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Your data matches 4 different statistics following compositions of up to 3 maps.
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Matching statistic: St000770
St000770: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[2]
=> 2
[1,1]
=> 1
[3]
=> 3
[2,1]
=> 4
[1,1,1]
=> 1
[4]
=> 4
[3,1]
=> 5
[2,2]
=> 2
[2,1,1]
=> 5
[1,1,1,1]
=> 1
[5]
=> 5
[4,1]
=> 6
[3,2]
=> 6
[3,1,1]
=> 6
[2,2,1]
=> 4
[2,1,1,1]
=> 6
[1,1,1,1,1]
=> 1
[6]
=> 6
[5,1]
=> 7
[4,2]
=> 7
[4,1,1]
=> 7
[3,3]
=> 3
[3,2,1]
=> 9
[3,1,1,1]
=> 7
[2,2,2]
=> 2
[2,2,1,1]
=> 5
[2,1,1,1,1]
=> 7
[1,1,1,1,1,1]
=> 1
[7]
=> 7
[6,1]
=> 8
[5,2]
=> 8
[5,1,1]
=> 8
[4,3]
=> 8
[4,2,1]
=> 10
[4,1,1,1]
=> 8
[3,3,1]
=> 5
[3,2,2]
=> 7
[3,2,1,1]
=> 11
[3,1,1,1,1]
=> 8
[2,2,2,1]
=> 4
[2,2,1,1,1]
=> 6
[2,1,1,1,1,1]
=> 8
[1,1,1,1,1,1,1]
=> 1
[8]
=> 8
[7,1]
=> 9
[6,2]
=> 9
[6,1,1]
=> 9
[5,3]
=> 9
[5,2,1]
=> 11
[5,1,1,1]
=> 9
Description
The major index of an integer partition when read from bottom to top.
This is the sum of the positions of the corners of the shape of an integer partition when reading from bottom to top.
For example, the partition λ=(8,6,6,4,3,3) has corners at positions 3,6,9, and 13, giving a major index of 31.
Matching statistic: St000290
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Mp00095: Integer partitions —to binary word⟶ Binary words
St000290: Binary words ⟶ ℤResult quality: 75% ●values known / values provided: 75%●distinct values known / distinct values provided: 91%
St000290: Binary words ⟶ ℤResult quality: 75% ●values known / values provided: 75%●distinct values known / distinct values provided: 91%
Values
[2]
=> 100 => 1
[1,1]
=> 110 => 2
[3]
=> 1000 => 1
[2,1]
=> 1010 => 4
[1,1,1]
=> 1110 => 3
[4]
=> 10000 => 1
[3,1]
=> 10010 => 5
[2,2]
=> 1100 => 2
[2,1,1]
=> 10110 => 5
[1,1,1,1]
=> 11110 => 4
[5]
=> 100000 => 1
[4,1]
=> 100010 => 6
[3,2]
=> 10100 => 4
[3,1,1]
=> 100110 => 6
[2,2,1]
=> 11010 => 6
[2,1,1,1]
=> 101110 => 6
[1,1,1,1,1]
=> 111110 => 5
[6]
=> 1000000 => 1
[5,1]
=> 1000010 => 7
[4,2]
=> 100100 => 5
[4,1,1]
=> 1000110 => 7
[3,3]
=> 11000 => 2
[3,2,1]
=> 101010 => 9
[3,1,1,1]
=> 1001110 => 7
[2,2,2]
=> 11100 => 3
[2,2,1,1]
=> 110110 => 7
[2,1,1,1,1]
=> 1011110 => 7
[1,1,1,1,1,1]
=> 1111110 => 6
[7]
=> 10000000 => 1
[6,1]
=> 10000010 => 8
[5,2]
=> 1000100 => 6
[5,1,1]
=> 10000110 => 8
[4,3]
=> 101000 => 4
[4,2,1]
=> 1001010 => 11
[4,1,1,1]
=> 10001110 => 8
[3,3,1]
=> 110010 => 7
[3,2,2]
=> 101100 => 5
[3,2,1,1]
=> 1010110 => 10
[3,1,1,1,1]
=> 10011110 => 8
[2,2,2,1]
=> 111010 => 8
[2,2,1,1,1]
=> 1101110 => 8
[2,1,1,1,1,1]
=> 10111110 => 8
[1,1,1,1,1,1,1]
=> 11111110 => 7
[8]
=> 100000000 => 1
[7,1]
=> 100000010 => 9
[6,2]
=> 10000100 => 7
[6,1,1]
=> 100000110 => 9
[5,3]
=> 1001000 => 5
[5,2,1]
=> 10001010 => 13
[5,1,1,1]
=> 100001110 => 9
[11]
=> 100000000000 => ? ∊ {1,8,9,11,11,12,12,12,12,12,12,12,12,12,13,14,15,15,16,17}
[10,1]
=> 100000000010 => ? ∊ {1,8,9,11,11,12,12,12,12,12,12,12,12,12,13,14,15,15,16,17}
[9,1,1]
=> 100000000110 => ? ∊ {1,8,9,11,11,12,12,12,12,12,12,12,12,12,13,14,15,15,16,17}
[8,3]
=> 1000001000 => ? ∊ {1,8,9,11,11,12,12,12,12,12,12,12,12,12,13,14,15,15,16,17}
[8,1,1,1]
=> 100000001110 => ? ∊ {1,8,9,11,11,12,12,12,12,12,12,12,12,12,13,14,15,15,16,17}
[7,3,1]
=> 1000010010 => ? ∊ {1,8,9,11,11,12,12,12,12,12,12,12,12,12,13,14,15,15,16,17}
[7,2,2]
=> 1000001100 => ? ∊ {1,8,9,11,11,12,12,12,12,12,12,12,12,12,13,14,15,15,16,17}
[7,1,1,1,1]
=> 100000011110 => ? ∊ {1,8,9,11,11,12,12,12,12,12,12,12,12,12,13,14,15,15,16,17}
[6,3,1,1]
=> 1000100110 => ? ∊ {1,8,9,11,11,12,12,12,12,12,12,12,12,12,13,14,15,15,16,17}
[6,2,2,1]
=> 1000011010 => ? ∊ {1,8,9,11,11,12,12,12,12,12,12,12,12,12,13,14,15,15,16,17}
[6,1,1,1,1,1]
=> 100000111110 => ? ∊ {1,8,9,11,11,12,12,12,12,12,12,12,12,12,13,14,15,15,16,17}
[5,1,1,1,1,1,1]
=> 100001111110 => ? ∊ {1,8,9,11,11,12,12,12,12,12,12,12,12,12,13,14,15,15,16,17}
[4,3,1,1,1,1]
=> 1010011110 => ? ∊ {1,8,9,11,11,12,12,12,12,12,12,12,12,12,13,14,15,15,16,17}
[4,2,2,1,1,1]
=> 1001101110 => ? ∊ {1,8,9,11,11,12,12,12,12,12,12,12,12,12,13,14,15,15,16,17}
[4,1,1,1,1,1,1,1]
=> 100011111110 => ? ∊ {1,8,9,11,11,12,12,12,12,12,12,12,12,12,13,14,15,15,16,17}
[3,3,1,1,1,1,1]
=> 1100111110 => ? ∊ {1,8,9,11,11,12,12,12,12,12,12,12,12,12,13,14,15,15,16,17}
[3,2,2,1,1,1,1]
=> 1011011110 => ? ∊ {1,8,9,11,11,12,12,12,12,12,12,12,12,12,13,14,15,15,16,17}
[3,1,1,1,1,1,1,1,1]
=> 100111111110 => ? ∊ {1,8,9,11,11,12,12,12,12,12,12,12,12,12,13,14,15,15,16,17}
[2,1,1,1,1,1,1,1,1,1]
=> 101111111110 => ? ∊ {1,8,9,11,11,12,12,12,12,12,12,12,12,12,13,14,15,15,16,17}
[1,1,1,1,1,1,1,1,1,1,1]
=> 111111111110 => ? ∊ {1,8,9,11,11,12,12,12,12,12,12,12,12,12,13,14,15,15,16,17}
[12]
=> 1000000000000 => ? ∊ {1,7,9,9,10,11,11,12,12,13,13,13,13,13,13,13,13,13,13,13,13,13,14,14,15,15,15,15,15,15,15,16,16,16,16,17,17,17,18,18,18,18,19,19,20,21,22}
[11,1]
=> 1000000000010 => ? ∊ {1,7,9,9,10,11,11,12,12,13,13,13,13,13,13,13,13,13,13,13,13,13,14,14,15,15,15,15,15,15,15,16,16,16,16,17,17,17,18,18,18,18,19,19,20,21,22}
[10,2]
=> 100000000100 => ? ∊ {1,7,9,9,10,11,11,12,12,13,13,13,13,13,13,13,13,13,13,13,13,13,14,14,15,15,15,15,15,15,15,16,16,16,16,17,17,17,18,18,18,18,19,19,20,21,22}
[10,1,1]
=> 1000000000110 => ? ∊ {1,7,9,9,10,11,11,12,12,13,13,13,13,13,13,13,13,13,13,13,13,13,14,14,15,15,15,15,15,15,15,16,16,16,16,17,17,17,18,18,18,18,19,19,20,21,22}
[9,3]
=> 10000001000 => ? ∊ {1,7,9,9,10,11,11,12,12,13,13,13,13,13,13,13,13,13,13,13,13,13,14,14,15,15,15,15,15,15,15,16,16,16,16,17,17,17,18,18,18,18,19,19,20,21,22}
[9,2,1]
=> 100000001010 => ? ∊ {1,7,9,9,10,11,11,12,12,13,13,13,13,13,13,13,13,13,13,13,13,13,14,14,15,15,15,15,15,15,15,16,16,16,16,17,17,17,18,18,18,18,19,19,20,21,22}
[9,1,1,1]
=> 1000000001110 => ? ∊ {1,7,9,9,10,11,11,12,12,13,13,13,13,13,13,13,13,13,13,13,13,13,14,14,15,15,15,15,15,15,15,16,16,16,16,17,17,17,18,18,18,18,19,19,20,21,22}
[8,4]
=> 1000010000 => ? ∊ {1,7,9,9,10,11,11,12,12,13,13,13,13,13,13,13,13,13,13,13,13,13,14,14,15,15,15,15,15,15,15,16,16,16,16,17,17,17,18,18,18,18,19,19,20,21,22}
[8,3,1]
=> 10000010010 => ? ∊ {1,7,9,9,10,11,11,12,12,13,13,13,13,13,13,13,13,13,13,13,13,13,14,14,15,15,15,15,15,15,15,16,16,16,16,17,17,17,18,18,18,18,19,19,20,21,22}
[8,2,2]
=> 10000001100 => ? ∊ {1,7,9,9,10,11,11,12,12,13,13,13,13,13,13,13,13,13,13,13,13,13,14,14,15,15,15,15,15,15,15,16,16,16,16,17,17,17,18,18,18,18,19,19,20,21,22}
[8,2,1,1]
=> 100000010110 => ? ∊ {1,7,9,9,10,11,11,12,12,13,13,13,13,13,13,13,13,13,13,13,13,13,14,14,15,15,15,15,15,15,15,16,16,16,16,17,17,17,18,18,18,18,19,19,20,21,22}
[8,1,1,1,1]
=> 1000000011110 => ? ∊ {1,7,9,9,10,11,11,12,12,13,13,13,13,13,13,13,13,13,13,13,13,13,14,14,15,15,15,15,15,15,15,16,16,16,16,17,17,17,18,18,18,18,19,19,20,21,22}
[7,4,1]
=> 1000100010 => ? ∊ {1,7,9,9,10,11,11,12,12,13,13,13,13,13,13,13,13,13,13,13,13,13,14,14,15,15,15,15,15,15,15,16,16,16,16,17,17,17,18,18,18,18,19,19,20,21,22}
[7,3,2]
=> 1000010100 => ? ∊ {1,7,9,9,10,11,11,12,12,13,13,13,13,13,13,13,13,13,13,13,13,13,14,14,15,15,15,15,15,15,15,16,16,16,16,17,17,17,18,18,18,18,19,19,20,21,22}
[7,3,1,1]
=> 10000100110 => ? ∊ {1,7,9,9,10,11,11,12,12,13,13,13,13,13,13,13,13,13,13,13,13,13,14,14,15,15,15,15,15,15,15,16,16,16,16,17,17,17,18,18,18,18,19,19,20,21,22}
[7,2,2,1]
=> 10000011010 => ? ∊ {1,7,9,9,10,11,11,12,12,13,13,13,13,13,13,13,13,13,13,13,13,13,14,14,15,15,15,15,15,15,15,16,16,16,16,17,17,17,18,18,18,18,19,19,20,21,22}
[7,2,1,1,1]
=> 100000101110 => ? ∊ {1,7,9,9,10,11,11,12,12,13,13,13,13,13,13,13,13,13,13,13,13,13,14,14,15,15,15,15,15,15,15,16,16,16,16,17,17,17,18,18,18,18,19,19,20,21,22}
[7,1,1,1,1,1]
=> 1000000111110 => ? ∊ {1,7,9,9,10,11,11,12,12,13,13,13,13,13,13,13,13,13,13,13,13,13,14,14,15,15,15,15,15,15,15,16,16,16,16,17,17,17,18,18,18,18,19,19,20,21,22}
[6,4,1,1]
=> 1001000110 => ? ∊ {1,7,9,9,10,11,11,12,12,13,13,13,13,13,13,13,13,13,13,13,13,13,14,14,15,15,15,15,15,15,15,16,16,16,16,17,17,17,18,18,18,18,19,19,20,21,22}
[6,3,2,1]
=> 1000101010 => ? ∊ {1,7,9,9,10,11,11,12,12,13,13,13,13,13,13,13,13,13,13,13,13,13,14,14,15,15,15,15,15,15,15,16,16,16,16,17,17,17,18,18,18,18,19,19,20,21,22}
[6,3,1,1,1]
=> 10001001110 => ? ∊ {1,7,9,9,10,11,11,12,12,13,13,13,13,13,13,13,13,13,13,13,13,13,14,14,15,15,15,15,15,15,15,16,16,16,16,17,17,17,18,18,18,18,19,19,20,21,22}
[6,2,2,2]
=> 1000011100 => ? ∊ {1,7,9,9,10,11,11,12,12,13,13,13,13,13,13,13,13,13,13,13,13,13,14,14,15,15,15,15,15,15,15,16,16,16,16,17,17,17,18,18,18,18,19,19,20,21,22}
[6,2,2,1,1]
=> 10000110110 => ? ∊ {1,7,9,9,10,11,11,12,12,13,13,13,13,13,13,13,13,13,13,13,13,13,14,14,15,15,15,15,15,15,15,16,16,16,16,17,17,17,18,18,18,18,19,19,20,21,22}
[6,2,1,1,1,1]
=> 100001011110 => ? ∊ {1,7,9,9,10,11,11,12,12,13,13,13,13,13,13,13,13,13,13,13,13,13,14,14,15,15,15,15,15,15,15,16,16,16,16,17,17,17,18,18,18,18,19,19,20,21,22}
[6,1,1,1,1,1,1]
=> 1000001111110 => ? ∊ {1,7,9,9,10,11,11,12,12,13,13,13,13,13,13,13,13,13,13,13,13,13,14,14,15,15,15,15,15,15,15,16,16,16,16,17,17,17,18,18,18,18,19,19,20,21,22}
[5,3,1,1,1,1]
=> 10010011110 => ? ∊ {1,7,9,9,10,11,11,12,12,13,13,13,13,13,13,13,13,13,13,13,13,13,14,14,15,15,15,15,15,15,15,16,16,16,16,17,17,17,18,18,18,18,19,19,20,21,22}
[5,2,2,1,1,1]
=> 10001101110 => ? ∊ {1,7,9,9,10,11,11,12,12,13,13,13,13,13,13,13,13,13,13,13,13,13,14,14,15,15,15,15,15,15,15,16,16,16,16,17,17,17,18,18,18,18,19,19,20,21,22}
[5,2,1,1,1,1,1]
=> 100010111110 => ? ∊ {1,7,9,9,10,11,11,12,12,13,13,13,13,13,13,13,13,13,13,13,13,13,14,14,15,15,15,15,15,15,15,16,16,16,16,17,17,17,18,18,18,18,19,19,20,21,22}
[5,1,1,1,1,1,1,1]
=> 1000011111110 => ? ∊ {1,7,9,9,10,11,11,12,12,13,13,13,13,13,13,13,13,13,13,13,13,13,14,14,15,15,15,15,15,15,15,16,16,16,16,17,17,17,18,18,18,18,19,19,20,21,22}
[4,4,1,1,1,1]
=> 1100011110 => ? ∊ {1,7,9,9,10,11,11,12,12,13,13,13,13,13,13,13,13,13,13,13,13,13,14,14,15,15,15,15,15,15,15,16,16,16,16,17,17,17,18,18,18,18,19,19,20,21,22}
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: St000293
Mp00095: Integer partitions —to binary word⟶ Binary words
Mp00096: Binary words —Foata bijection⟶ Binary words
St000293: Binary words ⟶ ℤResult quality: 73% ●values known / values provided: 73%●distinct values known / distinct values provided: 91%
Mp00096: Binary words —Foata bijection⟶ Binary words
St000293: Binary words ⟶ ℤResult quality: 73% ●values known / values provided: 73%●distinct values known / distinct values provided: 91%
Values
[2]
=> 100 => 010 => 1
[1,1]
=> 110 => 110 => 2
[3]
=> 1000 => 0010 => 1
[2,1]
=> 1010 => 1100 => 4
[1,1,1]
=> 1110 => 1110 => 3
[4]
=> 10000 => 00010 => 1
[3,1]
=> 10010 => 10100 => 5
[2,2]
=> 1100 => 0110 => 2
[2,1,1]
=> 10110 => 11010 => 5
[1,1,1,1]
=> 11110 => 11110 => 4
[5]
=> 100000 => 000010 => 1
[4,1]
=> 100010 => 100100 => 6
[3,2]
=> 10100 => 01100 => 4
[3,1,1]
=> 100110 => 101010 => 6
[2,2,1]
=> 11010 => 11100 => 6
[2,1,1,1]
=> 101110 => 110110 => 6
[1,1,1,1,1]
=> 111110 => 111110 => 5
[6]
=> 1000000 => 0000010 => 1
[5,1]
=> 1000010 => 1000100 => 7
[4,2]
=> 100100 => 010100 => 5
[4,1,1]
=> 1000110 => 1001010 => 7
[3,3]
=> 11000 => 00110 => 2
[3,2,1]
=> 101010 => 111000 => 9
[3,1,1,1]
=> 1001110 => 1010110 => 7
[2,2,2]
=> 11100 => 01110 => 3
[2,2,1,1]
=> 110110 => 111010 => 7
[2,1,1,1,1]
=> 1011110 => 1101110 => 7
[1,1,1,1,1,1]
=> 1111110 => 1111110 => 6
[7]
=> 10000000 => 00000010 => 1
[6,1]
=> 10000010 => 10000100 => 8
[5,2]
=> 1000100 => 0100100 => 6
[5,1,1]
=> 10000110 => 10001010 => 8
[4,3]
=> 101000 => 001100 => 4
[4,2,1]
=> 1001010 => 1101000 => 11
[4,1,1,1]
=> 10001110 => 10010110 => 8
[3,3,1]
=> 110010 => 101100 => 7
[3,2,2]
=> 101100 => 011010 => 5
[3,2,1,1]
=> 1010110 => 1110010 => 10
[3,1,1,1,1]
=> 10011110 => 10101110 => 8
[2,2,2,1]
=> 111010 => 111100 => 8
[2,2,1,1,1]
=> 1101110 => 1110110 => 8
[2,1,1,1,1,1]
=> 10111110 => 11011110 => 8
[1,1,1,1,1,1,1]
=> 11111110 => 11111110 => 7
[8]
=> 100000000 => 000000010 => 1
[7,1]
=> 100000010 => 100000100 => 9
[6,2]
=> 10000100 => 01000100 => 7
[6,1,1]
=> 100000110 => 100001010 => 9
[5,3]
=> 1001000 => 0010100 => 5
[5,2,1]
=> 10001010 => 11001000 => 13
[5,1,1,1]
=> 100001110 => 100010110 => 9
[11]
=> 100000000000 => ? => ? ∊ {1,9,10,11,11,12,12,12,12,12,12,12,12,12,12,12,13,14,14,15,15,15,16,16,17,17,18,19}
[10,1]
=> 100000000010 => ? => ? ∊ {1,9,10,11,11,12,12,12,12,12,12,12,12,12,12,12,13,14,14,15,15,15,16,16,17,17,18,19}
[9,2]
=> 10000000100 => 01000000100 => ? ∊ {1,9,10,11,11,12,12,12,12,12,12,12,12,12,12,12,13,14,14,15,15,15,16,16,17,17,18,19}
[9,1,1]
=> 100000000110 => ? => ? ∊ {1,9,10,11,11,12,12,12,12,12,12,12,12,12,12,12,13,14,14,15,15,15,16,16,17,17,18,19}
[8,2,1]
=> 10000001010 => 11000001000 => ? ∊ {1,9,10,11,11,12,12,12,12,12,12,12,12,12,12,12,13,14,14,15,15,15,16,16,17,17,18,19}
[8,1,1,1]
=> 100000001110 => ? => ? ∊ {1,9,10,11,11,12,12,12,12,12,12,12,12,12,12,12,13,14,14,15,15,15,16,16,17,17,18,19}
[7,3,1]
=> 1000010010 => ? => ? ∊ {1,9,10,11,11,12,12,12,12,12,12,12,12,12,12,12,13,14,14,15,15,15,16,16,17,17,18,19}
[7,2,2]
=> 1000001100 => ? => ? ∊ {1,9,10,11,11,12,12,12,12,12,12,12,12,12,12,12,13,14,14,15,15,15,16,16,17,17,18,19}
[7,2,1,1]
=> 10000010110 => 11000010010 => ? ∊ {1,9,10,11,11,12,12,12,12,12,12,12,12,12,12,12,13,14,14,15,15,15,16,16,17,17,18,19}
[7,1,1,1,1]
=> 100000011110 => ? => ? ∊ {1,9,10,11,11,12,12,12,12,12,12,12,12,12,12,12,13,14,14,15,15,15,16,16,17,17,18,19}
[6,3,1,1]
=> 1000100110 => ? => ? ∊ {1,9,10,11,11,12,12,12,12,12,12,12,12,12,12,12,13,14,14,15,15,15,16,16,17,17,18,19}
[6,2,2,1]
=> 1000011010 => ? => ? ∊ {1,9,10,11,11,12,12,12,12,12,12,12,12,12,12,12,13,14,14,15,15,15,16,16,17,17,18,19}
[6,2,1,1,1]
=> 10000101110 => 11000100110 => ? ∊ {1,9,10,11,11,12,12,12,12,12,12,12,12,12,12,12,13,14,14,15,15,15,16,16,17,17,18,19}
[6,1,1,1,1,1]
=> 100000111110 => 100001011110 => ? ∊ {1,9,10,11,11,12,12,12,12,12,12,12,12,12,12,12,13,14,14,15,15,15,16,16,17,17,18,19}
[5,2,1,1,1,1]
=> 10001011110 => 11001001110 => ? ∊ {1,9,10,11,11,12,12,12,12,12,12,12,12,12,12,12,13,14,14,15,15,15,16,16,17,17,18,19}
[5,1,1,1,1,1,1]
=> 100001111110 => ? => ? ∊ {1,9,10,11,11,12,12,12,12,12,12,12,12,12,12,12,13,14,14,15,15,15,16,16,17,17,18,19}
[4,3,1,1,1,1]
=> 1010011110 => ? => ? ∊ {1,9,10,11,11,12,12,12,12,12,12,12,12,12,12,12,13,14,14,15,15,15,16,16,17,17,18,19}
[4,2,2,1,1,1]
=> 1001101110 => ? => ? ∊ {1,9,10,11,11,12,12,12,12,12,12,12,12,12,12,12,13,14,14,15,15,15,16,16,17,17,18,19}
[4,2,1,1,1,1,1]
=> 10010111110 => 11010011110 => ? ∊ {1,9,10,11,11,12,12,12,12,12,12,12,12,12,12,12,13,14,14,15,15,15,16,16,17,17,18,19}
[4,1,1,1,1,1,1,1]
=> 100011111110 => ? => ? ∊ {1,9,10,11,11,12,12,12,12,12,12,12,12,12,12,12,13,14,14,15,15,15,16,16,17,17,18,19}
[3,3,1,1,1,1,1]
=> 1100111110 => ? => ? ∊ {1,9,10,11,11,12,12,12,12,12,12,12,12,12,12,12,13,14,14,15,15,15,16,16,17,17,18,19}
[3,2,2,1,1,1,1]
=> 1011011110 => ? => ? ∊ {1,9,10,11,11,12,12,12,12,12,12,12,12,12,12,12,13,14,14,15,15,15,16,16,17,17,18,19}
[3,2,1,1,1,1,1,1]
=> 10101111110 => 11100111110 => ? ∊ {1,9,10,11,11,12,12,12,12,12,12,12,12,12,12,12,13,14,14,15,15,15,16,16,17,17,18,19}
[3,1,1,1,1,1,1,1,1]
=> 100111111110 => ? => ? ∊ {1,9,10,11,11,12,12,12,12,12,12,12,12,12,12,12,13,14,14,15,15,15,16,16,17,17,18,19}
[2,2,2,1,1,1,1,1]
=> 1110111110 => 1111011110 => ? ∊ {1,9,10,11,11,12,12,12,12,12,12,12,12,12,12,12,13,14,14,15,15,15,16,16,17,17,18,19}
[2,2,1,1,1,1,1,1,1]
=> 11011111110 => 11101111110 => ? ∊ {1,9,10,11,11,12,12,12,12,12,12,12,12,12,12,12,13,14,14,15,15,15,16,16,17,17,18,19}
[2,1,1,1,1,1,1,1,1,1]
=> 101111111110 => ? => ? ∊ {1,9,10,11,11,12,12,12,12,12,12,12,12,12,12,12,13,14,14,15,15,15,16,16,17,17,18,19}
[1,1,1,1,1,1,1,1,1,1,1]
=> 111111111110 => ? => ? ∊ {1,9,10,11,11,12,12,12,12,12,12,12,12,12,12,12,13,14,14,15,15,15,16,16,17,17,18,19}
[12]
=> 1000000000000 => ? => ? ∊ {1,9,9,10,11,11,12,12,13,13,13,13,13,13,13,13,13,13,13,13,13,14,14,15,15,15,15,15,15,15,16,16,16,16,17,17,17,18,18,18,18,19,19,20,21,22}
[11,1]
=> 1000000000010 => ? => ? ∊ {1,9,9,10,11,11,12,12,13,13,13,13,13,13,13,13,13,13,13,13,13,14,14,15,15,15,15,15,15,15,16,16,16,16,17,17,17,18,18,18,18,19,19,20,21,22}
[10,2]
=> 100000000100 => 010000000100 => ? ∊ {1,9,9,10,11,11,12,12,13,13,13,13,13,13,13,13,13,13,13,13,13,14,14,15,15,15,15,15,15,15,16,16,16,16,17,17,17,18,18,18,18,19,19,20,21,22}
[10,1,1]
=> 1000000000110 => ? => ? ∊ {1,9,9,10,11,11,12,12,13,13,13,13,13,13,13,13,13,13,13,13,13,14,14,15,15,15,15,15,15,15,16,16,16,16,17,17,17,18,18,18,18,19,19,20,21,22}
[9,3]
=> 10000001000 => ? => ? ∊ {1,9,9,10,11,11,12,12,13,13,13,13,13,13,13,13,13,13,13,13,13,14,14,15,15,15,15,15,15,15,16,16,16,16,17,17,17,18,18,18,18,19,19,20,21,22}
[9,2,1]
=> 100000001010 => ? => ? ∊ {1,9,9,10,11,11,12,12,13,13,13,13,13,13,13,13,13,13,13,13,13,14,14,15,15,15,15,15,15,15,16,16,16,16,17,17,17,18,18,18,18,19,19,20,21,22}
[9,1,1,1]
=> 1000000001110 => ? => ? ∊ {1,9,9,10,11,11,12,12,13,13,13,13,13,13,13,13,13,13,13,13,13,14,14,15,15,15,15,15,15,15,16,16,16,16,17,17,17,18,18,18,18,19,19,20,21,22}
[8,3,1]
=> 10000010010 => ? => ? ∊ {1,9,9,10,11,11,12,12,13,13,13,13,13,13,13,13,13,13,13,13,13,14,14,15,15,15,15,15,15,15,16,16,16,16,17,17,17,18,18,18,18,19,19,20,21,22}
[8,2,2]
=> 10000001100 => ? => ? ∊ {1,9,9,10,11,11,12,12,13,13,13,13,13,13,13,13,13,13,13,13,13,14,14,15,15,15,15,15,15,15,16,16,16,16,17,17,17,18,18,18,18,19,19,20,21,22}
[8,2,1,1]
=> 100000010110 => ? => ? ∊ {1,9,9,10,11,11,12,12,13,13,13,13,13,13,13,13,13,13,13,13,13,14,14,15,15,15,15,15,15,15,16,16,16,16,17,17,17,18,18,18,18,19,19,20,21,22}
[8,1,1,1,1]
=> 1000000011110 => ? => ? ∊ {1,9,9,10,11,11,12,12,13,13,13,13,13,13,13,13,13,13,13,13,13,14,14,15,15,15,15,15,15,15,16,16,16,16,17,17,17,18,18,18,18,19,19,20,21,22}
[7,4,1]
=> 1000100010 => ? => ? ∊ {1,9,9,10,11,11,12,12,13,13,13,13,13,13,13,13,13,13,13,13,13,14,14,15,15,15,15,15,15,15,16,16,16,16,17,17,17,18,18,18,18,19,19,20,21,22}
[7,3,2]
=> 1000010100 => ? => ? ∊ {1,9,9,10,11,11,12,12,13,13,13,13,13,13,13,13,13,13,13,13,13,14,14,15,15,15,15,15,15,15,16,16,16,16,17,17,17,18,18,18,18,19,19,20,21,22}
[7,3,1,1]
=> 10000100110 => ? => ? ∊ {1,9,9,10,11,11,12,12,13,13,13,13,13,13,13,13,13,13,13,13,13,14,14,15,15,15,15,15,15,15,16,16,16,16,17,17,17,18,18,18,18,19,19,20,21,22}
[7,2,2,1]
=> 10000011010 => ? => ? ∊ {1,9,9,10,11,11,12,12,13,13,13,13,13,13,13,13,13,13,13,13,13,14,14,15,15,15,15,15,15,15,16,16,16,16,17,17,17,18,18,18,18,19,19,20,21,22}
[7,2,1,1,1]
=> 100000101110 => ? => ? ∊ {1,9,9,10,11,11,12,12,13,13,13,13,13,13,13,13,13,13,13,13,13,14,14,15,15,15,15,15,15,15,16,16,16,16,17,17,17,18,18,18,18,19,19,20,21,22}
[7,1,1,1,1,1]
=> 1000000111110 => ? => ? ∊ {1,9,9,10,11,11,12,12,13,13,13,13,13,13,13,13,13,13,13,13,13,14,14,15,15,15,15,15,15,15,16,16,16,16,17,17,17,18,18,18,18,19,19,20,21,22}
[6,4,1,1]
=> 1001000110 => 1001010010 => ? ∊ {1,9,9,10,11,11,12,12,13,13,13,13,13,13,13,13,13,13,13,13,13,14,14,15,15,15,15,15,15,15,16,16,16,16,17,17,17,18,18,18,18,19,19,20,21,22}
[6,3,2,1]
=> 1000101010 => 1110010000 => ? ∊ {1,9,9,10,11,11,12,12,13,13,13,13,13,13,13,13,13,13,13,13,13,14,14,15,15,15,15,15,15,15,16,16,16,16,17,17,17,18,18,18,18,19,19,20,21,22}
[6,3,1,1,1]
=> 10001001110 => ? => ? ∊ {1,9,9,10,11,11,12,12,13,13,13,13,13,13,13,13,13,13,13,13,13,14,14,15,15,15,15,15,15,15,16,16,16,16,17,17,17,18,18,18,18,19,19,20,21,22}
[6,2,2,2]
=> 1000011100 => 0100010110 => ? ∊ {1,9,9,10,11,11,12,12,13,13,13,13,13,13,13,13,13,13,13,13,13,14,14,15,15,15,15,15,15,15,16,16,16,16,17,17,17,18,18,18,18,19,19,20,21,22}
[6,2,2,1,1]
=> 10000110110 => ? => ? ∊ {1,9,9,10,11,11,12,12,13,13,13,13,13,13,13,13,13,13,13,13,13,14,14,15,15,15,15,15,15,15,16,16,16,16,17,17,17,18,18,18,18,19,19,20,21,22}
Description
The number of inversions of a binary word.
Matching statistic: St001232
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00032: Dyck paths —inverse zeta map⟶ Dyck paths
Mp00120: Dyck paths —Lalanne-Kreweras involution⟶ Dyck paths
St001232: Dyck paths ⟶ ℤResult quality: 7% ●values known / values provided: 7%●distinct values known / distinct values provided: 27%
Mp00032: Dyck paths —inverse zeta map⟶ Dyck paths
Mp00120: Dyck paths —Lalanne-Kreweras involution⟶ Dyck paths
St001232: Dyck paths ⟶ ℤResult quality: 7% ●values known / values provided: 7%●distinct values known / distinct values provided: 27%
Values
[2]
=> [1,1,0,0,1,0]
=> [1,1,0,1,0,0]
=> [1,1,0,1,0,0]
=> 2
[1,1]
=> [1,0,1,1,0,0]
=> [1,0,1,1,0,0]
=> [1,1,0,0,1,0]
=> 1
[3]
=> [1,1,1,0,0,0,1,0]
=> [1,1,0,1,0,1,0,0]
=> [1,1,1,0,1,0,0,0]
=> 3
[2,1]
=> [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> ? = 4
[1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,0,0]
=> [1,1,1,0,0,0,1,0]
=> 1
[4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> 4
[3,1]
=> [1,1,0,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> [1,0,1,1,0,1,0,0]
=> ? ∊ {5,5}
[2,2]
=> [1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,1,0,0]
=> [1,1,1,0,0,1,0,0]
=> 2
[2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,1,1,0,0,0,1,0]
=> [1,0,1,0,1,1,0,0]
=> ? ∊ {5,5}
[1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> 1
[5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> [1,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,1,1,1,0,1,0,0,0,0,0]
=> 5
[4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> [1,0,1,1,1,0,1,0,0,0]
=> ? ∊ {4,6,6,6,6}
[3,2]
=> [1,1,0,0,1,0,1,0]
=> [1,1,1,0,1,0,0,0]
=> [1,1,0,1,0,1,0,0]
=> ? ∊ {4,6,6,6,6}
[3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,0]
=> [1,1,0,1,0,0,1,0]
=> ? ∊ {4,6,6,6,6}
[2,2,1]
=> [1,0,1,0,1,1,0,0]
=> [1,0,1,1,1,0,0,0]
=> [1,1,0,0,1,0,1,0]
=> ? ∊ {4,6,6,6,6}
[2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> ? ∊ {4,6,6,6,6}
[1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> 1
[6]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,1,1,1,1,0,1,0,0,0,0,0,0]
=> 6
[5,1]
=> [1,1,1,1,0,1,0,0,0,0,1,0]
=> [1,1,1,0,0,1,0,1,0,1,0,0]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> ? ∊ {5,7,7,7,7,7,9}
[4,2]
=> [1,1,1,0,0,1,0,0,1,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> ? ∊ {5,7,7,7,7,7,9}
[4,1,1]
=> [1,1,0,1,1,0,0,0,1,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> ? ∊ {5,7,7,7,7,7,9}
[3,3]
=> [1,1,1,0,0,0,1,1,0,0]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> 3
[3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> ? ∊ {5,7,7,7,7,7,9}
[3,1,1,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,1,1,0,0,0,1,0]
=> [1,1,0,1,0,0,1,1,0,0]
=> ? ∊ {5,7,7,7,7,7,9}
[2,2,2]
=> [1,1,0,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> 2
[2,2,1,1]
=> [1,0,1,1,0,1,1,0,0,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> ? ∊ {5,7,7,7,7,7,9}
[2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> [1,1,1,0,0,0,1,0,1,0,1,0]
=> [1,0,1,0,1,1,1,1,0,0,0,0]
=> ? ∊ {5,7,7,7,7,7,9}
[1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> 1
[7]
=> [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]
=> [1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0]
=> ? ∊ {1,4,5,6,7,7,8,8,8,8,8,8,8,10,11}
[6,1]
=> [1,1,1,1,1,0,1,0,0,0,0,0,1,0]
=> [1,1,1,0,0,1,0,1,0,1,0,1,0,0]
=> [1,0,1,1,1,1,1,0,1,0,0,0,0,0]
=> ? ∊ {1,4,5,6,7,7,8,8,8,8,8,8,8,10,11}
[5,2]
=> [1,1,1,1,0,0,1,0,0,0,1,0]
=> [1,1,1,0,1,0,0,1,0,1,0,0]
=> [1,1,0,1,1,1,0,1,0,0,0,0]
=> ? ∊ {1,4,5,6,7,7,8,8,8,8,8,8,8,10,11}
[5,1,1]
=> [1,1,1,0,1,1,0,0,0,0,1,0]
=> [1,1,0,1,1,0,0,1,0,1,0,0]
=> [1,1,0,1,1,1,0,0,1,0,0,0]
=> ? ∊ {1,4,5,6,7,7,8,8,8,8,8,8,8,10,11}
[4,3]
=> [1,1,1,0,0,0,1,0,1,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> ? ∊ {1,4,5,6,7,7,8,8,8,8,8,8,8,10,11}
[4,2,1]
=> [1,1,0,1,0,1,0,0,1,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> [1,0,1,1,0,1,0,0,1,0]
=> ? ∊ {1,4,5,6,7,7,8,8,8,8,8,8,8,10,11}
[4,1,1,1]
=> [1,0,1,1,1,0,0,0,1,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> [1,1,1,0,1,0,0,0,1,0]
=> ? ∊ {1,4,5,6,7,7,8,8,8,8,8,8,8,10,11}
[3,3,1]
=> [1,1,0,1,0,0,1,1,0,0]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,0,1,1,0,1,0,0]
=> ? ∊ {1,4,5,6,7,7,8,8,8,8,8,8,8,10,11}
[3,2,2]
=> [1,1,0,0,1,1,0,1,0,0]
=> [1,1,1,0,1,0,0,0,1,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> ? ∊ {1,4,5,6,7,7,8,8,8,8,8,8,8,10,11}
[3,2,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> ? ∊ {1,4,5,6,7,7,8,8,8,8,8,8,8,10,11}
[3,1,1,1,1]
=> [1,0,1,1,1,1,0,0,1,0,0,0]
=> [1,1,0,1,1,0,0,0,1,0,1,0]
=> [1,1,0,1,0,0,1,1,1,0,0,0]
=> ? ∊ {1,4,5,6,7,7,8,8,8,8,8,8,8,10,11}
[2,2,2,1]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> ? ∊ {1,4,5,6,7,7,8,8,8,8,8,8,8,10,11}
[2,2,1,1,1]
=> [1,0,1,1,1,0,1,1,0,0,0,0]
=> [1,0,1,1,1,0,0,0,1,0,1,0]
=> [1,1,0,0,1,0,1,1,1,0,0,0]
=> ? ∊ {1,4,5,6,7,7,8,8,8,8,8,8,8,10,11}
[2,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,1,0,0,0,0,0]
=> [1,1,1,0,0,0,1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,1,1,1,1,0,0,0,0,0]
=> ? ∊ {1,4,5,6,7,7,8,8,8,8,8,8,8,10,11}
[1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> ? ∊ {1,4,5,6,7,7,8,8,8,8,8,8,8,10,11}
[8]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0]
=> ? ∊ {1,5,6,6,7,8,8,9,9,9,9,9,9,9,9,10,11,11,12,13}
[7,1]
=> [1,1,1,1,1,1,0,1,0,0,0,0,0,0,1,0]
=> [1,1,1,0,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,0,1,1,1,1,1,1,0,1,0,0,0,0,0,0]
=> ? ∊ {1,5,6,6,7,8,8,9,9,9,9,9,9,9,9,10,11,11,12,13}
[6,2]
=> [1,1,1,1,1,0,0,1,0,0,0,0,1,0]
=> [1,1,1,0,1,0,0,1,0,1,0,1,0,0]
=> [1,1,0,1,1,1,1,0,1,0,0,0,0,0]
=> ? ∊ {1,5,6,6,7,8,8,9,9,9,9,9,9,9,9,10,11,11,12,13}
[6,1,1]
=> [1,1,1,1,0,1,1,0,0,0,0,0,1,0]
=> [1,1,0,1,1,0,0,1,0,1,0,1,0,0]
=> [1,1,0,1,1,1,1,0,0,1,0,0,0,0]
=> ? ∊ {1,5,6,6,7,8,8,9,9,9,9,9,9,9,9,10,11,11,12,13}
[5,3]
=> [1,1,1,1,0,0,0,1,0,0,1,0]
=> [1,1,1,0,1,0,1,0,0,1,0,0]
=> [1,1,1,0,1,1,0,1,0,0,0,0]
=> ? ∊ {1,5,6,6,7,8,8,9,9,9,9,9,9,9,9,10,11,11,12,13}
[5,2,1]
=> [1,1,1,0,1,0,1,0,0,0,1,0]
=> [1,1,1,0,0,1,1,0,0,1,0,0]
=> [1,0,1,1,0,1,1,0,0,1,0,0]
=> ? ∊ {1,5,6,6,7,8,8,9,9,9,9,9,9,9,9,10,11,11,12,13}
[5,1,1,1]
=> [1,1,0,1,1,1,0,0,0,0,1,0]
=> [1,1,0,1,0,1,1,0,0,1,0,0]
=> [1,1,1,0,1,1,0,0,0,1,0,0]
=> ? ∊ {1,5,6,6,7,8,8,9,9,9,9,9,9,9,9,10,11,11,12,13}
[4,4]
=> [1,1,1,1,0,0,0,0,1,1,0,0]
=> [1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,1,1,1,1,0,0,1,0,0,0,0]
=> 4
[4,3,1]
=> [1,1,0,1,0,0,1,0,1,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> [1,0,1,1,0,1,0,1,0,0]
=> ? ∊ {1,5,6,6,7,8,8,9,9,9,9,9,9,9,9,10,11,11,12,13}
[4,2,2]
=> [1,1,0,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> ? ∊ {1,5,6,6,7,8,8,9,9,9,9,9,9,9,9,10,11,11,12,13}
[4,2,1,1]
=> [1,0,1,1,0,1,0,0,1,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> [1,0,1,0,1,1,0,1,0,0]
=> ? ∊ {1,5,6,6,7,8,8,9,9,9,9,9,9,9,9,10,11,11,12,13}
[4,1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,1,0,0]
=> [1,1,0,1,0,1,1,0,0,0,1,0]
=> [1,1,1,0,1,0,0,0,1,1,0,0]
=> ? ∊ {1,5,6,6,7,8,8,9,9,9,9,9,9,9,9,10,11,11,12,13}
[3,3,2]
=> [1,1,0,0,1,0,1,1,0,0]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> ? ∊ {1,5,6,6,7,8,8,9,9,9,9,9,9,9,9,10,11,11,12,13}
[3,3,1,1]
=> [1,0,1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,1,1,0,0,0]
=> [1,1,1,0,0,1,0,0,1,0]
=> ? ∊ {1,5,6,6,7,8,8,9,9,9,9,9,9,9,9,10,11,11,12,13}
[3,2,2,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> ? ∊ {1,5,6,6,7,8,8,9,9,9,9,9,9,9,9,10,11,11,12,13}
[3,2,1,1,1]
=> [1,0,1,1,1,0,1,0,1,0,0,0]
=> [1,1,0,0,1,1,1,0,0,0,1,0]
=> [1,0,1,1,0,0,1,0,1,1,0,0]
=> ? ∊ {1,5,6,6,7,8,8,9,9,9,9,9,9,9,9,10,11,11,12,13}
[3,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,1,0,0,0,0]
=> [1,1,0,1,1,0,0,0,1,0,1,0,1,0]
=> [1,1,0,1,0,0,1,1,1,1,0,0,0,0]
=> ? ∊ {1,5,6,6,7,8,8,9,9,9,9,9,9,9,9,10,11,11,12,13}
[2,2,2,2]
=> [1,1,0,0,1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,1,1,0,0,0,0,1,0,0]
=> 2
[2,2,2,1,1]
=> [1,0,1,1,0,1,1,1,0,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0,1,0]
=> [1,1,1,0,0,0,1,0,1,1,0,0]
=> ? ∊ {1,5,6,6,7,8,8,9,9,9,9,9,9,9,9,10,11,11,12,13}
[2,2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,1,0,0,0,0,0]
=> [1,0,1,1,1,0,0,0,1,0,1,0,1,0]
=> [1,1,0,0,1,0,1,1,1,1,0,0,0,0]
=> ? ∊ {1,5,6,6,7,8,8,9,9,9,9,9,9,9,9,10,11,11,12,13}
[2,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,1,0,0,0,0,0,0]
=> [1,1,1,0,0,0,1,0,1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> ? ∊ {1,5,6,6,7,8,8,9,9,9,9,9,9,9,9,10,11,11,12,13}
[1,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0]
=> ? ∊ {1,5,6,6,7,8,8,9,9,9,9,9,9,9,9,10,11,11,12,13}
[3,3,3]
=> [1,1,1,0,0,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,1,1,1,0,0,0,1,0,0,0]
=> 3
[5,5]
=> [1,1,1,1,1,0,0,0,0,0,1,1,0,0]
=> [1,0,1,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,1,1,1,1,0,0,1,0,0,0,0,0]
=> 5
[2,2,2,2,2]
=> [1,1,0,0,1,1,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,1,1,1,0,0,0,0,0,1,0,0]
=> 2
[4,4,4]
=> [1,1,1,1,0,0,0,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,1,1,1,1,1,0,0,0,1,0,0,0,0]
=> 4
[3,3,3,3]
=> [1,1,1,0,0,0,1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,1,1,1,1,0,0,0,0,1,0,0,0]
=> 3
Description
The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2.
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