Your data matches 34 different statistics following compositions of up to 3 maps.
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Matching statistic: St000382
(load all 4 compositions to match this statistic)
Mapping
Mp00105: Binary words —complement⟶ Binary words
Mp00178: Binary words —to composition⟶ Integer compositions
St000382: Integer compositions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
0 => 1 => [1,1] => 1 = 2 - 1
1 => 0 => [2] => 2 = 3 - 1
00 => 11 => [1,1,1] => 1 = 2 - 1
01 => 10 => [1,2] => 1 = 2 - 1
10 => 01 => [2,1] => 2 = 3 - 1
11 => 00 => [3] => 3 = 4 - 1
000 => 111 => [1,1,1,1] => 1 = 2 - 1
001 => 110 => [1,1,2] => 1 = 2 - 1
010 => 101 => [1,2,1] => 1 = 2 - 1
011 => 100 => [1,3] => 1 = 2 - 1
100 => 011 => [2,1,1] => 2 = 3 - 1
101 => 010 => [2,2] => 2 = 3 - 1
110 => 001 => [3,1] => 3 = 4 - 1
111 => 000 => [4] => 4 = 5 - 1
0000 => 1111 => [1,1,1,1,1] => 1 = 2 - 1
0001 => 1110 => [1,1,1,2] => 1 = 2 - 1
0010 => 1101 => [1,1,2,1] => 1 = 2 - 1
0011 => 1100 => [1,1,3] => 1 = 2 - 1
0100 => 1011 => [1,2,1,1] => 1 = 2 - 1
0101 => 1010 => [1,2,2] => 1 = 2 - 1
0110 => 1001 => [1,3,1] => 1 = 2 - 1
0111 => 1000 => [1,4] => 1 = 2 - 1
1000 => 0111 => [2,1,1,1] => 2 = 3 - 1
1001 => 0110 => [2,1,2] => 2 = 3 - 1
1010 => 0101 => [2,2,1] => 2 = 3 - 1
1011 => 0100 => [2,3] => 2 = 3 - 1
1100 => 0011 => [3,1,1] => 3 = 4 - 1
1101 => 0010 => [3,2] => 3 = 4 - 1
1110 => 0001 => [4,1] => 4 = 5 - 1
1111 => 0000 => [5] => 5 = 6 - 1
00000 => 11111 => [1,1,1,1,1,1] => 1 = 2 - 1
00001 => 11110 => [1,1,1,1,2] => 1 = 2 - 1
00010 => 11101 => [1,1,1,2,1] => 1 = 2 - 1
00011 => 11100 => [1,1,1,3] => 1 = 2 - 1
00100 => 11011 => [1,1,2,1,1] => 1 = 2 - 1
00101 => 11010 => [1,1,2,2] => 1 = 2 - 1
00110 => 11001 => [1,1,3,1] => 1 = 2 - 1
00111 => 11000 => [1,1,4] => 1 = 2 - 1
01000 => 10111 => [1,2,1,1,1] => 1 = 2 - 1
01001 => 10110 => [1,2,1,2] => 1 = 2 - 1
01010 => 10101 => [1,2,2,1] => 1 = 2 - 1
01011 => 10100 => [1,2,3] => 1 = 2 - 1
01100 => 10011 => [1,3,1,1] => 1 = 2 - 1
01101 => 10010 => [1,3,2] => 1 = 2 - 1
01110 => 10001 => [1,4,1] => 1 = 2 - 1
01111 => 10000 => [1,5] => 1 = 2 - 1
10000 => 01111 => [2,1,1,1,1] => 2 = 3 - 1
10001 => 01110 => [2,1,1,2] => 2 = 3 - 1
10010 => 01101 => [2,1,2,1] => 2 = 3 - 1
10011 => 01100 => [2,1,3] => 2 = 3 - 1
Description
The first part of an integer composition.
Matching statistic: St000439
Mapping
Mp00105: Binary words —complement⟶ Binary words
Mp00178: Binary words —to composition⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St000439: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
0 => 1 => [1,1] => [1,0,1,0]
 => 2
1 => 0 => [2] => [1,1,0,0]
 => 3
00 => 11 => [1,1,1] => [1,0,1,0,1,0]
 => 2
01 => 10 => [1,2] => [1,0,1,1,0,0]
 => 2
10 => 01 => [2,1] => [1,1,0,0,1,0]
 => 3
11 => 00 => [3] => [1,1,1,0,0,0]
 => 4
000 => 111 => [1,1,1,1] => [1,0,1,0,1,0,1,0]
 => 2
001 => 110 => [1,1,2] => [1,0,1,0,1,1,0,0]
 => 2
010 => 101 => [1,2,1] => [1,0,1,1,0,0,1,0]
 => 2
011 => 100 => [1,3] => [1,0,1,1,1,0,0,0]
 => 2
100 => 011 => [2,1,1] => [1,1,0,0,1,0,1,0]
 => 3
101 => 010 => [2,2] => [1,1,0,0,1,1,0,0]
 => 3
110 => 001 => [3,1] => [1,1,1,0,0,0,1,0]
 => 4
111 => 000 => [4] => [1,1,1,1,0,0,0,0]
 => 5
0000 => 1111 => [1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
 => 2
0001 => 1110 => [1,1,1,2] => [1,0,1,0,1,0,1,1,0,0]
 => 2
0010 => 1101 => [1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
 => 2
0011 => 1100 => [1,1,3] => [1,0,1,0,1,1,1,0,0,0]
 => 2
0100 => 1011 => [1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
 => 2
0101 => 1010 => [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
 => 2
0110 => 1001 => [1,3,1] => [1,0,1,1,1,0,0,0,1,0]
 => 2
0111 => 1000 => [1,4] => [1,0,1,1,1,1,0,0,0,0]
 => 2
1000 => 0111 => [2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
 => 3
1001 => 0110 => [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
 => 3
1010 => 0101 => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
 => 3
1011 => 0100 => [2,3] => [1,1,0,0,1,1,1,0,0,0]
 => 3
1100 => 0011 => [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
 => 4
1101 => 0010 => [3,2] => [1,1,1,0,0,0,1,1,0,0]
 => 4
1110 => 0001 => [4,1] => [1,1,1,1,0,0,0,0,1,0]
 => 5
1111 => 0000 => [5] => [1,1,1,1,1,0,0,0,0,0]
 => 6
00000 => 11111 => [1,1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0,1,0]
 => 2
00001 => 11110 => [1,1,1,1,2] => [1,0,1,0,1,0,1,0,1,1,0,0]
 => 2
00010 => 11101 => [1,1,1,2,1] => [1,0,1,0,1,0,1,1,0,0,1,0]
 => 2
00011 => 11100 => [1,1,1,3] => [1,0,1,0,1,0,1,1,1,0,0,0]
 => 2
00100 => 11011 => [1,1,2,1,1] => [1,0,1,0,1,1,0,0,1,0,1,0]
 => 2
00101 => 11010 => [1,1,2,2] => [1,0,1,0,1,1,0,0,1,1,0,0]
 => 2
00110 => 11001 => [1,1,3,1] => [1,0,1,0,1,1,1,0,0,0,1,0]
 => 2
00111 => 11000 => [1,1,4] => [1,0,1,0,1,1,1,1,0,0,0,0]
 => 2
01000 => 10111 => [1,2,1,1,1] => [1,0,1,1,0,0,1,0,1,0,1,0]
 => 2
01001 => 10110 => [1,2,1,2] => [1,0,1,1,0,0,1,0,1,1,0,0]
 => 2
01010 => 10101 => [1,2,2,1] => [1,0,1,1,0,0,1,1,0,0,1,0]
 => 2
01011 => 10100 => [1,2,3] => [1,0,1,1,0,0,1,1,1,0,0,0]
 => 2
01100 => 10011 => [1,3,1,1] => [1,0,1,1,1,0,0,0,1,0,1,0]
 => 2
01101 => 10010 => [1,3,2] => [1,0,1,1,1,0,0,0,1,1,0,0]
 => 2
01110 => 10001 => [1,4,1] => [1,0,1,1,1,1,0,0,0,0,1,0]
 => 2
01111 => 10000 => [1,5] => [1,0,1,1,1,1,1,0,0,0,0,0]
 => 2
10000 => 01111 => [2,1,1,1,1] => [1,1,0,0,1,0,1,0,1,0,1,0]
 => 3
10001 => 01110 => [2,1,1,2] => [1,1,0,0,1,0,1,0,1,1,0,0]
 => 3
10010 => 01101 => [2,1,2,1] => [1,1,0,0,1,0,1,1,0,0,1,0]
 => 3
10011 => 01100 => [2,1,3] => [1,1,0,0,1,0,1,1,1,0,0,0]
 => 3
Description
The position of the first down step of a Dyck path.
Matching statistic: St000297
(load all 10 compositions to match this statistic)
Mapping
St000297: Binary words ⟶ ℤResult quality: 99% ●values known / values provided: 99%●distinct values known / distinct values provided: 100%
Values
0 => 0 = 2 - 2
1 => 1 = 3 - 2
00 => 0 = 2 - 2
01 => 0 = 2 - 2
10 => 1 = 3 - 2
11 => 2 = 4 - 2
000 => 0 = 2 - 2
001 => 0 = 2 - 2
010 => 0 = 2 - 2
011 => 0 = 2 - 2
100 => 1 = 3 - 2
101 => 1 = 3 - 2
110 => 2 = 4 - 2
111 => 3 = 5 - 2
0000 => 0 = 2 - 2
0001 => 0 = 2 - 2
0010 => 0 = 2 - 2
0011 => 0 = 2 - 2
0100 => 0 = 2 - 2
0101 => 0 = 2 - 2
0110 => 0 = 2 - 2
0111 => 0 = 2 - 2
1000 => 1 = 3 - 2
1001 => 1 = 3 - 2
1010 => 1 = 3 - 2
1011 => 1 = 3 - 2
1100 => 2 = 4 - 2
1101 => 2 = 4 - 2
1110 => 3 = 5 - 2
1111 => 4 = 6 - 2
00000 => 0 = 2 - 2
00001 => 0 = 2 - 2
00010 => 0 = 2 - 2
00011 => 0 = 2 - 2
00100 => 0 = 2 - 2
00101 => 0 = 2 - 2
00110 => 0 = 2 - 2
00111 => 0 = 2 - 2
01000 => 0 = 2 - 2
01001 => 0 = 2 - 2
01010 => 0 = 2 - 2
01011 => 0 = 2 - 2
01100 => 0 = 2 - 2
01101 => 0 = 2 - 2
01110 => 0 = 2 - 2
01111 => 0 = 2 - 2
10000 => 1 = 3 - 2
10001 => 1 = 3 - 2
10010 => 1 = 3 - 2
10011 => 1 = 3 - 2
=> ? = 2 - 2
0011111110 => ? = 2 - 2
0111111100 => ? = 2 - 2
Description
The number of leading ones in a binary word.
Matching statistic: St000326
(load all 6 compositions to match this statistic)
Mapping
Mp00105: Binary words —complement⟶ Binary words
St000326: Binary words ⟶ ℤResult quality: 99% ●values known / values provided: 99%●distinct values known / distinct values provided: 100%
Values
0 => 1 => 1 = 2 - 1
1 => 0 => 2 = 3 - 1
00 => 11 => 1 = 2 - 1
01 => 10 => 1 = 2 - 1
10 => 01 => 2 = 3 - 1
11 => 00 => 3 = 4 - 1
000 => 111 => 1 = 2 - 1
001 => 110 => 1 = 2 - 1
010 => 101 => 1 = 2 - 1
011 => 100 => 1 = 2 - 1
100 => 011 => 2 = 3 - 1
101 => 010 => 2 = 3 - 1
110 => 001 => 3 = 4 - 1
111 => 000 => 4 = 5 - 1
0000 => 1111 => 1 = 2 - 1
0001 => 1110 => 1 = 2 - 1
0010 => 1101 => 1 = 2 - 1
0011 => 1100 => 1 = 2 - 1
0100 => 1011 => 1 = 2 - 1
0101 => 1010 => 1 = 2 - 1
0110 => 1001 => 1 = 2 - 1
0111 => 1000 => 1 = 2 - 1
1000 => 0111 => 2 = 3 - 1
1001 => 0110 => 2 = 3 - 1
1010 => 0101 => 2 = 3 - 1
1011 => 0100 => 2 = 3 - 1
1100 => 0011 => 3 = 4 - 1
1101 => 0010 => 3 = 4 - 1
1110 => 0001 => 4 = 5 - 1
1111 => 0000 => 5 = 6 - 1
00000 => 11111 => 1 = 2 - 1
00001 => 11110 => 1 = 2 - 1
00010 => 11101 => 1 = 2 - 1
00011 => 11100 => 1 = 2 - 1
00100 => 11011 => 1 = 2 - 1
00101 => 11010 => 1 = 2 - 1
00110 => 11001 => 1 = 2 - 1
00111 => 11000 => 1 = 2 - 1
01000 => 10111 => 1 = 2 - 1
01001 => 10110 => 1 = 2 - 1
01010 => 10101 => 1 = 2 - 1
01011 => 10100 => 1 = 2 - 1
01100 => 10011 => 1 = 2 - 1
01101 => 10010 => 1 = 2 - 1
01110 => 10001 => 1 = 2 - 1
01111 => 10000 => 1 = 2 - 1
10000 => 01111 => 2 = 3 - 1
10001 => 01110 => 2 = 3 - 1
10010 => 01101 => 2 = 3 - 1
10011 => 01100 => 2 = 3 - 1
=> => ? = 2 - 1
0011111110 => 1100000001 => ? = 2 - 1
0111111100 => 1000000011 => ? = 2 - 1
Description
The position of the first one in a binary word after appending a 1 at the end. Regarding the binary word as a subset of $\{1,\dots,n,n+1\}$ that contains $n+1$, this is the minimal element of the set.
Matching statistic: St000383
(load all 6 compositions to match this statistic)
Mapping
Mp00178: Binary words —to composition⟶ Integer compositions
Mp00041: Integer compositions —conjugate⟶ Integer compositions
St000383: Integer compositions ⟶ ℤResult quality: 84% ●values known / values provided: 84%●distinct values known / distinct values provided: 100%
Values
0 => [2] => [1,1] => 1 = 2 - 1
1 => [1,1] => [2] => 2 = 3 - 1
00 => [3] => [1,1,1] => 1 = 2 - 1
01 => [2,1] => [2,1] => 1 = 2 - 1
10 => [1,2] => [1,2] => 2 = 3 - 1
11 => [1,1,1] => [3] => 3 = 4 - 1
000 => [4] => [1,1,1,1] => 1 = 2 - 1
001 => [3,1] => [2,1,1] => 1 = 2 - 1
010 => [2,2] => [1,2,1] => 1 = 2 - 1
011 => [2,1,1] => [3,1] => 1 = 2 - 1
100 => [1,3] => [1,1,2] => 2 = 3 - 1
101 => [1,2,1] => [2,2] => 2 = 3 - 1
110 => [1,1,2] => [1,3] => 3 = 4 - 1
111 => [1,1,1,1] => [4] => 4 = 5 - 1
0000 => [5] => [1,1,1,1,1] => 1 = 2 - 1
0001 => [4,1] => [2,1,1,1] => 1 = 2 - 1
0010 => [3,2] => [1,2,1,1] => 1 = 2 - 1
0011 => [3,1,1] => [3,1,1] => 1 = 2 - 1
0100 => [2,3] => [1,1,2,1] => 1 = 2 - 1
0101 => [2,2,1] => [2,2,1] => 1 = 2 - 1
0110 => [2,1,2] => [1,3,1] => 1 = 2 - 1
0111 => [2,1,1,1] => [4,1] => 1 = 2 - 1
1000 => [1,4] => [1,1,1,2] => 2 = 3 - 1
1001 => [1,3,1] => [2,1,2] => 2 = 3 - 1
1010 => [1,2,2] => [1,2,2] => 2 = 3 - 1
1011 => [1,2,1,1] => [3,2] => 2 = 3 - 1
1100 => [1,1,3] => [1,1,3] => 3 = 4 - 1
1101 => [1,1,2,1] => [2,3] => 3 = 4 - 1
1110 => [1,1,1,2] => [1,4] => 4 = 5 - 1
1111 => [1,1,1,1,1] => [5] => 5 = 6 - 1
00000 => [6] => [1,1,1,1,1,1] => 1 = 2 - 1
00001 => [5,1] => [2,1,1,1,1] => 1 = 2 - 1
00010 => [4,2] => [1,2,1,1,1] => 1 = 2 - 1
00011 => [4,1,1] => [3,1,1,1] => 1 = 2 - 1
00100 => [3,3] => [1,1,2,1,1] => 1 = 2 - 1
00101 => [3,2,1] => [2,2,1,1] => 1 = 2 - 1
00110 => [3,1,2] => [1,3,1,1] => 1 = 2 - 1
00111 => [3,1,1,1] => [4,1,1] => 1 = 2 - 1
01000 => [2,4] => [1,1,1,2,1] => 1 = 2 - 1
01001 => [2,3,1] => [2,1,2,1] => 1 = 2 - 1
01010 => [2,2,2] => [1,2,2,1] => 1 = 2 - 1
01011 => [2,2,1,1] => [3,2,1] => 1 = 2 - 1
01100 => [2,1,3] => [1,1,3,1] => 1 = 2 - 1
01101 => [2,1,2,1] => [2,3,1] => 1 = 2 - 1
01110 => [2,1,1,2] => [1,4,1] => 1 = 2 - 1
01111 => [2,1,1,1,1] => [5,1] => 1 = 2 - 1
10000 => [1,5] => [1,1,1,1,2] => 2 = 3 - 1
10001 => [1,4,1] => [2,1,1,2] => 2 = 3 - 1
10010 => [1,3,2] => [1,2,1,2] => 2 = 3 - 1
10011 => [1,3,1,1] => [3,1,2] => 2 = 3 - 1
0000001 => [7,1] => [2,1,1,1,1,1,1] => ? = 2 - 1
0000011 => [6,1,1] => [3,1,1,1,1,1] => ? = 2 - 1
0000111 => [5,1,1,1] => [4,1,1,1,1] => ? = 2 - 1
0001011 => [4,2,1,1] => [3,2,1,1,1] => ? = 2 - 1
0001110 => [4,1,1,2] => [1,4,1,1,1] => ? = 2 - 1
0010011 => [3,3,1,1] => [3,1,2,1,1] => ? = 2 - 1
0010111 => [3,2,1,1,1] => [4,2,1,1] => ? = 2 - 1
0011100 => [3,1,1,3] => [1,1,4,1,1] => ? = 2 - 1
0011101 => [3,1,1,2,1] => [2,4,1,1] => ? = 2 - 1
0100011 => [2,4,1,1] => [3,1,1,2,1] => ? = 2 - 1
0111000 => [2,1,1,4] => [1,1,1,4,1] => ? = 2 - 1
0111101 => [2,1,1,1,2,1] => [2,5,1] => ? = 2 - 1
1000000 => [1,7] => [1,1,1,1,1,1,2] => ? = 3 - 1
1000011 => [1,5,1,1] => [3,1,1,1,2] => ? = 3 - 1
1000111 => [1,4,1,1,1] => [4,1,1,2] => ? = 3 - 1
1001111 => [1,3,1,1,1,1] => [5,1,2] => ? = 3 - 1
1010111 => [1,2,2,1,1,1] => [4,2,2] => ? = 3 - 1
1011100 => [1,2,1,1,3] => [1,1,4,2] => ? = 3 - 1
1011111 => [1,2,1,1,1,1,1] => [6,2] => ? = 3 - 1
1100000 => [1,1,6] => [1,1,1,1,1,3] => ? = 4 - 1
1100001 => [1,1,5,1] => [2,1,1,1,3] => ? = 4 - 1
1110101 => [1,1,1,2,2,1] => [2,2,4] => ? = 5 - 1
1111101 => [1,1,1,1,1,2,1] => [2,6] => ? = 7 - 1
1111111 => [1,1,1,1,1,1,1,1] => [8] => ? = 9 - 1
00000001 => [8,1] => [2,1,1,1,1,1,1,1] => ? = 2 - 1
00000101 => [6,2,1] => [2,2,1,1,1,1,1] => ? = 2 - 1
01000001 => [2,6,1] => [2,1,1,1,1,2,1] => ? = 2 - 1
01100000 => [2,1,6] => [1,1,1,1,1,3,1] => ? = 2 - 1
01111101 => [2,1,1,1,1,2,1] => [2,6,1] => ? = 2 - 1
10000000 => [1,8] => [1,1,1,1,1,1,1,2] => ? = 3 - 1
10000001 => [1,7,1] => [2,1,1,1,1,1,2] => ? = 3 - 1
10000010 => [1,6,2] => [1,2,1,1,1,1,2] => ? = 3 - 1
10111111 => [1,2,1,1,1,1,1,1] => [7,2] => ? = 3 - 1
11000000 => [1,1,7] => [1,1,1,1,1,1,3] => ? = 4 - 1
11111101 => [1,1,1,1,1,1,2,1] => [2,7] => ? = 8 - 1
11111111 => [1,1,1,1,1,1,1,1,1] => [9] => ? = 10 - 1
000000001 => [9,1] => [2,1,1,1,1,1,1,1,1] => ? = 2 - 1
100000000 => [1,9] => [1,1,1,1,1,1,1,1,2] => ? = 3 - 1
101111111 => [1,2,1,1,1,1,1,1,1] => [8,2] => ? = 3 - 1
110000000 => [1,1,8] => [1,1,1,1,1,1,1,3] => ? = 4 - 1
111111101 => [1,1,1,1,1,1,1,2,1] => [2,8] => ? = 9 - 1
111111111 => [1,1,1,1,1,1,1,1,1,1] => [10] => ? = 11 - 1
0000000001 => [10,1] => [2,1,1,1,1,1,1,1,1,1] => ? = 2 - 1
0111111100 => [2,1,1,1,1,1,1,3] => ? => ? = 2 - 1
Description
The last part of an integer composition.
Matching statistic: St000678
Mapping
Mp00104: Binary words —reverse⟶ Binary words
Mp00178: Binary words —to composition⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St000678: Dyck paths ⟶ ℤResult quality: 76% ●values known / values provided: 76%●distinct values known / distinct values provided: 80%
Values
0 => 0 => [2] => [1,1,0,0]
 => 1 = 2 - 1
1 => 1 => [1,1] => [1,0,1,0]
 => 2 = 3 - 1
00 => 00 => [3] => [1,1,1,0,0,0]
 => 1 = 2 - 1
01 => 10 => [1,2] => [1,0,1,1,0,0]
 => 1 = 2 - 1
10 => 01 => [2,1] => [1,1,0,0,1,0]
 => 2 = 3 - 1
11 => 11 => [1,1,1] => [1,0,1,0,1,0]
 => 3 = 4 - 1
000 => 000 => [4] => [1,1,1,1,0,0,0,0]
 => 1 = 2 - 1
001 => 100 => [1,3] => [1,0,1,1,1,0,0,0]
 => 1 = 2 - 1
010 => 010 => [2,2] => [1,1,0,0,1,1,0,0]
 => 1 = 2 - 1
011 => 110 => [1,1,2] => [1,0,1,0,1,1,0,0]
 => 1 = 2 - 1
100 => 001 => [3,1] => [1,1,1,0,0,0,1,0]
 => 2 = 3 - 1
101 => 101 => [1,2,1] => [1,0,1,1,0,0,1,0]
 => 2 = 3 - 1
110 => 011 => [2,1,1] => [1,1,0,0,1,0,1,0]
 => 3 = 4 - 1
111 => 111 => [1,1,1,1] => [1,0,1,0,1,0,1,0]
 => 4 = 5 - 1
0000 => 0000 => [5] => [1,1,1,1,1,0,0,0,0,0]
 => 1 = 2 - 1
0001 => 1000 => [1,4] => [1,0,1,1,1,1,0,0,0,0]
 => 1 = 2 - 1
0010 => 0100 => [2,3] => [1,1,0,0,1,1,1,0,0,0]
 => 1 = 2 - 1
0011 => 1100 => [1,1,3] => [1,0,1,0,1,1,1,0,0,0]
 => 1 = 2 - 1
0100 => 0010 => [3,2] => [1,1,1,0,0,0,1,1,0,0]
 => 1 = 2 - 1
0101 => 1010 => [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
 => 1 = 2 - 1
0110 => 0110 => [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
 => 1 = 2 - 1
0111 => 1110 => [1,1,1,2] => [1,0,1,0,1,0,1,1,0,0]
 => 1 = 2 - 1
1000 => 0001 => [4,1] => [1,1,1,1,0,0,0,0,1,0]
 => 2 = 3 - 1
1001 => 1001 => [1,3,1] => [1,0,1,1,1,0,0,0,1,0]
 => 2 = 3 - 1
1010 => 0101 => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
 => 2 = 3 - 1
1011 => 1101 => [1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
 => 2 = 3 - 1
1100 => 0011 => [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
 => 3 = 4 - 1
1101 => 1011 => [1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
 => 3 = 4 - 1
1110 => 0111 => [2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
 => 4 = 5 - 1
1111 => 1111 => [1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
 => 5 = 6 - 1
00000 => 00000 => [6] => [1,1,1,1,1,1,0,0,0,0,0,0]
 => 1 = 2 - 1
00001 => 10000 => [1,5] => [1,0,1,1,1,1,1,0,0,0,0,0]
 => 1 = 2 - 1
00010 => 01000 => [2,4] => [1,1,0,0,1,1,1,1,0,0,0,0]
 => 1 = 2 - 1
00011 => 11000 => [1,1,4] => [1,0,1,0,1,1,1,1,0,0,0,0]
 => 1 = 2 - 1
00100 => 00100 => [3,3] => [1,1,1,0,0,0,1,1,1,0,0,0]
 => 1 = 2 - 1
00101 => 10100 => [1,2,3] => [1,0,1,1,0,0,1,1,1,0,0,0]
 => 1 = 2 - 1
00110 => 01100 => [2,1,3] => [1,1,0,0,1,0,1,1,1,0,0,0]
 => 1 = 2 - 1
00111 => 11100 => [1,1,1,3] => [1,0,1,0,1,0,1,1,1,0,0,0]
 => 1 = 2 - 1
01000 => 00010 => [4,2] => [1,1,1,1,0,0,0,0,1,1,0,0]
 => 1 = 2 - 1
01001 => 10010 => [1,3,2] => [1,0,1,1,1,0,0,0,1,1,0,0]
 => 1 = 2 - 1
01010 => 01010 => [2,2,2] => [1,1,0,0,1,1,0,0,1,1,0,0]
 => 1 = 2 - 1
01011 => 11010 => [1,1,2,2] => [1,0,1,0,1,1,0,0,1,1,0,0]
 => 1 = 2 - 1
01100 => 00110 => [3,1,2] => [1,1,1,0,0,0,1,0,1,1,0,0]
 => 1 = 2 - 1
01101 => 10110 => [1,2,1,2] => [1,0,1,1,0,0,1,0,1,1,0,0]
 => 1 = 2 - 1
01110 => 01110 => [2,1,1,2] => [1,1,0,0,1,0,1,0,1,1,0,0]
 => 1 = 2 - 1
01111 => 11110 => [1,1,1,1,2] => [1,0,1,0,1,0,1,0,1,1,0,0]
 => 1 = 2 - 1
10000 => 00001 => [5,1] => [1,1,1,1,1,0,0,0,0,0,1,0]
 => 2 = 3 - 1
10001 => 10001 => [1,4,1] => [1,0,1,1,1,1,0,0,0,0,1,0]
 => 2 = 3 - 1
10010 => 01001 => [2,3,1] => [1,1,0,0,1,1,1,0,0,0,1,0]
 => 2 = 3 - 1
10011 => 11001 => [1,1,3,1] => [1,0,1,0,1,1,1,0,0,0,1,0]
 => 2 = 3 - 1
0000000 => 0000000 => [8] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
 => ? = 2 - 1
0000100 => 0010000 => [3,5] => [1,1,1,0,0,0,1,1,1,1,1,0,0,0,0,0]
 => ? = 2 - 1
0001000 => 0001000 => [4,4] => [1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0]
 => ? = 2 - 1
0001100 => 0011000 => [3,1,4] => [1,1,1,0,0,0,1,0,1,1,1,1,0,0,0,0]
 => ? = 2 - 1
0010000 => 0000100 => [5,3] => [1,1,1,1,1,0,0,0,0,0,1,1,1,0,0,0]
 => ? = 2 - 1
0010100 => 0010100 => [3,2,3] => [1,1,1,0,0,0,1,1,0,0,1,1,1,0,0,0]
 => ? = 2 - 1
0011000 => 0001100 => [4,1,3] => [1,1,1,1,0,0,0,0,1,0,1,1,1,0,0,0]
 => ? = 2 - 1
0011100 => 0011100 => [3,1,1,3] => [1,1,1,0,0,0,1,0,1,0,1,1,1,0,0,0]
 => ? = 2 - 1
0100000 => 0000010 => [6,2] => [1,1,1,1,1,1,0,0,0,0,0,0,1,1,0,0]
 => ? = 2 - 1
0100100 => 0010010 => [3,3,2] => [1,1,1,0,0,0,1,1,1,0,0,0,1,1,0,0]
 => ? = 2 - 1
0101000 => 0001010 => [4,2,2] => [1,1,1,1,0,0,0,0,1,1,0,0,1,1,0,0]
 => ? = 2 - 1
0101100 => 0011010 => [3,1,2,2] => [1,1,1,0,0,0,1,0,1,1,0,0,1,1,0,0]
 => ? = 2 - 1
0110000 => 0000110 => [5,1,2] => [1,1,1,1,1,0,0,0,0,0,1,0,1,1,0,0]
 => ? = 2 - 1
0110100 => 0010110 => [3,2,1,2] => [1,1,1,0,0,0,1,1,0,0,1,0,1,1,0,0]
 => ? = 2 - 1
0111000 => 0001110 => [4,1,1,2] => [1,1,1,1,0,0,0,0,1,0,1,0,1,1,0,0]
 => ? = 2 - 1
0111100 => 0011110 => [3,1,1,1,2] => [1,1,1,0,0,0,1,0,1,0,1,0,1,1,0,0]
 => ? = 2 - 1
1000000 => 0000001 => [7,1] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
 => ? = 3 - 1
1000100 => 0010001 => [3,4,1] => [1,1,1,0,0,0,1,1,1,1,0,0,0,0,1,0]
 => ? = 3 - 1
1001000 => 0001001 => [4,3,1] => [1,1,1,1,0,0,0,0,1,1,1,0,0,0,1,0]
 => ? = 3 - 1
1001100 => 0011001 => [3,1,3,1] => [1,1,1,0,0,0,1,0,1,1,1,0,0,0,1,0]
 => ? = 3 - 1
1010000 => 0000101 => [5,2,1] => [1,1,1,1,1,0,0,0,0,0,1,1,0,0,1,0]
 => ? = 3 - 1
1010100 => 0010101 => [3,2,2,1] => [1,1,1,0,0,0,1,1,0,0,1,1,0,0,1,0]
 => ? = 3 - 1
1011000 => 0001101 => [4,1,2,1] => [1,1,1,1,0,0,0,0,1,0,1,1,0,0,1,0]
 => ? = 3 - 1
1011100 => 0011101 => [3,1,1,2,1] => [1,1,1,0,0,0,1,0,1,0,1,1,0,0,1,0]
 => ? = 3 - 1
1100000 => 0000011 => [6,1,1] => [1,1,1,1,1,1,0,0,0,0,0,0,1,0,1,0]
 => ? = 4 - 1
00000001 => 10000000 => [1,8] => [1,0,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
 => ? = 2 - 1
00000101 => 10100000 => [1,2,6] => [1,0,1,1,0,0,1,1,1,1,1,1,0,0,0,0,0,0]
 => ? = 2 - 1
00111110 => 01111100 => [2,1,1,1,1,3] => [1,1,0,0,1,0,1,0,1,0,1,0,1,1,1,0,0,0]
 => ? = 2 - 1
00111111 => 11111100 => [1,1,1,1,1,1,3] => [1,0,1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,0]
 => ? = 2 - 1
01000001 => 10000010 => [1,6,2] => [1,0,1,1,1,1,1,1,0,0,0,0,0,0,1,1,0,0]
 => ? = 2 - 1
01011111 => 11111010 => [1,1,1,1,1,2,2] => [1,0,1,0,1,0,1,0,1,0,1,1,0,0,1,1,0,0]
 => ? = 2 - 1
01100000 => 00000110 => [6,1,2] => [1,1,1,1,1,1,0,0,0,0,0,0,1,0,1,1,0,0]
 => ? = 2 - 1
01111100 => 00111110 => [3,1,1,1,1,2] => [1,1,1,0,0,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => ? = 2 - 1
01111101 => 10111110 => [1,2,1,1,1,1,2] => [1,0,1,1,0,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => ? = 2 - 1
01111110 => 01111110 => [2,1,1,1,1,1,2] => [1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => ? = 2 - 1
01111111 => 11111110 => [1,1,1,1,1,1,1,2] => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => ? = 2 - 1
10000000 => 00000001 => [8,1] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0]
 => ? = 3 - 1
10000001 => 10000001 => [1,7,1] => [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
 => ? = 3 - 1
10000010 => 01000001 => [2,6,1] => [1,1,0,0,1,1,1,1,1,1,0,0,0,0,0,0,1,0]
 => ? = 3 - 1
10111110 => 01111101 => [2,1,1,1,1,2,1] => [1,1,0,0,1,0,1,0,1,0,1,0,1,1,0,0,1,0]
 => ? = 3 - 1
10111111 => 11111101 => [1,1,1,1,1,1,2,1] => [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0,1,0]
 => ? = 3 - 1
11000000 => 00000011 => [7,1,1] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0,1,0]
 => ? = 4 - 1
11111101 => 10111111 => [1,2,1,1,1,1,1,1] => [1,0,1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => ? = 8 - 1
11111110 => 01111111 => [2,1,1,1,1,1,1,1] => [1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => ? = 9 - 1
11111111 => 11111111 => [1,1,1,1,1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => ? = 10 - 1
000000001 => 100000000 => [1,9] => [1,0,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0]
 => ? = 2 - 1
001111110 => 011111100 => [2,1,1,1,1,1,3] => [1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,0]
 => ? = 2 - 1
001111111 => 111111100 => [1,1,1,1,1,1,1,3] => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,0]
 => ? = 2 - 1
011111100 => 001111110 => [3,1,1,1,1,1,2] => [1,1,1,0,0,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => ? = 2 - 1
011111110 => 011111110 => [2,1,1,1,1,1,1,2] => [1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => ? = 2 - 1
Description
The number of up steps after the last double rise of a Dyck path.
Matching statistic: St000617
Mapping
Mp00178: Binary words —to composition⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
Mp00132: Dyck paths —switch returns and last double rise⟶ Dyck paths
St000617: Dyck paths ⟶ ℤResult quality: 61% ●values known / values provided: 61%●distinct values known / distinct values provided: 100%
Values
0 => [2] => [1,1,0,0]
 => [1,1,0,0]
 => 1 = 2 - 1
1 => [1,1] => [1,0,1,0]
 => [1,0,1,0]
 => 2 = 3 - 1
00 => [3] => [1,1,1,0,0,0]
 => [1,1,1,0,0,0]
 => 1 = 2 - 1
01 => [2,1] => [1,1,0,0,1,0]
 => [1,1,0,0,1,0]
 => 1 = 2 - 1
10 => [1,2] => [1,0,1,1,0,0]
 => [1,1,0,1,0,0]
 => 2 = 3 - 1
11 => [1,1,1] => [1,0,1,0,1,0]
 => [1,0,1,0,1,0]
 => 3 = 4 - 1
000 => [4] => [1,1,1,1,0,0,0,0]
 => [1,1,1,1,0,0,0,0]
 => 1 = 2 - 1
001 => [3,1] => [1,1,1,0,0,0,1,0]
 => [1,1,1,0,0,0,1,0]
 => 1 = 2 - 1
010 => [2,2] => [1,1,0,0,1,1,0,0]
 => [1,1,1,0,0,1,0,0]
 => 1 = 2 - 1
011 => [2,1,1] => [1,1,0,0,1,0,1,0]
 => [1,1,0,0,1,0,1,0]
 => 1 = 2 - 1
100 => [1,3] => [1,0,1,1,1,0,0,0]
 => [1,1,1,0,1,0,0,0]
 => 2 = 3 - 1
101 => [1,2,1] => [1,0,1,1,0,0,1,0]
 => [1,1,0,1,0,0,1,0]
 => 2 = 3 - 1
110 => [1,1,2] => [1,0,1,0,1,1,0,0]
 => [1,1,0,1,0,1,0,0]
 => 3 = 4 - 1
111 => [1,1,1,1] => [1,0,1,0,1,0,1,0]
 => [1,0,1,0,1,0,1,0]
 => 4 = 5 - 1
0000 => [5] => [1,1,1,1,1,0,0,0,0,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => 1 = 2 - 1
0001 => [4,1] => [1,1,1,1,0,0,0,0,1,0]
 => [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]
 => [1,1,1,1,0,0,0,1,0,0]
 => 1 = 2 - 1
0011 => [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
 => [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]
 => [1,1,1,1,0,0,1,0,0,0]
 => 1 = 2 - 1
0101 => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
 => [1,1,1,0,0,1,0,0,1,0]
 => 1 = 2 - 1
0110 => [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
 => [1,1,1,0,0,1,0,1,0,0]
 => 1 = 2 - 1
0111 => [2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
 => [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]
 => [1,1,1,1,0,1,0,0,0,0]
 => 2 = 3 - 1
1001 => [1,3,1] => [1,0,1,1,1,0,0,0,1,0]
 => [1,1,1,0,1,0,0,0,1,0]
 => 2 = 3 - 1
1010 => [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
 => [1,1,1,0,1,0,0,1,0,0]
 => 2 = 3 - 1
1011 => [1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
 => [1,1,0,1,0,0,1,0,1,0]
 => 2 = 3 - 1
1100 => [1,1,3] => [1,0,1,0,1,1,1,0,0,0]
 => [1,1,1,0,1,0,1,0,0,0]
 => 3 = 4 - 1
1101 => [1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
 => [1,1,0,1,0,1,0,0,1,0]
 => 3 = 4 - 1
1110 => [1,1,1,2] => [1,0,1,0,1,0,1,1,0,0]
 => [1,1,0,1,0,1,0,1,0,0]
 => 4 = 5 - 1
1111 => [1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => 5 = 6 - 1
00000 => [6] => [1,1,1,1,1,1,0,0,0,0,0,0]
 => [1,1,1,1,1,1,0,0,0,0,0,0]
 => 1 = 2 - 1
00001 => [5,1] => [1,1,1,1,1,0,0,0,0,0,1,0]
 => [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]
 => [1,1,1,1,1,0,0,0,0,1,0,0]
 => 1 = 2 - 1
00011 => [4,1,1] => [1,1,1,1,0,0,0,0,1,0,1,0]
 => [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]
 => [1,1,1,1,1,0,0,0,1,0,0,0]
 => 1 = 2 - 1
00101 => [3,2,1] => [1,1,1,0,0,0,1,1,0,0,1,0]
 => [1,1,1,1,0,0,0,1,0,0,1,0]
 => 1 = 2 - 1
00110 => [3,1,2] => [1,1,1,0,0,0,1,0,1,1,0,0]
 => [1,1,1,1,0,0,0,1,0,1,0,0]
 => 1 = 2 - 1
00111 => [3,1,1,1] => [1,1,1,0,0,0,1,0,1,0,1,0]
 => [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]
 => [1,1,1,1,1,0,0,1,0,0,0,0]
 => 1 = 2 - 1
01001 => [2,3,1] => [1,1,0,0,1,1,1,0,0,0,1,0]
 => [1,1,1,1,0,0,1,0,0,0,1,0]
 => 1 = 2 - 1
01010 => [2,2,2] => [1,1,0,0,1,1,0,0,1,1,0,0]
 => [1,1,1,1,0,0,1,0,0,1,0,0]
 => 1 = 2 - 1
01011 => [2,2,1,1] => [1,1,0,0,1,1,0,0,1,0,1,0]
 => [1,1,1,0,0,1,0,0,1,0,1,0]
 => 1 = 2 - 1
01100 => [2,1,3] => [1,1,0,0,1,0,1,1,1,0,0,0]
 => [1,1,1,1,0,0,1,0,1,0,0,0]
 => 1 = 2 - 1
01101 => [2,1,2,1] => [1,1,0,0,1,0,1,1,0,0,1,0]
 => [1,1,1,0,0,1,0,1,0,0,1,0]
 => 1 = 2 - 1
01110 => [2,1,1,2] => [1,1,0,0,1,0,1,0,1,1,0,0]
 => [1,1,1,0,0,1,0,1,0,1,0,0]
 => 1 = 2 - 1
01111 => [2,1,1,1,1] => [1,1,0,0,1,0,1,0,1,0,1,0]
 => [1,1,0,0,1,0,1,0,1,0,1,0]
 => 1 = 2 - 1
10000 => [1,5] => [1,0,1,1,1,1,1,0,0,0,0,0]
 => [1,1,1,1,1,0,1,0,0,0,0,0]
 => 2 = 3 - 1
10001 => [1,4,1] => [1,0,1,1,1,1,0,0,0,0,1,0]
 => [1,1,1,1,0,1,0,0,0,0,1,0]
 => 2 = 3 - 1
10010 => [1,3,2] => [1,0,1,1,1,0,0,0,1,1,0,0]
 => [1,1,1,1,0,1,0,0,0,1,0,0]
 => 2 = 3 - 1
10011 => [1,3,1,1] => [1,0,1,1,1,0,0,0,1,0,1,0]
 => [1,1,1,0,1,0,0,0,1,0,1,0]
 => 2 = 3 - 1
000011 => [5,1,1] => [1,1,1,1,1,0,0,0,0,0,1,0,1,0]
 => [1,1,1,1,1,0,0,0,0,0,1,0,1,0]
 => ? = 2 - 1
000111 => [4,1,1,1] => [1,1,1,1,0,0,0,0,1,0,1,0,1,0]
 => [1,1,1,1,0,0,0,0,1,0,1,0,1,0]
 => ? = 2 - 1
001011 => [3,2,1,1] => [1,1,1,0,0,0,1,1,0,0,1,0,1,0]
 => [1,1,1,1,0,0,0,1,0,0,1,0,1,0]
 => ? = 2 - 1
001100 => [3,1,3] => [1,1,1,0,0,0,1,0,1,1,1,0,0,0]
 => [1,1,1,1,1,0,0,0,1,0,1,0,0,0]
 => ? = 2 - 1
001101 => [3,1,2,1] => [1,1,1,0,0,0,1,0,1,1,0,0,1,0]
 => [1,1,1,1,0,0,0,1,0,1,0,0,1,0]
 => ? = 2 - 1
001110 => [3,1,1,2] => [1,1,1,0,0,0,1,0,1,0,1,1,0,0]
 => [1,1,1,1,0,0,0,1,0,1,0,1,0,0]
 => ? = 2 - 1
010011 => [2,3,1,1] => [1,1,0,0,1,1,1,0,0,0,1,0,1,0]
 => [1,1,1,1,0,0,1,0,0,0,1,0,1,0]
 => ? = 2 - 1
010110 => [2,2,1,2] => [1,1,0,0,1,1,0,0,1,0,1,1,0,0]
 => [1,1,1,1,0,0,1,0,0,1,0,1,0,0]
 => ? = 2 - 1
011000 => [2,1,4] => [1,1,0,0,1,0,1,1,1,1,0,0,0,0]
 => [1,1,1,1,1,0,0,1,0,1,0,0,0,0]
 => ? = 2 - 1
011001 => [2,1,3,1] => [1,1,0,0,1,0,1,1,1,0,0,0,1,0]
 => [1,1,1,1,0,0,1,0,1,0,0,0,1,0]
 => ? = 2 - 1
100011 => [1,4,1,1] => [1,0,1,1,1,1,0,0,0,0,1,0,1,0]
 => [1,1,1,1,0,1,0,0,0,0,1,0,1,0]
 => ? = 3 - 1
100101 => [1,3,2,1] => [1,0,1,1,1,0,0,0,1,1,0,0,1,0]
 => [1,1,1,1,0,1,0,0,0,1,0,0,1,0]
 => ? = 3 - 1
100110 => [1,3,1,2] => [1,0,1,1,1,0,0,0,1,0,1,1,0,0]
 => [1,1,1,1,0,1,0,0,0,1,0,1,0,0]
 => ? = 3 - 1
101011 => [1,2,2,1,1] => [1,0,1,1,0,0,1,1,0,0,1,0,1,0]
 => [1,1,1,0,1,0,0,1,0,0,1,0,1,0]
 => ? = 3 - 1
101100 => [1,2,1,3] => [1,0,1,1,0,0,1,0,1,1,1,0,0,0]
 => [1,1,1,1,0,1,0,0,1,0,1,0,0,0]
 => ? = 3 - 1
101101 => [1,2,1,2,1] => [1,0,1,1,0,0,1,0,1,1,0,0,1,0]
 => [1,1,1,0,1,0,0,1,0,1,0,0,1,0]
 => ? = 3 - 1
110100 => [1,1,2,3] => [1,0,1,0,1,1,0,0,1,1,1,0,0,0]
 => [1,1,1,1,0,1,0,1,0,0,1,0,0,0]
 => ? = 4 - 1
110101 => [1,1,2,2,1] => [1,0,1,0,1,1,0,0,1,1,0,0,1,0]
 => [1,1,1,0,1,0,1,0,0,1,0,0,1,0]
 => ? = 4 - 1
111111 => [1,1,1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => ? = 8 - 1
0000011 => [6,1,1] => [1,1,1,1,1,1,0,0,0,0,0,0,1,0,1,0]
 => [1,1,1,1,1,1,0,0,0,0,0,0,1,0,1,0]
 => ? = 2 - 1
0000101 => [5,2,1] => [1,1,1,1,1,0,0,0,0,0,1,1,0,0,1,0]
 => [1,1,1,1,1,1,0,0,0,0,0,1,0,0,1,0]
 => ? = 2 - 1
0000110 => [5,1,2] => [1,1,1,1,1,0,0,0,0,0,1,0,1,1,0,0]
 => [1,1,1,1,1,1,0,0,0,0,0,1,0,1,0,0]
 => ? = 2 - 1
0000111 => [5,1,1,1] => [1,1,1,1,1,0,0,0,0,0,1,0,1,0,1,0]
 => [1,1,1,1,1,0,0,0,0,0,1,0,1,0,1,0]
 => ? = 2 - 1
0001001 => [4,3,1] => [1,1,1,1,0,0,0,0,1,1,1,0,0,0,1,0]
 => [1,1,1,1,1,1,0,0,0,0,1,0,0,0,1,0]
 => ? = 2 - 1
0001010 => [4,2,2] => [1,1,1,1,0,0,0,0,1,1,0,0,1,1,0,0]
 => [1,1,1,1,1,1,0,0,0,0,1,0,0,1,0,0]
 => ? = 2 - 1
0001011 => [4,2,1,1] => [1,1,1,1,0,0,0,0,1,1,0,0,1,0,1,0]
 => [1,1,1,1,1,0,0,0,0,1,0,0,1,0,1,0]
 => ? = 2 - 1
0001100 => [4,1,3] => [1,1,1,1,0,0,0,0,1,0,1,1,1,0,0,0]
 => [1,1,1,1,1,1,0,0,0,0,1,0,1,0,0,0]
 => ? = 2 - 1
0001101 => [4,1,2,1] => [1,1,1,1,0,0,0,0,1,0,1,1,0,0,1,0]
 => [1,1,1,1,1,0,0,0,0,1,0,1,0,0,1,0]
 => ? = 2 - 1
0001110 => [4,1,1,2] => [1,1,1,1,0,0,0,0,1,0,1,0,1,1,0,0]
 => [1,1,1,1,1,0,0,0,0,1,0,1,0,1,0,0]
 => ? = 2 - 1
0001111 => [4,1,1,1,1] => [1,1,1,1,0,0,0,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,0,0,0,0,1,0,1,0,1,0,1,0]
 => ? = 2 - 1
0010011 => [3,3,1,1] => [1,1,1,0,0,0,1,1,1,0,0,0,1,0,1,0]
 => [1,1,1,1,1,0,0,0,1,0,0,0,1,0,1,0]
 => ? = 2 - 1
0010100 => [3,2,3] => [1,1,1,0,0,0,1,1,0,0,1,1,1,0,0,0]
 => [1,1,1,1,1,1,0,0,0,1,0,0,1,0,0,0]
 => ? = 2 - 1
0010101 => [3,2,2,1] => [1,1,1,0,0,0,1,1,0,0,1,1,0,0,1,0]
 => [1,1,1,1,1,0,0,0,1,0,0,1,0,0,1,0]
 => ? = 2 - 1
0010110 => [3,2,1,2] => [1,1,1,0,0,0,1,1,0,0,1,0,1,1,0,0]
 => [1,1,1,1,1,0,0,0,1,0,0,1,0,1,0,0]
 => ? = 2 - 1
0010111 => [3,2,1,1,1] => [1,1,1,0,0,0,1,1,0,0,1,0,1,0,1,0]
 => [1,1,1,1,0,0,0,1,0,0,1,0,1,0,1,0]
 => ? = 2 - 1
0011001 => [3,1,3,1] => [1,1,1,0,0,0,1,0,1,1,1,0,0,0,1,0]
 => [1,1,1,1,1,0,0,0,1,0,1,0,0,0,1,0]
 => ? = 2 - 1
0011010 => [3,1,2,2] => [1,1,1,0,0,0,1,0,1,1,0,0,1,1,0,0]
 => [1,1,1,1,1,0,0,0,1,0,1,0,0,1,0,0]
 => ? = 2 - 1
0011011 => [3,1,2,1,1] => [1,1,1,0,0,0,1,0,1,1,0,0,1,0,1,0]
 => [1,1,1,1,0,0,0,1,0,1,0,0,1,0,1,0]
 => ? = 2 - 1
0011100 => [3,1,1,3] => [1,1,1,0,0,0,1,0,1,0,1,1,1,0,0,0]
 => [1,1,1,1,1,0,0,0,1,0,1,0,1,0,0,0]
 => ? = 2 - 1
0011101 => [3,1,1,2,1] => [1,1,1,0,0,0,1,0,1,0,1,1,0,0,1,0]
 => [1,1,1,1,0,0,0,1,0,1,0,1,0,0,1,0]
 => ? = 2 - 1
0011110 => [3,1,1,1,2] => [1,1,1,0,0,0,1,0,1,0,1,0,1,1,0,0]
 => [1,1,1,1,0,0,0,1,0,1,0,1,0,1,0,0]
 => ? = 2 - 1
0100011 => [2,4,1,1] => [1,1,0,0,1,1,1,1,0,0,0,0,1,0,1,0]
 => [1,1,1,1,1,0,0,1,0,0,0,0,1,0,1,0]
 => ? = 2 - 1
0100100 => [2,3,3] => [1,1,0,0,1,1,1,0,0,0,1,1,1,0,0,0]
 => [1,1,1,1,1,1,0,0,1,0,0,0,1,0,0,0]
 => ? = 2 - 1
0100101 => [2,3,2,1] => [1,1,0,0,1,1,1,0,0,0,1,1,0,0,1,0]
 => [1,1,1,1,1,0,0,1,0,0,0,1,0,0,1,0]
 => ? = 2 - 1
0100110 => [2,3,1,2] => [1,1,0,0,1,1,1,0,0,0,1,0,1,1,0,0]
 => [1,1,1,1,1,0,0,1,0,0,0,1,0,1,0,0]
 => ? = 2 - 1
0100111 => [2,3,1,1,1] => [1,1,0,0,1,1,1,0,0,0,1,0,1,0,1,0]
 => [1,1,1,1,0,0,1,0,0,0,1,0,1,0,1,0]
 => ? = 2 - 1
0101001 => [2,2,3,1] => [1,1,0,0,1,1,0,0,1,1,1,0,0,0,1,0]
 => [1,1,1,1,1,0,0,1,0,0,1,0,0,0,1,0]
 => ? = 2 - 1
0101010 => [2,2,2,2] => [1,1,0,0,1,1,0,0,1,1,0,0,1,1,0,0]
 => [1,1,1,1,1,0,0,1,0,0,1,0,0,1,0,0]
 => ? = 2 - 1
0101011 => [2,2,2,1,1] => [1,1,0,0,1,1,0,0,1,1,0,0,1,0,1,0]
 => [1,1,1,1,0,0,1,0,0,1,0,0,1,0,1,0]
 => ? = 2 - 1
0101100 => [2,2,1,3] => [1,1,0,0,1,1,0,0,1,0,1,1,1,0,0,0]
 => [1,1,1,1,1,0,0,1,0,0,1,0,1,0,0,0]
 => ? = 2 - 1
Description
The number of global maxima of a Dyck path.
Matching statistic: St000759
Mapping
Mp00178: Binary words —to composition⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
Mp00027: Dyck paths —to partition⟶ Integer partitions
St000759: Integer partitions ⟶ ℤResult quality: 55% ●values known / values provided: 55%●distinct values known / distinct values provided: 100%
Values
0 => [2] => [1,1,0,0]
 => []
 => 1 = 2 - 1
1 => [1,1] => [1,0,1,0]
 => [1]
 => 2 = 3 - 1
00 => [3] => [1,1,1,0,0,0]
 => []
 => 1 = 2 - 1
01 => [2,1] => [1,1,0,0,1,0]
 => [2]
 => 1 = 2 - 1
10 => [1,2] => [1,0,1,1,0,0]
 => [1,1]
 => 2 = 3 - 1
11 => [1,1,1] => [1,0,1,0,1,0]
 => [2,1]
 => 3 = 4 - 1
000 => [4] => [1,1,1,1,0,0,0,0]
 => []
 => 1 = 2 - 1
001 => [3,1] => [1,1,1,0,0,0,1,0]
 => [3]
 => 1 = 2 - 1
010 => [2,2] => [1,1,0,0,1,1,0,0]
 => [2,2]
 => 1 = 2 - 1
011 => [2,1,1] => [1,1,0,0,1,0,1,0]
 => [3,2]
 => 1 = 2 - 1
100 => [1,3] => [1,0,1,1,1,0,0,0]
 => [1,1,1]
 => 2 = 3 - 1
101 => [1,2,1] => [1,0,1,1,0,0,1,0]
 => [3,1,1]
 => 2 = 3 - 1
110 => [1,1,2] => [1,0,1,0,1,1,0,0]
 => [2,2,1]
 => 3 = 4 - 1
111 => [1,1,1,1] => [1,0,1,0,1,0,1,0]
 => [3,2,1]
 => 4 = 5 - 1
0000 => [5] => [1,1,1,1,1,0,0,0,0,0]
 => []
 => 1 = 2 - 1
0001 => [4,1] => [1,1,1,1,0,0,0,0,1,0]
 => [4]
 => 1 = 2 - 1
0010 => [3,2] => [1,1,1,0,0,0,1,1,0,0]
 => [3,3]
 => 1 = 2 - 1
0011 => [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
 => [4,3]
 => 1 = 2 - 1
0100 => [2,3] => [1,1,0,0,1,1,1,0,0,0]
 => [2,2,2]
 => 1 = 2 - 1
0101 => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
 => [4,2,2]
 => 1 = 2 - 1
0110 => [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
 => [3,3,2]
 => 1 = 2 - 1
0111 => [2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
 => [4,3,2]
 => 1 = 2 - 1
1000 => [1,4] => [1,0,1,1,1,1,0,0,0,0]
 => [1,1,1,1]
 => 2 = 3 - 1
1001 => [1,3,1] => [1,0,1,1,1,0,0,0,1,0]
 => [4,1,1,1]
 => 2 = 3 - 1
1010 => [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
 => [3,3,1,1]
 => 2 = 3 - 1
1011 => [1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
 => [4,3,1,1]
 => 2 = 3 - 1
1100 => [1,1,3] => [1,0,1,0,1,1,1,0,0,0]
 => [2,2,2,1]
 => 3 = 4 - 1
1101 => [1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
 => [4,2,2,1]
 => 3 = 4 - 1
1110 => [1,1,1,2] => [1,0,1,0,1,0,1,1,0,0]
 => [3,3,2,1]
 => 4 = 5 - 1
1111 => [1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
 => [4,3,2,1]
 => 5 = 6 - 1
00000 => [6] => [1,1,1,1,1,1,0,0,0,0,0,0]
 => []
 => 1 = 2 - 1
00001 => [5,1] => [1,1,1,1,1,0,0,0,0,0,1,0]
 => [5]
 => 1 = 2 - 1
00010 => [4,2] => [1,1,1,1,0,0,0,0,1,1,0,0]
 => [4,4]
 => 1 = 2 - 1
00011 => [4,1,1] => [1,1,1,1,0,0,0,0,1,0,1,0]
 => [5,4]
 => 1 = 2 - 1
00100 => [3,3] => [1,1,1,0,0,0,1,1,1,0,0,0]
 => [3,3,3]
 => 1 = 2 - 1
00101 => [3,2,1] => [1,1,1,0,0,0,1,1,0,0,1,0]
 => [5,3,3]
 => 1 = 2 - 1
00110 => [3,1,2] => [1,1,1,0,0,0,1,0,1,1,0,0]
 => [4,4,3]
 => 1 = 2 - 1
00111 => [3,1,1,1] => [1,1,1,0,0,0,1,0,1,0,1,0]
 => [5,4,3]
 => 1 = 2 - 1
01000 => [2,4] => [1,1,0,0,1,1,1,1,0,0,0,0]
 => [2,2,2,2]
 => 1 = 2 - 1
01001 => [2,3,1] => [1,1,0,0,1,1,1,0,0,0,1,0]
 => [5,2,2,2]
 => 1 = 2 - 1
01010 => [2,2,2] => [1,1,0,0,1,1,0,0,1,1,0,0]
 => [4,4,2,2]
 => 1 = 2 - 1
01011 => [2,2,1,1] => [1,1,0,0,1,1,0,0,1,0,1,0]
 => [5,4,2,2]
 => 1 = 2 - 1
01100 => [2,1,3] => [1,1,0,0,1,0,1,1,1,0,0,0]
 => [3,3,3,2]
 => 1 = 2 - 1
01101 => [2,1,2,1] => [1,1,0,0,1,0,1,1,0,0,1,0]
 => [5,3,3,2]
 => 1 = 2 - 1
01110 => [2,1,1,2] => [1,1,0,0,1,0,1,0,1,1,0,0]
 => [4,4,3,2]
 => 1 = 2 - 1
01111 => [2,1,1,1,1] => [1,1,0,0,1,0,1,0,1,0,1,0]
 => [5,4,3,2]
 => 1 = 2 - 1
10000 => [1,5] => [1,0,1,1,1,1,1,0,0,0,0,0]
 => [1,1,1,1,1]
 => 2 = 3 - 1
10001 => [1,4,1] => [1,0,1,1,1,1,0,0,0,0,1,0]
 => [5,1,1,1,1]
 => 2 = 3 - 1
10010 => [1,3,2] => [1,0,1,1,1,0,0,0,1,1,0,0]
 => [4,4,1,1,1]
 => 2 = 3 - 1
10011 => [1,3,1,1] => [1,0,1,1,1,0,0,0,1,0,1,0]
 => [5,4,1,1,1]
 => 2 = 3 - 1
0000011 => [6,1,1] => [1,1,1,1,1,1,0,0,0,0,0,0,1,0,1,0]
 => [7,6]
 => ? = 2 - 1
0000101 => [5,2,1] => [1,1,1,1,1,0,0,0,0,0,1,1,0,0,1,0]
 => [7,5,5]
 => ? = 2 - 1
0000110 => [5,1,2] => [1,1,1,1,1,0,0,0,0,0,1,0,1,1,0,0]
 => [6,6,5]
 => ? = 2 - 1
0000111 => [5,1,1,1] => [1,1,1,1,1,0,0,0,0,0,1,0,1,0,1,0]
 => [7,6,5]
 => ? = 2 - 1
0001001 => [4,3,1] => [1,1,1,1,0,0,0,0,1,1,1,0,0,0,1,0]
 => [7,4,4,4]
 => ? = 2 - 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 - 1
0001011 => [4,2,1,1] => [1,1,1,1,0,0,0,0,1,1,0,0,1,0,1,0]
 => [7,6,4,4]
 => ? = 2 - 1
0001100 => [4,1,3] => [1,1,1,1,0,0,0,0,1,0,1,1,1,0,0,0]
 => [5,5,5,4]
 => ? = 2 - 1
0001101 => [4,1,2,1] => [1,1,1,1,0,0,0,0,1,0,1,1,0,0,1,0]
 => [7,5,5,4]
 => ? = 2 - 1
0001110 => [4,1,1,2] => [1,1,1,1,0,0,0,0,1,0,1,0,1,1,0,0]
 => [6,6,5,4]
 => ? = 2 - 1
0001111 => [4,1,1,1,1] => [1,1,1,1,0,0,0,0,1,0,1,0,1,0,1,0]
 => [7,6,5,4]
 => ? = 2 - 1
0010001 => [3,4,1] => [1,1,1,0,0,0,1,1,1,1,0,0,0,0,1,0]
 => [7,3,3,3,3]
 => ? = 2 - 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 - 1
0010011 => [3,3,1,1] => [1,1,1,0,0,0,1,1,1,0,0,0,1,0,1,0]
 => [7,6,3,3,3]
 => ? = 2 - 1
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 - 1
0010101 => [3,2,2,1] => [1,1,1,0,0,0,1,1,0,0,1,1,0,0,1,0]
 => [7,5,5,3,3]
 => ? = 2 - 1
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 - 1
0010111 => [3,2,1,1,1] => [1,1,1,0,0,0,1,1,0,0,1,0,1,0,1,0]
 => [7,6,5,3,3]
 => ? = 2 - 1
0011000 => [3,1,4] => [1,1,1,0,0,0,1,0,1,1,1,1,0,0,0,0]
 => [4,4,4,4,3]
 => ? = 2 - 1
0011001 => [3,1,3,1] => [1,1,1,0,0,0,1,0,1,1,1,0,0,0,1,0]
 => [7,4,4,4,3]
 => ? = 2 - 1
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 - 1
0011011 => [3,1,2,1,1] => [1,1,1,0,0,0,1,0,1,1,0,0,1,0,1,0]
 => [7,6,4,4,3]
 => ? = 2 - 1
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]
 => ? = 2 - 1
0011101 => [3,1,1,2,1] => [1,1,1,0,0,0,1,0,1,0,1,1,0,0,1,0]
 => [7,5,5,4,3]
 => ? = 2 - 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]
 => ? = 2 - 1
0011111 => [3,1,1,1,1,1] => [1,1,1,0,0,0,1,0,1,0,1,0,1,0,1,0]
 => [7,6,5,4,3]
 => ? = 2 - 1
0100001 => [2,5,1] => [1,1,0,0,1,1,1,1,1,0,0,0,0,0,1,0]
 => [7,2,2,2,2,2]
 => ? = 2 - 1
0100010 => [2,4,2] => [1,1,0,0,1,1,1,1,0,0,0,0,1,1,0,0]
 => [6,6,2,2,2,2]
 => ? = 2 - 1
0100011 => [2,4,1,1] => [1,1,0,0,1,1,1,1,0,0,0,0,1,0,1,0]
 => [7,6,2,2,2,2]
 => ? = 2 - 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 - 1
0100101 => [2,3,2,1] => [1,1,0,0,1,1,1,0,0,0,1,1,0,0,1,0]
 => [7,5,5,2,2,2]
 => ? = 2 - 1
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 - 1
0100111 => [2,3,1,1,1] => [1,1,0,0,1,1,1,0,0,0,1,0,1,0,1,0]
 => [7,6,5,2,2,2]
 => ? = 2 - 1
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 - 1
0101001 => [2,2,3,1] => [1,1,0,0,1,1,0,0,1,1,1,0,0,0,1,0]
 => [7,4,4,4,2,2]
 => ? = 2 - 1
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]
 => ? = 2 - 1
0101011 => [2,2,2,1,1] => [1,1,0,0,1,1,0,0,1,1,0,0,1,0,1,0]
 => [7,6,4,4,2,2]
 => ? = 2 - 1
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 - 1
0101101 => [2,2,1,2,1] => [1,1,0,0,1,1,0,0,1,0,1,1,0,0,1,0]
 => [7,5,5,4,2,2]
 => ? = 2 - 1
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 - 1
0101111 => [2,2,1,1,1,1] => [1,1,0,0,1,1,0,0,1,0,1,0,1,0,1,0]
 => [7,6,5,4,2,2]
 => ? = 2 - 1
0110000 => [2,1,5] => [1,1,0,0,1,0,1,1,1,1,1,0,0,0,0,0]
 => [3,3,3,3,3,2]
 => ? = 2 - 1
0110001 => [2,1,4,1] => [1,1,0,0,1,0,1,1,1,1,0,0,0,0,1,0]
 => [7,3,3,3,3,2]
 => ? = 2 - 1
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 - 1
0110011 => [2,1,3,1,1] => [1,1,0,0,1,0,1,1,1,0,0,0,1,0,1,0]
 => [7,6,3,3,3,2]
 => ? = 2 - 1
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 - 1
0110101 => [2,1,2,2,1] => [1,1,0,0,1,0,1,1,0,0,1,1,0,0,1,0]
 => [7,5,5,3,3,2]
 => ? = 2 - 1
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 - 1
0110111 => [2,1,2,1,1,1] => [1,1,0,0,1,0,1,1,0,0,1,0,1,0,1,0]
 => [7,6,5,3,3,2]
 => ? = 2 - 1
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]
 => ? = 2 - 1
Description
The smallest missing part in an integer partition. In [3], this is referred to as the mex, the minimal excluded part of the partition. For compositions, this is studied in [sec.3.2., 1].
Matching statistic: St001733
Mapping
Mp00178: Binary words —to composition⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
Mp00132: Dyck paths —switch returns and last double rise⟶ Dyck paths
St001733: Dyck paths ⟶ ℤResult quality: 48% ●values known / values provided: 48%●distinct values known / distinct values provided: 80%
Values
0 => [2] => [1,1,0,0]
 => [1,1,0,0]
 => 1 = 2 - 1
1 => [1,1] => [1,0,1,0]
 => [1,0,1,0]
 => 2 = 3 - 1
00 => [3] => [1,1,1,0,0,0]
 => [1,1,1,0,0,0]
 => 1 = 2 - 1
01 => [2,1] => [1,1,0,0,1,0]
 => [1,1,0,0,1,0]
 => 1 = 2 - 1
10 => [1,2] => [1,0,1,1,0,0]
 => [1,1,0,1,0,0]
 => 2 = 3 - 1
11 => [1,1,1] => [1,0,1,0,1,0]
 => [1,0,1,0,1,0]
 => 3 = 4 - 1
000 => [4] => [1,1,1,1,0,0,0,0]
 => [1,1,1,1,0,0,0,0]
 => 1 = 2 - 1
001 => [3,1] => [1,1,1,0,0,0,1,0]
 => [1,1,1,0,0,0,1,0]
 => 1 = 2 - 1
010 => [2,2] => [1,1,0,0,1,1,0,0]
 => [1,1,1,0,0,1,0,0]
 => 1 = 2 - 1
011 => [2,1,1] => [1,1,0,0,1,0,1,0]
 => [1,1,0,0,1,0,1,0]
 => 1 = 2 - 1
100 => [1,3] => [1,0,1,1,1,0,0,0]
 => [1,1,1,0,1,0,0,0]
 => 2 = 3 - 1
101 => [1,2,1] => [1,0,1,1,0,0,1,0]
 => [1,1,0,1,0,0,1,0]
 => 2 = 3 - 1
110 => [1,1,2] => [1,0,1,0,1,1,0,0]
 => [1,1,0,1,0,1,0,0]
 => 3 = 4 - 1
111 => [1,1,1,1] => [1,0,1,0,1,0,1,0]
 => [1,0,1,0,1,0,1,0]
 => 4 = 5 - 1
0000 => [5] => [1,1,1,1,1,0,0,0,0,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => 1 = 2 - 1
0001 => [4,1] => [1,1,1,1,0,0,0,0,1,0]
 => [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]
 => [1,1,1,1,0,0,0,1,0,0]
 => 1 = 2 - 1
0011 => [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
 => [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]
 => [1,1,1,1,0,0,1,0,0,0]
 => 1 = 2 - 1
0101 => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
 => [1,1,1,0,0,1,0,0,1,0]
 => 1 = 2 - 1
0110 => [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
 => [1,1,1,0,0,1,0,1,0,0]
 => 1 = 2 - 1
0111 => [2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
 => [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]
 => [1,1,1,1,0,1,0,0,0,0]
 => 2 = 3 - 1
1001 => [1,3,1] => [1,0,1,1,1,0,0,0,1,0]
 => [1,1,1,0,1,0,0,0,1,0]
 => 2 = 3 - 1
1010 => [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
 => [1,1,1,0,1,0,0,1,0,0]
 => 2 = 3 - 1
1011 => [1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
 => [1,1,0,1,0,0,1,0,1,0]
 => 2 = 3 - 1
1100 => [1,1,3] => [1,0,1,0,1,1,1,0,0,0]
 => [1,1,1,0,1,0,1,0,0,0]
 => 3 = 4 - 1
1101 => [1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
 => [1,1,0,1,0,1,0,0,1,0]
 => 3 = 4 - 1
1110 => [1,1,1,2] => [1,0,1,0,1,0,1,1,0,0]
 => [1,1,0,1,0,1,0,1,0,0]
 => 4 = 5 - 1
1111 => [1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => 5 = 6 - 1
00000 => [6] => [1,1,1,1,1,1,0,0,0,0,0,0]
 => [1,1,1,1,1,1,0,0,0,0,0,0]
 => 1 = 2 - 1
00001 => [5,1] => [1,1,1,1,1,0,0,0,0,0,1,0]
 => [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]
 => [1,1,1,1,1,0,0,0,0,1,0,0]
 => 1 = 2 - 1
00011 => [4,1,1] => [1,1,1,1,0,0,0,0,1,0,1,0]
 => [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]
 => [1,1,1,1,1,0,0,0,1,0,0,0]
 => 1 = 2 - 1
00101 => [3,2,1] => [1,1,1,0,0,0,1,1,0,0,1,0]
 => [1,1,1,1,0,0,0,1,0,0,1,0]
 => 1 = 2 - 1
00110 => [3,1,2] => [1,1,1,0,0,0,1,0,1,1,0,0]
 => [1,1,1,1,0,0,0,1,0,1,0,0]
 => 1 = 2 - 1
00111 => [3,1,1,1] => [1,1,1,0,0,0,1,0,1,0,1,0]
 => [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]
 => [1,1,1,1,1,0,0,1,0,0,0,0]
 => 1 = 2 - 1
01001 => [2,3,1] => [1,1,0,0,1,1,1,0,0,0,1,0]
 => [1,1,1,1,0,0,1,0,0,0,1,0]
 => 1 = 2 - 1
01010 => [2,2,2] => [1,1,0,0,1,1,0,0,1,1,0,0]
 => [1,1,1,1,0,0,1,0,0,1,0,0]
 => 1 = 2 - 1
01011 => [2,2,1,1] => [1,1,0,0,1,1,0,0,1,0,1,0]
 => [1,1,1,0,0,1,0,0,1,0,1,0]
 => 1 = 2 - 1
01100 => [2,1,3] => [1,1,0,0,1,0,1,1,1,0,0,0]
 => [1,1,1,1,0,0,1,0,1,0,0,0]
 => 1 = 2 - 1
01101 => [2,1,2,1] => [1,1,0,0,1,0,1,1,0,0,1,0]
 => [1,1,1,0,0,1,0,1,0,0,1,0]
 => 1 = 2 - 1
01110 => [2,1,1,2] => [1,1,0,0,1,0,1,0,1,1,0,0]
 => [1,1,1,0,0,1,0,1,0,1,0,0]
 => 1 = 2 - 1
01111 => [2,1,1,1,1] => [1,1,0,0,1,0,1,0,1,0,1,0]
 => [1,1,0,0,1,0,1,0,1,0,1,0]
 => 1 = 2 - 1
10000 => [1,5] => [1,0,1,1,1,1,1,0,0,0,0,0]
 => [1,1,1,1,1,0,1,0,0,0,0,0]
 => 2 = 3 - 1
10001 => [1,4,1] => [1,0,1,1,1,1,0,0,0,0,1,0]
 => [1,1,1,1,0,1,0,0,0,0,1,0]
 => 2 = 3 - 1
10010 => [1,3,2] => [1,0,1,1,1,0,0,0,1,1,0,0]
 => [1,1,1,1,0,1,0,0,0,1,0,0]
 => 2 = 3 - 1
10011 => [1,3,1,1] => [1,0,1,1,1,0,0,0,1,0,1,0]
 => [1,1,1,0,1,0,0,0,1,0,1,0]
 => 2 = 3 - 1
0000000 => [8] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
 => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
 => ? = 2 - 1
0000001 => [7,1] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
 => [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
 => ? = 2 - 1
0000010 => [6,2] => [1,1,1,1,1,1,0,0,0,0,0,0,1,1,0,0]
 => [1,1,1,1,1,1,1,0,0,0,0,0,0,1,0,0]
 => ? = 2 - 1
0000011 => [6,1,1] => [1,1,1,1,1,1,0,0,0,0,0,0,1,0,1,0]
 => [1,1,1,1,1,1,0,0,0,0,0,0,1,0,1,0]
 => ? = 2 - 1
0000100 => [5,3] => [1,1,1,1,1,0,0,0,0,0,1,1,1,0,0,0]
 => [1,1,1,1,1,1,1,0,0,0,0,0,1,0,0,0]
 => ? = 2 - 1
0000101 => [5,2,1] => [1,1,1,1,1,0,0,0,0,0,1,1,0,0,1,0]
 => [1,1,1,1,1,1,0,0,0,0,0,1,0,0,1,0]
 => ? = 2 - 1
0000110 => [5,1,2] => [1,1,1,1,1,0,0,0,0,0,1,0,1,1,0,0]
 => [1,1,1,1,1,1,0,0,0,0,0,1,0,1,0,0]
 => ? = 2 - 1
0000111 => [5,1,1,1] => [1,1,1,1,1,0,0,0,0,0,1,0,1,0,1,0]
 => [1,1,1,1,1,0,0,0,0,0,1,0,1,0,1,0]
 => ? = 2 - 1
0001000 => [4,4] => [1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0]
 => [1,1,1,1,1,1,1,0,0,0,0,1,0,0,0,0]
 => ? = 2 - 1
0001001 => [4,3,1] => [1,1,1,1,0,0,0,0,1,1,1,0,0,0,1,0]
 => [1,1,1,1,1,1,0,0,0,0,1,0,0,0,1,0]
 => ? = 2 - 1
0001010 => [4,2,2] => [1,1,1,1,0,0,0,0,1,1,0,0,1,1,0,0]
 => [1,1,1,1,1,1,0,0,0,0,1,0,0,1,0,0]
 => ? = 2 - 1
0001011 => [4,2,1,1] => [1,1,1,1,0,0,0,0,1,1,0,0,1,0,1,0]
 => [1,1,1,1,1,0,0,0,0,1,0,0,1,0,1,0]
 => ? = 2 - 1
0001100 => [4,1,3] => [1,1,1,1,0,0,0,0,1,0,1,1,1,0,0,0]
 => [1,1,1,1,1,1,0,0,0,0,1,0,1,0,0,0]
 => ? = 2 - 1
0001101 => [4,1,2,1] => [1,1,1,1,0,0,0,0,1,0,1,1,0,0,1,0]
 => [1,1,1,1,1,0,0,0,0,1,0,1,0,0,1,0]
 => ? = 2 - 1
0001110 => [4,1,1,2] => [1,1,1,1,0,0,0,0,1,0,1,0,1,1,0,0]
 => [1,1,1,1,1,0,0,0,0,1,0,1,0,1,0,0]
 => ? = 2 - 1
0001111 => [4,1,1,1,1] => [1,1,1,1,0,0,0,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,0,0,0,0,1,0,1,0,1,0,1,0]
 => ? = 2 - 1
0010000 => [3,5] => [1,1,1,0,0,0,1,1,1,1,1,0,0,0,0,0]
 => [1,1,1,1,1,1,1,0,0,0,1,0,0,0,0,0]
 => ? = 2 - 1
0010001 => [3,4,1] => [1,1,1,0,0,0,1,1,1,1,0,0,0,0,1,0]
 => [1,1,1,1,1,1,0,0,0,1,0,0,0,0,1,0]
 => ? = 2 - 1
0010010 => [3,3,2] => [1,1,1,0,0,0,1,1,1,0,0,0,1,1,0,0]
 => [1,1,1,1,1,1,0,0,0,1,0,0,0,1,0,0]
 => ? = 2 - 1
0010011 => [3,3,1,1] => [1,1,1,0,0,0,1,1,1,0,0,0,1,0,1,0]
 => [1,1,1,1,1,0,0,0,1,0,0,0,1,0,1,0]
 => ? = 2 - 1
0010100 => [3,2,3] => [1,1,1,0,0,0,1,1,0,0,1,1,1,0,0,0]
 => [1,1,1,1,1,1,0,0,0,1,0,0,1,0,0,0]
 => ? = 2 - 1
0010101 => [3,2,2,1] => [1,1,1,0,0,0,1,1,0,0,1,1,0,0,1,0]
 => [1,1,1,1,1,0,0,0,1,0,0,1,0,0,1,0]
 => ? = 2 - 1
0010110 => [3,2,1,2] => [1,1,1,0,0,0,1,1,0,0,1,0,1,1,0,0]
 => [1,1,1,1,1,0,0,0,1,0,0,1,0,1,0,0]
 => ? = 2 - 1
0010111 => [3,2,1,1,1] => [1,1,1,0,0,0,1,1,0,0,1,0,1,0,1,0]
 => [1,1,1,1,0,0,0,1,0,0,1,0,1,0,1,0]
 => ? = 2 - 1
0011000 => [3,1,4] => [1,1,1,0,0,0,1,0,1,1,1,1,0,0,0,0]
 => [1,1,1,1,1,1,0,0,0,1,0,1,0,0,0,0]
 => ? = 2 - 1
0011001 => [3,1,3,1] => [1,1,1,0,0,0,1,0,1,1,1,0,0,0,1,0]
 => [1,1,1,1,1,0,0,0,1,0,1,0,0,0,1,0]
 => ? = 2 - 1
0011010 => [3,1,2,2] => [1,1,1,0,0,0,1,0,1,1,0,0,1,1,0,0]
 => [1,1,1,1,1,0,0,0,1,0,1,0,0,1,0,0]
 => ? = 2 - 1
0011011 => [3,1,2,1,1] => [1,1,1,0,0,0,1,0,1,1,0,0,1,0,1,0]
 => [1,1,1,1,0,0,0,1,0,1,0,0,1,0,1,0]
 => ? = 2 - 1
0011100 => [3,1,1,3] => [1,1,1,0,0,0,1,0,1,0,1,1,1,0,0,0]
 => [1,1,1,1,1,0,0,0,1,0,1,0,1,0,0,0]
 => ? = 2 - 1
0011101 => [3,1,1,2,1] => [1,1,1,0,0,0,1,0,1,0,1,1,0,0,1,0]
 => [1,1,1,1,0,0,0,1,0,1,0,1,0,0,1,0]
 => ? = 2 - 1
0011110 => [3,1,1,1,2] => [1,1,1,0,0,0,1,0,1,0,1,0,1,1,0,0]
 => [1,1,1,1,0,0,0,1,0,1,0,1,0,1,0,0]
 => ? = 2 - 1
0011111 => [3,1,1,1,1,1] => [1,1,1,0,0,0,1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,0,0,0,1,0,1,0,1,0,1,0,1,0]
 => ? = 2 - 1
0100000 => [2,6] => [1,1,0,0,1,1,1,1,1,1,0,0,0,0,0,0]
 => [1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0]
 => ? = 2 - 1
0100001 => [2,5,1] => [1,1,0,0,1,1,1,1,1,0,0,0,0,0,1,0]
 => [1,1,1,1,1,1,0,0,1,0,0,0,0,0,1,0]
 => ? = 2 - 1
0100010 => [2,4,2] => [1,1,0,0,1,1,1,1,0,0,0,0,1,1,0,0]
 => [1,1,1,1,1,1,0,0,1,0,0,0,0,1,0,0]
 => ? = 2 - 1
0100011 => [2,4,1,1] => [1,1,0,0,1,1,1,1,0,0,0,0,1,0,1,0]
 => [1,1,1,1,1,0,0,1,0,0,0,0,1,0,1,0]
 => ? = 2 - 1
0100100 => [2,3,3] => [1,1,0,0,1,1,1,0,0,0,1,1,1,0,0,0]
 => [1,1,1,1,1,1,0,0,1,0,0,0,1,0,0,0]
 => ? = 2 - 1
0100101 => [2,3,2,1] => [1,1,0,0,1,1,1,0,0,0,1,1,0,0,1,0]
 => [1,1,1,1,1,0,0,1,0,0,0,1,0,0,1,0]
 => ? = 2 - 1
0100110 => [2,3,1,2] => [1,1,0,0,1,1,1,0,0,0,1,0,1,1,0,0]
 => [1,1,1,1,1,0,0,1,0,0,0,1,0,1,0,0]
 => ? = 2 - 1
0100111 => [2,3,1,1,1] => [1,1,0,0,1,1,1,0,0,0,1,0,1,0,1,0]
 => [1,1,1,1,0,0,1,0,0,0,1,0,1,0,1,0]
 => ? = 2 - 1
0101000 => [2,2,4] => [1,1,0,0,1,1,0,0,1,1,1,1,0,0,0,0]
 => [1,1,1,1,1,1,0,0,1,0,0,1,0,0,0,0]
 => ? = 2 - 1
0101001 => [2,2,3,1] => [1,1,0,0,1,1,0,0,1,1,1,0,0,0,1,0]
 => [1,1,1,1,1,0,0,1,0,0,1,0,0,0,1,0]
 => ? = 2 - 1
0101010 => [2,2,2,2] => [1,1,0,0,1,1,0,0,1,1,0,0,1,1,0,0]
 => [1,1,1,1,1,0,0,1,0,0,1,0,0,1,0,0]
 => ? = 2 - 1
0101011 => [2,2,2,1,1] => [1,1,0,0,1,1,0,0,1,1,0,0,1,0,1,0]
 => [1,1,1,1,0,0,1,0,0,1,0,0,1,0,1,0]
 => ? = 2 - 1
0101100 => [2,2,1,3] => [1,1,0,0,1,1,0,0,1,0,1,1,1,0,0,0]
 => [1,1,1,1,1,0,0,1,0,0,1,0,1,0,0,0]
 => ? = 2 - 1
0101101 => [2,2,1,2,1] => [1,1,0,0,1,1,0,0,1,0,1,1,0,0,1,0]
 => [1,1,1,1,0,0,1,0,0,1,0,1,0,0,1,0]
 => ? = 2 - 1
0101110 => [2,2,1,1,2] => [1,1,0,0,1,1,0,0,1,0,1,0,1,1,0,0]
 => [1,1,1,1,0,0,1,0,0,1,0,1,0,1,0,0]
 => ? = 2 - 1
0101111 => [2,2,1,1,1,1] => [1,1,0,0,1,1,0,0,1,0,1,0,1,0,1,0]
 => [1,1,1,0,0,1,0,0,1,0,1,0,1,0,1,0]
 => ? = 2 - 1
0110000 => [2,1,5] => [1,1,0,0,1,0,1,1,1,1,1,0,0,0,0,0]
 => [1,1,1,1,1,1,0,0,1,0,1,0,0,0,0,0]
 => ? = 2 - 1
0110001 => [2,1,4,1] => [1,1,0,0,1,0,1,1,1,1,0,0,0,0,1,0]
 => [1,1,1,1,1,0,0,1,0,1,0,0,0,0,1,0]
 => ? = 2 - 1
Description
The number of weak left to right maxima of a Dyck path. A weak left to right maximum is a peak whose height is larger than or equal to the height of all peaks to its left.
Matching statistic: St000025
Mapping
Mp00105: Binary words —complement⟶ Binary words
Mp00178: Binary words —to composition⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St000025: Dyck paths ⟶ ℤResult quality: 47% ●values known / values provided: 47%●distinct values known / distinct values provided: 70%
Values
0 => 1 => [1,1] => [1,0,1,0]
 => 1 = 2 - 1
1 => 0 => [2] => [1,1,0,0]
 => 2 = 3 - 1
00 => 11 => [1,1,1] => [1,0,1,0,1,0]
 => 1 = 2 - 1
01 => 10 => [1,2] => [1,0,1,1,0,0]
 => 1 = 2 - 1
10 => 01 => [2,1] => [1,1,0,0,1,0]
 => 2 = 3 - 1
11 => 00 => [3] => [1,1,1,0,0,0]
 => 3 = 4 - 1
000 => 111 => [1,1,1,1] => [1,0,1,0,1,0,1,0]
 => 1 = 2 - 1
001 => 110 => [1,1,2] => [1,0,1,0,1,1,0,0]
 => 1 = 2 - 1
010 => 101 => [1,2,1] => [1,0,1,1,0,0,1,0]
 => 1 = 2 - 1
011 => 100 => [1,3] => [1,0,1,1,1,0,0,0]
 => 1 = 2 - 1
100 => 011 => [2,1,1] => [1,1,0,0,1,0,1,0]
 => 2 = 3 - 1
101 => 010 => [2,2] => [1,1,0,0,1,1,0,0]
 => 2 = 3 - 1
110 => 001 => [3,1] => [1,1,1,0,0,0,1,0]
 => 3 = 4 - 1
111 => 000 => [4] => [1,1,1,1,0,0,0,0]
 => 4 = 5 - 1
0000 => 1111 => [1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
 => 1 = 2 - 1
0001 => 1110 => [1,1,1,2] => [1,0,1,0,1,0,1,1,0,0]
 => 1 = 2 - 1
0010 => 1101 => [1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
 => 1 = 2 - 1
0011 => 1100 => [1,1,3] => [1,0,1,0,1,1,1,0,0,0]
 => 1 = 2 - 1
0100 => 1011 => [1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
 => 1 = 2 - 1
0101 => 1010 => [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
 => 1 = 2 - 1
0110 => 1001 => [1,3,1] => [1,0,1,1,1,0,0,0,1,0]
 => 1 = 2 - 1
0111 => 1000 => [1,4] => [1,0,1,1,1,1,0,0,0,0]
 => 1 = 2 - 1
1000 => 0111 => [2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
 => 2 = 3 - 1
1001 => 0110 => [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
 => 2 = 3 - 1
1010 => 0101 => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
 => 2 = 3 - 1
1011 => 0100 => [2,3] => [1,1,0,0,1,1,1,0,0,0]
 => 2 = 3 - 1
1100 => 0011 => [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
 => 3 = 4 - 1
1101 => 0010 => [3,2] => [1,1,1,0,0,0,1,1,0,0]
 => 3 = 4 - 1
1110 => 0001 => [4,1] => [1,1,1,1,0,0,0,0,1,0]
 => 4 = 5 - 1
1111 => 0000 => [5] => [1,1,1,1,1,0,0,0,0,0]
 => 5 = 6 - 1
00000 => 11111 => [1,1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0,1,0]
 => 1 = 2 - 1
00001 => 11110 => [1,1,1,1,2] => [1,0,1,0,1,0,1,0,1,1,0,0]
 => 1 = 2 - 1
00010 => 11101 => [1,1,1,2,1] => [1,0,1,0,1,0,1,1,0,0,1,0]
 => 1 = 2 - 1
00011 => 11100 => [1,1,1,3] => [1,0,1,0,1,0,1,1,1,0,0,0]
 => 1 = 2 - 1
00100 => 11011 => [1,1,2,1,1] => [1,0,1,0,1,1,0,0,1,0,1,0]
 => 1 = 2 - 1
00101 => 11010 => [1,1,2,2] => [1,0,1,0,1,1,0,0,1,1,0,0]
 => 1 = 2 - 1
00110 => 11001 => [1,1,3,1] => [1,0,1,0,1,1,1,0,0,0,1,0]
 => 1 = 2 - 1
00111 => 11000 => [1,1,4] => [1,0,1,0,1,1,1,1,0,0,0,0]
 => 1 = 2 - 1
01000 => 10111 => [1,2,1,1,1] => [1,0,1,1,0,0,1,0,1,0,1,0]
 => 1 = 2 - 1
01001 => 10110 => [1,2,1,2] => [1,0,1,1,0,0,1,0,1,1,0,0]
 => 1 = 2 - 1
01010 => 10101 => [1,2,2,1] => [1,0,1,1,0,0,1,1,0,0,1,0]
 => 1 = 2 - 1
01011 => 10100 => [1,2,3] => [1,0,1,1,0,0,1,1,1,0,0,0]
 => 1 = 2 - 1
01100 => 10011 => [1,3,1,1] => [1,0,1,1,1,0,0,0,1,0,1,0]
 => 1 = 2 - 1
01101 => 10010 => [1,3,2] => [1,0,1,1,1,0,0,0,1,1,0,0]
 => 1 = 2 - 1
01110 => 10001 => [1,4,1] => [1,0,1,1,1,1,0,0,0,0,1,0]
 => 1 = 2 - 1
01111 => 10000 => [1,5] => [1,0,1,1,1,1,1,0,0,0,0,0]
 => 1 = 2 - 1
10000 => 01111 => [2,1,1,1,1] => [1,1,0,0,1,0,1,0,1,0,1,0]
 => 2 = 3 - 1
10001 => 01110 => [2,1,1,2] => [1,1,0,0,1,0,1,0,1,1,0,0]
 => 2 = 3 - 1
10010 => 01101 => [2,1,2,1] => [1,1,0,0,1,0,1,1,0,0,1,0]
 => 2 = 3 - 1
10011 => 01100 => [2,1,3] => [1,1,0,0,1,0,1,1,1,0,0,0]
 => 2 = 3 - 1
0000000 => 1111111 => [1,1,1,1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => ? = 2 - 1
0000001 => 1111110 => [1,1,1,1,1,1,2] => [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => ? = 2 - 1
0000010 => 1111101 => [1,1,1,1,1,2,1] => [1,0,1,0,1,0,1,0,1,0,1,1,0,0,1,0]
 => ? = 2 - 1
0000011 => 1111100 => [1,1,1,1,1,3] => [1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,0]
 => ? = 2 - 1
0000100 => 1111011 => [1,1,1,1,2,1,1] => [1,0,1,0,1,0,1,0,1,1,0,0,1,0,1,0]
 => ? = 2 - 1
0000101 => 1111010 => [1,1,1,1,2,2] => [1,0,1,0,1,0,1,0,1,1,0,0,1,1,0,0]
 => ? = 2 - 1
0000110 => 1111001 => [1,1,1,1,3,1] => [1,0,1,0,1,0,1,0,1,1,1,0,0,0,1,0]
 => ? = 2 - 1
0000111 => 1111000 => [1,1,1,1,4] => [1,0,1,0,1,0,1,0,1,1,1,1,0,0,0,0]
 => ? = 2 - 1
0001000 => 1110111 => [1,1,1,2,1,1,1] => [1,0,1,0,1,0,1,1,0,0,1,0,1,0,1,0]
 => ? = 2 - 1
0001001 => 1110110 => [1,1,1,2,1,2] => [1,0,1,0,1,0,1,1,0,0,1,0,1,1,0,0]
 => ? = 2 - 1
0001010 => 1110101 => [1,1,1,2,2,1] => [1,0,1,0,1,0,1,1,0,0,1,1,0,0,1,0]
 => ? = 2 - 1
0001011 => 1110100 => [1,1,1,2,3] => [1,0,1,0,1,0,1,1,0,0,1,1,1,0,0,0]
 => ? = 2 - 1
0001100 => 1110011 => [1,1,1,3,1,1] => [1,0,1,0,1,0,1,1,1,0,0,0,1,0,1,0]
 => ? = 2 - 1
0001101 => 1110010 => [1,1,1,3,2] => [1,0,1,0,1,0,1,1,1,0,0,0,1,1,0,0]
 => ? = 2 - 1
0001110 => 1110001 => [1,1,1,4,1] => [1,0,1,0,1,0,1,1,1,1,0,0,0,0,1,0]
 => ? = 2 - 1
0001111 => 1110000 => [1,1,1,5] => [1,0,1,0,1,0,1,1,1,1,1,0,0,0,0,0]
 => ? = 2 - 1
0010000 => 1101111 => [1,1,2,1,1,1,1] => [1,0,1,0,1,1,0,0,1,0,1,0,1,0,1,0]
 => ? = 2 - 1
0010001 => 1101110 => [1,1,2,1,1,2] => [1,0,1,0,1,1,0,0,1,0,1,0,1,1,0,0]
 => ? = 2 - 1
0010010 => 1101101 => [1,1,2,1,2,1] => [1,0,1,0,1,1,0,0,1,0,1,1,0,0,1,0]
 => ? = 2 - 1
0010011 => 1101100 => [1,1,2,1,3] => [1,0,1,0,1,1,0,0,1,0,1,1,1,0,0,0]
 => ? = 2 - 1
0010100 => 1101011 => [1,1,2,2,1,1] => [1,0,1,0,1,1,0,0,1,1,0,0,1,0,1,0]
 => ? = 2 - 1
0010101 => 1101010 => [1,1,2,2,2] => [1,0,1,0,1,1,0,0,1,1,0,0,1,1,0,0]
 => ? = 2 - 1
0010110 => 1101001 => [1,1,2,3,1] => [1,0,1,0,1,1,0,0,1,1,1,0,0,0,1,0]
 => ? = 2 - 1
0010111 => 1101000 => [1,1,2,4] => [1,0,1,0,1,1,0,0,1,1,1,1,0,0,0,0]
 => ? = 2 - 1
0011000 => 1100111 => [1,1,3,1,1,1] => [1,0,1,0,1,1,1,0,0,0,1,0,1,0,1,0]
 => ? = 2 - 1
0011001 => 1100110 => [1,1,3,1,2] => [1,0,1,0,1,1,1,0,0,0,1,0,1,1,0,0]
 => ? = 2 - 1
0011010 => 1100101 => [1,1,3,2,1] => [1,0,1,0,1,1,1,0,0,0,1,1,0,0,1,0]
 => ? = 2 - 1
0011011 => 1100100 => [1,1,3,3] => [1,0,1,0,1,1,1,0,0,0,1,1,1,0,0,0]
 => ? = 2 - 1
0011100 => 1100011 => [1,1,4,1,1] => [1,0,1,0,1,1,1,1,0,0,0,0,1,0,1,0]
 => ? = 2 - 1
0011101 => 1100010 => [1,1,4,2] => [1,0,1,0,1,1,1,1,0,0,0,0,1,1,0,0]
 => ? = 2 - 1
0011110 => 1100001 => [1,1,5,1] => [1,0,1,0,1,1,1,1,1,0,0,0,0,0,1,0]
 => ? = 2 - 1
0011111 => 1100000 => [1,1,6] => [1,0,1,0,1,1,1,1,1,1,0,0,0,0,0,0]
 => ? = 2 - 1
0100000 => 1011111 => [1,2,1,1,1,1,1] => [1,0,1,1,0,0,1,0,1,0,1,0,1,0,1,0]
 => ? = 2 - 1
0100001 => 1011110 => [1,2,1,1,1,2] => [1,0,1,1,0,0,1,0,1,0,1,0,1,1,0,0]
 => ? = 2 - 1
0100010 => 1011101 => [1,2,1,1,2,1] => [1,0,1,1,0,0,1,0,1,0,1,1,0,0,1,0]
 => ? = 2 - 1
0100011 => 1011100 => [1,2,1,1,3] => [1,0,1,1,0,0,1,0,1,0,1,1,1,0,0,0]
 => ? = 2 - 1
0100100 => 1011011 => [1,2,1,2,1,1] => [1,0,1,1,0,0,1,0,1,1,0,0,1,0,1,0]
 => ? = 2 - 1
0100101 => 1011010 => [1,2,1,2,2] => [1,0,1,1,0,0,1,0,1,1,0,0,1,1,0,0]
 => ? = 2 - 1
0100110 => 1011001 => [1,2,1,3,1] => [1,0,1,1,0,0,1,0,1,1,1,0,0,0,1,0]
 => ? = 2 - 1
0100111 => 1011000 => [1,2,1,4] => [1,0,1,1,0,0,1,0,1,1,1,1,0,0,0,0]
 => ? = 2 - 1
0101000 => 1010111 => [1,2,2,1,1,1] => [1,0,1,1,0,0,1,1,0,0,1,0,1,0,1,0]
 => ? = 2 - 1
0101001 => 1010110 => [1,2,2,1,2] => [1,0,1,1,0,0,1,1,0,0,1,0,1,1,0,0]
 => ? = 2 - 1
0101010 => 1010101 => [1,2,2,2,1] => [1,0,1,1,0,0,1,1,0,0,1,1,0,0,1,0]
 => ? = 2 - 1
0101011 => 1010100 => [1,2,2,3] => [1,0,1,1,0,0,1,1,0,0,1,1,1,0,0,0]
 => ? = 2 - 1
0101100 => 1010011 => [1,2,3,1,1] => [1,0,1,1,0,0,1,1,1,0,0,0,1,0,1,0]
 => ? = 2 - 1
0101101 => 1010010 => [1,2,3,2] => [1,0,1,1,0,0,1,1,1,0,0,0,1,1,0,0]
 => ? = 2 - 1
0101110 => 1010001 => [1,2,4,1] => [1,0,1,1,0,0,1,1,1,1,0,0,0,0,1,0]
 => ? = 2 - 1
0101111 => 1010000 => [1,2,5] => [1,0,1,1,0,0,1,1,1,1,1,0,0,0,0,0]
 => ? = 2 - 1
0110000 => 1001111 => [1,3,1,1,1,1] => [1,0,1,1,1,0,0,0,1,0,1,0,1,0,1,0]
 => ? = 2 - 1
0110001 => 1001110 => [1,3,1,1,2] => [1,0,1,1,1,0,0,0,1,0,1,0,1,1,0,0]
 => ? = 2 - 1
Description
The number of initial rises of a Dyck path. In other words, this is the height of the first peak of $D$.
The following 24 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000026The position of the first return of a Dyck path. St001135The projective dimension of the first simple module in the Nakayama algebra corresponding to the Dyck path. St000363The number of minimal vertex covers of a graph. St000273The domination number of a graph. St000544The cop number of a graph. St000916The packing number of a graph. St001829The common independence number of a graph. St001322The size of a minimal independent dominating set in a graph. St001316The domatic number of a graph. St001339The irredundance number of a graph. St001363The Euler characteristic of a graph according to Knill. St000501The size of the first part in the decomposition of a permutation. St000654The first descent of a permutation. St001226The number of integers i such that the radical of the i-th indecomposable projective module has vanishing first extension group with the Jacobson radical J in the corresponding Nakayama algebra. St000286The number of connected components of the complement of a graph. St000287The number of connected components of a graph. St000335The difference of lower and upper interactions. St001184Number of indecomposable injective modules with grade at least 1 in the corresponding Nakayama algebra. St001201The grade of the simple module $S_0$ in the special CNakayama algebra corresponding to the Dyck path. St001481The minimal height of a peak of a Dyck path. St000261The edge connectivity of a graph. St000262The vertex connectivity of a graph. St000310The minimal degree of a vertex of a graph. St000054The first entry of the permutation.