Your data matches 34 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)
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: 100% ●values known / values provided: 100%●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
Description
The number of global maxima 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 = 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
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
1 => 0 => 2
00 => 11 => 1
01 => 10 => 1
10 => 01 => 2
11 => 00 => 3
000 => 111 => 1
001 => 110 => 1
010 => 101 => 1
011 => 100 => 1
100 => 011 => 2
101 => 010 => 2
110 => 001 => 3
111 => 000 => 4
0000 => 1111 => 1
0001 => 1110 => 1
0010 => 1101 => 1
0011 => 1100 => 1
0100 => 1011 => 1
0101 => 1010 => 1
0110 => 1001 => 1
0111 => 1000 => 1
1000 => 0111 => 2
1001 => 0110 => 2
1010 => 0101 => 2
1011 => 0100 => 2
1100 => 0011 => 3
1101 => 0010 => 3
1110 => 0001 => 4
1111 => 0000 => 5
00000 => 11111 => 1
00001 => 11110 => 1
00010 => 11101 => 1
00011 => 11100 => 1
00100 => 11011 => 1
00101 => 11010 => 1
00110 => 11001 => 1
00111 => 11000 => 1
01000 => 10111 => 1
01001 => 10110 => 1
01010 => 10101 => 1
01011 => 10100 => 1
01100 => 10011 => 1
01101 => 10010 => 1
01110 => 10001 => 1
01111 => 10000 => 1
10000 => 01111 => 2
10001 => 01110 => 2
10010 => 01101 => 2
10011 => 01100 => 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: 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: 98% ●values known / values provided: 98%●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
100000010 => 011111101 => [2,1,1,1,1,1,2,1] => ? = 2
111111001 => 000000110 => [7,1,2] => ? = 7
1000000000 => 0111111111 => [2,1,1,1,1,1,1,1,1,1] => ? = 2
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: 86% ●values known / values provided: 86%●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
1100010 => 0011101 => [3,1,1,2,1] => [1,1,1,0,0,0,1,0,1,0,1,1,0,0,1,0]
 => ? = 3 + 1
1100111 => 0011000 => [3,1,4] => [1,1,1,0,0,0,1,0,1,1,1,1,0,0,0,0]
 => ? = 3 + 1
1110000 => 0001111 => [4,1,1,1,1] => [1,1,1,1,0,0,0,0,1,0,1,0,1,0,1,0]
 => ? = 4 + 1
1110011 => 0001100 => [4,1,3] => [1,1,1,1,0,0,0,0,1,0,1,1,1,0,0,0]
 => ? = 4 + 1
1111001 => 0000110 => [5,1,2] => [1,1,1,1,1,0,0,0,0,0,1,0,1,1,0,0]
 => ? = 5 + 1
1111010 => 0000101 => [5,2,1] => [1,1,1,1,1,0,0,0,0,0,1,1,0,0,1,0]
 => ? = 5 + 1
1111011 => 0000100 => [5,3] => [1,1,1,1,1,0,0,0,0,0,1,1,1,0,0,0]
 => ? = 5 + 1
1111100 => 0000011 => [6,1,1] => [1,1,1,1,1,1,0,0,0,0,0,0,1,0,1,0]
 => ? = 6 + 1
00000000 => 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]
 => ? = 1 + 1
00000010 => 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]
 => ? = 1 + 1
00000100 => 11111011 => [1,1,1,1,1,2,1,1] => [1,0,1,0,1,0,1,0,1,0,1,1,0,0,1,0,1,0]
 => ? = 1 + 1
01000000 => 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]
 => ? = 1 + 1
01000010 => 10111101 => [1,2,1,1,1,2,1] => [1,0,1,1,0,0,1,0,1,0,1,0,1,1,0,0,1,0]
 => ? = 1 + 1
11110011 => 00001100 => [5,1,3] => [1,1,1,1,1,0,0,0,0,0,1,0,1,1,1,0,0,0]
 => ? = 5 + 1
11111001 => 00000110 => [6,1,2] => [1,1,1,1,1,1,0,0,0,0,0,0,1,0,1,1,0,0]
 => ? = 6 + 1
11111010 => 00000101 => [6,2,1] => [1,1,1,1,1,1,0,0,0,0,0,0,1,1,0,0,1,0]
 => ? = 6 + 1
11111011 => 00000100 => [6,3] => [1,1,1,1,1,1,0,0,0,0,0,0,1,1,1,0,0,0]
 => ? = 6 + 1
11111100 => 00000011 => [7,1,1] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0,1,0]
 => ? = 7 + 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
000000010 => 111111101 => [1,1,1,1,1,1,1,2,1] => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0,1,0]
 => ? = 1 + 1
010000000 => 101111111 => [1,2,1,1,1,1,1,1,1] => [1,0,1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => ? = 1 + 1
100000010 => 011111101 => [2,1,1,1,1,1,2,1] => [1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0,1,0]
 => ? = 2 + 1
101010101 => 010101010 => [2,2,2,2,2] => [1,1,0,0,1,1,0,0,1,1,0,0,1,1,0,0,1,1,0,0]
 => ? = 2 + 1
111111001 => 000000110 => [7,1,2] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0,1,1,0,0]
 => ? = 7 + 1
111111100 => 000000011 => [8,1,1] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0,1,0]
 => ? = 8 + 1
1000000000 => 0111111111 => [2,1,1,1,1,1,1,1,1,1] => ?
 => ? = 2 + 1
10101010101 => 01010101010 => [2,2,2,2,2,2] => [1,1,0,0,1,1,0,0,1,1,0,0,1,1,0,0,1,1,0,0,1,1,0,0]
 => ? = 2 + 1
Description
The position of the first down step of a Dyck path.
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: 78% ●values known / values provided: 78%●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
0000001 => [7,1] => [2,1,1,1,1,1,1] => ? = 1
0111101 => [2,1,1,1,2,1] => [2,5,1] => ? = 1
1000000 => [1,7] => [1,1,1,1,1,1,2] => ? = 2
1100000 => [1,1,6] => [1,1,1,1,1,3] => ? = 3
1100001 => [1,1,5,1] => [2,1,1,1,3] => ? = 3
1100010 => [1,1,4,2] => [1,2,1,1,3] => ? = 3
1100111 => [1,1,3,1,1,1] => [4,1,3] => ? = 3
1110000 => [1,1,1,5] => [1,1,1,1,4] => ? = 4
1110011 => [1,1,1,3,1,1] => [3,1,4] => ? = 4
1111001 => [1,1,1,1,3,1] => [2,1,5] => ? = 5
1111010 => [1,1,1,1,2,2] => [1,2,5] => ? = 5
1111011 => [1,1,1,1,2,1,1] => [3,5] => ? = 5
1111100 => [1,1,1,1,1,3] => [1,1,6] => ? = 6
1111101 => [1,1,1,1,1,2,1] => [2,6] => ? = 6
00000000 => [9] => [1,1,1,1,1,1,1,1,1] => ? = 1
00000001 => [8,1] => [2,1,1,1,1,1,1,1] => ? = 1
00000010 => [7,2] => [1,2,1,1,1,1,1,1] => ? = 1
00000100 => [6,3] => [1,1,2,1,1,1,1,1] => ? = 1
01000000 => [2,7] => [1,1,1,1,1,1,2,1] => ? = 1
01000001 => [2,6,1] => [2,1,1,1,1,2,1] => ? = 1
01000010 => [2,5,2] => [1,2,1,1,1,2,1] => ? = 1
01111101 => [2,1,1,1,1,2,1] => [2,6,1] => ? = 1
10000000 => [1,8] => [1,1,1,1,1,1,1,2] => ? = 2
10000001 => [1,7,1] => [2,1,1,1,1,1,2] => ? = 2
10000010 => [1,6,2] => [1,2,1,1,1,1,2] => ? = 2
11110011 => [1,1,1,1,3,1,1] => [3,1,5] => ? = 5
11111001 => [1,1,1,1,1,3,1] => [2,1,6] => ? = 6
11111010 => [1,1,1,1,1,2,2] => [1,2,6] => ? = 6
11111011 => [1,1,1,1,1,2,1,1] => [3,6] => ? = 6
11111100 => [1,1,1,1,1,1,3] => [1,1,7] => ? = 7
11111101 => [1,1,1,1,1,1,2,1] => [2,7] => ? = 7
000000001 => [9,1] => [2,1,1,1,1,1,1,1,1] => ? = 1
000000010 => [8,2] => [1,2,1,1,1,1,1,1,1] => ? = 1
010000000 => [2,8] => [1,1,1,1,1,1,1,2,1] => ? = 1
100000000 => [1,9] => [1,1,1,1,1,1,1,1,2] => ? = 2
100000010 => [1,7,2] => [1,2,1,1,1,1,1,2] => ? = 2
101010101 => [1,2,2,2,2,1] => [2,2,2,2,2] => ? = 2
111111001 => [1,1,1,1,1,1,3,1] => [2,1,7] => ? = 7
111111100 => [1,1,1,1,1,1,1,3] => [1,1,8] => ? = 8
111111101 => [1,1,1,1,1,1,1,2,1] => [2,8] => ? = 8
1000000000 => [1,10] => [1,1,1,1,1,1,1,1,1,2] => ? = 2
0000000001 => [10,1] => [2,1,1,1,1,1,1,1,1,1] => ? = 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: 70% ●values known / values provided: 70%●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
0010000 => 0000100 => [5,3] => [1,1,1,1,1,0,0,0,0,0,1,1,1,0,0,0]
 => ? = 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
0101000 => 0001010 => [4,2,2] => [1,1,1,1,0,0,0,0,1,1,0,0,1,1,0,0]
 => ? = 1
0110000 => 0000110 => [5,1,2] => [1,1,1,1,1,0,0,0,0,0,1,0,1,1,0,0]
 => ? = 1
0111100 => 0011110 => [3,1,1,1,2] => [1,1,1,0,0,0,1,0,1,0,1,0,1,1,0,0]
 => ? = 1
1000000 => 0000001 => [7,1] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
 => ? = 2
1001000 => 0001001 => [4,3,1] => [1,1,1,1,0,0,0,0,1,1,1,0,0,0,1,0]
 => ? = 2
1010000 => 0000101 => [5,2,1] => [1,1,1,1,1,0,0,0,0,0,1,1,0,0,1,0]
 => ? = 2
1100000 => 0000011 => [6,1,1] => [1,1,1,1,1,1,0,0,0,0,0,0,1,0,1,0]
 => ? = 3
1110000 => 0000111 => [5,1,1,1] => [1,1,1,1,1,0,0,0,0,0,1,0,1,0,1,0]
 => ? = 4
1111100 => 0011111 => [3,1,1,1,1,1] => [1,1,1,0,0,0,1,0,1,0,1,0,1,0,1,0]
 => ? = 6
00000000 => 00000000 => [9] => [1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0]
 => ? = 1
00000001 => 10000000 => [1,8] => [1,0,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
 => ? = 1
00000010 => 01000000 => [2,7] => [1,1,0,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
 => ? = 1
00000100 => 00100000 => [3,6] => [1,1,1,0,0,0,1,1,1,1,1,1,0,0,0,0,0,0]
 => ? = 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
01000000 => 00000010 => [7,2] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,1,0,0]
 => ? = 1
01000001 => 10000010 => [1,6,2] => [1,0,1,1,1,1,1,1,0,0,0,0,0,0,1,1,0,0]
 => ? = 1
01000010 => 01000010 => [2,5,2] => [1,1,0,0,1,1,1,1,1,0,0,0,0,0,1,1,0,0]
 => ? = 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]
 => ? = 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
10000000 => 00000001 => [8,1] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0]
 => ? = 2
10000001 => 10000001 => [1,7,1] => [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
 => ? = 2
10000010 => 01000001 => [2,6,1] => [1,1,0,0,1,1,1,1,1,1,0,0,0,0,0,0,1,0]
 => ? = 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
11110011 => 11001111 => [1,1,3,1,1,1,1] => [1,0,1,0,1,1,1,0,0,0,1,0,1,0,1,0,1,0]
 => ? = 5
11111001 => 10011111 => [1,3,1,1,1,1,1] => [1,0,1,1,1,0,0,0,1,0,1,0,1,0,1,0,1,0]
 => ? = 6
11111010 => 01011111 => [2,2,1,1,1,1,1] => [1,1,0,0,1,1,0,0,1,0,1,0,1,0,1,0,1,0]
 => ? = 6
11111011 => 11011111 => [1,1,2,1,1,1,1,1] => [1,0,1,0,1,1,0,0,1,0,1,0,1,0,1,0,1,0]
 => ? = 6
11111100 => 00111111 => [3,1,1,1,1,1,1] => [1,1,1,0,0,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => ? = 7
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]
 => ? = 7
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
000000001 => 100000000 => [1,9] => [1,0,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0]
 => ? = 1
000000010 => 010000000 => [2,8] => [1,1,0,0,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
 => ? = 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]
 => ? = 1
010000000 => 000000010 => [8,2] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,1,0,0]
 => ? = 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]
 => ? = 1
011111111 => 111111110 => [1,1,1,1,1,1,1,1,2] => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => ? = 1
100000000 => 000000001 => [9,1] => [1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,1,0]
 => ? = 2
100000001 => 100000001 => [1,8,1] => [1,0,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0]
 => ? = 2
100000010 => 010000001 => [2,7,1] => [1,1,0,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
 => ? = 2
101010101 => 101010101 => [1,2,2,2,2,1] => [1,0,1,1,0,0,1,1,0,0,1,1,0,0,1,1,0,0,1,0]
 => ? = 2
111111001 => 100111111 => [1,3,1,1,1,1,1,1] => [1,0,1,1,1,0,0,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => ? = 7
111111100 => 001111111 => [3,1,1,1,1,1,1,1] => [1,1,1,0,0,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => ? = 8
Description
The number of up steps after the last double rise 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: 68% ●values known / values provided: 68%●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
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
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
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
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
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
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
0111101 => [2,1,1,1,2,1] => [1,1,0,0,1,0,1,0,1,0,1,1,0,0,1,0]
 => [7,5,5,4,3,2]
 => ? = 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
1001000 => [1,3,4] => [1,0,1,1,1,0,0,0,1,1,1,1,0,0,0,0]
 => [4,4,4,4,1,1,1]
 => ? = 2
1010000 => [1,2,5] => [1,0,1,1,0,0,1,1,1,1,1,0,0,0,0,0]
 => [3,3,3,3,3,1,1]
 => ? = 2
1010101 => [1,2,2,2,1] => [1,0,1,1,0,0,1,1,0,0,1,1,0,0,1,0]
 => [7,5,5,3,3,1,1]
 => ? = 2
1011110 => [1,2,1,1,1,2] => [1,0,1,1,0,0,1,0,1,0,1,0,1,1,0,0]
 => [6,6,5,4,3,1,1]
 => ? = 2
1100000 => [1,1,6] => [1,0,1,0,1,1,1,1,1,1,0,0,0,0,0,0]
 => [2,2,2,2,2,2,1]
 => ? = 3
1100001 => [1,1,5,1] => [1,0,1,0,1,1,1,1,1,0,0,0,0,0,1,0]
 => [7,2,2,2,2,2,1]
 => ? = 3
1100010 => [1,1,4,2] => [1,0,1,0,1,1,1,1,0,0,0,0,1,1,0,0]
 => [6,6,2,2,2,2,1]
 => ? = 3
1100111 => [1,1,3,1,1,1] => [1,0,1,0,1,1,1,0,0,0,1,0,1,0,1,0]
 => [7,6,5,2,2,2,1]
 => ? = 3
1101110 => [1,1,2,1,1,2] => [1,0,1,0,1,1,0,0,1,0,1,0,1,1,0,0]
 => [6,6,5,4,2,2,1]
 => ? = 3
1110000 => [1,1,1,5] => [1,0,1,0,1,0,1,1,1,1,1,0,0,0,0,0]
 => [3,3,3,3,3,2,1]
 => ? = 4
1110011 => [1,1,1,3,1,1] => [1,0,1,0,1,0,1,1,1,0,0,0,1,0,1,0]
 => [7,6,3,3,3,2,1]
 => ? = 4
1110111 => [1,1,1,2,1,1,1] => [1,0,1,0,1,0,1,1,0,0,1,0,1,0,1,0]
 => [7,6,5,3,3,2,1]
 => ? = 4
1111001 => [1,1,1,1,3,1] => [1,0,1,0,1,0,1,0,1,1,1,0,0,0,1,0]
 => [7,4,4,4,3,2,1]
 => ? = 5
1111010 => [1,1,1,1,2,2] => [1,0,1,0,1,0,1,0,1,1,0,0,1,1,0,0]
 => [6,6,4,4,3,2,1]
 => ? = 5
1111011 => [1,1,1,1,2,1,1] => [1,0,1,0,1,0,1,0,1,1,0,0,1,0,1,0]
 => [7,6,4,4,3,2,1]
 => ? = 5
1111101 => [1,1,1,1,1,2,1] => [1,0,1,0,1,0,1,0,1,0,1,1,0,0,1,0]
 => [7,5,5,4,3,2,1]
 => ? = 6
00000010 => [7,2] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,1,0,0]
 => [7,7]
 => ? = 1
00000100 => [6,3] => [1,1,1,1,1,1,0,0,0,0,0,0,1,1,1,0,0,0]
 => [6,6,6]
 => ? = 1
00111111 => [3,1,1,1,1,1,1] => [1,1,1,0,0,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => [8,7,6,5,4,3]
 => ? = 1
01000000 => [2,7] => [1,1,0,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
 => [2,2,2,2,2,2,2]
 => ? = 1
01000001 => [2,6,1] => [1,1,0,0,1,1,1,1,1,1,0,0,0,0,0,0,1,0]
 => [8,2,2,2,2,2,2]
 => ? = 1
01000010 => [2,5,2] => [1,1,0,0,1,1,1,1,1,0,0,0,0,0,1,1,0,0]
 => [7,7,2,2,2,2,2]
 => ? = 1
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]
 => [8,6,6,5,4,3,2]
 => ? = 1
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]
 => [7,7,6,5,4,3,2]
 => ? = 1
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]
 => [8,7,6,5,4,3,2]
 => ? = 1
10000001 => [1,7,1] => [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
 => [8,1,1,1,1,1,1,1]
 => ? = 2
10000010 => [1,6,2] => [1,0,1,1,1,1,1,1,0,0,0,0,0,0,1,1,0,0]
 => [7,7,1,1,1,1,1,1]
 => ? = 2
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]
 => [7,7,6,5,4,3,1,1]
 => ? = 2
11110011 => [1,1,1,1,3,1,1] => [1,0,1,0,1,0,1,0,1,1,1,0,0,0,1,0,1,0]
 => [8,7,4,4,4,3,2,1]
 => ? = 5
11111001 => [1,1,1,1,1,3,1] => [1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,0,1,0]
 => [8,5,5,5,4,3,2,1]
 => ? = 6
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]
 => [7,7,5,5,4,3,2,1]
 => ? = 6
11111011 => [1,1,1,1,1,2,1,1] => [1,0,1,0,1,0,1,0,1,0,1,1,0,0,1,0,1,0]
 => [8,7,5,5,4,3,2,1]
 => ? = 6
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]
 => [8,6,6,5,4,3,2,1]
 => ? = 7
000000010 => [8,2] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,1,0,0]
 => [8,8]
 => ? = 1
001111111 => [3,1,1,1,1,1,1,1] => [1,1,1,0,0,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => [9,8,7,6,5,4,3]
 => ? = 1
010000000 => [2,8] => [1,1,0,0,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
 => [2,2,2,2,2,2,2,2]
 => ? = 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: 57% ●values known / values provided: 57%●distinct values known / distinct values provided: 60%
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
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]
 => ? = 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
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
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
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
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
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
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
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
0111101 => [2,1,1,1,2,1] => [1,1,0,0,1,0,1,0,1,0,1,1,0,0,1,0]
 => [1,1,1,0,0,1,0,1,0,1,0,1,0,0,1,0]
 => ? = 1
0111110 => [2,1,1,1,1,2] => [1,1,0,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => [1,1,1,0,0,1,0,1,0,1,0,1,0,1,0,0]
 => ? = 1
1000000 => [1,7] => [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
 => [1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0]
 => ? = 2
1000001 => [1,6,1] => [1,0,1,1,1,1,1,1,0,0,0,0,0,0,1,0]
 => [1,1,1,1,1,1,0,1,0,0,0,0,0,0,1,0]
 => ? = 2
1000010 => [1,5,2] => [1,0,1,1,1,1,1,0,0,0,0,0,1,1,0,0]
 => [1,1,1,1,1,1,0,1,0,0,0,0,0,1,0,0]
 => ? = 2
1001000 => [1,3,4] => [1,0,1,1,1,0,0,0,1,1,1,1,0,0,0,0]
 => [1,1,1,1,1,1,0,1,0,0,0,1,0,0,0,0]
 => ? = 2
1010000 => [1,2,5] => [1,0,1,1,0,0,1,1,1,1,1,0,0,0,0,0]
 => [1,1,1,1,1,1,0,1,0,0,1,0,0,0,0,0]
 => ? = 2
1010101 => [1,2,2,2,1] => [1,0,1,1,0,0,1,1,0,0,1,1,0,0,1,0]
 => [1,1,1,1,0,1,0,0,1,0,0,1,0,0,1,0]
 => ? = 2
1011110 => [1,2,1,1,1,2] => [1,0,1,1,0,0,1,0,1,0,1,0,1,1,0,0]
 => [1,1,1,0,1,0,0,1,0,1,0,1,0,1,0,0]
 => ? = 2
1100000 => [1,1,6] => [1,0,1,0,1,1,1,1,1,1,0,0,0,0,0,0]
 => [1,1,1,1,1,1,0,1,0,1,0,0,0,0,0,0]
 => ? = 3
1100001 => [1,1,5,1] => [1,0,1,0,1,1,1,1,1,0,0,0,0,0,1,0]
 => [1,1,1,1,1,0,1,0,1,0,0,0,0,0,1,0]
 => ? = 3
1100010 => [1,1,4,2] => [1,0,1,0,1,1,1,1,0,0,0,0,1,1,0,0]
 => [1,1,1,1,1,0,1,0,1,0,0,0,0,1,0,0]
 => ? = 3
1100111 => [1,1,3,1,1,1] => [1,0,1,0,1,1,1,0,0,0,1,0,1,0,1,0]
 => [1,1,1,0,1,0,1,0,0,0,1,0,1,0,1,0]
 => ? = 3
1101110 => [1,1,2,1,1,2] => [1,0,1,0,1,1,0,0,1,0,1,0,1,1,0,0]
 => [1,1,1,0,1,0,1,0,0,1,0,1,0,1,0,0]
 => ? = 3
1110000 => [1,1,1,5] => [1,0,1,0,1,0,1,1,1,1,1,0,0,0,0,0]
 => [1,1,1,1,1,0,1,0,1,0,1,0,0,0,0,0]
 => ? = 4
1110011 => [1,1,1,3,1,1] => [1,0,1,0,1,0,1,1,1,0,0,0,1,0,1,0]
 => [1,1,1,0,1,0,1,0,1,0,0,0,1,0,1,0]
 => ? = 4
1110111 => [1,1,1,2,1,1,1] => [1,0,1,0,1,0,1,1,0,0,1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,1,0,0,1,0,1,0,1,0]
 => ? = 4
1111001 => [1,1,1,1,3,1] => [1,0,1,0,1,0,1,0,1,1,1,0,0,0,1,0]
 => [1,1,1,0,1,0,1,0,1,0,1,0,0,0,1,0]
 => ? = 5
1111010 => [1,1,1,1,2,2] => [1,0,1,0,1,0,1,0,1,1,0,0,1,1,0,0]
 => [1,1,1,0,1,0,1,0,1,0,1,0,0,1,0,0]
 => ? = 5
1111011 => [1,1,1,1,2,1,1] => [1,0,1,0,1,0,1,0,1,1,0,0,1,0,1,0]
 => [1,1,0,1,0,1,0,1,0,1,0,0,1,0,1,0]
 => ? = 5
1111100 => [1,1,1,1,1,3] => [1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,0]
 => [1,1,1,0,1,0,1,0,1,0,1,0,1,0,0,0]
 => ? = 6
1111101 => [1,1,1,1,1,2,1] => [1,0,1,0,1,0,1,0,1,0,1,1,0,0,1,0]
 => [1,1,0,1,0,1,0,1,0,1,0,1,0,0,1,0]
 => ? = 6
1111110 => [1,1,1,1,1,1,2] => [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => ? = 7
00000000 => [9] => [1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0]
 => [1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0]
 => ? = 1
00000001 => [8,1] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0]
 => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0]
 => ? = 1
00000010 => [7,2] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,1,0,0]
 => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0,0]
 => ? = 1
00000100 => [6,3] => [1,1,1,1,1,1,0,0,0,0,0,0,1,1,1,0,0,0]
 => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,1,0,0,0]
 => ? = 1
00111111 => [3,1,1,1,1,1,1] => [1,1,1,0,0,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,0,0,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => ? = 1
01000000 => [2,7] => [1,1,0,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
 => [1,1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0,0]
 => ? = 1
01000001 => [2,6,1] => [1,1,0,0,1,1,1,1,1,1,0,0,0,0,0,0,1,0]
 => [1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0,1,0]
 => ? = 1
01000010 => [2,5,2] => [1,1,0,0,1,1,1,1,1,0,0,0,0,0,1,1,0,0]
 => [1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,1,0,0]
 => ? = 1
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]
 => [1,1,1,0,0,1,0,1,0,1,0,1,0,1,0,0,1,0]
 => ? = 1
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]
 => [1,1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => ? = 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: 57% ●values known / values provided: 57%●distinct values known / distinct values provided: 60%
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
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]
 => ? = 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
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
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
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
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
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
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
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
0111101 => 1000010 => [1,5,2] => [1,0,1,1,1,1,1,0,0,0,0,0,1,1,0,0]
 => ? = 1
0111110 => 1000001 => [1,6,1] => [1,0,1,1,1,1,1,1,0,0,0,0,0,0,1,0]
 => ? = 1
0111111 => 1000000 => [1,7] => [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
 => ? = 1
1000000 => 0111111 => [2,1,1,1,1,1,1] => [1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => ? = 2
1000001 => 0111110 => [2,1,1,1,1,2] => [1,1,0,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => ? = 2
1000010 => 0111101 => [2,1,1,1,2,1] => [1,1,0,0,1,0,1,0,1,0,1,1,0,0,1,0]
 => ? = 2
1001000 => 0110111 => [2,1,2,1,1,1] => [1,1,0,0,1,0,1,1,0,0,1,0,1,0,1,0]
 => ? = 2
1010000 => 0101111 => [2,2,1,1,1,1] => [1,1,0,0,1,1,0,0,1,0,1,0,1,0,1,0]
 => ? = 2
1010101 => 0101010 => [2,2,2,2] => [1,1,0,0,1,1,0,0,1,1,0,0,1,1,0,0]
 => ? = 2
1011110 => 0100001 => [2,5,1] => [1,1,0,0,1,1,1,1,1,0,0,0,0,0,1,0]
 => ? = 2
1100000 => 0011111 => [3,1,1,1,1,1] => [1,1,1,0,0,0,1,0,1,0,1,0,1,0,1,0]
 => ? = 3
1100001 => 0011110 => [3,1,1,1,2] => [1,1,1,0,0,0,1,0,1,0,1,0,1,1,0,0]
 => ? = 3
1100010 => 0011101 => [3,1,1,2,1] => [1,1,1,0,0,0,1,0,1,0,1,1,0,0,1,0]
 => ? = 3
1100111 => 0011000 => [3,1,4] => [1,1,1,0,0,0,1,0,1,1,1,1,0,0,0,0]
 => ? = 3
1101110 => 0010001 => [3,4,1] => [1,1,1,0,0,0,1,1,1,1,0,0,0,0,1,0]
 => ? = 3
1110000 => 0001111 => [4,1,1,1,1] => [1,1,1,1,0,0,0,0,1,0,1,0,1,0,1,0]
 => ? = 4
1110011 => 0001100 => [4,1,3] => [1,1,1,1,0,0,0,0,1,0,1,1,1,0,0,0]
 => ? = 4
1110111 => 0001000 => [4,4] => [1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0]
 => ? = 4
1111001 => 0000110 => [5,1,2] => [1,1,1,1,1,0,0,0,0,0,1,0,1,1,0,0]
 => ? = 5
1111010 => 0000101 => [5,2,1] => [1,1,1,1,1,0,0,0,0,0,1,1,0,0,1,0]
 => ? = 5
1111011 => 0000100 => [5,3] => [1,1,1,1,1,0,0,0,0,0,1,1,1,0,0,0]
 => ? = 5
1111100 => 0000011 => [6,1,1] => [1,1,1,1,1,1,0,0,0,0,0,0,1,0,1,0]
 => ? = 6
1111101 => 0000010 => [6,2] => [1,1,1,1,1,1,0,0,0,0,0,0,1,1,0,0]
 => ? = 6
1111110 => 0000001 => [7,1] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
 => ? = 7
00000000 => 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]
 => ? = 1
00000001 => 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
00000010 => 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]
 => ? = 1
00000100 => 11111011 => [1,1,1,1,1,2,1,1] => [1,0,1,0,1,0,1,0,1,0,1,1,0,0,1,0,1,0]
 => ? = 1
00111111 => 11000000 => [1,1,7] => [1,0,1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
 => ? = 1
01000000 => 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]
 => ? = 1
01000001 => 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]
 => ? = 1
01000010 => 10111101 => [1,2,1,1,1,2,1] => [1,0,1,1,0,0,1,0,1,0,1,0,1,1,0,0,1,0]
 => ? = 1
01111101 => 10000010 => [1,6,2] => [1,0,1,1,1,1,1,1,0,0,0,0,0,0,1,1,0,0]
 => ? = 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. St000273The domination number of a graph. St000544The cop number of a graph. St000916The packing number of a graph. St001135The projective dimension of the first simple module in the Nakayama algebra corresponding to the Dyck path. St001829The common independence number of a graph. St000363The number of minimal vertex covers of a graph. St001316The domatic number of a graph. St001322The size of a minimal independent dominating set in 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.