Your data matches 34 different statistics following compositions of up to 3 maps.
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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: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
0 => [2] => [1,1] => 1
1 => [1,1] => [2] => 2
00 => [3] => [1,1,1] => 1
01 => [2,1] => [2,1] => 1
10 => [1,2] => [1,2] => 2
11 => [1,1,1] => [3] => 3
000 => [4] => [1,1,1,1] => 1
001 => [3,1] => [2,1,1] => 1
010 => [2,2] => [1,2,1] => 1
011 => [2,1,1] => [3,1] => 1
100 => [1,3] => [1,1,2] => 2
101 => [1,2,1] => [2,2] => 2
110 => [1,1,2] => [1,3] => 3
111 => [1,1,1,1] => [4] => 4
0000 => [5] => [1,1,1,1,1] => 1
0001 => [4,1] => [2,1,1,1] => 1
0010 => [3,2] => [1,2,1,1] => 1
0011 => [3,1,1] => [3,1,1] => 1
0100 => [2,3] => [1,1,2,1] => 1
0101 => [2,2,1] => [2,2,1] => 1
0110 => [2,1,2] => [1,3,1] => 1
0111 => [2,1,1,1] => [4,1] => 1
1000 => [1,4] => [1,1,1,2] => 2
1001 => [1,3,1] => [2,1,2] => 2
1010 => [1,2,2] => [1,2,2] => 2
1011 => [1,2,1,1] => [3,2] => 2
1100 => [1,1,3] => [1,1,3] => 3
1101 => [1,1,2,1] => [2,3] => 3
1110 => [1,1,1,2] => [1,4] => 4
1111 => [1,1,1,1,1] => [5] => 5
00000 => [6] => [1,1,1,1,1,1] => 1
00001 => [5,1] => [2,1,1,1,1] => 1
00010 => [4,2] => [1,2,1,1,1] => 1
00011 => [4,1,1] => [3,1,1,1] => 1
00100 => [3,3] => [1,1,2,1,1] => 1
00101 => [3,2,1] => [2,2,1,1] => 1
00110 => [3,1,2] => [1,3,1,1] => 1
00111 => [3,1,1,1] => [4,1,1] => 1
01000 => [2,4] => [1,1,1,2,1] => 1
01001 => [2,3,1] => [2,1,2,1] => 1
01010 => [2,2,2] => [1,2,2,1] => 1
01011 => [2,2,1,1] => [3,2,1] => 1
01100 => [2,1,3] => [1,1,3,1] => 1
01101 => [2,1,2,1] => [2,3,1] => 1
01110 => [2,1,1,2] => [1,4,1] => 1
01111 => [2,1,1,1,1] => [5,1] => 1
10000 => [1,5] => [1,1,1,1,2] => 2
10001 => [1,4,1] => [2,1,1,2] => 2
10010 => [1,3,2] => [1,2,1,2] => 2
10011 => [1,3,1,1] => [3,1,2] => 2
Description
The last part of an integer composition.
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: 97% ●values known / values provided: 97%●distinct values known / distinct values provided: 100%
Values
0 => 1 => [1,1] => 1
1 => 0 => [2] => 2
00 => 11 => [1,1,1] => 1
01 => 10 => [1,2] => 1
10 => 01 => [2,1] => 2
11 => 00 => [3] => 3
000 => 111 => [1,1,1,1] => 1
001 => 110 => [1,1,2] => 1
010 => 101 => [1,2,1] => 1
011 => 100 => [1,3] => 1
100 => 011 => [2,1,1] => 2
101 => 010 => [2,2] => 2
110 => 001 => [3,1] => 3
111 => 000 => [4] => 4
0000 => 1111 => [1,1,1,1,1] => 1
0001 => 1110 => [1,1,1,2] => 1
0010 => 1101 => [1,1,2,1] => 1
0011 => 1100 => [1,1,3] => 1
0100 => 1011 => [1,2,1,1] => 1
0101 => 1010 => [1,2,2] => 1
0110 => 1001 => [1,3,1] => 1
0111 => 1000 => [1,4] => 1
1000 => 0111 => [2,1,1,1] => 2
1001 => 0110 => [2,1,2] => 2
1010 => 0101 => [2,2,1] => 2
1011 => 0100 => [2,3] => 2
1100 => 0011 => [3,1,1] => 3
1101 => 0010 => [3,2] => 3
1110 => 0001 => [4,1] => 4
1111 => 0000 => [5] => 5
00000 => 11111 => [1,1,1,1,1,1] => 1
00001 => 11110 => [1,1,1,1,2] => 1
00010 => 11101 => [1,1,1,2,1] => 1
00011 => 11100 => [1,1,1,3] => 1
00100 => 11011 => [1,1,2,1,1] => 1
00101 => 11010 => [1,1,2,2] => 1
00110 => 11001 => [1,1,3,1] => 1
00111 => 11000 => [1,1,4] => 1
01000 => 10111 => [1,2,1,1,1] => 1
01001 => 10110 => [1,2,1,2] => 1
01010 => 10101 => [1,2,2,1] => 1
01011 => 10100 => [1,2,3] => 1
01100 => 10011 => [1,3,1,1] => 1
01101 => 10010 => [1,3,2] => 1
01110 => 10001 => [1,4,1] => 1
01111 => 10000 => [1,5] => 1
10000 => 01111 => [2,1,1,1,1] => 2
10001 => 01110 => [2,1,1,2] => 2
10010 => 01101 => [2,1,2,1] => 2
10011 => 01100 => [2,1,3] => 2
10100010001 => ? => ? => ? = 2
10010001001 => ? => ? => ? = 2
10001000101 => ? => ? => ? = 2
10001010001 => ? => ? => ? = 2
00100000100 => ? => ? => ? = 1
00100101001 => ? => ? => ? = 1
00001010000 => ? => ? => ? = 1
00001010101 => 11110101010 => ? => ? = 1
10101101001 => 01010010110 => ? => ? = 2
11010101001 => 00101010110 => ? => ? = 3
11010000100 => 00101111011 => ? => ? = 3
11010001011 => ? => ? => ? = 3
Description
The first part of an integer composition.
Matching statistic: St000297
(load all 10 compositions to match this statistic)
Mapping
St000297: Binary words ⟶ ℤResult quality: 94% ●values known / values provided: 94%●distinct values known / distinct values provided: 100%
Values
0 => 0 = 1 - 1
1 => 1 = 2 - 1
00 => 0 = 1 - 1
01 => 0 = 1 - 1
10 => 1 = 2 - 1
11 => 2 = 3 - 1
000 => 0 = 1 - 1
001 => 0 = 1 - 1
010 => 0 = 1 - 1
011 => 0 = 1 - 1
100 => 1 = 2 - 1
101 => 1 = 2 - 1
110 => 2 = 3 - 1
111 => 3 = 4 - 1
0000 => 0 = 1 - 1
0001 => 0 = 1 - 1
0010 => 0 = 1 - 1
0011 => 0 = 1 - 1
0100 => 0 = 1 - 1
0101 => 0 = 1 - 1
0110 => 0 = 1 - 1
0111 => 0 = 1 - 1
1000 => 1 = 2 - 1
1001 => 1 = 2 - 1
1010 => 1 = 2 - 1
1011 => 1 = 2 - 1
1100 => 2 = 3 - 1
1101 => 2 = 3 - 1
1110 => 3 = 4 - 1
1111 => 4 = 5 - 1
00000 => 0 = 1 - 1
00001 => 0 = 1 - 1
00010 => 0 = 1 - 1
00011 => 0 = 1 - 1
00100 => 0 = 1 - 1
00101 => 0 = 1 - 1
00110 => 0 = 1 - 1
00111 => 0 = 1 - 1
01000 => 0 = 1 - 1
01001 => 0 = 1 - 1
01010 => 0 = 1 - 1
01011 => 0 = 1 - 1
01100 => 0 = 1 - 1
01101 => 0 = 1 - 1
01110 => 0 = 1 - 1
01111 => 0 = 1 - 1
10000 => 1 = 2 - 1
10001 => 1 = 2 - 1
10010 => 1 = 2 - 1
10011 => 1 = 2 - 1
=> ? = 1 - 1
11010101100 => ? = 3 - 1
11011010100 => ? = 3 - 1
00101011011 => ? = 1 - 1
00110101011 => ? = 1 - 1
0011111110 => ? = 1 - 1
00000000000 => ? = 1 - 1
0101111110 => ? = 1 - 1
10000000001 => ? = 2 - 1
10101110111 => ? = 2 - 1
11011011011 => ? = 3 - 1
11010110111 => ? = 3 - 1
11101110101 => ? = 4 - 1
11101101011 => ? = 4 - 1
11101010111 => ? = 4 - 1
11111011111 => ? = 6 - 1
0111111010 => ? = 1 - 1
00001010101 => ? = 1 - 1
10101101001 => ? = 2 - 1
11010101001 => ? = 3 - 1
11010000100 => ? = 3 - 1
11010001011 => ? = 3 - 1
Description
The number of leading ones in a binary word.
Matching statistic: St000326
(load all 6 compositions to match this statistic)
Mapping
Mp00104: Binary words —reverse⟶ Binary words
Mp00280: Binary words —path rowmotion⟶ Binary words
Mp00104: Binary words —reverse⟶ Binary words
St000326: Binary words ⟶ ℤResult quality: 90% ●values known / values provided: 90%●distinct values known / distinct values provided: 100%
Values
0 => 0 => 1 => 1 => 1
1 => 1 => 0 => 0 => 2
00 => 00 => 01 => 10 => 1
01 => 10 => 11 => 11 => 1
10 => 01 => 10 => 01 => 2
11 => 11 => 00 => 00 => 3
000 => 000 => 001 => 100 => 1
001 => 100 => 011 => 110 => 1
010 => 010 => 101 => 101 => 1
011 => 110 => 111 => 111 => 1
100 => 001 => 010 => 010 => 2
101 => 101 => 110 => 011 => 2
110 => 011 => 100 => 001 => 3
111 => 111 => 000 => 000 => 4
0000 => 0000 => 0001 => 1000 => 1
0001 => 1000 => 0011 => 1100 => 1
0010 => 0100 => 1001 => 1001 => 1
0011 => 1100 => 0111 => 1110 => 1
0100 => 0010 => 0101 => 1010 => 1
0101 => 1010 => 1101 => 1011 => 1
0110 => 0110 => 1011 => 1101 => 1
0111 => 1110 => 1111 => 1111 => 1
1000 => 0001 => 0010 => 0100 => 2
1001 => 1001 => 0110 => 0110 => 2
1010 => 0101 => 1010 => 0101 => 2
1011 => 1101 => 1110 => 0111 => 2
1100 => 0011 => 0100 => 0010 => 3
1101 => 1011 => 1100 => 0011 => 3
1110 => 0111 => 1000 => 0001 => 4
1111 => 1111 => 0000 => 0000 => 5
00000 => 00000 => 00001 => 10000 => 1
00001 => 10000 => 00011 => 11000 => 1
00010 => 01000 => 10001 => 10001 => 1
00011 => 11000 => 00111 => 11100 => 1
00100 => 00100 => 01001 => 10010 => 1
00101 => 10100 => 11001 => 10011 => 1
00110 => 01100 => 10011 => 11001 => 1
00111 => 11100 => 01111 => 11110 => 1
01000 => 00010 => 00101 => 10100 => 1
01001 => 10010 => 01101 => 10110 => 1
01010 => 01010 => 10101 => 10101 => 1
01011 => 11010 => 11101 => 10111 => 1
01100 => 00110 => 01011 => 11010 => 1
01101 => 10110 => 11011 => 11011 => 1
01110 => 01110 => 10111 => 11101 => 1
01111 => 11110 => 11111 => 11111 => 1
10000 => 00001 => 00010 => 01000 => 2
10001 => 10001 => 00110 => 01100 => 2
10010 => 01001 => 10010 => 01001 => 2
10011 => 11001 => 01110 => 01110 => 2
10000000100 => 00100000001 => ? => ? => ? = 2
=> => => => ? = 1
0111111111 => 1111111110 => 1111111111 => 1111111111 => ? = 1
11010101100 => 00110101011 => 01011010100 => 00101011010 => ? = 3
11011010100 => 00101011011 => 01010101100 => 00110101010 => ? = 3
00101011011 => 11011010100 => 11101101001 => 10010110111 => ? = 1
00110101011 => 11010101100 => 11101010011 => 11001010111 => ? = 1
0011111110 => 0111111100 => ? => ? => ? = 1
0101111110 => 0111111010 => ? => ? => ? = 1
10101010101 => 10101010101 => 11010101010 => 01010101011 => ? = 2
10100010001 => 10001000101 => 00110001010 => ? => ? = 2
10010001001 => 10010001001 => 01100010010 => ? => ? = 2
10001000101 => 10100010001 => 11000100010 => 01000100011 => ? = 2
10001010001 => 10001010001 => 00110100010 => ? => ? = 2
00100010101 => 10101000100 => 11010001001 => 10010001011 => ? = 1
00100000100 => 00100000100 => 01000001001 => 10010000010 => ? = 1
00100101001 => 10010100100 => 01101001001 => ? => ? = 1
00101000101 => 10100010100 => 11000101001 => 10010100011 => ? = 1
00101010001 => 10001010100 => 00110101001 => ? => ? = 1
00001000001 => 10000010000 => 00001100001 => ? => ? = 1
00000000101 => 10100000000 => ? => ? => ? = 1
10000000001 => 10000000001 => ? => ? => ? = 2
10101110111 => 11101110101 => ? => ? => ? = 2
11011011011 => 11011011011 => ? => ? => ? = 3
11010110111 => 11101101011 => ? => ? => ? = 3
11101110101 => 10101110111 => ? => ? => ? = 4
11101101011 => 11010110111 => ? => ? => ? = 4
11101010111 => 11101010111 => ? => ? => ? = 4
11111011111 => 11111011111 => ? => ? => ? = 6
0111111010 => 0101111110 => 1010111111 => 1111110101 => ? = 1
00001010101 => 10101010000 => ? => ? => ? = 1
10101101001 => 10010110101 => ? => ? => ? = 2
11010101001 => 10010101011 => ? => ? => ? = 3
11010000100 => 00100001011 => ? => ? => ? = 3
11010001011 => 11010001011 => ? => ? => ? = 3
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: 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: 64% ●values known / values provided: 64%●distinct values known / distinct values provided: 100%
Values
0 => 1 => [1,1] => [1,0,1,0]
 => 2 = 1 + 1
1 => 0 => [2] => [1,1,0,0]
 => 3 = 2 + 1
00 => 11 => [1,1,1] => [1,0,1,0,1,0]
 => 2 = 1 + 1
01 => 10 => [1,2] => [1,0,1,1,0,0]
 => 2 = 1 + 1
10 => 01 => [2,1] => [1,1,0,0,1,0]
 => 3 = 2 + 1
11 => 00 => [3] => [1,1,1,0,0,0]
 => 4 = 3 + 1
000 => 111 => [1,1,1,1] => [1,0,1,0,1,0,1,0]
 => 2 = 1 + 1
001 => 110 => [1,1,2] => [1,0,1,0,1,1,0,0]
 => 2 = 1 + 1
010 => 101 => [1,2,1] => [1,0,1,1,0,0,1,0]
 => 2 = 1 + 1
011 => 100 => [1,3] => [1,0,1,1,1,0,0,0]
 => 2 = 1 + 1
100 => 011 => [2,1,1] => [1,1,0,0,1,0,1,0]
 => 3 = 2 + 1
101 => 010 => [2,2] => [1,1,0,0,1,1,0,0]
 => 3 = 2 + 1
110 => 001 => [3,1] => [1,1,1,0,0,0,1,0]
 => 4 = 3 + 1
111 => 000 => [4] => [1,1,1,1,0,0,0,0]
 => 5 = 4 + 1
0000 => 1111 => [1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
 => 2 = 1 + 1
0001 => 1110 => [1,1,1,2] => [1,0,1,0,1,0,1,1,0,0]
 => 2 = 1 + 1
0010 => 1101 => [1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
 => 2 = 1 + 1
0011 => 1100 => [1,1,3] => [1,0,1,0,1,1,1,0,0,0]
 => 2 = 1 + 1
0100 => 1011 => [1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
 => 2 = 1 + 1
0101 => 1010 => [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
 => 2 = 1 + 1
0110 => 1001 => [1,3,1] => [1,0,1,1,1,0,0,0,1,0]
 => 2 = 1 + 1
0111 => 1000 => [1,4] => [1,0,1,1,1,1,0,0,0,0]
 => 2 = 1 + 1
1000 => 0111 => [2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
 => 3 = 2 + 1
1001 => 0110 => [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
 => 3 = 2 + 1
1010 => 0101 => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
 => 3 = 2 + 1
1011 => 0100 => [2,3] => [1,1,0,0,1,1,1,0,0,0]
 => 3 = 2 + 1
1100 => 0011 => [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
 => 4 = 3 + 1
1101 => 0010 => [3,2] => [1,1,1,0,0,0,1,1,0,0]
 => 4 = 3 + 1
1110 => 0001 => [4,1] => [1,1,1,1,0,0,0,0,1,0]
 => 5 = 4 + 1
1111 => 0000 => [5] => [1,1,1,1,1,0,0,0,0,0]
 => 6 = 5 + 1
00000 => 11111 => [1,1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0,1,0]
 => 2 = 1 + 1
00001 => 11110 => [1,1,1,1,2] => [1,0,1,0,1,0,1,0,1,1,0,0]
 => 2 = 1 + 1
00010 => 11101 => [1,1,1,2,1] => [1,0,1,0,1,0,1,1,0,0,1,0]
 => 2 = 1 + 1
00011 => 11100 => [1,1,1,3] => [1,0,1,0,1,0,1,1,1,0,0,0]
 => 2 = 1 + 1
00100 => 11011 => [1,1,2,1,1] => [1,0,1,0,1,1,0,0,1,0,1,0]
 => 2 = 1 + 1
00101 => 11010 => [1,1,2,2] => [1,0,1,0,1,1,0,0,1,1,0,0]
 => 2 = 1 + 1
00110 => 11001 => [1,1,3,1] => [1,0,1,0,1,1,1,0,0,0,1,0]
 => 2 = 1 + 1
00111 => 11000 => [1,1,4] => [1,0,1,0,1,1,1,1,0,0,0,0]
 => 2 = 1 + 1
01000 => 10111 => [1,2,1,1,1] => [1,0,1,1,0,0,1,0,1,0,1,0]
 => 2 = 1 + 1
01001 => 10110 => [1,2,1,2] => [1,0,1,1,0,0,1,0,1,1,0,0]
 => 2 = 1 + 1
01010 => 10101 => [1,2,2,1] => [1,0,1,1,0,0,1,1,0,0,1,0]
 => 2 = 1 + 1
01011 => 10100 => [1,2,3] => [1,0,1,1,0,0,1,1,1,0,0,0]
 => 2 = 1 + 1
01100 => 10011 => [1,3,1,1] => [1,0,1,1,1,0,0,0,1,0,1,0]
 => 2 = 1 + 1
01101 => 10010 => [1,3,2] => [1,0,1,1,1,0,0,0,1,1,0,0]
 => 2 = 1 + 1
01110 => 10001 => [1,4,1] => [1,0,1,1,1,1,0,0,0,0,1,0]
 => 2 = 1 + 1
01111 => 10000 => [1,5] => [1,0,1,1,1,1,1,0,0,0,0,0]
 => 2 = 1 + 1
10000 => 01111 => [2,1,1,1,1] => [1,1,0,0,1,0,1,0,1,0,1,0]
 => 3 = 2 + 1
10001 => 01110 => [2,1,1,2] => [1,1,0,0,1,0,1,0,1,1,0,0]
 => 3 = 2 + 1
10010 => 01101 => [2,1,2,1] => [1,1,0,0,1,0,1,1,0,0,1,0]
 => 3 = 2 + 1
10011 => 01100 => [2,1,3] => [1,1,0,0,1,0,1,1,1,0,0,0]
 => 3 = 2 + 1
1100011 => 0011100 => [3,1,1,3] => [1,1,1,0,0,0,1,0,1,0,1,1,1,0,0,0]
 => ? = 3 + 1
1100101 => 0011010 => [3,1,2,2] => [1,1,1,0,0,0,1,0,1,1,0,0,1,1,0,0]
 => ? = 3 + 1
1100110 => 0011001 => [3,1,3,1] => [1,1,1,0,0,0,1,0,1,1,1,0,0,0,1,0]
 => ? = 3 + 1
1101001 => 0010110 => [3,2,1,2] => [1,1,1,0,0,0,1,1,0,0,1,0,1,1,0,0]
 => ? = 3 + 1
1101010 => 0010101 => [3,2,2,1] => [1,1,1,0,0,0,1,1,0,0,1,1,0,0,1,0]
 => ? = 3 + 1
1101011 => 0010100 => [3,2,3] => [1,1,1,0,0,0,1,1,0,0,1,1,1,0,0,0]
 => ? = 3 + 1
1101100 => 0010011 => [3,3,1,1] => [1,1,1,0,0,0,1,1,1,0,0,0,1,0,1,0]
 => ? = 3 + 1
1101101 => 0010010 => [3,3,2] => [1,1,1,0,0,0,1,1,1,0,0,0,1,1,0,0]
 => ? = 3 + 1
1110010 => 0001101 => [4,1,2,1] => [1,1,1,1,0,0,0,0,1,0,1,1,0,0,1,0]
 => ? = 4 + 1
00001110 => 11110001 => [1,1,1,1,4,1] => [1,0,1,0,1,0,1,0,1,1,1,1,0,0,0,0,1,0]
 => ? = 1 + 1
00010110 => 11101001 => [1,1,1,2,3,1] => [1,0,1,0,1,0,1,1,0,0,1,1,1,0,0,0,1,0]
 => ? = 1 + 1
00100110 => 11011001 => [1,1,2,1,3,1] => [1,0,1,0,1,1,0,0,1,0,1,1,1,0,0,0,1,0]
 => ? = 1 + 1
01000110 => 10111001 => [1,2,1,1,3,1] => [1,0,1,1,0,0,1,0,1,0,1,1,1,0,0,0,1,0]
 => ? = 1 + 1
01001010 => 10110101 => [1,2,1,2,2,1] => [1,0,1,1,0,0,1,0,1,1,0,0,1,1,0,0,1,0]
 => ? = 1 + 1
01010010 => 10101101 => [1,2,2,1,2,1] => [1,0,1,1,0,0,1,1,0,0,1,0,1,1,0,0,1,0]
 => ? = 1 + 1
01011110 => 10100001 => [1,2,5,1] => [1,0,1,1,0,0,1,1,1,1,1,0,0,0,0,0,1,0]
 => ? = 1 + 1
01100010 => 10011101 => [1,3,1,1,2,1] => [1,0,1,1,1,0,0,0,1,0,1,0,1,1,0,0,1,0]
 => ? = 1 + 1
01100100 => 10011011 => [1,3,1,2,1,1] => [1,0,1,1,1,0,0,0,1,0,1,1,0,0,1,0,1,0]
 => ? = 1 + 1
01101000 => 10010111 => [1,3,2,1,1,1] => [1,0,1,1,1,0,0,0,1,1,0,0,1,0,1,0,1,0]
 => ? = 1 + 1
01101110 => 10010001 => [1,3,4,1] => [1,0,1,1,1,0,0,0,1,1,1,1,0,0,0,0,1,0]
 => ? = 1 + 1
01110000 => 10001111 => [1,4,1,1,1,1] => [1,0,1,1,1,1,0,0,0,0,1,0,1,0,1,0,1,0]
 => ? = 1 + 1
01110011 => 10001100 => [1,4,1,3] => [1,0,1,1,1,1,0,0,0,0,1,0,1,1,1,0,0,0]
 => ? = 1 + 1
01110101 => 10001010 => [1,4,2,2] => [1,0,1,1,1,1,0,0,0,0,1,1,0,0,1,1,0,0]
 => ? = 1 + 1
01110110 => 10001001 => [1,4,3,1] => [1,0,1,1,1,1,0,0,0,0,1,1,1,0,0,0,1,0]
 => ? = 1 + 1
01111001 => 10000110 => [1,5,1,2] => [1,0,1,1,1,1,1,0,0,0,0,0,1,0,1,1,0,0]
 => ? = 1 + 1
01111010 => 10000101 => [1,5,2,1] => [1,0,1,1,1,1,1,0,0,0,0,0,1,1,0,0,1,0]
 => ? = 1 + 1
10011110 => 01100001 => [2,1,5,1] => [1,1,0,0,1,0,1,1,1,1,1,0,0,0,0,0,1,0]
 => ? = 2 + 1
10101110 => 01010001 => [2,2,4,1] => [1,1,0,0,1,1,0,0,1,1,1,1,0,0,0,0,1,0]
 => ? = 2 + 1
11001110 => 00110001 => [3,1,4,1] => [1,1,1,0,0,0,1,0,1,1,1,1,0,0,0,0,1,0]
 => ? = 3 + 1
000000000 => 111111111 => [1,1,1,1,1,1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => ? = 1 + 1
000000101 => 111111010 => [1,1,1,1,1,1,2,2] => [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0,1,1,0,0]
 => ? = 1 + 1
000010001 => 111101110 => [1,1,1,1,2,1,1,2] => [1,0,1,0,1,0,1,0,1,1,0,0,1,0,1,0,1,1,0,0]
 => ? = 1 + 1
000010100 => 111101011 => [1,1,1,1,2,2,1,1] => [1,0,1,0,1,0,1,0,1,1,0,0,1,1,0,0,1,0,1,0]
 => ? = 1 + 1
000011011 => 111100100 => [1,1,1,1,3,3] => [1,0,1,0,1,0,1,0,1,1,1,0,0,0,1,1,1,0,0,0]
 => ? = 1 + 1
000011110 => 111100001 => [1,1,1,1,5,1] => [1,0,1,0,1,0,1,0,1,1,1,1,1,0,0,0,0,0,1,0]
 => ? = 1 + 1
001000001 => 110111110 => [1,1,2,1,1,1,1,2] => [1,0,1,0,1,1,0,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => ? = 1 + 1
001000100 => 110111011 => [1,1,2,1,1,2,1,1] => [1,0,1,0,1,1,0,0,1,0,1,0,1,1,0,0,1,0,1,0]
 => ? = 1 + 1
001001011 => 110110100 => [1,1,2,1,2,3] => [1,0,1,0,1,1,0,0,1,0,1,1,0,0,1,1,1,0,0,0]
 => ? = 1 + 1
001010000 => 110101111 => [1,1,2,2,1,1,1,1] => [1,0,1,0,1,1,0,0,1,1,0,0,1,0,1,0,1,0,1,0]
 => ? = 1 + 1
001010101 => 110101010 => [1,1,2,2,2,2] => [1,0,1,0,1,1,0,0,1,1,0,0,1,1,0,0,1,1,0,0]
 => ? = 1 + 1
001100011 => 110011100 => [1,1,3,1,1,3] => [1,0,1,0,1,1,1,0,0,0,1,0,1,0,1,1,1,0,0,0]
 => ? = 1 + 1
001101001 => 110010110 => [1,1,3,2,1,2] => [1,0,1,0,1,1,1,0,0,0,1,1,0,0,1,0,1,1,0,0]
 => ? = 1 + 1
001101100 => 110010011 => [1,1,3,3,1,1] => [1,0,1,0,1,1,1,0,0,0,1,1,1,0,0,0,1,0,1,0]
 => ? = 1 + 1
001110111 => 110001000 => [1,1,4,4] => [1,0,1,0,1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0]
 => ? = 1 + 1
010001110 => 101110001 => [1,2,1,1,4,1] => [1,0,1,1,0,0,1,0,1,0,1,1,1,1,0,0,0,0,1,0]
 => ? = 1 + 1
010111110 => 101000001 => [1,2,6,1] => [1,0,1,1,0,0,1,1,1,1,1,1,0,0,0,0,0,0,1,0]
 => ? = 1 + 1
011000110 => 100111001 => [1,3,1,1,3,1] => [1,0,1,1,1,0,0,0,1,0,1,0,1,1,1,0,0,0,1,0]
 => ? = 1 + 1
011011110 => 100100001 => [1,3,5,1] => [1,0,1,1,1,0,0,0,1,1,1,1,1,0,0,0,0,0,1,0]
 => ? = 1 + 1
011100010 => 100011101 => [1,4,1,1,2,1] => [1,0,1,1,1,1,0,0,0,0,1,0,1,0,1,1,0,0,1,0]
 => ? = 1 + 1
011101110 => 100010001 => [1,4,4,1] => [1,0,1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0,1,0]
 => ? = 1 + 1
Description
The position of the first down step of a Dyck path.
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: 56% ●values known / values provided: 56%●distinct values known / distinct values provided: 70%
Values
0 => 0 => [2] => [1,1,0,0]
 => 1
1 => 1 => [1,1] => [1,0,1,0]
 => 2
00 => 00 => [3] => [1,1,1,0,0,0]
 => 1
01 => 10 => [1,2] => [1,0,1,1,0,0]
 => 1
10 => 01 => [2,1] => [1,1,0,0,1,0]
 => 2
11 => 11 => [1,1,1] => [1,0,1,0,1,0]
 => 3
000 => 000 => [4] => [1,1,1,1,0,0,0,0]
 => 1
001 => 100 => [1,3] => [1,0,1,1,1,0,0,0]
 => 1
010 => 010 => [2,2] => [1,1,0,0,1,1,0,0]
 => 1
011 => 110 => [1,1,2] => [1,0,1,0,1,1,0,0]
 => 1
100 => 001 => [3,1] => [1,1,1,0,0,0,1,0]
 => 2
101 => 101 => [1,2,1] => [1,0,1,1,0,0,1,0]
 => 2
110 => 011 => [2,1,1] => [1,1,0,0,1,0,1,0]
 => 3
111 => 111 => [1,1,1,1] => [1,0,1,0,1,0,1,0]
 => 4
0000 => 0000 => [5] => [1,1,1,1,1,0,0,0,0,0]
 => 1
0001 => 1000 => [1,4] => [1,0,1,1,1,1,0,0,0,0]
 => 1
0010 => 0100 => [2,3] => [1,1,0,0,1,1,1,0,0,0]
 => 1
0011 => 1100 => [1,1,3] => [1,0,1,0,1,1,1,0,0,0]
 => 1
0100 => 0010 => [3,2] => [1,1,1,0,0,0,1,1,0,0]
 => 1
0101 => 1010 => [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
 => 1
0110 => 0110 => [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
 => 1
0111 => 1110 => [1,1,1,2] => [1,0,1,0,1,0,1,1,0,0]
 => 1
1000 => 0001 => [4,1] => [1,1,1,1,0,0,0,0,1,0]
 => 2
1001 => 1001 => [1,3,1] => [1,0,1,1,1,0,0,0,1,0]
 => 2
1010 => 0101 => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
 => 2
1011 => 1101 => [1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
 => 2
1100 => 0011 => [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
 => 3
1101 => 1011 => [1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
 => 3
1110 => 0111 => [2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
 => 4
1111 => 1111 => [1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
 => 5
00000 => 00000 => [6] => [1,1,1,1,1,1,0,0,0,0,0,0]
 => 1
00001 => 10000 => [1,5] => [1,0,1,1,1,1,1,0,0,0,0,0]
 => 1
00010 => 01000 => [2,4] => [1,1,0,0,1,1,1,1,0,0,0,0]
 => 1
00011 => 11000 => [1,1,4] => [1,0,1,0,1,1,1,1,0,0,0,0]
 => 1
00100 => 00100 => [3,3] => [1,1,1,0,0,0,1,1,1,0,0,0]
 => 1
00101 => 10100 => [1,2,3] => [1,0,1,1,0,0,1,1,1,0,0,0]
 => 1
00110 => 01100 => [2,1,3] => [1,1,0,0,1,0,1,1,1,0,0,0]
 => 1
00111 => 11100 => [1,1,1,3] => [1,0,1,0,1,0,1,1,1,0,0,0]
 => 1
01000 => 00010 => [4,2] => [1,1,1,1,0,0,0,0,1,1,0,0]
 => 1
01001 => 10010 => [1,3,2] => [1,0,1,1,1,0,0,0,1,1,0,0]
 => 1
01010 => 01010 => [2,2,2] => [1,1,0,0,1,1,0,0,1,1,0,0]
 => 1
01011 => 11010 => [1,1,2,2] => [1,0,1,0,1,1,0,0,1,1,0,0]
 => 1
01100 => 00110 => [3,1,2] => [1,1,1,0,0,0,1,0,1,1,0,0]
 => 1
01101 => 10110 => [1,2,1,2] => [1,0,1,1,0,0,1,0,1,1,0,0]
 => 1
01110 => 01110 => [2,1,1,2] => [1,1,0,0,1,0,1,0,1,1,0,0]
 => 1
01111 => 11110 => [1,1,1,1,2] => [1,0,1,0,1,0,1,0,1,1,0,0]
 => 1
10000 => 00001 => [5,1] => [1,1,1,1,1,0,0,0,0,0,1,0]
 => 2
10001 => 10001 => [1,4,1] => [1,0,1,1,1,1,0,0,0,0,1,0]
 => 2
10010 => 01001 => [2,3,1] => [1,1,0,0,1,1,1,0,0,0,1,0]
 => 2
10011 => 11001 => [1,1,3,1] => [1,0,1,0,1,1,1,0,0,0,1,0]
 => 2
0000000 => 0000000 => [8] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
 => ? = 1
0000100 => 0010000 => [3,5] => [1,1,1,0,0,0,1,1,1,1,1,0,0,0,0,0]
 => ? = 1
0001000 => 0001000 => [4,4] => [1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0]
 => ? = 1
0001100 => 0011000 => [3,1,4] => [1,1,1,0,0,0,1,0,1,1,1,1,0,0,0,0]
 => ? = 1
0010000 => 0000100 => [5,3] => [1,1,1,1,1,0,0,0,0,0,1,1,1,0,0,0]
 => ? = 1
0010100 => 0010100 => [3,2,3] => [1,1,1,0,0,0,1,1,0,0,1,1,1,0,0,0]
 => ? = 1
0011000 => 0001100 => [4,1,3] => [1,1,1,1,0,0,0,0,1,0,1,1,1,0,0,0]
 => ? = 1
0100000 => 0000010 => [6,2] => [1,1,1,1,1,1,0,0,0,0,0,0,1,1,0,0]
 => ? = 1
0100100 => 0010010 => [3,3,2] => [1,1,1,0,0,0,1,1,1,0,0,0,1,1,0,0]
 => ? = 1
0101000 => 0001010 => [4,2,2] => [1,1,1,1,0,0,0,0,1,1,0,0,1,1,0,0]
 => ? = 1
0101100 => 0011010 => [3,1,2,2] => [1,1,1,0,0,0,1,0,1,1,0,0,1,1,0,0]
 => ? = 1
0110000 => 0000110 => [5,1,2] => [1,1,1,1,1,0,0,0,0,0,1,0,1,1,0,0]
 => ? = 1
0110100 => 0010110 => [3,2,1,2] => [1,1,1,0,0,0,1,1,0,0,1,0,1,1,0,0]
 => ? = 1
0111100 => 0011110 => [3,1,1,1,2] => [1,1,1,0,0,0,1,0,1,0,1,0,1,1,0,0]
 => ? = 1
1000100 => 0010001 => [3,4,1] => [1,1,1,0,0,0,1,1,1,1,0,0,0,0,1,0]
 => ? = 2
1001000 => 0001001 => [4,3,1] => [1,1,1,1,0,0,0,0,1,1,1,0,0,0,1,0]
 => ? = 2
1001100 => 0011001 => [3,1,3,1] => [1,1,1,0,0,0,1,0,1,1,1,0,0,0,1,0]
 => ? = 2
1010000 => 0000101 => [5,2,1] => [1,1,1,1,1,0,0,0,0,0,1,1,0,0,1,0]
 => ? = 2
1010100 => 0010101 => [3,2,2,1] => [1,1,1,0,0,0,1,1,0,0,1,1,0,0,1,0]
 => ? = 2
1011000 => 0001101 => [4,1,2,1] => [1,1,1,1,0,0,0,0,1,0,1,1,0,0,1,0]
 => ? = 2
1101100 => 0011011 => [3,1,2,1,1] => [1,1,1,0,0,0,1,0,1,1,0,0,1,0,1,0]
 => ? = 3
00001110 => 01110000 => [2,1,1,5] => [1,1,0,0,1,0,1,0,1,1,1,1,1,0,0,0,0,0]
 => ? = 1
00010110 => 01101000 => [2,1,2,4] => [1,1,0,0,1,0,1,1,0,0,1,1,1,1,0,0,0,0]
 => ? = 1
00100110 => 01100100 => [2,1,3,3] => [1,1,0,0,1,0,1,1,1,0,0,0,1,1,1,0,0,0]
 => ? = 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]
 => ? = 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]
 => ? = 1
01000110 => 01100010 => [2,1,4,2] => [1,1,0,0,1,0,1,1,1,1,0,0,0,0,1,1,0,0]
 => ? = 1
01001010 => 01010010 => [2,2,3,2] => [1,1,0,0,1,1,0,0,1,1,1,0,0,0,1,1,0,0]
 => ? = 1
01010010 => 01001010 => [2,3,2,2] => [1,1,0,0,1,1,1,0,0,0,1,1,0,0,1,1,0,0]
 => ? = 1
01011110 => 01111010 => [2,1,1,1,2,2] => [1,1,0,0,1,0,1,0,1,0,1,1,0,0,1,1,0,0]
 => ? = 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]
 => ? = 1
01100010 => 01000110 => [2,4,1,2] => [1,1,0,0,1,1,1,1,0,0,0,0,1,0,1,1,0,0]
 => ? = 1
01100100 => 00100110 => [3,3,1,2] => [1,1,1,0,0,0,1,1,1,0,0,0,1,0,1,1,0,0]
 => ? = 1
01101000 => 00010110 => [4,2,1,2] => [1,1,1,1,0,0,0,0,1,1,0,0,1,0,1,1,0,0]
 => ? = 1
01101110 => 01110110 => [2,1,1,2,1,2] => [1,1,0,0,1,0,1,0,1,1,0,0,1,0,1,1,0,0]
 => ? = 1
01110000 => 00001110 => [5,1,1,2] => [1,1,1,1,1,0,0,0,0,0,1,0,1,0,1,1,0,0]
 => ? = 1
01110011 => 11001110 => [1,1,3,1,1,2] => [1,0,1,0,1,1,1,0,0,0,1,0,1,0,1,1,0,0]
 => ? = 1
01110101 => 10101110 => [1,2,2,1,1,2] => [1,0,1,1,0,0,1,1,0,0,1,0,1,0,1,1,0,0]
 => ? = 1
01110110 => 01101110 => [2,1,2,1,1,2] => [1,1,0,0,1,0,1,1,0,0,1,0,1,0,1,1,0,0]
 => ? = 1
01111001 => 10011110 => [1,3,1,1,1,2] => [1,0,1,1,1,0,0,0,1,0,1,0,1,0,1,1,0,0]
 => ? = 1
01111010 => 01011110 => [2,2,1,1,1,2] => [1,1,0,0,1,1,0,0,1,0,1,0,1,0,1,1,0,0]
 => ? = 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]
 => ? = 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]
 => ? = 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]
 => ? = 1
10011110 => 01111001 => [2,1,1,1,3,1] => [1,1,0,0,1,0,1,0,1,0,1,1,1,0,0,0,1,0]
 => ? = 2
10101110 => 01110101 => [2,1,1,2,2,1] => [1,1,0,0,1,0,1,0,1,1,0,0,1,1,0,0,1,0]
 => ? = 2
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]
 => ? = 2
11001110 => 01110011 => [2,1,1,3,1,1] => [1,1,0,0,1,0,1,0,1,1,1,0,0,0,1,0,1,0]
 => ? = 3
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]
 => ? = 8
000000000 => 000000000 => [10] => [1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0]
 => ? = 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: 42% ●values known / values provided: 42%●distinct values known / distinct values provided: 100%
Values
0 => [2] => [1,1,0,0]
 => [1,1,0,0]
 => 1
1 => [1,1] => [1,0,1,0]
 => [1,0,1,0]
 => 2
00 => [3] => [1,1,1,0,0,0]
 => [1,1,1,0,0,0]
 => 1
01 => [2,1] => [1,1,0,0,1,0]
 => [1,1,0,0,1,0]
 => 1
10 => [1,2] => [1,0,1,1,0,0]
 => [1,1,0,1,0,0]
 => 2
11 => [1,1,1] => [1,0,1,0,1,0]
 => [1,0,1,0,1,0]
 => 3
000 => [4] => [1,1,1,1,0,0,0,0]
 => [1,1,1,1,0,0,0,0]
 => 1
001 => [3,1] => [1,1,1,0,0,0,1,0]
 => [1,1,1,0,0,0,1,0]
 => 1
010 => [2,2] => [1,1,0,0,1,1,0,0]
 => [1,1,1,0,0,1,0,0]
 => 1
011 => [2,1,1] => [1,1,0,0,1,0,1,0]
 => [1,1,0,0,1,0,1,0]
 => 1
100 => [1,3] => [1,0,1,1,1,0,0,0]
 => [1,1,1,0,1,0,0,0]
 => 2
101 => [1,2,1] => [1,0,1,1,0,0,1,0]
 => [1,1,0,1,0,0,1,0]
 => 2
110 => [1,1,2] => [1,0,1,0,1,1,0,0]
 => [1,1,0,1,0,1,0,0]
 => 3
111 => [1,1,1,1] => [1,0,1,0,1,0,1,0]
 => [1,0,1,0,1,0,1,0]
 => 4
0000 => [5] => [1,1,1,1,1,0,0,0,0,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => 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
0010 => [3,2] => [1,1,1,0,0,0,1,1,0,0]
 => [1,1,1,1,0,0,0,1,0,0]
 => 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
0100 => [2,3] => [1,1,0,0,1,1,1,0,0,0]
 => [1,1,1,1,0,0,1,0,0,0]
 => 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
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
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
1000 => [1,4] => [1,0,1,1,1,1,0,0,0,0]
 => [1,1,1,1,0,1,0,0,0,0]
 => 2
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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]
 => ? = 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]
 => ? = 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]
 => ? = 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]
 => ? = 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]
 => ? = 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]
 => ? = 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]
 => ? = 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]
 => ? = 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]
 => ? = 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]
 => ? = 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]
 => ? = 2
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]
 => ? = 2
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]
 => ? = 2
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]
 => ? = 2
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]
 => ? = 2
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]
 => ? = 2
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]
 => ? = 3
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]
 => ? = 3
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]
 => ? = 7
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]
 => ? = 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]
 => ? = 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]
 => ? = 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]
 => ? = 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]
 => ? = 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]
 => ? = 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]
 => ? = 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]
 => ? = 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]
 => ? = 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]
 => ? = 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]
 => ? = 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]
 => ? = 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]
 => ? = 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]
 => ? = 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]
 => ? = 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]
 => ? = 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]
 => ? = 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]
 => ? = 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]
 => ? = 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]
 => ? = 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]
 => ? = 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]
 => ? = 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]
 => ? = 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]
 => ? = 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]
 => ? = 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]
 => ? = 1
0110010 => [2,1,3,2] => [1,1,0,0,1,0,1,1,1,0,0,0,1,1,0,0]
 => [1,1,1,1,1,0,0,1,0,1,0,0,0,1,0,0]
 => ? = 1
0110011 => [2,1,3,1,1] => [1,1,0,0,1,0,1,1,1,0,0,0,1,0,1,0]
 => [1,1,1,1,0,0,1,0,1,0,0,0,1,0,1,0]
 => ? = 1
0110100 => [2,1,2,3] => [1,1,0,0,1,0,1,1,0,0,1,1,1,0,0,0]
 => [1,1,1,1,1,0,0,1,0,1,0,0,1,0,0,0]
 => ? = 1
0110101 => [2,1,2,2,1] => [1,1,0,0,1,0,1,1,0,0,1,1,0,0,1,0]
 => [1,1,1,1,0,0,1,0,1,0,0,1,0,0,1,0]
 => ? = 1
0110110 => [2,1,2,1,2] => [1,1,0,0,1,0,1,1,0,0,1,0,1,1,0,0]
 => [1,1,1,1,0,0,1,0,1,0,0,1,0,1,0,0]
 => ? = 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: 40% ●values known / values provided: 40%●distinct values known / distinct values provided: 90%
Values
0 => [2] => [1,1,0,0]
 => []
 => 1
1 => [1,1] => [1,0,1,0]
 => [1]
 => 2
00 => [3] => [1,1,1,0,0,0]
 => []
 => 1
01 => [2,1] => [1,1,0,0,1,0]
 => [2]
 => 1
10 => [1,2] => [1,0,1,1,0,0]
 => [1,1]
 => 2
11 => [1,1,1] => [1,0,1,0,1,0]
 => [2,1]
 => 3
000 => [4] => [1,1,1,1,0,0,0,0]
 => []
 => 1
001 => [3,1] => [1,1,1,0,0,0,1,0]
 => [3]
 => 1
010 => [2,2] => [1,1,0,0,1,1,0,0]
 => [2,2]
 => 1
011 => [2,1,1] => [1,1,0,0,1,0,1,0]
 => [3,2]
 => 1
100 => [1,3] => [1,0,1,1,1,0,0,0]
 => [1,1,1]
 => 2
101 => [1,2,1] => [1,0,1,1,0,0,1,0]
 => [3,1,1]
 => 2
110 => [1,1,2] => [1,0,1,0,1,1,0,0]
 => [2,2,1]
 => 3
111 => [1,1,1,1] => [1,0,1,0,1,0,1,0]
 => [3,2,1]
 => 4
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]
 => [4]
 => 1
0010 => [3,2] => [1,1,1,0,0,0,1,1,0,0]
 => [3,3]
 => 1
0011 => [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
 => [4,3]
 => 1
0100 => [2,3] => [1,1,0,0,1,1,1,0,0,0]
 => [2,2,2]
 => 1
0101 => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
 => [4,2,2]
 => 1
0110 => [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
 => [3,3,2]
 => 1
0111 => [2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
 => [4,3,2]
 => 1
1000 => [1,4] => [1,0,1,1,1,1,0,0,0,0]
 => [1,1,1,1]
 => 2
1001 => [1,3,1] => [1,0,1,1,1,0,0,0,1,0]
 => [4,1,1,1]
 => 2
1010 => [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
 => [3,3,1,1]
 => 2
1011 => [1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
 => [4,3,1,1]
 => 2
1100 => [1,1,3] => [1,0,1,0,1,1,1,0,0,0]
 => [2,2,2,1]
 => 3
1101 => [1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
 => [4,2,2,1]
 => 3
1110 => [1,1,1,2] => [1,0,1,0,1,0,1,1,0,0]
 => [3,3,2,1]
 => 4
1111 => [1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
 => [4,3,2,1]
 => 5
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]
 => [5]
 => 1
00010 => [4,2] => [1,1,1,1,0,0,0,0,1,1,0,0]
 => [4,4]
 => 1
00011 => [4,1,1] => [1,1,1,1,0,0,0,0,1,0,1,0]
 => [5,4]
 => 1
00100 => [3,3] => [1,1,1,0,0,0,1,1,1,0,0,0]
 => [3,3,3]
 => 1
00101 => [3,2,1] => [1,1,1,0,0,0,1,1,0,0,1,0]
 => [5,3,3]
 => 1
00110 => [3,1,2] => [1,1,1,0,0,0,1,0,1,1,0,0]
 => [4,4,3]
 => 1
00111 => [3,1,1,1] => [1,1,1,0,0,0,1,0,1,0,1,0]
 => [5,4,3]
 => 1
01000 => [2,4] => [1,1,0,0,1,1,1,1,0,0,0,0]
 => [2,2,2,2]
 => 1
01001 => [2,3,1] => [1,1,0,0,1,1,1,0,0,0,1,0]
 => [5,2,2,2]
 => 1
01010 => [2,2,2] => [1,1,0,0,1,1,0,0,1,1,0,0]
 => [4,4,2,2]
 => 1
01011 => [2,2,1,1] => [1,1,0,0,1,1,0,0,1,0,1,0]
 => [5,4,2,2]
 => 1
01100 => [2,1,3] => [1,1,0,0,1,0,1,1,1,0,0,0]
 => [3,3,3,2]
 => 1
01101 => [2,1,2,1] => [1,1,0,0,1,0,1,1,0,0,1,0]
 => [5,3,3,2]
 => 1
01110 => [2,1,1,2] => [1,1,0,0,1,0,1,0,1,1,0,0]
 => [4,4,3,2]
 => 1
01111 => [2,1,1,1,1] => [1,1,0,0,1,0,1,0,1,0,1,0]
 => [5,4,3,2]
 => 1
10000 => [1,5] => [1,0,1,1,1,1,1,0,0,0,0,0]
 => [1,1,1,1,1]
 => 2
10001 => [1,4,1] => [1,0,1,1,1,1,0,0,0,0,1,0]
 => [5,1,1,1,1]
 => 2
10010 => [1,3,2] => [1,0,1,1,1,0,0,0,1,1,0,0]
 => [4,4,1,1,1]
 => 2
10011 => [1,3,1,1] => [1,0,1,1,1,0,0,0,1,0,1,0]
 => [5,4,1,1,1]
 => 2
0000101 => [5,2,1] => [1,1,1,1,1,0,0,0,0,0,1,1,0,0,1,0]
 => [7,5,5]
 => ? = 1
0000110 => [5,1,2] => [1,1,1,1,1,0,0,0,0,0,1,0,1,1,0,0]
 => [6,6,5]
 => ? = 1
0001001 => [4,3,1] => [1,1,1,1,0,0,0,0,1,1,1,0,0,0,1,0]
 => [7,4,4,4]
 => ? = 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]
 => ? = 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]
 => ? = 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]
 => ? = 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]
 => ? = 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]
 => ? = 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]
 => ? = 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]
 => ? = 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]
 => ? = 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]
 => ? = 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]
 => ? = 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]
 => ? = 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]
 => ? = 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]
 => ? = 1
0011110 => [3,1,1,1,2] => [1,1,1,0,0,0,1,0,1,0,1,0,1,1,0,0]
 => [6,6,5,4,3]
 => ? = 1
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]
 => ? = 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]
 => ? = 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]
 => ? = 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]
 => ? = 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]
 => ? = 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]
 => ? = 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]
 => ? = 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]
 => ? = 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]
 => ? = 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]
 => ? = 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]
 => ? = 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]
 => ? = 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]
 => ? = 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]
 => ? = 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]
 => ? = 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]
 => ? = 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]
 => ? = 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]
 => ? = 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]
 => ? = 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]
 => ? = 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]
 => ? = 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]
 => ? = 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]
 => ? = 1
0111001 => [2,1,1,3,1] => [1,1,0,0,1,0,1,0,1,1,1,0,0,0,1,0]
 => [7,4,4,4,3,2]
 => ? = 1
0111010 => [2,1,1,2,2] => [1,1,0,0,1,0,1,0,1,1,0,0,1,1,0,0]
 => [6,6,4,4,3,2]
 => ? = 1
0111011 => [2,1,1,2,1,1] => [1,1,0,0,1,0,1,0,1,1,0,0,1,0,1,0]
 => [7,6,4,4,3,2]
 => ? = 1
0111100 => [2,1,1,1,3] => [1,1,0,0,1,0,1,0,1,0,1,1,1,0,0,0]
 => [5,5,5,4,3,2]
 => ? = 1
0111110 => [2,1,1,1,1,2] => [1,1,0,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => [6,6,5,4,3,2]
 => ? = 1
1000001 => [1,6,1] => [1,0,1,1,1,1,1,1,0,0,0,0,0,0,1,0]
 => [7,1,1,1,1,1,1]
 => ? = 2
1000010 => [1,5,2] => [1,0,1,1,1,1,1,0,0,0,0,0,1,1,0,0]
 => [6,6,1,1,1,1,1]
 => ? = 2
1000100 => [1,4,3] => [1,0,1,1,1,1,0,0,0,0,1,1,1,0,0,0]
 => [5,5,5,1,1,1,1]
 => ? = 2
1000101 => [1,4,2,1] => [1,0,1,1,1,1,0,0,0,0,1,1,0,0,1,0]
 => [7,5,5,1,1,1,1]
 => ? = 2
1000110 => [1,4,1,2] => [1,0,1,1,1,1,0,0,0,0,1,0,1,1,0,0]
 => [6,6,5,1,1,1,1]
 => ? = 2
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: 37% ●values known / values provided: 37%●distinct values known / distinct values provided: 70%
Values
0 => [2] => [1,1,0,0]
 => [1,1,0,0]
 => 1
1 => [1,1] => [1,0,1,0]
 => [1,0,1,0]
 => 2
00 => [3] => [1,1,1,0,0,0]
 => [1,1,1,0,0,0]
 => 1
01 => [2,1] => [1,1,0,0,1,0]
 => [1,1,0,0,1,0]
 => 1
10 => [1,2] => [1,0,1,1,0,0]
 => [1,1,0,1,0,0]
 => 2
11 => [1,1,1] => [1,0,1,0,1,0]
 => [1,0,1,0,1,0]
 => 3
000 => [4] => [1,1,1,1,0,0,0,0]
 => [1,1,1,1,0,0,0,0]
 => 1
001 => [3,1] => [1,1,1,0,0,0,1,0]
 => [1,1,1,0,0,0,1,0]
 => 1
010 => [2,2] => [1,1,0,0,1,1,0,0]
 => [1,1,1,0,0,1,0,0]
 => 1
011 => [2,1,1] => [1,1,0,0,1,0,1,0]
 => [1,1,0,0,1,0,1,0]
 => 1
100 => [1,3] => [1,0,1,1,1,0,0,0]
 => [1,1,1,0,1,0,0,0]
 => 2
101 => [1,2,1] => [1,0,1,1,0,0,1,0]
 => [1,1,0,1,0,0,1,0]
 => 2
110 => [1,1,2] => [1,0,1,0,1,1,0,0]
 => [1,1,0,1,0,1,0,0]
 => 3
111 => [1,1,1,1] => [1,0,1,0,1,0,1,0]
 => [1,0,1,0,1,0,1,0]
 => 4
0000 => [5] => [1,1,1,1,1,0,0,0,0,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => 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
0010 => [3,2] => [1,1,1,0,0,0,1,1,0,0]
 => [1,1,1,1,0,0,0,1,0,0]
 => 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
0100 => [2,3] => [1,1,0,0,1,1,1,0,0,0]
 => [1,1,1,1,0,0,1,0,0,0]
 => 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
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
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
1000 => [1,4] => [1,0,1,1,1,1,0,0,0,0]
 => [1,1,1,1,0,1,0,0,0,0]
 => 2
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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]
 => ? = 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]
 => ? = 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]
 => ? = 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]
 => ? = 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]
 => ? = 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]
 => ? = 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]
 => ? = 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]
 => ? = 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]
 => ? = 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]
 => ? = 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]
 => ? = 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]
 => ? = 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]
 => ? = 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]
 => ? = 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]
 => ? = 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]
 => ? = 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]
 => ? = 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]
 => ? = 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]
 => ? = 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]
 => ? = 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]
 => ? = 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]
 => ? = 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]
 => ? = 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]
 => ? = 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]
 => ? = 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]
 => ? = 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]
 => ? = 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]
 => ? = 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]
 => ? = 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]
 => ? = 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]
 => ? = 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]
 => ? = 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]
 => ? = 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]
 => ? = 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]
 => ? = 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]
 => ? = 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]
 => ? = 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]
 => ? = 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]
 => ? = 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]
 => ? = 1
0110010 => [2,1,3,2] => [1,1,0,0,1,0,1,1,1,0,0,0,1,1,0,0]
 => [1,1,1,1,1,0,0,1,0,1,0,0,0,1,0,0]
 => ? = 1
0110011 => [2,1,3,1,1] => [1,1,0,0,1,0,1,1,1,0,0,0,1,0,1,0]
 => [1,1,1,1,0,0,1,0,1,0,0,0,1,0,1,0]
 => ? = 1
0110100 => [2,1,2,3] => [1,1,0,0,1,0,1,1,0,0,1,1,1,0,0,0]
 => [1,1,1,1,1,0,0,1,0,1,0,0,1,0,0,0]
 => ? = 1
0110101 => [2,1,2,2,1] => [1,1,0,0,1,0,1,1,0,0,1,1,0,0,1,0]
 => [1,1,1,1,0,0,1,0,1,0,0,1,0,0,1,0]
 => ? = 1
0110110 => [2,1,2,1,2] => [1,1,0,0,1,0,1,1,0,0,1,0,1,1,0,0]
 => [1,1,1,1,0,0,1,0,1,0,0,1,0,1,0,0]
 => ? = 1
0110111 => [2,1,2,1,1,1] => [1,1,0,0,1,0,1,1,0,0,1,0,1,0,1,0]
 => [1,1,1,0,0,1,0,1,0,0,1,0,1,0,1,0]
 => ? = 1
0111001 => [2,1,1,3,1] => [1,1,0,0,1,0,1,0,1,1,1,0,0,0,1,0]
 => [1,1,1,1,0,0,1,0,1,0,1,0,0,0,1,0]
 => ? = 1
0111010 => [2,1,1,2,2] => [1,1,0,0,1,0,1,0,1,1,0,0,1,1,0,0]
 => [1,1,1,1,0,0,1,0,1,0,1,0,0,1,0,0]
 => ? = 1
0111011 => [2,1,1,2,1,1] => [1,1,0,0,1,0,1,0,1,1,0,0,1,0,1,0]
 => [1,1,1,0,0,1,0,1,0,1,0,0,1,0,1,0]
 => ? = 1
0111100 => [2,1,1,1,3] => [1,1,0,0,1,0,1,0,1,0,1,1,1,0,0,0]
 => [1,1,1,1,0,0,1,0,1,0,1,0,1,0,0,0]
 => ? = 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: 36% ●values known / values provided: 36%●distinct values known / distinct values provided: 70%
Values
0 => 1 => [1,1] => [1,0,1,0]
 => 1
1 => 0 => [2] => [1,1,0,0]
 => 2
00 => 11 => [1,1,1] => [1,0,1,0,1,0]
 => 1
01 => 10 => [1,2] => [1,0,1,1,0,0]
 => 1
10 => 01 => [2,1] => [1,1,0,0,1,0]
 => 2
11 => 00 => [3] => [1,1,1,0,0,0]
 => 3
000 => 111 => [1,1,1,1] => [1,0,1,0,1,0,1,0]
 => 1
001 => 110 => [1,1,2] => [1,0,1,0,1,1,0,0]
 => 1
010 => 101 => [1,2,1] => [1,0,1,1,0,0,1,0]
 => 1
011 => 100 => [1,3] => [1,0,1,1,1,0,0,0]
 => 1
100 => 011 => [2,1,1] => [1,1,0,0,1,0,1,0]
 => 2
101 => 010 => [2,2] => [1,1,0,0,1,1,0,0]
 => 2
110 => 001 => [3,1] => [1,1,1,0,0,0,1,0]
 => 3
111 => 000 => [4] => [1,1,1,1,0,0,0,0]
 => 4
0000 => 1111 => [1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
 => 1
0001 => 1110 => [1,1,1,2] => [1,0,1,0,1,0,1,1,0,0]
 => 1
0010 => 1101 => [1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
 => 1
0011 => 1100 => [1,1,3] => [1,0,1,0,1,1,1,0,0,0]
 => 1
0100 => 1011 => [1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
 => 1
0101 => 1010 => [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
 => 1
0110 => 1001 => [1,3,1] => [1,0,1,1,1,0,0,0,1,0]
 => 1
0111 => 1000 => [1,4] => [1,0,1,1,1,1,0,0,0,0]
 => 1
1000 => 0111 => [2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
 => 2
1001 => 0110 => [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
 => 2
1010 => 0101 => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
 => 2
1011 => 0100 => [2,3] => [1,1,0,0,1,1,1,0,0,0]
 => 2
1100 => 0011 => [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
 => 3
1101 => 0010 => [3,2] => [1,1,1,0,0,0,1,1,0,0]
 => 3
1110 => 0001 => [4,1] => [1,1,1,1,0,0,0,0,1,0]
 => 4
1111 => 0000 => [5] => [1,1,1,1,1,0,0,0,0,0]
 => 5
00000 => 11111 => [1,1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0,1,0]
 => 1
00001 => 11110 => [1,1,1,1,2] => [1,0,1,0,1,0,1,0,1,1,0,0]
 => 1
00010 => 11101 => [1,1,1,2,1] => [1,0,1,0,1,0,1,1,0,0,1,0]
 => 1
00011 => 11100 => [1,1,1,3] => [1,0,1,0,1,0,1,1,1,0,0,0]
 => 1
00100 => 11011 => [1,1,2,1,1] => [1,0,1,0,1,1,0,0,1,0,1,0]
 => 1
00101 => 11010 => [1,1,2,2] => [1,0,1,0,1,1,0,0,1,1,0,0]
 => 1
00110 => 11001 => [1,1,3,1] => [1,0,1,0,1,1,1,0,0,0,1,0]
 => 1
00111 => 11000 => [1,1,4] => [1,0,1,0,1,1,1,1,0,0,0,0]
 => 1
01000 => 10111 => [1,2,1,1,1] => [1,0,1,1,0,0,1,0,1,0,1,0]
 => 1
01001 => 10110 => [1,2,1,2] => [1,0,1,1,0,0,1,0,1,1,0,0]
 => 1
01010 => 10101 => [1,2,2,1] => [1,0,1,1,0,0,1,1,0,0,1,0]
 => 1
01011 => 10100 => [1,2,3] => [1,0,1,1,0,0,1,1,1,0,0,0]
 => 1
01100 => 10011 => [1,3,1,1] => [1,0,1,1,1,0,0,0,1,0,1,0]
 => 1
01101 => 10010 => [1,3,2] => [1,0,1,1,1,0,0,0,1,1,0,0]
 => 1
01110 => 10001 => [1,4,1] => [1,0,1,1,1,1,0,0,0,0,1,0]
 => 1
01111 => 10000 => [1,5] => [1,0,1,1,1,1,1,0,0,0,0,0]
 => 1
10000 => 01111 => [2,1,1,1,1] => [1,1,0,0,1,0,1,0,1,0,1,0]
 => 2
10001 => 01110 => [2,1,1,2] => [1,1,0,0,1,0,1,0,1,1,0,0]
 => 2
10010 => 01101 => [2,1,2,1] => [1,1,0,0,1,0,1,1,0,0,1,0]
 => 2
10011 => 01100 => [2,1,3] => [1,1,0,0,1,0,1,1,1,0,0,0]
 => 2
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]
 => ? = 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]
 => ? = 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]
 => ? = 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]
 => ? = 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]
 => ? = 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]
 => ? = 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]
 => ? = 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]
 => ? = 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]
 => ? = 1
0001101 => 1110010 => [1,1,1,3,2] => [1,0,1,0,1,0,1,1,1,0,0,0,1,1,0,0]
 => ? = 1
0001111 => 1110000 => [1,1,1,5] => [1,0,1,0,1,0,1,1,1,1,1,0,0,0,0,0]
 => ? = 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]
 => ? = 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]
 => ? = 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]
 => ? = 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]
 => ? = 1
0010101 => 1101010 => [1,1,2,2,2] => [1,0,1,0,1,1,0,0,1,1,0,0,1,1,0,0]
 => ? = 1
0010110 => 1101001 => [1,1,2,3,1] => [1,0,1,0,1,1,0,0,1,1,1,0,0,0,1,0]
 => ? = 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]
 => ? = 1
0011001 => 1100110 => [1,1,3,1,2] => [1,0,1,0,1,1,1,0,0,0,1,0,1,1,0,0]
 => ? = 1
0011010 => 1100101 => [1,1,3,2,1] => [1,0,1,0,1,1,1,0,0,0,1,1,0,0,1,0]
 => ? = 1
0011011 => 1100100 => [1,1,3,3] => [1,0,1,0,1,1,1,0,0,0,1,1,1,0,0,0]
 => ? = 1
0011110 => 1100001 => [1,1,5,1] => [1,0,1,0,1,1,1,1,1,0,0,0,0,0,1,0]
 => ? = 1
0011111 => 1100000 => [1,1,6] => [1,0,1,0,1,1,1,1,1,1,0,0,0,0,0,0]
 => ? = 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]
 => ? = 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]
 => ? = 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]
 => ? = 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]
 => ? = 1
0100101 => 1011010 => [1,2,1,2,2] => [1,0,1,1,0,0,1,0,1,1,0,0,1,1,0,0]
 => ? = 1
0100110 => 1011001 => [1,2,1,3,1] => [1,0,1,1,0,0,1,0,1,1,1,0,0,0,1,0]
 => ? = 1
0100111 => 1011000 => [1,2,1,4] => [1,0,1,1,0,0,1,0,1,1,1,1,0,0,0,0]
 => ? = 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]
 => ? = 1
0101001 => 1010110 => [1,2,2,1,2] => [1,0,1,1,0,0,1,1,0,0,1,0,1,1,0,0]
 => ? = 1
0101010 => 1010101 => [1,2,2,2,1] => [1,0,1,1,0,0,1,1,0,0,1,1,0,0,1,0]
 => ? = 1
0101011 => 1010100 => [1,2,2,3] => [1,0,1,1,0,0,1,1,0,0,1,1,1,0,0,0]
 => ? = 1
0101100 => 1010011 => [1,2,3,1,1] => [1,0,1,1,0,0,1,1,1,0,0,0,1,0,1,0]
 => ? = 1
0101101 => 1010010 => [1,2,3,2] => [1,0,1,1,0,0,1,1,1,0,0,0,1,1,0,0]
 => ? = 1
0101110 => 1010001 => [1,2,4,1] => [1,0,1,1,0,0,1,1,1,1,0,0,0,0,1,0]
 => ? = 1
0101111 => 1010000 => [1,2,5] => [1,0,1,1,0,0,1,1,1,1,1,0,0,0,0,0]
 => ? = 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]
 => ? = 1
0110001 => 1001110 => [1,3,1,1,2] => [1,0,1,1,1,0,0,0,1,0,1,0,1,1,0,0]
 => ? = 1
0110010 => 1001101 => [1,3,1,2,1] => [1,0,1,1,1,0,0,0,1,0,1,1,0,0,1,0]
 => ? = 1
0110011 => 1001100 => [1,3,1,3] => [1,0,1,1,1,0,0,0,1,0,1,1,1,0,0,0]
 => ? = 1
0110100 => 1001011 => [1,3,2,1,1] => [1,0,1,1,1,0,0,0,1,1,0,0,1,0,1,0]
 => ? = 1
0110101 => 1001010 => [1,3,2,2] => [1,0,1,1,1,0,0,0,1,1,0,0,1,1,0,0]
 => ? = 1
0110110 => 1001001 => [1,3,3,1] => [1,0,1,1,1,0,0,0,1,1,1,0,0,0,1,0]
 => ? = 1
0110111 => 1001000 => [1,3,4] => [1,0,1,1,1,0,0,0,1,1,1,1,0,0,0,0]
 => ? = 1
0111001 => 1000110 => [1,4,1,2] => [1,0,1,1,1,1,0,0,0,0,1,0,1,1,0,0]
 => ? = 1
0111010 => 1000101 => [1,4,2,1] => [1,0,1,1,1,1,0,0,0,0,1,1,0,0,1,0]
 => ? = 1
0111011 => 1000100 => [1,4,3] => [1,0,1,1,1,1,0,0,0,0,1,1,1,0,0,0]
 => ? = 1
0111100 => 1000011 => [1,5,1,1] => [1,0,1,1,1,1,1,0,0,0,0,0,1,0,1,0]
 => ? = 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. 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. 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. St000054The first entry of the permutation.