Processing math: 100%

Your data matches 35 different statistics following compositions of up to 3 maps.
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Mp00178: Binary words to compositionInteger compositions
Mp00094: Integer compositions to binary wordBinary words
St000297: Binary words ⟶ ℤResult quality: 100% values known / values provided: 100%distinct values known / distinct values provided: 100%
Values
0 => [2] => 10 => 1
1 => [1,1] => 11 => 2
00 => [3] => 100 => 1
01 => [2,1] => 101 => 1
10 => [1,2] => 110 => 2
11 => [1,1,1] => 111 => 3
000 => [4] => 1000 => 1
001 => [3,1] => 1001 => 1
010 => [2,2] => 1010 => 1
011 => [2,1,1] => 1011 => 1
100 => [1,3] => 1100 => 2
101 => [1,2,1] => 1101 => 2
110 => [1,1,2] => 1110 => 3
111 => [1,1,1,1] => 1111 => 4
0000 => [5] => 10000 => 1
0001 => [4,1] => 10001 => 1
0010 => [3,2] => 10010 => 1
0011 => [3,1,1] => 10011 => 1
0100 => [2,3] => 10100 => 1
0101 => [2,2,1] => 10101 => 1
0110 => [2,1,2] => 10110 => 1
0111 => [2,1,1,1] => 10111 => 1
1000 => [1,4] => 11000 => 2
1001 => [1,3,1] => 11001 => 2
1010 => [1,2,2] => 11010 => 2
1011 => [1,2,1,1] => 11011 => 2
1100 => [1,1,3] => 11100 => 3
1101 => [1,1,2,1] => 11101 => 3
1110 => [1,1,1,2] => 11110 => 4
1111 => [1,1,1,1,1] => 11111 => 5
00000 => [6] => 100000 => 1
00001 => [5,1] => 100001 => 1
00010 => [4,2] => 100010 => 1
00011 => [4,1,1] => 100011 => 1
00100 => [3,3] => 100100 => 1
00101 => [3,2,1] => 100101 => 1
00110 => [3,1,2] => 100110 => 1
00111 => [3,1,1,1] => 100111 => 1
01000 => [2,4] => 101000 => 1
01001 => [2,3,1] => 101001 => 1
01010 => [2,2,2] => 101010 => 1
01011 => [2,2,1,1] => 101011 => 1
01100 => [2,1,3] => 101100 => 1
01101 => [2,1,2,1] => 101101 => 1
01110 => [2,1,1,2] => 101110 => 1
01111 => [2,1,1,1,1] => 101111 => 1
10000 => [1,5] => 110000 => 2
10001 => [1,4,1] => 110001 => 2
10010 => [1,3,2] => 110010 => 2
10011 => [1,3,1,1] => 110011 => 2
Description
The number of leading ones in a binary word.
Mp00105: Binary words complementBinary words
Mp00178: Binary words to compositionInteger compositions
St000382: Integer compositions ⟶ ℤResult quality: 100% values known / values provided: 100%distinct values known / distinct values provided: 100%
Values
0 => 1 => [1,1] => 1
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
Description
The first part of an integer composition.
Mp00178: Binary words to compositionInteger compositions
Mp00041: Integer compositions conjugateInteger 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: St000025
Mp00105: Binary words complementBinary words
Mp00178: Binary words to compositionInteger compositions
Mp00231: Integer compositions bounce pathDyck paths
St000025: Dyck paths ⟶ ℤResult quality: 100% values known / values provided: 100%distinct values known / distinct values provided: 100%
Values
0 => 1 => [1,1] => [1,0,1,0]
=> 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
Description
The number of initial rises of a Dyck path. In other words, this is the height of the first peak of D.
Mp00178: Binary words to compositionInteger compositions
Mp00231: Integer compositions bounce pathDyck paths
Mp00229: Dyck paths Delest-ViennotDyck paths
St000026: Dyck paths ⟶ ℤResult quality: 100% values known / values provided: 100%distinct values known / distinct values provided: 100%
Values
0 => [2] => [1,1,0,0]
=> [1,0,1,0]
=> 1
1 => [1,1] => [1,0,1,0]
=> [1,1,0,0]
=> 2
00 => [3] => [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> 1
01 => [2,1] => [1,1,0,0,1,0]
=> [1,0,1,1,0,0]
=> 1
10 => [1,2] => [1,0,1,1,0,0]
=> [1,1,0,0,1,0]
=> 2
11 => [1,1,1] => [1,0,1,0,1,0]
=> [1,1,0,1,0,0]
=> 3
000 => [4] => [1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> 1
001 => [3,1] => [1,1,1,0,0,0,1,0]
=> [1,0,1,0,1,1,0,0]
=> 1
010 => [2,2] => [1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,0,1,0]
=> 1
011 => [2,1,1] => [1,1,0,0,1,0,1,0]
=> [1,0,1,1,0,1,0,0]
=> 1
100 => [1,3] => [1,0,1,1,1,0,0,0]
=> [1,1,0,0,1,0,1,0]
=> 2
101 => [1,2,1] => [1,0,1,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0]
=> 2
110 => [1,1,2] => [1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> 3
111 => [1,1,1,1] => [1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,0]
=> 4
0000 => [5] => [1,1,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> 1
0001 => [4,1] => [1,1,1,1,0,0,0,0,1,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> 1
0010 => [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> 1
0011 => [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> [1,0,1,0,1,1,0,1,0,0]
=> 1
0100 => [2,3] => [1,1,0,0,1,1,1,0,0,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> 1
0101 => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> 1
0110 => [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
=> [1,0,1,1,0,1,0,0,1,0]
=> 1
0111 => [2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
=> [1,0,1,1,0,1,0,1,0,0]
=> 1
1000 => [1,4] => [1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> 2
1001 => [1,3,1] => [1,0,1,1,1,0,0,0,1,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> 2
1010 => [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> 2
1011 => [1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
=> [1,1,0,0,1,1,0,1,0,0]
=> 2
1100 => [1,1,3] => [1,0,1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,0,1,0,1,0]
=> 3
1101 => [1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,0,0,1,1,0,0]
=> 3
1110 => [1,1,1,2] => [1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,0,1,0]
=> 4
1111 => [1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> 5
00000 => [6] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> 1
00001 => [5,1] => [1,1,1,1,1,0,0,0,0,0,1,0]
=> [1,0,1,0,1,0,1,0,1,1,0,0]
=> 1
00010 => [4,2] => [1,1,1,1,0,0,0,0,1,1,0,0]
=> [1,0,1,0,1,0,1,1,0,0,1,0]
=> 1
00011 => [4,1,1] => [1,1,1,1,0,0,0,0,1,0,1,0]
=> [1,0,1,0,1,0,1,1,0,1,0,0]
=> 1
00100 => [3,3] => [1,1,1,0,0,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,0,0,1,0,1,0]
=> 1
00101 => [3,2,1] => [1,1,1,0,0,0,1,1,0,0,1,0]
=> [1,0,1,0,1,1,0,0,1,1,0,0]
=> 1
00110 => [3,1,2] => [1,1,1,0,0,0,1,0,1,1,0,0]
=> [1,0,1,0,1,1,0,1,0,0,1,0]
=> 1
00111 => [3,1,1,1] => [1,1,1,0,0,0,1,0,1,0,1,0]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> 1
01000 => [2,4] => [1,1,0,0,1,1,1,1,0,0,0,0]
=> [1,0,1,1,0,0,1,0,1,0,1,0]
=> 1
01001 => [2,3,1] => [1,1,0,0,1,1,1,0,0,0,1,0]
=> [1,0,1,1,0,0,1,0,1,1,0,0]
=> 1
01010 => [2,2,2] => [1,1,0,0,1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,0,1,1,0,0,1,0]
=> 1
01011 => [2,2,1,1] => [1,1,0,0,1,1,0,0,1,0,1,0]
=> [1,0,1,1,0,0,1,1,0,1,0,0]
=> 1
01100 => [2,1,3] => [1,1,0,0,1,0,1,1,1,0,0,0]
=> [1,0,1,1,0,1,0,0,1,0,1,0]
=> 1
01101 => [2,1,2,1] => [1,1,0,0,1,0,1,1,0,0,1,0]
=> [1,0,1,1,0,1,0,0,1,1,0,0]
=> 1
01110 => [2,1,1,2] => [1,1,0,0,1,0,1,0,1,1,0,0]
=> [1,0,1,1,0,1,0,1,0,0,1,0]
=> 1
01111 => [2,1,1,1,1] => [1,1,0,0,1,0,1,0,1,0,1,0]
=> [1,0,1,1,0,1,0,1,0,1,0,0]
=> 1
10000 => [1,5] => [1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,1,0,0,1,0,1,0,1,0,1,0]
=> 2
10001 => [1,4,1] => [1,0,1,1,1,1,0,0,0,0,1,0]
=> [1,1,0,0,1,0,1,0,1,1,0,0]
=> 2
10010 => [1,3,2] => [1,0,1,1,1,0,0,0,1,1,0,0]
=> [1,1,0,0,1,0,1,1,0,0,1,0]
=> 2
10011 => [1,3,1,1] => [1,0,1,1,1,0,0,0,1,0,1,0]
=> [1,1,0,0,1,0,1,1,0,1,0,0]
=> 2
Description
The position of the first return of a Dyck path.
Matching statistic: St000759
Mp00178: Binary words to compositionInteger compositions
Mp00231: Integer compositions bounce pathDyck paths
Mp00027: Dyck paths to partitionInteger partitions
St000759: Integer partitions ⟶ ℤResult quality: 100% values known / values provided: 100%distinct values known / distinct values provided: 100%
Values
0 => [2] => [1,1,0,0]
=> []
=> 1
1 => [1,1] => [1,0,1,0]
=> [1]
=> 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
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: St001135
Mp00104: Binary words reverseBinary words
Mp00178: Binary words to compositionInteger compositions
Mp00231: Integer compositions bounce pathDyck paths
St001135: Dyck paths ⟶ ℤResult quality: 100% values known / values provided: 100%distinct values known / distinct values provided: 100%
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
Description
The projective dimension of the first simple module in the Nakayama algebra corresponding to the Dyck path.
Matching statistic: St001733
Mp00178: Binary words to compositionInteger compositions
Mp00231: Integer compositions bounce pathDyck paths
Mp00132: Dyck paths switch returns and last double riseDyck paths
St001733: 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 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.
Mp00178: Binary words to compositionInteger compositions
Mp00094: Integer compositions to binary wordBinary words
Mp00105: Binary words complementBinary words
St000326: Binary words ⟶ ℤResult quality: 100% values known / values provided: 100%distinct values known / distinct values provided: 100%
Values
0 => [2] => 10 => 01 => 2 = 1 + 1
1 => [1,1] => 11 => 00 => 3 = 2 + 1
00 => [3] => 100 => 011 => 2 = 1 + 1
01 => [2,1] => 101 => 010 => 2 = 1 + 1
10 => [1,2] => 110 => 001 => 3 = 2 + 1
11 => [1,1,1] => 111 => 000 => 4 = 3 + 1
000 => [4] => 1000 => 0111 => 2 = 1 + 1
001 => [3,1] => 1001 => 0110 => 2 = 1 + 1
010 => [2,2] => 1010 => 0101 => 2 = 1 + 1
011 => [2,1,1] => 1011 => 0100 => 2 = 1 + 1
100 => [1,3] => 1100 => 0011 => 3 = 2 + 1
101 => [1,2,1] => 1101 => 0010 => 3 = 2 + 1
110 => [1,1,2] => 1110 => 0001 => 4 = 3 + 1
111 => [1,1,1,1] => 1111 => 0000 => 5 = 4 + 1
0000 => [5] => 10000 => 01111 => 2 = 1 + 1
0001 => [4,1] => 10001 => 01110 => 2 = 1 + 1
0010 => [3,2] => 10010 => 01101 => 2 = 1 + 1
0011 => [3,1,1] => 10011 => 01100 => 2 = 1 + 1
0100 => [2,3] => 10100 => 01011 => 2 = 1 + 1
0101 => [2,2,1] => 10101 => 01010 => 2 = 1 + 1
0110 => [2,1,2] => 10110 => 01001 => 2 = 1 + 1
0111 => [2,1,1,1] => 10111 => 01000 => 2 = 1 + 1
1000 => [1,4] => 11000 => 00111 => 3 = 2 + 1
1001 => [1,3,1] => 11001 => 00110 => 3 = 2 + 1
1010 => [1,2,2] => 11010 => 00101 => 3 = 2 + 1
1011 => [1,2,1,1] => 11011 => 00100 => 3 = 2 + 1
1100 => [1,1,3] => 11100 => 00011 => 4 = 3 + 1
1101 => [1,1,2,1] => 11101 => 00010 => 4 = 3 + 1
1110 => [1,1,1,2] => 11110 => 00001 => 5 = 4 + 1
1111 => [1,1,1,1,1] => 11111 => 00000 => 6 = 5 + 1
00000 => [6] => 100000 => 011111 => 2 = 1 + 1
00001 => [5,1] => 100001 => 011110 => 2 = 1 + 1
00010 => [4,2] => 100010 => 011101 => 2 = 1 + 1
00011 => [4,1,1] => 100011 => 011100 => 2 = 1 + 1
00100 => [3,3] => 100100 => 011011 => 2 = 1 + 1
00101 => [3,2,1] => 100101 => 011010 => 2 = 1 + 1
00110 => [3,1,2] => 100110 => 011001 => 2 = 1 + 1
00111 => [3,1,1,1] => 100111 => 011000 => 2 = 1 + 1
01000 => [2,4] => 101000 => 010111 => 2 = 1 + 1
01001 => [2,3,1] => 101001 => 010110 => 2 = 1 + 1
01010 => [2,2,2] => 101010 => 010101 => 2 = 1 + 1
01011 => [2,2,1,1] => 101011 => 010100 => 2 = 1 + 1
01100 => [2,1,3] => 101100 => 010011 => 2 = 1 + 1
01101 => [2,1,2,1] => 101101 => 010010 => 2 = 1 + 1
01110 => [2,1,1,2] => 101110 => 010001 => 2 = 1 + 1
01111 => [2,1,1,1,1] => 101111 => 010000 => 2 = 1 + 1
10000 => [1,5] => 110000 => 001111 => 3 = 2 + 1
10001 => [1,4,1] => 110001 => 001110 => 3 = 2 + 1
10010 => [1,3,2] => 110010 => 001101 => 3 = 2 + 1
10011 => [1,3,1,1] => 110011 => 001100 => 3 = 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,,n,n+1} that contains n+1, this is the minimal element of the set.
Matching statistic: St000439
Mp00105: Binary words complementBinary words
Mp00178: Binary words to compositionInteger compositions
Mp00231: Integer compositions bounce pathDyck paths
St000439: Dyck paths ⟶ ℤResult quality: 100% values known / values provided: 100%distinct values known / distinct values provided: 100%
Values
0 => 1 => [1,1] => [1,0,1,0]
=> 2 = 1 + 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
Description
The position of the first down step of a Dyck path.
The following 25 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000363The number of minimal vertex covers of a graph. St000273The domination number of a graph. St000544The cop number of a graph. St000678The number of up steps after the last double rise of a Dyck path. 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. St000617The number of global maxima of a Dyck path. 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 S0 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. St001330The hat guessing number of a graph.