Your data matches 60 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)
Matching statistic: St000291
(load all 9 compositions to match this statistic)
Mapping
St000291: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●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
Description
The number of descents of a binary word.
Matching statistic: St000292
(load all 6 compositions to match this statistic)
Mapping
St000292: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
0 => 0
1 => 0
00 => 0
01 => 1
10 => 0
11 => 0
000 => 0
001 => 1
010 => 1
011 => 1
100 => 0
101 => 1
110 => 0
111 => 0
0000 => 0
0001 => 1
0010 => 1
0011 => 1
0100 => 1
0101 => 2
0110 => 1
0111 => 1
1000 => 0
1001 => 1
1010 => 1
1011 => 1
1100 => 0
1101 => 1
1110 => 0
1111 => 0
00000 => 0
00001 => 1
00010 => 1
00011 => 1
00100 => 1
00101 => 2
00110 => 1
00111 => 1
01000 => 1
01001 => 2
01010 => 2
01011 => 2
01100 => 1
01101 => 2
01110 => 1
01111 => 1
10000 => 0
10001 => 1
10010 => 1
10011 => 1
Description
The number of ascents of a binary word.
Matching statistic: St000386
(load all 8 compositions to match this statistic)
Mapping
Mp00178: Binary words —to composition⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St000386: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
0 => [2] => [1,1,0,0]
 => 0
1 => [1,1] => [1,0,1,0]
 => 0
00 => [3] => [1,1,1,0,0,0]
 => 0
01 => [2,1] => [1,1,0,0,1,0]
 => 1
10 => [1,2] => [1,0,1,1,0,0]
 => 0
11 => [1,1,1] => [1,0,1,0,1,0]
 => 0
000 => [4] => [1,1,1,1,0,0,0,0]
 => 0
001 => [3,1] => [1,1,1,0,0,0,1,0]
 => 1
010 => [2,2] => [1,1,0,0,1,1,0,0]
 => 1
011 => [2,1,1] => [1,1,0,0,1,0,1,0]
 => 1
100 => [1,3] => [1,0,1,1,1,0,0,0]
 => 0
101 => [1,2,1] => [1,0,1,1,0,0,1,0]
 => 1
110 => [1,1,2] => [1,0,1,0,1,1,0,0]
 => 0
111 => [1,1,1,1] => [1,0,1,0,1,0,1,0]
 => 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]
 => 1
0010 => [3,2] => [1,1,1,0,0,0,1,1,0,0]
 => 1
0011 => [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
 => 1
0100 => [2,3] => [1,1,0,0,1,1,1,0,0,0]
 => 1
0101 => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
 => 2
0110 => [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
 => 1
0111 => [2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
 => 1
1000 => [1,4] => [1,0,1,1,1,1,0,0,0,0]
 => 0
1001 => [1,3,1] => [1,0,1,1,1,0,0,0,1,0]
 => 1
1010 => [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
 => 1
1011 => [1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
 => 1
1100 => [1,1,3] => [1,0,1,0,1,1,1,0,0,0]
 => 0
1101 => [1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
 => 1
1110 => [1,1,1,2] => [1,0,1,0,1,0,1,1,0,0]
 => 0
1111 => [1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
 => 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]
 => 1
00010 => [4,2] => [1,1,1,1,0,0,0,0,1,1,0,0]
 => 1
00011 => [4,1,1] => [1,1,1,1,0,0,0,0,1,0,1,0]
 => 1
00100 => [3,3] => [1,1,1,0,0,0,1,1,1,0,0,0]
 => 1
00101 => [3,2,1] => [1,1,1,0,0,0,1,1,0,0,1,0]
 => 2
00110 => [3,1,2] => [1,1,1,0,0,0,1,0,1,1,0,0]
 => 1
00111 => [3,1,1,1] => [1,1,1,0,0,0,1,0,1,0,1,0]
 => 1
01000 => [2,4] => [1,1,0,0,1,1,1,1,0,0,0,0]
 => 1
01001 => [2,3,1] => [1,1,0,0,1,1,1,0,0,0,1,0]
 => 2
01010 => [2,2,2] => [1,1,0,0,1,1,0,0,1,1,0,0]
 => 2
01011 => [2,2,1,1] => [1,1,0,0,1,1,0,0,1,0,1,0]
 => 2
01100 => [2,1,3] => [1,1,0,0,1,0,1,1,1,0,0,0]
 => 1
01101 => [2,1,2,1] => [1,1,0,0,1,0,1,1,0,0,1,0]
 => 2
01110 => [2,1,1,2] => [1,1,0,0,1,0,1,0,1,1,0,0]
 => 1
01111 => [2,1,1,1,1] => [1,1,0,0,1,0,1,0,1,0,1,0]
 => 1
10000 => [1,5] => [1,0,1,1,1,1,1,0,0,0,0,0]
 => 0
10001 => [1,4,1] => [1,0,1,1,1,1,0,0,0,0,1,0]
 => 1
10010 => [1,3,2] => [1,0,1,1,1,0,0,0,1,1,0,0]
 => 1
10011 => [1,3,1,1] => [1,0,1,1,1,0,0,0,1,0,1,0]
 => 1
Description
The number of factors DDU in a Dyck path.
Matching statistic: St001037
(load all 5 compositions to match this statistic)
Mapping
Mp00178: Binary words —to composition⟶ 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] => [1,1,0,0]
 => 0
1 => [1,1] => [1,0,1,0]
 => 0
00 => [3] => [1,1,1,0,0,0]
 => 0
01 => [2,1] => [1,1,0,0,1,0]
 => 0
10 => [1,2] => [1,0,1,1,0,0]
 => 1
11 => [1,1,1] => [1,0,1,0,1,0]
 => 0
000 => [4] => [1,1,1,1,0,0,0,0]
 => 0
001 => [3,1] => [1,1,1,0,0,0,1,0]
 => 0
010 => [2,2] => [1,1,0,0,1,1,0,0]
 => 1
011 => [2,1,1] => [1,1,0,0,1,0,1,0]
 => 0
100 => [1,3] => [1,0,1,1,1,0,0,0]
 => 1
101 => [1,2,1] => [1,0,1,1,0,0,1,0]
 => 1
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]
 => 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]
 => 0
0010 => [3,2] => [1,1,1,0,0,0,1,1,0,0]
 => 1
0011 => [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
 => 0
0100 => [2,3] => [1,1,0,0,1,1,1,0,0,0]
 => 1
0101 => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
 => 1
0110 => [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
 => 1
0111 => [2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
 => 0
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]
 => 1
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]
 => 1
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]
 => 1
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]
 => 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]
 => 0
00010 => [4,2] => [1,1,1,1,0,0,0,0,1,1,0,0]
 => 1
00011 => [4,1,1] => [1,1,1,1,0,0,0,0,1,0,1,0]
 => 0
00100 => [3,3] => [1,1,1,0,0,0,1,1,1,0,0,0]
 => 1
00101 => [3,2,1] => [1,1,1,0,0,0,1,1,0,0,1,0]
 => 1
00110 => [3,1,2] => [1,1,1,0,0,0,1,0,1,1,0,0]
 => 1
00111 => [3,1,1,1] => [1,1,1,0,0,0,1,0,1,0,1,0]
 => 0
01000 => [2,4] => [1,1,0,0,1,1,1,1,0,0,0,0]
 => 1
01001 => [2,3,1] => [1,1,0,0,1,1,1,0,0,0,1,0]
 => 1
01010 => [2,2,2] => [1,1,0,0,1,1,0,0,1,1,0,0]
 => 2
01011 => [2,2,1,1] => [1,1,0,0,1,1,0,0,1,0,1,0]
 => 1
01100 => [2,1,3] => [1,1,0,0,1,0,1,1,1,0,0,0]
 => 1
01101 => [2,1,2,1] => [1,1,0,0,1,0,1,1,0,0,1,0]
 => 1
01110 => [2,1,1,2] => [1,1,0,0,1,0,1,0,1,1,0,0]
 => 1
01111 => [2,1,1,1,1] => [1,1,0,0,1,0,1,0,1,0,1,0]
 => 0
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]
 => 1
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]
 => 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
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 => [2] => 10 => 1 = 0 + 1
1 => [1,1] => 11 => 1 = 0 + 1
00 => [3] => 100 => 1 = 0 + 1
01 => [2,1] => 101 => 2 = 1 + 1
10 => [1,2] => 110 => 1 = 0 + 1
11 => [1,1,1] => 111 => 1 = 0 + 1
000 => [4] => 1000 => 1 = 0 + 1
001 => [3,1] => 1001 => 2 = 1 + 1
010 => [2,2] => 1010 => 2 = 1 + 1
011 => [2,1,1] => 1011 => 2 = 1 + 1
100 => [1,3] => 1100 => 1 = 0 + 1
101 => [1,2,1] => 1101 => 2 = 1 + 1
110 => [1,1,2] => 1110 => 1 = 0 + 1
111 => [1,1,1,1] => 1111 => 1 = 0 + 1
0000 => [5] => 10000 => 1 = 0 + 1
0001 => [4,1] => 10001 => 2 = 1 + 1
0010 => [3,2] => 10010 => 2 = 1 + 1
0011 => [3,1,1] => 10011 => 2 = 1 + 1
0100 => [2,3] => 10100 => 2 = 1 + 1
0101 => [2,2,1] => 10101 => 3 = 2 + 1
0110 => [2,1,2] => 10110 => 2 = 1 + 1
0111 => [2,1,1,1] => 10111 => 2 = 1 + 1
1000 => [1,4] => 11000 => 1 = 0 + 1
1001 => [1,3,1] => 11001 => 2 = 1 + 1
1010 => [1,2,2] => 11010 => 2 = 1 + 1
1011 => [1,2,1,1] => 11011 => 2 = 1 + 1
1100 => [1,1,3] => 11100 => 1 = 0 + 1
1101 => [1,1,2,1] => 11101 => 2 = 1 + 1
1110 => [1,1,1,2] => 11110 => 1 = 0 + 1
1111 => [1,1,1,1,1] => 11111 => 1 = 0 + 1
00000 => [6] => 100000 => 1 = 0 + 1
00001 => [5,1] => 100001 => 2 = 1 + 1
00010 => [4,2] => 100010 => 2 = 1 + 1
00011 => [4,1,1] => 100011 => 2 = 1 + 1
00100 => [3,3] => 100100 => 2 = 1 + 1
00101 => [3,2,1] => 100101 => 3 = 2 + 1
00110 => [3,1,2] => 100110 => 2 = 1 + 1
00111 => [3,1,1,1] => 100111 => 2 = 1 + 1
01000 => [2,4] => 101000 => 2 = 1 + 1
01001 => [2,3,1] => 101001 => 3 = 2 + 1
01010 => [2,2,2] => 101010 => 3 = 2 + 1
01011 => [2,2,1,1] => 101011 => 3 = 2 + 1
01100 => [2,1,3] => 101100 => 2 = 1 + 1
01101 => [2,1,2,1] => 101101 => 3 = 2 + 1
01110 => [2,1,1,2] => 101110 => 2 = 1 + 1
01111 => [2,1,1,1,1] => 101111 => 2 = 1 + 1
10000 => [1,5] => 110000 => 1 = 0 + 1
10001 => [1,4,1] => 110001 => 2 = 1 + 1
10010 => [1,3,2] => 110010 => 2 = 1 + 1
10011 => [1,3,1,1] => 110011 => 2 = 1 + 1
Description
The number of runs of ones in a binary word.
Matching statistic: St000159
(load all 2 compositions to match this statistic)
Mapping
Mp00178: Binary words —to composition⟶ Integer compositions
Mp00180: Integer compositions —to ribbon⟶ Skew partitions
Mp00183: Skew partitions —inner shape⟶ Integer 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],[]]
 => []
 => 0
00 => [3] => [[3],[]]
 => []
 => 0
01 => [2,1] => [[2,2],[1]]
 => [1]
 => 1
10 => [1,2] => [[2,1],[]]
 => []
 => 0
11 => [1,1,1] => [[1,1,1],[]]
 => []
 => 0
000 => [4] => [[4],[]]
 => []
 => 0
001 => [3,1] => [[3,3],[2]]
 => [2]
 => 1
010 => [2,2] => [[3,2],[1]]
 => [1]
 => 1
011 => [2,1,1] => [[2,2,2],[1,1]]
 => [1,1]
 => 1
100 => [1,3] => [[3,1],[]]
 => []
 => 0
101 => [1,2,1] => [[2,2,1],[1]]
 => [1]
 => 1
110 => [1,1,2] => [[2,1,1],[]]
 => []
 => 0
111 => [1,1,1,1] => [[1,1,1,1],[]]
 => []
 => 0
0000 => [5] => [[5],[]]
 => []
 => 0
0001 => [4,1] => [[4,4],[3]]
 => [3]
 => 1
0010 => [3,2] => [[4,3],[2]]
 => [2]
 => 1
0011 => [3,1,1] => [[3,3,3],[2,2]]
 => [2,2]
 => 1
0100 => [2,3] => [[4,2],[1]]
 => [1]
 => 1
0101 => [2,2,1] => [[3,3,2],[2,1]]
 => [2,1]
 => 2
0110 => [2,1,2] => [[3,2,2],[1,1]]
 => [1,1]
 => 1
0111 => [2,1,1,1] => [[2,2,2,2],[1,1,1]]
 => [1,1,1]
 => 1
1000 => [1,4] => [[4,1],[]]
 => []
 => 0
1001 => [1,3,1] => [[3,3,1],[2]]
 => [2]
 => 1
1010 => [1,2,2] => [[3,2,1],[1]]
 => [1]
 => 1
1011 => [1,2,1,1] => [[2,2,2,1],[1,1]]
 => [1,1]
 => 1
1100 => [1,1,3] => [[3,1,1],[]]
 => []
 => 0
1101 => [1,1,2,1] => [[2,2,1,1],[1]]
 => [1]
 => 1
1110 => [1,1,1,2] => [[2,1,1,1],[]]
 => []
 => 0
1111 => [1,1,1,1,1] => [[1,1,1,1,1],[]]
 => []
 => 0
00000 => [6] => [[6],[]]
 => []
 => 0
00001 => [5,1] => [[5,5],[4]]
 => [4]
 => 1
00010 => [4,2] => [[5,4],[3]]
 => [3]
 => 1
00011 => [4,1,1] => [[4,4,4],[3,3]]
 => [3,3]
 => 1
00100 => [3,3] => [[5,3],[2]]
 => [2]
 => 1
00101 => [3,2,1] => [[4,4,3],[3,2]]
 => [3,2]
 => 2
00110 => [3,1,2] => [[4,3,3],[2,2]]
 => [2,2]
 => 1
00111 => [3,1,1,1] => [[3,3,3,3],[2,2,2]]
 => [2,2,2]
 => 1
01000 => [2,4] => [[5,2],[1]]
 => [1]
 => 1
01001 => [2,3,1] => [[4,4,2],[3,1]]
 => [3,1]
 => 2
01010 => [2,2,2] => [[4,3,2],[2,1]]
 => [2,1]
 => 2
01011 => [2,2,1,1] => [[3,3,3,2],[2,2,1]]
 => [2,2,1]
 => 2
01100 => [2,1,3] => [[4,2,2],[1,1]]
 => [1,1]
 => 1
01101 => [2,1,2,1] => [[3,3,2,2],[2,1,1]]
 => [2,1,1]
 => 2
01110 => [2,1,1,2] => [[3,2,2,2],[1,1,1]]
 => [1,1,1]
 => 1
01111 => [2,1,1,1,1] => [[2,2,2,2,2],[1,1,1,1]]
 => [1,1,1,1]
 => 1
10000 => [1,5] => [[5,1],[]]
 => []
 => 0
10001 => [1,4,1] => [[4,4,1],[3]]
 => [3]
 => 1
10010 => [1,3,2] => [[4,3,1],[2]]
 => [2]
 => 1
10011 => [1,3,1,1] => [[3,3,3,1],[2,2]]
 => [2,2]
 => 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.
Matching statistic: St000183
(load all 2 compositions to match this statistic)
Mapping
Mp00178: Binary words —to composition⟶ Integer compositions
Mp00180: Integer compositions —to ribbon⟶ Skew partitions
Mp00183: Skew partitions —inner shape⟶ Integer partitions
St000183: 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],[]]
 => []
 => 0
00 => [3] => [[3],[]]
 => []
 => 0
01 => [2,1] => [[2,2],[1]]
 => [1]
 => 1
10 => [1,2] => [[2,1],[]]
 => []
 => 0
11 => [1,1,1] => [[1,1,1],[]]
 => []
 => 0
000 => [4] => [[4],[]]
 => []
 => 0
001 => [3,1] => [[3,3],[2]]
 => [2]
 => 1
010 => [2,2] => [[3,2],[1]]
 => [1]
 => 1
011 => [2,1,1] => [[2,2,2],[1,1]]
 => [1,1]
 => 1
100 => [1,3] => [[3,1],[]]
 => []
 => 0
101 => [1,2,1] => [[2,2,1],[1]]
 => [1]
 => 1
110 => [1,1,2] => [[2,1,1],[]]
 => []
 => 0
111 => [1,1,1,1] => [[1,1,1,1],[]]
 => []
 => 0
0000 => [5] => [[5],[]]
 => []
 => 0
0001 => [4,1] => [[4,4],[3]]
 => [3]
 => 1
0010 => [3,2] => [[4,3],[2]]
 => [2]
 => 1
0011 => [3,1,1] => [[3,3,3],[2,2]]
 => [2,2]
 => 2
0100 => [2,3] => [[4,2],[1]]
 => [1]
 => 1
0101 => [2,2,1] => [[3,3,2],[2,1]]
 => [2,1]
 => 1
0110 => [2,1,2] => [[3,2,2],[1,1]]
 => [1,1]
 => 1
0111 => [2,1,1,1] => [[2,2,2,2],[1,1,1]]
 => [1,1,1]
 => 1
1000 => [1,4] => [[4,1],[]]
 => []
 => 0
1001 => [1,3,1] => [[3,3,1],[2]]
 => [2]
 => 1
1010 => [1,2,2] => [[3,2,1],[1]]
 => [1]
 => 1
1011 => [1,2,1,1] => [[2,2,2,1],[1,1]]
 => [1,1]
 => 1
1100 => [1,1,3] => [[3,1,1],[]]
 => []
 => 0
1101 => [1,1,2,1] => [[2,2,1,1],[1]]
 => [1]
 => 1
1110 => [1,1,1,2] => [[2,1,1,1],[]]
 => []
 => 0
1111 => [1,1,1,1,1] => [[1,1,1,1,1],[]]
 => []
 => 0
00000 => [6] => [[6],[]]
 => []
 => 0
00001 => [5,1] => [[5,5],[4]]
 => [4]
 => 1
00010 => [4,2] => [[5,4],[3]]
 => [3]
 => 1
00011 => [4,1,1] => [[4,4,4],[3,3]]
 => [3,3]
 => 2
00100 => [3,3] => [[5,3],[2]]
 => [2]
 => 1
00101 => [3,2,1] => [[4,4,3],[3,2]]
 => [3,2]
 => 2
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]
 => 1
01001 => [2,3,1] => [[4,4,2],[3,1]]
 => [3,1]
 => 1
01010 => [2,2,2] => [[4,3,2],[2,1]]
 => [2,1]
 => 1
01011 => [2,2,1,1] => [[3,3,3,2],[2,2,1]]
 => [2,2,1]
 => 2
01100 => [2,1,3] => [[4,2,2],[1,1]]
 => [1,1]
 => 1
01101 => [2,1,2,1] => [[3,3,2,2],[2,1,1]]
 => [2,1,1]
 => 1
01110 => [2,1,1,2] => [[3,2,2,2],[1,1,1]]
 => [1,1,1]
 => 1
01111 => [2,1,1,1,1] => [[2,2,2,2,2],[1,1,1,1]]
 => [1,1,1,1]
 => 1
10000 => [1,5] => [[5,1],[]]
 => []
 => 0
10001 => [1,4,1] => [[4,4,1],[3]]
 => [3]
 => 1
10010 => [1,3,2] => [[4,3,1],[2]]
 => [2]
 => 1
10011 => [1,3,1,1] => [[3,3,3,1],[2,2]]
 => [2,2]
 => 2
Description
The side length of the Durfee square of an integer partition. Given a partition $\lambda = (\lambda_1,\ldots,\lambda_n)$, the Durfee square is the largest partition $(s^s)$ whose diagram fits inside the diagram of $\lambda$. In symbols, $s = \max\{ i \mid \lambda_i \geq i \}$. This is also known as the Frobenius rank.
Matching statistic: St000196
(load all 2 compositions to match this statistic)
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
St000196: Binary trees ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
0 => [2] => [1,1,0,0]
 => [[.,.],.]
 => 0
1 => [1,1] => [1,0,1,0]
 => [.,[.,.]]
 => 0
00 => [3] => [1,1,1,0,0,0]
 => [[[.,.],.],.]
 => 0
01 => [2,1] => [1,1,0,0,1,0]
 => [[.,.],[.,.]]
 => 1
10 => [1,2] => [1,0,1,1,0,0]
 => [.,[[.,.],.]]
 => 0
11 => [1,1,1] => [1,0,1,0,1,0]
 => [.,[.,[.,.]]]
 => 0
000 => [4] => [1,1,1,1,0,0,0,0]
 => [[[[.,.],.],.],.]
 => 0
001 => [3,1] => [1,1,1,0,0,0,1,0]
 => [[[.,.],.],[.,.]]
 => 1
010 => [2,2] => [1,1,0,0,1,1,0,0]
 => [[.,.],[[.,.],.]]
 => 1
011 => [2,1,1] => [1,1,0,0,1,0,1,0]
 => [[.,.],[.,[.,.]]]
 => 1
100 => [1,3] => [1,0,1,1,1,0,0,0]
 => [.,[[[.,.],.],.]]
 => 0
101 => [1,2,1] => [1,0,1,1,0,0,1,0]
 => [.,[[.,.],[.,.]]]
 => 1
110 => [1,1,2] => [1,0,1,0,1,1,0,0]
 => [.,[.,[[.,.],.]]]
 => 0
111 => [1,1,1,1] => [1,0,1,0,1,0,1,0]
 => [.,[.,[.,[.,.]]]]
 => 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]
 => [[[[.,.],.],.],[.,.]]
 => 1
0010 => [3,2] => [1,1,1,0,0,0,1,1,0,0]
 => [[[.,.],.],[[.,.],.]]
 => 1
0011 => [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
 => [[[.,.],.],[.,[.,.]]]
 => 1
0100 => [2,3] => [1,1,0,0,1,1,1,0,0,0]
 => [[.,.],[[[.,.],.],.]]
 => 1
0101 => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
 => [[.,.],[[.,.],[.,.]]]
 => 2
0110 => [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
 => [[.,.],[.,[[.,.],.]]]
 => 1
0111 => [2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
 => [[.,.],[.,[.,[.,.]]]]
 => 1
1000 => [1,4] => [1,0,1,1,1,1,0,0,0,0]
 => [.,[[[[.,.],.],.],.]]
 => 0
1001 => [1,3,1] => [1,0,1,1,1,0,0,0,1,0]
 => [.,[[[.,.],.],[.,.]]]
 => 1
1010 => [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
 => [.,[[.,.],[[.,.],.]]]
 => 1
1011 => [1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
 => [.,[[.,.],[.,[.,.]]]]
 => 1
1100 => [1,1,3] => [1,0,1,0,1,1,1,0,0,0]
 => [.,[.,[[[.,.],.],.]]]
 => 0
1101 => [1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
 => [.,[.,[[.,.],[.,.]]]]
 => 1
1110 => [1,1,1,2] => [1,0,1,0,1,0,1,1,0,0]
 => [.,[.,[.,[[.,.],.]]]]
 => 0
1111 => [1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
 => [.,[.,[.,[.,[.,.]]]]]
 => 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]
 => [[[[[.,.],.],.],.],[.,.]]
 => 1
00010 => [4,2] => [1,1,1,1,0,0,0,0,1,1,0,0]
 => [[[[.,.],.],.],[[.,.],.]]
 => 1
00011 => [4,1,1] => [1,1,1,1,0,0,0,0,1,0,1,0]
 => [[[[.,.],.],.],[.,[.,.]]]
 => 1
00100 => [3,3] => [1,1,1,0,0,0,1,1,1,0,0,0]
 => [[[.,.],.],[[[.,.],.],.]]
 => 1
00101 => [3,2,1] => [1,1,1,0,0,0,1,1,0,0,1,0]
 => [[[.,.],.],[[.,.],[.,.]]]
 => 2
00110 => [3,1,2] => [1,1,1,0,0,0,1,0,1,1,0,0]
 => [[[.,.],.],[.,[[.,.],.]]]
 => 1
00111 => [3,1,1,1] => [1,1,1,0,0,0,1,0,1,0,1,0]
 => [[[.,.],.],[.,[.,[.,.]]]]
 => 1
01000 => [2,4] => [1,1,0,0,1,1,1,1,0,0,0,0]
 => [[.,.],[[[[.,.],.],.],.]]
 => 1
01001 => [2,3,1] => [1,1,0,0,1,1,1,0,0,0,1,0]
 => [[.,.],[[[.,.],.],[.,.]]]
 => 2
01010 => [2,2,2] => [1,1,0,0,1,1,0,0,1,1,0,0]
 => [[.,.],[[.,.],[[.,.],.]]]
 => 2
01011 => [2,2,1,1] => [1,1,0,0,1,1,0,0,1,0,1,0]
 => [[.,.],[[.,.],[.,[.,.]]]]
 => 2
01100 => [2,1,3] => [1,1,0,0,1,0,1,1,1,0,0,0]
 => [[.,.],[.,[[[.,.],.],.]]]
 => 1
01101 => [2,1,2,1] => [1,1,0,0,1,0,1,1,0,0,1,0]
 => [[.,.],[.,[[.,.],[.,.]]]]
 => 2
01110 => [2,1,1,2] => [1,1,0,0,1,0,1,0,1,1,0,0]
 => [[.,.],[.,[.,[[.,.],.]]]]
 => 1
01111 => [2,1,1,1,1] => [1,1,0,0,1,0,1,0,1,0,1,0]
 => [[.,.],[.,[.,[.,[.,.]]]]]
 => 1
10000 => [1,5] => [1,0,1,1,1,1,1,0,0,0,0,0]
 => [.,[[[[[.,.],.],.],.],.]]
 => 0
10001 => [1,4,1] => [1,0,1,1,1,1,0,0,0,0,1,0]
 => [.,[[[[.,.],.],.],[.,.]]]
 => 1
10010 => [1,3,2] => [1,0,1,1,1,0,0,0,1,1,0,0]
 => [.,[[[.,.],.],[[.,.],.]]]
 => 1
10011 => [1,3,1,1] => [1,0,1,1,1,0,0,0,1,0,1,0]
 => [.,[[[.,.],.],[.,[.,.]]]]
 => 1
Description
The number of occurrences of the contiguous pattern {{{[[.,.],[.,.]]}}} in a binary tree. Equivalently, this is the number of branches in the tree, i.e. the number of nodes with two children. Binary trees avoiding this pattern are counted by $2^{n-2}$.
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: 100% ●values known / values provided: 100%●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
Description
The number of distinct parts of a partition that occur at least twice. See Section 3.3.1 of [2].
Matching statistic: St000875
(load all 2 compositions to match this statistic)
Mapping
Mp00224: Binary words —runsort⟶ Binary words
Mp00104: Binary words —reverse⟶ Binary words
Mp00268: Binary words —zeros to flag zeros⟶ Binary words
St000875: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
0 => 0 => 0 => 0 => 0
1 => 1 => 1 => 1 => 0
00 => 00 => 00 => 10 => 1
01 => 01 => 10 => 01 => 0
10 => 01 => 10 => 01 => 0
11 => 11 => 11 => 11 => 0
000 => 000 => 000 => 010 => 1
001 => 001 => 100 => 101 => 1
010 => 001 => 100 => 101 => 1
011 => 011 => 110 => 011 => 0
100 => 001 => 100 => 101 => 1
101 => 011 => 110 => 011 => 0
110 => 011 => 110 => 011 => 0
111 => 111 => 111 => 111 => 0
0000 => 0000 => 0000 => 1010 => 2
0001 => 0001 => 1000 => 0101 => 1
0010 => 0001 => 1000 => 0101 => 1
0011 => 0011 => 1100 => 1011 => 1
0100 => 0001 => 1000 => 0101 => 1
0101 => 0101 => 1010 => 1001 => 1
0110 => 0011 => 1100 => 1011 => 1
0111 => 0111 => 1110 => 0111 => 0
1000 => 0001 => 1000 => 0101 => 1
1001 => 0011 => 1100 => 1011 => 1
1010 => 0011 => 1100 => 1011 => 1
1011 => 0111 => 1110 => 0111 => 0
1100 => 0011 => 1100 => 1011 => 1
1101 => 0111 => 1110 => 0111 => 0
1110 => 0111 => 1110 => 0111 => 0
1111 => 1111 => 1111 => 1111 => 0
00000 => 00000 => 00000 => 01010 => 2
00001 => 00001 => 10000 => 10101 => 2
00010 => 00001 => 10000 => 10101 => 2
00011 => 00011 => 11000 => 01011 => 1
00100 => 00001 => 10000 => 10101 => 2
00101 => 00101 => 10100 => 01001 => 1
00110 => 00011 => 11000 => 01011 => 1
00111 => 00111 => 11100 => 10111 => 1
01000 => 00001 => 10000 => 10101 => 2
01001 => 00101 => 10100 => 01001 => 1
01010 => 00101 => 10100 => 01001 => 1
01011 => 01011 => 11010 => 10011 => 1
01100 => 00011 => 11000 => 01011 => 1
01101 => 01011 => 11010 => 10011 => 1
01110 => 00111 => 11100 => 10111 => 1
01111 => 01111 => 11110 => 01111 => 0
10000 => 00001 => 10000 => 10101 => 2
10001 => 00011 => 11000 => 01011 => 1
10010 => 00011 => 11000 => 01011 => 1
10011 => 00111 => 11100 => 10111 => 1
Description
The semilength of the longest Dyck word in the Catalan factorisation of a binary word. Every binary word can be written in a unique way as $(\mathcal D 0)^\ell \mathcal D (1 \mathcal D)^m$, where $\mathcal D$ is the set of Dyck words. This is the Catalan factorisation, see [1, sec.9.1.2]. This statistic records the semilength of the longest Dyck word in this factorisation.
The following 50 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St001022Number of simple modules with projective dimension 3 in the Nakayama algebra corresponding to the Dyck path. St001036The number of inner corners of the parallelogram polyomino associated with the Dyck path. St001421Half the length of a longest factor which is its own reverse-complement and begins with a one of a binary word. St000068The number of minimal elements in a poset. St000069The number of maximal elements of a poset. St000093The cardinality of a maximal independent set of vertices of a graph. St000201The number of leaf nodes in a binary tree. St000318The number of addable cells of the Ferrers diagram of an integer partition. St000568The hook number of a binary tree. St000786The maximal number of occurrences of a colour in a proper colouring of a graph. St001716The 1-improper chromatic number of a graph. St001732The number of peaks visible from the left. St001337The upper domination number of a graph. St001338The upper irredundance number of a graph. St001086The number of occurrences of the consecutive pattern 132 in a permutation. St000356The number of occurrences of the pattern 13-2. St001124The multiplicity of the standard representation in the Kronecker square corresponding to a partition. 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. St001687The number of distinct positions of the pattern letter 2 in occurrences of 213 in a permutation. St001738The minimal order of a graph which is not an induced subgraph of the given graph. St001432The order dimension of the partition. St000779The tier of a permutation. St000023The number of inner peaks of a permutation. St001113Number of indecomposable projective non-injective modules with reflexive Auslander-Reiten sequences in the corresponding Nakayama algebra. St001186Number of simple modules with grade at least 3 in the corresponding Nakayama algebra. St001266The largest vector space dimension of an indecomposable non-projective module that is reflexive in the corresponding Nakayama algebra. 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. St001878The projective dimension of the simple modules corresponding to the minimum of L in the incidence algebra of the lattice L. St000035The number of left outer peaks of a permutation. St000647The number of big descents of a permutation. St000834The number of right outer peaks of a permutation. St000884The number of isolated descents 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. St001960The number of descents of a permutation minus one if its first entry is not one. St000021The number of descents of a permutation. St000333The dez statistic, the number of descents of a permutation after replacing fixed points by zeros. St000354The number of recoils of a permutation. 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. St001290The first natural number n such that the tensor product of n copies of D(A) is zero for the corresponding Nakayama algebra A. St001630The global dimension of the incidence algebra of the lattice over the rational numbers.