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Your data matches 43 different statistics following compositions of up to 3 maps.
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Matching statistic: St000291
(load all 11 compositions to match this statistic)
(load all 11 compositions to match this statistic)
St000291: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
0 => 0
1 => 0
00 => 0
01 => 0
10 => 1
11 => 0
000 => 0
001 => 0
010 => 1
011 => 0
100 => 1
101 => 1
110 => 1
111 => 0
0000 => 0
0001 => 0
0010 => 1
0011 => 0
0100 => 1
0101 => 1
0110 => 1
0111 => 0
1000 => 1
1001 => 1
1010 => 2
1011 => 1
1100 => 1
1101 => 1
1110 => 1
1111 => 0
00000 => 0
00001 => 0
00010 => 1
00011 => 0
00100 => 1
00101 => 1
00110 => 1
00111 => 0
01000 => 1
01001 => 1
01010 => 2
01011 => 1
01100 => 1
01101 => 1
01110 => 1
01111 => 0
10000 => 1
10001 => 1
10010 => 2
10011 => 1
Description
The number of descents of a binary word.
Matching statistic: St000292
(load all 13 compositions to match this statistic)
(load all 13 compositions to match this statistic)
Mp00104: Binary words —reverse⟶ Binary words
St000292: Binary words ⟶ ℤResult quality: 80% ●values known / values provided: 80%●distinct values known / distinct values provided: 100%
St000292: Binary words ⟶ ℤResult quality: 80% ●values known / values provided: 80%●distinct values known / distinct values provided: 100%
Values
0 => 0 => 0
1 => 1 => 0
00 => 00 => 0
01 => 10 => 0
10 => 01 => 1
11 => 11 => 0
000 => 000 => 0
001 => 100 => 0
010 => 010 => 1
011 => 110 => 0
100 => 001 => 1
101 => 101 => 1
110 => 011 => 1
111 => 111 => 0
0000 => 0000 => 0
0001 => 1000 => 0
0010 => 0100 => 1
0011 => 1100 => 0
0100 => 0010 => 1
0101 => 1010 => 1
0110 => 0110 => 1
0111 => 1110 => 0
1000 => 0001 => 1
1001 => 1001 => 1
1010 => 0101 => 2
1011 => 1101 => 1
1100 => 0011 => 1
1101 => 1011 => 1
1110 => 0111 => 1
1111 => 1111 => 0
00000 => 00000 => 0
00001 => 10000 => 0
00010 => 01000 => 1
00011 => 11000 => 0
00100 => 00100 => 1
00101 => 10100 => 1
00110 => 01100 => 1
00111 => 11100 => 0
01000 => 00010 => 1
01001 => 10010 => 1
01010 => 01010 => 2
01011 => 11010 => 1
01100 => 00110 => 1
01101 => 10110 => 1
01110 => 01110 => 1
01111 => 11110 => 0
10000 => 00001 => 1
10001 => 10001 => 1
10010 => 01001 => 2
10011 => 11001 => 1
10110111100000 => 00000111101101 => ? = 3
10111011010000 => 00001011011101 => ? = 4
10111011100000 => 00000111011101 => ? = 3
10111100110000 => 00001100111101 => ? = 3
10111101001000 => 00010010111101 => ? = 4
10111101010000 => 00001010111101 => ? = 4
10111101100000 => 00000110111101 => ? = 3
10111110000100 => 00100001111101 => ? = 3
10111110001000 => 00010001111101 => ? = 3
10111110010000 => 00001001111101 => ? = 3
10111110100000 => 00000101111101 => ? = 3
10111111000000 => 00000011111101 => ? = 2
11001111100000 => 00000111110011 => ? = 2
11011110000010 => 01000001111011 => ? = 3
11101101000010 => 01000010110111 => ? = 4
11101110000010 => 01000001110111 => ? = 3
11110011000010 => 01000011001111 => ? = 3
11110100100010 => 01000100101111 => ? = 4
11110101000010 => 01000010101111 => ? = 4
11110110000010 => 01000001101111 => ? = 3
11111000001100 => 00110000011111 => ? = 2
11111000010010 => 01001000011111 => ? = 3
11111000100010 => 01000100011111 => ? = 3
11111001000010 => 01000010011111 => ? = 3
11111010000010 => 01000001011111 => ? = 3
11111100000010 => 01000000111111 => ? = 2
1011110111000000 => 0000001110111101 => ? = 3
1011111010100000 => 0000010101111101 => ? = 4
1011111011000000 => 0000001101111101 => ? = 3
1011111100010000 => 0000100011111101 => ? = 3
1011111100100000 => 0000010011111101 => ? = 3
1011111101000000 => 0000001011111101 => ? = 3
1011111110000000 => 0000000111111101 => ? = 2
1111011100000010 => 0100000011101111 => ? = 3
1111101010000010 => 0100000101011111 => ? = 4
1111101100000010 => 0100000011011111 => ? = 3
1111110001000010 => 0100001000111111 => ? = 3
1111110010000010 => 0100000100111111 => ? = 3
1111110100000010 => 0100000010111111 => ? = 3
1111111000000010 => 0100000001111111 => ? = 2
1111111101 => 1011111111 => ? = 1
1111000001 => 1000001111 => ? = 1
1110111110 => 0111110111 => ? = 2
1110100001 => 1000010111 => ? = 2
1110011110 => 0111100111 => ? = 2
1110010001 => 1000100111 => ? = 2
1110001001 => 1001000111 => ? = 2
1110000101 => 1010000111 => ? = 2
1101100001 => 1000011011 => ? = 2
1101010001 => 1000101011 => ? = 3
Description
The number of ascents of a binary word.
Matching statistic: St000390
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Mp00104: Binary words —reverse⟶ Binary words
Mp00178: Binary words —to composition⟶ Integer compositions
Mp00094: Integer compositions —to binary word⟶ Binary words
St000390: Binary words ⟶ ℤResult quality: 29% ●values known / values provided: 29%●distinct values known / distinct values provided: 86%
Mp00178: Binary words —to composition⟶ Integer compositions
Mp00094: Integer compositions —to binary word⟶ Binary words
St000390: Binary words ⟶ ℤResult quality: 29% ●values known / values provided: 29%●distinct values known / distinct values provided: 86%
Values
0 => 0 => [2] => 10 => 1 = 0 + 1
1 => 1 => [1,1] => 11 => 1 = 0 + 1
00 => 00 => [3] => 100 => 1 = 0 + 1
01 => 10 => [1,2] => 110 => 1 = 0 + 1
10 => 01 => [2,1] => 101 => 2 = 1 + 1
11 => 11 => [1,1,1] => 111 => 1 = 0 + 1
000 => 000 => [4] => 1000 => 1 = 0 + 1
001 => 100 => [1,3] => 1100 => 1 = 0 + 1
010 => 010 => [2,2] => 1010 => 2 = 1 + 1
011 => 110 => [1,1,2] => 1110 => 1 = 0 + 1
100 => 001 => [3,1] => 1001 => 2 = 1 + 1
101 => 101 => [1,2,1] => 1101 => 2 = 1 + 1
110 => 011 => [2,1,1] => 1011 => 2 = 1 + 1
111 => 111 => [1,1,1,1] => 1111 => 1 = 0 + 1
0000 => 0000 => [5] => 10000 => 1 = 0 + 1
0001 => 1000 => [1,4] => 11000 => 1 = 0 + 1
0010 => 0100 => [2,3] => 10100 => 2 = 1 + 1
0011 => 1100 => [1,1,3] => 11100 => 1 = 0 + 1
0100 => 0010 => [3,2] => 10010 => 2 = 1 + 1
0101 => 1010 => [1,2,2] => 11010 => 2 = 1 + 1
0110 => 0110 => [2,1,2] => 10110 => 2 = 1 + 1
0111 => 1110 => [1,1,1,2] => 11110 => 1 = 0 + 1
1000 => 0001 => [4,1] => 10001 => 2 = 1 + 1
1001 => 1001 => [1,3,1] => 11001 => 2 = 1 + 1
1010 => 0101 => [2,2,1] => 10101 => 3 = 2 + 1
1011 => 1101 => [1,1,2,1] => 11101 => 2 = 1 + 1
1100 => 0011 => [3,1,1] => 10011 => 2 = 1 + 1
1101 => 1011 => [1,2,1,1] => 11011 => 2 = 1 + 1
1110 => 0111 => [2,1,1,1] => 10111 => 2 = 1 + 1
1111 => 1111 => [1,1,1,1,1] => 11111 => 1 = 0 + 1
00000 => 00000 => [6] => 100000 => 1 = 0 + 1
00001 => 10000 => [1,5] => 110000 => 1 = 0 + 1
00010 => 01000 => [2,4] => 101000 => 2 = 1 + 1
00011 => 11000 => [1,1,4] => 111000 => 1 = 0 + 1
00100 => 00100 => [3,3] => 100100 => 2 = 1 + 1
00101 => 10100 => [1,2,3] => 110100 => 2 = 1 + 1
00110 => 01100 => [2,1,3] => 101100 => 2 = 1 + 1
00111 => 11100 => [1,1,1,3] => 111100 => 1 = 0 + 1
01000 => 00010 => [4,2] => 100010 => 2 = 1 + 1
01001 => 10010 => [1,3,2] => 110010 => 2 = 1 + 1
01010 => 01010 => [2,2,2] => 101010 => 3 = 2 + 1
01011 => 11010 => [1,1,2,2] => 111010 => 2 = 1 + 1
01100 => 00110 => [3,1,2] => 100110 => 2 = 1 + 1
01101 => 10110 => [1,2,1,2] => 110110 => 2 = 1 + 1
01110 => 01110 => [2,1,1,2] => 101110 => 2 = 1 + 1
01111 => 11110 => [1,1,1,1,2] => 111110 => 1 = 0 + 1
10000 => 00001 => [5,1] => 100001 => 2 = 1 + 1
10001 => 10001 => [1,4,1] => 110001 => 2 = 1 + 1
10010 => 01001 => [2,3,1] => 101001 => 3 = 2 + 1
10011 => 11001 => [1,1,3,1] => 111001 => 2 = 1 + 1
000000001 => 100000000 => [1,9] => 1100000000 => ? = 0 + 1
000000010 => 010000000 => [2,8] => 1010000000 => ? = 1 + 1
000000011 => 110000000 => [1,1,8] => 1110000000 => ? = 0 + 1
000000100 => 001000000 => [3,7] => 1001000000 => ? = 1 + 1
000000101 => 101000000 => [1,2,7] => 1101000000 => ? = 1 + 1
000000110 => 011000000 => [2,1,7] => 1011000000 => ? = 1 + 1
000000111 => 111000000 => [1,1,1,7] => 1111000000 => ? = 0 + 1
000001000 => 000100000 => [4,6] => 1000100000 => ? = 1 + 1
000001001 => 100100000 => [1,3,6] => 1100100000 => ? = 1 + 1
000001010 => 010100000 => [2,2,6] => 1010100000 => ? = 2 + 1
000001011 => 110100000 => [1,1,2,6] => 1110100000 => ? = 1 + 1
000001100 => 001100000 => [3,1,6] => 1001100000 => ? = 1 + 1
000001101 => 101100000 => [1,2,1,6] => 1101100000 => ? = 1 + 1
000001110 => 011100000 => [2,1,1,6] => 1011100000 => ? = 1 + 1
000010000 => 000010000 => [5,5] => 1000010000 => ? = 1 + 1
000010001 => 100010000 => [1,4,5] => 1100010000 => ? = 1 + 1
000010010 => 010010000 => [2,3,5] => 1010010000 => ? = 2 + 1
000010011 => 110010000 => [1,1,3,5] => 1110010000 => ? = 1 + 1
000010100 => 001010000 => [3,2,5] => 1001010000 => ? = 2 + 1
000010101 => 101010000 => [1,2,2,5] => 1101010000 => ? = 2 + 1
000010110 => 011010000 => [2,1,2,5] => 1011010000 => ? = 2 + 1
000011000 => 000110000 => [4,1,5] => 1000110000 => ? = 1 + 1
000011001 => 100110000 => [1,3,1,5] => 1100110000 => ? = 1 + 1
000011010 => 010110000 => [2,2,1,5] => 1010110000 => ? = 2 + 1
000011100 => 001110000 => [3,1,1,5] => 1001110000 => ? = 1 + 1
000011111 => 111110000 => [1,1,1,1,1,5] => 1111110000 => ? = 0 + 1
000100000 => 000001000 => [6,4] => 1000001000 => ? = 1 + 1
000100010 => 010001000 => [2,4,4] => 1010001000 => ? = 2 + 1
000100011 => 110001000 => [1,1,4,4] => 1110001000 => ? = 1 + 1
000100100 => 001001000 => [3,3,4] => 1001001000 => ? = 2 + 1
000100101 => 101001000 => [1,2,3,4] => 1101001000 => ? = 2 + 1
000100110 => 011001000 => [2,1,3,4] => 1011001000 => ? = 2 + 1
000101000 => 000101000 => [4,2,4] => 1000101000 => ? = 2 + 1
000101001 => 100101000 => [1,3,2,4] => 1100101000 => ? = 2 + 1
000101010 => 010101000 => [2,2,2,4] => 1010101000 => ? = 3 + 1
000101100 => 001101000 => [3,1,2,4] => 1001101000 => ? = 2 + 1
000101111 => 111101000 => [1,1,1,1,2,4] => 1111101000 => ? = 1 + 1
000110000 => 000011000 => [5,1,4] => 1000011000 => ? = 1 + 1
000110001 => 100011000 => [1,4,1,4] => 1100011000 => ? = 1 + 1
000110010 => 010011000 => [2,3,1,4] => 1010011000 => ? = 2 + 1
000110100 => 001011000 => [3,2,1,4] => 1001011000 => ? = 2 + 1
000110111 => 111011000 => [1,1,1,2,1,4] => 1111011000 => ? = 1 + 1
000111000 => 000111000 => [4,1,1,4] => 1000111000 => ? = 1 + 1
000111011 => 110111000 => [1,1,2,1,1,4] => 1110111000 => ? = 1 + 1
000111101 => 101111000 => [1,2,1,1,1,4] => 1101111000 => ? = 1 + 1
000111110 => 011111000 => [2,1,1,1,1,4] => 1011111000 => ? = 1 + 1
000111111 => 111111000 => [1,1,1,1,1,1,4] => 1111111000 => ? = 0 + 1
001000001 => 100000100 => [1,6,3] => 1100000100 => ? = 1 + 1
001000010 => 010000100 => [2,5,3] => 1010000100 => ? = 2 + 1
001000011 => 110000100 => [1,1,5,3] => 1110000100 => ? = 1 + 1
Description
The number of runs of ones in a binary word.
Matching statistic: St000386
(load all 6 compositions to match this statistic)
(load all 6 compositions to match this statistic)
Mp00105: Binary words —complement⟶ Binary words
Mp00178: Binary words —to composition⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St000386: Dyck paths ⟶ ℤResult quality: 12% ●values known / values provided: 12%●distinct values known / distinct values provided: 57%
Mp00178: Binary words —to composition⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St000386: Dyck paths ⟶ ℤResult quality: 12% ●values known / values provided: 12%●distinct values known / distinct values provided: 57%
Values
0 => 1 => [1,1] => [1,0,1,0]
=> 0
1 => 0 => [2] => [1,1,0,0]
=> 0
00 => 11 => [1,1,1] => [1,0,1,0,1,0]
=> 0
01 => 10 => [1,2] => [1,0,1,1,0,0]
=> 0
10 => 01 => [2,1] => [1,1,0,0,1,0]
=> 1
11 => 00 => [3] => [1,1,1,0,0,0]
=> 0
000 => 111 => [1,1,1,1] => [1,0,1,0,1,0,1,0]
=> 0
001 => 110 => [1,1,2] => [1,0,1,0,1,1,0,0]
=> 0
010 => 101 => [1,2,1] => [1,0,1,1,0,0,1,0]
=> 1
011 => 100 => [1,3] => [1,0,1,1,1,0,0,0]
=> 0
100 => 011 => [2,1,1] => [1,1,0,0,1,0,1,0]
=> 1
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]
=> 1
111 => 000 => [4] => [1,1,1,1,0,0,0,0]
=> 0
0000 => 1111 => [1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
=> 0
0001 => 1110 => [1,1,1,2] => [1,0,1,0,1,0,1,1,0,0]
=> 0
0010 => 1101 => [1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
=> 1
0011 => 1100 => [1,1,3] => [1,0,1,0,1,1,1,0,0,0]
=> 0
0100 => 1011 => [1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
=> 1
0101 => 1010 => [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
=> 1
0110 => 1001 => [1,3,1] => [1,0,1,1,1,0,0,0,1,0]
=> 1
0111 => 1000 => [1,4] => [1,0,1,1,1,1,0,0,0,0]
=> 0
1000 => 0111 => [2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
=> 1
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]
=> 1
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]
=> 1
1111 => 0000 => [5] => [1,1,1,1,1,0,0,0,0,0]
=> 0
00000 => 11111 => [1,1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0,1,0]
=> 0
00001 => 11110 => [1,1,1,1,2] => [1,0,1,0,1,0,1,0,1,1,0,0]
=> 0
00010 => 11101 => [1,1,1,2,1] => [1,0,1,0,1,0,1,1,0,0,1,0]
=> 1
00011 => 11100 => [1,1,1,3] => [1,0,1,0,1,0,1,1,1,0,0,0]
=> 0
00100 => 11011 => [1,1,2,1,1] => [1,0,1,0,1,1,0,0,1,0,1,0]
=> 1
00101 => 11010 => [1,1,2,2] => [1,0,1,0,1,1,0,0,1,1,0,0]
=> 1
00110 => 11001 => [1,1,3,1] => [1,0,1,0,1,1,1,0,0,0,1,0]
=> 1
00111 => 11000 => [1,1,4] => [1,0,1,0,1,1,1,1,0,0,0,0]
=> 0
01000 => 10111 => [1,2,1,1,1] => [1,0,1,1,0,0,1,0,1,0,1,0]
=> 1
01001 => 10110 => [1,2,1,2] => [1,0,1,1,0,0,1,0,1,1,0,0]
=> 1
01010 => 10101 => [1,2,2,1] => [1,0,1,1,0,0,1,1,0,0,1,0]
=> 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]
=> 1
01101 => 10010 => [1,3,2] => [1,0,1,1,1,0,0,0,1,1,0,0]
=> 1
01110 => 10001 => [1,4,1] => [1,0,1,1,1,1,0,0,0,0,1,0]
=> 1
01111 => 10000 => [1,5] => [1,0,1,1,1,1,1,0,0,0,0,0]
=> 0
10000 => 01111 => [2,1,1,1,1] => [1,1,0,0,1,0,1,0,1,0,1,0]
=> 1
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
1100010 => 0011101 => [3,1,1,2,1] => [1,1,1,0,0,0,1,0,1,0,1,1,0,0,1,0]
=> ? = 2
1100011 => 0011100 => [3,1,1,3] => [1,1,1,0,0,0,1,0,1,0,1,1,1,0,0,0]
=> ? = 1
1100100 => 0011011 => [3,1,2,1,1] => [1,1,1,0,0,0,1,0,1,1,0,0,1,0,1,0]
=> ? = 2
1100101 => 0011010 => [3,1,2,2] => [1,1,1,0,0,0,1,0,1,1,0,0,1,1,0,0]
=> ? = 2
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
1101000 => 0010111 => [3,2,1,1,1] => [1,1,1,0,0,0,1,1,0,0,1,0,1,0,1,0]
=> ? = 2
1101001 => 0010110 => [3,2,1,2] => [1,1,1,0,0,0,1,1,0,0,1,0,1,1,0,0]
=> ? = 2
1101010 => 0010101 => [3,2,2,1] => [1,1,1,0,0,0,1,1,0,0,1,1,0,0,1,0]
=> ? = 3
1101011 => 0010100 => [3,2,3] => [1,1,1,0,0,0,1,1,0,0,1,1,1,0,0,0]
=> ? = 2
1101100 => 0010011 => [3,3,1,1] => [1,1,1,0,0,0,1,1,1,0,0,0,1,0,1,0]
=> ? = 2
1101101 => 0010010 => [3,3,2] => [1,1,1,0,0,0,1,1,1,0,0,0,1,1,0,0]
=> ? = 2
1101111 => 0010000 => [3,5] => [1,1,1,0,0,0,1,1,1,1,1,0,0,0,0,0]
=> ? = 1
1110000 => 0001111 => [4,1,1,1,1] => [1,1,1,1,0,0,0,0,1,0,1,0,1,0,1,0]
=> ? = 1
1110001 => 0001110 => [4,1,1,2] => [1,1,1,1,0,0,0,0,1,0,1,0,1,1,0,0]
=> ? = 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
1110100 => 0001011 => [4,2,1,1] => [1,1,1,1,0,0,0,0,1,1,0,0,1,0,1,0]
=> ? = 2
1110110 => 0001001 => [4,3,1] => [1,1,1,1,0,0,0,0,1,1,1,0,0,0,1,0]
=> ? = 2
1111000 => 0000111 => [5,1,1,1] => [1,1,1,1,1,0,0,0,0,0,1,0,1,0,1,0]
=> ? = 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
1111100 => 0000011 => [6,1,1] => [1,1,1,1,1,1,0,0,0,0,0,0,1,0,1,0]
=> ? = 1
00000000 => 11111111 => [1,1,1,1,1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 0
00000010 => 11111101 => [1,1,1,1,1,1,2,1] => [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0,1,0]
=> ? = 1
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]
=> ? = 0
00000100 => 11111011 => [1,1,1,1,1,2,1,1] => [1,0,1,0,1,0,1,0,1,0,1,1,0,0,1,0,1,0]
=> ? = 1
00000110 => 11111001 => [1,1,1,1,1,3,1] => [1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,0,1,0]
=> ? = 1
00000111 => 11111000 => [1,1,1,1,1,4] => [1,0,1,0,1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> ? = 0
00001000 => 11110111 => [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]
=> ? = 1
00001001 => 11110110 => [1,1,1,1,2,1,2] => [1,0,1,0,1,0,1,0,1,1,0,0,1,0,1,1,0,0]
=> ? = 1
00001010 => 11110101 => [1,1,1,1,2,2,1] => [1,0,1,0,1,0,1,0,1,1,0,0,1,1,0,0,1,0]
=> ? = 2
00001011 => 11110100 => [1,1,1,1,2,3] => [1,0,1,0,1,0,1,0,1,1,0,0,1,1,1,0,0,0]
=> ? = 1
00001100 => 11110011 => [1,1,1,1,3,1,1] => [1,0,1,0,1,0,1,0,1,1,1,0,0,0,1,0,1,0]
=> ? = 1
00001101 => 11110010 => [1,1,1,1,3,2] => [1,0,1,0,1,0,1,0,1,1,1,0,0,0,1,1,0,0]
=> ? = 1
00001110 => 11110001 => [1,1,1,1,4,1] => [1,0,1,0,1,0,1,0,1,1,1,1,0,0,0,0,1,0]
=> ? = 1
00001111 => 11110000 => [1,1,1,1,5] => [1,0,1,0,1,0,1,0,1,1,1,1,1,0,0,0,0,0]
=> ? = 0
00010000 => 11101111 => [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]
=> ? = 1
00010001 => 11101110 => [1,1,1,2,1,1,2] => [1,0,1,0,1,0,1,1,0,0,1,0,1,0,1,1,0,0]
=> ? = 1
00010010 => 11101101 => [1,1,1,2,1,2,1] => [1,0,1,0,1,0,1,1,0,0,1,0,1,1,0,0,1,0]
=> ? = 2
00010011 => 11101100 => [1,1,1,2,1,3] => [1,0,1,0,1,0,1,1,0,0,1,0,1,1,1,0,0,0]
=> ? = 1
00010100 => 11101011 => [1,1,1,2,2,1,1] => [1,0,1,0,1,0,1,1,0,0,1,1,0,0,1,0,1,0]
=> ? = 2
00010101 => 11101010 => [1,1,1,2,2,2] => [1,0,1,0,1,0,1,1,0,0,1,1,0,0,1,1,0,0]
=> ? = 2
00010110 => 11101001 => [1,1,1,2,3,1] => [1,0,1,0,1,0,1,1,0,0,1,1,1,0,0,0,1,0]
=> ? = 2
00010111 => 11101000 => [1,1,1,2,4] => [1,0,1,0,1,0,1,1,0,0,1,1,1,1,0,0,0,0]
=> ? = 1
00011000 => 11100111 => [1,1,1,3,1,1,1] => [1,0,1,0,1,0,1,1,1,0,0,0,1,0,1,0,1,0]
=> ? = 1
00011001 => 11100110 => [1,1,1,3,1,2] => [1,0,1,0,1,0,1,1,1,0,0,0,1,0,1,1,0,0]
=> ? = 1
00011010 => 11100101 => [1,1,1,3,2,1] => [1,0,1,0,1,0,1,1,1,0,0,0,1,1,0,0,1,0]
=> ? = 2
00011011 => 11100100 => [1,1,1,3,3] => [1,0,1,0,1,0,1,1,1,0,0,0,1,1,1,0,0,0]
=> ? = 1
Description
The number of factors DDU in a Dyck path.
Matching statistic: St001037
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00178: Binary words —to composition⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St001037: Dyck paths ⟶ ℤResult quality: 10% ●values known / values provided: 10%●distinct values known / distinct values provided: 57%
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St001037: Dyck paths ⟶ ℤResult quality: 10% ●values known / values provided: 10%●distinct values known / distinct values provided: 57%
Values
0 => [2] => [1,1,0,0]
=> 0
1 => [1,1] => [1,0,1,0]
=> 0
00 => [3] => [1,1,1,0,0,0]
=> 0
01 => [2,1] => [1,1,0,0,1,0]
=> 0
10 => [1,2] => [1,0,1,1,0,0]
=> 1
11 => [1,1,1] => [1,0,1,0,1,0]
=> 0
000 => [4] => [1,1,1,1,0,0,0,0]
=> 0
001 => [3,1] => [1,1,1,0,0,0,1,0]
=> 0
010 => [2,2] => [1,1,0,0,1,1,0,0]
=> 1
011 => [2,1,1] => [1,1,0,0,1,0,1,0]
=> 0
100 => [1,3] => [1,0,1,1,1,0,0,0]
=> 1
101 => [1,2,1] => [1,0,1,1,0,0,1,0]
=> 1
110 => [1,1,2] => [1,0,1,0,1,1,0,0]
=> 1
111 => [1,1,1,1] => [1,0,1,0,1,0,1,0]
=> 0
0000 => [5] => [1,1,1,1,1,0,0,0,0,0]
=> 0
0001 => [4,1] => [1,1,1,1,0,0,0,0,1,0]
=> 0
0010 => [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> 1
0011 => [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> 0
0100 => [2,3] => [1,1,0,0,1,1,1,0,0,0]
=> 1
0101 => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> 1
0110 => [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
=> 1
0111 => [2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
=> 0
1000 => [1,4] => [1,0,1,1,1,1,0,0,0,0]
=> 1
1001 => [1,3,1] => [1,0,1,1,1,0,0,0,1,0]
=> 1
1010 => [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
=> 2
1011 => [1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
=> 1
1100 => [1,1,3] => [1,0,1,0,1,1,1,0,0,0]
=> 1
1101 => [1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
=> 1
1110 => [1,1,1,2] => [1,0,1,0,1,0,1,1,0,0]
=> 1
1111 => [1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
=> 0
00000 => [6] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> 0
00001 => [5,1] => [1,1,1,1,1,0,0,0,0,0,1,0]
=> 0
00010 => [4,2] => [1,1,1,1,0,0,0,0,1,1,0,0]
=> 1
00011 => [4,1,1] => [1,1,1,1,0,0,0,0,1,0,1,0]
=> 0
00100 => [3,3] => [1,1,1,0,0,0,1,1,1,0,0,0]
=> 1
00101 => [3,2,1] => [1,1,1,0,0,0,1,1,0,0,1,0]
=> 1
00110 => [3,1,2] => [1,1,1,0,0,0,1,0,1,1,0,0]
=> 1
00111 => [3,1,1,1] => [1,1,1,0,0,0,1,0,1,0,1,0]
=> 0
01000 => [2,4] => [1,1,0,0,1,1,1,1,0,0,0,0]
=> 1
01001 => [2,3,1] => [1,1,0,0,1,1,1,0,0,0,1,0]
=> 1
01010 => [2,2,2] => [1,1,0,0,1,1,0,0,1,1,0,0]
=> 2
01011 => [2,2,1,1] => [1,1,0,0,1,1,0,0,1,0,1,0]
=> 1
01100 => [2,1,3] => [1,1,0,0,1,0,1,1,1,0,0,0]
=> 1
01101 => [2,1,2,1] => [1,1,0,0,1,0,1,1,0,0,1,0]
=> 1
01110 => [2,1,1,2] => [1,1,0,0,1,0,1,0,1,1,0,0]
=> 1
01111 => [2,1,1,1,1] => [1,1,0,0,1,0,1,0,1,0,1,0]
=> 0
10000 => [1,5] => [1,0,1,1,1,1,1,0,0,0,0,0]
=> 1
10001 => [1,4,1] => [1,0,1,1,1,1,0,0,0,0,1,0]
=> 1
10010 => [1,3,2] => [1,0,1,1,1,0,0,0,1,1,0,0]
=> 2
10011 => [1,3,1,1] => [1,0,1,1,1,0,0,0,1,0,1,0]
=> 1
0000000 => [8] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 0
0000001 => [7,1] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> ? = 0
0000010 => [6,2] => [1,1,1,1,1,1,0,0,0,0,0,0,1,1,0,0]
=> ? = 1
0000011 => [6,1,1] => [1,1,1,1,1,1,0,0,0,0,0,0,1,0,1,0]
=> ? = 0
0000100 => [5,3] => [1,1,1,1,1,0,0,0,0,0,1,1,1,0,0,0]
=> ? = 1
0000101 => [5,2,1] => [1,1,1,1,1,0,0,0,0,0,1,1,0,0,1,0]
=> ? = 1
0000110 => [5,1,2] => [1,1,1,1,1,0,0,0,0,0,1,0,1,1,0,0]
=> ? = 1
0000111 => [5,1,1,1] => [1,1,1,1,1,0,0,0,0,0,1,0,1,0,1,0]
=> ? = 0
0001000 => [4,4] => [1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0]
=> ? = 1
0001001 => [4,3,1] => [1,1,1,1,0,0,0,0,1,1,1,0,0,0,1,0]
=> ? = 1
0001010 => [4,2,2] => [1,1,1,1,0,0,0,0,1,1,0,0,1,1,0,0]
=> ? = 2
0001011 => [4,2,1,1] => [1,1,1,1,0,0,0,0,1,1,0,0,1,0,1,0]
=> ? = 1
0001100 => [4,1,3] => [1,1,1,1,0,0,0,0,1,0,1,1,1,0,0,0]
=> ? = 1
0001101 => [4,1,2,1] => [1,1,1,1,0,0,0,0,1,0,1,1,0,0,1,0]
=> ? = 1
0001110 => [4,1,1,2] => [1,1,1,1,0,0,0,0,1,0,1,0,1,1,0,0]
=> ? = 1
0001111 => [4,1,1,1,1] => [1,1,1,1,0,0,0,0,1,0,1,0,1,0,1,0]
=> ? = 0
0010000 => [3,5] => [1,1,1,0,0,0,1,1,1,1,1,0,0,0,0,0]
=> ? = 1
0010001 => [3,4,1] => [1,1,1,0,0,0,1,1,1,1,0,0,0,0,1,0]
=> ? = 1
0010010 => [3,3,2] => [1,1,1,0,0,0,1,1,1,0,0,0,1,1,0,0]
=> ? = 2
0010011 => [3,3,1,1] => [1,1,1,0,0,0,1,1,1,0,0,0,1,0,1,0]
=> ? = 1
0010100 => [3,2,3] => [1,1,1,0,0,0,1,1,0,0,1,1,1,0,0,0]
=> ? = 2
0010101 => [3,2,2,1] => [1,1,1,0,0,0,1,1,0,0,1,1,0,0,1,0]
=> ? = 2
0010110 => [3,2,1,2] => [1,1,1,0,0,0,1,1,0,0,1,0,1,1,0,0]
=> ? = 2
0010111 => [3,2,1,1,1] => [1,1,1,0,0,0,1,1,0,0,1,0,1,0,1,0]
=> ? = 1
0011000 => [3,1,4] => [1,1,1,0,0,0,1,0,1,1,1,1,0,0,0,0]
=> ? = 1
0011001 => [3,1,3,1] => [1,1,1,0,0,0,1,0,1,1,1,0,0,0,1,0]
=> ? = 1
0011010 => [3,1,2,2] => [1,1,1,0,0,0,1,0,1,1,0,0,1,1,0,0]
=> ? = 2
0011011 => [3,1,2,1,1] => [1,1,1,0,0,0,1,0,1,1,0,0,1,0,1,0]
=> ? = 1
0011100 => [3,1,1,3] => [1,1,1,0,0,0,1,0,1,0,1,1,1,0,0,0]
=> ? = 1
0011101 => [3,1,1,2,1] => [1,1,1,0,0,0,1,0,1,0,1,1,0,0,1,0]
=> ? = 1
0011110 => [3,1,1,1,2] => [1,1,1,0,0,0,1,0,1,0,1,0,1,1,0,0]
=> ? = 1
0011111 => [3,1,1,1,1,1] => [1,1,1,0,0,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 0
00000000 => [9] => [1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0]
=> ? = 0
00000001 => [8,1] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0]
=> ? = 0
00000010 => [7,2] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,1,0,0]
=> ? = 1
00000011 => [7,1,1] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0,1,0]
=> ? = 0
00000100 => [6,3] => [1,1,1,1,1,1,0,0,0,0,0,0,1,1,1,0,0,0]
=> ? = 1
00000101 => [6,2,1] => [1,1,1,1,1,1,0,0,0,0,0,0,1,1,0,0,1,0]
=> ? = 1
00000110 => [6,1,2] => [1,1,1,1,1,1,0,0,0,0,0,0,1,0,1,1,0,0]
=> ? = 1
00000111 => [6,1,1,1] => [1,1,1,1,1,1,0,0,0,0,0,0,1,0,1,0,1,0]
=> ? = 0
00001000 => [5,4] => [1,1,1,1,1,0,0,0,0,0,1,1,1,1,0,0,0,0]
=> ? = 1
00001001 => [5,3,1] => [1,1,1,1,1,0,0,0,0,0,1,1,1,0,0,0,1,0]
=> ? = 1
00001010 => [5,2,2] => [1,1,1,1,1,0,0,0,0,0,1,1,0,0,1,1,0,0]
=> ? = 2
00001011 => [5,2,1,1] => [1,1,1,1,1,0,0,0,0,0,1,1,0,0,1,0,1,0]
=> ? = 1
00001100 => [5,1,3] => [1,1,1,1,1,0,0,0,0,0,1,0,1,1,1,0,0,0]
=> ? = 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
00001110 => [5,1,1,2] => [1,1,1,1,1,0,0,0,0,0,1,0,1,0,1,1,0,0]
=> ? = 1
00001111 => [5,1,1,1,1] => [1,1,1,1,1,0,0,0,0,0,1,0,1,0,1,0,1,0]
=> ? = 0
00010000 => [4,5] => [1,1,1,1,0,0,0,0,1,1,1,1,1,0,0,0,0,0]
=> ? = 1
00010001 => [4,4,1] => [1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0,1,0]
=> ? = 1
Description
The number of inner corners of the upper path of the parallelogram polyomino associated with the Dyck path.
Matching statistic: St000568
Mp00178: Binary words —to composition⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
Mp00034: Dyck paths —to binary tree: up step, left tree, down step, right tree⟶ Binary trees
St000568: Binary trees ⟶ ℤResult quality: 10% ●values known / values provided: 10%●distinct values known / distinct values provided: 57%
Mp00231: Integer compositions —bounce path⟶ Dyck paths
Mp00034: Dyck paths —to binary tree: up step, left tree, down step, right tree⟶ Binary trees
St000568: Binary trees ⟶ ℤResult quality: 10% ●values known / values provided: 10%●distinct values known / distinct values provided: 57%
Values
0 => [2] => [1,1,0,0]
=> [[.,.],.]
=> 1 = 0 + 1
1 => [1,1] => [1,0,1,0]
=> [.,[.,.]]
=> 1 = 0 + 1
00 => [3] => [1,1,1,0,0,0]
=> [[[.,.],.],.]
=> 1 = 0 + 1
01 => [2,1] => [1,1,0,0,1,0]
=> [[.,.],[.,.]]
=> 1 = 0 + 1
10 => [1,2] => [1,0,1,1,0,0]
=> [.,[[.,.],.]]
=> 2 = 1 + 1
11 => [1,1,1] => [1,0,1,0,1,0]
=> [.,[.,[.,.]]]
=> 1 = 0 + 1
000 => [4] => [1,1,1,1,0,0,0,0]
=> [[[[.,.],.],.],.]
=> 1 = 0 + 1
001 => [3,1] => [1,1,1,0,0,0,1,0]
=> [[[.,.],.],[.,.]]
=> 1 = 0 + 1
010 => [2,2] => [1,1,0,0,1,1,0,0]
=> [[.,.],[[.,.],.]]
=> 2 = 1 + 1
011 => [2,1,1] => [1,1,0,0,1,0,1,0]
=> [[.,.],[.,[.,.]]]
=> 1 = 0 + 1
100 => [1,3] => [1,0,1,1,1,0,0,0]
=> [.,[[[.,.],.],.]]
=> 2 = 1 + 1
101 => [1,2,1] => [1,0,1,1,0,0,1,0]
=> [.,[[.,.],[.,.]]]
=> 2 = 1 + 1
110 => [1,1,2] => [1,0,1,0,1,1,0,0]
=> [.,[.,[[.,.],.]]]
=> 2 = 1 + 1
111 => [1,1,1,1] => [1,0,1,0,1,0,1,0]
=> [.,[.,[.,[.,.]]]]
=> 1 = 0 + 1
0000 => [5] => [1,1,1,1,1,0,0,0,0,0]
=> [[[[[.,.],.],.],.],.]
=> 1 = 0 + 1
0001 => [4,1] => [1,1,1,1,0,0,0,0,1,0]
=> [[[[.,.],.],.],[.,.]]
=> 1 = 0 + 1
0010 => [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> [[[.,.],.],[[.,.],.]]
=> 2 = 1 + 1
0011 => [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> [[[.,.],.],[.,[.,.]]]
=> 1 = 0 + 1
0100 => [2,3] => [1,1,0,0,1,1,1,0,0,0]
=> [[.,.],[[[.,.],.],.]]
=> 2 = 1 + 1
0101 => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> [[.,.],[[.,.],[.,.]]]
=> 2 = 1 + 1
0110 => [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
=> [[.,.],[.,[[.,.],.]]]
=> 2 = 1 + 1
0111 => [2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
=> [[.,.],[.,[.,[.,.]]]]
=> 1 = 0 + 1
1000 => [1,4] => [1,0,1,1,1,1,0,0,0,0]
=> [.,[[[[.,.],.],.],.]]
=> 2 = 1 + 1
1001 => [1,3,1] => [1,0,1,1,1,0,0,0,1,0]
=> [.,[[[.,.],.],[.,.]]]
=> 2 = 1 + 1
1010 => [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
=> [.,[[.,.],[[.,.],.]]]
=> 3 = 2 + 1
1011 => [1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
=> [.,[[.,.],[.,[.,.]]]]
=> 2 = 1 + 1
1100 => [1,1,3] => [1,0,1,0,1,1,1,0,0,0]
=> [.,[.,[[[.,.],.],.]]]
=> 2 = 1 + 1
1101 => [1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
=> [.,[.,[[.,.],[.,.]]]]
=> 2 = 1 + 1
1110 => [1,1,1,2] => [1,0,1,0,1,0,1,1,0,0]
=> [.,[.,[.,[[.,.],.]]]]
=> 2 = 1 + 1
1111 => [1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
=> [.,[.,[.,[.,[.,.]]]]]
=> 1 = 0 + 1
00000 => [6] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> [[[[[[.,.],.],.],.],.],.]
=> 1 = 0 + 1
00001 => [5,1] => [1,1,1,1,1,0,0,0,0,0,1,0]
=> [[[[[.,.],.],.],.],[.,.]]
=> 1 = 0 + 1
00010 => [4,2] => [1,1,1,1,0,0,0,0,1,1,0,0]
=> [[[[.,.],.],.],[[.,.],.]]
=> 2 = 1 + 1
00011 => [4,1,1] => [1,1,1,1,0,0,0,0,1,0,1,0]
=> [[[[.,.],.],.],[.,[.,.]]]
=> 1 = 0 + 1
00100 => [3,3] => [1,1,1,0,0,0,1,1,1,0,0,0]
=> [[[.,.],.],[[[.,.],.],.]]
=> 2 = 1 + 1
00101 => [3,2,1] => [1,1,1,0,0,0,1,1,0,0,1,0]
=> [[[.,.],.],[[.,.],[.,.]]]
=> 2 = 1 + 1
00110 => [3,1,2] => [1,1,1,0,0,0,1,0,1,1,0,0]
=> [[[.,.],.],[.,[[.,.],.]]]
=> 2 = 1 + 1
00111 => [3,1,1,1] => [1,1,1,0,0,0,1,0,1,0,1,0]
=> [[[.,.],.],[.,[.,[.,.]]]]
=> 1 = 0 + 1
01000 => [2,4] => [1,1,0,0,1,1,1,1,0,0,0,0]
=> [[.,.],[[[[.,.],.],.],.]]
=> 2 = 1 + 1
01001 => [2,3,1] => [1,1,0,0,1,1,1,0,0,0,1,0]
=> [[.,.],[[[.,.],.],[.,.]]]
=> 2 = 1 + 1
01010 => [2,2,2] => [1,1,0,0,1,1,0,0,1,1,0,0]
=> [[.,.],[[.,.],[[.,.],.]]]
=> 3 = 2 + 1
01011 => [2,2,1,1] => [1,1,0,0,1,1,0,0,1,0,1,0]
=> [[.,.],[[.,.],[.,[.,.]]]]
=> 2 = 1 + 1
01100 => [2,1,3] => [1,1,0,0,1,0,1,1,1,0,0,0]
=> [[.,.],[.,[[[.,.],.],.]]]
=> 2 = 1 + 1
01101 => [2,1,2,1] => [1,1,0,0,1,0,1,1,0,0,1,0]
=> [[.,.],[.,[[.,.],[.,.]]]]
=> 2 = 1 + 1
01110 => [2,1,1,2] => [1,1,0,0,1,0,1,0,1,1,0,0]
=> [[.,.],[.,[.,[[.,.],.]]]]
=> 2 = 1 + 1
01111 => [2,1,1,1,1] => [1,1,0,0,1,0,1,0,1,0,1,0]
=> [[.,.],[.,[.,[.,[.,.]]]]]
=> 1 = 0 + 1
10000 => [1,5] => [1,0,1,1,1,1,1,0,0,0,0,0]
=> [.,[[[[[.,.],.],.],.],.]]
=> 2 = 1 + 1
10001 => [1,4,1] => [1,0,1,1,1,1,0,0,0,0,1,0]
=> [.,[[[[.,.],.],.],[.,.]]]
=> 2 = 1 + 1
10010 => [1,3,2] => [1,0,1,1,1,0,0,0,1,1,0,0]
=> [.,[[[.,.],.],[[.,.],.]]]
=> 3 = 2 + 1
10011 => [1,3,1,1] => [1,0,1,1,1,0,0,0,1,0,1,0]
=> [.,[[[.,.],.],[.,[.,.]]]]
=> 2 = 1 + 1
0000000 => [8] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [[[[[[[[.,.],.],.],.],.],.],.],.]
=> ? = 0 + 1
0000001 => [7,1] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> [[[[[[[.,.],.],.],.],.],.],[.,.]]
=> ? = 0 + 1
0000010 => [6,2] => [1,1,1,1,1,1,0,0,0,0,0,0,1,1,0,0]
=> [[[[[[.,.],.],.],.],.],[[.,.],.]]
=> ? = 1 + 1
0000011 => [6,1,1] => [1,1,1,1,1,1,0,0,0,0,0,0,1,0,1,0]
=> [[[[[[.,.],.],.],.],.],[.,[.,.]]]
=> ? = 0 + 1
0000100 => [5,3] => [1,1,1,1,1,0,0,0,0,0,1,1,1,0,0,0]
=> [[[[[.,.],.],.],.],[[[.,.],.],.]]
=> ? = 1 + 1
0000101 => [5,2,1] => [1,1,1,1,1,0,0,0,0,0,1,1,0,0,1,0]
=> [[[[[.,.],.],.],.],[[.,.],[.,.]]]
=> ? = 1 + 1
0000110 => [5,1,2] => [1,1,1,1,1,0,0,0,0,0,1,0,1,1,0,0]
=> [[[[[.,.],.],.],.],[.,[[.,.],.]]]
=> ? = 1 + 1
0000111 => [5,1,1,1] => [1,1,1,1,1,0,0,0,0,0,1,0,1,0,1,0]
=> [[[[[.,.],.],.],.],[.,[.,[.,.]]]]
=> ? = 0 + 1
0001000 => [4,4] => [1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0]
=> [[[[.,.],.],.],[[[[.,.],.],.],.]]
=> ? = 1 + 1
0001001 => [4,3,1] => [1,1,1,1,0,0,0,0,1,1,1,0,0,0,1,0]
=> [[[[.,.],.],.],[[[.,.],.],[.,.]]]
=> ? = 1 + 1
0001010 => [4,2,2] => [1,1,1,1,0,0,0,0,1,1,0,0,1,1,0,0]
=> [[[[.,.],.],.],[[.,.],[[.,.],.]]]
=> ? = 2 + 1
0001011 => [4,2,1,1] => [1,1,1,1,0,0,0,0,1,1,0,0,1,0,1,0]
=> [[[[.,.],.],.],[[.,.],[.,[.,.]]]]
=> ? = 1 + 1
0001100 => [4,1,3] => [1,1,1,1,0,0,0,0,1,0,1,1,1,0,0,0]
=> [[[[.,.],.],.],[.,[[[.,.],.],.]]]
=> ? = 1 + 1
0001101 => [4,1,2,1] => [1,1,1,1,0,0,0,0,1,0,1,1,0,0,1,0]
=> [[[[.,.],.],.],[.,[[.,.],[.,.]]]]
=> ? = 1 + 1
0001110 => [4,1,1,2] => [1,1,1,1,0,0,0,0,1,0,1,0,1,1,0,0]
=> [[[[.,.],.],.],[.,[.,[[.,.],.]]]]
=> ? = 1 + 1
0001111 => [4,1,1,1,1] => [1,1,1,1,0,0,0,0,1,0,1,0,1,0,1,0]
=> [[[[.,.],.],.],[.,[.,[.,[.,.]]]]]
=> ? = 0 + 1
0010000 => [3,5] => [1,1,1,0,0,0,1,1,1,1,1,0,0,0,0,0]
=> [[[.,.],.],[[[[[.,.],.],.],.],.]]
=> ? = 1 + 1
0010001 => [3,4,1] => [1,1,1,0,0,0,1,1,1,1,0,0,0,0,1,0]
=> [[[.,.],.],[[[[.,.],.],.],[.,.]]]
=> ? = 1 + 1
0010010 => [3,3,2] => [1,1,1,0,0,0,1,1,1,0,0,0,1,1,0,0]
=> [[[.,.],.],[[[.,.],.],[[.,.],.]]]
=> ? = 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
0010100 => [3,2,3] => [1,1,1,0,0,0,1,1,0,0,1,1,1,0,0,0]
=> [[[.,.],.],[[.,.],[[[.,.],.],.]]]
=> ? = 2 + 1
0010101 => [3,2,2,1] => [1,1,1,0,0,0,1,1,0,0,1,1,0,0,1,0]
=> [[[.,.],.],[[.,.],[[.,.],[.,.]]]]
=> ? = 2 + 1
0010110 => [3,2,1,2] => [1,1,1,0,0,0,1,1,0,0,1,0,1,1,0,0]
=> [[[.,.],.],[[.,.],[.,[[.,.],.]]]]
=> ? = 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
0011000 => [3,1,4] => [1,1,1,0,0,0,1,0,1,1,1,1,0,0,0,0]
=> [[[.,.],.],[.,[[[[.,.],.],.],.]]]
=> ? = 1 + 1
0011001 => [3,1,3,1] => [1,1,1,0,0,0,1,0,1,1,1,0,0,0,1,0]
=> [[[.,.],.],[.,[[[.,.],.],[.,.]]]]
=> ? = 1 + 1
0011010 => [3,1,2,2] => [1,1,1,0,0,0,1,0,1,1,0,0,1,1,0,0]
=> [[[.,.],.],[.,[[.,.],[[.,.],.]]]]
=> ? = 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
0011100 => [3,1,1,3] => [1,1,1,0,0,0,1,0,1,0,1,1,1,0,0,0]
=> [[[.,.],.],[.,[.,[[[.,.],.],.]]]]
=> ? = 1 + 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
0011110 => [3,1,1,1,2] => [1,1,1,0,0,0,1,0,1,0,1,0,1,1,0,0]
=> [[[.,.],.],[.,[.,[.,[[.,.],.]]]]]
=> ? = 1 + 1
0011111 => [3,1,1,1,1,1] => [1,1,1,0,0,0,1,0,1,0,1,0,1,0,1,0]
=> [[[.,.],.],[.,[.,[.,[.,[.,.]]]]]]
=> ? = 0 + 1
00000000 => [9] => [1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0]
=> [[[[[[[[[.,.],.],.],.],.],.],.],.],.]
=> ? = 0 + 1
00000001 => [8,1] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0]
=> [[[[[[[[.,.],.],.],.],.],.],.],[.,.]]
=> ? = 0 + 1
00000010 => [7,2] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,1,0,0]
=> [[[[[[[.,.],.],.],.],.],.],[[.,.],.]]
=> ? = 1 + 1
00000011 => [7,1,1] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0,1,0]
=> [[[[[[[.,.],.],.],.],.],.],[.,[.,.]]]
=> ? = 0 + 1
00000100 => [6,3] => [1,1,1,1,1,1,0,0,0,0,0,0,1,1,1,0,0,0]
=> [[[[[[.,.],.],.],.],.],[[[.,.],.],.]]
=> ? = 1 + 1
00000101 => [6,2,1] => [1,1,1,1,1,1,0,0,0,0,0,0,1,1,0,0,1,0]
=> [[[[[[.,.],.],.],.],.],[[.,.],[.,.]]]
=> ? = 1 + 1
00000110 => [6,1,2] => [1,1,1,1,1,1,0,0,0,0,0,0,1,0,1,1,0,0]
=> [[[[[[.,.],.],.],.],.],[.,[[.,.],.]]]
=> ? = 1 + 1
00000111 => [6,1,1,1] => [1,1,1,1,1,1,0,0,0,0,0,0,1,0,1,0,1,0]
=> [[[[[[.,.],.],.],.],.],[.,[.,[.,.]]]]
=> ? = 0 + 1
00001000 => [5,4] => [1,1,1,1,1,0,0,0,0,0,1,1,1,1,0,0,0,0]
=> [[[[[.,.],.],.],.],[[[[.,.],.],.],.]]
=> ? = 1 + 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
00001010 => [5,2,2] => [1,1,1,1,1,0,0,0,0,0,1,1,0,0,1,1,0,0]
=> ?
=> ? = 2 + 1
00001011 => [5,2,1,1] => [1,1,1,1,1,0,0,0,0,0,1,1,0,0,1,0,1,0]
=> [[[[[.,.],.],.],.],[[.,.],[.,[.,.]]]]
=> ? = 1 + 1
00001100 => [5,1,3] => [1,1,1,1,1,0,0,0,0,0,1,0,1,1,1,0,0,0]
=> [[[[[.,.],.],.],.],[.,[[[.,.],.],.]]]
=> ? = 1 + 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
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
00001111 => [5,1,1,1,1] => [1,1,1,1,1,0,0,0,0,0,1,0,1,0,1,0,1,0]
=> [[[[[.,.],.],.],.],[.,[.,[.,[.,.]]]]]
=> ? = 0 + 1
00010000 => [4,5] => [1,1,1,1,0,0,0,0,1,1,1,1,1,0,0,0,0,0]
=> [[[[.,.],.],.],[[[[[.,.],.],.],.],.]]
=> ? = 1 + 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
Description
The hook number of a binary tree.
A hook of a binary tree is a vertex together with is left- and its right-most branch. Then there is a unique decomposition of the tree into hooks and the hook number is the number of hooks in this decomposition.
Matching statistic: St001036
Mp00178: Binary words —to composition⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
Mp00032: Dyck paths —inverse zeta map⟶ Dyck paths
St001036: Dyck paths ⟶ ℤResult quality: 9% ●values known / values provided: 9%●distinct values known / distinct values provided: 57%
Mp00231: Integer compositions —bounce path⟶ Dyck paths
Mp00032: Dyck paths —inverse zeta map⟶ Dyck paths
St001036: Dyck paths ⟶ ℤResult quality: 9% ●values known / values provided: 9%●distinct values known / distinct values provided: 57%
Values
0 => [2] => [1,1,0,0]
=> [1,0,1,0]
=> 0
1 => [1,1] => [1,0,1,0]
=> [1,1,0,0]
=> 0
00 => [3] => [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> 0
01 => [2,1] => [1,1,0,0,1,0]
=> [1,1,0,1,0,0]
=> 0
10 => [1,2] => [1,0,1,1,0,0]
=> [1,0,1,1,0,0]
=> 1
11 => [1,1,1] => [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> 0
000 => [4] => [1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> 0
001 => [3,1] => [1,1,1,0,0,0,1,0]
=> [1,1,0,1,0,1,0,0]
=> 0
010 => [2,2] => [1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,1,0,0]
=> 1
011 => [2,1,1] => [1,1,0,0,1,0,1,0]
=> [1,1,1,0,1,0,0,0]
=> 0
100 => [1,3] => [1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,0,0]
=> 1
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]
=> 1
111 => [1,1,1,1] => [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> 0
0000 => [5] => [1,1,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> 0
0001 => [4,1] => [1,1,1,1,0,0,0,0,1,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> 0
0010 => [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> [1,0,1,1,0,1,0,1,0,0]
=> 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]
=> 0
0100 => [2,3] => [1,1,0,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,0,1,0,0]
=> 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]
=> 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]
=> 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]
=> 0
1000 => [1,4] => [1,0,1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> 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
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]
=> 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
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
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]
=> 0
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]
=> 0
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]
=> 0
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]
=> 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]
=> 0
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]
=> 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]
=> 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]
=> 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]
=> 0
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]
=> 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]
=> 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]
=> 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]
=> 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]
=> 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]
=> 0
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
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
0000001 => [7,1] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 0
0000011 => [6,1,1] => [1,1,1,1,1,1,0,0,0,0,0,0,1,0,1,0]
=> [1,1,1,0,1,0,1,0,1,0,1,0,1,0,0,0]
=> ? = 0
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]
=> ? = 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]
=> ? = 0
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]
=> ? = 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]
=> ? = 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]
=> ? = 1
0001111 => [4,1,1,1,1] => [1,1,1,1,0,0,0,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,1,0,1,0,1,0,0,0,0,0]
=> ? = 0
0010001 => [3,4,1] => [1,1,1,0,0,0,1,1,1,1,0,0,0,0,1,0]
=> [1,1,0,1,0,1,0,1,1,0,1,0,1,0,0,0]
=> ? = 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]
=> ? = 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
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]
=> ? = 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]
=> ? = 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]
=> ? = 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]
=> ? = 1
0011111 => [3,1,1,1,1,1] => [1,1,1,0,0,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,0,1,0,1,0,0,0,0,0,0]
=> ? = 0
0100001 => [2,5,1] => [1,1,0,0,1,1,1,1,1,0,0,0,0,0,1,0]
=> [1,1,0,1,0,1,0,1,0,1,1,0,1,0,0,0]
=> ? = 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
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]
=> ? = 2
0100111 => [2,3,1,1,1] => [1,1,0,0,1,1,1,0,0,0,1,0,1,0,1,0]
=> [1,1,1,1,0,1,0,1,1,0,1,0,0,0,0,0]
=> ? = 1
0101001 => [2,2,3,1] => [1,1,0,0,1,1,0,0,1,1,1,0,0,0,1,0]
=> [1,1,0,1,0,1,1,0,1,1,0,1,0,0,0,0]
=> ? = 2
0101011 => [2,2,2,1,1] => [1,1,0,0,1,1,0,0,1,1,0,0,1,0,1,0]
=> [1,1,1,0,1,1,0,1,1,0,1,0,0,0,0,0]
=> ? = 2
0101101 => [2,2,1,2,1] => [1,1,0,0,1,1,0,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,1,0,1,1,0,1,0,0,0,0,0]
=> ? = 2
0101111 => [2,2,1,1,1,1] => [1,1,0,0,1,1,0,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,1,1,0,1,0,0,0,0,0,0]
=> ? = 1
0110001 => [2,1,4,1] => [1,1,0,0,1,0,1,1,1,1,0,0,0,0,1,0]
=> [1,1,0,1,0,1,0,1,1,1,0,1,0,0,0,0]
=> ? = 1
0110011 => [2,1,3,1,1] => [1,1,0,0,1,0,1,1,1,0,0,0,1,0,1,0]
=> [1,1,1,0,1,0,1,1,1,0,1,0,0,0,0,0]
=> ? = 1
0110101 => [2,1,2,2,1] => [1,1,0,0,1,0,1,1,0,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,1,1,1,0,1,0,0,0,0,0]
=> ? = 2
0110111 => [2,1,2,1,1,1] => [1,1,0,0,1,0,1,1,0,0,1,0,1,0,1,0]
=> [1,1,1,1,0,1,1,1,0,1,0,0,0,0,0,0]
=> ? = 1
0111001 => [2,1,1,3,1] => [1,1,0,0,1,0,1,0,1,1,1,0,0,0,1,0]
=> [1,1,0,1,0,1,1,1,1,0,1,0,0,0,0,0]
=> ? = 1
0111011 => [2,1,1,2,1,1] => [1,1,0,0,1,0,1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,0,1,1,1,1,0,1,0,0,0,0,0,0]
=> ? = 1
0111101 => [2,1,1,1,2,1] => [1,1,0,0,1,0,1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,1,1,1,0,1,0,0,0,0,0,0]
=> ? = 1
0111111 => [2,1,1,1,1,1,1] => [1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0]
=> ? = 0
1000001 => [1,6,1] => [1,0,1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> [1,1,0,1,0,1,0,1,0,1,0,1,1,0,0,0]
=> ? = 1
1000011 => [1,5,1,1] => [1,0,1,1,1,1,1,0,0,0,0,0,1,0,1,0]
=> [1,1,1,0,1,0,1,0,1,0,1,1,0,0,0,0]
=> ? = 1
1000101 => [1,4,2,1] => [1,0,1,1,1,1,0,0,0,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,1,0,1,0,1,1,0,0,0,0]
=> ? = 2
1000111 => [1,4,1,1,1] => [1,0,1,1,1,1,0,0,0,0,1,0,1,0,1,0]
=> [1,1,1,1,0,1,0,1,0,1,1,0,0,0,0,0]
=> ? = 1
1001001 => [1,3,3,1] => [1,0,1,1,1,0,0,0,1,1,1,0,0,0,1,0]
=> [1,1,0,1,0,1,1,0,1,0,1,1,0,0,0,0]
=> ? = 2
1001011 => [1,3,2,1,1] => [1,0,1,1,1,0,0,0,1,1,0,0,1,0,1,0]
=> [1,1,1,0,1,1,0,1,0,1,1,0,0,0,0,0]
=> ? = 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]
=> ? = 2
1001111 => [1,3,1,1,1,1] => [1,0,1,1,1,0,0,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,1,0,1,1,0,0,0,0,0,0]
=> ? = 1
1010001 => [1,2,4,1] => [1,0,1,1,0,0,1,1,1,1,0,0,0,0,1,0]
=> [1,1,0,1,0,1,0,1,1,0,1,1,0,0,0,0]
=> ? = 2
1010011 => [1,2,3,1,1] => [1,0,1,1,0,0,1,1,1,0,0,0,1,0,1,0]
=> [1,1,1,0,1,0,1,1,0,1,1,0,0,0,0,0]
=> ? = 2
1010101 => [1,2,2,2,1] => [1,0,1,1,0,0,1,1,0,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,1,1,0,1,1,0,0,0,0,0]
=> ? = 3
1010111 => [1,2,2,1,1,1] => [1,0,1,1,0,0,1,1,0,0,1,0,1,0,1,0]
=> [1,1,1,1,0,1,1,0,1,1,0,0,0,0,0,0]
=> ? = 2
1011001 => [1,2,1,3,1] => [1,0,1,1,0,0,1,0,1,1,1,0,0,0,1,0]
=> [1,1,0,1,0,1,1,1,0,1,1,0,0,0,0,0]
=> ? = 2
1011011 => [1,2,1,2,1,1] => [1,0,1,1,0,0,1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,0,1,1,1,0,1,1,0,0,0,0,0,0]
=> ? = 2
1011101 => [1,2,1,1,2,1] => [1,0,1,1,0,0,1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,1,1,0,1,1,0,0,0,0,0,0]
=> ? = 2
1011111 => [1,2,1,1,1,1,1] => [1,0,1,1,0,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0]
=> ? = 1
1100001 => [1,1,5,1] => [1,0,1,0,1,1,1,1,1,0,0,0,0,0,1,0]
=> [1,1,0,1,0,1,0,1,0,1,1,1,0,0,0,0]
=> ? = 1
1100011 => [1,1,4,1,1] => [1,0,1,0,1,1,1,1,0,0,0,0,1,0,1,0]
=> [1,1,1,0,1,0,1,0,1,1,1,0,0,0,0,0]
=> ? = 1
Description
The number of inner corners of the parallelogram polyomino associated with the Dyck path.
Matching statistic: St001732
Mp00178: Binary words —to composition⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
Mp00032: Dyck paths —inverse zeta map⟶ Dyck paths
St001732: Dyck paths ⟶ ℤResult quality: 9% ●values known / values provided: 9%●distinct values known / distinct values provided: 57%
Mp00231: Integer compositions —bounce path⟶ Dyck paths
Mp00032: Dyck paths —inverse zeta map⟶ Dyck paths
St001732: Dyck paths ⟶ ℤResult quality: 9% ●values known / values provided: 9%●distinct values known / distinct values provided: 57%
Values
0 => [2] => [1,1,0,0]
=> [1,0,1,0]
=> 1 = 0 + 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]
=> 1 = 0 + 1
01 => [2,1] => [1,1,0,0,1,0]
=> [1,1,0,1,0,0]
=> 1 = 0 + 1
10 => [1,2] => [1,0,1,1,0,0]
=> [1,0,1,1,0,0]
=> 2 = 1 + 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]
=> 1 = 0 + 1
001 => [3,1] => [1,1,1,0,0,0,1,0]
=> [1,1,0,1,0,1,0,0]
=> 1 = 0 + 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]
=> 1 = 0 + 1
100 => [1,3] => [1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,0,0]
=> 2 = 1 + 1
101 => [1,2,1] => [1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,0]
=> 2 = 1 + 1
110 => [1,1,2] => [1,0,1,0,1,1,0,0]
=> [1,0,1,1,1,0,0,0]
=> 2 = 1 + 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]
=> 1 = 0 + 1
0001 => [4,1] => [1,1,1,1,0,0,0,0,1,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> 1 = 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 = 1 + 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]
=> 1 = 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]
=> 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]
=> 1 = 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]
=> 2 = 1 + 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]
=> 2 = 1 + 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]
=> 3 = 2 + 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]
=> 2 = 1 + 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]
=> 2 = 1 + 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]
=> 2 = 1 + 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 = 1 + 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]
=> 1 = 0 + 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]
=> 1 = 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 = 1 + 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]
=> 1 = 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]
=> 2 = 1 + 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]
=> 2 = 1 + 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 = 1 + 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]
=> 1 = 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]
=> 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]
=> 3 = 2 + 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]
=> 1 = 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]
=> 2 = 1 + 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]
=> 2 = 1 + 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]
=> 3 = 2 + 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]
=> 2 = 1 + 1
0000001 => [7,1] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 0 + 1
0000011 => [6,1,1] => [1,1,1,1,1,1,0,0,0,0,0,0,1,0,1,0]
=> [1,1,1,0,1,0,1,0,1,0,1,0,1,0,0,0]
=> ? = 0 + 1
0000101 => [5,2,1] => [1,1,1,1,1,0,0,0,0,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,1,0,1,0,1,0,1,0,0,0]
=> ? = 1 + 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]
=> ? = 0 + 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]
=> ? = 1 + 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]
=> ? = 1 + 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]
=> ? = 1 + 1
0001111 => [4,1,1,1,1] => [1,1,1,1,0,0,0,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,1,0,1,0,1,0,0,0,0,0]
=> ? = 0 + 1
0010001 => [3,4,1] => [1,1,1,0,0,0,1,1,1,1,0,0,0,0,1,0]
=> [1,1,0,1,0,1,0,1,1,0,1,0,1,0,0,0]
=> ? = 1 + 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]
=> ? = 1 + 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
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]
=> ? = 1 + 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]
=> ? = 1 + 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]
=> ? = 1 + 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]
=> ? = 1 + 1
0011111 => [3,1,1,1,1,1] => [1,1,1,0,0,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,0,1,0,1,0,0,0,0,0,0]
=> ? = 0 + 1
0100001 => [2,5,1] => [1,1,0,0,1,1,1,1,1,0,0,0,0,0,1,0]
=> [1,1,0,1,0,1,0,1,0,1,1,0,1,0,0,0]
=> ? = 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
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]
=> ? = 2 + 1
0100111 => [2,3,1,1,1] => [1,1,0,0,1,1,1,0,0,0,1,0,1,0,1,0]
=> [1,1,1,1,0,1,0,1,1,0,1,0,0,0,0,0]
=> ? = 1 + 1
0101001 => [2,2,3,1] => [1,1,0,0,1,1,0,0,1,1,1,0,0,0,1,0]
=> [1,1,0,1,0,1,1,0,1,1,0,1,0,0,0,0]
=> ? = 2 + 1
0101011 => [2,2,2,1,1] => [1,1,0,0,1,1,0,0,1,1,0,0,1,0,1,0]
=> [1,1,1,0,1,1,0,1,1,0,1,0,0,0,0,0]
=> ? = 2 + 1
0101101 => [2,2,1,2,1] => [1,1,0,0,1,1,0,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,1,0,1,1,0,1,0,0,0,0,0]
=> ? = 2 + 1
0101111 => [2,2,1,1,1,1] => [1,1,0,0,1,1,0,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,1,1,0,1,0,0,0,0,0,0]
=> ? = 1 + 1
0110001 => [2,1,4,1] => [1,1,0,0,1,0,1,1,1,1,0,0,0,0,1,0]
=> [1,1,0,1,0,1,0,1,1,1,0,1,0,0,0,0]
=> ? = 1 + 1
0110011 => [2,1,3,1,1] => [1,1,0,0,1,0,1,1,1,0,0,0,1,0,1,0]
=> [1,1,1,0,1,0,1,1,1,0,1,0,0,0,0,0]
=> ? = 1 + 1
0110101 => [2,1,2,2,1] => [1,1,0,0,1,0,1,1,0,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,1,1,1,0,1,0,0,0,0,0]
=> ? = 2 + 1
0110111 => [2,1,2,1,1,1] => [1,1,0,0,1,0,1,1,0,0,1,0,1,0,1,0]
=> [1,1,1,1,0,1,1,1,0,1,0,0,0,0,0,0]
=> ? = 1 + 1
0111001 => [2,1,1,3,1] => [1,1,0,0,1,0,1,0,1,1,1,0,0,0,1,0]
=> [1,1,0,1,0,1,1,1,1,0,1,0,0,0,0,0]
=> ? = 1 + 1
0111011 => [2,1,1,2,1,1] => [1,1,0,0,1,0,1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,0,1,1,1,1,0,1,0,0,0,0,0,0]
=> ? = 1 + 1
0111101 => [2,1,1,1,2,1] => [1,1,0,0,1,0,1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,1,1,1,0,1,0,0,0,0,0,0]
=> ? = 1 + 1
0111111 => [2,1,1,1,1,1,1] => [1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0]
=> ? = 0 + 1
1000001 => [1,6,1] => [1,0,1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> [1,1,0,1,0,1,0,1,0,1,0,1,1,0,0,0]
=> ? = 1 + 1
1000011 => [1,5,1,1] => [1,0,1,1,1,1,1,0,0,0,0,0,1,0,1,0]
=> [1,1,1,0,1,0,1,0,1,0,1,1,0,0,0,0]
=> ? = 1 + 1
1000101 => [1,4,2,1] => [1,0,1,1,1,1,0,0,0,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,1,0,1,0,1,1,0,0,0,0]
=> ? = 2 + 1
1000111 => [1,4,1,1,1] => [1,0,1,1,1,1,0,0,0,0,1,0,1,0,1,0]
=> [1,1,1,1,0,1,0,1,0,1,1,0,0,0,0,0]
=> ? = 1 + 1
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]
=> ? = 2 + 1
1001011 => [1,3,2,1,1] => [1,0,1,1,1,0,0,0,1,1,0,0,1,0,1,0]
=> [1,1,1,0,1,1,0,1,0,1,1,0,0,0,0,0]
=> ? = 2 + 1
1001101 => [1,3,1,2,1] => [1,0,1,1,1,0,0,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,1,0,1,0,1,1,0,0,0,0,0]
=> ? = 2 + 1
1001111 => [1,3,1,1,1,1] => [1,0,1,1,1,0,0,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,1,0,1,1,0,0,0,0,0,0]
=> ? = 1 + 1
1010001 => [1,2,4,1] => [1,0,1,1,0,0,1,1,1,1,0,0,0,0,1,0]
=> [1,1,0,1,0,1,0,1,1,0,1,1,0,0,0,0]
=> ? = 2 + 1
1010011 => [1,2,3,1,1] => [1,0,1,1,0,0,1,1,1,0,0,0,1,0,1,0]
=> [1,1,1,0,1,0,1,1,0,1,1,0,0,0,0,0]
=> ? = 2 + 1
1010101 => [1,2,2,2,1] => [1,0,1,1,0,0,1,1,0,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,1,1,0,1,1,0,0,0,0,0]
=> ? = 3 + 1
1010111 => [1,2,2,1,1,1] => [1,0,1,1,0,0,1,1,0,0,1,0,1,0,1,0]
=> [1,1,1,1,0,1,1,0,1,1,0,0,0,0,0,0]
=> ? = 2 + 1
1011001 => [1,2,1,3,1] => [1,0,1,1,0,0,1,0,1,1,1,0,0,0,1,0]
=> [1,1,0,1,0,1,1,1,0,1,1,0,0,0,0,0]
=> ? = 2 + 1
1011011 => [1,2,1,2,1,1] => [1,0,1,1,0,0,1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,0,1,1,1,0,1,1,0,0,0,0,0,0]
=> ? = 2 + 1
1011101 => [1,2,1,1,2,1] => [1,0,1,1,0,0,1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,1,1,0,1,1,0,0,0,0,0,0]
=> ? = 2 + 1
1011111 => [1,2,1,1,1,1,1] => [1,0,1,1,0,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0]
=> ? = 1 + 1
1100001 => [1,1,5,1] => [1,0,1,0,1,1,1,1,1,0,0,0,0,0,1,0]
=> [1,1,0,1,0,1,0,1,0,1,1,1,0,0,0,0]
=> ? = 1 + 1
1100011 => [1,1,4,1,1] => [1,0,1,0,1,1,1,1,0,0,0,0,1,0,1,0]
=> [1,1,1,0,1,0,1,0,1,1,1,0,0,0,0,0]
=> ? = 1 + 1
Description
The number of peaks visible from the left.
This is, the number of left-to-right maxima of the heights of the peaks of a Dyck path.
Matching statistic: St000257
Mp00178: Binary words —to composition⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
Mp00027: Dyck paths —to partition⟶ Integer partitions
St000257: Integer partitions ⟶ ℤResult quality: 8% ●values known / values provided: 8%●distinct values known / distinct values provided: 57%
Mp00231: Integer compositions —bounce path⟶ Dyck paths
Mp00027: Dyck paths —to partition⟶ Integer partitions
St000257: Integer partitions ⟶ ℤResult quality: 8% ●values known / values provided: 8%●distinct values known / distinct values provided: 57%
Values
0 => [2] => [1,1,0,0]
=> []
=> 0
1 => [1,1] => [1,0,1,0]
=> [1]
=> 0
00 => [3] => [1,1,1,0,0,0]
=> []
=> 0
01 => [2,1] => [1,1,0,0,1,0]
=> [2]
=> 0
10 => [1,2] => [1,0,1,1,0,0]
=> [1,1]
=> 1
11 => [1,1,1] => [1,0,1,0,1,0]
=> [2,1]
=> 0
000 => [4] => [1,1,1,1,0,0,0,0]
=> []
=> 0
001 => [3,1] => [1,1,1,0,0,0,1,0]
=> [3]
=> 0
010 => [2,2] => [1,1,0,0,1,1,0,0]
=> [2,2]
=> 1
011 => [2,1,1] => [1,1,0,0,1,0,1,0]
=> [3,2]
=> 0
100 => [1,3] => [1,0,1,1,1,0,0,0]
=> [1,1,1]
=> 1
101 => [1,2,1] => [1,0,1,1,0,0,1,0]
=> [3,1,1]
=> 1
110 => [1,1,2] => [1,0,1,0,1,1,0,0]
=> [2,2,1]
=> 1
111 => [1,1,1,1] => [1,0,1,0,1,0,1,0]
=> [3,2,1]
=> 0
0000 => [5] => [1,1,1,1,1,0,0,0,0,0]
=> []
=> 0
0001 => [4,1] => [1,1,1,1,0,0,0,0,1,0]
=> [4]
=> 0
0010 => [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> [3,3]
=> 1
0011 => [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> [4,3]
=> 0
0100 => [2,3] => [1,1,0,0,1,1,1,0,0,0]
=> [2,2,2]
=> 1
0101 => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> [4,2,2]
=> 1
0110 => [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
=> [3,3,2]
=> 1
0111 => [2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
=> [4,3,2]
=> 0
1000 => [1,4] => [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1]
=> 1
1001 => [1,3,1] => [1,0,1,1,1,0,0,0,1,0]
=> [4,1,1,1]
=> 1
1010 => [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
=> [3,3,1,1]
=> 2
1011 => [1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
=> [4,3,1,1]
=> 1
1100 => [1,1,3] => [1,0,1,0,1,1,1,0,0,0]
=> [2,2,2,1]
=> 1
1101 => [1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
=> [4,2,2,1]
=> 1
1110 => [1,1,1,2] => [1,0,1,0,1,0,1,1,0,0]
=> [3,3,2,1]
=> 1
1111 => [1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
=> [4,3,2,1]
=> 0
00000 => [6] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> []
=> 0
00001 => [5,1] => [1,1,1,1,1,0,0,0,0,0,1,0]
=> [5]
=> 0
00010 => [4,2] => [1,1,1,1,0,0,0,0,1,1,0,0]
=> [4,4]
=> 1
00011 => [4,1,1] => [1,1,1,1,0,0,0,0,1,0,1,0]
=> [5,4]
=> 0
00100 => [3,3] => [1,1,1,0,0,0,1,1,1,0,0,0]
=> [3,3,3]
=> 1
00101 => [3,2,1] => [1,1,1,0,0,0,1,1,0,0,1,0]
=> [5,3,3]
=> 1
00110 => [3,1,2] => [1,1,1,0,0,0,1,0,1,1,0,0]
=> [4,4,3]
=> 1
00111 => [3,1,1,1] => [1,1,1,0,0,0,1,0,1,0,1,0]
=> [5,4,3]
=> 0
01000 => [2,4] => [1,1,0,0,1,1,1,1,0,0,0,0]
=> [2,2,2,2]
=> 1
01001 => [2,3,1] => [1,1,0,0,1,1,1,0,0,0,1,0]
=> [5,2,2,2]
=> 1
01010 => [2,2,2] => [1,1,0,0,1,1,0,0,1,1,0,0]
=> [4,4,2,2]
=> 2
01011 => [2,2,1,1] => [1,1,0,0,1,1,0,0,1,0,1,0]
=> [5,4,2,2]
=> 1
01100 => [2,1,3] => [1,1,0,0,1,0,1,1,1,0,0,0]
=> [3,3,3,2]
=> 1
01101 => [2,1,2,1] => [1,1,0,0,1,0,1,1,0,0,1,0]
=> [5,3,3,2]
=> 1
01110 => [2,1,1,2] => [1,1,0,0,1,0,1,0,1,1,0,0]
=> [4,4,3,2]
=> 1
01111 => [2,1,1,1,1] => [1,1,0,0,1,0,1,0,1,0,1,0]
=> [5,4,3,2]
=> 0
10000 => [1,5] => [1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1]
=> 1
10001 => [1,4,1] => [1,0,1,1,1,1,0,0,0,0,1,0]
=> [5,1,1,1,1]
=> 1
10010 => [1,3,2] => [1,0,1,1,1,0,0,0,1,1,0,0]
=> [4,4,1,1,1]
=> 2
10011 => [1,3,1,1] => [1,0,1,1,1,0,0,0,1,0,1,0]
=> [5,4,1,1,1]
=> 1
0000101 => [5,2,1] => [1,1,1,1,1,0,0,0,0,0,1,1,0,0,1,0]
=> [7,5,5]
=> ? = 1
0000110 => [5,1,2] => [1,1,1,1,1,0,0,0,0,0,1,0,1,1,0,0]
=> [6,6,5]
=> ? = 1
0000111 => [5,1,1,1] => [1,1,1,1,1,0,0,0,0,0,1,0,1,0,1,0]
=> [7,6,5]
=> ? = 0
0001001 => [4,3,1] => [1,1,1,1,0,0,0,0,1,1,1,0,0,0,1,0]
=> [7,4,4,4]
=> ? = 1
0001010 => [4,2,2] => [1,1,1,1,0,0,0,0,1,1,0,0,1,1,0,0]
=> [6,6,4,4]
=> ? = 2
0001011 => [4,2,1,1] => [1,1,1,1,0,0,0,0,1,1,0,0,1,0,1,0]
=> [7,6,4,4]
=> ? = 1
0001100 => [4,1,3] => [1,1,1,1,0,0,0,0,1,0,1,1,1,0,0,0]
=> [5,5,5,4]
=> ? = 1
0001101 => [4,1,2,1] => [1,1,1,1,0,0,0,0,1,0,1,1,0,0,1,0]
=> [7,5,5,4]
=> ? = 1
0001111 => [4,1,1,1,1] => [1,1,1,1,0,0,0,0,1,0,1,0,1,0,1,0]
=> [7,6,5,4]
=> ? = 0
0010001 => [3,4,1] => [1,1,1,0,0,0,1,1,1,1,0,0,0,0,1,0]
=> [7,3,3,3,3]
=> ? = 1
0010010 => [3,3,2] => [1,1,1,0,0,0,1,1,1,0,0,0,1,1,0,0]
=> [6,6,3,3,3]
=> ? = 2
0010011 => [3,3,1,1] => [1,1,1,0,0,0,1,1,1,0,0,0,1,0,1,0]
=> [7,6,3,3,3]
=> ? = 1
0010100 => [3,2,3] => [1,1,1,0,0,0,1,1,0,0,1,1,1,0,0,0]
=> [5,5,5,3,3]
=> ? = 2
0010101 => [3,2,2,1] => [1,1,1,0,0,0,1,1,0,0,1,1,0,0,1,0]
=> [7,5,5,3,3]
=> ? = 2
0010110 => [3,2,1,2] => [1,1,1,0,0,0,1,1,0,0,1,0,1,1,0,0]
=> [6,6,5,3,3]
=> ? = 2
0010111 => [3,2,1,1,1] => [1,1,1,0,0,0,1,1,0,0,1,0,1,0,1,0]
=> [7,6,5,3,3]
=> ? = 1
0011001 => [3,1,3,1] => [1,1,1,0,0,0,1,0,1,1,1,0,0,0,1,0]
=> [7,4,4,4,3]
=> ? = 1
0011010 => [3,1,2,2] => [1,1,1,0,0,0,1,0,1,1,0,0,1,1,0,0]
=> [6,6,4,4,3]
=> ? = 2
0011011 => [3,1,2,1,1] => [1,1,1,0,0,0,1,0,1,1,0,0,1,0,1,0]
=> [7,6,4,4,3]
=> ? = 1
0011100 => [3,1,1,3] => [1,1,1,0,0,0,1,0,1,0,1,1,1,0,0,0]
=> [5,5,5,4,3]
=> ? = 1
0011101 => [3,1,1,2,1] => [1,1,1,0,0,0,1,0,1,0,1,1,0,0,1,0]
=> [7,5,5,4,3]
=> ? = 1
0011110 => [3,1,1,1,2] => [1,1,1,0,0,0,1,0,1,0,1,0,1,1,0,0]
=> [6,6,5,4,3]
=> ? = 1
0011111 => [3,1,1,1,1,1] => [1,1,1,0,0,0,1,0,1,0,1,0,1,0,1,0]
=> [7,6,5,4,3]
=> ? = 0
0100001 => [2,5,1] => [1,1,0,0,1,1,1,1,1,0,0,0,0,0,1,0]
=> [7,2,2,2,2,2]
=> ? = 1
0100100 => [2,3,3] => [1,1,0,0,1,1,1,0,0,0,1,1,1,0,0,0]
=> [5,5,5,2,2,2]
=> ? = 2
0100101 => [2,3,2,1] => [1,1,0,0,1,1,1,0,0,0,1,1,0,0,1,0]
=> [7,5,5,2,2,2]
=> ? = 2
0100110 => [2,3,1,2] => [1,1,0,0,1,1,1,0,0,0,1,0,1,1,0,0]
=> [6,6,5,2,2,2]
=> ? = 2
0100111 => [2,3,1,1,1] => [1,1,0,0,1,1,1,0,0,0,1,0,1,0,1,0]
=> [7,6,5,2,2,2]
=> ? = 1
0101000 => [2,2,4] => [1,1,0,0,1,1,0,0,1,1,1,1,0,0,0,0]
=> [4,4,4,4,2,2]
=> ? = 2
0101001 => [2,2,3,1] => [1,1,0,0,1,1,0,0,1,1,1,0,0,0,1,0]
=> [7,4,4,4,2,2]
=> ? = 2
0101010 => [2,2,2,2] => [1,1,0,0,1,1,0,0,1,1,0,0,1,1,0,0]
=> [6,6,4,4,2,2]
=> ? = 3
0101011 => [2,2,2,1,1] => [1,1,0,0,1,1,0,0,1,1,0,0,1,0,1,0]
=> [7,6,4,4,2,2]
=> ? = 2
0101100 => [2,2,1,3] => [1,1,0,0,1,1,0,0,1,0,1,1,1,0,0,0]
=> [5,5,5,4,2,2]
=> ? = 2
0101101 => [2,2,1,2,1] => [1,1,0,0,1,1,0,0,1,0,1,1,0,0,1,0]
=> [7,5,5,4,2,2]
=> ? = 2
0101110 => [2,2,1,1,2] => [1,1,0,0,1,1,0,0,1,0,1,0,1,1,0,0]
=> [6,6,5,4,2,2]
=> ? = 2
0101111 => [2,2,1,1,1,1] => [1,1,0,0,1,1,0,0,1,0,1,0,1,0,1,0]
=> [7,6,5,4,2,2]
=> ? = 1
0110001 => [2,1,4,1] => [1,1,0,0,1,0,1,1,1,1,0,0,0,0,1,0]
=> [7,3,3,3,3,2]
=> ? = 1
0110010 => [2,1,3,2] => [1,1,0,0,1,0,1,1,1,0,0,0,1,1,0,0]
=> [6,6,3,3,3,2]
=> ? = 2
0110011 => [2,1,3,1,1] => [1,1,0,0,1,0,1,1,1,0,0,0,1,0,1,0]
=> [7,6,3,3,3,2]
=> ? = 1
0110100 => [2,1,2,3] => [1,1,0,0,1,0,1,1,0,0,1,1,1,0,0,0]
=> [5,5,5,3,3,2]
=> ? = 2
0110101 => [2,1,2,2,1] => [1,1,0,0,1,0,1,1,0,0,1,1,0,0,1,0]
=> [7,5,5,3,3,2]
=> ? = 2
0110110 => [2,1,2,1,2] => [1,1,0,0,1,0,1,1,0,0,1,0,1,1,0,0]
=> [6,6,5,3,3,2]
=> ? = 2
0110111 => [2,1,2,1,1,1] => [1,1,0,0,1,0,1,1,0,0,1,0,1,0,1,0]
=> [7,6,5,3,3,2]
=> ? = 1
0111000 => [2,1,1,4] => [1,1,0,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> [4,4,4,4,3,2]
=> ? = 1
0111001 => [2,1,1,3,1] => [1,1,0,0,1,0,1,0,1,1,1,0,0,0,1,0]
=> [7,4,4,4,3,2]
=> ? = 1
0111010 => [2,1,1,2,2] => [1,1,0,0,1,0,1,0,1,1,0,0,1,1,0,0]
=> [6,6,4,4,3,2]
=> ? = 2
0111011 => [2,1,1,2,1,1] => [1,1,0,0,1,0,1,0,1,1,0,0,1,0,1,0]
=> [7,6,4,4,3,2]
=> ? = 1
0111100 => [2,1,1,1,3] => [1,1,0,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [5,5,5,4,3,2]
=> ? = 1
0111101 => [2,1,1,1,2,1] => [1,1,0,0,1,0,1,0,1,0,1,1,0,0,1,0]
=> [7,5,5,4,3,2]
=> ? = 1
0111110 => [2,1,1,1,1,2] => [1,1,0,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [6,6,5,4,3,2]
=> ? = 1
Description
The number of distinct parts of a partition that occur at least twice.
See Section 3.3.1 of [2].
Matching statistic: St001086
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00178: Binary words —to composition⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
Mp00119: Dyck paths —to 321-avoiding permutation (Krattenthaler)⟶ Permutations
St001086: Permutations ⟶ ℤResult quality: 8% ●values known / values provided: 8%●distinct values known / distinct values provided: 86%
Mp00231: Integer compositions —bounce path⟶ Dyck paths
Mp00119: Dyck paths —to 321-avoiding permutation (Krattenthaler)⟶ Permutations
St001086: Permutations ⟶ ℤResult quality: 8% ●values known / values provided: 8%●distinct values known / distinct values provided: 86%
Values
0 => [2] => [1,1,0,0]
=> [2,1] => 0
1 => [1,1] => [1,0,1,0]
=> [1,2] => 0
00 => [3] => [1,1,1,0,0,0]
=> [3,1,2] => 0
01 => [2,1] => [1,1,0,0,1,0]
=> [2,1,3] => 0
10 => [1,2] => [1,0,1,1,0,0]
=> [1,3,2] => 1
11 => [1,1,1] => [1,0,1,0,1,0]
=> [1,2,3] => 0
000 => [4] => [1,1,1,1,0,0,0,0]
=> [4,1,2,3] => 0
001 => [3,1] => [1,1,1,0,0,0,1,0]
=> [3,1,2,4] => 0
010 => [2,2] => [1,1,0,0,1,1,0,0]
=> [2,1,4,3] => 1
011 => [2,1,1] => [1,1,0,0,1,0,1,0]
=> [2,1,3,4] => 0
100 => [1,3] => [1,0,1,1,1,0,0,0]
=> [1,4,2,3] => 1
101 => [1,2,1] => [1,0,1,1,0,0,1,0]
=> [1,3,2,4] => 1
110 => [1,1,2] => [1,0,1,0,1,1,0,0]
=> [1,2,4,3] => 1
111 => [1,1,1,1] => [1,0,1,0,1,0,1,0]
=> [1,2,3,4] => 0
0000 => [5] => [1,1,1,1,1,0,0,0,0,0]
=> [5,1,2,3,4] => 0
0001 => [4,1] => [1,1,1,1,0,0,0,0,1,0]
=> [4,1,2,3,5] => 0
0010 => [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> [3,1,2,5,4] => 1
0011 => [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> [3,1,2,4,5] => 0
0100 => [2,3] => [1,1,0,0,1,1,1,0,0,0]
=> [2,1,5,3,4] => 1
0101 => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> [2,1,4,3,5] => 1
0110 => [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
=> [2,1,3,5,4] => 1
0111 => [2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
=> [2,1,3,4,5] => 0
1000 => [1,4] => [1,0,1,1,1,1,0,0,0,0]
=> [1,5,2,3,4] => 1
1001 => [1,3,1] => [1,0,1,1,1,0,0,0,1,0]
=> [1,4,2,3,5] => 1
1010 => [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
=> [1,3,2,5,4] => 2
1011 => [1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
=> [1,3,2,4,5] => 1
1100 => [1,1,3] => [1,0,1,0,1,1,1,0,0,0]
=> [1,2,5,3,4] => 1
1101 => [1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
=> [1,2,4,3,5] => 1
1110 => [1,1,1,2] => [1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,5,4] => 1
1111 => [1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5] => 0
00000 => [6] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> [6,1,2,3,4,5] => 0
00001 => [5,1] => [1,1,1,1,1,0,0,0,0,0,1,0]
=> [5,1,2,3,4,6] => 0
00010 => [4,2] => [1,1,1,1,0,0,0,0,1,1,0,0]
=> [4,1,2,3,6,5] => 1
00011 => [4,1,1] => [1,1,1,1,0,0,0,0,1,0,1,0]
=> [4,1,2,3,5,6] => 0
00100 => [3,3] => [1,1,1,0,0,0,1,1,1,0,0,0]
=> [3,1,2,6,4,5] => 1
00101 => [3,2,1] => [1,1,1,0,0,0,1,1,0,0,1,0]
=> [3,1,2,5,4,6] => 1
00110 => [3,1,2] => [1,1,1,0,0,0,1,0,1,1,0,0]
=> [3,1,2,4,6,5] => 1
00111 => [3,1,1,1] => [1,1,1,0,0,0,1,0,1,0,1,0]
=> [3,1,2,4,5,6] => 0
01000 => [2,4] => [1,1,0,0,1,1,1,1,0,0,0,0]
=> [2,1,6,3,4,5] => 1
01001 => [2,3,1] => [1,1,0,0,1,1,1,0,0,0,1,0]
=> [2,1,5,3,4,6] => 1
01010 => [2,2,2] => [1,1,0,0,1,1,0,0,1,1,0,0]
=> [2,1,4,3,6,5] => 2
01011 => [2,2,1,1] => [1,1,0,0,1,1,0,0,1,0,1,0]
=> [2,1,4,3,5,6] => 1
01100 => [2,1,3] => [1,1,0,0,1,0,1,1,1,0,0,0]
=> [2,1,3,6,4,5] => 1
01101 => [2,1,2,1] => [1,1,0,0,1,0,1,1,0,0,1,0]
=> [2,1,3,5,4,6] => 1
01110 => [2,1,1,2] => [1,1,0,0,1,0,1,0,1,1,0,0]
=> [2,1,3,4,6,5] => 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] => 0
10000 => [1,5] => [1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,6,2,3,4,5] => 1
10001 => [1,4,1] => [1,0,1,1,1,1,0,0,0,0,1,0]
=> [1,5,2,3,4,6] => 1
10010 => [1,3,2] => [1,0,1,1,1,0,0,0,1,1,0,0]
=> [1,4,2,3,6,5] => 2
10011 => [1,3,1,1] => [1,0,1,1,1,0,0,0,1,0,1,0]
=> [1,4,2,3,5,6] => 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] => ? = 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] => ? = 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] => ? = 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] => ? = 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
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] => ? = 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] => ? = 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] => ? = 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] => ? = 2
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
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] => ? = 2
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] => ? = 2
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] => ? = 2
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
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
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] => ? = 2
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
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
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
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] => ? = 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] => ? = 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] => ? = 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] => ? = 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] => ? = 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] => ? = 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] => ? = 1
0001101 => [4,1,2,1] => [1,1,1,1,0,0,0,0,1,0,1,1,0,0,1,0]
=> [4,1,2,3,5,7,6,8] => ? = 1
0001110 => [4,1,1,2] => [1,1,1,1,0,0,0,0,1,0,1,0,1,1,0,0]
=> [4,1,2,3,5,6,8,7] => ? = 1
0001111 => [4,1,1,1,1] => [1,1,1,1,0,0,0,0,1,0,1,0,1,0,1,0]
=> [4,1,2,3,5,6,7,8] => ? = 0
0010000 => [3,5] => [1,1,1,0,0,0,1,1,1,1,1,0,0,0,0,0]
=> [3,1,2,8,4,5,6,7] => ? = 1
0010001 => [3,4,1] => [1,1,1,0,0,0,1,1,1,1,0,0,0,0,1,0]
=> [3,1,2,7,4,5,6,8] => ? = 1
0010010 => [3,3,2] => [1,1,1,0,0,0,1,1,1,0,0,0,1,1,0,0]
=> [3,1,2,6,4,5,8,7] => ? = 2
0010011 => [3,3,1,1] => [1,1,1,0,0,0,1,1,1,0,0,0,1,0,1,0]
=> [3,1,2,6,4,5,7,8] => ? = 1
0010100 => [3,2,3] => [1,1,1,0,0,0,1,1,0,0,1,1,1,0,0,0]
=> [3,1,2,5,4,8,6,7] => ? = 2
0010101 => [3,2,2,1] => [1,1,1,0,0,0,1,1,0,0,1,1,0,0,1,0]
=> [3,1,2,5,4,7,6,8] => ? = 2
0010110 => [3,2,1,2] => [1,1,1,0,0,0,1,1,0,0,1,0,1,1,0,0]
=> [3,1,2,5,4,6,8,7] => ? = 2
0010111 => [3,2,1,1,1] => [1,1,1,0,0,0,1,1,0,0,1,0,1,0,1,0]
=> [3,1,2,5,4,6,7,8] => ? = 1
0011000 => [3,1,4] => [1,1,1,0,0,0,1,0,1,1,1,1,0,0,0,0]
=> [3,1,2,4,8,5,6,7] => ? = 1
0011001 => [3,1,3,1] => [1,1,1,0,0,0,1,0,1,1,1,0,0,0,1,0]
=> [3,1,2,4,7,5,6,8] => ? = 1
0011010 => [3,1,2,2] => [1,1,1,0,0,0,1,0,1,1,0,0,1,1,0,0]
=> [3,1,2,4,6,5,8,7] => ? = 2
0011011 => [3,1,2,1,1] => [1,1,1,0,0,0,1,0,1,1,0,0,1,0,1,0]
=> [3,1,2,4,6,5,7,8] => ? = 1
0011101 => [3,1,1,2,1] => [1,1,1,0,0,0,1,0,1,0,1,1,0,0,1,0]
=> [3,1,2,4,5,7,6,8] => ? = 1
0011110 => [3,1,1,1,2] => [1,1,1,0,0,0,1,0,1,0,1,0,1,1,0,0]
=> [3,1,2,4,5,6,8,7] => ? = 1
0011111 => [3,1,1,1,1,1] => [1,1,1,0,0,0,1,0,1,0,1,0,1,0,1,0]
=> [3,1,2,4,5,6,7,8] => ? = 0
0100001 => [2,5,1] => [1,1,0,0,1,1,1,1,1,0,0,0,0,0,1,0]
=> [2,1,7,3,4,5,6,8] => ? = 1
0100011 => [2,4,1,1] => [1,1,0,0,1,1,1,1,0,0,0,0,1,0,1,0]
=> [2,1,6,3,4,5,7,8] => ? = 1
0100100 => [2,3,3] => [1,1,0,0,1,1,1,0,0,0,1,1,1,0,0,0]
=> [2,1,5,3,4,8,6,7] => ? = 2
0100101 => [2,3,2,1] => [1,1,0,0,1,1,1,0,0,0,1,1,0,0,1,0]
=> [2,1,5,3,4,7,6,8] => ? = 2
0100110 => [2,3,1,2] => [1,1,0,0,1,1,1,0,0,0,1,0,1,1,0,0]
=> [2,1,5,3,4,6,8,7] => ? = 2
0100111 => [2,3,1,1,1] => [1,1,0,0,1,1,1,0,0,0,1,0,1,0,1,0]
=> [2,1,5,3,4,6,7,8] => ? = 1
Description
The number of occurrences of the consecutive pattern 132 in a permutation.
This is the number of occurrences of the pattern $132$, where the matched entries are all adjacent.
The following 33 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000201The number of leaf nodes in a binary tree. St000159The number of distinct parts of the integer partition. St001124The multiplicity of the standard representation in the Kronecker square corresponding to a partition. St000069The number of maximal elements of a poset. St000318The number of addable cells of the Ferrers diagram of an integer partition. St000093The cardinality of a maximal independent set of vertices of a graph. St000786The maximal number of occurrences of a colour in a proper colouring of a graph. St000196The number of occurrences of the contiguous pattern [[.,.],[.,. St001337The upper domination number of a graph. St001338The upper irredundance number of a graph. St000353The number of inner valleys of a permutation. St001683The number of distinct positions of the pattern letter 3 in occurrences of 132 in a permutation. St001738The minimal order of a graph which is not an induced subgraph of the given graph. St000779The tier of a permutation. St000023The number of inner peaks of a permutation. St000092The number of outer peaks of a permutation. St000099The number of valleys of a permutation, including the boundary. St000659The number of rises of length at least 2 of a Dyck path. St000985The number of positive eigenvalues of the adjacency matrix of the graph. St000455The second largest eigenvalue of a graph if it is integral. St000035The number of left outer peaks of a permutation. St000647The number of big descents of a permutation. St000884The number of isolated descents of a permutation. St001354The number of series nodes in the modular decomposition of a graph. St001960The number of descents of a permutation minus one if its first entry is not one. St000021The number of descents of a permutation. St000333The dez statistic, the number of descents of a permutation after replacing fixed points by zeros. St001487The number of inner corners of a skew partition. St001729The number of visible descents of a permutation. St001737The number of descents of type 2 in a permutation. St001928The number of non-overlapping descents in a permutation. St000325The width of the tree associated to a permutation. St000470The number of runs in a permutation.
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