Your data matches 52 different statistics following compositions of up to 3 maps.
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Matching statistic: St000297
(load all 5 compositions to match this statistic)
Mapping
Mp00105: Binary words —complement⟶ Binary words
St000297: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
0 => 1 => 1
1 => 0 => 0
00 => 11 => 2
01 => 10 => 1
10 => 01 => 0
11 => 00 => 0
000 => 111 => 3
001 => 110 => 2
010 => 101 => 1
011 => 100 => 1
100 => 011 => 0
101 => 010 => 0
110 => 001 => 0
111 => 000 => 0
0000 => 1111 => 4
0001 => 1110 => 3
0010 => 1101 => 2
0011 => 1100 => 2
0100 => 1011 => 1
0101 => 1010 => 1
0110 => 1001 => 1
0111 => 1000 => 1
1000 => 0111 => 0
1001 => 0110 => 0
1010 => 0101 => 0
1011 => 0100 => 0
1100 => 0011 => 0
1101 => 0010 => 0
1110 => 0001 => 0
1111 => 0000 => 0
00000 => 11111 => 5
00001 => 11110 => 4
00010 => 11101 => 3
00011 => 11100 => 3
00100 => 11011 => 2
00101 => 11010 => 2
00110 => 11001 => 2
00111 => 11000 => 2
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 => 0
10001 => 01110 => 0
10010 => 01101 => 0
10011 => 01100 => 0
Description
The number of leading ones in a binary word.
Matching statistic: St000326
(load all 5 compositions to match this statistic)
Mapping
St000326: Binary words ⟶ ℤResult quality: 76% ●values known / values provided: 76%●distinct values known / distinct values provided: 100%
Values
0 => 2 = 1 + 1
1 => 1 = 0 + 1
00 => 3 = 2 + 1
01 => 2 = 1 + 1
10 => 1 = 0 + 1
11 => 1 = 0 + 1
000 => 4 = 3 + 1
001 => 3 = 2 + 1
010 => 2 = 1 + 1
011 => 2 = 1 + 1
100 => 1 = 0 + 1
101 => 1 = 0 + 1
110 => 1 = 0 + 1
111 => 1 = 0 + 1
0000 => 5 = 4 + 1
0001 => 4 = 3 + 1
0010 => 3 = 2 + 1
0011 => 3 = 2 + 1
0100 => 2 = 1 + 1
0101 => 2 = 1 + 1
0110 => 2 = 1 + 1
0111 => 2 = 1 + 1
1000 => 1 = 0 + 1
1001 => 1 = 0 + 1
1010 => 1 = 0 + 1
1011 => 1 = 0 + 1
1100 => 1 = 0 + 1
1101 => 1 = 0 + 1
1110 => 1 = 0 + 1
1111 => 1 = 0 + 1
00000 => 6 = 5 + 1
00001 => 5 = 4 + 1
00010 => 4 = 3 + 1
00011 => 4 = 3 + 1
00100 => 3 = 2 + 1
00101 => 3 = 2 + 1
00110 => 3 = 2 + 1
00111 => 3 = 2 + 1
01000 => 2 = 1 + 1
01001 => 2 = 1 + 1
01010 => 2 = 1 + 1
01011 => 2 = 1 + 1
01100 => 2 = 1 + 1
01101 => 2 = 1 + 1
01110 => 2 = 1 + 1
01111 => 2 = 1 + 1
10000 => 1 = 0 + 1
10001 => 1 = 0 + 1
10010 => 1 = 0 + 1
10011 => 1 = 0 + 1
1111111111 => ? = 0 + 1
1111111100 => ? = 0 + 1
1111111011 => ? = 0 + 1
1111111010 => ? = 0 + 1
1111111001 => ? = 0 + 1
1111111000 => ? = 0 + 1
1111110111 => ? = 0 + 1
1111110110 => ? = 0 + 1
1111110101 => ? = 0 + 1
1111110100 => ? = 0 + 1
1111110011 => ? = 0 + 1
1111110010 => ? = 0 + 1
1111110001 => ? = 0 + 1
1111110000 => ? = 0 + 1
1111101111 => ? = 0 + 1
1111101110 => ? = 0 + 1
1111101101 => ? = 0 + 1
1111101100 => ? = 0 + 1
1111101011 => ? = 0 + 1
1111101010 => ? = 0 + 1
1111101001 => ? = 0 + 1
1111101000 => ? = 0 + 1
1111100111 => ? = 0 + 1
1111100100 => ? = 0 + 1
1111100011 => ? = 0 + 1
1111100010 => ? = 0 + 1
1111100001 => ? = 0 + 1
1111011111 => ? = 0 + 1
1111011110 => ? = 0 + 1
1111011101 => ? = 0 + 1
1111011100 => ? = 0 + 1
1111011011 => ? = 0 + 1
1111011010 => ? = 0 + 1
1111011001 => ? = 0 + 1
1111011000 => ? = 0 + 1
1111010111 => ? = 0 + 1
1111010110 => ? = 0 + 1
1111010101 => ? = 0 + 1
1111010100 => ? = 0 + 1
1111010011 => ? = 0 + 1
1111010010 => ? = 0 + 1
1111010001 => ? = 0 + 1
1111001110 => ? = 0 + 1
1111001100 => ? = 0 + 1
1111001011 => ? = 0 + 1
1111001010 => ? = 0 + 1
1111001001 => ? = 0 + 1
1111000111 => ? = 0 + 1
1111000110 => ? = 0 + 1
1111000101 => ? = 0 + 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 2 compositions to match this statistic)
Mapping
Mp00178: Binary words —to composition⟶ Integer compositions
St000382: Integer compositions ⟶ ℤResult quality: 46% ●values known / values provided: 46%●distinct values known / distinct values provided: 91%
Values
0 => [2] => 2 = 1 + 1
1 => [1,1] => 1 = 0 + 1
00 => [3] => 3 = 2 + 1
01 => [2,1] => 2 = 1 + 1
10 => [1,2] => 1 = 0 + 1
11 => [1,1,1] => 1 = 0 + 1
000 => [4] => 4 = 3 + 1
001 => [3,1] => 3 = 2 + 1
010 => [2,2] => 2 = 1 + 1
011 => [2,1,1] => 2 = 1 + 1
100 => [1,3] => 1 = 0 + 1
101 => [1,2,1] => 1 = 0 + 1
110 => [1,1,2] => 1 = 0 + 1
111 => [1,1,1,1] => 1 = 0 + 1
0000 => [5] => 5 = 4 + 1
0001 => [4,1] => 4 = 3 + 1
0010 => [3,2] => 3 = 2 + 1
0011 => [3,1,1] => 3 = 2 + 1
0100 => [2,3] => 2 = 1 + 1
0101 => [2,2,1] => 2 = 1 + 1
0110 => [2,1,2] => 2 = 1 + 1
0111 => [2,1,1,1] => 2 = 1 + 1
1000 => [1,4] => 1 = 0 + 1
1001 => [1,3,1] => 1 = 0 + 1
1010 => [1,2,2] => 1 = 0 + 1
1011 => [1,2,1,1] => 1 = 0 + 1
1100 => [1,1,3] => 1 = 0 + 1
1101 => [1,1,2,1] => 1 = 0 + 1
1110 => [1,1,1,2] => 1 = 0 + 1
1111 => [1,1,1,1,1] => 1 = 0 + 1
00000 => [6] => 6 = 5 + 1
00001 => [5,1] => 5 = 4 + 1
00010 => [4,2] => 4 = 3 + 1
00011 => [4,1,1] => 4 = 3 + 1
00100 => [3,3] => 3 = 2 + 1
00101 => [3,2,1] => 3 = 2 + 1
00110 => [3,1,2] => 3 = 2 + 1
00111 => [3,1,1,1] => 3 = 2 + 1
01000 => [2,4] => 2 = 1 + 1
01001 => [2,3,1] => 2 = 1 + 1
01010 => [2,2,2] => 2 = 1 + 1
01011 => [2,2,1,1] => 2 = 1 + 1
01100 => [2,1,3] => 2 = 1 + 1
01101 => [2,1,2,1] => 2 = 1 + 1
01110 => [2,1,1,2] => 2 = 1 + 1
01111 => [2,1,1,1,1] => 2 = 1 + 1
10000 => [1,5] => 1 = 0 + 1
10001 => [1,4,1] => 1 = 0 + 1
10010 => [1,3,2] => 1 = 0 + 1
10011 => [1,3,1,1] => 1 = 0 + 1
000000110 => [7,1,2] => ? = 6 + 1
000001100 => [6,1,3] => ? = 5 + 1
000001101 => [6,1,2,1] => ? = 5 + 1
000001110 => [6,1,1,2] => ? = 5 + 1
000010110 => [5,2,1,2] => ? = 4 + 1
000011000 => [5,1,4] => ? = 4 + 1
000011001 => [5,1,3,1] => ? = 4 + 1
000011010 => [5,1,2,2] => ? = 4 + 1
000011011 => [5,1,2,1,1] => ? = 4 + 1
000011100 => [5,1,1,3] => ? = 4 + 1
000011101 => [5,1,1,2,1] => ? = 4 + 1
000011110 => [5,1,1,1,2] => ? = 4 + 1
000101101 => [4,2,1,2,1] => ? = 3 + 1
000101110 => [4,2,1,1,2] => ? = 3 + 1
000110000 => [4,1,5] => ? = 3 + 1
000110001 => [4,1,4,1] => ? = 3 + 1
000110011 => [4,1,3,1,1] => ? = 3 + 1
000110101 => [4,1,2,2,1] => ? = 3 + 1
000110110 => [4,1,2,1,2] => ? = 3 + 1
000110111 => [4,1,2,1,1,1] => ? = 3 + 1
000111001 => [4,1,1,3,1] => ? = 3 + 1
000111010 => [4,1,1,2,2] => ? = 3 + 1
000111011 => [4,1,1,2,1,1] => ? = 3 + 1
000111100 => [4,1,1,1,3] => ? = 3 + 1
000111101 => [4,1,1,1,2,1] => ? = 3 + 1
000111110 => [4,1,1,1,1,2] => ? = 3 + 1
001000001 => [3,6,1] => ? = 2 + 1
001000011 => [3,5,1,1] => ? = 2 + 1
001000111 => [3,4,1,1,1] => ? = 2 + 1
001001100 => [3,3,1,3] => ? = 2 + 1
001001101 => [3,3,1,2,1] => ? = 2 + 1
001001110 => [3,3,1,1,2] => ? = 2 + 1
001010011 => [3,2,3,1,1] => ? = 2 + 1
001010110 => [3,2,2,1,2] => ? = 2 + 1
001011001 => [3,2,1,3,1] => ? = 2 + 1
001011010 => [3,2,1,2,2] => ? = 2 + 1
001011100 => [3,2,1,1,3] => ? = 2 + 1
001011101 => [3,2,1,1,2,1] => ? = 2 + 1
001100000 => [3,1,6] => ? = 2 + 1
001100001 => [3,1,5,1] => ? = 2 + 1
001100011 => [3,1,4,1,1] => ? = 2 + 1
001100100 => [3,1,3,3] => ? = 2 + 1
001100101 => [3,1,3,2,1] => ? = 2 + 1
001100110 => [3,1,3,1,2] => ? = 2 + 1
001101001 => [3,1,2,3,1] => ? = 2 + 1
001101010 => [3,1,2,2,2] => ? = 2 + 1
001101011 => [3,1,2,2,1,1] => ? = 2 + 1
001101100 => [3,1,2,1,3] => ? = 2 + 1
001101101 => [3,1,2,1,2,1] => ? = 2 + 1
001101110 => [3,1,2,1,1,2] => ? = 2 + 1
Description
The first part of an integer composition.
Matching statistic: St000383
(load all 7 compositions to match this statistic)
Mapping
Mp00178: Binary words —to composition⟶ Integer compositions
Mp00173: Integer compositions —rotate front to back⟶ Integer compositions
St000383: Integer compositions ⟶ ℤResult quality: 15% ●values known / values provided: 15%●distinct values known / distinct values provided: 91%
Values
0 => [2] => [2] => 2 = 1 + 1
1 => [1,1] => [1,1] => 1 = 0 + 1
00 => [3] => [3] => 3 = 2 + 1
01 => [2,1] => [1,2] => 2 = 1 + 1
10 => [1,2] => [2,1] => 1 = 0 + 1
11 => [1,1,1] => [1,1,1] => 1 = 0 + 1
000 => [4] => [4] => 4 = 3 + 1
001 => [3,1] => [1,3] => 3 = 2 + 1
010 => [2,2] => [2,2] => 2 = 1 + 1
011 => [2,1,1] => [1,1,2] => 2 = 1 + 1
100 => [1,3] => [3,1] => 1 = 0 + 1
101 => [1,2,1] => [2,1,1] => 1 = 0 + 1
110 => [1,1,2] => [1,2,1] => 1 = 0 + 1
111 => [1,1,1,1] => [1,1,1,1] => 1 = 0 + 1
0000 => [5] => [5] => 5 = 4 + 1
0001 => [4,1] => [1,4] => 4 = 3 + 1
0010 => [3,2] => [2,3] => 3 = 2 + 1
0011 => [3,1,1] => [1,1,3] => 3 = 2 + 1
0100 => [2,3] => [3,2] => 2 = 1 + 1
0101 => [2,2,1] => [2,1,2] => 2 = 1 + 1
0110 => [2,1,2] => [1,2,2] => 2 = 1 + 1
0111 => [2,1,1,1] => [1,1,1,2] => 2 = 1 + 1
1000 => [1,4] => [4,1] => 1 = 0 + 1
1001 => [1,3,1] => [3,1,1] => 1 = 0 + 1
1010 => [1,2,2] => [2,2,1] => 1 = 0 + 1
1011 => [1,2,1,1] => [2,1,1,1] => 1 = 0 + 1
1100 => [1,1,3] => [1,3,1] => 1 = 0 + 1
1101 => [1,1,2,1] => [1,2,1,1] => 1 = 0 + 1
1110 => [1,1,1,2] => [1,1,2,1] => 1 = 0 + 1
1111 => [1,1,1,1,1] => [1,1,1,1,1] => 1 = 0 + 1
00000 => [6] => [6] => 6 = 5 + 1
00001 => [5,1] => [1,5] => 5 = 4 + 1
00010 => [4,2] => [2,4] => 4 = 3 + 1
00011 => [4,1,1] => [1,1,4] => 4 = 3 + 1
00100 => [3,3] => [3,3] => 3 = 2 + 1
00101 => [3,2,1] => [2,1,3] => 3 = 2 + 1
00110 => [3,1,2] => [1,2,3] => 3 = 2 + 1
00111 => [3,1,1,1] => [1,1,1,3] => 3 = 2 + 1
01000 => [2,4] => [4,2] => 2 = 1 + 1
01001 => [2,3,1] => [3,1,2] => 2 = 1 + 1
01010 => [2,2,2] => [2,2,2] => 2 = 1 + 1
01011 => [2,2,1,1] => [2,1,1,2] => 2 = 1 + 1
01100 => [2,1,3] => [1,3,2] => 2 = 1 + 1
01101 => [2,1,2,1] => [1,2,1,2] => 2 = 1 + 1
01110 => [2,1,1,2] => [1,1,2,2] => 2 = 1 + 1
01111 => [2,1,1,1,1] => [1,1,1,1,2] => 2 = 1 + 1
10000 => [1,5] => [5,1] => 1 = 0 + 1
10001 => [1,4,1] => [4,1,1] => 1 = 0 + 1
10010 => [1,3,2] => [3,2,1] => 1 = 0 + 1
10011 => [1,3,1,1] => [3,1,1,1] => 1 = 0 + 1
0000000 => [8] => [8] => ? = 7 + 1
0000010 => [6,2] => [2,6] => ? = 5 + 1
0000011 => [6,1,1] => [1,1,6] => ? = 5 + 1
0000100 => [5,3] => [3,5] => ? = 4 + 1
0000101 => [5,2,1] => [2,1,5] => ? = 4 + 1
0000110 => [5,1,2] => [1,2,5] => ? = 4 + 1
0000111 => [5,1,1,1] => [1,1,1,5] => ? = 4 + 1
0001001 => [4,3,1] => [3,1,4] => ? = 3 + 1
0001010 => [4,2,2] => [2,2,4] => ? = 3 + 1
0001011 => [4,2,1,1] => [2,1,1,4] => ? = 3 + 1
0001100 => [4,1,3] => [1,3,4] => ? = 3 + 1
0001110 => [4,1,1,2] => [1,1,2,4] => ? = 3 + 1
0001111 => [4,1,1,1,1] => [1,1,1,1,4] => ? = 3 + 1
0010000 => [3,5] => [5,3] => ? = 2 + 1
0010001 => [3,4,1] => [4,1,3] => ? = 2 + 1
0010111 => [3,2,1,1,1] => [2,1,1,1,3] => ? = 2 + 1
0011011 => [3,1,2,1,1] => [1,2,1,1,3] => ? = 2 + 1
0011101 => [3,1,1,2,1] => [1,1,2,1,3] => ? = 2 + 1
0011110 => [3,1,1,1,2] => [1,1,1,2,3] => ? = 2 + 1
0011111 => [3,1,1,1,1,1] => [1,1,1,1,1,3] => ? = 2 + 1
0100000 => [2,6] => [6,2] => ? = 1 + 1
0100001 => [2,5,1] => [5,1,2] => ? = 1 + 1
0100010 => [2,4,2] => [4,2,2] => ? = 1 + 1
0100011 => [2,4,1,1] => [4,1,1,2] => ? = 1 + 1
0100111 => [2,3,1,1,1] => [3,1,1,1,2] => ? = 1 + 1
0111000 => [2,1,1,4] => [1,1,4,2] => ? = 1 + 1
0111111 => [2,1,1,1,1,1,1] => [1,1,1,1,1,1,2] => ? = 1 + 1
1000101 => [1,4,2,1] => [4,2,1,1] => ? = 0 + 1
1000111 => [1,4,1,1,1] => [4,1,1,1,1] => ? = 0 + 1
1001011 => [1,3,2,1,1] => [3,2,1,1,1] => ? = 0 + 1
1001101 => [1,3,1,2,1] => [3,1,2,1,1] => ? = 0 + 1
1001110 => [1,3,1,1,2] => [3,1,1,2,1] => ? = 0 + 1
1001111 => [1,3,1,1,1,1] => [3,1,1,1,1,1] => ? = 0 + 1
1010000 => [1,2,5] => [2,5,1] => ? = 0 + 1
1010001 => [1,2,4,1] => [2,4,1,1] => ? = 0 + 1
1011111 => [1,2,1,1,1,1,1] => [2,1,1,1,1,1,1] => ? = 0 + 1
1100011 => [1,1,4,1,1] => [1,4,1,1,1] => ? = 0 + 1
1110001 => [1,1,1,4,1] => [1,1,4,1,1] => ? = 0 + 1
1111000 => [1,1,1,1,4] => [1,1,1,4,1] => ? = 0 + 1
00000000 => [9] => [9] => ? = 8 + 1
00000010 => [7,2] => [2,7] => ? = 6 + 1
00000011 => [7,1,1] => [1,1,7] => ? = 6 + 1
00000100 => [6,3] => [3,6] => ? = 5 + 1
00000101 => [6,2,1] => [2,1,6] => ? = 5 + 1
00000110 => [6,1,2] => [1,2,6] => ? = 5 + 1
00000111 => [6,1,1,1] => [1,1,1,6] => ? = 5 + 1
00001000 => [5,4] => [4,5] => ? = 4 + 1
00001001 => [5,3,1] => [3,1,5] => ? = 4 + 1
00001010 => [5,2,2] => [2,2,5] => ? = 4 + 1
00001011 => [5,2,1,1] => [2,1,1,5] => ? = 4 + 1
Description
The last part of an integer composition.
Matching statistic: St000011
Mapping
Mp00178: Binary words —to composition⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
Mp00101: Dyck paths —decomposition reverse⟶ Dyck paths
St000011: Dyck paths ⟶ ℤResult quality: 15% ●values known / values provided: 15%●distinct values known / distinct values provided: 91%
Values
0 => [2] => [1,1,0,0]
 => [1,0,1,0]
 => 2 = 1 + 1
1 => [1,1] => [1,0,1,0]
 => [1,1,0,0]
 => 1 = 0 + 1
00 => [3] => [1,1,1,0,0,0]
 => [1,0,1,0,1,0]
 => 3 = 2 + 1
01 => [2,1] => [1,1,0,0,1,0]
 => [1,1,0,0,1,0]
 => 2 = 1 + 1
10 => [1,2] => [1,0,1,1,0,0]
 => [1,1,0,1,0,0]
 => 1 = 0 + 1
11 => [1,1,1] => [1,0,1,0,1,0]
 => [1,1,1,0,0,0]
 => 1 = 0 + 1
000 => [4] => [1,1,1,1,0,0,0,0]
 => [1,0,1,0,1,0,1,0]
 => 4 = 3 + 1
001 => [3,1] => [1,1,1,0,0,0,1,0]
 => [1,1,0,0,1,0,1,0]
 => 3 = 2 + 1
010 => [2,2] => [1,1,0,0,1,1,0,0]
 => [1,1,0,1,0,0,1,0]
 => 2 = 1 + 1
011 => [2,1,1] => [1,1,0,0,1,0,1,0]
 => [1,1,1,0,0,0,1,0]
 => 2 = 1 + 1
100 => [1,3] => [1,0,1,1,1,0,0,0]
 => [1,1,0,1,0,1,0,0]
 => 1 = 0 + 1
101 => [1,2,1] => [1,0,1,1,0,0,1,0]
 => [1,1,1,0,0,1,0,0]
 => 1 = 0 + 1
110 => [1,1,2] => [1,0,1,0,1,1,0,0]
 => [1,1,1,0,1,0,0,0]
 => 1 = 0 + 1
111 => [1,1,1,1] => [1,0,1,0,1,0,1,0]
 => [1,1,1,1,0,0,0,0]
 => 1 = 0 + 1
0000 => [5] => [1,1,1,1,1,0,0,0,0,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => 5 = 4 + 1
0001 => [4,1] => [1,1,1,1,0,0,0,0,1,0]
 => [1,1,0,0,1,0,1,0,1,0]
 => 4 = 3 + 1
0010 => [3,2] => [1,1,1,0,0,0,1,1,0,0]
 => [1,1,0,1,0,0,1,0,1,0]
 => 3 = 2 + 1
0011 => [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
 => [1,1,1,0,0,0,1,0,1,0]
 => 3 = 2 + 1
0100 => [2,3] => [1,1,0,0,1,1,1,0,0,0]
 => [1,1,0,1,0,1,0,0,1,0]
 => 2 = 1 + 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]
 => 2 = 1 + 1
0110 => [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
 => [1,1,1,0,1,0,0,0,1,0]
 => 2 = 1 + 1
0111 => [2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
 => [1,1,1,1,0,0,0,0,1,0]
 => 2 = 1 + 1
1000 => [1,4] => [1,0,1,1,1,1,0,0,0,0]
 => [1,1,0,1,0,1,0,1,0,0]
 => 1 = 0 + 1
1001 => [1,3,1] => [1,0,1,1,1,0,0,0,1,0]
 => [1,1,1,0,0,1,0,1,0,0]
 => 1 = 0 + 1
1010 => [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
 => [1,1,1,0,1,0,0,1,0,0]
 => 1 = 0 + 1
1011 => [1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
 => [1,1,1,1,0,0,0,1,0,0]
 => 1 = 0 + 1
1100 => [1,1,3] => [1,0,1,0,1,1,1,0,0,0]
 => [1,1,1,0,1,0,1,0,0,0]
 => 1 = 0 + 1
1101 => [1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
 => [1,1,1,1,0,0,1,0,0,0]
 => 1 = 0 + 1
1110 => [1,1,1,2] => [1,0,1,0,1,0,1,1,0,0]
 => [1,1,1,1,0,1,0,0,0,0]
 => 1 = 0 + 1
1111 => [1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => 1 = 0 + 1
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]
 => 6 = 5 + 1
00001 => [5,1] => [1,1,1,1,1,0,0,0,0,0,1,0]
 => [1,1,0,0,1,0,1,0,1,0,1,0]
 => 5 = 4 + 1
00010 => [4,2] => [1,1,1,1,0,0,0,0,1,1,0,0]
 => [1,1,0,1,0,0,1,0,1,0,1,0]
 => 4 = 3 + 1
00011 => [4,1,1] => [1,1,1,1,0,0,0,0,1,0,1,0]
 => [1,1,1,0,0,0,1,0,1,0,1,0]
 => 4 = 3 + 1
00100 => [3,3] => [1,1,1,0,0,0,1,1,1,0,0,0]
 => [1,1,0,1,0,1,0,0,1,0,1,0]
 => 3 = 2 + 1
00101 => [3,2,1] => [1,1,1,0,0,0,1,1,0,0,1,0]
 => [1,1,1,0,0,1,0,0,1,0,1,0]
 => 3 = 2 + 1
00110 => [3,1,2] => [1,1,1,0,0,0,1,0,1,1,0,0]
 => [1,1,1,0,1,0,0,0,1,0,1,0]
 => 3 = 2 + 1
00111 => [3,1,1,1] => [1,1,1,0,0,0,1,0,1,0,1,0]
 => [1,1,1,1,0,0,0,0,1,0,1,0]
 => 3 = 2 + 1
01000 => [2,4] => [1,1,0,0,1,1,1,1,0,0,0,0]
 => [1,1,0,1,0,1,0,1,0,0,1,0]
 => 2 = 1 + 1
01001 => [2,3,1] => [1,1,0,0,1,1,1,0,0,0,1,0]
 => [1,1,1,0,0,1,0,1,0,0,1,0]
 => 2 = 1 + 1
01010 => [2,2,2] => [1,1,0,0,1,1,0,0,1,1,0,0]
 => [1,1,1,0,1,0,0,1,0,0,1,0]
 => 2 = 1 + 1
01011 => [2,2,1,1] => [1,1,0,0,1,1,0,0,1,0,1,0]
 => [1,1,1,1,0,0,0,1,0,0,1,0]
 => 2 = 1 + 1
01100 => [2,1,3] => [1,1,0,0,1,0,1,1,1,0,0,0]
 => [1,1,1,0,1,0,1,0,0,0,1,0]
 => 2 = 1 + 1
01101 => [2,1,2,1] => [1,1,0,0,1,0,1,1,0,0,1,0]
 => [1,1,1,1,0,0,1,0,0,0,1,0]
 => 2 = 1 + 1
01110 => [2,1,1,2] => [1,1,0,0,1,0,1,0,1,1,0,0]
 => [1,1,1,1,0,1,0,0,0,0,1,0]
 => 2 = 1 + 1
01111 => [2,1,1,1,1] => [1,1,0,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,0,0,0,0,0,1,0]
 => 2 = 1 + 1
10000 => [1,5] => [1,0,1,1,1,1,1,0,0,0,0,0]
 => [1,1,0,1,0,1,0,1,0,1,0,0]
 => 1 = 0 + 1
10001 => [1,4,1] => [1,0,1,1,1,1,0,0,0,0,1,0]
 => [1,1,1,0,0,1,0,1,0,1,0,0]
 => 1 = 0 + 1
10010 => [1,3,2] => [1,0,1,1,1,0,0,0,1,1,0,0]
 => [1,1,1,0,1,0,0,1,0,1,0,0]
 => 1 = 0 + 1
10011 => [1,3,1,1] => [1,0,1,1,1,0,0,0,1,0,1,0]
 => [1,1,1,1,0,0,0,1,0,1,0,0]
 => 1 = 0 + 1
0001010 => [4,2,2] => [1,1,1,1,0,0,0,0,1,1,0,0,1,1,0,0]
 => [1,1,1,0,1,0,0,1,0,0,1,0,1,0,1,0]
 => ? = 3 + 1
0001110 => [4,1,1,2] => [1,1,1,1,0,0,0,0,1,0,1,0,1,1,0,0]
 => [1,1,1,1,0,1,0,0,0,0,1,0,1,0,1,0]
 => ? = 3 + 1
0010010 => [3,3,2] => [1,1,1,0,0,0,1,1,1,0,0,0,1,1,0,0]
 => [1,1,1,0,1,0,0,1,0,1,0,0,1,0,1,0]
 => ? = 2 + 1
0010101 => [3,2,2,1] => [1,1,1,0,0,0,1,1,0,0,1,1,0,0,1,0]
 => [1,1,1,1,0,0,1,0,0,1,0,0,1,0,1,0]
 => ? = 2 + 1
0010110 => [3,2,1,2] => [1,1,1,0,0,0,1,1,0,0,1,0,1,1,0,0]
 => [1,1,1,1,0,1,0,0,0,1,0,0,1,0,1,0]
 => ? = 2 + 1
0011001 => [3,1,3,1] => [1,1,1,0,0,0,1,0,1,1,1,0,0,0,1,0]
 => [1,1,1,1,0,0,1,0,1,0,0,0,1,0,1,0]
 => ? = 2 + 1
0011010 => [3,1,2,2] => [1,1,1,0,0,0,1,0,1,1,0,0,1,1,0,0]
 => [1,1,1,1,0,1,0,0,1,0,0,0,1,0,1,0]
 => ? = 2 + 1
0011100 => [3,1,1,3] => [1,1,1,0,0,0,1,0,1,0,1,1,1,0,0,0]
 => [1,1,1,1,0,1,0,1,0,0,0,0,1,0,1,0]
 => ? = 2 + 1
0011101 => [3,1,1,2,1] => [1,1,1,0,0,0,1,0,1,0,1,1,0,0,1,0]
 => [1,1,1,1,1,0,0,1,0,0,0,0,1,0,1,0]
 => ? = 2 + 1
0011110 => [3,1,1,1,2] => [1,1,1,0,0,0,1,0,1,0,1,0,1,1,0,0]
 => [1,1,1,1,1,0,1,0,0,0,0,0,1,0,1,0]
 => ? = 2 + 1
0100101 => [2,3,2,1] => [1,1,0,0,1,1,1,0,0,0,1,1,0,0,1,0]
 => [1,1,1,1,0,0,1,0,0,1,0,1,0,0,1,0]
 => ? = 1 + 1
0100110 => [2,3,1,2] => [1,1,0,0,1,1,1,0,0,0,1,0,1,1,0,0]
 => [1,1,1,1,0,1,0,0,0,1,0,1,0,0,1,0]
 => ? = 1 + 1
0101000 => [2,2,4] => [1,1,0,0,1,1,0,0,1,1,1,1,0,0,0,0]
 => [1,1,1,0,1,0,1,0,1,0,0,1,0,0,1,0]
 => ? = 1 + 1
0101001 => [2,2,3,1] => [1,1,0,0,1,1,0,0,1,1,1,0,0,0,1,0]
 => [1,1,1,1,0,0,1,0,1,0,0,1,0,0,1,0]
 => ? = 1 + 1
0101011 => [2,2,2,1,1] => [1,1,0,0,1,1,0,0,1,1,0,0,1,0,1,0]
 => [1,1,1,1,1,0,0,0,1,0,0,1,0,0,1,0]
 => ? = 1 + 1
0101100 => [2,2,1,3] => [1,1,0,0,1,1,0,0,1,0,1,1,1,0,0,0]
 => [1,1,1,1,0,1,0,1,0,0,0,1,0,0,1,0]
 => ? = 1 + 1
0101101 => [2,2,1,2,1] => [1,1,0,0,1,1,0,0,1,0,1,1,0,0,1,0]
 => [1,1,1,1,1,0,0,1,0,0,0,1,0,0,1,0]
 => ? = 1 + 1
0101110 => [2,2,1,1,2] => [1,1,0,0,1,1,0,0,1,0,1,0,1,1,0,0]
 => [1,1,1,1,1,0,1,0,0,0,0,1,0,0,1,0]
 => ? = 1 + 1
0110001 => [2,1,4,1] => [1,1,0,0,1,0,1,1,1,1,0,0,0,0,1,0]
 => [1,1,1,1,0,0,1,0,1,0,1,0,0,0,1,0]
 => ? = 1 + 1
0110010 => [2,1,3,2] => [1,1,0,0,1,0,1,1,1,0,0,0,1,1,0,0]
 => [1,1,1,1,0,1,0,0,1,0,1,0,0,0,1,0]
 => ? = 1 + 1
0110011 => [2,1,3,1,1] => [1,1,0,0,1,0,1,1,1,0,0,0,1,0,1,0]
 => [1,1,1,1,1,0,0,0,1,0,1,0,0,0,1,0]
 => ? = 1 + 1
0110100 => [2,1,2,3] => [1,1,0,0,1,0,1,1,0,0,1,1,1,0,0,0]
 => [1,1,1,1,0,1,0,1,0,0,1,0,0,0,1,0]
 => ? = 1 + 1
0110101 => [2,1,2,2,1] => [1,1,0,0,1,0,1,1,0,0,1,1,0,0,1,0]
 => [1,1,1,1,1,0,0,1,0,0,1,0,0,0,1,0]
 => ? = 1 + 1
0110110 => [2,1,2,1,2] => [1,1,0,0,1,0,1,1,0,0,1,0,1,1,0,0]
 => [1,1,1,1,1,0,1,0,0,0,1,0,0,0,1,0]
 => ? = 1 + 1
0111000 => [2,1,1,4] => [1,1,0,0,1,0,1,0,1,1,1,1,0,0,0,0]
 => [1,1,1,1,0,1,0,1,0,1,0,0,0,0,1,0]
 => ? = 1 + 1
0111001 => [2,1,1,3,1] => [1,1,0,0,1,0,1,0,1,1,1,0,0,0,1,0]
 => [1,1,1,1,1,0,0,1,0,1,0,0,0,0,1,0]
 => ? = 1 + 1
0111010 => [2,1,1,2,2] => [1,1,0,0,1,0,1,0,1,1,0,0,1,1,0,0]
 => [1,1,1,1,1,0,1,0,0,1,0,0,0,0,1,0]
 => ? = 1 + 1
1000101 => [1,4,2,1] => [1,0,1,1,1,1,0,0,0,0,1,1,0,0,1,0]
 => [1,1,1,1,0,0,1,0,0,1,0,1,0,1,0,0]
 => ? = 0 + 1
1001000 => [1,3,4] => [1,0,1,1,1,0,0,0,1,1,1,1,0,0,0,0]
 => [1,1,1,0,1,0,1,0,1,0,0,1,0,1,0,0]
 => ? = 0 + 1
1001001 => [1,3,3,1] => [1,0,1,1,1,0,0,0,1,1,1,0,0,0,1,0]
 => [1,1,1,1,0,0,1,0,1,0,0,1,0,1,0,0]
 => ? = 0 + 1
1001010 => [1,3,2,2] => [1,0,1,1,1,0,0,0,1,1,0,0,1,1,0,0]
 => [1,1,1,1,0,1,0,0,1,0,0,1,0,1,0,0]
 => ? = 0 + 1
1001011 => [1,3,2,1,1] => [1,0,1,1,1,0,0,0,1,1,0,0,1,0,1,0]
 => [1,1,1,1,1,0,0,0,1,0,0,1,0,1,0,0]
 => ? = 0 + 1
1001101 => [1,3,1,2,1] => [1,0,1,1,1,0,0,0,1,0,1,1,0,0,1,0]
 => [1,1,1,1,1,0,0,1,0,0,0,1,0,1,0,0]
 => ? = 0 + 1
1001110 => [1,3,1,1,2] => [1,0,1,1,1,0,0,0,1,0,1,0,1,1,0,0]
 => [1,1,1,1,1,0,1,0,0,0,0,1,0,1,0,0]
 => ? = 0 + 1
1010010 => [1,2,3,2] => [1,0,1,1,0,0,1,1,1,0,0,0,1,1,0,0]
 => [1,1,1,1,0,1,0,0,1,0,1,0,0,1,0,0]
 => ? = 0 + 1
1010011 => [1,2,3,1,1] => [1,0,1,1,0,0,1,1,1,0,0,0,1,0,1,0]
 => [1,1,1,1,1,0,0,0,1,0,1,0,0,1,0,0]
 => ? = 0 + 1
1010100 => [1,2,2,3] => [1,0,1,1,0,0,1,1,0,0,1,1,1,0,0,0]
 => [1,1,1,1,0,1,0,1,0,0,1,0,0,1,0,0]
 => ? = 0 + 1
1010101 => [1,2,2,2,1] => [1,0,1,1,0,0,1,1,0,0,1,1,0,0,1,0]
 => [1,1,1,1,1,0,0,1,0,0,1,0,0,1,0,0]
 => ? = 0 + 1
1011001 => [1,2,1,3,1] => [1,0,1,1,0,0,1,0,1,1,1,0,0,0,1,0]
 => [1,1,1,1,1,0,0,1,0,1,0,0,0,1,0,0]
 => ? = 0 + 1
00001001 => [5,3,1] => [1,1,1,1,1,0,0,0,0,0,1,1,1,0,0,0,1,0]
 => [1,1,1,0,0,1,0,1,0,0,1,0,1,0,1,0,1,0]
 => ? = 4 + 1
00001010 => [5,2,2] => [1,1,1,1,1,0,0,0,0,0,1,1,0,0,1,1,0,0]
 => [1,1,1,0,1,0,0,1,0,0,1,0,1,0,1,0,1,0]
 => ? = 4 + 1
00001101 => [5,1,2,1] => [1,1,1,1,1,0,0,0,0,0,1,0,1,1,0,0,1,0]
 => [1,1,1,1,0,0,1,0,0,0,1,0,1,0,1,0,1,0]
 => ? = 4 + 1
00001110 => [5,1,1,2] => [1,1,1,1,1,0,0,0,0,0,1,0,1,0,1,1,0,0]
 => [1,1,1,1,0,1,0,0,0,0,1,0,1,0,1,0,1,0]
 => ? = 4 + 1
00010001 => [4,4,1] => [1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0,1,0]
 => [1,1,1,0,0,1,0,1,0,1,0,0,1,0,1,0,1,0]
 => ? = 3 + 1
00010010 => [4,3,2] => [1,1,1,1,0,0,0,0,1,1,1,0,0,0,1,1,0,0]
 => [1,1,1,0,1,0,0,1,0,1,0,0,1,0,1,0,1,0]
 => ? = 3 + 1
00010011 => [4,3,1,1] => [1,1,1,1,0,0,0,0,1,1,1,0,0,0,1,0,1,0]
 => [1,1,1,1,0,0,0,1,0,1,0,0,1,0,1,0,1,0]
 => ? = 3 + 1
00010100 => [4,2,3] => [1,1,1,1,0,0,0,0,1,1,0,0,1,1,1,0,0,0]
 => [1,1,1,0,1,0,1,0,0,1,0,0,1,0,1,0,1,0]
 => ? = 3 + 1
00010101 => [4,2,2,1] => [1,1,1,1,0,0,0,0,1,1,0,0,1,1,0,0,1,0]
 => [1,1,1,1,0,0,1,0,0,1,0,0,1,0,1,0,1,0]
 => ? = 3 + 1
00010110 => [4,2,1,2] => [1,1,1,1,0,0,0,0,1,1,0,0,1,0,1,1,0,0]
 => ?
 => ? = 3 + 1
00011001 => [4,1,3,1] => [1,1,1,1,0,0,0,0,1,0,1,1,1,0,0,0,1,0]
 => [1,1,1,1,0,0,1,0,1,0,0,0,1,0,1,0,1,0]
 => ? = 3 + 1
Description
The number of touch points (or returns) of a Dyck path. This is the number of points, excluding the origin, where the Dyck path has height 0.
Matching statistic: St000439
(load all 2 compositions to match this statistic)
Mapping
Mp00178: Binary words —to composition⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St000439: Dyck paths ⟶ ℤResult quality: 13% ●values known / values provided: 13%●distinct values known / distinct values provided: 91%
Values
0 => [2] => [1,1,0,0]
 => 3 = 1 + 2
1 => [1,1] => [1,0,1,0]
 => 2 = 0 + 2
00 => [3] => [1,1,1,0,0,0]
 => 4 = 2 + 2
01 => [2,1] => [1,1,0,0,1,0]
 => 3 = 1 + 2
10 => [1,2] => [1,0,1,1,0,0]
 => 2 = 0 + 2
11 => [1,1,1] => [1,0,1,0,1,0]
 => 2 = 0 + 2
000 => [4] => [1,1,1,1,0,0,0,0]
 => 5 = 3 + 2
001 => [3,1] => [1,1,1,0,0,0,1,0]
 => 4 = 2 + 2
010 => [2,2] => [1,1,0,0,1,1,0,0]
 => 3 = 1 + 2
011 => [2,1,1] => [1,1,0,0,1,0,1,0]
 => 3 = 1 + 2
100 => [1,3] => [1,0,1,1,1,0,0,0]
 => 2 = 0 + 2
101 => [1,2,1] => [1,0,1,1,0,0,1,0]
 => 2 = 0 + 2
110 => [1,1,2] => [1,0,1,0,1,1,0,0]
 => 2 = 0 + 2
111 => [1,1,1,1] => [1,0,1,0,1,0,1,0]
 => 2 = 0 + 2
0000 => [5] => [1,1,1,1,1,0,0,0,0,0]
 => 6 = 4 + 2
0001 => [4,1] => [1,1,1,1,0,0,0,0,1,0]
 => 5 = 3 + 2
0010 => [3,2] => [1,1,1,0,0,0,1,1,0,0]
 => 4 = 2 + 2
0011 => [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
 => 4 = 2 + 2
0100 => [2,3] => [1,1,0,0,1,1,1,0,0,0]
 => 3 = 1 + 2
0101 => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
 => 3 = 1 + 2
0110 => [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
 => 3 = 1 + 2
0111 => [2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
 => 3 = 1 + 2
1000 => [1,4] => [1,0,1,1,1,1,0,0,0,0]
 => 2 = 0 + 2
1001 => [1,3,1] => [1,0,1,1,1,0,0,0,1,0]
 => 2 = 0 + 2
1010 => [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
 => 2 = 0 + 2
1011 => [1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
 => 2 = 0 + 2
1100 => [1,1,3] => [1,0,1,0,1,1,1,0,0,0]
 => 2 = 0 + 2
1101 => [1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
 => 2 = 0 + 2
1110 => [1,1,1,2] => [1,0,1,0,1,0,1,1,0,0]
 => 2 = 0 + 2
1111 => [1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
 => 2 = 0 + 2
00000 => [6] => [1,1,1,1,1,1,0,0,0,0,0,0]
 => 7 = 5 + 2
00001 => [5,1] => [1,1,1,1,1,0,0,0,0,0,1,0]
 => 6 = 4 + 2
00010 => [4,2] => [1,1,1,1,0,0,0,0,1,1,0,0]
 => 5 = 3 + 2
00011 => [4,1,1] => [1,1,1,1,0,0,0,0,1,0,1,0]
 => 5 = 3 + 2
00100 => [3,3] => [1,1,1,0,0,0,1,1,1,0,0,0]
 => 4 = 2 + 2
00101 => [3,2,1] => [1,1,1,0,0,0,1,1,0,0,1,0]
 => 4 = 2 + 2
00110 => [3,1,2] => [1,1,1,0,0,0,1,0,1,1,0,0]
 => 4 = 2 + 2
00111 => [3,1,1,1] => [1,1,1,0,0,0,1,0,1,0,1,0]
 => 4 = 2 + 2
01000 => [2,4] => [1,1,0,0,1,1,1,1,0,0,0,0]
 => 3 = 1 + 2
01001 => [2,3,1] => [1,1,0,0,1,1,1,0,0,0,1,0]
 => 3 = 1 + 2
01010 => [2,2,2] => [1,1,0,0,1,1,0,0,1,1,0,0]
 => 3 = 1 + 2
01011 => [2,2,1,1] => [1,1,0,0,1,1,0,0,1,0,1,0]
 => 3 = 1 + 2
01100 => [2,1,3] => [1,1,0,0,1,0,1,1,1,0,0,0]
 => 3 = 1 + 2
01101 => [2,1,2,1] => [1,1,0,0,1,0,1,1,0,0,1,0]
 => 3 = 1 + 2
01110 => [2,1,1,2] => [1,1,0,0,1,0,1,0,1,1,0,0]
 => 3 = 1 + 2
01111 => [2,1,1,1,1] => [1,1,0,0,1,0,1,0,1,0,1,0]
 => 3 = 1 + 2
10000 => [1,5] => [1,0,1,1,1,1,1,0,0,0,0,0]
 => 2 = 0 + 2
10001 => [1,4,1] => [1,0,1,1,1,1,0,0,0,0,1,0]
 => 2 = 0 + 2
10010 => [1,3,2] => [1,0,1,1,1,0,0,0,1,1,0,0]
 => 2 = 0 + 2
10011 => [1,3,1,1] => [1,0,1,1,1,0,0,0,1,0,1,0]
 => 2 = 0 + 2
0000011 => [6,1,1] => [1,1,1,1,1,1,0,0,0,0,0,0,1,0,1,0]
 => ? = 5 + 2
0000100 => [5,3] => [1,1,1,1,1,0,0,0,0,0,1,1,1,0,0,0]
 => ? = 4 + 2
0000101 => [5,2,1] => [1,1,1,1,1,0,0,0,0,0,1,1,0,0,1,0]
 => ? = 4 + 2
0000110 => [5,1,2] => [1,1,1,1,1,0,0,0,0,0,1,0,1,1,0,0]
 => ? = 4 + 2
0000111 => [5,1,1,1] => [1,1,1,1,1,0,0,0,0,0,1,0,1,0,1,0]
 => ? = 4 + 2
0001001 => [4,3,1] => [1,1,1,1,0,0,0,0,1,1,1,0,0,0,1,0]
 => ? = 3 + 2
0001011 => [4,2,1,1] => [1,1,1,1,0,0,0,0,1,1,0,0,1,0,1,0]
 => ? = 3 + 2
0001100 => [4,1,3] => [1,1,1,1,0,0,0,0,1,0,1,1,1,0,0,0]
 => ? = 3 + 2
0001101 => [4,1,2,1] => [1,1,1,1,0,0,0,0,1,0,1,1,0,0,1,0]
 => ? = 3 + 2
0001110 => [4,1,1,2] => [1,1,1,1,0,0,0,0,1,0,1,0,1,1,0,0]
 => ? = 3 + 2
0001111 => [4,1,1,1,1] => [1,1,1,1,0,0,0,0,1,0,1,0,1,0,1,0]
 => ? = 3 + 2
0010000 => [3,5] => [1,1,1,0,0,0,1,1,1,1,1,0,0,0,0,0]
 => ? = 2 + 2
0010010 => [3,3,2] => [1,1,1,0,0,0,1,1,1,0,0,0,1,1,0,0]
 => ? = 2 + 2
0010011 => [3,3,1,1] => [1,1,1,0,0,0,1,1,1,0,0,0,1,0,1,0]
 => ? = 2 + 2
0010100 => [3,2,3] => [1,1,1,0,0,0,1,1,0,0,1,1,1,0,0,0]
 => ? = 2 + 2
0010101 => [3,2,2,1] => [1,1,1,0,0,0,1,1,0,0,1,1,0,0,1,0]
 => ? = 2 + 2
0010110 => [3,2,1,2] => [1,1,1,0,0,0,1,1,0,0,1,0,1,1,0,0]
 => ? = 2 + 2
0010111 => [3,2,1,1,1] => [1,1,1,0,0,0,1,1,0,0,1,0,1,0,1,0]
 => ? = 2 + 2
0011000 => [3,1,4] => [1,1,1,0,0,0,1,0,1,1,1,1,0,0,0,0]
 => ? = 2 + 2
0011001 => [3,1,3,1] => [1,1,1,0,0,0,1,0,1,1,1,0,0,0,1,0]
 => ? = 2 + 2
0011010 => [3,1,2,2] => [1,1,1,0,0,0,1,0,1,1,0,0,1,1,0,0]
 => ? = 2 + 2
0011011 => [3,1,2,1,1] => [1,1,1,0,0,0,1,0,1,1,0,0,1,0,1,0]
 => ? = 2 + 2
0011100 => [3,1,1,3] => [1,1,1,0,0,0,1,0,1,0,1,1,1,0,0,0]
 => ? = 2 + 2
0011101 => [3,1,1,2,1] => [1,1,1,0,0,0,1,0,1,0,1,1,0,0,1,0]
 => ? = 2 + 2
00000011 => [7,1,1] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0,1,0]
 => ? = 6 + 2
00000100 => [6,3] => [1,1,1,1,1,1,0,0,0,0,0,0,1,1,1,0,0,0]
 => ? = 5 + 2
00000101 => [6,2,1] => [1,1,1,1,1,1,0,0,0,0,0,0,1,1,0,0,1,0]
 => ? = 5 + 2
00000110 => [6,1,2] => [1,1,1,1,1,1,0,0,0,0,0,0,1,0,1,1,0,0]
 => ? = 5 + 2
00000111 => [6,1,1,1] => [1,1,1,1,1,1,0,0,0,0,0,0,1,0,1,0,1,0]
 => ? = 5 + 2
00001000 => [5,4] => [1,1,1,1,1,0,0,0,0,0,1,1,1,1,0,0,0,0]
 => ? = 4 + 2
00001001 => [5,3,1] => [1,1,1,1,1,0,0,0,0,0,1,1,1,0,0,0,1,0]
 => ? = 4 + 2
00001010 => [5,2,2] => [1,1,1,1,1,0,0,0,0,0,1,1,0,0,1,1,0,0]
 => ? = 4 + 2
00001011 => [5,2,1,1] => [1,1,1,1,1,0,0,0,0,0,1,1,0,0,1,0,1,0]
 => ? = 4 + 2
00001100 => [5,1,3] => [1,1,1,1,1,0,0,0,0,0,1,0,1,1,1,0,0,0]
 => ? = 4 + 2
00001101 => [5,1,2,1] => [1,1,1,1,1,0,0,0,0,0,1,0,1,1,0,0,1,0]
 => ? = 4 + 2
00001110 => [5,1,1,2] => [1,1,1,1,1,0,0,0,0,0,1,0,1,0,1,1,0,0]
 => ? = 4 + 2
00001111 => [5,1,1,1,1] => [1,1,1,1,1,0,0,0,0,0,1,0,1,0,1,0,1,0]
 => ? = 4 + 2
00010000 => [4,5] => [1,1,1,1,0,0,0,0,1,1,1,1,1,0,0,0,0,0]
 => ? = 3 + 2
00010001 => [4,4,1] => [1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0,1,0]
 => ? = 3 + 2
00010010 => [4,3,2] => [1,1,1,1,0,0,0,0,1,1,1,0,0,0,1,1,0,0]
 => ? = 3 + 2
00010011 => [4,3,1,1] => [1,1,1,1,0,0,0,0,1,1,1,0,0,0,1,0,1,0]
 => ? = 3 + 2
00010100 => [4,2,3] => [1,1,1,1,0,0,0,0,1,1,0,0,1,1,1,0,0,0]
 => ? = 3 + 2
00010101 => [4,2,2,1] => [1,1,1,1,0,0,0,0,1,1,0,0,1,1,0,0,1,0]
 => ? = 3 + 2
00010110 => [4,2,1,2] => [1,1,1,1,0,0,0,0,1,1,0,0,1,0,1,1,0,0]
 => ? = 3 + 2
00010111 => [4,2,1,1,1] => [1,1,1,1,0,0,0,0,1,1,0,0,1,0,1,0,1,0]
 => ? = 3 + 2
00011000 => [4,1,4] => [1,1,1,1,0,0,0,0,1,0,1,1,1,1,0,0,0,0]
 => ? = 3 + 2
00011001 => [4,1,3,1] => [1,1,1,1,0,0,0,0,1,0,1,1,1,0,0,0,1,0]
 => ? = 3 + 2
00011010 => [4,1,2,2] => [1,1,1,1,0,0,0,0,1,0,1,1,0,0,1,1,0,0]
 => ? = 3 + 2
00011011 => [4,1,2,1,1] => [1,1,1,1,0,0,0,0,1,0,1,1,0,0,1,0,1,0]
 => ? = 3 + 2
00011100 => [4,1,1,3] => [1,1,1,1,0,0,0,0,1,0,1,0,1,1,1,0,0,0]
 => ? = 3 + 2
Description
The position of the first down step of a Dyck path.
Matching statistic: St000678
(load all 2 compositions to match this statistic)
Mapping
Mp00178: Binary words —to composition⟶ Integer compositions
Mp00041: Integer compositions —conjugate⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St000678: Dyck paths ⟶ ℤResult quality: 11% ●values known / values provided: 11%●distinct values known / distinct values provided: 73%
Values
0 => [2] => [1,1] => [1,0,1,0]
 => 2 = 1 + 1
1 => [1,1] => [2] => [1,1,0,0]
 => 1 = 0 + 1
00 => [3] => [1,1,1] => [1,0,1,0,1,0]
 => 3 = 2 + 1
01 => [2,1] => [2,1] => [1,1,0,0,1,0]
 => 2 = 1 + 1
10 => [1,2] => [1,2] => [1,0,1,1,0,0]
 => 1 = 0 + 1
11 => [1,1,1] => [3] => [1,1,1,0,0,0]
 => 1 = 0 + 1
000 => [4] => [1,1,1,1] => [1,0,1,0,1,0,1,0]
 => 4 = 3 + 1
001 => [3,1] => [2,1,1] => [1,1,0,0,1,0,1,0]
 => 3 = 2 + 1
010 => [2,2] => [1,2,1] => [1,0,1,1,0,0,1,0]
 => 2 = 1 + 1
011 => [2,1,1] => [3,1] => [1,1,1,0,0,0,1,0]
 => 2 = 1 + 1
100 => [1,3] => [1,1,2] => [1,0,1,0,1,1,0,0]
 => 1 = 0 + 1
101 => [1,2,1] => [2,2] => [1,1,0,0,1,1,0,0]
 => 1 = 0 + 1
110 => [1,1,2] => [1,3] => [1,0,1,1,1,0,0,0]
 => 1 = 0 + 1
111 => [1,1,1,1] => [4] => [1,1,1,1,0,0,0,0]
 => 1 = 0 + 1
0000 => [5] => [1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
 => 5 = 4 + 1
0001 => [4,1] => [2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
 => 4 = 3 + 1
0010 => [3,2] => [1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
 => 3 = 2 + 1
0011 => [3,1,1] => [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
 => 3 = 2 + 1
0100 => [2,3] => [1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
 => 2 = 1 + 1
0101 => [2,2,1] => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
 => 2 = 1 + 1
0110 => [2,1,2] => [1,3,1] => [1,0,1,1,1,0,0,0,1,0]
 => 2 = 1 + 1
0111 => [2,1,1,1] => [4,1] => [1,1,1,1,0,0,0,0,1,0]
 => 2 = 1 + 1
1000 => [1,4] => [1,1,1,2] => [1,0,1,0,1,0,1,1,0,0]
 => 1 = 0 + 1
1001 => [1,3,1] => [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
 => 1 = 0 + 1
1010 => [1,2,2] => [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
 => 1 = 0 + 1
1011 => [1,2,1,1] => [3,2] => [1,1,1,0,0,0,1,1,0,0]
 => 1 = 0 + 1
1100 => [1,1,3] => [1,1,3] => [1,0,1,0,1,1,1,0,0,0]
 => 1 = 0 + 1
1101 => [1,1,2,1] => [2,3] => [1,1,0,0,1,1,1,0,0,0]
 => 1 = 0 + 1
1110 => [1,1,1,2] => [1,4] => [1,0,1,1,1,1,0,0,0,0]
 => 1 = 0 + 1
1111 => [1,1,1,1,1] => [5] => [1,1,1,1,1,0,0,0,0,0]
 => 1 = 0 + 1
00000 => [6] => [1,1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0,1,0]
 => 6 = 5 + 1
00001 => [5,1] => [2,1,1,1,1] => [1,1,0,0,1,0,1,0,1,0,1,0]
 => 5 = 4 + 1
00010 => [4,2] => [1,2,1,1,1] => [1,0,1,1,0,0,1,0,1,0,1,0]
 => 4 = 3 + 1
00011 => [4,1,1] => [3,1,1,1] => [1,1,1,0,0,0,1,0,1,0,1,0]
 => 4 = 3 + 1
00100 => [3,3] => [1,1,2,1,1] => [1,0,1,0,1,1,0,0,1,0,1,0]
 => 3 = 2 + 1
00101 => [3,2,1] => [2,2,1,1] => [1,1,0,0,1,1,0,0,1,0,1,0]
 => 3 = 2 + 1
00110 => [3,1,2] => [1,3,1,1] => [1,0,1,1,1,0,0,0,1,0,1,0]
 => 3 = 2 + 1
00111 => [3,1,1,1] => [4,1,1] => [1,1,1,1,0,0,0,0,1,0,1,0]
 => 3 = 2 + 1
01000 => [2,4] => [1,1,1,2,1] => [1,0,1,0,1,0,1,1,0,0,1,0]
 => 2 = 1 + 1
01001 => [2,3,1] => [2,1,2,1] => [1,1,0,0,1,0,1,1,0,0,1,0]
 => 2 = 1 + 1
01010 => [2,2,2] => [1,2,2,1] => [1,0,1,1,0,0,1,1,0,0,1,0]
 => 2 = 1 + 1
01011 => [2,2,1,1] => [3,2,1] => [1,1,1,0,0,0,1,1,0,0,1,0]
 => 2 = 1 + 1
01100 => [2,1,3] => [1,1,3,1] => [1,0,1,0,1,1,1,0,0,0,1,0]
 => 2 = 1 + 1
01101 => [2,1,2,1] => [2,3,1] => [1,1,0,0,1,1,1,0,0,0,1,0]
 => 2 = 1 + 1
01110 => [2,1,1,2] => [1,4,1] => [1,0,1,1,1,1,0,0,0,0,1,0]
 => 2 = 1 + 1
01111 => [2,1,1,1,1] => [5,1] => [1,1,1,1,1,0,0,0,0,0,1,0]
 => 2 = 1 + 1
10000 => [1,5] => [1,1,1,1,2] => [1,0,1,0,1,0,1,0,1,1,0,0]
 => 1 = 0 + 1
10001 => [1,4,1] => [2,1,1,2] => [1,1,0,0,1,0,1,0,1,1,0,0]
 => 1 = 0 + 1
10010 => [1,3,2] => [1,2,1,2] => [1,0,1,1,0,0,1,0,1,1,0,0]
 => 1 = 0 + 1
10011 => [1,3,1,1] => [3,1,2] => [1,1,1,0,0,0,1,0,1,1,0,0]
 => 1 = 0 + 1
0000011 => [6,1,1] => [3,1,1,1,1,1] => [1,1,1,0,0,0,1,0,1,0,1,0,1,0,1,0]
 => ? = 5 + 1
0000111 => [5,1,1,1] => [4,1,1,1,1] => [1,1,1,1,0,0,0,0,1,0,1,0,1,0,1,0]
 => ? = 4 + 1
0001011 => [4,2,1,1] => [3,2,1,1,1] => [1,1,1,0,0,0,1,1,0,0,1,0,1,0,1,0]
 => ? = 3 + 1
0001111 => [4,1,1,1,1] => [5,1,1,1] => [1,1,1,1,1,0,0,0,0,0,1,0,1,0,1,0]
 => ? = 3 + 1
0010011 => [3,3,1,1] => [3,1,2,1,1] => [1,1,1,0,0,0,1,0,1,1,0,0,1,0,1,0]
 => ? = 2 + 1
0010111 => [3,2,1,1,1] => [4,2,1,1] => [1,1,1,1,0,0,0,0,1,1,0,0,1,0,1,0]
 => ? = 2 + 1
0011011 => [3,1,2,1,1] => [3,3,1,1] => [1,1,1,0,0,0,1,1,1,0,0,0,1,0,1,0]
 => ? = 2 + 1
0011111 => [3,1,1,1,1,1] => [6,1,1] => [1,1,1,1,1,1,0,0,0,0,0,0,1,0,1,0]
 => ? = 2 + 1
0100011 => [2,4,1,1] => [3,1,1,2,1] => [1,1,1,0,0,0,1,0,1,0,1,1,0,0,1,0]
 => ? = 1 + 1
0100111 => [2,3,1,1,1] => [4,1,2,1] => [1,1,1,1,0,0,0,0,1,0,1,1,0,0,1,0]
 => ? = 1 + 1
0101011 => [2,2,2,1,1] => [3,2,2,1] => [1,1,1,0,0,0,1,1,0,0,1,1,0,0,1,0]
 => ? = 1 + 1
0101111 => [2,2,1,1,1,1] => [5,2,1] => [1,1,1,1,1,0,0,0,0,0,1,1,0,0,1,0]
 => ? = 1 + 1
0110011 => [2,1,3,1,1] => [3,1,3,1] => [1,1,1,0,0,0,1,0,1,1,1,0,0,0,1,0]
 => ? = 1 + 1
0110111 => [2,1,2,1,1,1] => [4,3,1] => [1,1,1,1,0,0,0,0,1,1,1,0,0,0,1,0]
 => ? = 1 + 1
0111011 => [2,1,1,2,1,1] => [3,4,1] => [1,1,1,0,0,0,1,1,1,1,0,0,0,0,1,0]
 => ? = 1 + 1
0111111 => [2,1,1,1,1,1,1] => [7,1] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
 => ? = 1 + 1
1000011 => [1,5,1,1] => [3,1,1,1,2] => [1,1,1,0,0,0,1,0,1,0,1,0,1,1,0,0]
 => ? = 0 + 1
1000111 => [1,4,1,1,1] => [4,1,1,2] => [1,1,1,1,0,0,0,0,1,0,1,0,1,1,0,0]
 => ? = 0 + 1
1001011 => [1,3,2,1,1] => [3,2,1,2] => [1,1,1,0,0,0,1,1,0,0,1,0,1,1,0,0]
 => ? = 0 + 1
1001111 => [1,3,1,1,1,1] => [5,1,2] => [1,1,1,1,1,0,0,0,0,0,1,0,1,1,0,0]
 => ? = 0 + 1
1010011 => [1,2,3,1,1] => [3,1,2,2] => [1,1,1,0,0,0,1,0,1,1,0,0,1,1,0,0]
 => ? = 0 + 1
1010111 => [1,2,2,1,1,1] => [4,2,2] => [1,1,1,1,0,0,0,0,1,1,0,0,1,1,0,0]
 => ? = 0 + 1
1011011 => [1,2,1,2,1,1] => [3,3,2] => [1,1,1,0,0,0,1,1,1,0,0,0,1,1,0,0]
 => ? = 0 + 1
1011111 => [1,2,1,1,1,1,1] => [6,2] => [1,1,1,1,1,1,0,0,0,0,0,0,1,1,0,0]
 => ? = 0 + 1
1100011 => [1,1,4,1,1] => [3,1,1,3] => [1,1,1,0,0,0,1,0,1,0,1,1,1,0,0,0]
 => ? = 0 + 1
1100111 => [1,1,3,1,1,1] => [4,1,3] => [1,1,1,1,0,0,0,0,1,0,1,1,1,0,0,0]
 => ? = 0 + 1
1101011 => [1,1,2,2,1,1] => [3,2,3] => [1,1,1,0,0,0,1,1,0,0,1,1,1,0,0,0]
 => ? = 0 + 1
1101111 => [1,1,2,1,1,1,1] => [5,3] => [1,1,1,1,1,0,0,0,0,0,1,1,1,0,0,0]
 => ? = 0 + 1
1110011 => [1,1,1,3,1,1] => [3,1,4] => [1,1,1,0,0,0,1,0,1,1,1,1,0,0,0,0]
 => ? = 0 + 1
1110111 => [1,1,1,2,1,1,1] => [4,4] => [1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0]
 => ? = 0 + 1
1111011 => [1,1,1,1,2,1,1] => [3,5] => [1,1,1,0,0,0,1,1,1,1,1,0,0,0,0,0]
 => ? = 0 + 1
1111111 => [1,1,1,1,1,1,1,1] => [8] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
 => ? = 0 + 1
00000000 => [9] => [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]
 => ? = 8 + 1
00000001 => [8,1] => [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]
 => ? = 7 + 1
00000010 => [7,2] => [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]
 => ? = 6 + 1
00000011 => [7,1,1] => [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]
 => ? = 6 + 1
00000100 => [6,3] => [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]
 => ? = 5 + 1
00000101 => [6,2,1] => [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]
 => ? = 5 + 1
00000110 => [6,1,2] => [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]
 => ? = 5 + 1
00000111 => [6,1,1,1] => [4,1,1,1,1,1] => [1,1,1,1,0,0,0,0,1,0,1,0,1,0,1,0,1,0]
 => ? = 5 + 1
00001000 => [5,4] => [1,1,1,2,1,1,1,1] => [1,0,1,0,1,0,1,1,0,0,1,0,1,0,1,0,1,0]
 => ? = 4 + 1
00001001 => [5,3,1] => [2,1,2,1,1,1,1] => [1,1,0,0,1,0,1,1,0,0,1,0,1,0,1,0,1,0]
 => ? = 4 + 1
00001010 => [5,2,2] => [1,2,2,1,1,1,1] => [1,0,1,1,0,0,1,1,0,0,1,0,1,0,1,0,1,0]
 => ? = 4 + 1
00001011 => [5,2,1,1] => [3,2,1,1,1,1] => [1,1,1,0,0,0,1,1,0,0,1,0,1,0,1,0,1,0]
 => ? = 4 + 1
00001100 => [5,1,3] => [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]
 => ? = 4 + 1
00001101 => [5,1,2,1] => [2,3,1,1,1,1] => [1,1,0,0,1,1,1,0,0,0,1,0,1,0,1,0,1,0]
 => ? = 4 + 1
00001110 => [5,1,1,2] => [1,4,1,1,1,1] => [1,0,1,1,1,1,0,0,0,0,1,0,1,0,1,0,1,0]
 => ? = 4 + 1
00001111 => [5,1,1,1,1] => [5,1,1,1,1] => [1,1,1,1,1,0,0,0,0,0,1,0,1,0,1,0,1,0]
 => ? = 4 + 1
00010000 => [4,5] => [1,1,1,1,2,1,1,1] => [1,0,1,0,1,0,1,0,1,1,0,0,1,0,1,0,1,0]
 => ? = 3 + 1
00010001 => [4,4,1] => [2,1,1,2,1,1,1] => [1,1,0,0,1,0,1,0,1,1,0,0,1,0,1,0,1,0]
 => ? = 3 + 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
Mp00032: Dyck paths —inverse zeta map⟶ Dyck paths
St000617: Dyck paths ⟶ ℤResult quality: 9% ●values known / values provided: 9%●distinct values known / distinct values provided: 91%
Values
0 => [2] => [1,1,0,0]
 => [1,0,1,0]
 => 2 = 1 + 1
1 => [1,1] => [1,0,1,0]
 => [1,1,0,0]
 => 1 = 0 + 1
00 => [3] => [1,1,1,0,0,0]
 => [1,0,1,0,1,0]
 => 3 = 2 + 1
01 => [2,1] => [1,1,0,0,1,0]
 => [1,1,0,1,0,0]
 => 2 = 1 + 1
10 => [1,2] => [1,0,1,1,0,0]
 => [1,0,1,1,0,0]
 => 1 = 0 + 1
11 => [1,1,1] => [1,0,1,0,1,0]
 => [1,1,1,0,0,0]
 => 1 = 0 + 1
000 => [4] => [1,1,1,1,0,0,0,0]
 => [1,0,1,0,1,0,1,0]
 => 4 = 3 + 1
001 => [3,1] => [1,1,1,0,0,0,1,0]
 => [1,1,0,1,0,1,0,0]
 => 3 = 2 + 1
010 => [2,2] => [1,1,0,0,1,1,0,0]
 => [1,0,1,1,0,1,0,0]
 => 2 = 1 + 1
011 => [2,1,1] => [1,1,0,0,1,0,1,0]
 => [1,1,1,0,1,0,0,0]
 => 2 = 1 + 1
100 => [1,3] => [1,0,1,1,1,0,0,0]
 => [1,0,1,0,1,1,0,0]
 => 1 = 0 + 1
101 => [1,2,1] => [1,0,1,1,0,0,1,0]
 => [1,1,0,1,1,0,0,0]
 => 1 = 0 + 1
110 => [1,1,2] => [1,0,1,0,1,1,0,0]
 => [1,0,1,1,1,0,0,0]
 => 1 = 0 + 1
111 => [1,1,1,1] => [1,0,1,0,1,0,1,0]
 => [1,1,1,1,0,0,0,0]
 => 1 = 0 + 1
0000 => [5] => [1,1,1,1,1,0,0,0,0,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => 5 = 4 + 1
0001 => [4,1] => [1,1,1,1,0,0,0,0,1,0]
 => [1,1,0,1,0,1,0,1,0,0]
 => 4 = 3 + 1
0010 => [3,2] => [1,1,1,0,0,0,1,1,0,0]
 => [1,0,1,1,0,1,0,1,0,0]
 => 3 = 2 + 1
0011 => [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
 => [1,1,1,0,1,0,1,0,0,0]
 => 3 = 2 + 1
0100 => [2,3] => [1,1,0,0,1,1,1,0,0,0]
 => [1,0,1,0,1,1,0,1,0,0]
 => 2 = 1 + 1
0101 => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
 => [1,1,0,1,1,0,1,0,0,0]
 => 2 = 1 + 1
0110 => [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
 => [1,0,1,1,1,0,1,0,0,0]
 => 2 = 1 + 1
0111 => [2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
 => [1,1,1,1,0,1,0,0,0,0]
 => 2 = 1 + 1
1000 => [1,4] => [1,0,1,1,1,1,0,0,0,0]
 => [1,0,1,0,1,0,1,1,0,0]
 => 1 = 0 + 1
1001 => [1,3,1] => [1,0,1,1,1,0,0,0,1,0]
 => [1,1,0,1,0,1,1,0,0,0]
 => 1 = 0 + 1
1010 => [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
 => [1,0,1,1,0,1,1,0,0,0]
 => 1 = 0 + 1
1011 => [1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
 => [1,1,1,0,1,1,0,0,0,0]
 => 1 = 0 + 1
1100 => [1,1,3] => [1,0,1,0,1,1,1,0,0,0]
 => [1,0,1,0,1,1,1,0,0,0]
 => 1 = 0 + 1
1101 => [1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
 => [1,1,0,1,1,1,0,0,0,0]
 => 1 = 0 + 1
1110 => [1,1,1,2] => [1,0,1,0,1,0,1,1,0,0]
 => [1,0,1,1,1,1,0,0,0,0]
 => 1 = 0 + 1
1111 => [1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => 1 = 0 + 1
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]
 => 6 = 5 + 1
00001 => [5,1] => [1,1,1,1,1,0,0,0,0,0,1,0]
 => [1,1,0,1,0,1,0,1,0,1,0,0]
 => 5 = 4 + 1
00010 => [4,2] => [1,1,1,1,0,0,0,0,1,1,0,0]
 => [1,0,1,1,0,1,0,1,0,1,0,0]
 => 4 = 3 + 1
00011 => [4,1,1] => [1,1,1,1,0,0,0,0,1,0,1,0]
 => [1,1,1,0,1,0,1,0,1,0,0,0]
 => 4 = 3 + 1
00100 => [3,3] => [1,1,1,0,0,0,1,1,1,0,0,0]
 => [1,0,1,0,1,1,0,1,0,1,0,0]
 => 3 = 2 + 1
00101 => [3,2,1] => [1,1,1,0,0,0,1,1,0,0,1,0]
 => [1,1,0,1,1,0,1,0,1,0,0,0]
 => 3 = 2 + 1
00110 => [3,1,2] => [1,1,1,0,0,0,1,0,1,1,0,0]
 => [1,0,1,1,1,0,1,0,1,0,0,0]
 => 3 = 2 + 1
00111 => [3,1,1,1] => [1,1,1,0,0,0,1,0,1,0,1,0]
 => [1,1,1,1,0,1,0,1,0,0,0,0]
 => 3 = 2 + 1
01000 => [2,4] => [1,1,0,0,1,1,1,1,0,0,0,0]
 => [1,0,1,0,1,0,1,1,0,1,0,0]
 => 2 = 1 + 1
01001 => [2,3,1] => [1,1,0,0,1,1,1,0,0,0,1,0]
 => [1,1,0,1,0,1,1,0,1,0,0,0]
 => 2 = 1 + 1
01010 => [2,2,2] => [1,1,0,0,1,1,0,0,1,1,0,0]
 => [1,0,1,1,0,1,1,0,1,0,0,0]
 => 2 = 1 + 1
01011 => [2,2,1,1] => [1,1,0,0,1,1,0,0,1,0,1,0]
 => [1,1,1,0,1,1,0,1,0,0,0,0]
 => 2 = 1 + 1
01100 => [2,1,3] => [1,1,0,0,1,0,1,1,1,0,0,0]
 => [1,0,1,0,1,1,1,0,1,0,0,0]
 => 2 = 1 + 1
01101 => [2,1,2,1] => [1,1,0,0,1,0,1,1,0,0,1,0]
 => [1,1,0,1,1,1,0,1,0,0,0,0]
 => 2 = 1 + 1
01110 => [2,1,1,2] => [1,1,0,0,1,0,1,0,1,1,0,0]
 => [1,0,1,1,1,1,0,1,0,0,0,0]
 => 2 = 1 + 1
01111 => [2,1,1,1,1] => [1,1,0,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,0,1,0,0,0,0,0]
 => 2 = 1 + 1
10000 => [1,5] => [1,0,1,1,1,1,1,0,0,0,0,0]
 => [1,0,1,0,1,0,1,0,1,1,0,0]
 => 1 = 0 + 1
10001 => [1,4,1] => [1,0,1,1,1,1,0,0,0,0,1,0]
 => [1,1,0,1,0,1,0,1,1,0,0,0]
 => 1 = 0 + 1
10010 => [1,3,2] => [1,0,1,1,1,0,0,0,1,1,0,0]
 => [1,0,1,1,0,1,0,1,1,0,0,0]
 => 1 = 0 + 1
10011 => [1,3,1,1] => [1,0,1,1,1,0,0,0,1,0,1,0]
 => [1,1,1,0,1,0,1,1,0,0,0,0]
 => 1 = 0 + 1
000000 => [7] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => ? = 6 + 1
000110 => [4,1,2] => [1,1,1,1,0,0,0,0,1,0,1,1,0,0]
 => [1,0,1,1,1,0,1,0,1,0,1,0,0,0]
 => ? = 3 + 1
001010 => [3,2,2] => [1,1,1,0,0,0,1,1,0,0,1,1,0,0]
 => [1,0,1,1,0,1,1,0,1,0,1,0,0,0]
 => ? = 2 + 1
001011 => [3,2,1,1] => [1,1,1,0,0,0,1,1,0,0,1,0,1,0]
 => [1,1,1,0,1,1,0,1,0,1,0,0,0,0]
 => ? = 2 + 1
001100 => [3,1,3] => [1,1,1,0,0,0,1,0,1,1,1,0,0,0]
 => [1,0,1,0,1,1,1,0,1,0,1,0,0,0]
 => ? = 2 + 1
001101 => [3,1,2,1] => [1,1,1,0,0,0,1,0,1,1,0,0,1,0]
 => [1,1,0,1,1,1,0,1,0,1,0,0,0,0]
 => ? = 2 + 1
010010 => [2,3,2] => [1,1,0,0,1,1,1,0,0,0,1,1,0,0]
 => [1,0,1,1,0,1,0,1,1,0,1,0,0,0]
 => ? = 1 + 1
010100 => [2,2,3] => [1,1,0,0,1,1,0,0,1,1,1,0,0,0]
 => [1,0,1,0,1,1,0,1,1,0,1,0,0,0]
 => ? = 1 + 1
011000 => [2,1,4] => [1,1,0,0,1,0,1,1,1,1,0,0,0,0]
 => [1,0,1,0,1,0,1,1,1,0,1,0,0,0]
 => ? = 1 + 1
100110 => [1,3,1,2] => [1,0,1,1,1,0,0,0,1,0,1,1,0,0]
 => [1,0,1,1,1,0,1,0,1,1,0,0,0,0]
 => ? = 0 + 1
100111 => [1,3,1,1,1] => [1,0,1,1,1,0,0,0,1,0,1,0,1,0]
 => [1,1,1,1,0,1,0,1,1,0,0,0,0,0]
 => ? = 0 + 1
101011 => [1,2,2,1,1] => [1,0,1,1,0,0,1,1,0,0,1,0,1,0]
 => [1,1,1,0,1,1,0,1,1,0,0,0,0,0]
 => ? = 0 + 1
101100 => [1,2,1,3] => [1,0,1,1,0,0,1,0,1,1,1,0,0,0]
 => [1,0,1,0,1,1,1,0,1,1,0,0,0,0]
 => ? = 0 + 1
101101 => [1,2,1,2,1] => [1,0,1,1,0,0,1,0,1,1,0,0,1,0]
 => [1,1,0,1,1,1,0,1,1,0,0,0,0,0]
 => ? = 0 + 1
110000 => [1,1,5] => [1,0,1,0,1,1,1,1,1,0,0,0,0,0]
 => [1,0,1,0,1,0,1,0,1,1,1,0,0,0]
 => ? = 0 + 1
110010 => [1,1,3,2] => [1,0,1,0,1,1,1,0,0,0,1,1,0,0]
 => [1,0,1,1,0,1,0,1,1,1,0,0,0,0]
 => ? = 0 + 1
110100 => [1,1,2,3] => [1,0,1,0,1,1,0,0,1,1,1,0,0,0]
 => [1,0,1,0,1,1,0,1,1,1,0,0,0,0]
 => ? = 0 + 1
110101 => [1,1,2,2,1] => [1,0,1,0,1,1,0,0,1,1,0,0,1,0]
 => [1,1,0,1,1,0,1,1,1,0,0,0,0,0]
 => ? = 0 + 1
111000 => [1,1,1,4] => [1,0,1,0,1,0,1,1,1,1,0,0,0,0]
 => [1,0,1,0,1,0,1,1,1,1,0,0,0,0]
 => ? = 0 + 1
111001 => [1,1,1,3,1] => [1,0,1,0,1,0,1,1,1,0,0,0,1,0]
 => [1,1,0,1,0,1,1,1,1,0,0,0,0,0]
 => ? = 0 + 1
0000000 => [8] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => ? = 7 + 1
0000010 => [6,2] => [1,1,1,1,1,1,0,0,0,0,0,0,1,1,0,0]
 => [1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => ? = 5 + 1
0000100 => [5,3] => [1,1,1,1,1,0,0,0,0,0,1,1,1,0,0,0]
 => [1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,0]
 => ? = 4 + 1
0000101 => [5,2,1] => [1,1,1,1,1,0,0,0,0,0,1,1,0,0,1,0]
 => [1,1,0,1,1,0,1,0,1,0,1,0,1,0,0,0]
 => ? = 4 + 1
0000110 => [5,1,2] => [1,1,1,1,1,0,0,0,0,0,1,0,1,1,0,0]
 => [1,0,1,1,1,0,1,0,1,0,1,0,1,0,0,0]
 => ? = 4 + 1
0000111 => [5,1,1,1] => [1,1,1,1,1,0,0,0,0,0,1,0,1,0,1,0]
 => [1,1,1,1,0,1,0,1,0,1,0,1,0,0,0,0]
 => ? = 4 + 1
0001000 => [4,4] => [1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0]
 => [1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,0]
 => ? = 3 + 1
0001001 => [4,3,1] => [1,1,1,1,0,0,0,0,1,1,1,0,0,0,1,0]
 => [1,1,0,1,0,1,1,0,1,0,1,0,1,0,0,0]
 => ? = 3 + 1
0001010 => [4,2,2] => [1,1,1,1,0,0,0,0,1,1,0,0,1,1,0,0]
 => [1,0,1,1,0,1,1,0,1,0,1,0,1,0,0,0]
 => ? = 3 + 1
0001011 => [4,2,1,1] => [1,1,1,1,0,0,0,0,1,1,0,0,1,0,1,0]
 => [1,1,1,0,1,1,0,1,0,1,0,1,0,0,0,0]
 => ? = 3 + 1
0001100 => [4,1,3] => [1,1,1,1,0,0,0,0,1,0,1,1,1,0,0,0]
 => [1,0,1,0,1,1,1,0,1,0,1,0,1,0,0,0]
 => ? = 3 + 1
0001101 => [4,1,2,1] => [1,1,1,1,0,0,0,0,1,0,1,1,0,0,1,0]
 => [1,1,0,1,1,1,0,1,0,1,0,1,0,0,0,0]
 => ? = 3 + 1
0001110 => [4,1,1,2] => [1,1,1,1,0,0,0,0,1,0,1,0,1,1,0,0]
 => [1,0,1,1,1,1,0,1,0,1,0,1,0,0,0,0]
 => ? = 3 + 1
0010010 => [3,3,2] => [1,1,1,0,0,0,1,1,1,0,0,0,1,1,0,0]
 => [1,0,1,1,0,1,0,1,1,0,1,0,1,0,0,0]
 => ? = 2 + 1
0010011 => [3,3,1,1] => [1,1,1,0,0,0,1,1,1,0,0,0,1,0,1,0]
 => [1,1,1,0,1,0,1,1,0,1,0,1,0,0,0,0]
 => ? = 2 + 1
0010100 => [3,2,3] => [1,1,1,0,0,0,1,1,0,0,1,1,1,0,0,0]
 => [1,0,1,0,1,1,0,1,1,0,1,0,1,0,0,0]
 => ? = 2 + 1
0010101 => [3,2,2,1] => [1,1,1,0,0,0,1,1,0,0,1,1,0,0,1,0]
 => [1,1,0,1,1,0,1,1,0,1,0,1,0,0,0,0]
 => ? = 2 + 1
0010110 => [3,2,1,2] => [1,1,1,0,0,0,1,1,0,0,1,0,1,1,0,0]
 => [1,0,1,1,1,0,1,1,0,1,0,1,0,0,0,0]
 => ? = 2 + 1
0010111 => [3,2,1,1,1] => [1,1,1,0,0,0,1,1,0,0,1,0,1,0,1,0]
 => [1,1,1,1,0,1,1,0,1,0,1,0,0,0,0,0]
 => ? = 2 + 1
0011000 => [3,1,4] => [1,1,1,0,0,0,1,0,1,1,1,1,0,0,0,0]
 => [1,0,1,0,1,0,1,1,1,0,1,0,1,0,0,0]
 => ? = 2 + 1
0011001 => [3,1,3,1] => [1,1,1,0,0,0,1,0,1,1,1,0,0,0,1,0]
 => [1,1,0,1,0,1,1,1,0,1,0,1,0,0,0,0]
 => ? = 2 + 1
0011010 => [3,1,2,2] => [1,1,1,0,0,0,1,0,1,1,0,0,1,1,0,0]
 => [1,0,1,1,0,1,1,1,0,1,0,1,0,0,0,0]
 => ? = 2 + 1
0011011 => [3,1,2,1,1] => [1,1,1,0,0,0,1,0,1,1,0,0,1,0,1,0]
 => [1,1,1,0,1,1,1,0,1,0,1,0,0,0,0,0]
 => ? = 2 + 1
0011100 => [3,1,1,3] => [1,1,1,0,0,0,1,0,1,0,1,1,1,0,0,0]
 => [1,0,1,0,1,1,1,1,0,1,0,1,0,0,0,0]
 => ? = 2 + 1
0011101 => [3,1,1,2,1] => [1,1,1,0,0,0,1,0,1,0,1,1,0,0,1,0]
 => [1,1,0,1,1,1,1,0,1,0,1,0,0,0,0,0]
 => ? = 2 + 1
0100010 => [2,4,2] => [1,1,0,0,1,1,1,1,0,0,0,0,1,1,0,0]
 => [1,0,1,1,0,1,0,1,0,1,1,0,1,0,0,0]
 => ? = 1 + 1
0100011 => [2,4,1,1] => [1,1,0,0,1,1,1,1,0,0,0,0,1,0,1,0]
 => [1,1,1,0,1,0,1,0,1,1,0,1,0,0,0,0]
 => ? = 1 + 1
0100100 => [2,3,3] => [1,1,0,0,1,1,1,0,0,0,1,1,1,0,0,0]
 => [1,0,1,0,1,1,0,1,0,1,1,0,1,0,0,0]
 => ? = 1 + 1
0100101 => [2,3,2,1] => [1,1,0,0,1,1,1,0,0,0,1,1,0,0,1,0]
 => [1,1,0,1,1,0,1,0,1,1,0,1,0,0,0,0]
 => ? = 1 + 1
0100110 => [2,3,1,2] => [1,1,0,0,1,1,1,0,0,0,1,0,1,1,0,0]
 => [1,0,1,1,1,0,1,0,1,1,0,1,0,0,0,0]
 => ? = 1 + 1
Description
The number of global maxima of a Dyck path.
Matching statistic: St000054
(load all 2 compositions to match this statistic)
Mapping
Mp00178: Binary words —to composition⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
Mp00119: Dyck paths —to 321-avoiding permutation (Krattenthaler)⟶ Permutations
St000054: Permutations ⟶ ℤResult quality: 7% ●values known / values provided: 7%●distinct values known / distinct values provided: 91%
Values
0 => [2] => [1,1,0,0]
 => [2,1] => 2 = 1 + 1
1 => [1,1] => [1,0,1,0]
 => [1,2] => 1 = 0 + 1
00 => [3] => [1,1,1,0,0,0]
 => [3,1,2] => 3 = 2 + 1
01 => [2,1] => [1,1,0,0,1,0]
 => [2,1,3] => 2 = 1 + 1
10 => [1,2] => [1,0,1,1,0,0]
 => [1,3,2] => 1 = 0 + 1
11 => [1,1,1] => [1,0,1,0,1,0]
 => [1,2,3] => 1 = 0 + 1
000 => [4] => [1,1,1,1,0,0,0,0]
 => [4,1,2,3] => 4 = 3 + 1
001 => [3,1] => [1,1,1,0,0,0,1,0]
 => [3,1,2,4] => 3 = 2 + 1
010 => [2,2] => [1,1,0,0,1,1,0,0]
 => [2,1,4,3] => 2 = 1 + 1
011 => [2,1,1] => [1,1,0,0,1,0,1,0]
 => [2,1,3,4] => 2 = 1 + 1
100 => [1,3] => [1,0,1,1,1,0,0,0]
 => [1,4,2,3] => 1 = 0 + 1
101 => [1,2,1] => [1,0,1,1,0,0,1,0]
 => [1,3,2,4] => 1 = 0 + 1
110 => [1,1,2] => [1,0,1,0,1,1,0,0]
 => [1,2,4,3] => 1 = 0 + 1
111 => [1,1,1,1] => [1,0,1,0,1,0,1,0]
 => [1,2,3,4] => 1 = 0 + 1
0000 => [5] => [1,1,1,1,1,0,0,0,0,0]
 => [5,1,2,3,4] => 5 = 4 + 1
0001 => [4,1] => [1,1,1,1,0,0,0,0,1,0]
 => [4,1,2,3,5] => 4 = 3 + 1
0010 => [3,2] => [1,1,1,0,0,0,1,1,0,0]
 => [3,1,2,5,4] => 3 = 2 + 1
0011 => [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
 => [3,1,2,4,5] => 3 = 2 + 1
0100 => [2,3] => [1,1,0,0,1,1,1,0,0,0]
 => [2,1,5,3,4] => 2 = 1 + 1
0101 => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
 => [2,1,4,3,5] => 2 = 1 + 1
0110 => [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
 => [2,1,3,5,4] => 2 = 1 + 1
0111 => [2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
 => [2,1,3,4,5] => 2 = 1 + 1
1000 => [1,4] => [1,0,1,1,1,1,0,0,0,0]
 => [1,5,2,3,4] => 1 = 0 + 1
1001 => [1,3,1] => [1,0,1,1,1,0,0,0,1,0]
 => [1,4,2,3,5] => 1 = 0 + 1
1010 => [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
 => [1,3,2,5,4] => 1 = 0 + 1
1011 => [1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
 => [1,3,2,4,5] => 1 = 0 + 1
1100 => [1,1,3] => [1,0,1,0,1,1,1,0,0,0]
 => [1,2,5,3,4] => 1 = 0 + 1
1101 => [1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
 => [1,2,4,3,5] => 1 = 0 + 1
1110 => [1,1,1,2] => [1,0,1,0,1,0,1,1,0,0]
 => [1,2,3,5,4] => 1 = 0 + 1
1111 => [1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
 => [1,2,3,4,5] => 1 = 0 + 1
00000 => [6] => [1,1,1,1,1,1,0,0,0,0,0,0]
 => [6,1,2,3,4,5] => 6 = 5 + 1
00001 => [5,1] => [1,1,1,1,1,0,0,0,0,0,1,0]
 => [5,1,2,3,4,6] => 5 = 4 + 1
00010 => [4,2] => [1,1,1,1,0,0,0,0,1,1,0,0]
 => [4,1,2,3,6,5] => 4 = 3 + 1
00011 => [4,1,1] => [1,1,1,1,0,0,0,0,1,0,1,0]
 => [4,1,2,3,5,6] => 4 = 3 + 1
00100 => [3,3] => [1,1,1,0,0,0,1,1,1,0,0,0]
 => [3,1,2,6,4,5] => 3 = 2 + 1
00101 => [3,2,1] => [1,1,1,0,0,0,1,1,0,0,1,0]
 => [3,1,2,5,4,6] => 3 = 2 + 1
00110 => [3,1,2] => [1,1,1,0,0,0,1,0,1,1,0,0]
 => [3,1,2,4,6,5] => 3 = 2 + 1
00111 => [3,1,1,1] => [1,1,1,0,0,0,1,0,1,0,1,0]
 => [3,1,2,4,5,6] => 3 = 2 + 1
01000 => [2,4] => [1,1,0,0,1,1,1,1,0,0,0,0]
 => [2,1,6,3,4,5] => 2 = 1 + 1
01001 => [2,3,1] => [1,1,0,0,1,1,1,0,0,0,1,0]
 => [2,1,5,3,4,6] => 2 = 1 + 1
01010 => [2,2,2] => [1,1,0,0,1,1,0,0,1,1,0,0]
 => [2,1,4,3,6,5] => 2 = 1 + 1
01011 => [2,2,1,1] => [1,1,0,0,1,1,0,0,1,0,1,0]
 => [2,1,4,3,5,6] => 2 = 1 + 1
01100 => [2,1,3] => [1,1,0,0,1,0,1,1,1,0,0,0]
 => [2,1,3,6,4,5] => 2 = 1 + 1
01101 => [2,1,2,1] => [1,1,0,0,1,0,1,1,0,0,1,0]
 => [2,1,3,5,4,6] => 2 = 1 + 1
01110 => [2,1,1,2] => [1,1,0,0,1,0,1,0,1,1,0,0]
 => [2,1,3,4,6,5] => 2 = 1 + 1
01111 => [2,1,1,1,1] => [1,1,0,0,1,0,1,0,1,0,1,0]
 => [2,1,3,4,5,6] => 2 = 1 + 1
10000 => [1,5] => [1,0,1,1,1,1,1,0,0,0,0,0]
 => [1,6,2,3,4,5] => 1 = 0 + 1
10001 => [1,4,1] => [1,0,1,1,1,1,0,0,0,0,1,0]
 => [1,5,2,3,4,6] => 1 = 0 + 1
10010 => [1,3,2] => [1,0,1,1,1,0,0,0,1,1,0,0]
 => [1,4,2,3,6,5] => 1 = 0 + 1
10011 => [1,3,1,1] => [1,0,1,1,1,0,0,0,1,0,1,0]
 => [1,4,2,3,5,6] => 1 = 0 + 1
000100 => [4,3] => [1,1,1,1,0,0,0,0,1,1,1,0,0,0]
 => [4,1,2,3,7,5,6] => ? = 3 + 1
000101 => [4,2,1] => [1,1,1,1,0,0,0,0,1,1,0,0,1,0]
 => [4,1,2,3,6,5,7] => ? = 3 + 1
001000 => [3,4] => [1,1,1,0,0,0,1,1,1,1,0,0,0,0]
 => [3,1,2,7,4,5,6] => ? = 2 + 1
001001 => [3,3,1] => [1,1,1,0,0,0,1,1,1,0,0,0,1,0]
 => [3,1,2,6,4,5,7] => ? = 2 + 1
001010 => [3,2,2] => [1,1,1,0,0,0,1,1,0,0,1,1,0,0]
 => [3,1,2,5,4,7,6] => ? = 2 + 1
001011 => [3,2,1,1] => [1,1,1,0,0,0,1,1,0,0,1,0,1,0]
 => [3,1,2,5,4,6,7] => ? = 2 + 1
001100 => [3,1,3] => [1,1,1,0,0,0,1,0,1,1,1,0,0,0]
 => [3,1,2,4,7,5,6] => ? = 2 + 1
001101 => [3,1,2,1] => [1,1,1,0,0,0,1,0,1,1,0,0,1,0]
 => [3,1,2,4,6,5,7] => ? = 2 + 1
010010 => [2,3,2] => [1,1,0,0,1,1,1,0,0,0,1,1,0,0]
 => [2,1,5,3,4,7,6] => ? = 1 + 1
010011 => [2,3,1,1] => [1,1,0,0,1,1,1,0,0,0,1,0,1,0]
 => [2,1,5,3,4,6,7] => ? = 1 + 1
010100 => [2,2,3] => [1,1,0,0,1,1,0,0,1,1,1,0,0,0]
 => [2,1,4,3,7,5,6] => ? = 1 + 1
010101 => [2,2,2,1] => [1,1,0,0,1,1,0,0,1,1,0,0,1,0]
 => [2,1,4,3,6,5,7] => ? = 1 + 1
010110 => [2,2,1,2] => [1,1,0,0,1,1,0,0,1,0,1,1,0,0]
 => [2,1,4,3,5,7,6] => ? = 1 + 1
010111 => [2,2,1,1,1] => [1,1,0,0,1,1,0,0,1,0,1,0,1,0]
 => [2,1,4,3,5,6,7] => ? = 1 + 1
011001 => [2,1,3,1] => [1,1,0,0,1,0,1,1,1,0,0,0,1,0]
 => [2,1,3,6,4,5,7] => ? = 1 + 1
011010 => [2,1,2,2] => [1,1,0,0,1,0,1,1,0,0,1,1,0,0]
 => [2,1,3,5,4,7,6] => ? = 1 + 1
011011 => [2,1,2,1,1] => [1,1,0,0,1,0,1,1,0,0,1,0,1,0]
 => [2,1,3,5,4,6,7] => ? = 1 + 1
011101 => [2,1,1,2,1] => [1,1,0,0,1,0,1,0,1,1,0,0,1,0]
 => [2,1,3,4,6,5,7] => ? = 1 + 1
011110 => [2,1,1,1,2] => [1,1,0,0,1,0,1,0,1,0,1,1,0,0]
 => [2,1,3,4,5,7,6] => ? = 1 + 1
100001 => [1,5,1] => [1,0,1,1,1,1,1,0,0,0,0,0,1,0]
 => [1,6,2,3,4,5,7] => ? = 0 + 1
100011 => [1,4,1,1] => [1,0,1,1,1,1,0,0,0,0,1,0,1,0]
 => [1,5,2,3,4,6,7] => ? = 0 + 1
100100 => [1,3,3] => [1,0,1,1,1,0,0,0,1,1,1,0,0,0]
 => [1,4,2,3,7,5,6] => ? = 0 + 1
100101 => [1,3,2,1] => [1,0,1,1,1,0,0,0,1,1,0,0,1,0]
 => [1,4,2,3,6,5,7] => ? = 0 + 1
100110 => [1,3,1,2] => [1,0,1,1,1,0,0,0,1,0,1,1,0,0]
 => [1,4,2,3,5,7,6] => ? = 0 + 1
100111 => [1,3,1,1,1] => [1,0,1,1,1,0,0,0,1,0,1,0,1,0]
 => [1,4,2,3,5,6,7] => ? = 0 + 1
101000 => [1,2,4] => [1,0,1,1,0,0,1,1,1,1,0,0,0,0]
 => [1,3,2,7,4,5,6] => ? = 0 + 1
101001 => [1,2,3,1] => [1,0,1,1,0,0,1,1,1,0,0,0,1,0]
 => [1,3,2,6,4,5,7] => ? = 0 + 1
101010 => [1,2,2,2] => [1,0,1,1,0,0,1,1,0,0,1,1,0,0]
 => [1,3,2,5,4,7,6] => ? = 0 + 1
101011 => [1,2,2,1,1] => [1,0,1,1,0,0,1,1,0,0,1,0,1,0]
 => [1,3,2,5,4,6,7] => ? = 0 + 1
101100 => [1,2,1,3] => [1,0,1,1,0,0,1,0,1,1,1,0,0,0]
 => [1,3,2,4,7,5,6] => ? = 0 + 1
101101 => [1,2,1,2,1] => [1,0,1,1,0,0,1,0,1,1,0,0,1,0]
 => [1,3,2,4,6,5,7] => ? = 0 + 1
101110 => [1,2,1,1,2] => [1,0,1,1,0,0,1,0,1,0,1,1,0,0]
 => [1,3,2,4,5,7,6] => ? = 0 + 1
110000 => [1,1,5] => [1,0,1,0,1,1,1,1,1,0,0,0,0,0]
 => [1,2,7,3,4,5,6] => ? = 0 + 1
110001 => [1,1,4,1] => [1,0,1,0,1,1,1,1,0,0,0,0,1,0]
 => [1,2,6,3,4,5,7] => ? = 0 + 1
110010 => [1,1,3,2] => [1,0,1,0,1,1,1,0,0,0,1,1,0,0]
 => [1,2,5,3,4,7,6] => ? = 0 + 1
110011 => [1,1,3,1,1] => [1,0,1,0,1,1,1,0,0,0,1,0,1,0]
 => [1,2,5,3,4,6,7] => ? = 0 + 1
110100 => [1,1,2,3] => [1,0,1,0,1,1,0,0,1,1,1,0,0,0]
 => [1,2,4,3,7,5,6] => ? = 0 + 1
110101 => [1,1,2,2,1] => [1,0,1,0,1,1,0,0,1,1,0,0,1,0]
 => [1,2,4,3,6,5,7] => ? = 0 + 1
110110 => [1,1,2,1,2] => [1,0,1,0,1,1,0,0,1,0,1,1,0,0]
 => [1,2,4,3,5,7,6] => ? = 0 + 1
110111 => [1,1,2,1,1,1] => [1,0,1,0,1,1,0,0,1,0,1,0,1,0]
 => [1,2,4,3,5,6,7] => ? = 0 + 1
111000 => [1,1,1,4] => [1,0,1,0,1,0,1,1,1,1,0,0,0,0]
 => [1,2,3,7,4,5,6] => ? = 0 + 1
111001 => [1,1,1,3,1] => [1,0,1,0,1,0,1,1,1,0,0,0,1,0]
 => [1,2,3,6,4,5,7] => ? = 0 + 1
111010 => [1,1,1,2,2] => [1,0,1,0,1,0,1,1,0,0,1,1,0,0]
 => [1,2,3,5,4,7,6] => ? = 0 + 1
111011 => [1,1,1,2,1,1] => [1,0,1,0,1,0,1,1,0,0,1,0,1,0]
 => [1,2,3,5,4,6,7] => ? = 0 + 1
111100 => [1,1,1,1,3] => [1,0,1,0,1,0,1,0,1,1,1,0,0,0]
 => [1,2,3,4,7,5,6] => ? = 0 + 1
0000100 => [5,3] => [1,1,1,1,1,0,0,0,0,0,1,1,1,0,0,0]
 => [5,1,2,3,4,8,6,7] => ? = 4 + 1
0000101 => [5,2,1] => [1,1,1,1,1,0,0,0,0,0,1,1,0,0,1,0]
 => [5,1,2,3,4,7,6,8] => ? = 4 + 1
0001001 => [4,3,1] => [1,1,1,1,0,0,0,0,1,1,1,0,0,0,1,0]
 => [4,1,2,3,7,5,6,8] => ? = 3 + 1
0001011 => [4,2,1,1] => [1,1,1,1,0,0,0,0,1,1,0,0,1,0,1,0]
 => [4,1,2,3,6,5,7,8] => ? = 3 + 1
0001100 => [4,1,3] => [1,1,1,1,0,0,0,0,1,0,1,1,1,0,0,0]
 => [4,1,2,3,5,8,6,7] => ? = 3 + 1
Description
The first entry of the permutation. This can be described as 1 plus the number of occurrences of the vincular pattern ([2,1], {(0,0),(0,1),(0,2)}), i.e., the first column is shaded, see [1]. This statistic is related to the number of deficiencies [[St000703]] as follows: consider the arc diagram of a permutation $\pi$ of $n$, together with its rotations, obtained by conjugating with the long cycle $(1,\dots,n)$. Drawing the labels $1$ to $n$ in this order on a circle, and the arcs $(i, \pi(i))$ as straight lines, the rotation of $\pi$ is obtained by replacing each number $i$ by $(i\bmod n) +1$. Then, $\pi(1)-1$ is the number of rotations of $\pi$ where the arc $(1, \pi(1))$ is a deficiency. In particular, if $O(\pi)$ is the orbit of rotations of $\pi$, then the number of deficiencies of $\pi$ equals $$ \frac{1}{|O(\pi)|}\sum_{\sigma\in O(\pi)} (\sigma(1)-1). $$
Matching statistic: St000505
Mapping
Mp00178: Binary words —to composition⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
Mp00138: Dyck paths —to noncrossing partition⟶ Set partitions
St000505: Set partitions ⟶ ℤResult quality: 7% ●values known / values provided: 7%●distinct values known / distinct values provided: 91%
Values
0 => [2] => [1,1,0,0]
 => {{1,2}}
 => 2 = 1 + 1
1 => [1,1] => [1,0,1,0]
 => {{1},{2}}
 => 1 = 0 + 1
00 => [3] => [1,1,1,0,0,0]
 => {{1,2,3}}
 => 3 = 2 + 1
01 => [2,1] => [1,1,0,0,1,0]
 => {{1,2},{3}}
 => 2 = 1 + 1
10 => [1,2] => [1,0,1,1,0,0]
 => {{1},{2,3}}
 => 1 = 0 + 1
11 => [1,1,1] => [1,0,1,0,1,0]
 => {{1},{2},{3}}
 => 1 = 0 + 1
000 => [4] => [1,1,1,1,0,0,0,0]
 => {{1,2,3,4}}
 => 4 = 3 + 1
001 => [3,1] => [1,1,1,0,0,0,1,0]
 => {{1,2,3},{4}}
 => 3 = 2 + 1
010 => [2,2] => [1,1,0,0,1,1,0,0]
 => {{1,2},{3,4}}
 => 2 = 1 + 1
011 => [2,1,1] => [1,1,0,0,1,0,1,0]
 => {{1,2},{3},{4}}
 => 2 = 1 + 1
100 => [1,3] => [1,0,1,1,1,0,0,0]
 => {{1},{2,3,4}}
 => 1 = 0 + 1
101 => [1,2,1] => [1,0,1,1,0,0,1,0]
 => {{1},{2,3},{4}}
 => 1 = 0 + 1
110 => [1,1,2] => [1,0,1,0,1,1,0,0]
 => {{1},{2},{3,4}}
 => 1 = 0 + 1
111 => [1,1,1,1] => [1,0,1,0,1,0,1,0]
 => {{1},{2},{3},{4}}
 => 1 = 0 + 1
0000 => [5] => [1,1,1,1,1,0,0,0,0,0]
 => {{1,2,3,4,5}}
 => 5 = 4 + 1
0001 => [4,1] => [1,1,1,1,0,0,0,0,1,0]
 => {{1,2,3,4},{5}}
 => 4 = 3 + 1
0010 => [3,2] => [1,1,1,0,0,0,1,1,0,0]
 => {{1,2,3},{4,5}}
 => 3 = 2 + 1
0011 => [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
 => {{1,2,3},{4},{5}}
 => 3 = 2 + 1
0100 => [2,3] => [1,1,0,0,1,1,1,0,0,0]
 => {{1,2},{3,4,5}}
 => 2 = 1 + 1
0101 => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
 => {{1,2},{3,4},{5}}
 => 2 = 1 + 1
0110 => [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
 => {{1,2},{3},{4,5}}
 => 2 = 1 + 1
0111 => [2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
 => {{1,2},{3},{4},{5}}
 => 2 = 1 + 1
1000 => [1,4] => [1,0,1,1,1,1,0,0,0,0]
 => {{1},{2,3,4,5}}
 => 1 = 0 + 1
1001 => [1,3,1] => [1,0,1,1,1,0,0,0,1,0]
 => {{1},{2,3,4},{5}}
 => 1 = 0 + 1
1010 => [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
 => {{1},{2,3},{4,5}}
 => 1 = 0 + 1
1011 => [1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
 => {{1},{2,3},{4},{5}}
 => 1 = 0 + 1
1100 => [1,1,3] => [1,0,1,0,1,1,1,0,0,0]
 => {{1},{2},{3,4,5}}
 => 1 = 0 + 1
1101 => [1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
 => {{1},{2},{3,4},{5}}
 => 1 = 0 + 1
1110 => [1,1,1,2] => [1,0,1,0,1,0,1,1,0,0]
 => {{1},{2},{3},{4,5}}
 => 1 = 0 + 1
1111 => [1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
 => {{1},{2},{3},{4},{5}}
 => 1 = 0 + 1
00000 => [6] => [1,1,1,1,1,1,0,0,0,0,0,0]
 => {{1,2,3,4,5,6}}
 => 6 = 5 + 1
00001 => [5,1] => [1,1,1,1,1,0,0,0,0,0,1,0]
 => {{1,2,3,4,5},{6}}
 => 5 = 4 + 1
00010 => [4,2] => [1,1,1,1,0,0,0,0,1,1,0,0]
 => {{1,2,3,4},{5,6}}
 => 4 = 3 + 1
00011 => [4,1,1] => [1,1,1,1,0,0,0,0,1,0,1,0]
 => {{1,2,3,4},{5},{6}}
 => 4 = 3 + 1
00100 => [3,3] => [1,1,1,0,0,0,1,1,1,0,0,0]
 => {{1,2,3},{4,5,6}}
 => 3 = 2 + 1
00101 => [3,2,1] => [1,1,1,0,0,0,1,1,0,0,1,0]
 => {{1,2,3},{4,5},{6}}
 => 3 = 2 + 1
00110 => [3,1,2] => [1,1,1,0,0,0,1,0,1,1,0,0]
 => {{1,2,3},{4},{5,6}}
 => 3 = 2 + 1
00111 => [3,1,1,1] => [1,1,1,0,0,0,1,0,1,0,1,0]
 => {{1,2,3},{4},{5},{6}}
 => 3 = 2 + 1
01000 => [2,4] => [1,1,0,0,1,1,1,1,0,0,0,0]
 => {{1,2},{3,4,5,6}}
 => 2 = 1 + 1
01001 => [2,3,1] => [1,1,0,0,1,1,1,0,0,0,1,0]
 => {{1,2},{3,4,5},{6}}
 => 2 = 1 + 1
01010 => [2,2,2] => [1,1,0,0,1,1,0,0,1,1,0,0]
 => {{1,2},{3,4},{5,6}}
 => 2 = 1 + 1
01011 => [2,2,1,1] => [1,1,0,0,1,1,0,0,1,0,1,0]
 => {{1,2},{3,4},{5},{6}}
 => 2 = 1 + 1
01100 => [2,1,3] => [1,1,0,0,1,0,1,1,1,0,0,0]
 => {{1,2},{3},{4,5,6}}
 => 2 = 1 + 1
01101 => [2,1,2,1] => [1,1,0,0,1,0,1,1,0,0,1,0]
 => {{1,2},{3},{4,5},{6}}
 => 2 = 1 + 1
01110 => [2,1,1,2] => [1,1,0,0,1,0,1,0,1,1,0,0]
 => {{1,2},{3},{4},{5,6}}
 => 2 = 1 + 1
01111 => [2,1,1,1,1] => [1,1,0,0,1,0,1,0,1,0,1,0]
 => {{1,2},{3},{4},{5},{6}}
 => 2 = 1 + 1
10000 => [1,5] => [1,0,1,1,1,1,1,0,0,0,0,0]
 => {{1},{2,3,4,5,6}}
 => 1 = 0 + 1
10001 => [1,4,1] => [1,0,1,1,1,1,0,0,0,0,1,0]
 => {{1},{2,3,4,5},{6}}
 => 1 = 0 + 1
10010 => [1,3,2] => [1,0,1,1,1,0,0,0,1,1,0,0]
 => {{1},{2,3,4},{5,6}}
 => 1 = 0 + 1
10011 => [1,3,1,1] => [1,0,1,1,1,0,0,0,1,0,1,0]
 => {{1},{2,3,4},{5},{6}}
 => 1 = 0 + 1
0000011 => [6,1,1] => [1,1,1,1,1,1,0,0,0,0,0,0,1,0,1,0]
 => {{1,2,3,4,5,6},{7},{8}}
 => ? = 5 + 1
0000100 => [5,3] => [1,1,1,1,1,0,0,0,0,0,1,1,1,0,0,0]
 => {{1,2,3,4,5},{6,7,8}}
 => ? = 4 + 1
0000101 => [5,2,1] => [1,1,1,1,1,0,0,0,0,0,1,1,0,0,1,0]
 => {{1,2,3,4,5},{6,7},{8}}
 => ? = 4 + 1
0000110 => [5,1,2] => [1,1,1,1,1,0,0,0,0,0,1,0,1,1,0,0]
 => {{1,2,3,4,5},{6},{7,8}}
 => ? = 4 + 1
0000111 => [5,1,1,1] => [1,1,1,1,1,0,0,0,0,0,1,0,1,0,1,0]
 => {{1,2,3,4,5},{6},{7},{8}}
 => ? = 4 + 1
0001000 => [4,4] => [1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0]
 => {{1,2,3,4},{5,6,7,8}}
 => ? = 3 + 1
0001001 => [4,3,1] => [1,1,1,1,0,0,0,0,1,1,1,0,0,0,1,0]
 => {{1,2,3,4},{5,6,7},{8}}
 => ? = 3 + 1
0001010 => [4,2,2] => [1,1,1,1,0,0,0,0,1,1,0,0,1,1,0,0]
 => {{1,2,3,4},{5,6},{7,8}}
 => ? = 3 + 1
0001011 => [4,2,1,1] => [1,1,1,1,0,0,0,0,1,1,0,0,1,0,1,0]
 => {{1,2,3,4},{5,6},{7},{8}}
 => ? = 3 + 1
0001100 => [4,1,3] => [1,1,1,1,0,0,0,0,1,0,1,1,1,0,0,0]
 => {{1,2,3,4},{5},{6,7,8}}
 => ? = 3 + 1
0001101 => [4,1,2,1] => [1,1,1,1,0,0,0,0,1,0,1,1,0,0,1,0]
 => {{1,2,3,4},{5},{6,7},{8}}
 => ? = 3 + 1
0001110 => [4,1,1,2] => [1,1,1,1,0,0,0,0,1,0,1,0,1,1,0,0]
 => {{1,2,3,4},{5},{6},{7,8}}
 => ? = 3 + 1
0001111 => [4,1,1,1,1] => [1,1,1,1,0,0,0,0,1,0,1,0,1,0,1,0]
 => {{1,2,3,4},{5},{6},{7},{8}}
 => ? = 3 + 1
0010000 => [3,5] => [1,1,1,0,0,0,1,1,1,1,1,0,0,0,0,0]
 => {{1,2,3},{4,5,6,7,8}}
 => ? = 2 + 1
0010001 => [3,4,1] => [1,1,1,0,0,0,1,1,1,1,0,0,0,0,1,0]
 => {{1,2,3},{4,5,6,7},{8}}
 => ? = 2 + 1
0010010 => [3,3,2] => [1,1,1,0,0,0,1,1,1,0,0,0,1,1,0,0]
 => {{1,2,3},{4,5,6},{7,8}}
 => ? = 2 + 1
0010011 => [3,3,1,1] => [1,1,1,0,0,0,1,1,1,0,0,0,1,0,1,0]
 => {{1,2,3},{4,5,6},{7},{8}}
 => ? = 2 + 1
0010100 => [3,2,3] => [1,1,1,0,0,0,1,1,0,0,1,1,1,0,0,0]
 => {{1,2,3},{4,5},{6,7,8}}
 => ? = 2 + 1
0010101 => [3,2,2,1] => [1,1,1,0,0,0,1,1,0,0,1,1,0,0,1,0]
 => {{1,2,3},{4,5},{6,7},{8}}
 => ? = 2 + 1
0010110 => [3,2,1,2] => [1,1,1,0,0,0,1,1,0,0,1,0,1,1,0,0]
 => {{1,2,3},{4,5},{6},{7,8}}
 => ? = 2 + 1
0010111 => [3,2,1,1,1] => [1,1,1,0,0,0,1,1,0,0,1,0,1,0,1,0]
 => {{1,2,3},{4,5},{6},{7},{8}}
 => ? = 2 + 1
0011000 => [3,1,4] => [1,1,1,0,0,0,1,0,1,1,1,1,0,0,0,0]
 => {{1,2,3},{4},{5,6,7,8}}
 => ? = 2 + 1
0011001 => [3,1,3,1] => [1,1,1,0,0,0,1,0,1,1,1,0,0,0,1,0]
 => {{1,2,3},{4},{5,6,7},{8}}
 => ? = 2 + 1
0011010 => [3,1,2,2] => [1,1,1,0,0,0,1,0,1,1,0,0,1,1,0,0]
 => {{1,2,3},{4},{5,6},{7,8}}
 => ? = 2 + 1
0011011 => [3,1,2,1,1] => [1,1,1,0,0,0,1,0,1,1,0,0,1,0,1,0]
 => {{1,2,3},{4},{5,6},{7},{8}}
 => ? = 2 + 1
0011100 => [3,1,1,3] => [1,1,1,0,0,0,1,0,1,0,1,1,1,0,0,0]
 => {{1,2,3},{4},{5},{6,7,8}}
 => ? = 2 + 1
0011101 => [3,1,1,2,1] => [1,1,1,0,0,0,1,0,1,0,1,1,0,0,1,0]
 => {{1,2,3},{4},{5},{6,7},{8}}
 => ? = 2 + 1
0011110 => [3,1,1,1,2] => [1,1,1,0,0,0,1,0,1,0,1,0,1,1,0,0]
 => {{1,2,3},{4},{5},{6},{7,8}}
 => ? = 2 + 1
0011111 => [3,1,1,1,1,1] => [1,1,1,0,0,0,1,0,1,0,1,0,1,0,1,0]
 => {{1,2,3},{4},{5},{6},{7},{8}}
 => ? = 2 + 1
0100001 => [2,5,1] => [1,1,0,0,1,1,1,1,1,0,0,0,0,0,1,0]
 => {{1,2},{3,4,5,6,7},{8}}
 => ? = 1 + 1
0100010 => [2,4,2] => [1,1,0,0,1,1,1,1,0,0,0,0,1,1,0,0]
 => {{1,2},{3,4,5,6},{7,8}}
 => ? = 1 + 1
0100011 => [2,4,1,1] => [1,1,0,0,1,1,1,1,0,0,0,0,1,0,1,0]
 => {{1,2},{3,4,5,6},{7},{8}}
 => ? = 1 + 1
0100100 => [2,3,3] => [1,1,0,0,1,1,1,0,0,0,1,1,1,0,0,0]
 => {{1,2},{3,4,5},{6,7,8}}
 => ? = 1 + 1
0100101 => [2,3,2,1] => [1,1,0,0,1,1,1,0,0,0,1,1,0,0,1,0]
 => {{1,2},{3,4,5},{6,7},{8}}
 => ? = 1 + 1
0100110 => [2,3,1,2] => [1,1,0,0,1,1,1,0,0,0,1,0,1,1,0,0]
 => {{1,2},{3,4,5},{6},{7,8}}
 => ? = 1 + 1
0100111 => [2,3,1,1,1] => [1,1,0,0,1,1,1,0,0,0,1,0,1,0,1,0]
 => {{1,2},{3,4,5},{6},{7},{8}}
 => ? = 1 + 1
0101000 => [2,2,4] => [1,1,0,0,1,1,0,0,1,1,1,1,0,0,0,0]
 => {{1,2},{3,4},{5,6,7,8}}
 => ? = 1 + 1
0101001 => [2,2,3,1] => [1,1,0,0,1,1,0,0,1,1,1,0,0,0,1,0]
 => {{1,2},{3,4},{5,6,7},{8}}
 => ? = 1 + 1
0101011 => [2,2,2,1,1] => [1,1,0,0,1,1,0,0,1,1,0,0,1,0,1,0]
 => {{1,2},{3,4},{5,6},{7},{8}}
 => ? = 1 + 1
0101100 => [2,2,1,3] => [1,1,0,0,1,1,0,0,1,0,1,1,1,0,0,0]
 => {{1,2},{3,4},{5},{6,7,8}}
 => ? = 1 + 1
0101101 => [2,2,1,2,1] => [1,1,0,0,1,1,0,0,1,0,1,1,0,0,1,0]
 => {{1,2},{3,4},{5},{6,7},{8}}
 => ? = 1 + 1
0101110 => [2,2,1,1,2] => [1,1,0,0,1,1,0,0,1,0,1,0,1,1,0,0]
 => {{1,2},{3,4},{5},{6},{7,8}}
 => ? = 1 + 1
0101111 => [2,2,1,1,1,1] => [1,1,0,0,1,1,0,0,1,0,1,0,1,0,1,0]
 => {{1,2},{3,4},{5},{6},{7},{8}}
 => ? = 1 + 1
0110000 => [2,1,5] => [1,1,0,0,1,0,1,1,1,1,1,0,0,0,0,0]
 => {{1,2},{3},{4,5,6,7,8}}
 => ? = 1 + 1
0110001 => [2,1,4,1] => [1,1,0,0,1,0,1,1,1,1,0,0,0,0,1,0]
 => {{1,2},{3},{4,5,6,7},{8}}
 => ? = 1 + 1
0110010 => [2,1,3,2] => [1,1,0,0,1,0,1,1,1,0,0,0,1,1,0,0]
 => {{1,2},{3},{4,5,6},{7,8}}
 => ? = 1 + 1
0110011 => [2,1,3,1,1] => [1,1,0,0,1,0,1,1,1,0,0,0,1,0,1,0]
 => {{1,2},{3},{4,5,6},{7},{8}}
 => ? = 1 + 1
0110100 => [2,1,2,3] => [1,1,0,0,1,0,1,1,0,0,1,1,1,0,0,0]
 => {{1,2},{3},{4,5},{6,7,8}}
 => ? = 1 + 1
0110101 => [2,1,2,2,1] => [1,1,0,0,1,0,1,1,0,0,1,1,0,0,1,0]
 => {{1,2},{3},{4,5},{6,7},{8}}
 => ? = 1 + 1
0110110 => [2,1,2,1,2] => [1,1,0,0,1,0,1,1,0,0,1,0,1,1,0,0]
 => {{1,2},{3},{4,5},{6},{7,8}}
 => ? = 1 + 1
Description
The biggest entry in the block containing the 1.
The following 42 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000971The smallest closer of a set partition. St000504The cardinality of the first block of a set partition. St000823The number of unsplittable factors of the set partition. St000025The number of initial rises of a Dyck path. 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. St000273The domination number of a graph. St000363The number of minimal vertex covers of a graph. St000544The cop number of a graph. St000916The packing number of a graph. St001829The common independence number of a graph. St001498The normalised height of a Nakayama algebra with magnitude 1. 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. St000541The number of indices greater than or equal to 2 of a permutation such that all smaller indices appear to its right. St000501The size of the first part in the decomposition of a permutation. St000542The number of left-to-right-minima of a permutation. St000990The first ascent of a permutation. St000066The column of the unique '1' in the first row of the alternating sign matrix. St000738The first entry in the last row of a standard tableau. St000989The number of final rises of a permutation. St001640The number of ascent tops in the permutation such that all smaller elements appear before. St000260The radius of a connected graph. St000740The last entry of a permutation. St000051The size of the left subtree of a binary tree. St000261The edge connectivity of a graph. St000262The vertex connectivity of a graph. St000310The minimal degree of a vertex of a graph. St000061The number of nodes on the left branch of a binary tree. 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. St000991The number of right-to-left minima of a permutation. 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. St001390The number of bumps occurring when Schensted-inserting the letter 1 of a permutation. St001226The number of integers i such that the radical of the i-th indecomposable projective module has vanishing first extension group with the Jacobson radical J in the corresponding Nakayama algebra. St000338The number of pixed points of a permutation. St000193The row of the unique '1' in the first column of the alternating sign matrix. St000654The first descent of a permutation.