0,1,0]
 => [5,4]
 => 0
00100 => [3,3] => [1,1,1,0,0,0,1,1,1,0,0,0]
 => [3,3,3]
 => 1
00101 => [3,2,1] => [1,1,1,0,0,0,1,1,0,0,1,0]
 => [5,3,3]
 => 1
00110 => [3,1,2] => [1,1,1,0,0,0,1,0,1,1,0,0]
 => [4,4,3]
 => 1
00111 => [3,1,1,1] => [1,1,1,0,0,0,1,0,1,0,1,0]
 => [5,4,3]
 => 0
01000 => [2,4] => [1,1,0,0,1,1,1,1,0,0,0,0]
 => [2,2,2,2]
 => 1
01001 => [2,3,1] => [1,1,0,0,1,1,1,0,0,0,1,0]
 => [5,2,2,2]
 => 1
01010 => [2,2,2] => [1,1,0,0,1,1,0,0,1,1,0,0]
 => [4,4,2,2]
 => 2
01011 => [2,2,1,1] => [1,1,0,0,1,1,0,0,1,0,1,0]
 => [5,4,2,2]
 => 1
01100 => [2,1,3] => [1,1,0,0,1,0,1,1,1,0,0,0]
 => [3,3,3,2]
 => 1
01101 => [2,1,2,1] => [1,1,0,0,1,0,1,1,0,0,1,0]
 => [5,3,3,2]
 => 1
01110 => [2,1,1,2] => [1,1,0,0,1,0,1,0,1,1,0,0]
 => [4,4,3,2]
 => 1
01111 => [2,1,1,1,1] => [1,1,0,0,1,0,1,0,1,0,1,0]
 => [5,4,3,2]
 => 0
10000 => [1,5] => [1,0,1,1,1,1,1,0,0,0,0,0]
 => [1,1,1,1,1]
 => 1
10001 => [1,4,1] => [1,0,1,1,1,1,0,0,0,0,1,0]
 => [5,1,1,1,1]
 => 1
10010 => [1,3,2] => [1,0,1,1,1,0,0,0,1,1,0,0]
 => [4,4,1,1,1]
 => 2
10011 => [1,3,1,1] => [1,0,1,1,1,0,0,0,1,0,1,0]
 => [5,4,1,1,1]
 => 1
0000110 => [5,1,2] => [1,1,1,1,1,0,0,0,0,0,1,0,1,1,0,0]
 => [6,6,5]
 => ? = 1
0001010 => [4,2,2] => [1,1,1,1,0,0,0,0,1,1,0,0,1,1,0,0]
 => [6,6,4,4]
 => ? = 2
0001100 => [4,1,3] => [1,1,1,1,0,0,0,0,1,0,1,1,1,0,0,0]
 => [5,5,5,4]
 => ? = 1
0010010 => [3,3,2] => [1,1,1,0,0,0,1,1,1,0,0,0,1,1,0,0]
 => [6,6,3,3,3]
 => ? = 2
0010100 => [3,2,3] => [1,1,1,0,0,0,1,1,0,0,1,1,1,0,0,0]
 => [5,5,5,3,3]
 => ? = 2
0010110 => [3,2,1,2] => [1,1,1,0,0,0,1,1,0,0,1,0,1,1,0,0]
 => [6,6,5,3,3]
 => ? = 2
0011010 => [3,1,2,2] => [1,1,1,0,0,0,1,0,1,1,0,0,1,1,0,0]
 => [6,6,4,4,3]
 => ? = 2
0011100 => [3,1,1,3] => [1,1,1,0,0,0,1,0,1,0,1,1,1,0,0,0]
 => [5,5,5,4,3]
 => ? = 1
0011110 => [3,1,1,1,2] => [1,1,1,0,0,0,1,0,1,0,1,0,1,1,0,0]
 => [6,6,5,4,3]
 => ? = 1
0100100 => [2,3,3] => [1,1,0,0,1,1,1,0,0,0,1,1,1,0,0,0]
 => [5,5,5,2,2,2]
 => ? = 2
0100110 => [2,3,1,2] => [1,1,0,0,1,1,1,0,0,0,1,0,1,1,0,0]
 => [6,6,5,2,2,2]
 => ? = 2
0101000 => [2,2,4] => [1,1,0,0,1,1,0,0,1,1,1,1,0,0,0,0]
 => [4,4,4,4,2,2]
 => ? = 2
0101010 => [2,2,2,2] => [1,1,0,0,1,1,0,0,1,1,0,0,1,1,0,0]
 => [6,6,4,4,2,2]
 => ? = 3
0101100 => [2,2,1,3] => [1,1,0,0,1,1,0,0,1,0,1,1,1,0,0,0]
 => [5,5,5,4,2,2]
 => ? = 2
0101110 => [2,2,1,1,2] => [1,1,0,0,1,1,0,0,1,0,1,0,1,1,0,0]
 => [6,6,5,4,2,2]
 => ? = 2
0110010 => [2,1,3,2] => [1,1,0,0,1,0,1,1,1,0,0,0,1,1,0,0]
 => [6,6,3,3,3,2]
 => ? = 2
0110100 => [2,1,2,3] => [1,1,0,0,1,0,1,1,0,0,1,1,1,0,0,0]
 => [5,5,5,3,3,2]
 => ? = 2
0110110 => [2,1,2,1,2] => [1,1,0,0,1,0,1,1,0,0,1,0,1,1,0,0]
 => [6,6,5,3,3,2]
 => ? = 2
0111000 => [2,1,1,4] => [1,1,0,0,1,0,1,0,1,1,1,1,0,0,0,0]
 => [4,4,4,4,3,2]
 => ? = 1
0111010 => [2,1,1,2,2] => [1,1,0,0,1,0,1,0,1,1,0,0,1,1,0,0]
 => [6,6,4,4,3,2]
 => ? = 2
0111100 => [2,1,1,1,3] => [1,1,0,0,1,0,1,0,1,0,1,1,1,0,0,0]
 => [5,5,5,4,3,2]
 => ? = 1
0111110 => [2,1,1,1,1,2] => [1,1,0,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => [6,6,5,4,3,2]
 => ? = 1
1000010 => [1,5,2] => [1,0,1,1,1,1,1,0,0,0,0,0,1,1,0,0]
 => [6,6,1,1,1,1,1]
 => ? = 2
1000100 => [1,4,3] => [1,0,1,1,1,1,0,0,0,0,1,1,1,0,0,0]
 => [5,5,5,1,1,1,1]
 => ? = 2
1000110 => [1,4,1,2] => [1,0,1,1,1,1,0,0,0,0,1,0,1,1,0,0]
 => [6,6,5,1,1,1,1]
 => ? = 2
1001000 => [1,3,4] => [1,0,1,1,1,0,0,0,1,1,1,1,0,0,0,0]
 => [4,4,4,4,1,1,1]
 => ? = 2
1001010 => [1,3,2,2] => [1,0,1,1,1,0,0,0,1,1,0,0,1,1,0,0]
 => [6,6,4,4,1,1,1]
 => ? = 3
1001100 => [1,3,1,3] => [1,0,1,1,1,0,0,0,1,0,1,1,1,0,0,0]
 => [5,5,5,4,1,1,1]
 => ? = 2
1001110 => [1,3,1,1,2] => [1,0,1,1,1,0,0,0,1,0,1,0,1,1,0,0]
 => [6,6,5,4,1,1,1]
 => ? = 2
1010000 => [1,2,5] => [1,0,1,1,0,0,1,1,1,1,1,0,0,0,0,0]
 => [3,3,3,3,3,1,1]
 => ? = 2
1010010 => [1,2,3,2] => [1,0,1,1,0,0,1,1,1,0,0,0,1,1,0,0]
 => [6,6,3,3,3,1,1]
 => ? = 3
1010100 => [1,2,2,3] => [1,0,1,1,0,0,1,1,0,0,1,1,1,0,0,0]
 => [5,5,5,3,3,1,1]
 => ? = 3
1010110 => [1,2,2,1,2] => [1,0,1,1,0,0,1,1,0,0,1,0,1,1,0,0]
 => [6,6,5,3,3,1,1]
 => ? = 3
1011000 => [1,2,1,4] => [1,0,1,1,0,0,1,0,1,1,1,1,0,0,0,0]
 => [4,4,4,4,3,1,1]
 => ? = 2
1011010 => [1,2,1,2,2] => [1,0,1,1,0,0,1,0,1,1,0,0,1,1,0,0]
 => [6,6,4,4,3,1,1]
 => ? = 3
1011100 => [1,2,1,1,3] => [1,0,1,1,0,0,1,0,1,0,1,1,1,0,0,0]
 => [5,5,5,4,3,1,1]
 => ? = 2
1011110 => [1,2,1,1,1,2] => [1,0,1,1,0,0,1,0,1,0,1,0,1,1,0,0]
 => [6,6,5,4,3,1,1]
 => ? = 2
1100010 => [1,1,4,2] => [1,0,1,0,1,1,1,1,0,0,0,0,1,1,0,0]
 => [6,6,2,2,2,2,1]
 => ? = 2
1100100 => [1,1,3,3] => [1,0,1,0,1,1,1,0,0,0,1,1,1,0,0,0]
 => [5,5,5,2,2,2,1]
 => ? = 2
1100110 => [1,1,3,1,2] => [1,0,1,0,1,1,1,0,0,0,1,0,1,1,0,0]
 => [6,6,5,2,2,2,1]
 => ? = 2
1101000 => [1,1,2,4] => [1,0,1,0,1,1,0,0,1,1,1,1,0,0,0,0]
 => [4,4,4,4,2,2,1]
 => ? = 2
1101010 => [1,1,2,2,2] => [1,0,1,0,1,1,0,0,1,1,0,0,1,1,0,0]
 => [6,6,4,4,2,2,1]
 => ? = 3
1101100 => [1,1,2,1,3] => [1,0,1,0,1,1,0,0,1,0,1,1,1,0,0,0]
 => [5,5,5,4,2,2,1]
 => ? = 2
1101110 => [1,1,2,1,1,2] => [1,0,1,0,1,1,0,0,1,0,1,0,1,1,0,0]
 => [6,6,5,4,2,2,1]
 => ? = 2
1110000 => [1,1,1,5] => [1,0,1,0,1,0,1,1,1,1,1,0,0,0,0,0]
 => [3,3,3,3,3,2,1]
 => ? = 1
1110010 => [1,1,1,3,2] => [1,0,1,0,1,0,1,1,1,0,0,0,1,1,0,0]
 => [6,6,3,3,3,2,1]
 => ? = 2
1110100 => [1,1,1,2,3] => [1,0,1,0,1,0,1,1,0,0,1,1,1,0,0,0]
 => [5,5,5,3,3,2,1]
 => ? = 2
1110110 => [1,1,1,2,1,2] => [1,0,1,0,1,0,1,1,0,0,1,0,1,1,0,0]
 => [6,6,5,3,3,2,1]
 => ? = 2
1111010 => [1,1,1,1,2,2] => [1,0,1,0,1,0,1,0,1,1,0,0,1,1,0,0]
 => [6,6,4,4,3,2,1]
 => ? = 2
Description
The number of distinct parts of a partition that occur at least twice. See Section 3.3.1 of [2].
The following 32 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000159The number of distinct parts of the integer partition. St000069The number of maximal elements of a poset. St001086The number of occurrences of the consecutive pattern 132 in a permutation. St000196The number of occurrences of the contiguous pattern [[.,.],[.,. St000093The cardinality of a maximal independent set of vertices of a graph. St000786The maximal number of occurrences of a colour in a proper colouring of a graph. St000318The number of addable cells of the Ferrers diagram of an integer partition. St001124The multiplicity of the standard representation in the Kronecker square corresponding to a partition. St001337The upper domination number of a graph. St001338The upper irredundance number of a graph. St000353The number of inner valleys of a permutation. St001683The number of distinct positions of the pattern letter 3 in occurrences of 132 in a permutation. St001738The minimal order of a graph which is not an induced subgraph of the given graph. St000023The number of inner peaks of a permutation. St000779The tier of a permutation. St000092The number of outer peaks of a permutation. St000099The number of valleys of a permutation, including the boundary. St000659The number of rises of length at least 2 of a Dyck path. St000985The number of positive eigenvalues of the adjacency matrix of the graph. St001354The number of series nodes in the modular decomposition of a graph. St000455The second largest eigenvalue of a graph if it is integral. St000035The number of left outer peaks of a permutation. St000884The number of isolated descents of a permutation. St000021The number of descents of a permutation. St000333The dez statistic, the number of descents of a permutation after replacing fixed points by zeros. St001487The number of inner corners of a skew partition. St001729The number of visible descents of a permutation. St001737The number of descents of type 2 in a permutation. St001928The number of non-overlapping descents in a permutation. St000325The width of the tree associated to a permutation. St000470The number of runs in a permutation. St001960The number of descents of a permutation minus one if its first entry is not one.