searching the database
Your data matches 7 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)
(click to perform a complete search on your data)
Matching statistic: St000293
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
St000293: 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 => 2
101 => 1
110 => 2
111 => 0
0000 => 0
0001 => 0
0010 => 1
0011 => 0
0100 => 2
0101 => 1
0110 => 2
0111 => 0
1000 => 3
1001 => 2
1010 => 3
1011 => 1
1100 => 4
1101 => 2
1110 => 3
1111 => 0
00000 => 0
00001 => 0
00010 => 1
00011 => 0
00100 => 2
00101 => 1
00110 => 2
00111 => 0
01000 => 3
01001 => 2
01010 => 3
01011 => 1
01100 => 4
01101 => 2
01110 => 3
01111 => 0
10000 => 4
10001 => 3
10010 => 4
10011 => 2
Description
The number of inversions of a binary word.
Matching statistic: St000290
(load all 4 compositions to match this statistic)
(load all 4 compositions to match this statistic)
Mp00316: Binary words —inverse Foata bijection⟶ Binary words
St000290: Binary words ⟶ ℤResult quality: 40% ●values known / values provided: 48%●distinct values known / distinct values provided: 40%
St000290: Binary words ⟶ ℤResult quality: 40% ●values known / values provided: 48%●distinct values known / distinct values provided: 40%
Values
0 => 0 => 0
1 => 1 => 0
00 => 00 => 0
01 => 01 => 0
10 => 10 => 1
11 => 11 => 0
000 => 000 => 0
001 => 001 => 0
010 => 100 => 1
011 => 011 => 0
100 => 010 => 2
101 => 101 => 1
110 => 110 => 2
111 => 111 => 0
0000 => 0000 => 0
0001 => 0001 => 0
0010 => 1000 => 1
0011 => 0011 => 0
0100 => 0100 => 2
0101 => 1001 => 1
0110 => 1100 => 2
0111 => 0111 => 0
1000 => 0010 => 3
1001 => 0101 => 2
1010 => 0110 => 3
1011 => 1011 => 1
1100 => 1010 => 4
1101 => 1101 => 2
1110 => 1110 => 3
1111 => 1111 => 0
00000 => 00000 => 0
00001 => 00001 => 0
00010 => 10000 => 1
00011 => 00011 => 0
00100 => 01000 => 2
00101 => 10001 => 1
00110 => 11000 => 2
00111 => 00111 => 0
01000 => 00100 => 3
01001 => 01001 => 2
01010 => 01100 => 3
01011 => 10011 => 1
01100 => 10100 => 4
01101 => 11001 => 2
01110 => 11100 => 3
01111 => 01111 => 0
10000 => 00010 => 4
10001 => 00101 => 3
10010 => 00110 => 4
10011 => 01011 => 2
1010101010 => ? => ? = 15
1010101100 => ? => ? = 16
1010110010 => ? => ? = 16
1010110100 => ? => ? = 17
1010111000 => ? => ? = 18
1011001010 => ? => ? = 16
1011001100 => ? => ? = 17
1011010010 => ? => ? = 17
1011010100 => ? => ? = 18
1011011000 => ? => ? = 19
1011100010 => ? => ? = 18
1011100100 => ? => ? = 19
1011101000 => ? => ? = 20
1011110000 => ? => ? = 21
1100101010 => ? => ? = 16
1100101100 => ? => ? = 17
1100110010 => ? => ? = 17
1100110100 => ? => ? = 18
1100111000 => ? => ? = 19
1101001010 => ? => ? = 17
1101001100 => ? => ? = 18
1101010010 => ? => ? = 18
1101010100 => ? => ? = 19
1101011000 => ? => ? = 20
1101100010 => ? => ? = 19
1101100100 => ? => ? = 20
1101101000 => ? => ? = 21
1101110000 => ? => ? = 22
1110001010 => ? => ? = 18
1110001100 => ? => ? = 19
1110010010 => ? => ? = 19
1110010100 => ? => ? = 20
1110011000 => ? => ? = 21
1110100010 => ? => ? = 20
1110100100 => ? => ? = 21
1110101000 => ? => ? = 22
1110110000 => ? => ? = 23
1111000010 => ? => ? = 21
1111000100 => ? => ? = 22
1111001000 => ? => ? = 23
1111010000 => ? => ? = 24
1111100000 => ? => ? = 25
101010101010 => ? => ? = 21
101010101100 => ? => ? = 22
101010110010 => ? => ? = 22
101010110100 => ? => ? = 23
101010111000 => ? => ? = 24
101011001010 => ? => ? = 22
101011001100 => ? => ? = 23
101011010010 => ? => ? = 23
Description
The major index of a binary word.
This is the sum of the positions of descents, i.e., a one followed by a zero.
For words of length $n$ with $a$ zeros, the generating function for the major index is the $q$-binomial coefficient $\binom{n}{a}_q$.
Matching statistic: St000589
Mp00178: Binary words —to composition⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
Mp00138: Dyck paths —to noncrossing partition⟶ Set partitions
St000589: Set partitions ⟶ ℤResult quality: 6% ●values known / values provided: 6%●distinct values known / distinct values provided: 21%
Mp00231: Integer compositions —bounce path⟶ Dyck paths
Mp00138: Dyck paths —to noncrossing partition⟶ Set partitions
St000589: Set partitions ⟶ ℤResult quality: 6% ●values known / values provided: 6%●distinct values known / distinct values provided: 21%
Values
0 => [2] => [1,1,0,0]
=> {{1,2}}
=> 0
1 => [1,1] => [1,0,1,0]
=> {{1},{2}}
=> 0
00 => [3] => [1,1,1,0,0,0]
=> {{1,2,3}}
=> 0
01 => [2,1] => [1,1,0,0,1,0]
=> {{1,2},{3}}
=> 0
10 => [1,2] => [1,0,1,1,0,0]
=> {{1},{2,3}}
=> 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]
=> {{1,2,3,4}}
=> 0
001 => [3,1] => [1,1,1,0,0,0,1,0]
=> {{1,2,3},{4}}
=> 0
010 => [2,2] => [1,1,0,0,1,1,0,0]
=> {{1,2},{3,4}}
=> 1
011 => [2,1,1] => [1,1,0,0,1,0,1,0]
=> {{1,2},{3},{4}}
=> 0
100 => [1,3] => [1,0,1,1,1,0,0,0]
=> {{1},{2,3,4}}
=> 2
101 => [1,2,1] => [1,0,1,1,0,0,1,0]
=> {{1},{2,3},{4}}
=> 1
110 => [1,1,2] => [1,0,1,0,1,1,0,0]
=> {{1},{2},{3,4}}
=> 2
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]
=> {{1,2,3,4,5}}
=> 0
0001 => [4,1] => [1,1,1,1,0,0,0,0,1,0]
=> {{1,2,3,4},{5}}
=> 0
0010 => [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> {{1,2,3},{4,5}}
=> 1
0011 => [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> {{1,2,3},{4},{5}}
=> 0
0100 => [2,3] => [1,1,0,0,1,1,1,0,0,0]
=> {{1,2},{3,4,5}}
=> 2
0101 => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> {{1,2},{3,4},{5}}
=> 1
0110 => [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
=> {{1,2},{3},{4,5}}
=> 2
0111 => [2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
=> {{1,2},{3},{4},{5}}
=> 0
1000 => [1,4] => [1,0,1,1,1,1,0,0,0,0]
=> {{1},{2,3,4,5}}
=> 3
1001 => [1,3,1] => [1,0,1,1,1,0,0,0,1,0]
=> {{1},{2,3,4},{5}}
=> 2
1010 => [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
=> {{1},{2,3},{4,5}}
=> 3
1011 => [1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
=> {{1},{2,3},{4},{5}}
=> 1
1100 => [1,1,3] => [1,0,1,0,1,1,1,0,0,0]
=> {{1},{2},{3,4,5}}
=> 4
1101 => [1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
=> {{1},{2},{3,4},{5}}
=> 2
1110 => [1,1,1,2] => [1,0,1,0,1,0,1,1,0,0]
=> {{1},{2},{3},{4,5}}
=> 3
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]
=> {{1,2,3,4,5,6}}
=> 0
00001 => [5,1] => [1,1,1,1,1,0,0,0,0,0,1,0]
=> {{1,2,3,4,5},{6}}
=> 0
00010 => [4,2] => [1,1,1,1,0,0,0,0,1,1,0,0]
=> {{1,2,3,4},{5,6}}
=> 1
00011 => [4,1,1] => [1,1,1,1,0,0,0,0,1,0,1,0]
=> {{1,2,3,4},{5},{6}}
=> 0
00100 => [3,3] => [1,1,1,0,0,0,1,1,1,0,0,0]
=> {{1,2,3},{4,5,6}}
=> 2
00101 => [3,2,1] => [1,1,1,0,0,0,1,1,0,0,1,0]
=> {{1,2,3},{4,5},{6}}
=> 1
00110 => [3,1,2] => [1,1,1,0,0,0,1,0,1,1,0,0]
=> {{1,2,3},{4},{5,6}}
=> 2
00111 => [3,1,1,1] => [1,1,1,0,0,0,1,0,1,0,1,0]
=> {{1,2,3},{4},{5},{6}}
=> 0
01000 => [2,4] => [1,1,0,0,1,1,1,1,0,0,0,0]
=> {{1,2},{3,4,5,6}}
=> 3
01001 => [2,3,1] => [1,1,0,0,1,1,1,0,0,0,1,0]
=> {{1,2},{3,4,5},{6}}
=> 2
01010 => [2,2,2] => [1,1,0,0,1,1,0,0,1,1,0,0]
=> {{1,2},{3,4},{5,6}}
=> 3
01011 => [2,2,1,1] => [1,1,0,0,1,1,0,0,1,0,1,0]
=> {{1,2},{3,4},{5},{6}}
=> 1
01100 => [2,1,3] => [1,1,0,0,1,0,1,1,1,0,0,0]
=> {{1,2},{3},{4,5,6}}
=> 4
01101 => [2,1,2,1] => [1,1,0,0,1,0,1,1,0,0,1,0]
=> {{1,2},{3},{4,5},{6}}
=> 2
01110 => [2,1,1,2] => [1,1,0,0,1,0,1,0,1,1,0,0]
=> {{1,2},{3},{4},{5,6}}
=> 3
01111 => [2,1,1,1,1] => [1,1,0,0,1,0,1,0,1,0,1,0]
=> {{1,2},{3},{4},{5},{6}}
=> 0
10000 => [1,5] => [1,0,1,1,1,1,1,0,0,0,0,0]
=> {{1},{2,3,4,5,6}}
=> 4
10001 => [1,4,1] => [1,0,1,1,1,1,0,0,0,0,1,0]
=> {{1},{2,3,4,5},{6}}
=> 3
10010 => [1,3,2] => [1,0,1,1,1,0,0,0,1,1,0,0]
=> {{1},{2,3,4},{5,6}}
=> 4
10011 => [1,3,1,1] => [1,0,1,1,1,0,0,0,1,0,1,0]
=> {{1},{2,3,4},{5},{6}}
=> 2
0000011 => [6,1,1] => [1,1,1,1,1,1,0,0,0,0,0,0,1,0,1,0]
=> {{1,2,3,4,5,6},{7},{8}}
=> ? = 0
0000100 => [5,3] => [1,1,1,1,1,0,0,0,0,0,1,1,1,0,0,0]
=> {{1,2,3,4,5},{6,7,8}}
=> ? = 2
0000101 => [5,2,1] => [1,1,1,1,1,0,0,0,0,0,1,1,0,0,1,0]
=> {{1,2,3,4,5},{6,7},{8}}
=> ? = 1
0000110 => [5,1,2] => [1,1,1,1,1,0,0,0,0,0,1,0,1,1,0,0]
=> {{1,2,3,4,5},{6},{7,8}}
=> ? = 2
0000111 => [5,1,1,1] => [1,1,1,1,1,0,0,0,0,0,1,0,1,0,1,0]
=> {{1,2,3,4,5},{6},{7},{8}}
=> ? = 0
0001000 => [4,4] => [1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0]
=> {{1,2,3,4},{5,6,7,8}}
=> ? = 3
0001001 => [4,3,1] => [1,1,1,1,0,0,0,0,1,1,1,0,0,0,1,0]
=> {{1,2,3,4},{5,6,7},{8}}
=> ? = 2
0001010 => [4,2,2] => [1,1,1,1,0,0,0,0,1,1,0,0,1,1,0,0]
=> {{1,2,3,4},{5,6},{7,8}}
=> ? = 3
0001011 => [4,2,1,1] => [1,1,1,1,0,0,0,0,1,1,0,0,1,0,1,0]
=> {{1,2,3,4},{5,6},{7},{8}}
=> ? = 1
0001100 => [4,1,3] => [1,1,1,1,0,0,0,0,1,0,1,1,1,0,0,0]
=> {{1,2,3,4},{5},{6,7,8}}
=> ? = 4
0001101 => [4,1,2,1] => [1,1,1,1,0,0,0,0,1,0,1,1,0,0,1,0]
=> {{1,2,3,4},{5},{6,7},{8}}
=> ? = 2
0001110 => [4,1,1,2] => [1,1,1,1,0,0,0,0,1,0,1,0,1,1,0,0]
=> {{1,2,3,4},{5},{6},{7,8}}
=> ? = 3
0001111 => [4,1,1,1,1] => [1,1,1,1,0,0,0,0,1,0,1,0,1,0,1,0]
=> {{1,2,3,4},{5},{6},{7},{8}}
=> ? = 0
0010000 => [3,5] => [1,1,1,0,0,0,1,1,1,1,1,0,0,0,0,0]
=> {{1,2,3},{4,5,6,7,8}}
=> ? = 4
0010001 => [3,4,1] => [1,1,1,0,0,0,1,1,1,1,0,0,0,0,1,0]
=> {{1,2,3},{4,5,6,7},{8}}
=> ? = 3
0010010 => [3,3,2] => [1,1,1,0,0,0,1,1,1,0,0,0,1,1,0,0]
=> {{1,2,3},{4,5,6},{7,8}}
=> ? = 4
0010011 => [3,3,1,1] => [1,1,1,0,0,0,1,1,1,0,0,0,1,0,1,0]
=> {{1,2,3},{4,5,6},{7},{8}}
=> ? = 2
0010100 => [3,2,3] => [1,1,1,0,0,0,1,1,0,0,1,1,1,0,0,0]
=> {{1,2,3},{4,5},{6,7,8}}
=> ? = 5
0010101 => [3,2,2,1] => [1,1,1,0,0,0,1,1,0,0,1,1,0,0,1,0]
=> {{1,2,3},{4,5},{6,7},{8}}
=> ? = 3
0010110 => [3,2,1,2] => [1,1,1,0,0,0,1,1,0,0,1,0,1,1,0,0]
=> {{1,2,3},{4,5},{6},{7,8}}
=> ? = 4
0010111 => [3,2,1,1,1] => [1,1,1,0,0,0,1,1,0,0,1,0,1,0,1,0]
=> {{1,2,3},{4,5},{6},{7},{8}}
=> ? = 1
0011000 => [3,1,4] => [1,1,1,0,0,0,1,0,1,1,1,1,0,0,0,0]
=> {{1,2,3},{4},{5,6,7,8}}
=> ? = 6
0011001 => [3,1,3,1] => [1,1,1,0,0,0,1,0,1,1,1,0,0,0,1,0]
=> {{1,2,3},{4},{5,6,7},{8}}
=> ? = 4
0011010 => [3,1,2,2] => [1,1,1,0,0,0,1,0,1,1,0,0,1,1,0,0]
=> {{1,2,3},{4},{5,6},{7,8}}
=> ? = 5
0011011 => [3,1,2,1,1] => [1,1,1,0,0,0,1,0,1,1,0,0,1,0,1,0]
=> {{1,2,3},{4},{5,6},{7},{8}}
=> ? = 2
0011100 => [3,1,1,3] => [1,1,1,0,0,0,1,0,1,0,1,1,1,0,0,0]
=> {{1,2,3},{4},{5},{6,7,8}}
=> ? = 6
0011101 => [3,1,1,2,1] => [1,1,1,0,0,0,1,0,1,0,1,1,0,0,1,0]
=> {{1,2,3},{4},{5},{6,7},{8}}
=> ? = 3
0011110 => [3,1,1,1,2] => [1,1,1,0,0,0,1,0,1,0,1,0,1,1,0,0]
=> {{1,2,3},{4},{5},{6},{7,8}}
=> ? = 4
0011111 => [3,1,1,1,1,1] => [1,1,1,0,0,0,1,0,1,0,1,0,1,0,1,0]
=> {{1,2,3},{4},{5},{6},{7},{8}}
=> ? = 0
0100001 => [2,5,1] => [1,1,0,0,1,1,1,1,1,0,0,0,0,0,1,0]
=> {{1,2},{3,4,5,6,7},{8}}
=> ? = 4
0100010 => [2,4,2] => [1,1,0,0,1,1,1,1,0,0,0,0,1,1,0,0]
=> {{1,2},{3,4,5,6},{7,8}}
=> ? = 5
0100011 => [2,4,1,1] => [1,1,0,0,1,1,1,1,0,0,0,0,1,0,1,0]
=> {{1,2},{3,4,5,6},{7},{8}}
=> ? = 3
0100100 => [2,3,3] => [1,1,0,0,1,1,1,0,0,0,1,1,1,0,0,0]
=> {{1,2},{3,4,5},{6,7,8}}
=> ? = 6
0100101 => [2,3,2,1] => [1,1,0,0,1,1,1,0,0,0,1,1,0,0,1,0]
=> {{1,2},{3,4,5},{6,7},{8}}
=> ? = 4
0100110 => [2,3,1,2] => [1,1,0,0,1,1,1,0,0,0,1,0,1,1,0,0]
=> {{1,2},{3,4,5},{6},{7,8}}
=> ? = 5
0100111 => [2,3,1,1,1] => [1,1,0,0,1,1,1,0,0,0,1,0,1,0,1,0]
=> {{1,2},{3,4,5},{6},{7},{8}}
=> ? = 2
0101000 => [2,2,4] => [1,1,0,0,1,1,0,0,1,1,1,1,0,0,0,0]
=> {{1,2},{3,4},{5,6,7,8}}
=> ? = 7
0101001 => [2,2,3,1] => [1,1,0,0,1,1,0,0,1,1,1,0,0,0,1,0]
=> {{1,2},{3,4},{5,6,7},{8}}
=> ? = 5
0101010 => [2,2,2,2] => [1,1,0,0,1,1,0,0,1,1,0,0,1,1,0,0]
=> {{1,2},{3,4},{5,6},{7,8}}
=> ? = 6
0101011 => [2,2,2,1,1] => [1,1,0,0,1,1,0,0,1,1,0,0,1,0,1,0]
=> {{1,2},{3,4},{5,6},{7},{8}}
=> ? = 3
0101100 => [2,2,1,3] => [1,1,0,0,1,1,0,0,1,0,1,1,1,0,0,0]
=> {{1,2},{3,4},{5},{6,7,8}}
=> ? = 7
0101101 => [2,2,1,2,1] => [1,1,0,0,1,1,0,0,1,0,1,1,0,0,1,0]
=> {{1,2},{3,4},{5},{6,7},{8}}
=> ? = 4
0101110 => [2,2,1,1,2] => [1,1,0,0,1,1,0,0,1,0,1,0,1,1,0,0]
=> {{1,2},{3,4},{5},{6},{7,8}}
=> ? = 5
0101111 => [2,2,1,1,1,1] => [1,1,0,0,1,1,0,0,1,0,1,0,1,0,1,0]
=> {{1,2},{3,4},{5},{6},{7},{8}}
=> ? = 1
0110000 => [2,1,5] => [1,1,0,0,1,0,1,1,1,1,1,0,0,0,0,0]
=> {{1,2},{3},{4,5,6,7,8}}
=> ? = 8
0110001 => [2,1,4,1] => [1,1,0,0,1,0,1,1,1,1,0,0,0,0,1,0]
=> {{1,2},{3},{4,5,6,7},{8}}
=> ? = 6
0110010 => [2,1,3,2] => [1,1,0,0,1,0,1,1,1,0,0,0,1,1,0,0]
=> {{1,2},{3},{4,5,6},{7,8}}
=> ? = 7
0110011 => [2,1,3,1,1] => [1,1,0,0,1,0,1,1,1,0,0,0,1,0,1,0]
=> {{1,2},{3},{4,5,6},{7},{8}}
=> ? = 4
0110100 => [2,1,2,3] => [1,1,0,0,1,0,1,1,0,0,1,1,1,0,0,0]
=> {{1,2},{3},{4,5},{6,7,8}}
=> ? = 8
0110101 => [2,1,2,2,1] => [1,1,0,0,1,0,1,1,0,0,1,1,0,0,1,0]
=> {{1,2},{3},{4,5},{6,7},{8}}
=> ? = 5
Description
The number of occurrences of the pattern {{1},{2,3}} such that 1 is maximal, (2,3) are consecutive in a block.
Matching statistic: St000609
Mp00178: Binary words —to composition⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
Mp00138: Dyck paths —to noncrossing partition⟶ Set partitions
St000609: Set partitions ⟶ ℤResult quality: 6% ●values known / values provided: 6%●distinct values known / distinct values provided: 21%
Mp00231: Integer compositions —bounce path⟶ Dyck paths
Mp00138: Dyck paths —to noncrossing partition⟶ Set partitions
St000609: Set partitions ⟶ ℤResult quality: 6% ●values known / values provided: 6%●distinct values known / distinct values provided: 21%
Values
0 => [2] => [1,1,0,0]
=> {{1,2}}
=> 0
1 => [1,1] => [1,0,1,0]
=> {{1},{2}}
=> 0
00 => [3] => [1,1,1,0,0,0]
=> {{1,2,3}}
=> 0
01 => [2,1] => [1,1,0,0,1,0]
=> {{1,2},{3}}
=> 0
10 => [1,2] => [1,0,1,1,0,0]
=> {{1},{2,3}}
=> 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]
=> {{1,2,3,4}}
=> 0
001 => [3,1] => [1,1,1,0,0,0,1,0]
=> {{1,2,3},{4}}
=> 0
010 => [2,2] => [1,1,0,0,1,1,0,0]
=> {{1,2},{3,4}}
=> 1
011 => [2,1,1] => [1,1,0,0,1,0,1,0]
=> {{1,2},{3},{4}}
=> 0
100 => [1,3] => [1,0,1,1,1,0,0,0]
=> {{1},{2,3,4}}
=> 2
101 => [1,2,1] => [1,0,1,1,0,0,1,0]
=> {{1},{2,3},{4}}
=> 1
110 => [1,1,2] => [1,0,1,0,1,1,0,0]
=> {{1},{2},{3,4}}
=> 2
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]
=> {{1,2,3,4,5}}
=> 0
0001 => [4,1] => [1,1,1,1,0,0,0,0,1,0]
=> {{1,2,3,4},{5}}
=> 0
0010 => [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> {{1,2,3},{4,5}}
=> 1
0011 => [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> {{1,2,3},{4},{5}}
=> 0
0100 => [2,3] => [1,1,0,0,1,1,1,0,0,0]
=> {{1,2},{3,4,5}}
=> 2
0101 => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> {{1,2},{3,4},{5}}
=> 1
0110 => [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
=> {{1,2},{3},{4,5}}
=> 2
0111 => [2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
=> {{1,2},{3},{4},{5}}
=> 0
1000 => [1,4] => [1,0,1,1,1,1,0,0,0,0]
=> {{1},{2,3,4,5}}
=> 3
1001 => [1,3,1] => [1,0,1,1,1,0,0,0,1,0]
=> {{1},{2,3,4},{5}}
=> 2
1010 => [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
=> {{1},{2,3},{4,5}}
=> 3
1011 => [1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
=> {{1},{2,3},{4},{5}}
=> 1
1100 => [1,1,3] => [1,0,1,0,1,1,1,0,0,0]
=> {{1},{2},{3,4,5}}
=> 4
1101 => [1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
=> {{1},{2},{3,4},{5}}
=> 2
1110 => [1,1,1,2] => [1,0,1,0,1,0,1,1,0,0]
=> {{1},{2},{3},{4,5}}
=> 3
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]
=> {{1,2,3,4,5,6}}
=> 0
00001 => [5,1] => [1,1,1,1,1,0,0,0,0,0,1,0]
=> {{1,2,3,4,5},{6}}
=> 0
00010 => [4,2] => [1,1,1,1,0,0,0,0,1,1,0,0]
=> {{1,2,3,4},{5,6}}
=> 1
00011 => [4,1,1] => [1,1,1,1,0,0,0,0,1,0,1,0]
=> {{1,2,3,4},{5},{6}}
=> 0
00100 => [3,3] => [1,1,1,0,0,0,1,1,1,0,0,0]
=> {{1,2,3},{4,5,6}}
=> 2
00101 => [3,2,1] => [1,1,1,0,0,0,1,1,0,0,1,0]
=> {{1,2,3},{4,5},{6}}
=> 1
00110 => [3,1,2] => [1,1,1,0,0,0,1,0,1,1,0,0]
=> {{1,2,3},{4},{5,6}}
=> 2
00111 => [3,1,1,1] => [1,1,1,0,0,0,1,0,1,0,1,0]
=> {{1,2,3},{4},{5},{6}}
=> 0
01000 => [2,4] => [1,1,0,0,1,1,1,1,0,0,0,0]
=> {{1,2},{3,4,5,6}}
=> 3
01001 => [2,3,1] => [1,1,0,0,1,1,1,0,0,0,1,0]
=> {{1,2},{3,4,5},{6}}
=> 2
01010 => [2,2,2] => [1,1,0,0,1,1,0,0,1,1,0,0]
=> {{1,2},{3,4},{5,6}}
=> 3
01011 => [2,2,1,1] => [1,1,0,0,1,1,0,0,1,0,1,0]
=> {{1,2},{3,4},{5},{6}}
=> 1
01100 => [2,1,3] => [1,1,0,0,1,0,1,1,1,0,0,0]
=> {{1,2},{3},{4,5,6}}
=> 4
01101 => [2,1,2,1] => [1,1,0,0,1,0,1,1,0,0,1,0]
=> {{1,2},{3},{4,5},{6}}
=> 2
01110 => [2,1,1,2] => [1,1,0,0,1,0,1,0,1,1,0,0]
=> {{1,2},{3},{4},{5,6}}
=> 3
01111 => [2,1,1,1,1] => [1,1,0,0,1,0,1,0,1,0,1,0]
=> {{1,2},{3},{4},{5},{6}}
=> 0
10000 => [1,5] => [1,0,1,1,1,1,1,0,0,0,0,0]
=> {{1},{2,3,4,5,6}}
=> 4
10001 => [1,4,1] => [1,0,1,1,1,1,0,0,0,0,1,0]
=> {{1},{2,3,4,5},{6}}
=> 3
10010 => [1,3,2] => [1,0,1,1,1,0,0,0,1,1,0,0]
=> {{1},{2,3,4},{5,6}}
=> 4
10011 => [1,3,1,1] => [1,0,1,1,1,0,0,0,1,0,1,0]
=> {{1},{2,3,4},{5},{6}}
=> 2
0000011 => [6,1,1] => [1,1,1,1,1,1,0,0,0,0,0,0,1,0,1,0]
=> {{1,2,3,4,5,6},{7},{8}}
=> ? = 0
0000100 => [5,3] => [1,1,1,1,1,0,0,0,0,0,1,1,1,0,0,0]
=> {{1,2,3,4,5},{6,7,8}}
=> ? = 2
0000101 => [5,2,1] => [1,1,1,1,1,0,0,0,0,0,1,1,0,0,1,0]
=> {{1,2,3,4,5},{6,7},{8}}
=> ? = 1
0000110 => [5,1,2] => [1,1,1,1,1,0,0,0,0,0,1,0,1,1,0,0]
=> {{1,2,3,4,5},{6},{7,8}}
=> ? = 2
0000111 => [5,1,1,1] => [1,1,1,1,1,0,0,0,0,0,1,0,1,0,1,0]
=> {{1,2,3,4,5},{6},{7},{8}}
=> ? = 0
0001000 => [4,4] => [1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0]
=> {{1,2,3,4},{5,6,7,8}}
=> ? = 3
0001001 => [4,3,1] => [1,1,1,1,0,0,0,0,1,1,1,0,0,0,1,0]
=> {{1,2,3,4},{5,6,7},{8}}
=> ? = 2
0001010 => [4,2,2] => [1,1,1,1,0,0,0,0,1,1,0,0,1,1,0,0]
=> {{1,2,3,4},{5,6},{7,8}}
=> ? = 3
0001011 => [4,2,1,1] => [1,1,1,1,0,0,0,0,1,1,0,0,1,0,1,0]
=> {{1,2,3,4},{5,6},{7},{8}}
=> ? = 1
0001100 => [4,1,3] => [1,1,1,1,0,0,0,0,1,0,1,1,1,0,0,0]
=> {{1,2,3,4},{5},{6,7,8}}
=> ? = 4
0001101 => [4,1,2,1] => [1,1,1,1,0,0,0,0,1,0,1,1,0,0,1,0]
=> {{1,2,3,4},{5},{6,7},{8}}
=> ? = 2
0001110 => [4,1,1,2] => [1,1,1,1,0,0,0,0,1,0,1,0,1,1,0,0]
=> {{1,2,3,4},{5},{6},{7,8}}
=> ? = 3
0001111 => [4,1,1,1,1] => [1,1,1,1,0,0,0,0,1,0,1,0,1,0,1,0]
=> {{1,2,3,4},{5},{6},{7},{8}}
=> ? = 0
0010000 => [3,5] => [1,1,1,0,0,0,1,1,1,1,1,0,0,0,0,0]
=> {{1,2,3},{4,5,6,7,8}}
=> ? = 4
0010001 => [3,4,1] => [1,1,1,0,0,0,1,1,1,1,0,0,0,0,1,0]
=> {{1,2,3},{4,5,6,7},{8}}
=> ? = 3
0010010 => [3,3,2] => [1,1,1,0,0,0,1,1,1,0,0,0,1,1,0,0]
=> {{1,2,3},{4,5,6},{7,8}}
=> ? = 4
0010011 => [3,3,1,1] => [1,1,1,0,0,0,1,1,1,0,0,0,1,0,1,0]
=> {{1,2,3},{4,5,6},{7},{8}}
=> ? = 2
0010100 => [3,2,3] => [1,1,1,0,0,0,1,1,0,0,1,1,1,0,0,0]
=> {{1,2,3},{4,5},{6,7,8}}
=> ? = 5
0010101 => [3,2,2,1] => [1,1,1,0,0,0,1,1,0,0,1,1,0,0,1,0]
=> {{1,2,3},{4,5},{6,7},{8}}
=> ? = 3
0010110 => [3,2,1,2] => [1,1,1,0,0,0,1,1,0,0,1,0,1,1,0,0]
=> {{1,2,3},{4,5},{6},{7,8}}
=> ? = 4
0010111 => [3,2,1,1,1] => [1,1,1,0,0,0,1,1,0,0,1,0,1,0,1,0]
=> {{1,2,3},{4,5},{6},{7},{8}}
=> ? = 1
0011000 => [3,1,4] => [1,1,1,0,0,0,1,0,1,1,1,1,0,0,0,0]
=> {{1,2,3},{4},{5,6,7,8}}
=> ? = 6
0011001 => [3,1,3,1] => [1,1,1,0,0,0,1,0,1,1,1,0,0,0,1,0]
=> {{1,2,3},{4},{5,6,7},{8}}
=> ? = 4
0011010 => [3,1,2,2] => [1,1,1,0,0,0,1,0,1,1,0,0,1,1,0,0]
=> {{1,2,3},{4},{5,6},{7,8}}
=> ? = 5
0011011 => [3,1,2,1,1] => [1,1,1,0,0,0,1,0,1,1,0,0,1,0,1,0]
=> {{1,2,3},{4},{5,6},{7},{8}}
=> ? = 2
0011100 => [3,1,1,3] => [1,1,1,0,0,0,1,0,1,0,1,1,1,0,0,0]
=> {{1,2,3},{4},{5},{6,7,8}}
=> ? = 6
0011101 => [3,1,1,2,1] => [1,1,1,0,0,0,1,0,1,0,1,1,0,0,1,0]
=> {{1,2,3},{4},{5},{6,7},{8}}
=> ? = 3
0011110 => [3,1,1,1,2] => [1,1,1,0,0,0,1,0,1,0,1,0,1,1,0,0]
=> {{1,2,3},{4},{5},{6},{7,8}}
=> ? = 4
0011111 => [3,1,1,1,1,1] => [1,1,1,0,0,0,1,0,1,0,1,0,1,0,1,0]
=> {{1,2,3},{4},{5},{6},{7},{8}}
=> ? = 0
0100001 => [2,5,1] => [1,1,0,0,1,1,1,1,1,0,0,0,0,0,1,0]
=> {{1,2},{3,4,5,6,7},{8}}
=> ? = 4
0100010 => [2,4,2] => [1,1,0,0,1,1,1,1,0,0,0,0,1,1,0,0]
=> {{1,2},{3,4,5,6},{7,8}}
=> ? = 5
0100011 => [2,4,1,1] => [1,1,0,0,1,1,1,1,0,0,0,0,1,0,1,0]
=> {{1,2},{3,4,5,6},{7},{8}}
=> ? = 3
0100100 => [2,3,3] => [1,1,0,0,1,1,1,0,0,0,1,1,1,0,0,0]
=> {{1,2},{3,4,5},{6,7,8}}
=> ? = 6
0100101 => [2,3,2,1] => [1,1,0,0,1,1,1,0,0,0,1,1,0,0,1,0]
=> {{1,2},{3,4,5},{6,7},{8}}
=> ? = 4
0100110 => [2,3,1,2] => [1,1,0,0,1,1,1,0,0,0,1,0,1,1,0,0]
=> {{1,2},{3,4,5},{6},{7,8}}
=> ? = 5
0100111 => [2,3,1,1,1] => [1,1,0,0,1,1,1,0,0,0,1,0,1,0,1,0]
=> {{1,2},{3,4,5},{6},{7},{8}}
=> ? = 2
0101000 => [2,2,4] => [1,1,0,0,1,1,0,0,1,1,1,1,0,0,0,0]
=> {{1,2},{3,4},{5,6,7,8}}
=> ? = 7
0101001 => [2,2,3,1] => [1,1,0,0,1,1,0,0,1,1,1,0,0,0,1,0]
=> {{1,2},{3,4},{5,6,7},{8}}
=> ? = 5
0101010 => [2,2,2,2] => [1,1,0,0,1,1,0,0,1,1,0,0,1,1,0,0]
=> {{1,2},{3,4},{5,6},{7,8}}
=> ? = 6
0101011 => [2,2,2,1,1] => [1,1,0,0,1,1,0,0,1,1,0,0,1,0,1,0]
=> {{1,2},{3,4},{5,6},{7},{8}}
=> ? = 3
0101100 => [2,2,1,3] => [1,1,0,0,1,1,0,0,1,0,1,1,1,0,0,0]
=> {{1,2},{3,4},{5},{6,7,8}}
=> ? = 7
0101101 => [2,2,1,2,1] => [1,1,0,0,1,1,0,0,1,0,1,1,0,0,1,0]
=> {{1,2},{3,4},{5},{6,7},{8}}
=> ? = 4
0101110 => [2,2,1,1,2] => [1,1,0,0,1,1,0,0,1,0,1,0,1,1,0,0]
=> {{1,2},{3,4},{5},{6},{7,8}}
=> ? = 5
0101111 => [2,2,1,1,1,1] => [1,1,0,0,1,1,0,0,1,0,1,0,1,0,1,0]
=> {{1,2},{3,4},{5},{6},{7},{8}}
=> ? = 1
0110000 => [2,1,5] => [1,1,0,0,1,0,1,1,1,1,1,0,0,0,0,0]
=> {{1,2},{3},{4,5,6,7,8}}
=> ? = 8
0110001 => [2,1,4,1] => [1,1,0,0,1,0,1,1,1,1,0,0,0,0,1,0]
=> {{1,2},{3},{4,5,6,7},{8}}
=> ? = 6
0110010 => [2,1,3,2] => [1,1,0,0,1,0,1,1,1,0,0,0,1,1,0,0]
=> {{1,2},{3},{4,5,6},{7,8}}
=> ? = 7
0110011 => [2,1,3,1,1] => [1,1,0,0,1,0,1,1,1,0,0,0,1,0,1,0]
=> {{1,2},{3},{4,5,6},{7},{8}}
=> ? = 4
0110100 => [2,1,2,3] => [1,1,0,0,1,0,1,1,0,0,1,1,1,0,0,0]
=> {{1,2},{3},{4,5},{6,7,8}}
=> ? = 8
0110101 => [2,1,2,2,1] => [1,1,0,0,1,0,1,1,0,0,1,1,0,0,1,0]
=> {{1,2},{3},{4,5},{6,7},{8}}
=> ? = 5
Description
The number of occurrences of the pattern {{1},{2,3}} such that 1,2 are minimal.
Matching statistic: St000612
Mp00178: Binary words —to composition⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
Mp00138: Dyck paths —to noncrossing partition⟶ Set partitions
St000612: Set partitions ⟶ ℤResult quality: 6% ●values known / values provided: 6%●distinct values known / distinct values provided: 21%
Mp00231: Integer compositions —bounce path⟶ Dyck paths
Mp00138: Dyck paths —to noncrossing partition⟶ Set partitions
St000612: Set partitions ⟶ ℤResult quality: 6% ●values known / values provided: 6%●distinct values known / distinct values provided: 21%
Values
0 => [2] => [1,1,0,0]
=> {{1,2}}
=> 0
1 => [1,1] => [1,0,1,0]
=> {{1},{2}}
=> 0
00 => [3] => [1,1,1,0,0,0]
=> {{1,2,3}}
=> 0
01 => [2,1] => [1,1,0,0,1,0]
=> {{1,2},{3}}
=> 0
10 => [1,2] => [1,0,1,1,0,0]
=> {{1},{2,3}}
=> 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]
=> {{1,2,3,4}}
=> 0
001 => [3,1] => [1,1,1,0,0,0,1,0]
=> {{1,2,3},{4}}
=> 0
010 => [2,2] => [1,1,0,0,1,1,0,0]
=> {{1,2},{3,4}}
=> 1
011 => [2,1,1] => [1,1,0,0,1,0,1,0]
=> {{1,2},{3},{4}}
=> 0
100 => [1,3] => [1,0,1,1,1,0,0,0]
=> {{1},{2,3,4}}
=> 2
101 => [1,2,1] => [1,0,1,1,0,0,1,0]
=> {{1},{2,3},{4}}
=> 1
110 => [1,1,2] => [1,0,1,0,1,1,0,0]
=> {{1},{2},{3,4}}
=> 2
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]
=> {{1,2,3,4,5}}
=> 0
0001 => [4,1] => [1,1,1,1,0,0,0,0,1,0]
=> {{1,2,3,4},{5}}
=> 0
0010 => [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> {{1,2,3},{4,5}}
=> 1
0011 => [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> {{1,2,3},{4},{5}}
=> 0
0100 => [2,3] => [1,1,0,0,1,1,1,0,0,0]
=> {{1,2},{3,4,5}}
=> 2
0101 => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> {{1,2},{3,4},{5}}
=> 1
0110 => [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
=> {{1,2},{3},{4,5}}
=> 2
0111 => [2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
=> {{1,2},{3},{4},{5}}
=> 0
1000 => [1,4] => [1,0,1,1,1,1,0,0,0,0]
=> {{1},{2,3,4,5}}
=> 3
1001 => [1,3,1] => [1,0,1,1,1,0,0,0,1,0]
=> {{1},{2,3,4},{5}}
=> 2
1010 => [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
=> {{1},{2,3},{4,5}}
=> 3
1011 => [1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
=> {{1},{2,3},{4},{5}}
=> 1
1100 => [1,1,3] => [1,0,1,0,1,1,1,0,0,0]
=> {{1},{2},{3,4,5}}
=> 4
1101 => [1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
=> {{1},{2},{3,4},{5}}
=> 2
1110 => [1,1,1,2] => [1,0,1,0,1,0,1,1,0,0]
=> {{1},{2},{3},{4,5}}
=> 3
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]
=> {{1,2,3,4,5,6}}
=> 0
00001 => [5,1] => [1,1,1,1,1,0,0,0,0,0,1,0]
=> {{1,2,3,4,5},{6}}
=> 0
00010 => [4,2] => [1,1,1,1,0,0,0,0,1,1,0,0]
=> {{1,2,3,4},{5,6}}
=> 1
00011 => [4,1,1] => [1,1,1,1,0,0,0,0,1,0,1,0]
=> {{1,2,3,4},{5},{6}}
=> 0
00100 => [3,3] => [1,1,1,0,0,0,1,1,1,0,0,0]
=> {{1,2,3},{4,5,6}}
=> 2
00101 => [3,2,1] => [1,1,1,0,0,0,1,1,0,0,1,0]
=> {{1,2,3},{4,5},{6}}
=> 1
00110 => [3,1,2] => [1,1,1,0,0,0,1,0,1,1,0,0]
=> {{1,2,3},{4},{5,6}}
=> 2
00111 => [3,1,1,1] => [1,1,1,0,0,0,1,0,1,0,1,0]
=> {{1,2,3},{4},{5},{6}}
=> 0
01000 => [2,4] => [1,1,0,0,1,1,1,1,0,0,0,0]
=> {{1,2},{3,4,5,6}}
=> 3
01001 => [2,3,1] => [1,1,0,0,1,1,1,0,0,0,1,0]
=> {{1,2},{3,4,5},{6}}
=> 2
01010 => [2,2,2] => [1,1,0,0,1,1,0,0,1,1,0,0]
=> {{1,2},{3,4},{5,6}}
=> 3
01011 => [2,2,1,1] => [1,1,0,0,1,1,0,0,1,0,1,0]
=> {{1,2},{3,4},{5},{6}}
=> 1
01100 => [2,1,3] => [1,1,0,0,1,0,1,1,1,0,0,0]
=> {{1,2},{3},{4,5,6}}
=> 4
01101 => [2,1,2,1] => [1,1,0,0,1,0,1,1,0,0,1,0]
=> {{1,2},{3},{4,5},{6}}
=> 2
01110 => [2,1,1,2] => [1,1,0,0,1,0,1,0,1,1,0,0]
=> {{1,2},{3},{4},{5,6}}
=> 3
01111 => [2,1,1,1,1] => [1,1,0,0,1,0,1,0,1,0,1,0]
=> {{1,2},{3},{4},{5},{6}}
=> 0
10000 => [1,5] => [1,0,1,1,1,1,1,0,0,0,0,0]
=> {{1},{2,3,4,5,6}}
=> 4
10001 => [1,4,1] => [1,0,1,1,1,1,0,0,0,0,1,0]
=> {{1},{2,3,4,5},{6}}
=> 3
10010 => [1,3,2] => [1,0,1,1,1,0,0,0,1,1,0,0]
=> {{1},{2,3,4},{5,6}}
=> 4
10011 => [1,3,1,1] => [1,0,1,1,1,0,0,0,1,0,1,0]
=> {{1},{2,3,4},{5},{6}}
=> 2
0000011 => [6,1,1] => [1,1,1,1,1,1,0,0,0,0,0,0,1,0,1,0]
=> {{1,2,3,4,5,6},{7},{8}}
=> ? = 0
0000100 => [5,3] => [1,1,1,1,1,0,0,0,0,0,1,1,1,0,0,0]
=> {{1,2,3,4,5},{6,7,8}}
=> ? = 2
0000101 => [5,2,1] => [1,1,1,1,1,0,0,0,0,0,1,1,0,0,1,0]
=> {{1,2,3,4,5},{6,7},{8}}
=> ? = 1
0000110 => [5,1,2] => [1,1,1,1,1,0,0,0,0,0,1,0,1,1,0,0]
=> {{1,2,3,4,5},{6},{7,8}}
=> ? = 2
0000111 => [5,1,1,1] => [1,1,1,1,1,0,0,0,0,0,1,0,1,0,1,0]
=> {{1,2,3,4,5},{6},{7},{8}}
=> ? = 0
0001000 => [4,4] => [1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0]
=> {{1,2,3,4},{5,6,7,8}}
=> ? = 3
0001001 => [4,3,1] => [1,1,1,1,0,0,0,0,1,1,1,0,0,0,1,0]
=> {{1,2,3,4},{5,6,7},{8}}
=> ? = 2
0001010 => [4,2,2] => [1,1,1,1,0,0,0,0,1,1,0,0,1,1,0,0]
=> {{1,2,3,4},{5,6},{7,8}}
=> ? = 3
0001011 => [4,2,1,1] => [1,1,1,1,0,0,0,0,1,1,0,0,1,0,1,0]
=> {{1,2,3,4},{5,6},{7},{8}}
=> ? = 1
0001100 => [4,1,3] => [1,1,1,1,0,0,0,0,1,0,1,1,1,0,0,0]
=> {{1,2,3,4},{5},{6,7,8}}
=> ? = 4
0001101 => [4,1,2,1] => [1,1,1,1,0,0,0,0,1,0,1,1,0,0,1,0]
=> {{1,2,3,4},{5},{6,7},{8}}
=> ? = 2
0001110 => [4,1,1,2] => [1,1,1,1,0,0,0,0,1,0,1,0,1,1,0,0]
=> {{1,2,3,4},{5},{6},{7,8}}
=> ? = 3
0001111 => [4,1,1,1,1] => [1,1,1,1,0,0,0,0,1,0,1,0,1,0,1,0]
=> {{1,2,3,4},{5},{6},{7},{8}}
=> ? = 0
0010000 => [3,5] => [1,1,1,0,0,0,1,1,1,1,1,0,0,0,0,0]
=> {{1,2,3},{4,5,6,7,8}}
=> ? = 4
0010001 => [3,4,1] => [1,1,1,0,0,0,1,1,1,1,0,0,0,0,1,0]
=> {{1,2,3},{4,5,6,7},{8}}
=> ? = 3
0010010 => [3,3,2] => [1,1,1,0,0,0,1,1,1,0,0,0,1,1,0,0]
=> {{1,2,3},{4,5,6},{7,8}}
=> ? = 4
0010011 => [3,3,1,1] => [1,1,1,0,0,0,1,1,1,0,0,0,1,0,1,0]
=> {{1,2,3},{4,5,6},{7},{8}}
=> ? = 2
0010100 => [3,2,3] => [1,1,1,0,0,0,1,1,0,0,1,1,1,0,0,0]
=> {{1,2,3},{4,5},{6,7,8}}
=> ? = 5
0010101 => [3,2,2,1] => [1,1,1,0,0,0,1,1,0,0,1,1,0,0,1,0]
=> {{1,2,3},{4,5},{6,7},{8}}
=> ? = 3
0010110 => [3,2,1,2] => [1,1,1,0,0,0,1,1,0,0,1,0,1,1,0,0]
=> {{1,2,3},{4,5},{6},{7,8}}
=> ? = 4
0010111 => [3,2,1,1,1] => [1,1,1,0,0,0,1,1,0,0,1,0,1,0,1,0]
=> {{1,2,3},{4,5},{6},{7},{8}}
=> ? = 1
0011000 => [3,1,4] => [1,1,1,0,0,0,1,0,1,1,1,1,0,0,0,0]
=> {{1,2,3},{4},{5,6,7,8}}
=> ? = 6
0011001 => [3,1,3,1] => [1,1,1,0,0,0,1,0,1,1,1,0,0,0,1,0]
=> {{1,2,3},{4},{5,6,7},{8}}
=> ? = 4
0011010 => [3,1,2,2] => [1,1,1,0,0,0,1,0,1,1,0,0,1,1,0,0]
=> {{1,2,3},{4},{5,6},{7,8}}
=> ? = 5
0011011 => [3,1,2,1,1] => [1,1,1,0,0,0,1,0,1,1,0,0,1,0,1,0]
=> {{1,2,3},{4},{5,6},{7},{8}}
=> ? = 2
0011100 => [3,1,1,3] => [1,1,1,0,0,0,1,0,1,0,1,1,1,0,0,0]
=> {{1,2,3},{4},{5},{6,7,8}}
=> ? = 6
0011101 => [3,1,1,2,1] => [1,1,1,0,0,0,1,0,1,0,1,1,0,0,1,0]
=> {{1,2,3},{4},{5},{6,7},{8}}
=> ? = 3
0011110 => [3,1,1,1,2] => [1,1,1,0,0,0,1,0,1,0,1,0,1,1,0,0]
=> {{1,2,3},{4},{5},{6},{7,8}}
=> ? = 4
0011111 => [3,1,1,1,1,1] => [1,1,1,0,0,0,1,0,1,0,1,0,1,0,1,0]
=> {{1,2,3},{4},{5},{6},{7},{8}}
=> ? = 0
0100001 => [2,5,1] => [1,1,0,0,1,1,1,1,1,0,0,0,0,0,1,0]
=> {{1,2},{3,4,5,6,7},{8}}
=> ? = 4
0100010 => [2,4,2] => [1,1,0,0,1,1,1,1,0,0,0,0,1,1,0,0]
=> {{1,2},{3,4,5,6},{7,8}}
=> ? = 5
0100011 => [2,4,1,1] => [1,1,0,0,1,1,1,1,0,0,0,0,1,0,1,0]
=> {{1,2},{3,4,5,6},{7},{8}}
=> ? = 3
0100100 => [2,3,3] => [1,1,0,0,1,1,1,0,0,0,1,1,1,0,0,0]
=> {{1,2},{3,4,5},{6,7,8}}
=> ? = 6
0100101 => [2,3,2,1] => [1,1,0,0,1,1,1,0,0,0,1,1,0,0,1,0]
=> {{1,2},{3,4,5},{6,7},{8}}
=> ? = 4
0100110 => [2,3,1,2] => [1,1,0,0,1,1,1,0,0,0,1,0,1,1,0,0]
=> {{1,2},{3,4,5},{6},{7,8}}
=> ? = 5
0100111 => [2,3,1,1,1] => [1,1,0,0,1,1,1,0,0,0,1,0,1,0,1,0]
=> {{1,2},{3,4,5},{6},{7},{8}}
=> ? = 2
0101000 => [2,2,4] => [1,1,0,0,1,1,0,0,1,1,1,1,0,0,0,0]
=> {{1,2},{3,4},{5,6,7,8}}
=> ? = 7
0101001 => [2,2,3,1] => [1,1,0,0,1,1,0,0,1,1,1,0,0,0,1,0]
=> {{1,2},{3,4},{5,6,7},{8}}
=> ? = 5
0101010 => [2,2,2,2] => [1,1,0,0,1,1,0,0,1,1,0,0,1,1,0,0]
=> {{1,2},{3,4},{5,6},{7,8}}
=> ? = 6
0101011 => [2,2,2,1,1] => [1,1,0,0,1,1,0,0,1,1,0,0,1,0,1,0]
=> {{1,2},{3,4},{5,6},{7},{8}}
=> ? = 3
0101100 => [2,2,1,3] => [1,1,0,0,1,1,0,0,1,0,1,1,1,0,0,0]
=> {{1,2},{3,4},{5},{6,7,8}}
=> ? = 7
0101101 => [2,2,1,2,1] => [1,1,0,0,1,1,0,0,1,0,1,1,0,0,1,0]
=> {{1,2},{3,4},{5},{6,7},{8}}
=> ? = 4
0101110 => [2,2,1,1,2] => [1,1,0,0,1,1,0,0,1,0,1,0,1,1,0,0]
=> {{1,2},{3,4},{5},{6},{7,8}}
=> ? = 5
0101111 => [2,2,1,1,1,1] => [1,1,0,0,1,1,0,0,1,0,1,0,1,0,1,0]
=> {{1,2},{3,4},{5},{6},{7},{8}}
=> ? = 1
0110000 => [2,1,5] => [1,1,0,0,1,0,1,1,1,1,1,0,0,0,0,0]
=> {{1,2},{3},{4,5,6,7,8}}
=> ? = 8
0110001 => [2,1,4,1] => [1,1,0,0,1,0,1,1,1,1,0,0,0,0,1,0]
=> {{1,2},{3},{4,5,6,7},{8}}
=> ? = 6
0110010 => [2,1,3,2] => [1,1,0,0,1,0,1,1,1,0,0,0,1,1,0,0]
=> {{1,2},{3},{4,5,6},{7,8}}
=> ? = 7
0110011 => [2,1,3,1,1] => [1,1,0,0,1,0,1,1,1,0,0,0,1,0,1,0]
=> {{1,2},{3},{4,5,6},{7},{8}}
=> ? = 4
0110100 => [2,1,2,3] => [1,1,0,0,1,0,1,1,0,0,1,1,1,0,0,0]
=> {{1,2},{3},{4,5},{6,7,8}}
=> ? = 8
0110101 => [2,1,2,2,1] => [1,1,0,0,1,0,1,1,0,0,1,1,0,0,1,0]
=> {{1,2},{3},{4,5},{6,7},{8}}
=> ? = 5
Description
The number of occurrences of the pattern {{1},{2,3}} such that 1 is minimal, (2,3) are consecutive in a block.
Matching statistic: St000516
Mp00178: Binary words —to composition⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
Mp00201: Dyck paths —Ringel⟶ Permutations
St000516: Permutations ⟶ ℤResult quality: 1% ●values known / values provided: 1%●distinct values known / distinct values provided: 10%
Mp00231: Integer compositions —bounce path⟶ Dyck paths
Mp00201: Dyck paths —Ringel⟶ Permutations
St000516: Permutations ⟶ ℤResult quality: 1% ●values known / values provided: 1%●distinct values known / distinct values provided: 10%
Values
0 => [2] => [1,1,0,0]
=> [2,3,1] => 0
1 => [1,1] => [1,0,1,0]
=> [3,1,2] => 0
00 => [3] => [1,1,1,0,0,0]
=> [2,3,4,1] => 0
01 => [2,1] => [1,1,0,0,1,0]
=> [2,4,1,3] => 0
10 => [1,2] => [1,0,1,1,0,0]
=> [3,1,4,2] => 1
11 => [1,1,1] => [1,0,1,0,1,0]
=> [4,1,2,3] => 0
000 => [4] => [1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => 0
001 => [3,1] => [1,1,1,0,0,0,1,0]
=> [2,3,5,1,4] => 0
010 => [2,2] => [1,1,0,0,1,1,0,0]
=> [2,4,1,5,3] => 1
011 => [2,1,1] => [1,1,0,0,1,0,1,0]
=> [2,5,1,3,4] => 0
100 => [1,3] => [1,0,1,1,1,0,0,0]
=> [3,1,4,5,2] => 2
101 => [1,2,1] => [1,0,1,1,0,0,1,0]
=> [3,1,5,2,4] => 1
110 => [1,1,2] => [1,0,1,0,1,1,0,0]
=> [4,1,2,5,3] => 2
111 => [1,1,1,1] => [1,0,1,0,1,0,1,0]
=> [5,1,2,3,4] => 0
0000 => [5] => [1,1,1,1,1,0,0,0,0,0]
=> [2,3,4,5,6,1] => 0
0001 => [4,1] => [1,1,1,1,0,0,0,0,1,0]
=> [2,3,4,6,1,5] => 0
0010 => [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> [2,3,5,1,6,4] => 1
0011 => [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> [2,3,6,1,4,5] => 0
0100 => [2,3] => [1,1,0,0,1,1,1,0,0,0]
=> [2,4,1,5,6,3] => 2
0101 => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> [2,4,1,6,3,5] => 1
0110 => [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
=> [2,5,1,3,6,4] => 2
0111 => [2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
=> [2,6,1,3,4,5] => 0
1000 => [1,4] => [1,0,1,1,1,1,0,0,0,0]
=> [3,1,4,5,6,2] => 3
1001 => [1,3,1] => [1,0,1,1,1,0,0,0,1,0]
=> [3,1,4,6,2,5] => 2
1010 => [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
=> [3,1,5,2,6,4] => 3
1011 => [1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
=> [3,1,6,2,4,5] => 1
1100 => [1,1,3] => [1,0,1,0,1,1,1,0,0,0]
=> [4,1,2,5,6,3] => 4
1101 => [1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
=> [4,1,2,6,3,5] => 2
1110 => [1,1,1,2] => [1,0,1,0,1,0,1,1,0,0]
=> [5,1,2,3,6,4] => 3
1111 => [1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
=> [6,1,2,3,4,5] => 0
00000 => [6] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> [2,3,4,5,6,7,1] => ? = 0
00001 => [5,1] => [1,1,1,1,1,0,0,0,0,0,1,0]
=> [2,3,4,5,7,1,6] => ? = 0
00010 => [4,2] => [1,1,1,1,0,0,0,0,1,1,0,0]
=> [2,3,4,6,1,7,5] => ? = 1
00011 => [4,1,1] => [1,1,1,1,0,0,0,0,1,0,1,0]
=> [2,3,4,7,1,5,6] => ? = 0
00100 => [3,3] => [1,1,1,0,0,0,1,1,1,0,0,0]
=> [2,3,5,1,6,7,4] => ? = 2
00101 => [3,2,1] => [1,1,1,0,0,0,1,1,0,0,1,0]
=> [2,3,5,1,7,4,6] => ? = 1
00110 => [3,1,2] => [1,1,1,0,0,0,1,0,1,1,0,0]
=> [2,3,6,1,4,7,5] => ? = 2
00111 => [3,1,1,1] => [1,1,1,0,0,0,1,0,1,0,1,0]
=> [2,3,7,1,4,5,6] => ? = 0
01000 => [2,4] => [1,1,0,0,1,1,1,1,0,0,0,0]
=> [2,4,1,5,6,7,3] => ? = 3
01001 => [2,3,1] => [1,1,0,0,1,1,1,0,0,0,1,0]
=> [2,4,1,5,7,3,6] => ? = 2
01010 => [2,2,2] => [1,1,0,0,1,1,0,0,1,1,0,0]
=> [2,4,1,6,3,7,5] => ? = 3
01011 => [2,2,1,1] => [1,1,0,0,1,1,0,0,1,0,1,0]
=> [2,4,1,7,3,5,6] => ? = 1
01100 => [2,1,3] => [1,1,0,0,1,0,1,1,1,0,0,0]
=> [2,5,1,3,6,7,4] => ? = 4
01101 => [2,1,2,1] => [1,1,0,0,1,0,1,1,0,0,1,0]
=> [2,5,1,3,7,4,6] => ? = 2
01110 => [2,1,1,2] => [1,1,0,0,1,0,1,0,1,1,0,0]
=> [2,6,1,3,4,7,5] => ? = 3
01111 => [2,1,1,1,1] => [1,1,0,0,1,0,1,0,1,0,1,0]
=> [2,7,1,3,4,5,6] => ? = 0
10000 => [1,5] => [1,0,1,1,1,1,1,0,0,0,0,0]
=> [3,1,4,5,6,7,2] => ? = 4
10001 => [1,4,1] => [1,0,1,1,1,1,0,0,0,0,1,0]
=> [3,1,4,5,7,2,6] => ? = 3
10010 => [1,3,2] => [1,0,1,1,1,0,0,0,1,1,0,0]
=> [3,1,4,6,2,7,5] => ? = 4
10011 => [1,3,1,1] => [1,0,1,1,1,0,0,0,1,0,1,0]
=> [3,1,4,7,2,5,6] => ? = 2
10100 => [1,2,3] => [1,0,1,1,0,0,1,1,1,0,0,0]
=> [3,1,5,2,6,7,4] => ? = 5
10101 => [1,2,2,1] => [1,0,1,1,0,0,1,1,0,0,1,0]
=> [3,1,5,2,7,4,6] => ? = 3
10110 => [1,2,1,2] => [1,0,1,1,0,0,1,0,1,1,0,0]
=> [3,1,6,2,4,7,5] => ? = 4
10111 => [1,2,1,1,1] => [1,0,1,1,0,0,1,0,1,0,1,0]
=> [3,1,7,2,4,5,6] => ? = 1
11000 => [1,1,4] => [1,0,1,0,1,1,1,1,0,0,0,0]
=> [4,1,2,5,6,7,3] => ? = 6
11001 => [1,1,3,1] => [1,0,1,0,1,1,1,0,0,0,1,0]
=> [4,1,2,5,7,3,6] => ? = 4
11010 => [1,1,2,2] => [1,0,1,0,1,1,0,0,1,1,0,0]
=> [4,1,2,6,3,7,5] => ? = 5
11011 => [1,1,2,1,1] => [1,0,1,0,1,1,0,0,1,0,1,0]
=> [4,1,2,7,3,5,6] => ? = 2
11100 => [1,1,1,3] => [1,0,1,0,1,0,1,1,1,0,0,0]
=> [5,1,2,3,6,7,4] => ? = 6
11101 => [1,1,1,2,1] => [1,0,1,0,1,0,1,1,0,0,1,0]
=> [5,1,2,3,7,4,6] => ? = 3
11110 => [1,1,1,1,2] => [1,0,1,0,1,0,1,0,1,1,0,0]
=> [6,1,2,3,4,7,5] => ? = 4
11111 => [1,1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0,1,0]
=> [7,1,2,3,4,5,6] => ? = 0
000000 => [7] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [2,3,4,5,6,7,8,1] => ? = 0
000001 => [6,1] => [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> [2,3,4,5,6,8,1,7] => ? = 0
000010 => [5,2] => [1,1,1,1,1,0,0,0,0,0,1,1,0,0]
=> [2,3,4,5,7,1,8,6] => ? = 1
000011 => [5,1,1] => [1,1,1,1,1,0,0,0,0,0,1,0,1,0]
=> [2,3,4,5,8,1,6,7] => ? = 0
000100 => [4,3] => [1,1,1,1,0,0,0,0,1,1,1,0,0,0]
=> [2,3,4,6,1,7,8,5] => ? = 2
000101 => [4,2,1] => [1,1,1,1,0,0,0,0,1,1,0,0,1,0]
=> [2,3,4,6,1,8,5,7] => ? = 1
000110 => [4,1,2] => [1,1,1,1,0,0,0,0,1,0,1,1,0,0]
=> [2,3,4,7,1,5,8,6] => ? = 2
000111 => [4,1,1,1] => [1,1,1,1,0,0,0,0,1,0,1,0,1,0]
=> [2,3,4,8,1,5,6,7] => ? = 0
001000 => [3,4] => [1,1,1,0,0,0,1,1,1,1,0,0,0,0]
=> [2,3,5,1,6,7,8,4] => ? = 3
001001 => [3,3,1] => [1,1,1,0,0,0,1,1,1,0,0,0,1,0]
=> [2,3,5,1,6,8,4,7] => ? = 2
001010 => [3,2,2] => [1,1,1,0,0,0,1,1,0,0,1,1,0,0]
=> [2,3,5,1,7,4,8,6] => ? = 3
001011 => [3,2,1,1] => [1,1,1,0,0,0,1,1,0,0,1,0,1,0]
=> [2,3,5,1,8,4,6,7] => ? = 1
001100 => [3,1,3] => [1,1,1,0,0,0,1,0,1,1,1,0,0,0]
=> [2,3,6,1,4,7,8,5] => ? = 4
001101 => [3,1,2,1] => [1,1,1,0,0,0,1,0,1,1,0,0,1,0]
=> [2,3,6,1,4,8,5,7] => ? = 2
001110 => [3,1,1,2] => [1,1,1,0,0,0,1,0,1,0,1,1,0,0]
=> [2,3,7,1,4,5,8,6] => ? = 3
001111 => [3,1,1,1,1] => [1,1,1,0,0,0,1,0,1,0,1,0,1,0]
=> [2,3,8,1,4,5,6,7] => ? = 0
010000 => [2,5] => [1,1,0,0,1,1,1,1,1,0,0,0,0,0]
=> [2,4,1,5,6,7,8,3] => ? = 4
010001 => [2,4,1] => [1,1,0,0,1,1,1,1,0,0,0,0,1,0]
=> [2,4,1,5,6,8,3,7] => ? = 3
Description
The number of stretching pairs of a permutation.
This is the number of pairs $(i,j)$ with $\pi(i) < i < j < \pi(j)$.
Matching statistic: St001438
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00104: Binary words —reverse⟶ Binary words
Mp00178: Binary words —to composition⟶ Integer compositions
Mp00180: Integer compositions —to ribbon⟶ Skew partitions
St001438: Skew partitions ⟶ ℤResult quality: 1% ●values known / values provided: 1%●distinct values known / distinct values provided: 10%
Mp00178: Binary words —to composition⟶ Integer compositions
Mp00180: Integer compositions —to ribbon⟶ Skew partitions
St001438: Skew partitions ⟶ ℤResult quality: 1% ●values known / values provided: 1%●distinct values known / distinct values provided: 10%
Values
0 => 0 => [2] => [[2],[]]
=> 0
1 => 1 => [1,1] => [[1,1],[]]
=> 0
00 => 00 => [3] => [[3],[]]
=> 0
01 => 10 => [1,2] => [[2,1],[]]
=> 0
10 => 01 => [2,1] => [[2,2],[1]]
=> 1
11 => 11 => [1,1,1] => [[1,1,1],[]]
=> 0
000 => 000 => [4] => [[4],[]]
=> 0
001 => 100 => [1,3] => [[3,1],[]]
=> 0
010 => 010 => [2,2] => [[3,2],[1]]
=> 1
011 => 110 => [1,1,2] => [[2,1,1],[]]
=> 0
100 => 001 => [3,1] => [[3,3],[2]]
=> 2
101 => 101 => [1,2,1] => [[2,2,1],[1]]
=> 1
110 => 011 => [2,1,1] => [[2,2,2],[1,1]]
=> 2
111 => 111 => [1,1,1,1] => [[1,1,1,1],[]]
=> 0
0000 => 0000 => [5] => [[5],[]]
=> 0
0001 => 1000 => [1,4] => [[4,1],[]]
=> 0
0010 => 0100 => [2,3] => [[4,2],[1]]
=> 1
0011 => 1100 => [1,1,3] => [[3,1,1],[]]
=> 0
0100 => 0010 => [3,2] => [[4,3],[2]]
=> 2
0101 => 1010 => [1,2,2] => [[3,2,1],[1]]
=> 1
0110 => 0110 => [2,1,2] => [[3,2,2],[1,1]]
=> 2
0111 => 1110 => [1,1,1,2] => [[2,1,1,1],[]]
=> 0
1000 => 0001 => [4,1] => [[4,4],[3]]
=> 3
1001 => 1001 => [1,3,1] => [[3,3,1],[2]]
=> 2
1010 => 0101 => [2,2,1] => [[3,3,2],[2,1]]
=> 3
1011 => 1101 => [1,1,2,1] => [[2,2,1,1],[1]]
=> 1
1100 => 0011 => [3,1,1] => [[3,3,3],[2,2]]
=> 4
1101 => 1011 => [1,2,1,1] => [[2,2,2,1],[1,1]]
=> 2
1110 => 0111 => [2,1,1,1] => [[2,2,2,2],[1,1,1]]
=> 3
1111 => 1111 => [1,1,1,1,1] => [[1,1,1,1,1],[]]
=> 0
00000 => 00000 => [6] => [[6],[]]
=> ? = 0
00001 => 10000 => [1,5] => [[5,1],[]]
=> ? = 0
00010 => 01000 => [2,4] => [[5,2],[1]]
=> ? = 1
00011 => 11000 => [1,1,4] => [[4,1,1],[]]
=> ? = 0
00100 => 00100 => [3,3] => [[5,3],[2]]
=> ? = 2
00101 => 10100 => [1,2,3] => [[4,2,1],[1]]
=> ? = 1
00110 => 01100 => [2,1,3] => [[4,2,2],[1,1]]
=> ? = 2
00111 => 11100 => [1,1,1,3] => [[3,1,1,1],[]]
=> ? = 0
01000 => 00010 => [4,2] => [[5,4],[3]]
=> ? = 3
01001 => 10010 => [1,3,2] => [[4,3,1],[2]]
=> ? = 2
01010 => 01010 => [2,2,2] => [[4,3,2],[2,1]]
=> ? = 3
01011 => 11010 => [1,1,2,2] => [[3,2,1,1],[1]]
=> ? = 1
01100 => 00110 => [3,1,2] => [[4,3,3],[2,2]]
=> ? = 4
01101 => 10110 => [1,2,1,2] => [[3,2,2,1],[1,1]]
=> ? = 2
01110 => 01110 => [2,1,1,2] => [[3,2,2,2],[1,1,1]]
=> ? = 3
01111 => 11110 => [1,1,1,1,2] => [[2,1,1,1,1],[]]
=> ? = 0
10000 => 00001 => [5,1] => [[5,5],[4]]
=> ? = 4
10001 => 10001 => [1,4,1] => [[4,4,1],[3]]
=> ? = 3
10010 => 01001 => [2,3,1] => [[4,4,2],[3,1]]
=> ? = 4
10011 => 11001 => [1,1,3,1] => [[3,3,1,1],[2]]
=> ? = 2
10100 => 00101 => [3,2,1] => [[4,4,3],[3,2]]
=> ? = 5
10101 => 10101 => [1,2,2,1] => [[3,3,2,1],[2,1]]
=> ? = 3
10110 => 01101 => [2,1,2,1] => [[3,3,2,2],[2,1,1]]
=> ? = 4
10111 => 11101 => [1,1,1,2,1] => [[2,2,1,1,1],[1]]
=> ? = 1
11000 => 00011 => [4,1,1] => [[4,4,4],[3,3]]
=> ? = 6
11001 => 10011 => [1,3,1,1] => [[3,3,3,1],[2,2]]
=> ? = 4
11010 => 01011 => [2,2,1,1] => [[3,3,3,2],[2,2,1]]
=> ? = 5
11011 => 11011 => [1,1,2,1,1] => [[2,2,2,1,1],[1,1]]
=> ? = 2
11100 => 00111 => [3,1,1,1] => [[3,3,3,3],[2,2,2]]
=> ? = 6
11101 => 10111 => [1,2,1,1,1] => [[2,2,2,2,1],[1,1,1]]
=> ? = 3
11110 => 01111 => [2,1,1,1,1] => [[2,2,2,2,2],[1,1,1,1]]
=> ? = 4
11111 => 11111 => [1,1,1,1,1,1] => [[1,1,1,1,1,1],[]]
=> ? = 0
000000 => 000000 => [7] => [[7],[]]
=> ? = 0
000001 => 100000 => [1,6] => [[6,1],[]]
=> ? = 0
000010 => 010000 => [2,5] => [[6,2],[1]]
=> ? = 1
000011 => 110000 => [1,1,5] => [[5,1,1],[]]
=> ? = 0
000100 => 001000 => [3,4] => [[6,3],[2]]
=> ? = 2
000101 => 101000 => [1,2,4] => [[5,2,1],[1]]
=> ? = 1
000110 => 011000 => [2,1,4] => [[5,2,2],[1,1]]
=> ? = 2
000111 => 111000 => [1,1,1,4] => [[4,1,1,1],[]]
=> ? = 0
001000 => 000100 => [4,3] => [[6,4],[3]]
=> ? = 3
001001 => 100100 => [1,3,3] => [[5,3,1],[2]]
=> ? = 2
001010 => 010100 => [2,2,3] => [[5,3,2],[2,1]]
=> ? = 3
001011 => 110100 => [1,1,2,3] => [[4,2,1,1],[1]]
=> ? = 1
001100 => 001100 => [3,1,3] => [[5,3,3],[2,2]]
=> ? = 4
001101 => 101100 => [1,2,1,3] => [[4,2,2,1],[1,1]]
=> ? = 2
001110 => 011100 => [2,1,1,3] => [[4,2,2,2],[1,1,1]]
=> ? = 3
001111 => 111100 => [1,1,1,1,3] => [[3,1,1,1,1],[]]
=> ? = 0
010000 => 000010 => [5,2] => [[6,5],[4]]
=> ? = 4
010001 => 100010 => [1,4,2] => [[5,4,1],[3]]
=> ? = 3
Description
The number of missing boxes of a skew partition.
Sorry, this statistic was not found in the database
or
add this statistic to the database – it's very simple and we need your support!