Your data matches 41 different statistics following compositions of up to 3 maps.
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Matching statistic: St000383
(load all 9 compositions to match this statistic)
Mapping
Mp00178: Binary words —to composition⟶ Integer compositions
St000383: Integer compositions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
0 => [2] => 2
1 => [1,1] => 1
00 => [3] => 3
01 => [2,1] => 1
10 => [1,2] => 2
11 => [1,1,1] => 1
000 => [4] => 4
001 => [3,1] => 1
010 => [2,2] => 2
011 => [2,1,1] => 1
100 => [1,3] => 3
101 => [1,2,1] => 1
110 => [1,1,2] => 2
111 => [1,1,1,1] => 1
0000 => [5] => 5
0001 => [4,1] => 1
0010 => [3,2] => 2
0011 => [3,1,1] => 1
0100 => [2,3] => 3
0101 => [2,2,1] => 1
0110 => [2,1,2] => 2
0111 => [2,1,1,1] => 1
1000 => [1,4] => 4
1001 => [1,3,1] => 1
1010 => [1,2,2] => 2
1011 => [1,2,1,1] => 1
1100 => [1,1,3] => 3
1101 => [1,1,2,1] => 1
1110 => [1,1,1,2] => 2
1111 => [1,1,1,1,1] => 1
00000 => [6] => 6
00001 => [5,1] => 1
00010 => [4,2] => 2
00011 => [4,1,1] => 1
00100 => [3,3] => 3
00101 => [3,2,1] => 1
00110 => [3,1,2] => 2
00111 => [3,1,1,1] => 1
01000 => [2,4] => 4
01001 => [2,3,1] => 1
01010 => [2,2,2] => 2
01011 => [2,2,1,1] => 1
01100 => [2,1,3] => 3
01101 => [2,1,2,1] => 1
01110 => [2,1,1,2] => 2
01111 => [2,1,1,1,1] => 1
10000 => [1,5] => 5
10001 => [1,4,1] => 1
10010 => [1,3,2] => 2
10011 => [1,3,1,1] => 1
Description
The last part of an integer composition.
Matching statistic: St000382
(load all 6 compositions to match this statistic)
Mapping
Mp00104: Binary words —reverse⟶ Binary words
Mp00178: Binary words —to composition⟶ Integer compositions
St000382: Integer compositions ⟶ ℤResult quality: 98% ●values known / values provided: 98%●distinct values known / distinct values provided: 100%
Values
0 => 0 => [2] => 2
1 => 1 => [1,1] => 1
00 => 00 => [3] => 3
01 => 10 => [1,2] => 1
10 => 01 => [2,1] => 2
11 => 11 => [1,1,1] => 1
000 => 000 => [4] => 4
001 => 100 => [1,3] => 1
010 => 010 => [2,2] => 2
011 => 110 => [1,1,2] => 1
100 => 001 => [3,1] => 3
101 => 101 => [1,2,1] => 1
110 => 011 => [2,1,1] => 2
111 => 111 => [1,1,1,1] => 1
0000 => 0000 => [5] => 5
0001 => 1000 => [1,4] => 1
0010 => 0100 => [2,3] => 2
0011 => 1100 => [1,1,3] => 1
0100 => 0010 => [3,2] => 3
0101 => 1010 => [1,2,2] => 1
0110 => 0110 => [2,1,2] => 2
0111 => 1110 => [1,1,1,2] => 1
1000 => 0001 => [4,1] => 4
1001 => 1001 => [1,3,1] => 1
1010 => 0101 => [2,2,1] => 2
1011 => 1101 => [1,1,2,1] => 1
1100 => 0011 => [3,1,1] => 3
1101 => 1011 => [1,2,1,1] => 1
1110 => 0111 => [2,1,1,1] => 2
1111 => 1111 => [1,1,1,1,1] => 1
00000 => 00000 => [6] => 6
00001 => 10000 => [1,5] => 1
00010 => 01000 => [2,4] => 2
00011 => 11000 => [1,1,4] => 1
00100 => 00100 => [3,3] => 3
00101 => 10100 => [1,2,3] => 1
00110 => 01100 => [2,1,3] => 2
00111 => 11100 => [1,1,1,3] => 1
01000 => 00010 => [4,2] => 4
01001 => 10010 => [1,3,2] => 1
01010 => 01010 => [2,2,2] => 2
01011 => 11010 => [1,1,2,2] => 1
01100 => 00110 => [3,1,2] => 3
01101 => 10110 => [1,2,1,2] => 1
01110 => 01110 => [2,1,1,2] => 2
01111 => 11110 => [1,1,1,1,2] => 1
10000 => 00001 => [5,1] => 5
10001 => 10001 => [1,4,1] => 1
10010 => 01001 => [2,3,1] => 2
10011 => 11001 => [1,1,3,1] => 1
1000001001 => 1001000001 => ? => ? = 1
00101111110 => 01111110100 => ? => ? = 2
00111011110 => 01111011100 => ? => ? = 2
00111110110 => 01101111100 => ? => ? = 2
01101010100 => 00101010110 => ? => ? = 3
01101001010 => ? => ? => ? = 2
01010101111 => 11110101010 => ? => ? = 1
11011110100 => 00101111011 => ? => ? = 3
01010110100 => 00101101010 => ? => ? = 3
Description
The first part of an integer composition.
Matching statistic: St000326
(load all 5 compositions to match this statistic)
Mapping
Mp00104: Binary words —reverse⟶ Binary words
St000326: Binary words ⟶ ℤResult quality: 86% ●values known / values provided: 86%●distinct values known / distinct values provided: 100%
Values
0 => 0 => 2
1 => 1 => 1
00 => 00 => 3
01 => 10 => 1
10 => 01 => 2
11 => 11 => 1
000 => 000 => 4
001 => 100 => 1
010 => 010 => 2
011 => 110 => 1
100 => 001 => 3
101 => 101 => 1
110 => 011 => 2
111 => 111 => 1
0000 => 0000 => 5
0001 => 1000 => 1
0010 => 0100 => 2
0011 => 1100 => 1
0100 => 0010 => 3
0101 => 1010 => 1
0110 => 0110 => 2
0111 => 1110 => 1
1000 => 0001 => 4
1001 => 1001 => 1
1010 => 0101 => 2
1011 => 1101 => 1
1100 => 0011 => 3
1101 => 1011 => 1
1110 => 0111 => 2
1111 => 1111 => 1
00000 => 00000 => 6
00001 => 10000 => 1
00010 => 01000 => 2
00011 => 11000 => 1
00100 => 00100 => 3
00101 => 10100 => 1
00110 => 01100 => 2
00111 => 11100 => 1
01000 => 00010 => 4
01001 => 10010 => 1
01010 => 01010 => 2
01011 => 11010 => 1
01100 => 00110 => 3
01101 => 10110 => 1
01110 => 01110 => 2
01111 => 11110 => 1
10000 => 00001 => 5
10001 => 10001 => 1
10010 => 01001 => 2
10011 => 11001 => 1
1100000001 => 1000000011 => ? = 1
1010000001 => 1000000101 => ? = 1
1000001001 => 1001000001 => ? = 1
1000000101 => 1010000001 => ? = 1
1000000011 => 1100000001 => ? = 1
11011111110 => 01111111011 => ? = 2
11110111110 => 01111101111 => ? = 2
11111101110 => 01110111111 => ? = 2
11010101110 => 01110101011 => ? = 2
=> => ? = 1
11010100100 => 00100101011 => ? = 3
11001010100 => 00101010011 => ? = 3
00101010011 => 11001010100 => ? = 1
00100101011 => 11010100100 => ? = 1
01111111110 => 01111111110 => ? = 2
01011111111 => 11111111010 => ? = 1
11111111111 => 11111111111 => ? = 1
01010111011 => 11011101010 => ? = 1
01011101110 => 01110111010 => ? = 2
01011101011 => 11010111010 => ? = 1
01101110110 => 01101110110 => ? = 2
01101011011 => 11011010110 => ? = 1
01110111010 => 01011101110 => ? = 2
01110101110 => 01110101110 => ? = 2
01110101011 => 11010101110 => ? = 1
01111101111 => 11110111110 => ? = 1
11011101010 => 01010111011 => ? = 2
11011111011 => 11011111011 => ? = 1
11011010110 => 01101011011 => ? = 2
11010111010 => 01011101011 => ? = 2
11010101011 => 11010101011 => ? = 1
11011101111 => 11110111011 => ? = 1
11110111011 => 11011101111 => ? = 1
11110101111 => 11110101111 => ? = 1
00101111110 => 01111110100 => ? = 2
00111011110 => 01111011100 => ? = 2
00111110110 => 01101111100 => ? = 2
00111111100 => 00111111100 => ? = 3
01101010100 => 00101010110 => ? = 3
01101001010 => ? => ? = 2
01010101111 => 11110101010 => ? = 1
11011110100 => 00101111011 => ? = 3
01010110100 => 00101101010 => ? = 3
01110111111 => 11111101110 => ? = 1
11110001000 => 00010001111 => ? = 4
11000111000 => 00011100011 => ? = 4
11000100011 => 11000100011 => ? = 1
00011111000 => 00011111000 => ? = 4
00011100011 => 11000111000 => ? = 1
00010001111 => 11110001000 => ? = 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: St000297
(load all 7 compositions to match this statistic)
Mapping
Mp00104: Binary words —reverse⟶ Binary words
Mp00105: Binary words —complement⟶ Binary words
St000297: Binary words ⟶ ℤResult quality: 86% ●values known / values provided: 86%●distinct values known / distinct values provided: 100%
Values
0 => 0 => 1 => 1 = 2 - 1
1 => 1 => 0 => 0 = 1 - 1
00 => 00 => 11 => 2 = 3 - 1
01 => 10 => 01 => 0 = 1 - 1
10 => 01 => 10 => 1 = 2 - 1
11 => 11 => 00 => 0 = 1 - 1
000 => 000 => 111 => 3 = 4 - 1
001 => 100 => 011 => 0 = 1 - 1
010 => 010 => 101 => 1 = 2 - 1
011 => 110 => 001 => 0 = 1 - 1
100 => 001 => 110 => 2 = 3 - 1
101 => 101 => 010 => 0 = 1 - 1
110 => 011 => 100 => 1 = 2 - 1
111 => 111 => 000 => 0 = 1 - 1
0000 => 0000 => 1111 => 4 = 5 - 1
0001 => 1000 => 0111 => 0 = 1 - 1
0010 => 0100 => 1011 => 1 = 2 - 1
0011 => 1100 => 0011 => 0 = 1 - 1
0100 => 0010 => 1101 => 2 = 3 - 1
0101 => 1010 => 0101 => 0 = 1 - 1
0110 => 0110 => 1001 => 1 = 2 - 1
0111 => 1110 => 0001 => 0 = 1 - 1
1000 => 0001 => 1110 => 3 = 4 - 1
1001 => 1001 => 0110 => 0 = 1 - 1
1010 => 0101 => 1010 => 1 = 2 - 1
1011 => 1101 => 0010 => 0 = 1 - 1
1100 => 0011 => 1100 => 2 = 3 - 1
1101 => 1011 => 0100 => 0 = 1 - 1
1110 => 0111 => 1000 => 1 = 2 - 1
1111 => 1111 => 0000 => 0 = 1 - 1
00000 => 00000 => 11111 => 5 = 6 - 1
00001 => 10000 => 01111 => 0 = 1 - 1
00010 => 01000 => 10111 => 1 = 2 - 1
00011 => 11000 => 00111 => 0 = 1 - 1
00100 => 00100 => 11011 => 2 = 3 - 1
00101 => 10100 => 01011 => 0 = 1 - 1
00110 => 01100 => 10011 => 1 = 2 - 1
00111 => 11100 => 00011 => 0 = 1 - 1
01000 => 00010 => 11101 => 3 = 4 - 1
01001 => 10010 => 01101 => 0 = 1 - 1
01010 => 01010 => 10101 => 1 = 2 - 1
01011 => 11010 => 00101 => 0 = 1 - 1
01100 => 00110 => 11001 => 2 = 3 - 1
01101 => 10110 => 01001 => 0 = 1 - 1
01110 => 01110 => 10001 => 1 = 2 - 1
01111 => 11110 => 00001 => 0 = 1 - 1
10000 => 00001 => 11110 => 4 = 5 - 1
10001 => 10001 => 01110 => 0 = 1 - 1
10010 => 01001 => 10110 => 1 = 2 - 1
10011 => 11001 => 00110 => 0 = 1 - 1
1100000001 => 1000000011 => 0111111100 => ? = 1 - 1
1010000001 => 1000000101 => 0111111010 => ? = 1 - 1
1000001001 => 1001000001 => 0110111110 => ? = 1 - 1
1000000101 => 1010000001 => 0101111110 => ? = 1 - 1
1000000011 => 1100000001 => 0011111110 => ? = 1 - 1
11111101110 => 01110111111 => 10001000000 => ? = 2 - 1
11010101110 => 01110101011 => ? => ? = 2 - 1
11111111010 => 01011111111 => 10100000000 => ? = 2 - 1
=> => => ? = 1 - 1
11010100100 => 00100101011 => 11011010100 => ? = 3 - 1
11001010100 => 00101010011 => 11010101100 => ? = 3 - 1
00101010011 => 11001010100 => 00110101011 => ? = 1 - 1
00100101011 => 11010100100 => 00101011011 => ? = 1 - 1
01111111110 => 01111111110 => 10000000001 => ? = 2 - 1
01010001000 => 00010001010 => 11101110101 => ? = 4 - 1
00100100100 => 00100100100 => ? => ? = 3 - 1
00101001000 => 00010010100 => 11101101011 => ? = 4 - 1
00010001010 => 01010001000 => 10101110111 => ? = 2 - 1
00010010100 => 00101001000 => 11010110111 => ? = 3 - 1
00010101000 => 00010101000 => 11101010111 => ? = 4 - 1
00000100000 => 00000100000 => 11111011111 => ? = 6 - 1
01111111011 => 11011111110 => 00100000001 => ? = 1 - 1
11111111111 => 11111111111 => 00000000000 => ? = 1 - 1
01010111011 => 11011101010 => ? => ? = 1 - 1
01011101110 => 01110111010 => ? => ? = 2 - 1
01011101011 => 11010111010 => ? => ? = 1 - 1
01101110110 => 01101110110 => ? => ? = 2 - 1
01101011011 => 11011010110 => ? => ? = 1 - 1
01110111010 => 01011101110 => ? => ? = 2 - 1
01110101110 => 01110101110 => ? => ? = 2 - 1
01110101011 => 11010101110 => ? => ? = 1 - 1
11011101010 => 01010111011 => ? => ? = 2 - 1
11011111011 => 11011111011 => ? => ? = 1 - 1
11011010110 => 01101011011 => ? => ? = 2 - 1
11010111010 => 01011101011 => ? => ? = 2 - 1
11010101011 => 11010101011 => ? => ? = 1 - 1
00101111110 => 01111110100 => 10000001011 => ? = 2 - 1
00111011110 => 01111011100 => 10000100011 => ? = 2 - 1
00111110110 => 01101111100 => 10010000011 => ? = 2 - 1
00111111100 => 00111111100 => 11000000011 => ? = 3 - 1
01101010100 => 00101010110 => ? => ? = 3 - 1
01101001010 => ? => ? => ? = 2 - 1
01010101111 => 11110101010 => 00001010101 => ? = 1 - 1
11011110100 => 00101111011 => ? => ? = 3 - 1
01010110100 => 00101101010 => ? => ? = 3 - 1
01110111111 => 11111101110 => 00000010001 => ? = 1 - 1
11110001000 => 00010001111 => 11101110000 => ? = 4 - 1
11000111000 => 00011100011 => ? => ? = 4 - 1
11000100011 => 11000100011 => ? => ? = 1 - 1
00011111000 => 00011111000 => ? => ? = 4 - 1
Description
The number of leading ones in a binary word.
Matching statistic: St000439
(load all 2 compositions to match this statistic)
Mapping
Mp00104: Binary words —reverse⟶ Binary words
Mp00178: Binary words —to composition⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St000439: Dyck paths ⟶ ℤResult quality: 62% ●values known / values provided: 62%●distinct values known / distinct values provided: 100%
Values
0 => 0 => [2] => [1,1,0,0]
 => 3 = 2 + 1
1 => 1 => [1,1] => [1,0,1,0]
 => 2 = 1 + 1
00 => 00 => [3] => [1,1,1,0,0,0]
 => 4 = 3 + 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 => 11 => [1,1,1] => [1,0,1,0,1,0]
 => 2 = 1 + 1
000 => 000 => [4] => [1,1,1,1,0,0,0,0]
 => 5 = 4 + 1
001 => 100 => [1,3] => [1,0,1,1,1,0,0,0]
 => 2 = 1 + 1
010 => 010 => [2,2] => [1,1,0,0,1,1,0,0]
 => 3 = 2 + 1
011 => 110 => [1,1,2] => [1,0,1,0,1,1,0,0]
 => 2 = 1 + 1
100 => 001 => [3,1] => [1,1,1,0,0,0,1,0]
 => 4 = 3 + 1
101 => 101 => [1,2,1] => [1,0,1,1,0,0,1,0]
 => 2 = 1 + 1
110 => 011 => [2,1,1] => [1,1,0,0,1,0,1,0]
 => 3 = 2 + 1
111 => 111 => [1,1,1,1] => [1,0,1,0,1,0,1,0]
 => 2 = 1 + 1
0000 => 0000 => [5] => [1,1,1,1,1,0,0,0,0,0]
 => 6 = 5 + 1
0001 => 1000 => [1,4] => [1,0,1,1,1,1,0,0,0,0]
 => 2 = 1 + 1
0010 => 0100 => [2,3] => [1,1,0,0,1,1,1,0,0,0]
 => 3 = 2 + 1
0011 => 1100 => [1,1,3] => [1,0,1,0,1,1,1,0,0,0]
 => 2 = 1 + 1
0100 => 0010 => [3,2] => [1,1,1,0,0,0,1,1,0,0]
 => 4 = 3 + 1
0101 => 1010 => [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
 => 2 = 1 + 1
0110 => 0110 => [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
 => 3 = 2 + 1
0111 => 1110 => [1,1,1,2] => [1,0,1,0,1,0,1,1,0,0]
 => 2 = 1 + 1
1000 => 0001 => [4,1] => [1,1,1,1,0,0,0,0,1,0]
 => 5 = 4 + 1
1001 => 1001 => [1,3,1] => [1,0,1,1,1,0,0,0,1,0]
 => 2 = 1 + 1
1010 => 0101 => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
 => 3 = 2 + 1
1011 => 1101 => [1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
 => 2 = 1 + 1
1100 => 0011 => [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
 => 4 = 3 + 1
1101 => 1011 => [1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
 => 2 = 1 + 1
1110 => 0111 => [2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
 => 3 = 2 + 1
1111 => 1111 => [1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
 => 2 = 1 + 1
00000 => 00000 => [6] => [1,1,1,1,1,1,0,0,0,0,0,0]
 => 7 = 6 + 1
00001 => 10000 => [1,5] => [1,0,1,1,1,1,1,0,0,0,0,0]
 => 2 = 1 + 1
00010 => 01000 => [2,4] => [1,1,0,0,1,1,1,1,0,0,0,0]
 => 3 = 2 + 1
00011 => 11000 => [1,1,4] => [1,0,1,0,1,1,1,1,0,0,0,0]
 => 2 = 1 + 1
00100 => 00100 => [3,3] => [1,1,1,0,0,0,1,1,1,0,0,0]
 => 4 = 3 + 1
00101 => 10100 => [1,2,3] => [1,0,1,1,0,0,1,1,1,0,0,0]
 => 2 = 1 + 1
00110 => 01100 => [2,1,3] => [1,1,0,0,1,0,1,1,1,0,0,0]
 => 3 = 2 + 1
00111 => 11100 => [1,1,1,3] => [1,0,1,0,1,0,1,1,1,0,0,0]
 => 2 = 1 + 1
01000 => 00010 => [4,2] => [1,1,1,1,0,0,0,0,1,1,0,0]
 => 5 = 4 + 1
01001 => 10010 => [1,3,2] => [1,0,1,1,1,0,0,0,1,1,0,0]
 => 2 = 1 + 1
01010 => 01010 => [2,2,2] => [1,1,0,0,1,1,0,0,1,1,0,0]
 => 3 = 2 + 1
01011 => 11010 => [1,1,2,2] => [1,0,1,0,1,1,0,0,1,1,0,0]
 => 2 = 1 + 1
01100 => 00110 => [3,1,2] => [1,1,1,0,0,0,1,0,1,1,0,0]
 => 4 = 3 + 1
01101 => 10110 => [1,2,1,2] => [1,0,1,1,0,0,1,0,1,1,0,0]
 => 2 = 1 + 1
01110 => 01110 => [2,1,1,2] => [1,1,0,0,1,0,1,0,1,1,0,0]
 => 3 = 2 + 1
01111 => 11110 => [1,1,1,1,2] => [1,0,1,0,1,0,1,0,1,1,0,0]
 => 2 = 1 + 1
10000 => 00001 => [5,1] => [1,1,1,1,1,0,0,0,0,0,1,0]
 => 6 = 5 + 1
10001 => 10001 => [1,4,1] => [1,0,1,1,1,1,0,0,0,0,1,0]
 => 2 = 1 + 1
10010 => 01001 => [2,3,1] => [1,1,0,0,1,1,1,0,0,0,1,0]
 => 3 = 2 + 1
10011 => 11001 => [1,1,3,1] => [1,0,1,0,1,1,1,0,0,0,1,0]
 => 2 = 1 + 1
0010100 => 0010100 => [3,2,3] => [1,1,1,0,0,0,1,1,0,0,1,1,1,0,0,0]
 => ? = 3 + 1
0011100 => 0011100 => [3,1,1,3] => [1,1,1,0,0,0,1,0,1,0,1,1,1,0,0,0]
 => ? = 3 + 1
0100100 => 0010010 => [3,3,2] => [1,1,1,0,0,0,1,1,1,0,0,0,1,1,0,0]
 => ? = 3 + 1
0101100 => 0011010 => [3,1,2,2] => [1,1,1,0,0,0,1,0,1,1,0,0,1,1,0,0]
 => ? = 3 + 1
0110100 => 0010110 => [3,2,1,2] => [1,1,1,0,0,0,1,1,0,0,1,0,1,1,0,0]
 => ? = 3 + 1
1001100 => 0011001 => [3,1,3,1] => [1,1,1,0,0,0,1,0,1,1,1,0,0,0,1,0]
 => ? = 3 + 1
1010100 => 0010101 => [3,2,2,1] => [1,1,1,0,0,0,1,1,0,0,1,1,0,0,1,0]
 => ? = 3 + 1
1011000 => 0001101 => [4,1,2,1] => [1,1,1,1,0,0,0,0,1,0,1,1,0,0,1,0]
 => ? = 4 + 1
1100100 => 0010011 => [3,3,1,1] => [1,1,1,0,0,0,1,1,1,0,0,0,1,0,1,0]
 => ? = 3 + 1
00110001 => 10001100 => [1,4,1,3] => [1,0,1,1,1,1,0,0,0,0,1,0,1,1,1,0,0,0]
 => ? = 1 + 1
01010001 => 10001010 => [1,4,2,2] => [1,0,1,1,1,1,0,0,0,0,1,1,0,0,1,1,0,0]
 => ? = 1 + 1
01100001 => 10000110 => [1,5,1,2] => [1,0,1,1,1,1,1,0,0,0,0,0,1,0,1,1,0,0]
 => ? = 1 + 1
10000101 => 10100001 => [1,2,5,1] => [1,0,1,1,0,0,1,1,1,1,1,0,0,0,0,0,1,0]
 => ? = 1 + 1
10000110 => 01100001 => [2,1,5,1] => [1,1,0,0,1,0,1,1,1,1,1,0,0,0,0,0,1,0]
 => ? = 2 + 1
10001001 => 10010001 => [1,3,4,1] => [1,0,1,1,1,0,0,0,1,1,1,1,0,0,0,0,1,0]
 => ? = 1 + 1
10001010 => 01010001 => [2,2,4,1] => [1,1,0,0,1,1,0,0,1,1,1,1,0,0,0,0,1,0]
 => ? = 2 + 1
10001100 => 00110001 => [3,1,4,1] => [1,1,1,0,0,0,1,0,1,1,1,1,0,0,0,0,1,0]
 => ? = 3 + 1
10001111 => 11110001 => [1,1,1,1,4,1] => [1,0,1,0,1,0,1,0,1,1,1,1,0,0,0,0,1,0]
 => ? = 1 + 1
10010001 => 10001001 => [1,4,3,1] => [1,0,1,1,1,1,0,0,0,0,1,1,1,0,0,0,1,0]
 => ? = 1 + 1
10010111 => 11101001 => [1,1,1,2,3,1] => [1,0,1,0,1,0,1,1,0,0,1,1,1,0,0,0,1,0]
 => ? = 1 + 1
10011011 => 11011001 => [1,1,2,1,3,1] => [1,0,1,0,1,1,0,0,1,0,1,1,1,0,0,0,1,0]
 => ? = 1 + 1
10011101 => 10111001 => [1,2,1,1,3,1] => [1,0,1,1,0,0,1,0,1,0,1,1,1,0,0,0,1,0]
 => ? = 1 + 1
10100001 => 10000101 => [1,5,2,1] => [1,0,1,1,1,1,1,0,0,0,0,0,1,1,0,0,1,0]
 => ? = 1 + 1
10101101 => 10110101 => [1,2,1,2,2,1] => [1,0,1,1,0,0,1,0,1,1,0,0,1,1,0,0,1,0]
 => ? = 1 + 1
10110101 => 10101101 => [1,2,2,1,2,1] => [1,0,1,1,0,0,1,1,0,0,1,0,1,1,0,0,1,0]
 => ? = 1 + 1
10111001 => 10011101 => [1,3,1,1,2,1] => [1,0,1,1,1,0,0,0,1,0,1,0,1,1,0,0,1,0]
 => ? = 1 + 1
11011001 => 10011011 => [1,3,1,2,1,1] => [1,0,1,1,1,0,0,0,1,0,1,1,0,0,1,0,1,0]
 => ? = 1 + 1
11101001 => 10010111 => [1,3,2,1,1,1] => [1,0,1,1,1,0,0,0,1,1,0,0,1,0,1,0,1,0]
 => ? = 1 + 1
11110001 => 10001111 => [1,4,1,1,1,1] => [1,0,1,1,1,1,0,0,0,0,1,0,1,0,1,0,1,0]
 => ? = 1 + 1
000010000 => 000010000 => [5,5] => [1,1,1,1,1,0,0,0,0,0,1,1,1,1,1,0,0,0,0,0]
 => ? = 5 + 1
000100011 => 110001000 => [1,1,4,4] => [1,0,1,0,1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0]
 => ? = 1 + 1
000100110 => 011001000 => [2,1,3,4] => [1,1,0,0,1,0,1,1,1,0,0,0,1,1,1,1,0,0,0,0]
 => ? = 2 + 1
000101100 => 001101000 => [3,1,2,4] => [1,1,1,0,0,0,1,0,1,1,0,0,1,1,1,1,0,0,0,0]
 => ? = 3 + 1
000111000 => 000111000 => [4,1,1,4] => [1,1,1,1,0,0,0,0,1,0,1,0,1,1,1,1,0,0,0,0]
 => ? = 4 + 1
001001010 => 010100100 => [2,2,3,3] => [1,1,0,0,1,1,0,0,1,1,1,0,0,0,1,1,1,0,0,0]
 => ? = 2 + 1
001001111 => 111100100 => [1,1,1,1,3,3] => [1,0,1,0,1,0,1,0,1,1,1,0,0,0,1,1,1,0,0,0]
 => ? = 1 + 1
001010100 => 001010100 => [3,2,2,3] => [1,1,1,0,0,0,1,1,0,0,1,1,0,0,1,1,1,0,0,0]
 => ? = 3 + 1
001011011 => 110110100 => [1,1,2,1,2,3] => [1,0,1,0,1,1,0,0,1,0,1,1,0,0,1,1,1,0,0,0]
 => ? = 1 + 1
001011110 => 011110100 => [2,1,1,1,2,3] => [1,1,0,0,1,0,1,0,1,0,1,1,0,0,1,1,1,0,0,0]
 => ? = 2 + 1
001101000 => 000101100 => [4,2,1,3] => [1,1,1,1,0,0,0,0,1,1,0,0,1,0,1,1,1,0,0,0]
 => ? = 4 + 1
001110011 => 110011100 => [1,1,3,1,1,3] => [1,0,1,0,1,1,1,0,0,0,1,0,1,0,1,1,1,0,0,0]
 => ? = 1 + 1
001110110 => 011011100 => [2,1,2,1,1,3] => [1,1,0,0,1,0,1,1,0,0,1,0,1,0,1,1,1,0,0,0]
 => ? = 2 + 1
001111100 => 001111100 => [3,1,1,1,1,3] => [1,1,1,0,0,0,1,0,1,0,1,0,1,0,1,1,1,0,0,0]
 => ? = 3 + 1
010100100 => 001001010 => [3,3,2,2] => [1,1,1,0,0,0,1,1,1,0,0,0,1,1,0,0,1,1,0,0]
 => ? = 3 + 1
010101011 => 110101010 => [1,1,2,2,2,2] => [1,0,1,0,1,1,0,0,1,1,0,0,1,1,0,0,1,1,0,0]
 => ? = 1 + 1
010101110 => 011101010 => [2,1,1,2,2,2] => [1,1,0,0,1,0,1,0,1,1,0,0,1,1,0,0,1,1,0,0]
 => ? = 2 + 1
010111010 => 010111010 => [2,2,1,1,2,2] => [1,1,0,0,1,1,0,0,1,0,1,0,1,1,0,0,1,1,0,0]
 => ? = 2 + 1
010111111 => 111111010 => [1,1,1,1,1,1,2,2] => [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0,1,1,0,0]
 => ? = 1 + 1
011000001 => 100000110 => [1,6,1,2] => [1,0,1,1,1,1,1,1,0,0,0,0,0,0,1,0,1,1,0,0]
 => ? = 1 + 1
011001000 => 000100110 => [4,3,1,2] => [1,1,1,1,0,0,0,0,1,1,1,0,0,0,1,0,1,1,0,0]
 => ? = 4 + 1
Description
The position of the first down step of a Dyck path.
Matching statistic: St000011
(load all 2 compositions to match this statistic)
Mapping
Mp00178: Binary words —to composition⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
Mp00032: Dyck paths —inverse zeta map⟶ Dyck paths
St000011: Dyck paths ⟶ ℤResult quality: 57% ●values known / values provided: 57%●distinct values known / distinct values provided: 100%
Values
0 => [2] => [1,1,0,0]
 => [1,0,1,0]
 => 2
1 => [1,1] => [1,0,1,0]
 => [1,1,0,0]
 => 1
00 => [3] => [1,1,1,0,0,0]
 => [1,0,1,0,1,0]
 => 3
01 => [2,1] => [1,1,0,0,1,0]
 => [1,1,0,1,0,0]
 => 1
10 => [1,2] => [1,0,1,1,0,0]
 => [1,0,1,1,0,0]
 => 2
11 => [1,1,1] => [1,0,1,0,1,0]
 => [1,1,1,0,0,0]
 => 1
000 => [4] => [1,1,1,1,0,0,0,0]
 => [1,0,1,0,1,0,1,0]
 => 4
001 => [3,1] => [1,1,1,0,0,0,1,0]
 => [1,1,0,1,0,1,0,0]
 => 1
010 => [2,2] => [1,1,0,0,1,1,0,0]
 => [1,0,1,1,0,1,0,0]
 => 2
011 => [2,1,1] => [1,1,0,0,1,0,1,0]
 => [1,1,1,0,1,0,0,0]
 => 1
100 => [1,3] => [1,0,1,1,1,0,0,0]
 => [1,0,1,0,1,1,0,0]
 => 3
101 => [1,2,1] => [1,0,1,1,0,0,1,0]
 => [1,1,0,1,1,0,0,0]
 => 1
110 => [1,1,2] => [1,0,1,0,1,1,0,0]
 => [1,0,1,1,1,0,0,0]
 => 2
111 => [1,1,1,1] => [1,0,1,0,1,0,1,0]
 => [1,1,1,1,0,0,0,0]
 => 1
0000 => [5] => [1,1,1,1,1,0,0,0,0,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => 5
0001 => [4,1] => [1,1,1,1,0,0,0,0,1,0]
 => [1,1,0,1,0,1,0,1,0,0]
 => 1
0010 => [3,2] => [1,1,1,0,0,0,1,1,0,0]
 => [1,0,1,1,0,1,0,1,0,0]
 => 2
0011 => [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
 => [1,1,1,0,1,0,1,0,0,0]
 => 1
0100 => [2,3] => [1,1,0,0,1,1,1,0,0,0]
 => [1,0,1,0,1,1,0,1,0,0]
 => 3
0101 => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
 => [1,1,0,1,1,0,1,0,0,0]
 => 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
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]
 => 1
1000 => [1,4] => [1,0,1,1,1,1,0,0,0,0]
 => [1,0,1,0,1,0,1,1,0,0]
 => 4
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
1010 => [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
 => [1,0,1,1,0,1,1,0,0,0]
 => 2
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
1100 => [1,1,3] => [1,0,1,0,1,1,1,0,0,0]
 => [1,0,1,0,1,1,1,0,0,0]
 => 3
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
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]
 => 2
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
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
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]
 => 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]
 => 2
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]
 => 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
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]
 => 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]
 => 2
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]
 => 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]
 => 4
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]
 => 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
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]
 => 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]
 => 3
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]
 => 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
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]
 => 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]
 => 5
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
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]
 => 2
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
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
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]
 => ? = 3
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
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
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]
 => ? = 2
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]
 => ? = 3
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
0101110 => [2,2,1,1,2] => [1,1,0,0,1,1,0,0,1,0,1,0,1,1,0,0]
 => [1,0,1,1,1,1,0,1,1,0,1,0,0,0,0,0]
 => ? = 2
0110010 => [2,1,3,2] => [1,1,0,0,1,0,1,1,1,0,0,0,1,1,0,0]
 => [1,0,1,1,0,1,0,1,1,1,0,1,0,0,0,0]
 => ? = 2
0110100 => [2,1,2,3] => [1,1,0,0,1,0,1,1,0,0,1,1,1,0,0,0]
 => [1,0,1,0,1,1,0,1,1,1,0,1,0,0,0,0]
 => ? = 3
0110110 => [2,1,2,1,2] => [1,1,0,0,1,0,1,1,0,0,1,0,1,1,0,0]
 => [1,0,1,1,1,0,1,1,1,0,1,0,0,0,0,0]
 => ? = 2
0111010 => [2,1,1,2,2] => [1,1,0,0,1,0,1,0,1,1,0,0,1,1,0,0]
 => [1,0,1,1,0,1,1,1,1,0,1,0,0,0,0,0]
 => ? = 2
1000110 => [1,4,1,2] => [1,0,1,1,1,1,0,0,0,0,1,0,1,1,0,0]
 => [1,0,1,1,1,0,1,0,1,0,1,1,0,0,0,0]
 => ? = 2
1001001 => [1,3,3,1] => [1,0,1,1,1,0,0,0,1,1,1,0,0,0,1,0]
 => [1,1,0,1,0,1,1,0,1,0,1,1,0,0,0,0]
 => ? = 1
1001010 => [1,3,2,2] => [1,0,1,1,1,0,0,0,1,1,0,0,1,1,0,0]
 => [1,0,1,1,0,1,1,0,1,0,1,1,0,0,0,0]
 => ? = 2
1001101 => [1,3,1,2,1] => [1,0,1,1,1,0,0,0,1,0,1,1,0,0,1,0]
 => [1,1,0,1,1,1,0,1,0,1,1,0,0,0,0,0]
 => ? = 1
1001110 => [1,3,1,1,2] => [1,0,1,1,1,0,0,0,1,0,1,0,1,1,0,0]
 => [1,0,1,1,1,1,0,1,0,1,1,0,0,0,0,0]
 => ? = 2
1010010 => [1,2,3,2] => [1,0,1,1,0,0,1,1,1,0,0,0,1,1,0,0]
 => [1,0,1,1,0,1,0,1,1,0,1,1,0,0,0,0]
 => ? = 2
1010100 => [1,2,2,3] => [1,0,1,1,0,0,1,1,0,0,1,1,1,0,0,0]
 => [1,0,1,0,1,1,0,1,1,0,1,1,0,0,0,0]
 => ? = 3
1010101 => [1,2,2,2,1] => [1,0,1,1,0,0,1,1,0,0,1,1,0,0,1,0]
 => [1,1,0,1,1,0,1,1,0,1,1,0,0,0,0,0]
 => ? = 1
1010110 => [1,2,2,1,2] => [1,0,1,1,0,0,1,1,0,0,1,0,1,1,0,0]
 => [1,0,1,1,1,0,1,1,0,1,1,0,0,0,0,0]
 => ? = 2
1011000 => [1,2,1,4] => [1,0,1,1,0,0,1,0,1,1,1,1,0,0,0,0]
 => [1,0,1,0,1,0,1,1,1,0,1,1,0,0,0,0]
 => ? = 4
1011010 => [1,2,1,2,2] => [1,0,1,1,0,0,1,0,1,1,0,0,1,1,0,0]
 => [1,0,1,1,0,1,1,1,0,1,1,0,0,0,0,0]
 => ? = 2
1100100 => [1,1,3,3] => [1,0,1,0,1,1,1,0,0,0,1,1,1,0,0,0]
 => [1,0,1,0,1,1,0,1,0,1,1,1,0,0,0,0]
 => ? = 3
1100101 => [1,1,3,2,1] => [1,0,1,0,1,1,1,0,0,0,1,1,0,0,1,0]
 => [1,1,0,1,1,0,1,0,1,1,1,0,0,0,0,0]
 => ? = 1
1100110 => [1,1,3,1,2] => [1,0,1,0,1,1,1,0,0,0,1,0,1,1,0,0]
 => [1,0,1,1,1,0,1,0,1,1,1,0,0,0,0,0]
 => ? = 2
1101001 => [1,1,2,3,1] => [1,0,1,0,1,1,0,0,1,1,1,0,0,0,1,0]
 => [1,1,0,1,0,1,1,0,1,1,1,0,0,0,0,0]
 => ? = 1
1101010 => [1,1,2,2,2] => [1,0,1,0,1,1,0,0,1,1,0,0,1,1,0,0]
 => [1,0,1,1,0,1,1,0,1,1,1,0,0,0,0,0]
 => ? = 2
1110010 => [1,1,1,3,2] => [1,0,1,0,1,0,1,1,1,0,0,0,1,1,0,0]
 => [1,0,1,1,0,1,0,1,1,1,1,0,0,0,0,0]
 => ? = 2
1110110 => [1,1,1,2,1,2] => [1,0,1,0,1,0,1,1,0,0,1,0,1,1,0,0]
 => [1,0,1,1,1,0,1,1,1,1,0,0,0,0,0,0]
 => ? = 2
1111010 => [1,1,1,1,2,2] => [1,0,1,0,1,0,1,0,1,1,0,0,1,1,0,0]
 => [1,0,1,1,0,1,1,1,1,1,0,0,0,0,0,0]
 => ? = 2
01010001 => [2,2,4,1] => [1,1,0,0,1,1,0,0,1,1,1,1,0,0,0,0,1,0]
 => ?
 => ? = 1
01100001 => [2,1,5,1] => [1,1,0,0,1,0,1,1,1,1,1,0,0,0,0,0,1,0]
 => ?
 => ? = 1
10000011 => [1,6,1,1] => [1,0,1,1,1,1,1,1,0,0,0,0,0,0,1,0,1,0]
 => [1,1,1,0,1,0,1,0,1,0,1,0,1,1,0,0,0,0]
 => ? = 1
10000101 => [1,5,2,1] => [1,0,1,1,1,1,1,0,0,0,0,0,1,1,0,0,1,0]
 => [1,1,0,1,1,0,1,0,1,0,1,0,1,1,0,0,0,0]
 => ? = 1
10000110 => [1,5,1,2] => [1,0,1,1,1,1,1,0,0,0,0,0,1,0,1,1,0,0]
 => [1,0,1,1,1,0,1,0,1,0,1,0,1,1,0,0,0,0]
 => ? = 2
10001001 => [1,4,3,1] => [1,0,1,1,1,1,0,0,0,0,1,1,1,0,0,0,1,0]
 => ?
 => ? = 1
10001010 => [1,4,2,2] => [1,0,1,1,1,1,0,0,0,0,1,1,0,0,1,1,0,0]
 => ?
 => ? = 2
10001100 => [1,4,1,3] => [1,0,1,1,1,1,0,0,0,0,1,0,1,1,1,0,0,0]
 => [1,0,1,0,1,1,1,0,1,0,1,0,1,1,0,0,0,0]
 => ? = 3
10010001 => [1,3,4,1] => [1,0,1,1,1,0,0,0,1,1,1,1,0,0,0,0,1,0]
 => ?
 => ? = 1
10010111 => [1,3,2,1,1,1] => [1,0,1,1,1,0,0,0,1,1,0,0,1,0,1,0,1,0]
 => ?
 => ? = 1
10011011 => [1,3,1,2,1,1] => [1,0,1,1,1,0,0,0,1,0,1,1,0,0,1,0,1,0]
 => ?
 => ? = 1
10011101 => [1,3,1,1,2,1] => [1,0,1,1,1,0,0,0,1,0,1,0,1,1,0,0,1,0]
 => ?
 => ? = 1
10100001 => [1,2,5,1] => [1,0,1,1,0,0,1,1,1,1,1,0,0,0,0,0,1,0]
 => ?
 => ? = 1
10101101 => [1,2,2,1,2,1] => [1,0,1,1,0,0,1,1,0,0,1,0,1,1,0,0,1,0]
 => [1,1,0,1,1,1,0,1,1,0,1,1,0,0,0,0,0,0]
 => ? = 1
10110101 => [1,2,1,2,2,1] => [1,0,1,1,0,0,1,0,1,1,0,0,1,1,0,0,1,0]
 => [1,1,0,1,1,0,1,1,1,0,1,1,0,0,0,0,0,0]
 => ? = 1
10111001 => [1,2,1,1,3,1] => [1,0,1,1,0,0,1,0,1,0,1,1,1,0,0,0,1,0]
 => ?
 => ? = 1
11011001 => [1,1,2,1,3,1] => [1,0,1,0,1,1,0,0,1,0,1,1,1,0,0,0,1,0]
 => ?
 => ? = 1
11101001 => [1,1,1,2,3,1] => [1,0,1,0,1,0,1,1,0,0,1,1,1,0,0,0,1,0]
 => ?
 => ? = 1
11110001 => [1,1,1,1,4,1] => [1,0,1,0,1,0,1,0,1,1,1,1,0,0,0,0,1,0]
 => [1,1,0,1,0,1,0,1,1,1,1,1,0,0,0,0,0,0]
 => ? = 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: St000010
Mapping
Mp00178: Binary words —to composition⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
Mp00037: Graphs —to partition of connected components⟶ Integer partitions
St000010: Integer partitions ⟶ ℤResult quality: 55% ●values known / values provided: 55%●distinct values known / distinct values provided: 70%
Values
0 => [2] => ([],2)
 => [1,1]
 => 2
1 => [1,1] => ([(0,1)],2)
 => [2]
 => 1
00 => [3] => ([],3)
 => [1,1,1]
 => 3
01 => [2,1] => ([(0,2),(1,2)],3)
 => [3]
 => 1
10 => [1,2] => ([(1,2)],3)
 => [2,1]
 => 2
11 => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
 => [3]
 => 1
000 => [4] => ([],4)
 => [1,1,1,1]
 => 4
001 => [3,1] => ([(0,3),(1,3),(2,3)],4)
 => [4]
 => 1
010 => [2,2] => ([(1,3),(2,3)],4)
 => [3,1]
 => 2
011 => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => [4]
 => 1
100 => [1,3] => ([(2,3)],4)
 => [2,1,1]
 => 3
101 => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => [4]
 => 1
110 => [1,1,2] => ([(1,2),(1,3),(2,3)],4)
 => [3,1]
 => 2
111 => [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => [4]
 => 1
0000 => [5] => ([],5)
 => [1,1,1,1,1]
 => 5
0001 => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => [5]
 => 1
0010 => [3,2] => ([(1,4),(2,4),(3,4)],5)
 => [4,1]
 => 2
0011 => [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [5]
 => 1
0100 => [2,3] => ([(2,4),(3,4)],5)
 => [3,1,1]
 => 3
0101 => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [5]
 => 1
0110 => [2,1,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [4,1]
 => 2
0111 => [2,1,1,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [5]
 => 1
1000 => [1,4] => ([(3,4)],5)
 => [2,1,1,1]
 => 4
1001 => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => [5]
 => 1
1010 => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
 => [4,1]
 => 2
1011 => [1,2,1,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [5]
 => 1
1100 => [1,1,3] => ([(2,3),(2,4),(3,4)],5)
 => [3,1,1]
 => 3
1101 => [1,1,2,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [5]
 => 1
1110 => [1,1,1,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [4,1]
 => 2
1111 => [1,1,1,1,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [5]
 => 1
00000 => [6] => ([],6)
 => [1,1,1,1,1,1]
 => 6
00001 => [5,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
 => [6]
 => 1
00010 => [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6)
 => [5,1]
 => 2
00011 => [4,1,1] => ([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [6]
 => 1
00100 => [3,3] => ([(2,5),(3,5),(4,5)],6)
 => [4,1,1]
 => 3
00101 => [3,2,1] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [6]
 => 1
00110 => [3,1,2] => ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [5,1]
 => 2
00111 => [3,1,1,1] => ([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [6]
 => 1
01000 => [2,4] => ([(3,5),(4,5)],6)
 => [3,1,1,1]
 => 4
01001 => [2,3,1] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [6]
 => 1
01010 => [2,2,2] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [5,1]
 => 2
01011 => [2,2,1,1] => ([(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [6]
 => 1
01100 => [2,1,3] => ([(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [4,1,1]
 => 3
01101 => [2,1,2,1] => ([(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [6]
 => 1
01110 => [2,1,1,2] => ([(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [5,1]
 => 2
01111 => [2,1,1,1,1] => ([(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [6]
 => 1
10000 => [1,5] => ([(4,5)],6)
 => [2,1,1,1,1]
 => 5
10001 => [1,4,1] => ([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
 => [6]
 => 1
10010 => [1,3,2] => ([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
 => [5,1]
 => 2
10011 => [1,3,1,1] => ([(0,4),(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [6]
 => 1
0000001 => [7,1] => ([(0,7),(1,7),(2,7),(3,7),(4,7),(5,7),(6,7)],8)
 => ?
 => ? = 1
0000011 => [6,1,1] => ([(0,6),(0,7),(1,6),(1,7),(2,6),(2,7),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ?
 => ? = 1
0000101 => [5,2,1] => ([(0,7),(1,6),(1,7),(2,6),(2,7),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ?
 => ? = 1
0110001 => [2,1,4,1] => ([(0,7),(1,7),(2,7),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ?
 => ? = 1
1000000 => [1,7] => ([(6,7)],8)
 => ?
 => ? = 7
1000001 => [1,6,1] => ([(0,7),(1,7),(2,7),(3,7),(4,7),(5,6),(5,7),(6,7)],8)
 => ?
 => ? = 1
1000010 => [1,5,2] => ([(1,7),(2,7),(3,7),(4,7),(5,6),(5,7),(6,7)],8)
 => ?
 => ? = 2
1000011 => [1,5,1,1] => ([(0,6),(0,7),(1,6),(1,7),(2,6),(2,7),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ?
 => ? = 1
1001111 => [1,3,1,1,1,1] => ([(0,4),(0,5),(0,6),(0,7),(1,4),(1,5),(1,6),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ?
 => ? = 1
1010001 => [1,2,4,1] => ([(0,7),(1,7),(2,7),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ?
 => ? = 1
1011000 => [1,2,1,4] => ([(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ?
 => ? = 4
1100001 => [1,1,5,1] => ([(0,7),(1,7),(2,7),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ?
 => ? = 1
1110011 => [1,1,1,3,1,1] => ([(0,6),(0,7),(1,6),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ?
 => ? = 1
1111001 => [1,1,1,1,3,1] => ([(0,7),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ?
 => ? = 1
00000001 => [8,1] => ([(0,8),(1,8),(2,8),(3,8),(4,8),(5,8),(6,8),(7,8)],9)
 => ?
 => ? = 1
00000011 => [7,1,1] => ([(0,7),(0,8),(1,7),(1,8),(2,7),(2,8),(3,7),(3,8),(4,7),(4,8),(5,7),(5,8),(6,7),(6,8),(7,8)],9)
 => ?
 => ? = 1
00000101 => [6,2,1] => ([(0,8),(1,7),(1,8),(2,7),(2,8),(3,7),(3,8),(4,7),(4,8),(5,7),(5,8),(6,7),(6,8),(7,8)],9)
 => ?
 => ? = 1
00110001 => [3,1,4,1] => ([(0,8),(1,8),(2,8),(3,6),(3,7),(3,8),(4,6),(4,7),(4,8),(5,6),(5,7),(5,8),(6,7),(6,8),(7,8)],9)
 => ?
 => ? = 1
01010001 => [2,2,4,1] => ([(0,8),(1,8),(2,8),(3,7),(3,8),(4,6),(4,7),(4,8),(5,6),(5,7),(5,8),(6,7),(6,8),(7,8)],9)
 => ?
 => ? = 1
01100001 => [2,1,5,1] => ([(0,8),(1,8),(2,8),(3,8),(4,6),(4,7),(4,8),(5,6),(5,7),(5,8),(6,7),(6,8),(7,8)],9)
 => ?
 => ? = 1
10000000 => [1,8] => ([(7,8)],9)
 => ?
 => ? = 8
10000001 => [1,7,1] => ([(0,8),(1,8),(2,8),(3,8),(4,8),(5,8),(6,7),(6,8),(7,8)],9)
 => ?
 => ? = 1
10000010 => [1,6,2] => ([(1,8),(2,8),(3,8),(4,8),(5,8),(6,7),(6,8),(7,8)],9)
 => ?
 => ? = 2
10000011 => [1,6,1,1] => ([(0,7),(0,8),(1,7),(1,8),(2,7),(2,8),(3,7),(3,8),(4,7),(4,8),(5,6),(5,7),(5,8),(6,7),(6,8),(7,8)],9)
 => ?
 => ? = 1
10000101 => [1,5,2,1] => ([(0,8),(1,7),(1,8),(2,7),(2,8),(3,7),(3,8),(4,7),(4,8),(5,6),(5,7),(5,8),(6,7),(6,8),(7,8)],9)
 => ?
 => ? = 1
10000110 => [1,5,1,2] => ([(1,7),(1,8),(2,7),(2,8),(3,7),(3,8),(4,7),(4,8),(5,6),(5,7),(5,8),(6,7),(6,8),(7,8)],9)
 => ?
 => ? = 2
10001001 => [1,4,3,1] => ([(0,8),(1,8),(2,7),(2,8),(3,7),(3,8),(4,7),(4,8),(5,6),(5,7),(5,8),(6,7),(6,8),(7,8)],9)
 => ?
 => ? = 1
10001010 => [1,4,2,2] => ([(1,8),(2,7),(2,8),(3,7),(3,8),(4,7),(4,8),(5,6),(5,7),(5,8),(6,7),(6,8),(7,8)],9)
 => ?
 => ? = 2
10001100 => [1,4,1,3] => ([(2,7),(2,8),(3,7),(3,8),(4,7),(4,8),(5,6),(5,7),(5,8),(6,7),(6,8),(7,8)],9)
 => ?
 => ? = 3
10001111 => [1,4,1,1,1,1] => ([(0,5),(0,6),(0,7),(0,8),(1,5),(1,6),(1,7),(1,8),(2,5),(2,6),(2,7),(2,8),(3,4),(3,5),(3,6),(3,7),(3,8),(4,5),(4,6),(4,7),(4,8),(5,6),(5,7),(5,8),(6,7),(6,8),(7,8)],9)
 => ?
 => ? = 1
10010001 => [1,3,4,1] => ([(0,8),(1,8),(2,8),(3,7),(3,8),(4,7),(4,8),(5,6),(5,7),(5,8),(6,7),(6,8),(7,8)],9)
 => ?
 => ? = 1
10010111 => [1,3,2,1,1,1] => ([(0,6),(0,7),(0,8),(1,5),(1,6),(1,7),(1,8),(2,5),(2,6),(2,7),(2,8),(3,4),(3,5),(3,6),(3,7),(3,8),(4,5),(4,6),(4,7),(4,8),(5,6),(5,7),(5,8),(6,7),(6,8),(7,8)],9)
 => ?
 => ? = 1
10011011 => [1,3,1,2,1,1] => ([(0,7),(0,8),(1,5),(1,6),(1,7),(1,8),(2,5),(2,6),(2,7),(2,8),(3,4),(3,5),(3,6),(3,7),(3,8),(4,5),(4,6),(4,7),(4,8),(5,6),(5,7),(5,8),(6,7),(6,8),(7,8)],9)
 => ?
 => ? = 1
10011101 => [1,3,1,1,2,1] => ([(0,8),(1,5),(1,6),(1,7),(1,8),(2,5),(2,6),(2,7),(2,8),(3,4),(3,5),(3,6),(3,7),(3,8),(4,5),(4,6),(4,7),(4,8),(5,6),(5,7),(5,8),(6,7),(6,8),(7,8)],9)
 => ?
 => ? = 1
10100001 => [1,2,5,1] => ([(0,8),(1,8),(2,8),(3,8),(4,7),(4,8),(5,6),(5,7),(5,8),(6,7),(6,8),(7,8)],9)
 => ?
 => ? = 1
10101101 => [1,2,2,1,2,1] => ([(0,8),(1,6),(1,7),(1,8),(2,5),(2,6),(2,7),(2,8),(3,4),(3,5),(3,6),(3,7),(3,8),(4,5),(4,6),(4,7),(4,8),(5,6),(5,7),(5,8),(6,7),(6,8),(7,8)],9)
 => ?
 => ? = 1
10110101 => [1,2,1,2,2,1] => ([(0,8),(1,7),(1,8),(2,5),(2,6),(2,7),(2,8),(3,4),(3,5),(3,6),(3,7),(3,8),(4,5),(4,6),(4,7),(4,8),(5,6),(5,7),(5,8),(6,7),(6,8),(7,8)],9)
 => ?
 => ? = 1
10111001 => [1,2,1,1,3,1] => ([(0,8),(1,8),(2,5),(2,6),(2,7),(2,8),(3,4),(3,5),(3,6),(3,7),(3,8),(4,5),(4,6),(4,7),(4,8),(5,6),(5,7),(5,8),(6,7),(6,8),(7,8)],9)
 => ?
 => ? = 1
11000001 => [1,1,6,1] => ([(0,8),(1,8),(2,8),(3,8),(4,8),(5,6),(5,7),(5,8),(6,7),(6,8),(7,8)],9)
 => ?
 => ? = 1
11011001 => [1,1,2,1,3,1] => ([(0,8),(1,8),(2,6),(2,7),(2,8),(3,4),(3,5),(3,6),(3,7),(3,8),(4,5),(4,6),(4,7),(4,8),(5,6),(5,7),(5,8),(6,7),(6,8),(7,8)],9)
 => ?
 => ? = 1
11101001 => [1,1,1,2,3,1] => ([(0,8),(1,8),(2,7),(2,8),(3,4),(3,5),(3,6),(3,7),(3,8),(4,5),(4,6),(4,7),(4,8),(5,6),(5,7),(5,8),(6,7),(6,8),(7,8)],9)
 => ?
 => ? = 1
11110001 => [1,1,1,1,4,1] => ([(0,8),(1,8),(2,8),(3,4),(3,5),(3,6),(3,7),(3,8),(4,5),(4,6),(4,7),(4,8),(5,6),(5,7),(5,8),(6,7),(6,8),(7,8)],9)
 => ?
 => ? = 1
000000001 => [9,1] => ([(0,9),(1,9),(2,9),(3,9),(4,9),(5,9),(6,9),(7,9),(8,9)],10)
 => ?
 => ? = 1
000000011 => [8,1,1] => ([(0,8),(0,9),(1,8),(1,9),(2,8),(2,9),(3,8),(3,9),(4,8),(4,9),(5,8),(5,9),(6,8),(6,9),(7,8),(7,9),(8,9)],10)
 => ?
 => ? = 1
000010000 => [5,5] => ([(4,9),(5,9),(6,9),(7,9),(8,9)],10)
 => ?
 => ? = 5
000100011 => [4,4,1,1] => ([(0,8),(0,9),(1,8),(1,9),(2,8),(2,9),(3,7),(3,8),(3,9),(4,7),(4,8),(4,9),(5,7),(5,8),(5,9),(6,7),(6,8),(6,9),(7,8),(7,9),(8,9)],10)
 => ?
 => ? = 1
000100110 => [4,3,1,2] => ([(1,8),(1,9),(2,8),(2,9),(3,7),(3,8),(3,9),(4,7),(4,8),(4,9),(5,7),(5,8),(5,9),(6,7),(6,8),(6,9),(7,8),(7,9),(8,9)],10)
 => ?
 => ? = 2
000101100 => [4,2,1,3] => ([(2,8),(2,9),(3,7),(3,8),(3,9),(4,7),(4,8),(4,9),(5,7),(5,8),(5,9),(6,7),(6,8),(6,9),(7,8),(7,9),(8,9)],10)
 => ?
 => ? = 3
000111000 => [4,1,1,4] => ([(3,7),(3,8),(3,9),(4,7),(4,8),(4,9),(5,7),(5,8),(5,9),(6,7),(6,8),(6,9),(7,8),(7,9),(8,9)],10)
 => ?
 => ? = 4
001001010 => [3,3,2,2] => ([(1,9),(2,8),(2,9),(3,8),(3,9),(4,7),(4,8),(4,9),(5,7),(5,8),(5,9),(6,7),(6,8),(6,9),(7,8),(7,9),(8,9)],10)
 => ?
 => ? = 2
Description
The length of the partition.
Matching statistic: St001176
Mapping
Mp00178: Binary words —to composition⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
Mp00037: Graphs —to partition of connected components⟶ Integer partitions
St001176: Integer partitions ⟶ ℤResult quality: 55% ●values known / values provided: 55%●distinct values known / distinct values provided: 70%
Values
0 => [2] => ([],2)
 => [1,1]
 => 1 = 2 - 1
1 => [1,1] => ([(0,1)],2)
 => [2]
 => 0 = 1 - 1
00 => [3] => ([],3)
 => [1,1,1]
 => 2 = 3 - 1
01 => [2,1] => ([(0,2),(1,2)],3)
 => [3]
 => 0 = 1 - 1
10 => [1,2] => ([(1,2)],3)
 => [2,1]
 => 1 = 2 - 1
11 => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
 => [3]
 => 0 = 1 - 1
000 => [4] => ([],4)
 => [1,1,1,1]
 => 3 = 4 - 1
001 => [3,1] => ([(0,3),(1,3),(2,3)],4)
 => [4]
 => 0 = 1 - 1
010 => [2,2] => ([(1,3),(2,3)],4)
 => [3,1]
 => 1 = 2 - 1
011 => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => [4]
 => 0 = 1 - 1
100 => [1,3] => ([(2,3)],4)
 => [2,1,1]
 => 2 = 3 - 1
101 => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => [4]
 => 0 = 1 - 1
110 => [1,1,2] => ([(1,2),(1,3),(2,3)],4)
 => [3,1]
 => 1 = 2 - 1
111 => [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => [4]
 => 0 = 1 - 1
0000 => [5] => ([],5)
 => [1,1,1,1,1]
 => 4 = 5 - 1
0001 => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => [5]
 => 0 = 1 - 1
0010 => [3,2] => ([(1,4),(2,4),(3,4)],5)
 => [4,1]
 => 1 = 2 - 1
0011 => [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [5]
 => 0 = 1 - 1
0100 => [2,3] => ([(2,4),(3,4)],5)
 => [3,1,1]
 => 2 = 3 - 1
0101 => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [5]
 => 0 = 1 - 1
0110 => [2,1,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [4,1]
 => 1 = 2 - 1
0111 => [2,1,1,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [5]
 => 0 = 1 - 1
1000 => [1,4] => ([(3,4)],5)
 => [2,1,1,1]
 => 3 = 4 - 1
1001 => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => [5]
 => 0 = 1 - 1
1010 => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
 => [4,1]
 => 1 = 2 - 1
1011 => [1,2,1,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [5]
 => 0 = 1 - 1
1100 => [1,1,3] => ([(2,3),(2,4),(3,4)],5)
 => [3,1,1]
 => 2 = 3 - 1
1101 => [1,1,2,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [5]
 => 0 = 1 - 1
1110 => [1,1,1,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [4,1]
 => 1 = 2 - 1
1111 => [1,1,1,1,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [5]
 => 0 = 1 - 1
00000 => [6] => ([],6)
 => [1,1,1,1,1,1]
 => 5 = 6 - 1
00001 => [5,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
 => [6]
 => 0 = 1 - 1
00010 => [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6)
 => [5,1]
 => 1 = 2 - 1
00011 => [4,1,1] => ([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [6]
 => 0 = 1 - 1
00100 => [3,3] => ([(2,5),(3,5),(4,5)],6)
 => [4,1,1]
 => 2 = 3 - 1
00101 => [3,2,1] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [6]
 => 0 = 1 - 1
00110 => [3,1,2] => ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [5,1]
 => 1 = 2 - 1
00111 => [3,1,1,1] => ([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [6]
 => 0 = 1 - 1
01000 => [2,4] => ([(3,5),(4,5)],6)
 => [3,1,1,1]
 => 3 = 4 - 1
01001 => [2,3,1] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [6]
 => 0 = 1 - 1
01010 => [2,2,2] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [5,1]
 => 1 = 2 - 1
01011 => [2,2,1,1] => ([(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [6]
 => 0 = 1 - 1
01100 => [2,1,3] => ([(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [4,1,1]
 => 2 = 3 - 1
01101 => [2,1,2,1] => ([(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [6]
 => 0 = 1 - 1
01110 => [2,1,1,2] => ([(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [5,1]
 => 1 = 2 - 1
01111 => [2,1,1,1,1] => ([(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [6]
 => 0 = 1 - 1
10000 => [1,5] => ([(4,5)],6)
 => [2,1,1,1,1]
 => 4 = 5 - 1
10001 => [1,4,1] => ([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
 => [6]
 => 0 = 1 - 1
10010 => [1,3,2] => ([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
 => [5,1]
 => 1 = 2 - 1
10011 => [1,3,1,1] => ([(0,4),(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [6]
 => 0 = 1 - 1
0000001 => [7,1] => ([(0,7),(1,7),(2,7),(3,7),(4,7),(5,7),(6,7)],8)
 => ?
 => ? = 1 - 1
0000011 => [6,1,1] => ([(0,6),(0,7),(1,6),(1,7),(2,6),(2,7),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ?
 => ? = 1 - 1
0000101 => [5,2,1] => ([(0,7),(1,6),(1,7),(2,6),(2,7),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ?
 => ? = 1 - 1
0110001 => [2,1,4,1] => ([(0,7),(1,7),(2,7),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ?
 => ? = 1 - 1
1000000 => [1,7] => ([(6,7)],8)
 => ?
 => ? = 7 - 1
1000001 => [1,6,1] => ([(0,7),(1,7),(2,7),(3,7),(4,7),(5,6),(5,7),(6,7)],8)
 => ?
 => ? = 1 - 1
1000010 => [1,5,2] => ([(1,7),(2,7),(3,7),(4,7),(5,6),(5,7),(6,7)],8)
 => ?
 => ? = 2 - 1
1000011 => [1,5,1,1] => ([(0,6),(0,7),(1,6),(1,7),(2,6),(2,7),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ?
 => ? = 1 - 1
1001111 => [1,3,1,1,1,1] => ([(0,4),(0,5),(0,6),(0,7),(1,4),(1,5),(1,6),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ?
 => ? = 1 - 1
1010001 => [1,2,4,1] => ([(0,7),(1,7),(2,7),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ?
 => ? = 1 - 1
1011000 => [1,2,1,4] => ([(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ?
 => ? = 4 - 1
1100001 => [1,1,5,1] => ([(0,7),(1,7),(2,7),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ?
 => ? = 1 - 1
1110011 => [1,1,1,3,1,1] => ([(0,6),(0,7),(1,6),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ?
 => ? = 1 - 1
1111001 => [1,1,1,1,3,1] => ([(0,7),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ?
 => ? = 1 - 1
00000001 => [8,1] => ([(0,8),(1,8),(2,8),(3,8),(4,8),(5,8),(6,8),(7,8)],9)
 => ?
 => ? = 1 - 1
00000011 => [7,1,1] => ([(0,7),(0,8),(1,7),(1,8),(2,7),(2,8),(3,7),(3,8),(4,7),(4,8),(5,7),(5,8),(6,7),(6,8),(7,8)],9)
 => ?
 => ? = 1 - 1
00000101 => [6,2,1] => ([(0,8),(1,7),(1,8),(2,7),(2,8),(3,7),(3,8),(4,7),(4,8),(5,7),(5,8),(6,7),(6,8),(7,8)],9)
 => ?
 => ? = 1 - 1
00110001 => [3,1,4,1] => ([(0,8),(1,8),(2,8),(3,6),(3,7),(3,8),(4,6),(4,7),(4,8),(5,6),(5,7),(5,8),(6,7),(6,8),(7,8)],9)
 => ?
 => ? = 1 - 1
01010001 => [2,2,4,1] => ([(0,8),(1,8),(2,8),(3,7),(3,8),(4,6),(4,7),(4,8),(5,6),(5,7),(5,8),(6,7),(6,8),(7,8)],9)
 => ?
 => ? = 1 - 1
01100001 => [2,1,5,1] => ([(0,8),(1,8),(2,8),(3,8),(4,6),(4,7),(4,8),(5,6),(5,7),(5,8),(6,7),(6,8),(7,8)],9)
 => ?
 => ? = 1 - 1
10000000 => [1,8] => ([(7,8)],9)
 => ?
 => ? = 8 - 1
10000001 => [1,7,1] => ([(0,8),(1,8),(2,8),(3,8),(4,8),(5,8),(6,7),(6,8),(7,8)],9)
 => ?
 => ? = 1 - 1
10000010 => [1,6,2] => ([(1,8),(2,8),(3,8),(4,8),(5,8),(6,7),(6,8),(7,8)],9)
 => ?
 => ? = 2 - 1
10000011 => [1,6,1,1] => ([(0,7),(0,8),(1,7),(1,8),(2,7),(2,8),(3,7),(3,8),(4,7),(4,8),(5,6),(5,7),(5,8),(6,7),(6,8),(7,8)],9)
 => ?
 => ? = 1 - 1
10000101 => [1,5,2,1] => ([(0,8),(1,7),(1,8),(2,7),(2,8),(3,7),(3,8),(4,7),(4,8),(5,6),(5,7),(5,8),(6,7),(6,8),(7,8)],9)
 => ?
 => ? = 1 - 1
10000110 => [1,5,1,2] => ([(1,7),(1,8),(2,7),(2,8),(3,7),(3,8),(4,7),(4,8),(5,6),(5,7),(5,8),(6,7),(6,8),(7,8)],9)
 => ?
 => ? = 2 - 1
10001001 => [1,4,3,1] => ([(0,8),(1,8),(2,7),(2,8),(3,7),(3,8),(4,7),(4,8),(5,6),(5,7),(5,8),(6,7),(6,8),(7,8)],9)
 => ?
 => ? = 1 - 1
10001010 => [1,4,2,2] => ([(1,8),(2,7),(2,8),(3,7),(3,8),(4,7),(4,8),(5,6),(5,7),(5,8),(6,7),(6,8),(7,8)],9)
 => ?
 => ? = 2 - 1
10001100 => [1,4,1,3] => ([(2,7),(2,8),(3,7),(3,8),(4,7),(4,8),(5,6),(5,7),(5,8),(6,7),(6,8),(7,8)],9)
 => ?
 => ? = 3 - 1
10001111 => [1,4,1,1,1,1] => ([(0,5),(0,6),(0,7),(0,8),(1,5),(1,6),(1,7),(1,8),(2,5),(2,6),(2,7),(2,8),(3,4),(3,5),(3,6),(3,7),(3,8),(4,5),(4,6),(4,7),(4,8),(5,6),(5,7),(5,8),(6,7),(6,8),(7,8)],9)
 => ?
 => ? = 1 - 1
10010001 => [1,3,4,1] => ([(0,8),(1,8),(2,8),(3,7),(3,8),(4,7),(4,8),(5,6),(5,7),(5,8),(6,7),(6,8),(7,8)],9)
 => ?
 => ? = 1 - 1
10010111 => [1,3,2,1,1,1] => ([(0,6),(0,7),(0,8),(1,5),(1,6),(1,7),(1,8),(2,5),(2,6),(2,7),(2,8),(3,4),(3,5),(3,6),(3,7),(3,8),(4,5),(4,6),(4,7),(4,8),(5,6),(5,7),(5,8),(6,7),(6,8),(7,8)],9)
 => ?
 => ? = 1 - 1
10011011 => [1,3,1,2,1,1] => ([(0,7),(0,8),(1,5),(1,6),(1,7),(1,8),(2,5),(2,6),(2,7),(2,8),(3,4),(3,5),(3,6),(3,7),(3,8),(4,5),(4,6),(4,7),(4,8),(5,6),(5,7),(5,8),(6,7),(6,8),(7,8)],9)
 => ?
 => ? = 1 - 1
10011101 => [1,3,1,1,2,1] => ([(0,8),(1,5),(1,6),(1,7),(1,8),(2,5),(2,6),(2,7),(2,8),(3,4),(3,5),(3,6),(3,7),(3,8),(4,5),(4,6),(4,7),(4,8),(5,6),(5,7),(5,8),(6,7),(6,8),(7,8)],9)
 => ?
 => ? = 1 - 1
10100001 => [1,2,5,1] => ([(0,8),(1,8),(2,8),(3,8),(4,7),(4,8),(5,6),(5,7),(5,8),(6,7),(6,8),(7,8)],9)
 => ?
 => ? = 1 - 1
10101101 => [1,2,2,1,2,1] => ([(0,8),(1,6),(1,7),(1,8),(2,5),(2,6),(2,7),(2,8),(3,4),(3,5),(3,6),(3,7),(3,8),(4,5),(4,6),(4,7),(4,8),(5,6),(5,7),(5,8),(6,7),(6,8),(7,8)],9)
 => ?
 => ? = 1 - 1
10110101 => [1,2,1,2,2,1] => ([(0,8),(1,7),(1,8),(2,5),(2,6),(2,7),(2,8),(3,4),(3,5),(3,6),(3,7),(3,8),(4,5),(4,6),(4,7),(4,8),(5,6),(5,7),(5,8),(6,7),(6,8),(7,8)],9)
 => ?
 => ? = 1 - 1
10111001 => [1,2,1,1,3,1] => ([(0,8),(1,8),(2,5),(2,6),(2,7),(2,8),(3,4),(3,5),(3,6),(3,7),(3,8),(4,5),(4,6),(4,7),(4,8),(5,6),(5,7),(5,8),(6,7),(6,8),(7,8)],9)
 => ?
 => ? = 1 - 1
11000001 => [1,1,6,1] => ([(0,8),(1,8),(2,8),(3,8),(4,8),(5,6),(5,7),(5,8),(6,7),(6,8),(7,8)],9)
 => ?
 => ? = 1 - 1
11011001 => [1,1,2,1,3,1] => ([(0,8),(1,8),(2,6),(2,7),(2,8),(3,4),(3,5),(3,6),(3,7),(3,8),(4,5),(4,6),(4,7),(4,8),(5,6),(5,7),(5,8),(6,7),(6,8),(7,8)],9)
 => ?
 => ? = 1 - 1
11101001 => [1,1,1,2,3,1] => ([(0,8),(1,8),(2,7),(2,8),(3,4),(3,5),(3,6),(3,7),(3,8),(4,5),(4,6),(4,7),(4,8),(5,6),(5,7),(5,8),(6,7),(6,8),(7,8)],9)
 => ?
 => ? = 1 - 1
11110001 => [1,1,1,1,4,1] => ([(0,8),(1,8),(2,8),(3,4),(3,5),(3,6),(3,7),(3,8),(4,5),(4,6),(4,7),(4,8),(5,6),(5,7),(5,8),(6,7),(6,8),(7,8)],9)
 => ?
 => ? = 1 - 1
000000001 => [9,1] => ([(0,9),(1,9),(2,9),(3,9),(4,9),(5,9),(6,9),(7,9),(8,9)],10)
 => ?
 => ? = 1 - 1
000000011 => [8,1,1] => ([(0,8),(0,9),(1,8),(1,9),(2,8),(2,9),(3,8),(3,9),(4,8),(4,9),(5,8),(5,9),(6,8),(6,9),(7,8),(7,9),(8,9)],10)
 => ?
 => ? = 1 - 1
000010000 => [5,5] => ([(4,9),(5,9),(6,9),(7,9),(8,9)],10)
 => ?
 => ? = 5 - 1
000100011 => [4,4,1,1] => ([(0,8),(0,9),(1,8),(1,9),(2,8),(2,9),(3,7),(3,8),(3,9),(4,7),(4,8),(4,9),(5,7),(5,8),(5,9),(6,7),(6,8),(6,9),(7,8),(7,9),(8,9)],10)
 => ?
 => ? = 1 - 1
000100110 => [4,3,1,2] => ([(1,8),(1,9),(2,8),(2,9),(3,7),(3,8),(3,9),(4,7),(4,8),(4,9),(5,7),(5,8),(5,9),(6,7),(6,8),(6,9),(7,8),(7,9),(8,9)],10)
 => ?
 => ? = 2 - 1
000101100 => [4,2,1,3] => ([(2,8),(2,9),(3,7),(3,8),(3,9),(4,7),(4,8),(4,9),(5,7),(5,8),(5,9),(6,7),(6,8),(6,9),(7,8),(7,9),(8,9)],10)
 => ?
 => ? = 3 - 1
000111000 => [4,1,1,4] => ([(3,7),(3,8),(3,9),(4,7),(4,8),(4,9),(5,7),(5,8),(5,9),(6,7),(6,8),(6,9),(7,8),(7,9),(8,9)],10)
 => ?
 => ? = 4 - 1
001001010 => [3,3,2,2] => ([(1,9),(2,8),(2,9),(3,8),(3,9),(4,7),(4,8),(4,9),(5,7),(5,8),(5,9),(6,7),(6,8),(6,9),(7,8),(7,9),(8,9)],10)
 => ?
 => ? = 2 - 1
Description
The size of a partition minus its first part. This is the number of boxes in its diagram that are not in the first row.
Matching statistic: St000678
Mapping
Mp00105: Binary words —complement⟶ Binary words
Mp00178: Binary words —to composition⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St000678: Dyck paths ⟶ ℤResult quality: 53% ●values known / values provided: 53%●distinct values known / distinct values provided: 70%
Values
0 => 1 => [1,1] => [1,0,1,0]
 => 2
1 => 0 => [2] => [1,1,0,0]
 => 1
00 => 11 => [1,1,1] => [1,0,1,0,1,0]
 => 3
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]
 => 1
000 => 111 => [1,1,1,1] => [1,0,1,0,1,0,1,0]
 => 4
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]
 => 2
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]
 => 3
101 => 010 => [2,2] => [1,1,0,0,1,1,0,0]
 => 1
110 => 001 => [3,1] => [1,1,1,0,0,0,1,0]
 => 2
111 => 000 => [4] => [1,1,1,1,0,0,0,0]
 => 1
0000 => 1111 => [1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
 => 5
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]
 => 2
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]
 => 3
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]
 => 2
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]
 => 4
1001 => 0110 => [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
 => 1
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]
 => 1
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]
 => 1
1110 => 0001 => [4,1] => [1,1,1,1,0,0,0,0,1,0]
 => 2
1111 => 0000 => [5] => [1,1,1,1,1,0,0,0,0,0]
 => 1
00000 => 11111 => [1,1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0,1,0]
 => 6
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]
 => 2
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]
 => 3
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]
 => 2
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]
 => 4
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]
 => 2
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]
 => 3
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]
 => 2
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]
 => 5
10001 => 01110 => [2,1,1,2] => [1,1,0,0,1,0,1,0,1,1,0,0]
 => 1
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]
 => 1
1100001 => 0011110 => [3,1,1,1,2] => [1,1,1,0,0,0,1,0,1,0,1,0,1,1,0,0]
 => ? = 1
1100100 => 0011011 => [3,1,2,1,1] => [1,1,1,0,0,0,1,0,1,1,0,0,1,0,1,0]
 => ? = 3
1100101 => 0011010 => [3,1,2,2] => [1,1,1,0,0,0,1,0,1,1,0,0,1,1,0,0]
 => ? = 1
1100110 => 0011001 => [3,1,3,1] => [1,1,1,0,0,0,1,0,1,1,1,0,0,0,1,0]
 => ? = 2
1100111 => 0011000 => [3,1,4] => [1,1,1,0,0,0,1,0,1,1,1,1,0,0,0,0]
 => ? = 1
1101001 => 0010110 => [3,2,1,2] => [1,1,1,0,0,0,1,1,0,0,1,0,1,1,0,0]
 => ? = 1
1101010 => 0010101 => [3,2,2,1] => [1,1,1,0,0,0,1,1,0,0,1,1,0,0,1,0]
 => ? = 2
1101011 => 0010100 => [3,2,3] => [1,1,1,0,0,0,1,1,0,0,1,1,1,0,0,0]
 => ? = 1
1101101 => 0010010 => [3,3,2] => [1,1,1,0,0,0,1,1,1,0,0,0,1,1,0,0]
 => ? = 1
1101110 => 0010001 => [3,4,1] => [1,1,1,0,0,0,1,1,1,1,0,0,0,0,1,0]
 => ? = 2
1101111 => 0010000 => [3,5] => [1,1,1,0,0,0,1,1,1,1,1,0,0,0,0,0]
 => ? = 1
1110010 => 0001101 => [4,1,2,1] => [1,1,1,1,0,0,0,0,1,0,1,1,0,0,1,0]
 => ? = 2
1110011 => 0001100 => [4,1,3] => [1,1,1,1,0,0,0,0,1,0,1,1,1,0,0,0]
 => ? = 1
1110101 => 0001010 => [4,2,2] => [1,1,1,1,0,0,0,0,1,1,0,0,1,1,0,0]
 => ? = 1
1110110 => 0001001 => [4,3,1] => [1,1,1,1,0,0,0,0,1,1,1,0,0,0,1,0]
 => ? = 2
1110111 => 0001000 => [4,4] => [1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0]
 => ? = 1
1111001 => 0000110 => [5,1,2] => [1,1,1,1,1,0,0,0,0,0,1,0,1,1,0,0]
 => ? = 1
1111010 => 0000101 => [5,2,1] => [1,1,1,1,1,0,0,0,0,0,1,1,0,0,1,0]
 => ? = 2
1111011 => 0000100 => [5,3] => [1,1,1,1,1,0,0,0,0,0,1,1,1,0,0,0]
 => ? = 1
1111101 => 0000010 => [6,2] => [1,1,1,1,1,1,0,0,0,0,0,0,1,1,0,0]
 => ? = 1
1111111 => 0000000 => [8] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
 => ? = 1
00000001 => 11111110 => [1,1,1,1,1,1,1,2] => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => ? = 1
00000011 => 11111100 => [1,1,1,1,1,1,3] => [1,0,1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,0]
 => ? = 1
00000101 => 11111010 => [1,1,1,1,1,2,2] => [1,0,1,0,1,0,1,0,1,0,1,1,0,0,1,1,0,0]
 => ? = 1
00110001 => 11001110 => [1,1,3,1,1,2] => [1,0,1,0,1,1,1,0,0,0,1,0,1,0,1,1,0,0]
 => ? = 1
01010001 => 10101110 => [1,2,2,1,1,2] => [1,0,1,1,0,0,1,1,0,0,1,0,1,0,1,1,0,0]
 => ? = 1
01100001 => 10011110 => [1,3,1,1,1,2] => [1,0,1,1,1,0,0,0,1,0,1,0,1,0,1,1,0,0]
 => ? = 1
10000000 => 01111111 => [2,1,1,1,1,1,1,1] => [1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => ? = 8
10000001 => 01111110 => [2,1,1,1,1,1,2] => [1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => ? = 1
10000010 => 01111101 => [2,1,1,1,1,2,1] => [1,1,0,0,1,0,1,0,1,0,1,0,1,1,0,0,1,0]
 => ? = 2
10000011 => 01111100 => [2,1,1,1,1,3] => [1,1,0,0,1,0,1,0,1,0,1,0,1,1,1,0,0,0]
 => ? = 1
10000101 => 01111010 => [2,1,1,1,2,2] => [1,1,0,0,1,0,1,0,1,0,1,1,0,0,1,1,0,0]
 => ? = 1
10000110 => 01111001 => [2,1,1,1,3,1] => [1,1,0,0,1,0,1,0,1,0,1,1,1,0,0,0,1,0]
 => ? = 2
10001001 => 01110110 => [2,1,1,2,1,2] => [1,1,0,0,1,0,1,0,1,1,0,0,1,0,1,1,0,0]
 => ? = 1
10001010 => 01110101 => [2,1,1,2,2,1] => [1,1,0,0,1,0,1,0,1,1,0,0,1,1,0,0,1,0]
 => ? = 2
10001100 => 01110011 => [2,1,1,3,1,1] => [1,1,0,0,1,0,1,0,1,1,1,0,0,0,1,0,1,0]
 => ? = 3
10001111 => 01110000 => [2,1,1,5] => [1,1,0,0,1,0,1,0,1,1,1,1,1,0,0,0,0,0]
 => ? = 1
10010001 => 01101110 => [2,1,2,1,1,2] => [1,1,0,0,1,0,1,1,0,0,1,0,1,0,1,1,0,0]
 => ? = 1
10010111 => 01101000 => [2,1,2,4] => [1,1,0,0,1,0,1,1,0,0,1,1,1,1,0,0,0,0]
 => ? = 1
10011011 => 01100100 => [2,1,3,3] => [1,1,0,0,1,0,1,1,1,0,0,0,1,1,1,0,0,0]
 => ? = 1
10011101 => 01100010 => [2,1,4,2] => [1,1,0,0,1,0,1,1,1,1,0,0,0,0,1,1,0,0]
 => ? = 1
10100001 => 01011110 => [2,2,1,1,1,2] => [1,1,0,0,1,1,0,0,1,0,1,0,1,0,1,1,0,0]
 => ? = 1
10101101 => 01010010 => [2,2,3,2] => [1,1,0,0,1,1,0,0,1,1,1,0,0,0,1,1,0,0]
 => ? = 1
10110101 => 01001010 => [2,3,2,2] => [1,1,0,0,1,1,1,0,0,0,1,1,0,0,1,1,0,0]
 => ? = 1
10111001 => 01000110 => [2,4,1,2] => [1,1,0,0,1,1,1,1,0,0,0,0,1,0,1,1,0,0]
 => ? = 1
11000001 => 00111110 => [3,1,1,1,1,2] => [1,1,1,0,0,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => ? = 1
11011001 => 00100110 => [3,3,1,2] => [1,1,1,0,0,0,1,1,1,0,0,0,1,0,1,1,0,0]
 => ? = 1
11101001 => 00010110 => [4,2,1,2] => [1,1,1,1,0,0,0,0,1,1,0,0,1,0,1,1,0,0]
 => ? = 1
11110001 => 00001110 => [5,1,1,2] => [1,1,1,1,1,0,0,0,0,0,1,0,1,0,1,1,0,0]
 => ? = 1
000000001 => 111111110 => [1,1,1,1,1,1,1,1,2] => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => ? = 1
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
Mp00101: Dyck paths —decomposition reverse⟶ Dyck paths
St000617: Dyck paths ⟶ ℤResult quality: 41% ●values known / values provided: 41%●distinct values known / distinct values provided: 100%
Values
0 => [2] => [1,1,0,0]
 => [1,0,1,0]
 => 2
1 => [1,1] => [1,0,1,0]
 => [1,1,0,0]
 => 1
00 => [3] => [1,1,1,0,0,0]
 => [1,0,1,0,1,0]
 => 3
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]
 => [1,1,1,0,0,0]
 => 1
000 => [4] => [1,1,1,1,0,0,0,0]
 => [1,0,1,0,1,0,1,0]
 => 4
001 => [3,1] => [1,1,1,0,0,0,1,0]
 => [1,1,0,0,1,0,1,0]
 => 1
010 => [2,2] => [1,1,0,0,1,1,0,0]
 => [1,1,0,1,0,0,1,0]
 => 2
011 => [2,1,1] => [1,1,0,0,1,0,1,0]
 => [1,1,1,0,0,0,1,0]
 => 1
100 => [1,3] => [1,0,1,1,1,0,0,0]
 => [1,1,0,1,0,1,0,0]
 => 3
101 => [1,2,1] => [1,0,1,1,0,0,1,0]
 => [1,1,1,0,0,1,0,0]
 => 1
110 => [1,1,2] => [1,0,1,0,1,1,0,0]
 => [1,1,1,0,1,0,0,0]
 => 2
111 => [1,1,1,1] => [1,0,1,0,1,0,1,0]
 => [1,1,1,1,0,0,0,0]
 => 1
0000 => [5] => [1,1,1,1,1,0,0,0,0,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => 5
0001 => [4,1] => [1,1,1,1,0,0,0,0,1,0]
 => [1,1,0,0,1,0,1,0,1,0]
 => 1
0010 => [3,2] => [1,1,1,0,0,0,1,1,0,0]
 => [1,1,0,1,0,0,1,0,1,0]
 => 2
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]
 => [1,1,0,1,0,1,0,0,1,0]
 => 3
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]
 => [1,1,1,0,1,0,0,0,1,0]
 => 2
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]
 => 1
1000 => [1,4] => [1,0,1,1,1,1,0,0,0,0]
 => [1,1,0,1,0,1,0,1,0,0]
 => 4
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
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]
 => [1,1,1,1,0,0,0,1,0,0]
 => 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]
 => 3
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
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]
 => 2
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
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
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]
 => 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]
 => 2
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]
 => 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
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]
 => 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]
 => 2
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]
 => 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]
 => 4
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]
 => 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
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]
 => 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]
 => 3
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]
 => 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
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]
 => 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]
 => 5
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
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]
 => 2
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
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]
 => ? = 7
000111 => [4,1,1,1] => [1,1,1,1,0,0,0,0,1,0,1,0,1,0]
 => [1,1,1,1,0,0,0,0,1,0,1,0,1,0]
 => ? = 1
001010 => [3,2,2] => [1,1,1,0,0,0,1,1,0,0,1,1,0,0]
 => [1,1,1,0,1,0,0,1,0,0,1,0,1,0]
 => ? = 2
001011 => [3,2,1,1] => [1,1,1,0,0,0,1,1,0,0,1,0,1,0]
 => [1,1,1,1,0,0,0,1,0,0,1,0,1,0]
 => ? = 1
001101 => [3,1,2,1] => [1,1,1,0,0,0,1,0,1,1,0,0,1,0]
 => [1,1,1,1,0,0,1,0,0,0,1,0,1,0]
 => ? = 1
001110 => [3,1,1,2] => [1,1,1,0,0,0,1,0,1,0,1,1,0,0]
 => [1,1,1,1,0,1,0,0,0,0,1,0,1,0]
 => ? = 2
001111 => [3,1,1,1,1] => [1,1,1,0,0,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,0,0,0,0,0,1,0,1,0]
 => ? = 1
010010 => [2,3,2] => [1,1,0,0,1,1,1,0,0,0,1,1,0,0]
 => [1,1,1,0,1,0,0,1,0,1,0,0,1,0]
 => ? = 2
010011 => [2,3,1,1] => [1,1,0,0,1,1,1,0,0,0,1,0,1,0]
 => [1,1,1,1,0,0,0,1,0,1,0,0,1,0]
 => ? = 1
010100 => [2,2,3] => [1,1,0,0,1,1,0,0,1,1,1,0,0,0]
 => [1,1,1,0,1,0,1,0,0,1,0,0,1,0]
 => ? = 3
010110 => [2,2,1,2] => [1,1,0,0,1,1,0,0,1,0,1,1,0,0]
 => [1,1,1,1,0,1,0,0,0,1,0,0,1,0]
 => ? = 2
011001 => [2,1,3,1] => [1,1,0,0,1,0,1,1,1,0,0,0,1,0]
 => [1,1,1,1,0,0,1,0,1,0,0,0,1,0]
 => ? = 1
100011 => [1,4,1,1] => [1,0,1,1,1,1,0,0,0,0,1,0,1,0]
 => [1,1,1,1,0,0,0,1,0,1,0,1,0,0]
 => ? = 1
100101 => [1,3,2,1] => [1,0,1,1,1,0,0,0,1,1,0,0,1,0]
 => [1,1,1,1,0,0,1,0,0,1,0,1,0,0]
 => ? = 1
100110 => [1,3,1,2] => [1,0,1,1,1,0,0,0,1,0,1,1,0,0]
 => [1,1,1,1,0,1,0,0,0,1,0,1,0,0]
 => ? = 2
110010 => [1,1,3,2] => [1,0,1,0,1,1,1,0,0,0,1,1,0,0]
 => [1,1,1,1,0,1,0,0,1,0,1,0,0,0]
 => ? = 2
110011 => [1,1,3,1,1] => [1,0,1,0,1,1,1,0,0,0,1,0,1,0]
 => [1,1,1,1,1,0,0,0,1,0,1,0,0,0]
 => ? = 1
110100 => [1,1,2,3] => [1,0,1,0,1,1,0,0,1,1,1,0,0,0]
 => [1,1,1,1,0,1,0,1,0,0,1,0,0,0]
 => ? = 3
111001 => [1,1,1,3,1] => [1,0,1,0,1,0,1,1,1,0,0,0,1,0]
 => [1,1,1,1,1,0,0,1,0,1,0,0,0,0]
 => ? = 1
0000101 => [5,2,1] => [1,1,1,1,1,0,0,0,0,0,1,1,0,0,1,0]
 => [1,1,1,0,0,1,0,0,1,0,1,0,1,0,1,0]
 => ? = 1
0000111 => [5,1,1,1] => [1,1,1,1,1,0,0,0,0,0,1,0,1,0,1,0]
 => [1,1,1,1,0,0,0,0,1,0,1,0,1,0,1,0]
 => ? = 1
0001001 => [4,3,1] => [1,1,1,1,0,0,0,0,1,1,1,0,0,0,1,0]
 => [1,1,1,0,0,1,0,1,0,0,1,0,1,0,1,0]
 => ? = 1
0001101 => [4,1,2,1] => [1,1,1,1,0,0,0,0,1,0,1,1,0,0,1,0]
 => [1,1,1,1,0,0,1,0,0,0,1,0,1,0,1,0]
 => ? = 1
0010010 => [3,3,2] => [1,1,1,0,0,0,1,1,1,0,0,0,1,1,0,0]
 => [1,1,1,0,1,0,0,1,0,1,0,0,1,0,1,0]
 => ? = 2
0010011 => [3,3,1,1] => [1,1,1,0,0,0,1,1,1,0,0,0,1,0,1,0]
 => [1,1,1,1,0,0,0,1,0,1,0,0,1,0,1,0]
 => ? = 1
0010100 => [3,2,3] => [1,1,1,0,0,0,1,1,0,0,1,1,1,0,0,0]
 => [1,1,1,0,1,0,1,0,0,1,0,0,1,0,1,0]
 => ? = 3
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]
 => ? = 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
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]
 => ? = 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
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]
 => ? = 3
0100010 => [2,4,2] => [1,1,0,0,1,1,1,1,0,0,0,0,1,1,0,0]
 => [1,1,1,0,1,0,0,1,0,1,0,1,0,0,1,0]
 => ? = 2
0100100 => [2,3,3] => [1,1,0,0,1,1,1,0,0,0,1,1,1,0,0,0]
 => [1,1,1,0,1,0,1,0,0,1,0,1,0,0,1,0]
 => ? = 3
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
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]
 => ? = 2
0100111 => [2,3,1,1,1] => [1,1,0,0,1,1,1,0,0,0,1,0,1,0,1,0]
 => [1,1,1,1,1,0,0,0,0,1,0,1,0,0,1,0]
 => ? = 1
0101001 => [2,2,3,1] => [1,1,0,0,1,1,0,0,1,1,1,0,0,0,1,0]
 => [1,1,1,1,0,0,1,0,1,0,0,1,0,0,1,0]
 => ? = 1
0101011 => [2,2,2,1,1] => [1,1,0,0,1,1,0,0,1,1,0,0,1,0,1,0]
 => [1,1,1,1,1,0,0,0,1,0,0,1,0,0,1,0]
 => ? = 1
0101100 => [2,2,1,3] => [1,1,0,0,1,1,0,0,1,0,1,1,1,0,0,0]
 => [1,1,1,1,0,1,0,1,0,0,0,1,0,0,1,0]
 => ? = 3
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
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]
 => ? = 2
0101111 => [2,2,1,1,1,1] => [1,1,0,0,1,1,0,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,1,0,0,0,0,0,1,0,0,1,0]
 => ? = 1
0110001 => [2,1,4,1] => [1,1,0,0,1,0,1,1,1,1,0,0,0,0,1,0]
 => [1,1,1,1,0,0,1,0,1,0,1,0,0,0,1,0]
 => ? = 1
0110010 => [2,1,3,2] => [1,1,0,0,1,0,1,1,1,0,0,0,1,1,0,0]
 => [1,1,1,1,0,1,0,0,1,0,1,0,0,0,1,0]
 => ? = 2
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
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]
 => ? = 3
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
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]
 => ? = 2
0110111 => [2,1,2,1,1,1] => [1,1,0,0,1,0,1,1,0,0,1,0,1,0,1,0]
 => [1,1,1,1,1,1,0,0,0,0,1,0,0,0,1,0]
 => ? = 1
0111001 => [2,1,1,3,1] => [1,1,0,0,1,0,1,0,1,1,1,0,0,0,1,0]
 => [1,1,1,1,1,0,0,1,0,1,0,0,0,0,1,0]
 => ? = 1
Description
The number of global maxima of a Dyck path.
The following 31 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St001733The number of weak left to right maxima of a Dyck path. 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. St000544The cop number of a graph. St000916The packing number of a graph. St001829The common independence number of a graph. St000363The number of minimal vertex covers of a graph. St001322The size of a minimal independent dominating set in a graph. St001339The irredundance number of a graph. St001363The Euler characteristic of a graph according to Knill. St001316The domatic number of a graph. St000007The number of saliances of the permutation. St000989The number of final rises of a permutation. St000838The number of terminal right-hand endpoints when the vertices are written in order. St000287The number of connected components of a graph. St001184Number of indecomposable injective modules with grade at least 1 in the corresponding Nakayama algebra. St000260The radius of a connected graph. St000286The number of connected components of the complement of a graph. St000314The number of left-to-right-maxima of a permutation. St000553The number of blocks of a graph. St000654The first descent of a permutation. St001201The grade of the simple module $S_0$ in the special CNakayama algebra corresponding to the Dyck path. St001481The minimal height of a peak of a Dyck path. St000090The variation of a composition. St000261The edge connectivity of a graph. St000262The vertex connectivity of a graph. St000310The minimal degree of a vertex of a graph. St001185The number of indecomposable injective modules of grade at least 2 in the corresponding Nakayama algebra. St000993The multiplicity of the largest part of an integer partition.