Your data matches 58 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 - 1
1 => 0 = 1 - 1
00 => 0 = 1 - 1
01 => 0 = 1 - 1
10 => 1 = 2 - 1
11 => 0 = 1 - 1
000 => 0 = 1 - 1
001 => 0 = 1 - 1
010 => 1 = 2 - 1
011 => 0 = 1 - 1
100 => 1 = 2 - 1
101 => 1 = 2 - 1
110 => 1 = 2 - 1
111 => 0 = 1 - 1
0000 => 0 = 1 - 1
0001 => 0 = 1 - 1
0010 => 1 = 2 - 1
0011 => 0 = 1 - 1
0100 => 1 = 2 - 1
0101 => 1 = 2 - 1
0110 => 1 = 2 - 1
0111 => 0 = 1 - 1
1000 => 1 = 2 - 1
1001 => 1 = 2 - 1
1010 => 2 = 3 - 1
1011 => 1 = 2 - 1
1100 => 1 = 2 - 1
1101 => 1 = 2 - 1
1110 => 1 = 2 - 1
1111 => 0 = 1 - 1
00000 => 0 = 1 - 1
00001 => 0 = 1 - 1
00010 => 1 = 2 - 1
00011 => 0 = 1 - 1
00100 => 1 = 2 - 1
00101 => 1 = 2 - 1
00110 => 1 = 2 - 1
00111 => 0 = 1 - 1
01000 => 1 = 2 - 1
01001 => 1 = 2 - 1
01010 => 2 = 3 - 1
01011 => 1 = 2 - 1
01100 => 1 = 2 - 1
01101 => 1 = 2 - 1
01110 => 1 = 2 - 1
01111 => 0 = 1 - 1
10000 => 1 = 2 - 1
10001 => 1 = 2 - 1
10010 => 2 = 3 - 1
10011 => 1 = 2 - 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 - 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
Description
The number of ascents of a binary word.
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
1 => [1,1] => 11 => 1
00 => [3] => 100 => 1
01 => [2,1] => 101 => 2
10 => [1,2] => 110 => 1
11 => [1,1,1] => 111 => 1
000 => [4] => 1000 => 1
001 => [3,1] => 1001 => 2
010 => [2,2] => 1010 => 2
011 => [2,1,1] => 1011 => 2
100 => [1,3] => 1100 => 1
101 => [1,2,1] => 1101 => 2
110 => [1,1,2] => 1110 => 1
111 => [1,1,1,1] => 1111 => 1
0000 => [5] => 10000 => 1
0001 => [4,1] => 10001 => 2
0010 => [3,2] => 10010 => 2
0011 => [3,1,1] => 10011 => 2
0100 => [2,3] => 10100 => 2
0101 => [2,2,1] => 10101 => 3
0110 => [2,1,2] => 10110 => 2
0111 => [2,1,1,1] => 10111 => 2
1000 => [1,4] => 11000 => 1
1001 => [1,3,1] => 11001 => 2
1010 => [1,2,2] => 11010 => 2
1011 => [1,2,1,1] => 11011 => 2
1100 => [1,1,3] => 11100 => 1
1101 => [1,1,2,1] => 11101 => 2
1110 => [1,1,1,2] => 11110 => 1
1111 => [1,1,1,1,1] => 11111 => 1
00000 => [6] => 100000 => 1
00001 => [5,1] => 100001 => 2
00010 => [4,2] => 100010 => 2
00011 => [4,1,1] => 100011 => 2
00100 => [3,3] => 100100 => 2
00101 => [3,2,1] => 100101 => 3
00110 => [3,1,2] => 100110 => 2
00111 => [3,1,1,1] => 100111 => 2
01000 => [2,4] => 101000 => 2
01001 => [2,3,1] => 101001 => 3
01010 => [2,2,2] => 101010 => 3
01011 => [2,2,1,1] => 101011 => 3
01100 => [2,1,3] => 101100 => 2
01101 => [2,1,2,1] => 101101 => 3
01110 => [2,1,1,2] => 101110 => 2
01111 => [2,1,1,1,1] => 101111 => 2
10000 => [1,5] => 110000 => 1
10001 => [1,4,1] => 110001 => 2
10010 => [1,3,2] => 110010 => 2
10011 => [1,3,1,1] => 110011 => 2
Description
The number of runs of ones in 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 => [1,1] => [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
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 => [1,1] => [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]
 => 0 = 1 - 1
10 => [1,2] => [1,0,1,1,0,0]
 => 1 = 2 - 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]
 => 0 = 1 - 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]
 => 0 = 1 - 1
100 => [1,3] => [1,0,1,1,1,0,0,0]
 => 1 = 2 - 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]
 => 1 = 2 - 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]
 => 0 = 1 - 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]
 => 0 = 1 - 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]
 => 1 = 2 - 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]
 => 0 = 1 - 1
1000 => [1,4] => [1,0,1,1,1,1,0,0,0,0]
 => 1 = 2 - 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]
 => 2 = 3 - 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]
 => 1 = 2 - 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]
 => 1 = 2 - 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]
 => 0 = 1 - 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]
 => 0 = 1 - 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]
 => 1 = 2 - 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]
 => 0 = 1 - 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]
 => 1 = 2 - 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]
 => 1 = 2 - 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]
 => 1 = 2 - 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]
 => 0 = 1 - 1
10000 => [1,5] => [1,0,1,1,1,1,1,0,0,0,0,0]
 => 1 = 2 - 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]
 => 2 = 3 - 1
10011 => [1,3,1,1] => [1,0,1,1,1,0,0,0,1,0,1,0]
 => 1 = 2 - 1
Description
The number of inner corners of the upper path of the parallelogram polyomino associated with the Dyck path.
Matching statistic: St000068
Mapping
Mp00178: Binary words —to composition⟶ Integer compositions
Mp00180: Integer compositions —to ribbon⟶ Skew partitions
Mp00185: Skew partitions —cell poset⟶ Posets
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: St000069
Mapping
Mp00178: Binary words —to composition⟶ Integer compositions
Mp00180: Integer compositions —to ribbon⟶ Skew partitions
Mp00185: Skew partitions —cell poset⟶ Posets
St000069: 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)
 => 1
10 => [1,2] => [[2,1],[]]
 => ([(0,1),(0,2)],3)
 => 2
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)
 => 1
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)
 => 1
100 => [1,3] => [[3,1],[]]
 => ([(0,2),(0,3),(3,1)],4)
 => 2
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)
 => 2
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)
 => 1
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)
 => 1
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)
 => 2
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)
 => 1
1000 => [1,4] => [[4,1],[]]
 => ([(0,2),(0,4),(3,1),(4,3)],5)
 => 2
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)
 => 3
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)
 => 2
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)
 => 2
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)
 => 1
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)
 => 1
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)
 => 2
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)
 => 1
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)
 => 2
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)
 => 2
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)
 => 2
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)
 => 1
10000 => [1,5] => [[5,1],[]]
 => ([(0,2),(0,5),(3,4),(4,1),(5,3)],6)
 => 2
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)
 => 3
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 maximal elements of a poset.
Matching statistic: St000093
Mapping
Mp00178: Binary words —to composition⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
Mp00247: Graphs —de-duplicate⟶ Graphs
St000093: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
0 => [2] => ([],2)
 => ([],1)
 => 1
1 => [1,1] => ([(0,1)],2)
 => ([(0,1)],2)
 => 1
00 => [3] => ([],3)
 => ([],1)
 => 1
01 => [2,1] => ([(0,2),(1,2)],3)
 => ([(0,1)],2)
 => 1
10 => [1,2] => ([(1,2)],3)
 => ([(1,2)],3)
 => 2
11 => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
 => ([(0,1),(0,2),(1,2)],3)
 => 1
000 => [4] => ([],4)
 => ([],1)
 => 1
001 => [3,1] => ([(0,3),(1,3),(2,3)],4)
 => ([(0,1)],2)
 => 1
010 => [2,2] => ([(1,3),(2,3)],4)
 => ([(1,2)],3)
 => 2
011 => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => ([(0,1),(0,2),(1,2)],3)
 => 1
100 => [1,3] => ([(2,3)],4)
 => ([(1,2)],3)
 => 2
101 => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => ([(0,3),(1,2),(1,3),(2,3)],4)
 => 2
110 => [1,1,2] => ([(1,2),(1,3),(2,3)],4)
 => ([(1,2),(1,3),(2,3)],4)
 => 2
111 => [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 1
0000 => [5] => ([],5)
 => ([],1)
 => 1
0001 => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => ([(0,1)],2)
 => 1
0010 => [3,2] => ([(1,4),(2,4),(3,4)],5)
 => ([(1,2)],3)
 => 2
0011 => [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(1,2)],3)
 => 1
0100 => [2,3] => ([(2,4),(3,4)],5)
 => ([(1,2)],3)
 => 2
0101 => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,3),(1,2),(1,3),(2,3)],4)
 => 2
0110 => [2,1,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(1,2),(1,3),(2,3)],4)
 => 2
0111 => [2,1,1,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 1
1000 => [1,4] => ([(3,4)],5)
 => ([(1,2)],3)
 => 2
1001 => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,3),(1,2),(1,3),(2,3)],4)
 => 2
1010 => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
 => ([(1,4),(2,3),(2,4),(3,4)],5)
 => 3
1011 => [1,2,1,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 2
1100 => [1,1,3] => ([(2,3),(2,4),(3,4)],5)
 => ([(1,2),(1,3),(2,3)],4)
 => 2
1101 => [1,1,2,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 2
1110 => [1,1,1,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 2
1111 => [1,1,1,1,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 1
00000 => [6] => ([],6)
 => ([],1)
 => 1
00001 => [5,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
 => ([(0,1)],2)
 => 1
00010 => [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6)
 => ([(1,2)],3)
 => 2
00011 => [4,1,1] => ([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,1),(0,2),(1,2)],3)
 => 1
00100 => [3,3] => ([(2,5),(3,5),(4,5)],6)
 => ([(1,2)],3)
 => 2
00101 => [3,2,1] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,3),(1,2),(1,3),(2,3)],4)
 => 2
00110 => [3,1,2] => ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(1,2),(1,3),(2,3)],4)
 => 2
00111 => [3,1,1,1] => ([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 1
01000 => [2,4] => ([(3,5),(4,5)],6)
 => ([(1,2)],3)
 => 2
01001 => [2,3,1] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,3),(1,2),(1,3),(2,3)],4)
 => 2
01010 => [2,2,2] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(1,4),(2,3),(2,4),(3,4)],5)
 => 3
01011 => [2,2,1,1] => ([(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 2
01100 => [2,1,3] => ([(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(1,2),(1,3),(2,3)],4)
 => 2
01101 => [2,1,2,1] => ([(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 2
01110 => [2,1,1,2] => ([(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 2
01111 => [2,1,1,1,1] => ([(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 1
10000 => [1,5] => ([(4,5)],6)
 => ([(1,2)],3)
 => 2
10001 => [1,4,1] => ([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,3),(1,2),(1,3),(2,3)],4)
 => 2
10010 => [1,3,2] => ([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(1,4),(2,3),(2,4),(3,4)],5)
 => 3
10011 => [1,3,1,1] => ([(0,4),(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 2
Description
The cardinality of a maximal independent set of vertices of a graph. An independent set of a graph is a set of pairwise non-adjacent vertices. A maximum independent set is an independent set of maximum cardinality. This statistic is also called the independence number or stability number $\alpha(G)$ of $G$.
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
Mp00182: Skew partitions —outer 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],[]]
 => [2]
 => 1
1 => [1,1] => [[1,1],[]]
 => [1,1]
 => 1
00 => [3] => [[3],[]]
 => [3]
 => 1
01 => [2,1] => [[2,2],[1]]
 => [2,2]
 => 2
10 => [1,2] => [[2,1],[]]
 => [2,1]
 => 1
11 => [1,1,1] => [[1,1,1],[]]
 => [1,1,1]
 => 1
000 => [4] => [[4],[]]
 => [4]
 => 1
001 => [3,1] => [[3,3],[2]]
 => [3,3]
 => 2
010 => [2,2] => [[3,2],[1]]
 => [3,2]
 => 2
011 => [2,1,1] => [[2,2,2],[1,1]]
 => [2,2,2]
 => 2
100 => [1,3] => [[3,1],[]]
 => [3,1]
 => 1
101 => [1,2,1] => [[2,2,1],[1]]
 => [2,2,1]
 => 2
110 => [1,1,2] => [[2,1,1],[]]
 => [2,1,1]
 => 1
111 => [1,1,1,1] => [[1,1,1,1],[]]
 => [1,1,1,1]
 => 1
0000 => [5] => [[5],[]]
 => [5]
 => 1
0001 => [4,1] => [[4,4],[3]]
 => [4,4]
 => 2
0010 => [3,2] => [[4,3],[2]]
 => [4,3]
 => 2
0011 => [3,1,1] => [[3,3,3],[2,2]]
 => [3,3,3]
 => 3
0100 => [2,3] => [[4,2],[1]]
 => [4,2]
 => 2
0101 => [2,2,1] => [[3,3,2],[2,1]]
 => [3,3,2]
 => 2
0110 => [2,1,2] => [[3,2,2],[1,1]]
 => [3,2,2]
 => 2
0111 => [2,1,1,1] => [[2,2,2,2],[1,1,1]]
 => [2,2,2,2]
 => 2
1000 => [1,4] => [[4,1],[]]
 => [4,1]
 => 1
1001 => [1,3,1] => [[3,3,1],[2]]
 => [3,3,1]
 => 2
1010 => [1,2,2] => [[3,2,1],[1]]
 => [3,2,1]
 => 2
1011 => [1,2,1,1] => [[2,2,2,1],[1,1]]
 => [2,2,2,1]
 => 2
1100 => [1,1,3] => [[3,1,1],[]]
 => [3,1,1]
 => 1
1101 => [1,1,2,1] => [[2,2,1,1],[1]]
 => [2,2,1,1]
 => 2
1110 => [1,1,1,2] => [[2,1,1,1],[]]
 => [2,1,1,1]
 => 1
1111 => [1,1,1,1,1] => [[1,1,1,1,1],[]]
 => [1,1,1,1,1]
 => 1
00000 => [6] => [[6],[]]
 => [6]
 => 1
00001 => [5,1] => [[5,5],[4]]
 => [5,5]
 => 2
00010 => [4,2] => [[5,4],[3]]
 => [5,4]
 => 2
00011 => [4,1,1] => [[4,4,4],[3,3]]
 => [4,4,4]
 => 3
00100 => [3,3] => [[5,3],[2]]
 => [5,3]
 => 2
00101 => [3,2,1] => [[4,4,3],[3,2]]
 => [4,4,3]
 => 3
00110 => [3,1,2] => [[4,3,3],[2,2]]
 => [4,3,3]
 => 3
00111 => [3,1,1,1] => [[3,3,3,3],[2,2,2]]
 => [3,3,3,3]
 => 3
01000 => [2,4] => [[5,2],[1]]
 => [5,2]
 => 2
01001 => [2,3,1] => [[4,4,2],[3,1]]
 => [4,4,2]
 => 2
01010 => [2,2,2] => [[4,3,2],[2,1]]
 => [4,3,2]
 => 2
01011 => [2,2,1,1] => [[3,3,3,2],[2,2,1]]
 => [3,3,3,2]
 => 3
01100 => [2,1,3] => [[4,2,2],[1,1]]
 => [4,2,2]
 => 2
01101 => [2,1,2,1] => [[3,3,2,2],[2,1,1]]
 => [3,3,2,2]
 => 2
01110 => [2,1,1,2] => [[3,2,2,2],[1,1,1]]
 => [3,2,2,2]
 => 2
01111 => [2,1,1,1,1] => [[2,2,2,2,2],[1,1,1,1]]
 => [2,2,2,2,2]
 => 2
10000 => [1,5] => [[5,1],[]]
 => [5,1]
 => 1
10001 => [1,4,1] => [[4,4,1],[3]]
 => [4,4,1]
 => 2
10010 => [1,3,2] => [[4,3,1],[2]]
 => [4,3,1]
 => 2
10011 => [1,3,1,1] => [[3,3,3,1],[2,2]]
 => [3,3,3,1]
 => 3
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: St000201
(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
St000201: Binary trees ⟶ ℤResult quality: 100% ●values known / values provided: 100%●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
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].
The following 48 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
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. St000159The number of distinct parts of the integer partition. St000196The number of occurrences of the contiguous pattern [[.,.],[.,. St000257The number of distinct parts of a partition that occur at least twice. St000875The semilength of the longest Dyck word in the Catalan factorisation of a binary word. 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. 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. St000259The diameter of a connected graph. 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. 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. St000985The number of positive eigenvalues of the adjacency matrix of the graph. St001354The number of series nodes in the modular decomposition of a graph. 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. 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. St001960The number of descents of a permutation minus one if its first entry is not one.