Your data matches 4 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)
Matching statistic: St000495
(load all 6 compositions to match this statistic)
Mapping
Mp00099: Dyck paths —bounce path⟶ Dyck paths
Mp00032: Dyck paths —inverse zeta map⟶ Dyck paths
Mp00031: Dyck paths —to 312-avoiding permutation⟶ Permutations
St000495: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0,1,0]
 => [1,0,1,0]
 => [1,1,0,0]
 => [2,1] => 1
[1,1,0,0]
 => [1,1,0,0]
 => [1,0,1,0]
 => [1,2] => 0
[1,0,1,0,1,0]
 => [1,0,1,0,1,0]
 => [1,1,1,0,0,0]
 => [3,2,1] => 3
[1,0,1,1,0,0]
 => [1,0,1,1,0,0]
 => [1,0,1,1,0,0]
 => [1,3,2] => 1
[1,1,0,0,1,0]
 => [1,1,0,0,1,0]
 => [1,1,0,1,0,0]
 => [2,3,1] => 2
[1,1,0,1,0,0]
 => [1,0,1,1,0,0]
 => [1,0,1,1,0,0]
 => [1,3,2] => 1
[1,1,1,0,0,0]
 => [1,1,1,0,0,0]
 => [1,0,1,0,1,0]
 => [1,2,3] => 0
[1,0,1,0,1,0,1,0]
 => [1,0,1,0,1,0,1,0]
 => [1,1,1,1,0,0,0,0]
 => [4,3,2,1] => 5
[1,0,1,0,1,1,0,0]
 => [1,0,1,0,1,1,0,0]
 => [1,0,1,1,1,0,0,0]
 => [1,4,3,2] => 3
[1,0,1,1,0,0,1,0]
 => [1,0,1,1,0,0,1,0]
 => [1,1,0,1,1,0,0,0]
 => [2,4,3,1] => 3
[1,0,1,1,0,1,0,0]
 => [1,0,1,0,1,1,0,0]
 => [1,0,1,1,1,0,0,0]
 => [1,4,3,2] => 3
[1,0,1,1,1,0,0,0]
 => [1,0,1,1,1,0,0,0]
 => [1,0,1,0,1,1,0,0]
 => [1,2,4,3] => 1
[1,1,0,0,1,0,1,0]
 => [1,1,0,0,1,0,1,0]
 => [1,1,1,0,1,0,0,0]
 => [3,4,2,1] => 4
[1,1,0,0,1,1,0,0]
 => [1,1,0,0,1,1,0,0]
 => [1,0,1,1,0,1,0,0]
 => [1,3,4,2] => 2
[1,1,0,1,0,0,1,0]
 => [1,0,1,1,0,0,1,0]
 => [1,1,0,1,1,0,0,0]
 => [2,4,3,1] => 3
[1,1,0,1,0,1,0,0]
 => [1,1,0,0,1,1,0,0]
 => [1,0,1,1,0,1,0,0]
 => [1,3,4,2] => 2
[1,1,0,1,1,0,0,0]
 => [1,0,1,1,1,0,0,0]
 => [1,0,1,0,1,1,0,0]
 => [1,2,4,3] => 1
[1,1,1,0,0,0,1,0]
 => [1,1,1,0,0,0,1,0]
 => [1,1,0,1,0,1,0,0]
 => [2,3,4,1] => 2
[1,1,1,0,0,1,0,0]
 => [1,1,0,0,1,1,0,0]
 => [1,0,1,1,0,1,0,0]
 => [1,3,4,2] => 2
[1,1,1,0,1,0,0,0]
 => [1,0,1,1,1,0,0,0]
 => [1,0,1,0,1,1,0,0]
 => [1,2,4,3] => 1
[1,1,1,1,0,0,0,0]
 => [1,1,1,1,0,0,0,0]
 => [1,0,1,0,1,0,1,0]
 => [1,2,3,4] => 0
[1,0,1,0,1,0,1,0,1,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => [5,4,3,2,1] => 7
[1,0,1,0,1,0,1,1,0,0]
 => [1,0,1,0,1,0,1,1,0,0]
 => [1,0,1,1,1,1,0,0,0,0]
 => [1,5,4,3,2] => 5
[1,0,1,0,1,1,0,0,1,0]
 => [1,0,1,0,1,1,0,0,1,0]
 => [1,1,0,1,1,1,0,0,0,0]
 => [2,5,4,3,1] => 5
[1,0,1,0,1,1,0,1,0,0]
 => [1,0,1,0,1,0,1,1,0,0]
 => [1,0,1,1,1,1,0,0,0,0]
 => [1,5,4,3,2] => 5
[1,0,1,0,1,1,1,0,0,0]
 => [1,0,1,0,1,1,1,0,0,0]
 => [1,0,1,0,1,1,1,0,0,0]
 => [1,2,5,4,3] => 3
[1,0,1,1,0,0,1,0,1,0]
 => [1,0,1,1,0,0,1,0,1,0]
 => [1,1,1,0,1,1,0,0,0,0]
 => [3,5,4,2,1] => 5
[1,0,1,1,0,0,1,1,0,0]
 => [1,0,1,1,0,0,1,1,0,0]
 => [1,0,1,1,0,1,1,0,0,0]
 => [1,3,5,4,2] => 3
[1,0,1,1,0,1,0,0,1,0]
 => [1,0,1,0,1,1,0,0,1,0]
 => [1,1,0,1,1,1,0,0,0,0]
 => [2,5,4,3,1] => 5
[1,0,1,1,0,1,0,1,0,0]
 => [1,0,1,1,0,0,1,1,0,0]
 => [1,0,1,1,0,1,1,0,0,0]
 => [1,3,5,4,2] => 3
[1,0,1,1,0,1,1,0,0,0]
 => [1,0,1,0,1,1,1,0,0,0]
 => [1,0,1,0,1,1,1,0,0,0]
 => [1,2,5,4,3] => 3
[1,0,1,1,1,0,0,0,1,0]
 => [1,0,1,1,1,0,0,0,1,0]
 => [1,1,0,1,0,1,1,0,0,0]
 => [2,3,5,4,1] => 3
[1,0,1,1,1,0,0,1,0,0]
 => [1,0,1,1,0,0,1,1,0,0]
 => [1,0,1,1,0,1,1,0,0,0]
 => [1,3,5,4,2] => 3
[1,0,1,1,1,0,1,0,0,0]
 => [1,0,1,0,1,1,1,0,0,0]
 => [1,0,1,0,1,1,1,0,0,0]
 => [1,2,5,4,3] => 3
[1,0,1,1,1,1,0,0,0,0]
 => [1,0,1,1,1,1,0,0,0,0]
 => [1,0,1,0,1,0,1,1,0,0]
 => [1,2,3,5,4] => 1
[1,1,0,0,1,0,1,0,1,0]
 => [1,1,0,0,1,0,1,0,1,0]
 => [1,1,1,1,0,1,0,0,0,0]
 => [4,5,3,2,1] => 6
[1,1,0,0,1,0,1,1,0,0]
 => [1,1,0,0,1,0,1,1,0,0]
 => [1,0,1,1,1,0,1,0,0,0]
 => [1,4,5,3,2] => 4
[1,1,0,0,1,1,0,0,1,0]
 => [1,1,0,0,1,1,0,0,1,0]
 => [1,1,0,1,1,0,1,0,0,0]
 => [2,4,5,3,1] => 4
[1,1,0,0,1,1,0,1,0,0]
 => [1,1,0,0,1,0,1,1,0,0]
 => [1,0,1,1,1,0,1,0,0,0]
 => [1,4,5,3,2] => 4
[1,1,0,0,1,1,1,0,0,0]
 => [1,1,0,0,1,1,1,0,0,0]
 => [1,0,1,0,1,1,0,1,0,0]
 => [1,2,4,5,3] => 2
[1,1,0,1,0,0,1,0,1,0]
 => [1,0,1,1,0,0,1,0,1,0]
 => [1,1,1,0,1,1,0,0,0,0]
 => [3,5,4,2,1] => 5
[1,1,0,1,0,0,1,1,0,0]
 => [1,0,1,1,0,0,1,1,0,0]
 => [1,0,1,1,0,1,1,0,0,0]
 => [1,3,5,4,2] => 3
[1,1,0,1,0,1,0,0,1,0]
 => [1,1,0,0,1,1,0,0,1,0]
 => [1,1,0,1,1,0,1,0,0,0]
 => [2,4,5,3,1] => 4
[1,1,0,1,0,1,0,1,0,0]
 => [1,0,1,1,0,0,1,1,0,0]
 => [1,0,1,1,0,1,1,0,0,0]
 => [1,3,5,4,2] => 3
[1,1,0,1,0,1,1,0,0,0]
 => [1,1,0,0,1,1,1,0,0,0]
 => [1,0,1,0,1,1,0,1,0,0]
 => [1,2,4,5,3] => 2
[1,1,0,1,1,0,0,0,1,0]
 => [1,0,1,1,1,0,0,0,1,0]
 => [1,1,0,1,0,1,1,0,0,0]
 => [2,3,5,4,1] => 3
[1,1,0,1,1,0,0,1,0,0]
 => [1,0,1,1,0,0,1,1,0,0]
 => [1,0,1,1,0,1,1,0,0,0]
 => [1,3,5,4,2] => 3
[1,1,0,1,1,0,1,0,0,0]
 => [1,1,0,0,1,1,1,0,0,0]
 => [1,0,1,0,1,1,0,1,0,0]
 => [1,2,4,5,3] => 2
[1,1,0,1,1,1,0,0,0,0]
 => [1,0,1,1,1,1,0,0,0,0]
 => [1,0,1,0,1,0,1,1,0,0]
 => [1,2,3,5,4] => 1
[1,1,1,0,0,0,1,0,1,0]
 => [1,1,1,0,0,0,1,0,1,0]
 => [1,1,1,0,1,0,1,0,0,0]
 => [3,4,5,2,1] => 4
Description
The number of inversions of distance at most 2 of a permutation. An inversion of a permutation $\pi$ is a pair $i < j$ such that $\sigma(i) > \sigma(j)$. Let $j-i$ be the distance of such an inversion. Then inversions of distance at most 1 are then exactly the descents of $\pi$, see [[St000021]]. This statistic counts the number of inversions of distance at most 2.
Matching statistic: St000876
(load all 2 compositions to match this statistic)
Mapping
Mp00032: Dyck paths —inverse zeta map⟶ Dyck paths
Mp00093: Dyck paths —to binary word⟶ Binary words
Mp00105: Binary words —complement⟶ Binary words
St000876: Binary words ⟶ ℤResult quality: 41% ●values known / values provided: 41%●distinct values known / distinct values provided: 100%
Values
[1,0,1,0]
 => [1,1,0,0]
 => 1100 => 0011 => 4 = 1 + 3
[1,1,0,0]
 => [1,0,1,0]
 => 1010 => 0101 => 3 = 0 + 3
[1,0,1,0,1,0]
 => [1,1,1,0,0,0]
 => 111000 => 000111 => 6 = 3 + 3
[1,0,1,1,0,0]
 => [1,0,1,1,0,0]
 => 101100 => 010011 => 4 = 1 + 3
[1,1,0,0,1,0]
 => [1,1,0,1,0,0]
 => 110100 => 001011 => 5 = 2 + 3
[1,1,0,1,0,0]
 => [1,1,0,0,1,0]
 => 110010 => 001101 => 4 = 1 + 3
[1,1,1,0,0,0]
 => [1,0,1,0,1,0]
 => 101010 => 010101 => 3 = 0 + 3
[1,0,1,0,1,0,1,0]
 => [1,1,1,1,0,0,0,0]
 => 11110000 => 00001111 => 8 = 5 + 3
[1,0,1,0,1,1,0,0]
 => [1,0,1,1,1,0,0,0]
 => 10111000 => 01000111 => 6 = 3 + 3
[1,0,1,1,0,0,1,0]
 => [1,1,0,1,1,0,0,0]
 => 11011000 => 00100111 => 6 = 3 + 3
[1,0,1,1,0,1,0,0]
 => [1,1,1,0,0,0,1,0]
 => 11100010 => 00011101 => 6 = 3 + 3
[1,0,1,1,1,0,0,0]
 => [1,0,1,0,1,1,0,0]
 => 10101100 => 01010011 => 4 = 1 + 3
[1,1,0,0,1,0,1,0]
 => [1,1,1,0,1,0,0,0]
 => 11101000 => 00010111 => 7 = 4 + 3
[1,1,0,0,1,1,0,0]
 => [1,0,1,1,0,1,0,0]
 => 10110100 => 01001011 => 5 = 2 + 3
[1,1,0,1,0,0,1,0]
 => [1,1,1,0,0,1,0,0]
 => 11100100 => 00011011 => 6 = 3 + 3
[1,1,0,1,0,1,0,0]
 => [1,1,0,0,1,1,0,0]
 => 11001100 => 00110011 => 5 = 2 + 3
[1,1,0,1,1,0,0,0]
 => [1,0,1,1,0,0,1,0]
 => 10110010 => 01001101 => 4 = 1 + 3
[1,1,1,0,0,0,1,0]
 => [1,1,0,1,0,1,0,0]
 => 11010100 => 00101011 => 5 = 2 + 3
[1,1,1,0,0,1,0,0]
 => [1,1,0,1,0,0,1,0]
 => 11010010 => 00101101 => 5 = 2 + 3
[1,1,1,0,1,0,0,0]
 => [1,1,0,0,1,0,1,0]
 => 11001010 => 00110101 => 4 = 1 + 3
[1,1,1,1,0,0,0,0]
 => [1,0,1,0,1,0,1,0]
 => 10101010 => 01010101 => 3 = 0 + 3
[1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => 1111100000 => 0000011111 => 10 = 7 + 3
[1,0,1,0,1,0,1,1,0,0]
 => [1,0,1,1,1,1,0,0,0,0]
 => 1011110000 => 0100001111 => 8 = 5 + 3
[1,0,1,0,1,1,0,0,1,0]
 => [1,1,0,1,1,1,0,0,0,0]
 => 1101110000 => 0010001111 => 8 = 5 + 3
[1,0,1,0,1,1,0,1,0,0]
 => [1,1,1,1,0,0,0,0,1,0]
 => 1111000010 => 0000111101 => 8 = 5 + 3
[1,0,1,0,1,1,1,0,0,0]
 => [1,0,1,0,1,1,1,0,0,0]
 => 1010111000 => 0101000111 => 6 = 3 + 3
[1,0,1,1,0,0,1,0,1,0]
 => [1,1,1,0,1,1,0,0,0,0]
 => 1110110000 => 0001001111 => 8 = 5 + 3
[1,0,1,1,0,0,1,1,0,0]
 => [1,0,1,1,0,1,1,0,0,0]
 => 1011011000 => 0100100111 => 6 = 3 + 3
[1,0,1,1,0,1,0,0,1,0]
 => [1,1,1,1,0,0,0,1,0,0]
 => 1111000100 => 0000111011 => 8 = 5 + 3
[1,0,1,1,0,1,0,1,0,0]
 => [1,1,0,0,1,1,1,0,0,0]
 => 1100111000 => 0011000111 => 6 = 3 + 3
[1,0,1,1,0,1,1,0,0,0]
 => [1,0,1,1,1,0,0,0,1,0]
 => 1011100010 => 0100011101 => 6 = 3 + 3
[1,0,1,1,1,0,0,0,1,0]
 => [1,1,0,1,0,1,1,0,0,0]
 => 1101011000 => 0010100111 => 6 = 3 + 3
[1,0,1,1,1,0,0,1,0,0]
 => [1,1,0,1,1,0,0,0,1,0]
 => 1101100010 => 0010011101 => 6 = 3 + 3
[1,0,1,1,1,0,1,0,0,0]
 => [1,1,1,0,0,0,1,0,1,0]
 => 1110001010 => 0001110101 => 6 = 3 + 3
[1,0,1,1,1,1,0,0,0,0]
 => [1,0,1,0,1,0,1,1,0,0]
 => 1010101100 => 0101010011 => 4 = 1 + 3
[1,1,0,0,1,0,1,0,1,0]
 => [1,1,1,1,0,1,0,0,0,0]
 => 1111010000 => 0000101111 => 9 = 6 + 3
[1,1,0,0,1,0,1,1,0,0]
 => [1,0,1,1,1,0,1,0,0,0]
 => 1011101000 => 0100010111 => 7 = 4 + 3
[1,1,0,0,1,1,0,0,1,0]
 => [1,1,0,1,1,0,1,0,0,0]
 => 1101101000 => 0010010111 => 7 = 4 + 3
[1,1,0,0,1,1,0,1,0,0]
 => [1,1,1,0,1,0,0,0,1,0]
 => 1110100010 => 0001011101 => 7 = 4 + 3
[1,1,0,0,1,1,1,0,0,0]
 => [1,0,1,0,1,1,0,1,0,0]
 => 1010110100 => 0101001011 => 5 = 2 + 3
[1,1,0,1,0,0,1,0,1,0]
 => [1,1,1,1,0,0,1,0,0,0]
 => 1111001000 => 0000110111 => 8 = 5 + 3
[1,1,0,1,0,0,1,1,0,0]
 => [1,0,1,1,1,0,0,1,0,0]
 => 1011100100 => 0100011011 => 6 = 3 + 3
[1,1,0,1,0,1,0,0,1,0]
 => [1,1,1,0,0,1,1,0,0,0]
 => 1110011000 => 0001100111 => 7 = 4 + 3
[1,1,0,1,0,1,0,1,0,0]
 => [1,1,1,0,0,0,1,1,0,0]
 => 1110001100 => 0001110011 => 6 = 3 + 3
[1,1,0,1,0,1,1,0,0,0]
 => [1,0,1,1,0,0,1,1,0,0]
 => 1011001100 => 0100110011 => 5 = 2 + 3
[1,1,0,1,1,0,0,0,1,0]
 => [1,1,0,1,1,0,0,1,0,0]
 => 1101100100 => 0010011011 => 6 = 3 + 3
[1,1,0,1,1,0,0,1,0,0]
 => [1,1,1,0,0,1,0,0,1,0]
 => 1110010010 => 0001101101 => 6 = 3 + 3
[1,1,0,1,1,0,1,0,0,0]
 => [1,1,0,0,1,0,1,1,0,0]
 => 1100101100 => 0011010011 => 5 = 2 + 3
[1,1,0,1,1,1,0,0,0,0]
 => [1,0,1,0,1,1,0,0,1,0]
 => 1010110010 => 0101001101 => 4 = 1 + 3
[1,1,1,0,0,0,1,0,1,0]
 => [1,1,1,0,1,0,1,0,0,0]
 => 1110101000 => 0001010111 => 7 = 4 + 3
[1,0,1,0,1,0,1,0,1,1,1,0,0,0]
 => [1,0,1,0,1,1,1,1,1,0,0,0,0,0]
 => 10101111100000 => 01010000011111 => ? = 7 + 3
[1,0,1,0,1,0,1,1,0,0,1,1,0,0]
 => [1,0,1,1,0,1,1,1,1,0,0,0,0,0]
 => 10110111100000 => 01001000011111 => ? = 7 + 3
[1,0,1,0,1,0,1,1,0,1,0,1,0,0]
 => [1,1,0,0,1,1,1,1,1,0,0,0,0,0]
 => 11001111100000 => 00110000011111 => ? = 7 + 3
[1,0,1,0,1,0,1,1,0,1,1,0,0,0]
 => [1,0,1,1,1,1,1,0,0,0,0,0,1,0]
 => 10111110000010 => 01000001111101 => ? = 7 + 3
[1,0,1,0,1,0,1,1,1,0,0,1,0,0]
 => [1,1,0,1,1,1,1,0,0,0,0,0,1,0]
 => 11011110000010 => 00100001111101 => ? = 7 + 3
[1,0,1,0,1,0,1,1,1,0,1,0,0,0]
 => [1,1,1,1,1,0,0,0,0,0,1,0,1,0]
 => 11111000001010 => 00000111110101 => ? = 7 + 3
[1,0,1,0,1,0,1,1,1,1,0,0,0,0]
 => [1,0,1,0,1,0,1,1,1,1,0,0,0,0]
 => 10101011110000 => 01010100001111 => ? = 5 + 3
[1,0,1,0,1,1,0,0,1,1,1,0,0,0]
 => [1,0,1,0,1,1,0,1,1,1,0,0,0,0]
 => 10101101110000 => 01010010001111 => ? = 5 + 3
[1,0,1,0,1,1,0,1,0,0,1,1,0,0]
 => [1,0,1,1,1,1,1,0,0,0,0,1,0,0]
 => 10111110000100 => 01000001111011 => ? = 7 + 3
[1,0,1,0,1,1,0,1,0,1,0,1,0,0]
 => [1,1,1,1,1,0,0,0,0,0,1,1,0,0]
 => 11111000001100 => 00000111110011 => ? = 7 + 3
[1,0,1,0,1,1,0,1,0,1,1,0,0,0]
 => [1,0,1,1,0,0,1,1,1,1,0,0,0,0]
 => 10110011110000 => 01001100001111 => ? = 5 + 3
[1,0,1,0,1,1,0,1,1,0,0,1,0,0]
 => [1,1,1,1,1,0,0,0,0,1,0,0,1,0]
 => 11111000010010 => 00000111101101 => ? = 7 + 3
[1,0,1,0,1,1,0,1,1,0,1,0,0,0]
 => [1,1,0,0,1,0,1,1,1,1,0,0,0,0]
 => 11001011110000 => 00110100001111 => ? = 5 + 3
[1,0,1,0,1,1,0,1,1,1,0,0,0,0]
 => [1,0,1,0,1,1,1,1,0,0,0,0,1,0]
 => 10101111000010 => 01010000111101 => ? = 5 + 3
[1,0,1,0,1,1,1,0,0,0,1,1,0,0]
 => [1,0,1,1,0,1,0,1,1,1,0,0,0,0]
 => 10110101110000 => 01001010001111 => ? = 5 + 3
[1,0,1,0,1,1,1,0,0,1,0,1,0,0]
 => [1,1,0,0,1,1,0,1,1,1,0,0,0,0]
 => 11001101110000 => 00110010001111 => ? = 5 + 3
[1,0,1,0,1,1,1,0,0,1,1,0,0,0]
 => [1,0,1,1,0,1,1,1,0,0,0,0,1,0]
 => 10110111000010 => 01001000111101 => ? = 5 + 3
[1,0,1,0,1,1,1,0,1,0,0,1,0,0]
 => [1,1,0,1,0,0,1,1,1,1,0,0,0,0]
 => 11010011110000 => 00101100001111 => ? = 5 + 3
[1,0,1,0,1,1,1,0,1,0,1,0,0,0]
 => [1,1,0,0,1,1,1,1,0,0,0,0,1,0]
 => 11001111000010 => 00110000111101 => ? = 5 + 3
[1,0,1,0,1,1,1,0,1,1,0,0,0,0]
 => [1,0,1,1,1,1,0,0,0,0,1,0,1,0]
 => 10111100001010 => 01000011110101 => ? = 5 + 3
[1,0,1,0,1,1,1,1,0,0,0,1,0,0]
 => [1,1,0,1,0,1,1,1,0,0,0,0,1,0]
 => 11010111000010 => 00101000111101 => ? = 5 + 3
[1,0,1,0,1,1,1,1,0,0,1,0,0,0]
 => [1,1,0,1,1,1,0,0,0,0,1,0,1,0]
 => 11011100001010 => 00100011110101 => ? = 5 + 3
[1,0,1,0,1,1,1,1,0,1,0,0,0,0]
 => [1,1,1,1,0,0,0,0,1,0,1,0,1,0]
 => 11110000101010 => 00001111010101 => ? = 5 + 3
[1,0,1,0,1,1,1,1,1,0,0,0,0,0]
 => [1,0,1,0,1,0,1,0,1,1,1,0,0,0]
 => 10101010111000 => 01010101000111 => ? = 3 + 3
[1,0,1,1,0,0,1,0,1,1,1,0,0,0]
 => [1,0,1,0,1,1,1,0,1,1,0,0,0,0]
 => 10101110110000 => 01010001001111 => ? = 5 + 3
[1,0,1,1,0,0,1,1,0,0,1,1,0,0]
 => [1,0,1,1,0,1,1,0,1,1,0,0,0,0]
 => 10110110110000 => 01001001001111 => ? = 5 + 3
[1,0,1,1,0,0,1,1,0,1,0,1,0,0]
 => [1,1,0,0,1,1,1,0,1,1,0,0,0,0]
 => 11001110110000 => 00110001001111 => ? = 5 + 3
[1,0,1,1,0,0,1,1,0,1,1,0,0,0]
 => [1,0,1,1,1,0,1,1,0,0,0,0,1,0]
 => 10111011000010 => 01000100111101 => ? = 5 + 3
[1,0,1,1,0,0,1,1,1,0,0,1,0,0]
 => [1,1,0,1,1,0,1,1,0,0,0,0,1,0]
 => 11011011000010 => 00100100111101 => ? = 5 + 3
[1,0,1,1,0,0,1,1,1,0,1,0,0,0]
 => [1,1,1,0,1,1,0,0,0,0,1,0,1,0]
 => 11101100001010 => 00010011110101 => ? = 5 + 3
[1,0,1,1,0,0,1,1,1,1,0,0,0,0]
 => [1,0,1,0,1,0,1,1,0,1,1,0,0,0]
 => 10101011011000 => 01010100100111 => ? = 3 + 3
[1,0,1,1,0,1,0,0,1,1,1,0,0,0]
 => [1,0,1,0,1,1,1,1,0,0,0,1,0,0]
 => 10101111000100 => 01010000111011 => ? = 5 + 3
[1,0,1,1,0,1,0,1,0,0,1,1,0,0]
 => [1,0,1,1,1,0,0,1,1,1,0,0,0,0]
 => 10111001110000 => 01000110001111 => ? = 5 + 3
[1,0,1,1,0,1,0,1,0,1,0,1,0,0]
 => [1,1,1,0,0,0,1,1,1,1,0,0,0,0]
 => 11100011110000 => 00011100001111 => ? = 5 + 3
[1,0,1,1,0,1,0,1,0,1,1,0,0,0]
 => [1,0,1,1,1,1,0,0,0,0,1,1,0,0]
 => 10111100001100 => 01000011110011 => ? = 5 + 3
[1,0,1,1,0,1,0,1,1,0,0,1,0,0]
 => [1,1,1,0,0,1,1,1,0,0,0,0,1,0]
 => 11100111000010 => 00011000111101 => ? = 5 + 3
[1,0,1,1,0,1,0,1,1,0,1,0,0,0]
 => [1,1,1,1,0,0,0,0,1,0,1,1,0,0]
 => 11110000101100 => 00001111010011 => ? = 5 + 3
[1,0,1,1,0,1,0,1,1,1,0,0,0,0]
 => [1,0,1,0,1,1,0,0,1,1,1,0,0,0]
 => 10101100111000 => 01010011000111 => ? = 3 + 3
[1,0,1,1,0,1,1,0,0,0,1,1,0,0]
 => [1,0,1,1,0,1,1,1,0,0,0,1,0,0]
 => 10110111000100 => 01001000111011 => ? = 5 + 3
[1,0,1,1,0,1,1,0,0,1,0,1,0,0]
 => [1,1,0,0,1,1,1,1,0,0,0,1,0,0]
 => 11001111000100 => 00110000111011 => ? = 5 + 3
[1,0,1,1,0,1,1,0,0,1,1,0,0,0]
 => [1,0,1,1,1,1,0,0,0,1,0,0,1,0]
 => 10111100010010 => 01000011101101 => ? = 5 + 3
[1,0,1,1,0,1,1,0,1,0,0,1,0,0]
 => [1,1,0,1,1,1,0,0,0,0,1,1,0,0]
 => 11011100001100 => 00100011110011 => ? = 5 + 3
[1,0,1,1,0,1,1,0,1,0,1,0,0,0]
 => [1,1,1,1,0,0,0,0,1,1,0,0,1,0]
 => 11110000110010 => 00001111001101 => ? = 5 + 3
[1,0,1,1,0,1,1,0,1,1,0,0,0,0]
 => [1,0,1,1,0,0,1,0,1,1,1,0,0,0]
 => 10110010111000 => 01001101000111 => ? = 3 + 3
[1,0,1,1,0,1,1,1,0,0,0,1,0,0]
 => [1,1,0,1,1,1,0,0,0,1,0,0,1,0]
 => 11011100010010 => 00100011101101 => ? = 5 + 3
[1,0,1,1,0,1,1,1,0,0,1,0,0,0]
 => [1,1,1,1,0,0,0,1,0,0,1,0,1,0]
 => 11110001001010 => 00001110110101 => ? = 5 + 3
[1,0,1,1,0,1,1,1,0,1,0,0,0,0]
 => [1,1,0,0,1,0,1,0,1,1,1,0,0,0]
 => 11001010111000 => 00110101000111 => ? = 3 + 3
[1,0,1,1,0,1,1,1,1,0,0,0,0,0]
 => [1,0,1,0,1,0,1,1,1,0,0,0,1,0]
 => 10101011100010 => 01010100011101 => ? = 3 + 3
[1,0,1,1,1,0,0,0,1,0,1,1,0,0]
 => [1,0,1,1,1,0,1,0,1,1,0,0,0,0]
 => 10111010110000 => 01000101001111 => ? = 5 + 3
[1,0,1,1,1,0,0,0,1,1,0,1,0,0]
 => [1,1,1,0,1,0,1,1,0,0,0,0,1,0]
 => 11101011000010 => 00010100111101 => ? = 5 + 3
Description
The number of factors in the Catalan decomposition of a binary word. Every binary word can be written in a unique way as $(\mathcal D 0)^\ell \mathcal D (1 \mathcal D)^m$, where $\mathcal D$ is the set of Dyck words. This is the Catalan factorisation, see [1, sec.9.1.2]. This statistic records the number of factors in the Catalan factorisation, that is, $\ell + m$ if the middle Dyck word is empty and $\ell + 1 + m$ otherwise.
Matching statistic: St000691
(load all 2 compositions to match this statistic)
Mapping
Mp00099: Dyck paths —bounce path⟶ Dyck paths
Mp00093: Dyck paths —to binary word⟶ Binary words
Mp00280: Binary words —path rowmotion⟶ Binary words
St000691: Binary words ⟶ ℤResult quality: 5% ●values known / values provided: 5%●distinct values known / distinct values provided: 60%
Values
[1,0,1,0]
 => [1,0,1,0]
 => 1010 => 1101 => 2 = 1 + 1
[1,1,0,0]
 => [1,1,0,0]
 => 1100 => 0111 => 1 = 0 + 1
[1,0,1,0,1,0]
 => [1,0,1,0,1,0]
 => 101010 => 110101 => 4 = 3 + 1
[1,0,1,1,0,0]
 => [1,0,1,1,0,0]
 => 101100 => 110011 => 2 = 1 + 1
[1,1,0,0,1,0]
 => [1,1,0,0,1,0]
 => 110010 => 011101 => 3 = 2 + 1
[1,1,0,1,0,0]
 => [1,0,1,1,0,0]
 => 101100 => 110011 => 2 = 1 + 1
[1,1,1,0,0,0]
 => [1,1,1,0,0,0]
 => 111000 => 001111 => 1 = 0 + 1
[1,0,1,0,1,0,1,0]
 => [1,0,1,0,1,0,1,0]
 => 10101010 => 11010101 => 6 = 5 + 1
[1,0,1,0,1,1,0,0]
 => [1,0,1,0,1,1,0,0]
 => 10101100 => 11010011 => 4 = 3 + 1
[1,0,1,1,0,0,1,0]
 => [1,0,1,1,0,0,1,0]
 => 10110010 => 11001101 => 4 = 3 + 1
[1,0,1,1,0,1,0,0]
 => [1,0,1,0,1,1,0,0]
 => 10101100 => 11010011 => 4 = 3 + 1
[1,0,1,1,1,0,0,0]
 => [1,0,1,1,1,0,0,0]
 => 10111000 => 11000111 => 2 = 1 + 1
[1,1,0,0,1,0,1,0]
 => [1,1,0,0,1,0,1,0]
 => 11001010 => 01110101 => 5 = 4 + 1
[1,1,0,0,1,1,0,0]
 => [1,1,0,0,1,1,0,0]
 => 11001100 => 01110011 => 3 = 2 + 1
[1,1,0,1,0,0,1,0]
 => [1,0,1,1,0,0,1,0]
 => 10110010 => 11001101 => 4 = 3 + 1
[1,1,0,1,0,1,0,0]
 => [1,1,0,0,1,1,0,0]
 => 11001100 => 01110011 => 3 = 2 + 1
[1,1,0,1,1,0,0,0]
 => [1,0,1,1,1,0,0,0]
 => 10111000 => 11000111 => 2 = 1 + 1
[1,1,1,0,0,0,1,0]
 => [1,1,1,0,0,0,1,0]
 => 11100010 => 00111101 => 3 = 2 + 1
[1,1,1,0,0,1,0,0]
 => [1,1,0,0,1,1,0,0]
 => 11001100 => 01110011 => 3 = 2 + 1
[1,1,1,0,1,0,0,0]
 => [1,0,1,1,1,0,0,0]
 => 10111000 => 11000111 => 2 = 1 + 1
[1,1,1,1,0,0,0,0]
 => [1,1,1,1,0,0,0,0]
 => 11110000 => 00011111 => 1 = 0 + 1
[1,0,1,0,1,0,1,0,1,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => 1010101010 => 1101010101 => ? = 7 + 1
[1,0,1,0,1,0,1,1,0,0]
 => [1,0,1,0,1,0,1,1,0,0]
 => 1010101100 => 1101010011 => ? = 5 + 1
[1,0,1,0,1,1,0,0,1,0]
 => [1,0,1,0,1,1,0,0,1,0]
 => 1010110010 => 1101001101 => ? = 5 + 1
[1,0,1,0,1,1,0,1,0,0]
 => [1,0,1,0,1,0,1,1,0,0]
 => 1010101100 => 1101010011 => ? = 5 + 1
[1,0,1,0,1,1,1,0,0,0]
 => [1,0,1,0,1,1,1,0,0,0]
 => 1010111000 => 1101000111 => ? = 3 + 1
[1,0,1,1,0,0,1,0,1,0]
 => [1,0,1,1,0,0,1,0,1,0]
 => 1011001010 => 1100110101 => ? = 5 + 1
[1,0,1,1,0,0,1,1,0,0]
 => [1,0,1,1,0,0,1,1,0,0]
 => 1011001100 => 1100110011 => ? = 3 + 1
[1,0,1,1,0,1,0,0,1,0]
 => [1,0,1,0,1,1,0,0,1,0]
 => 1010110010 => 1101001101 => ? = 5 + 1
[1,0,1,1,0,1,0,1,0,0]
 => [1,0,1,1,0,0,1,1,0,0]
 => 1011001100 => 1100110011 => ? = 3 + 1
[1,0,1,1,0,1,1,0,0,0]
 => [1,0,1,0,1,1,1,0,0,0]
 => 1010111000 => 1101000111 => ? = 3 + 1
[1,0,1,1,1,0,0,0,1,0]
 => [1,0,1,1,1,0,0,0,1,0]
 => 1011100010 => 1100011101 => ? = 3 + 1
[1,0,1,1,1,0,0,1,0,0]
 => [1,0,1,1,0,0,1,1,0,0]
 => 1011001100 => 1100110011 => ? = 3 + 1
[1,0,1,1,1,0,1,0,0,0]
 => [1,0,1,0,1,1,1,0,0,0]
 => 1010111000 => 1101000111 => ? = 3 + 1
[1,0,1,1,1,1,0,0,0,0]
 => [1,0,1,1,1,1,0,0,0,0]
 => 1011110000 => 1100001111 => ? = 1 + 1
[1,1,0,0,1,0,1,0,1,0]
 => [1,1,0,0,1,0,1,0,1,0]
 => 1100101010 => 0111010101 => ? = 6 + 1
[1,1,0,0,1,0,1,1,0,0]
 => [1,1,0,0,1,0,1,1,0,0]
 => 1100101100 => 0111010011 => ? = 4 + 1
[1,1,0,0,1,1,0,0,1,0]
 => [1,1,0,0,1,1,0,0,1,0]
 => 1100110010 => 0111001101 => ? = 4 + 1
[1,1,0,0,1,1,0,1,0,0]
 => [1,1,0,0,1,0,1,1,0,0]
 => 1100101100 => 0111010011 => ? = 4 + 1
[1,1,0,0,1,1,1,0,0,0]
 => [1,1,0,0,1,1,1,0,0,0]
 => 1100111000 => 0111000111 => ? = 2 + 1
[1,1,0,1,0,0,1,0,1,0]
 => [1,0,1,1,0,0,1,0,1,0]
 => 1011001010 => 1100110101 => ? = 5 + 1
[1,1,0,1,0,0,1,1,0,0]
 => [1,0,1,1,0,0,1,1,0,0]
 => 1011001100 => 1100110011 => ? = 3 + 1
[1,1,0,1,0,1,0,0,1,0]
 => [1,1,0,0,1,1,0,0,1,0]
 => 1100110010 => 0111001101 => ? = 4 + 1
[1,1,0,1,0,1,0,1,0,0]
 => [1,0,1,1,0,0,1,1,0,0]
 => 1011001100 => 1100110011 => ? = 3 + 1
[1,1,0,1,0,1,1,0,0,0]
 => [1,1,0,0,1,1,1,0,0,0]
 => 1100111000 => 0111000111 => ? = 2 + 1
[1,1,0,1,1,0,0,0,1,0]
 => [1,0,1,1,1,0,0,0,1,0]
 => 1011100010 => 1100011101 => ? = 3 + 1
[1,1,0,1,1,0,0,1,0,0]
 => [1,0,1,1,0,0,1,1,0,0]
 => 1011001100 => 1100110011 => ? = 3 + 1
[1,1,0,1,1,0,1,0,0,0]
 => [1,1,0,0,1,1,1,0,0,0]
 => 1100111000 => 0111000111 => ? = 2 + 1
[1,1,0,1,1,1,0,0,0,0]
 => [1,0,1,1,1,1,0,0,0,0]
 => 1011110000 => 1100001111 => ? = 1 + 1
[1,1,1,0,0,0,1,0,1,0]
 => [1,1,1,0,0,0,1,0,1,0]
 => 1110001010 => 0011110101 => ? = 4 + 1
[1,1,1,0,0,0,1,1,0,0]
 => [1,1,1,0,0,0,1,1,0,0]
 => 1110001100 => 0011110011 => ? = 2 + 1
[1,1,1,0,0,1,0,0,1,0]
 => [1,1,0,0,1,1,0,0,1,0]
 => 1100110010 => 0111001101 => ? = 4 + 1
[1,1,1,0,0,1,0,1,0,0]
 => [1,1,1,0,0,0,1,1,0,0]
 => 1110001100 => 0011110011 => ? = 2 + 1
[1,1,1,0,0,1,1,0,0,0]
 => [1,1,0,0,1,1,1,0,0,0]
 => 1100111000 => 0111000111 => ? = 2 + 1
[1,1,1,0,1,0,0,0,1,0]
 => [1,0,1,1,1,0,0,0,1,0]
 => 1011100010 => 1100011101 => ? = 3 + 1
[1,1,1,0,1,0,0,1,0,0]
 => [1,1,1,0,0,0,1,1,0,0]
 => 1110001100 => 0011110011 => ? = 2 + 1
[1,1,1,0,1,0,1,0,0,0]
 => [1,1,0,0,1,1,1,0,0,0]
 => 1100111000 => 0111000111 => ? = 2 + 1
[1,1,1,0,1,1,0,0,0,0]
 => [1,0,1,1,1,1,0,0,0,0]
 => 1011110000 => 1100001111 => ? = 1 + 1
[1,1,1,1,0,0,0,0,1,0]
 => [1,1,1,1,0,0,0,0,1,0]
 => 1111000010 => 0001111101 => 3 = 2 + 1
[1,1,1,1,0,0,0,1,0,0]
 => [1,1,1,0,0,0,1,1,0,0]
 => 1110001100 => 0011110011 => ? = 2 + 1
[1,1,1,1,0,0,1,0,0,0]
 => [1,1,0,0,1,1,1,0,0,0]
 => 1100111000 => 0111000111 => ? = 2 + 1
[1,1,1,1,0,1,0,0,0,0]
 => [1,0,1,1,1,1,0,0,0,0]
 => 1011110000 => 1100001111 => ? = 1 + 1
[1,1,1,1,1,0,0,0,0,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => 1111100000 => 0000111111 => 1 = 0 + 1
[1,0,1,0,1,0,1,0,1,0,1,0]
 => [1,0,1,0,1,0,1,0,1,0,1,0]
 => 101010101010 => 110101010101 => ? = 9 + 1
[1,0,1,0,1,0,1,0,1,1,0,0]
 => [1,0,1,0,1,0,1,0,1,1,0,0]
 => 101010101100 => 110101010011 => ? = 7 + 1
[1,0,1,0,1,0,1,1,0,0,1,0]
 => [1,0,1,0,1,0,1,1,0,0,1,0]
 => 101010110010 => 110101001101 => ? = 7 + 1
[1,0,1,0,1,0,1,1,0,1,0,0]
 => [1,0,1,0,1,0,1,0,1,1,0,0]
 => 101010101100 => 110101010011 => ? = 7 + 1
[1,0,1,0,1,0,1,1,1,0,0,0]
 => [1,0,1,0,1,0,1,1,1,0,0,0]
 => 101010111000 => 110101000111 => ? = 5 + 1
[1,0,1,0,1,1,0,0,1,0,1,0]
 => [1,0,1,0,1,1,0,0,1,0,1,0]
 => 101011001010 => 110100110101 => ? = 7 + 1
[1,0,1,0,1,1,0,0,1,1,0,0]
 => [1,0,1,0,1,1,0,0,1,1,0,0]
 => 101011001100 => 110100110011 => ? = 5 + 1
[1,0,1,0,1,1,0,1,0,0,1,0]
 => [1,0,1,0,1,0,1,1,0,0,1,0]
 => 101010110010 => 110101001101 => ? = 7 + 1
[1,0,1,0,1,1,0,1,0,1,0,0]
 => [1,0,1,0,1,1,0,0,1,1,0,0]
 => 101011001100 => 110100110011 => ? = 5 + 1
[1,0,1,0,1,1,0,1,1,0,0,0]
 => [1,0,1,0,1,0,1,1,1,0,0,0]
 => 101010111000 => 110101000111 => ? = 5 + 1
Description
The number of changes of a binary word. This is the number of indices $i$ such that $w_i \neq w_{i+1}$.
Matching statistic: St001232
Mapping
Mp00199: Dyck paths —prime Dyck path⟶ Dyck paths
Mp00099: Dyck paths —bounce path⟶ Dyck paths
Mp00296: Dyck paths —Knuth-Krattenthaler⟶ Dyck paths
St001232: Dyck paths ⟶ ℤResult quality: 4% ●values known / values provided: 4%●distinct values known / distinct values provided: 20%
Values
[1,0,1,0]
 => [1,1,0,1,0,0]
 => [1,0,1,1,0,0]
 => [1,1,0,1,0,0]
 => 2 = 1 + 1
[1,1,0,0]
 => [1,1,1,0,0,0]
 => [1,1,1,0,0,0]
 => [1,1,0,0,1,0]
 => 1 = 0 + 1
[1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,0]
 => [1,1,0,0,1,1,0,0]
 => [1,0,1,0,1,0,1,0]
 => ? = 3 + 1
[1,0,1,1,0,0]
 => [1,1,0,1,1,0,0,0]
 => [1,0,1,1,1,0,0,0]
 => [1,1,1,0,0,1,0,0]
 => 2 = 1 + 1
[1,1,0,0,1,0]
 => [1,1,1,0,0,1,0,0]
 => [1,1,0,0,1,1,0,0]
 => [1,0,1,0,1,0,1,0]
 => ? = 2 + 1
[1,1,0,1,0,0]
 => [1,1,1,0,1,0,0,0]
 => [1,0,1,1,1,0,0,0]
 => [1,1,1,0,0,1,0,0]
 => 2 = 1 + 1
[1,1,1,0,0,0]
 => [1,1,1,1,0,0,0,0]
 => [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]
 => [1,1,0,1,0,1,0,1,0,0]
 => [1,0,1,1,0,0,1,1,0,0]
 => [1,1,0,1,0,1,0,1,0,0]
 => ? = 5 + 1
[1,0,1,0,1,1,0,0]
 => [1,1,0,1,0,1,1,0,0,0]
 => [1,1,0,0,1,1,1,0,0,0]
 => [1,0,1,1,0,0,1,0,1,0]
 => ? = 3 + 1
[1,0,1,1,0,0,1,0]
 => [1,1,0,1,1,0,0,1,0,0]
 => [1,0,1,1,0,0,1,1,0,0]
 => [1,1,0,1,0,1,0,1,0,0]
 => ? = 3 + 1
[1,0,1,1,0,1,0,0]
 => [1,1,0,1,1,0,1,0,0,0]
 => [1,1,0,0,1,1,1,0,0,0]
 => [1,0,1,1,0,0,1,0,1,0]
 => ? = 3 + 1
[1,0,1,1,1,0,0,0]
 => [1,1,0,1,1,1,0,0,0,0]
 => [1,0,1,1,1,1,0,0,0,0]
 => [1,1,1,1,0,0,0,1,0,0]
 => 2 = 1 + 1
[1,1,0,0,1,0,1,0]
 => [1,1,1,0,0,1,0,1,0,0]
 => [1,1,1,0,0,0,1,1,0,0]
 => [1,1,0,1,0,1,0,0,1,0]
 => ? = 4 + 1
[1,1,0,0,1,1,0,0]
 => [1,1,1,0,0,1,1,0,0,0]
 => [1,1,0,0,1,1,1,0,0,0]
 => [1,0,1,1,0,0,1,0,1,0]
 => ? = 2 + 1
[1,1,0,1,0,0,1,0]
 => [1,1,1,0,1,0,0,1,0,0]
 => [1,1,1,0,0,0,1,1,0,0]
 => [1,1,0,1,0,1,0,0,1,0]
 => ? = 3 + 1
[1,1,0,1,0,1,0,0]
 => [1,1,1,0,1,0,1,0,0,0]
 => [1,1,0,0,1,1,1,0,0,0]
 => [1,0,1,1,0,0,1,0,1,0]
 => ? = 2 + 1
[1,1,0,1,1,0,0,0]
 => [1,1,1,0,1,1,0,0,0,0]
 => [1,0,1,1,1,1,0,0,0,0]
 => [1,1,1,1,0,0,0,1,0,0]
 => 2 = 1 + 1
[1,1,1,0,0,0,1,0]
 => [1,1,1,1,0,0,0,1,0,0]
 => [1,1,1,0,0,0,1,1,0,0]
 => [1,1,0,1,0,1,0,0,1,0]
 => ? = 2 + 1
[1,1,1,0,0,1,0,0]
 => [1,1,1,1,0,0,1,0,0,0]
 => [1,1,0,0,1,1,1,0,0,0]
 => [1,0,1,1,0,0,1,0,1,0]
 => ? = 2 + 1
[1,1,1,0,1,0,0,0]
 => [1,1,1,1,0,1,0,0,0,0]
 => [1,0,1,1,1,1,0,0,0,0]
 => [1,1,1,1,0,0,0,1,0,0]
 => 2 = 1 + 1
[1,1,1,1,0,0,0,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => [1,1,1,1,0,0,0,0,1,0]
 => 1 = 0 + 1
[1,0,1,0,1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,1,0,1,0,0]
 => [1,1,0,0,1,1,0,0,1,1,0,0]
 => [1,0,1,0,1,0,1,0,1,0,1,0]
 => ? = 7 + 1
[1,0,1,0,1,0,1,1,0,0]
 => [1,1,0,1,0,1,0,1,1,0,0,0]
 => [1,0,1,1,0,0,1,1,1,0,0,0]
 => [1,1,0,1,1,0,0,1,0,1,0,0]
 => ? = 5 + 1
[1,0,1,0,1,1,0,0,1,0]
 => [1,1,0,1,0,1,1,0,0,1,0,0]
 => [1,1,0,0,1,1,0,0,1,1,0,0]
 => [1,0,1,0,1,0,1,0,1,0,1,0]
 => ? = 5 + 1
[1,0,1,0,1,1,0,1,0,0]
 => [1,1,0,1,0,1,1,0,1,0,0,0]
 => [1,0,1,1,0,0,1,1,1,0,0,0]
 => [1,1,0,1,1,0,0,1,0,1,0,0]
 => ? = 5 + 1
[1,0,1,0,1,1,1,0,0,0]
 => [1,1,0,1,0,1,1,1,0,0,0,0]
 => [1,1,0,0,1,1,1,1,0,0,0,0]
 => [1,0,1,1,1,0,0,0,1,0,1,0]
 => ? = 3 + 1
[1,0,1,1,0,0,1,0,1,0]
 => [1,1,0,1,1,0,0,1,0,1,0,0]
 => [1,0,1,1,1,0,0,0,1,1,0,0]
 => [1,1,1,0,1,0,1,0,0,1,0,0]
 => ? = 5 + 1
[1,0,1,1,0,0,1,1,0,0]
 => [1,1,0,1,1,0,0,1,1,0,0,0]
 => [1,0,1,1,0,0,1,1,1,0,0,0]
 => [1,1,0,1,1,0,0,1,0,1,0,0]
 => ? = 3 + 1
[1,0,1,1,0,1,0,0,1,0]
 => [1,1,0,1,1,0,1,0,0,1,0,0]
 => [1,0,1,1,1,0,0,0,1,1,0,0]
 => [1,1,1,0,1,0,1,0,0,1,0,0]
 => ? = 5 + 1
[1,0,1,1,0,1,0,1,0,0]
 => [1,1,0,1,1,0,1,0,1,0,0,0]
 => [1,0,1,1,0,0,1,1,1,0,0,0]
 => [1,1,0,1,1,0,0,1,0,1,0,0]
 => ? = 3 + 1
[1,0,1,1,0,1,1,0,0,0]
 => [1,1,0,1,1,0,1,1,0,0,0,0]
 => [1,1,0,0,1,1,1,1,0,0,0,0]
 => [1,0,1,1,1,0,0,0,1,0,1,0]
 => ? = 3 + 1
[1,0,1,1,1,0,0,0,1,0]
 => [1,1,0,1,1,1,0,0,0,1,0,0]
 => [1,0,1,1,1,0,0,0,1,1,0,0]
 => [1,1,1,0,1,0,1,0,0,1,0,0]
 => ? = 3 + 1
[1,0,1,1,1,0,0,1,0,0]
 => [1,1,0,1,1,1,0,0,1,0,0,0]
 => [1,0,1,1,0,0,1,1,1,0,0,0]
 => [1,1,0,1,1,0,0,1,0,1,0,0]
 => ? = 3 + 1
[1,0,1,1,1,0,1,0,0,0]
 => [1,1,0,1,1,1,0,1,0,0,0,0]
 => [1,1,0,0,1,1,1,1,0,0,0,0]
 => [1,0,1,1,1,0,0,0,1,0,1,0]
 => ? = 3 + 1
[1,0,1,1,1,1,0,0,0,0]
 => [1,1,0,1,1,1,1,0,0,0,0,0]
 => [1,0,1,1,1,1,1,0,0,0,0,0]
 => [1,1,1,1,1,0,0,0,0,1,0,0]
 => 2 = 1 + 1
[1,1,0,0,1,0,1,0,1,0]
 => [1,1,1,0,0,1,0,1,0,1,0,0]
 => [1,1,0,0,1,1,0,0,1,1,0,0]
 => [1,0,1,0,1,0,1,0,1,0,1,0]
 => ? = 6 + 1
[1,1,0,0,1,0,1,1,0,0]
 => [1,1,1,0,0,1,0,1,1,0,0,0]
 => [1,1,1,0,0,0,1,1,1,0,0,0]
 => [1,1,0,1,1,0,0,1,0,0,1,0]
 => ? = 4 + 1
[1,1,0,0,1,1,0,0,1,0]
 => [1,1,1,0,0,1,1,0,0,1,0,0]
 => [1,1,0,0,1,1,0,0,1,1,0,0]
 => [1,0,1,0,1,0,1,0,1,0,1,0]
 => ? = 4 + 1
[1,1,0,0,1,1,0,1,0,0]
 => [1,1,1,0,0,1,1,0,1,0,0,0]
 => [1,1,1,0,0,0,1,1,1,0,0,0]
 => [1,1,0,1,1,0,0,1,0,0,1,0]
 => ? = 4 + 1
[1,1,0,0,1,1,1,0,0,0]
 => [1,1,1,0,0,1,1,1,0,0,0,0]
 => [1,1,0,0,1,1,1,1,0,0,0,0]
 => [1,0,1,1,1,0,0,0,1,0,1,0]
 => ? = 2 + 1
[1,1,0,1,0,0,1,0,1,0]
 => [1,1,1,0,1,0,0,1,0,1,0,0]
 => [1,0,1,1,1,0,0,0,1,1,0,0]
 => [1,1,1,0,1,0,1,0,0,1,0,0]
 => ? = 5 + 1
[1,1,0,1,0,0,1,1,0,0]
 => [1,1,1,0,1,0,0,1,1,0,0,0]
 => [1,1,1,0,0,0,1,1,1,0,0,0]
 => [1,1,0,1,1,0,0,1,0,0,1,0]
 => ? = 3 + 1
[1,1,0,1,0,1,0,0,1,0]
 => [1,1,1,0,1,0,1,0,0,1,0,0]
 => [1,0,1,1,1,0,0,0,1,1,0,0]
 => [1,1,1,0,1,0,1,0,0,1,0,0]
 => ? = 4 + 1
[1,1,0,1,0,1,0,1,0,0]
 => [1,1,1,0,1,0,1,0,1,0,0,0]
 => [1,1,1,0,0,0,1,1,1,0,0,0]
 => [1,1,0,1,1,0,0,1,0,0,1,0]
 => ? = 3 + 1
[1,1,0,1,0,1,1,0,0,0]
 => [1,1,1,0,1,0,1,1,0,0,0,0]
 => [1,1,0,0,1,1,1,1,0,0,0,0]
 => [1,0,1,1,1,0,0,0,1,0,1,0]
 => ? = 2 + 1
[1,1,0,1,1,0,0,0,1,0]
 => [1,1,1,0,1,1,0,0,0,1,0,0]
 => [1,0,1,1,1,0,0,0,1,1,0,0]
 => [1,1,1,0,1,0,1,0,0,1,0,0]
 => ? = 3 + 1
[1,1,0,1,1,0,0,1,0,0]
 => [1,1,1,0,1,1,0,0,1,0,0,0]
 => [1,1,1,0,0,0,1,1,1,0,0,0]
 => [1,1,0,1,1,0,0,1,0,0,1,0]
 => ? = 3 + 1
[1,1,0,1,1,0,1,0,0,0]
 => [1,1,1,0,1,1,0,1,0,0,0,0]
 => [1,1,0,0,1,1,1,1,0,0,0,0]
 => [1,0,1,1,1,0,0,0,1,0,1,0]
 => ? = 2 + 1
[1,1,0,1,1,1,0,0,0,0]
 => [1,1,1,0,1,1,1,0,0,0,0,0]
 => [1,0,1,1,1,1,1,0,0,0,0,0]
 => [1,1,1,1,1,0,0,0,0,1,0,0]
 => 2 = 1 + 1
[1,1,1,0,0,0,1,0,1,0]
 => [1,1,1,1,0,0,0,1,0,1,0,0]
 => [1,1,1,1,0,0,0,0,1,1,0,0]
 => [1,1,1,0,1,0,1,0,0,0,1,0]
 => ? = 4 + 1
[1,1,1,0,0,0,1,1,0,0]
 => [1,1,1,1,0,0,0,1,1,0,0,0]
 => [1,1,1,0,0,0,1,1,1,0,0,0]
 => [1,1,0,1,1,0,0,1,0,0,1,0]
 => ? = 2 + 1
[1,1,1,0,0,1,0,0,1,0]
 => [1,1,1,1,0,0,1,0,0,1,0,0]
 => [1,1,1,1,0,0,0,0,1,1,0,0]
 => [1,1,1,0,1,0,1,0,0,0,1,0]
 => ? = 4 + 1
[1,1,1,0,0,1,0,1,0,0]
 => [1,1,1,1,0,0,1,0,1,0,0,0]
 => [1,1,1,0,0,0,1,1,1,0,0,0]
 => [1,1,0,1,1,0,0,1,0,0,1,0]
 => ? = 2 + 1
[1,1,1,0,0,1,1,0,0,0]
 => [1,1,1,1,0,0,1,1,0,0,0,0]
 => [1,1,0,0,1,1,1,1,0,0,0,0]
 => [1,0,1,1,1,0,0,0,1,0,1,0]
 => ? = 2 + 1
[1,1,1,0,1,0,0,0,1,0]
 => [1,1,1,1,0,1,0,0,0,1,0,0]
 => [1,1,1,1,0,0,0,0,1,1,0,0]
 => [1,1,1,0,1,0,1,0,0,0,1,0]
 => ? = 3 + 1
[1,1,1,0,1,0,0,1,0,0]
 => [1,1,1,1,0,1,0,0,1,0,0,0]
 => [1,1,1,0,0,0,1,1,1,0,0,0]
 => [1,1,0,1,1,0,0,1,0,0,1,0]
 => ? = 2 + 1
[1,1,1,0,1,0,1,0,0,0]
 => [1,1,1,1,0,1,0,1,0,0,0,0]
 => [1,1,0,0,1,1,1,1,0,0,0,0]
 => [1,0,1,1,1,0,0,0,1,0,1,0]
 => ? = 2 + 1
[1,1,1,0,1,1,0,0,0,0]
 => [1,1,1,1,0,1,1,0,0,0,0,0]
 => [1,0,1,1,1,1,1,0,0,0,0,0]
 => [1,1,1,1,1,0,0,0,0,1,0,0]
 => 2 = 1 + 1
[1,1,1,1,0,0,0,0,1,0]
 => [1,1,1,1,1,0,0,0,0,1,0,0]
 => [1,1,1,1,0,0,0,0,1,1,0,0]
 => [1,1,1,0,1,0,1,0,0,0,1,0]
 => ? = 2 + 1
[1,1,1,1,0,0,0,1,0,0]
 => [1,1,1,1,1,0,0,0,1,0,0,0]
 => [1,1,1,0,0,0,1,1,1,0,0,0]
 => [1,1,0,1,1,0,0,1,0,0,1,0]
 => ? = 2 + 1
[1,1,1,1,0,0,1,0,0,0]
 => [1,1,1,1,1,0,0,1,0,0,0,0]
 => [1,1,0,0,1,1,1,1,0,0,0,0]
 => [1,0,1,1,1,0,0,0,1,0,1,0]
 => ? = 2 + 1
[1,1,1,1,0,1,0,0,0,0]
 => [1,1,1,1,1,0,1,0,0,0,0,0]
 => [1,0,1,1,1,1,1,0,0,0,0,0]
 => [1,1,1,1,1,0,0,0,0,1,0,0]
 => 2 = 1 + 1
[1,1,1,1,1,0,0,0,0,0]
 => [1,1,1,1,1,1,0,0,0,0,0,0]
 => [1,1,1,1,1,1,0,0,0,0,0,0]
 => [1,1,1,1,1,0,0,0,0,0,1,0]
 => 1 = 0 + 1
[1,0,1,0,1,0,1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => [1,0,1,1,0,0,1,1,0,0,1,1,0,0]
 => [1,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => ? = 9 + 1
[1,0,1,1,1,1,1,0,0,0,0,0]
 => [1,1,0,1,1,1,1,1,0,0,0,0,0,0]
 => [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
 => [1,1,1,1,1,1,0,0,0,0,0,1,0,0]
 => 2 = 1 + 1
[1,1,0,1,1,1,1,0,0,0,0,0]
 => [1,1,1,0,1,1,1,1,0,0,0,0,0,0]
 => [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
 => [1,1,1,1,1,1,0,0,0,0,0,1,0,0]
 => 2 = 1 + 1
[1,1,1,0,1,1,1,0,0,0,0,0]
 => [1,1,1,1,0,1,1,1,0,0,0,0,0,0]
 => [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
 => [1,1,1,1,1,1,0,0,0,0,0,1,0,0]
 => 2 = 1 + 1
[1,1,1,1,0,1,1,0,0,0,0,0]
 => [1,1,1,1,1,0,1,1,0,0,0,0,0,0]
 => [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
 => [1,1,1,1,1,1,0,0,0,0,0,1,0,0]
 => 2 = 1 + 1
[1,1,1,1,1,0,1,0,0,0,0,0]
 => [1,1,1,1,1,1,0,1,0,0,0,0,0,0]
 => [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
 => [1,1,1,1,1,1,0,0,0,0,0,1,0,0]
 => 2 = 1 + 1
[1,1,1,1,1,1,0,0,0,0,0,0]
 => [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
 => [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
 => [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
 => 1 = 0 + 1
Description
The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2.