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Your data matches 7 different statistics following compositions of up to 3 maps.
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Matching statistic: St000420
(load all 8 compositions to match this statistic)
(load all 8 compositions to match this statistic)
St000420: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0,1,0]
=> 2
[1,1,0,0]
=> 1
[1,0,1,0,1,0]
=> 5
[1,0,1,1,0,0]
=> 3
[1,1,0,0,1,0]
=> 3
[1,1,0,1,0,0]
=> 2
[1,1,1,0,0,0]
=> 1
[1,0,1,0,1,0,1,0]
=> 14
[1,0,1,0,1,1,0,0]
=> 9
[1,0,1,1,0,0,1,0]
=> 10
[1,0,1,1,0,1,0,0]
=> 7
[1,0,1,1,1,0,0,0]
=> 4
[1,1,0,0,1,0,1,0]
=> 9
[1,1,0,0,1,1,0,0]
=> 6
[1,1,0,1,0,0,1,0]
=> 7
[1,1,0,1,0,1,0,0]
=> 5
[1,1,0,1,1,0,0,0]
=> 3
[1,1,1,0,0,0,1,0]
=> 4
[1,1,1,0,0,1,0,0]
=> 3
[1,1,1,0,1,0,0,0]
=> 2
[1,1,1,1,0,0,0,0]
=> 1
[1,0,1,0,1,0,1,0,1,0]
=> 42
[1,0,1,0,1,0,1,1,0,0]
=> 28
[1,0,1,0,1,1,0,0,1,0]
=> 32
[1,0,1,0,1,1,0,1,0,0]
=> 23
[1,0,1,0,1,1,1,0,0,0]
=> 14
[1,0,1,1,0,0,1,0,1,0]
=> 32
[1,0,1,1,0,0,1,1,0,0]
=> 22
[1,0,1,1,0,1,0,0,1,0]
=> 26
[1,0,1,1,0,1,0,1,0,0]
=> 19
[1,0,1,1,0,1,1,0,0,0]
=> 12
[1,0,1,1,1,0,0,0,1,0]
=> 17
[1,0,1,1,1,0,0,1,0,0]
=> 13
[1,0,1,1,1,0,1,0,0,0]
=> 9
[1,0,1,1,1,1,0,0,0,0]
=> 5
[1,1,0,0,1,0,1,0,1,0]
=> 28
[1,1,0,0,1,0,1,1,0,0]
=> 19
[1,1,0,0,1,1,0,0,1,0]
=> 22
[1,1,0,0,1,1,0,1,0,0]
=> 16
[1,1,0,0,1,1,1,0,0,0]
=> 10
[1,1,0,1,0,0,1,0,1,0]
=> 23
[1,1,0,1,0,0,1,1,0,0]
=> 16
[1,1,0,1,0,1,0,0,1,0]
=> 19
[1,1,0,1,0,1,0,1,0,0]
=> 14
[1,1,0,1,0,1,1,0,0,0]
=> 9
[1,1,0,1,1,0,0,0,1,0]
=> 13
[1,1,0,1,1,0,0,1,0,0]
=> 10
[1,1,0,1,1,0,1,0,0,0]
=> 7
[1,1,0,1,1,1,0,0,0,0]
=> 4
[1,1,1,0,0,0,1,0,1,0]
=> 14
Description
The number of Dyck paths that are weakly above a Dyck path.
Matching statistic: St000419
(load all 6 compositions to match this statistic)
(load all 6 compositions to match this statistic)
St000419: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0,1,0]
=> 1 = 2 - 1
[1,1,0,0]
=> 0 = 1 - 1
[1,0,1,0,1,0]
=> 4 = 5 - 1
[1,0,1,1,0,0]
=> 2 = 3 - 1
[1,1,0,0,1,0]
=> 2 = 3 - 1
[1,1,0,1,0,0]
=> 1 = 2 - 1
[1,1,1,0,0,0]
=> 0 = 1 - 1
[1,0,1,0,1,0,1,0]
=> 13 = 14 - 1
[1,0,1,0,1,1,0,0]
=> 8 = 9 - 1
[1,0,1,1,0,0,1,0]
=> 9 = 10 - 1
[1,0,1,1,0,1,0,0]
=> 6 = 7 - 1
[1,0,1,1,1,0,0,0]
=> 3 = 4 - 1
[1,1,0,0,1,0,1,0]
=> 8 = 9 - 1
[1,1,0,0,1,1,0,0]
=> 5 = 6 - 1
[1,1,0,1,0,0,1,0]
=> 6 = 7 - 1
[1,1,0,1,0,1,0,0]
=> 4 = 5 - 1
[1,1,0,1,1,0,0,0]
=> 2 = 3 - 1
[1,1,1,0,0,0,1,0]
=> 3 = 4 - 1
[1,1,1,0,0,1,0,0]
=> 2 = 3 - 1
[1,1,1,0,1,0,0,0]
=> 1 = 2 - 1
[1,1,1,1,0,0,0,0]
=> 0 = 1 - 1
[1,0,1,0,1,0,1,0,1,0]
=> 41 = 42 - 1
[1,0,1,0,1,0,1,1,0,0]
=> 27 = 28 - 1
[1,0,1,0,1,1,0,0,1,0]
=> 31 = 32 - 1
[1,0,1,0,1,1,0,1,0,0]
=> 22 = 23 - 1
[1,0,1,0,1,1,1,0,0,0]
=> 13 = 14 - 1
[1,0,1,1,0,0,1,0,1,0]
=> 31 = 32 - 1
[1,0,1,1,0,0,1,1,0,0]
=> 21 = 22 - 1
[1,0,1,1,0,1,0,0,1,0]
=> 25 = 26 - 1
[1,0,1,1,0,1,0,1,0,0]
=> 18 = 19 - 1
[1,0,1,1,0,1,1,0,0,0]
=> 11 = 12 - 1
[1,0,1,1,1,0,0,0,1,0]
=> 16 = 17 - 1
[1,0,1,1,1,0,0,1,0,0]
=> 12 = 13 - 1
[1,0,1,1,1,0,1,0,0,0]
=> 8 = 9 - 1
[1,0,1,1,1,1,0,0,0,0]
=> 4 = 5 - 1
[1,1,0,0,1,0,1,0,1,0]
=> 27 = 28 - 1
[1,1,0,0,1,0,1,1,0,0]
=> 18 = 19 - 1
[1,1,0,0,1,1,0,0,1,0]
=> 21 = 22 - 1
[1,1,0,0,1,1,0,1,0,0]
=> 15 = 16 - 1
[1,1,0,0,1,1,1,0,0,0]
=> 9 = 10 - 1
[1,1,0,1,0,0,1,0,1,0]
=> 22 = 23 - 1
[1,1,0,1,0,0,1,1,0,0]
=> 15 = 16 - 1
[1,1,0,1,0,1,0,0,1,0]
=> 18 = 19 - 1
[1,1,0,1,0,1,0,1,0,0]
=> 13 = 14 - 1
[1,1,0,1,0,1,1,0,0,0]
=> 8 = 9 - 1
[1,1,0,1,1,0,0,0,1,0]
=> 12 = 13 - 1
[1,1,0,1,1,0,0,1,0,0]
=> 9 = 10 - 1
[1,1,0,1,1,0,1,0,0,0]
=> 6 = 7 - 1
[1,1,0,1,1,1,0,0,0,0]
=> 3 = 4 - 1
[1,1,1,0,0,0,1,0,1,0]
=> 13 = 14 - 1
Description
The number of Dyck paths that are weakly above the Dyck path, except for the path itself.
Matching statistic: St001313
(load all 8 compositions to match this statistic)
(load all 8 compositions to match this statistic)
Mp00027: Dyck paths —to partition⟶ Integer partitions
Mp00095: Integer partitions —to binary word⟶ Binary words
Mp00104: Binary words —reverse⟶ Binary words
St001313: Binary words ⟶ ℤResult quality: 47% ●values known / values provided: 61%●distinct values known / distinct values provided: 47%
Mp00095: Integer partitions —to binary word⟶ Binary words
Mp00104: Binary words —reverse⟶ Binary words
St001313: Binary words ⟶ ℤResult quality: 47% ●values known / values provided: 61%●distinct values known / distinct values provided: 47%
Values
[1,0,1,0]
=> [1]
=> 10 => 01 => 2
[1,1,0,0]
=> []
=> => => ? = 1
[1,0,1,0,1,0]
=> [2,1]
=> 1010 => 0101 => 5
[1,0,1,1,0,0]
=> [1,1]
=> 110 => 011 => 3
[1,1,0,0,1,0]
=> [2]
=> 100 => 001 => 3
[1,1,0,1,0,0]
=> [1]
=> 10 => 01 => 2
[1,1,1,0,0,0]
=> []
=> => => ? = 1
[1,0,1,0,1,0,1,0]
=> [3,2,1]
=> 101010 => 010101 => 14
[1,0,1,0,1,1,0,0]
=> [2,2,1]
=> 11010 => 01011 => 9
[1,0,1,1,0,0,1,0]
=> [3,1,1]
=> 100110 => 011001 => 10
[1,0,1,1,0,1,0,0]
=> [2,1,1]
=> 10110 => 01101 => 7
[1,0,1,1,1,0,0,0]
=> [1,1,1]
=> 1110 => 0111 => 4
[1,1,0,0,1,0,1,0]
=> [3,2]
=> 10100 => 00101 => 9
[1,1,0,0,1,1,0,0]
=> [2,2]
=> 1100 => 0011 => 6
[1,1,0,1,0,0,1,0]
=> [3,1]
=> 10010 => 01001 => 7
[1,1,0,1,0,1,0,0]
=> [2,1]
=> 1010 => 0101 => 5
[1,1,0,1,1,0,0,0]
=> [1,1]
=> 110 => 011 => 3
[1,1,1,0,0,0,1,0]
=> [3]
=> 1000 => 0001 => 4
[1,1,1,0,0,1,0,0]
=> [2]
=> 100 => 001 => 3
[1,1,1,0,1,0,0,0]
=> [1]
=> 10 => 01 => 2
[1,1,1,1,0,0,0,0]
=> []
=> => => ? = 1
[1,0,1,0,1,0,1,0,1,0]
=> [4,3,2,1]
=> 10101010 => 01010101 => 42
[1,0,1,0,1,0,1,1,0,0]
=> [3,3,2,1]
=> 1101010 => 0101011 => 28
[1,0,1,0,1,1,0,0,1,0]
=> [4,2,2,1]
=> 10011010 => 01011001 => 32
[1,0,1,0,1,1,0,1,0,0]
=> [3,2,2,1]
=> 1011010 => 0101101 => 23
[1,0,1,0,1,1,1,0,0,0]
=> [2,2,2,1]
=> 111010 => 010111 => 14
[1,0,1,1,0,0,1,0,1,0]
=> [4,3,1,1]
=> 10100110 => 01100101 => 32
[1,0,1,1,0,0,1,1,0,0]
=> [3,3,1,1]
=> 1100110 => 0110011 => 22
[1,0,1,1,0,1,0,0,1,0]
=> [4,2,1,1]
=> 10010110 => 01101001 => 26
[1,0,1,1,0,1,0,1,0,0]
=> [3,2,1,1]
=> 1010110 => 0110101 => 19
[1,0,1,1,0,1,1,0,0,0]
=> [2,2,1,1]
=> 110110 => 011011 => 12
[1,0,1,1,1,0,0,0,1,0]
=> [4,1,1,1]
=> 10001110 => 01110001 => 17
[1,0,1,1,1,0,0,1,0,0]
=> [3,1,1,1]
=> 1001110 => 0111001 => 13
[1,0,1,1,1,0,1,0,0,0]
=> [2,1,1,1]
=> 101110 => 011101 => 9
[1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1]
=> 11110 => 01111 => 5
[1,1,0,0,1,0,1,0,1,0]
=> [4,3,2]
=> 1010100 => 0010101 => 28
[1,1,0,0,1,0,1,1,0,0]
=> [3,3,2]
=> 110100 => 001011 => 19
[1,1,0,0,1,1,0,0,1,0]
=> [4,2,2]
=> 1001100 => 0011001 => 22
[1,1,0,0,1,1,0,1,0,0]
=> [3,2,2]
=> 101100 => 001101 => 16
[1,1,0,0,1,1,1,0,0,0]
=> [2,2,2]
=> 11100 => 00111 => 10
[1,1,0,1,0,0,1,0,1,0]
=> [4,3,1]
=> 1010010 => 0100101 => 23
[1,1,0,1,0,0,1,1,0,0]
=> [3,3,1]
=> 110010 => 010011 => 16
[1,1,0,1,0,1,0,0,1,0]
=> [4,2,1]
=> 1001010 => 0101001 => 19
[1,1,0,1,0,1,0,1,0,0]
=> [3,2,1]
=> 101010 => 010101 => 14
[1,1,0,1,0,1,1,0,0,0]
=> [2,2,1]
=> 11010 => 01011 => 9
[1,1,0,1,1,0,0,0,1,0]
=> [4,1,1]
=> 1000110 => 0110001 => 13
[1,1,0,1,1,0,0,1,0,0]
=> [3,1,1]
=> 100110 => 011001 => 10
[1,1,0,1,1,0,1,0,0,0]
=> [2,1,1]
=> 10110 => 01101 => 7
[1,1,0,1,1,1,0,0,0,0]
=> [1,1,1]
=> 1110 => 0111 => 4
[1,1,1,0,0,0,1,0,1,0]
=> [4,3]
=> 101000 => 000101 => 14
[1,1,1,0,0,0,1,1,0,0]
=> [3,3]
=> 11000 => 00011 => 10
[1,1,1,0,0,1,0,0,1,0]
=> [4,2]
=> 100100 => 001001 => 12
[1,1,1,0,0,1,0,1,0,0]
=> [3,2]
=> 10100 => 00101 => 9
[1,1,1,1,1,0,0,0,0,0]
=> []
=> => => ? = 1
[1,0,1,0,1,0,1,0,1,0,1,0]
=> [5,4,3,2,1]
=> 1010101010 => 0101010101 => ? ∊ {1,26,42,54,54,62,62,75,84,89,89,104,104,107,132}
[1,0,1,0,1,0,1,1,0,0,1,0]
=> [5,3,3,2,1]
=> 1001101010 => 0101011001 => ? ∊ {1,26,42,54,54,62,62,75,84,89,89,104,104,107,132}
[1,0,1,0,1,1,0,0,1,0,1,0]
=> [5,4,2,2,1]
=> 1010011010 => 0101100101 => ? ∊ {1,26,42,54,54,62,62,75,84,89,89,104,104,107,132}
[1,0,1,0,1,1,0,1,0,0,1,0]
=> [5,3,2,2,1]
=> 1001011010 => 0101101001 => ? ∊ {1,26,42,54,54,62,62,75,84,89,89,104,104,107,132}
[1,0,1,0,1,1,1,0,0,0,1,0]
=> [5,2,2,2,1]
=> 1000111010 => 0101110001 => ? ∊ {1,26,42,54,54,62,62,75,84,89,89,104,104,107,132}
[1,0,1,1,0,0,1,0,1,0,1,0]
=> [5,4,3,1,1]
=> 1010100110 => 0110010101 => ? ∊ {1,26,42,54,54,62,62,75,84,89,89,104,104,107,132}
[1,0,1,1,0,0,1,1,0,0,1,0]
=> [5,3,3,1,1]
=> 1001100110 => 0110011001 => ? ∊ {1,26,42,54,54,62,62,75,84,89,89,104,104,107,132}
[1,0,1,1,0,1,0,0,1,0,1,0]
=> [5,4,2,1,1]
=> 1010010110 => 0110100101 => ? ∊ {1,26,42,54,54,62,62,75,84,89,89,104,104,107,132}
[1,0,1,1,0,1,0,1,0,0,1,0]
=> [5,3,2,1,1]
=> 1001010110 => 0110101001 => ? ∊ {1,26,42,54,54,62,62,75,84,89,89,104,104,107,132}
[1,0,1,1,0,1,1,0,0,0,1,0]
=> [5,2,2,1,1]
=> 1000110110 => 0110110001 => ? ∊ {1,26,42,54,54,62,62,75,84,89,89,104,104,107,132}
[1,0,1,1,1,0,0,0,1,0,1,0]
=> [5,4,1,1,1]
=> 1010001110 => 0111000101 => ? ∊ {1,26,42,54,54,62,62,75,84,89,89,104,104,107,132}
[1,0,1,1,1,0,0,1,0,0,1,0]
=> [5,3,1,1,1]
=> 1001001110 => 0111001001 => ? ∊ {1,26,42,54,54,62,62,75,84,89,89,104,104,107,132}
[1,0,1,1,1,0,1,0,0,0,1,0]
=> [5,2,1,1,1]
=> 1000101110 => 0111010001 => ? ∊ {1,26,42,54,54,62,62,75,84,89,89,104,104,107,132}
[1,0,1,1,1,1,0,0,0,0,1,0]
=> [5,1,1,1,1]
=> 1000011110 => 0111100001 => ? ∊ {1,26,42,54,54,62,62,75,84,89,89,104,104,107,132}
[1,1,1,1,1,1,0,0,0,0,0,0]
=> []
=> => => ? ∊ {1,26,42,54,54,62,62,75,84,89,89,104,104,107,132}
[1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [6,5,4,3,2,1]
=> 101010101010 => 010101010101 => ? ∊ {1,25,25,26,31,31,37,40,40,42,50,50,51,51,52,52,54,54,55,55,61,61,62,62,62,66,66,66,66,67,67,67,67,70,70,75,76,76,78,78,79,79,79,79,81,81,82,82,84,85,85,87,87,89,89,91,91,94,94,95,95,95,95,96,96,97,97,97,97,99,99,104,104,105,105,106,106,107,107,107,109,109,110,110,112,112,113,113,114,114,115,115,116,116,118,118,118,121,121,123,123,123,123,125,125,126,127,127,129,129,132,132,132,134,136,136,136,136,137,137,139,139,140,140,142,142,145,145,145,145,145,145,145,145,146,146,151,151,151,151,156,156,156,157,157,157,157,158,158,161,161,162,163,163,165,165,171,171,171,171,173,173,174,174,179,179,180,180,181,185,185,188,188,189,189,193,193,199,199,202,202,205,205,207,209,209,211,211,214,214,219,219,220,220,225,225,227,227,230,233,241,241,251,251,252,252,255,255,255,255,268,268,281,295,295,297,297,303,303,314,345,345,359,359,429}
[1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [5,5,4,3,2,1]
=> 11010101010 => 01010101011 => ? ∊ {1,25,25,26,31,31,37,40,40,42,50,50,51,51,52,52,54,54,55,55,61,61,62,62,62,66,66,66,66,67,67,67,67,70,70,75,76,76,78,78,79,79,79,79,81,81,82,82,84,85,85,87,87,89,89,91,91,94,94,95,95,95,95,96,96,97,97,97,97,99,99,104,104,105,105,106,106,107,107,107,109,109,110,110,112,112,113,113,114,114,115,115,116,116,118,118,118,121,121,123,123,123,123,125,125,126,127,127,129,129,132,132,132,134,136,136,136,136,137,137,139,139,140,140,142,142,145,145,145,145,145,145,145,145,146,146,151,151,151,151,156,156,156,157,157,157,157,158,158,161,161,162,163,163,165,165,171,171,171,171,173,173,174,174,179,179,180,180,181,185,185,188,188,189,189,193,193,199,199,202,202,205,205,207,209,209,211,211,214,214,219,219,220,220,225,225,227,227,230,233,241,241,251,251,252,252,255,255,255,255,268,268,281,295,295,297,297,303,303,314,345,345,359,359,429}
[1,0,1,0,1,0,1,0,1,1,0,0,1,0]
=> [6,4,4,3,2,1]
=> 100110101010 => 010101011001 => ? ∊ {1,25,25,26,31,31,37,40,40,42,50,50,51,51,52,52,54,54,55,55,61,61,62,62,62,66,66,66,66,67,67,67,67,70,70,75,76,76,78,78,79,79,79,79,81,81,82,82,84,85,85,87,87,89,89,91,91,94,94,95,95,95,95,96,96,97,97,97,97,99,99,104,104,105,105,106,106,107,107,107,109,109,110,110,112,112,113,113,114,114,115,115,116,116,118,118,118,121,121,123,123,123,123,125,125,126,127,127,129,129,132,132,132,134,136,136,136,136,137,137,139,139,140,140,142,142,145,145,145,145,145,145,145,145,146,146,151,151,151,151,156,156,156,157,157,157,157,158,158,161,161,162,163,163,165,165,171,171,171,171,173,173,174,174,179,179,180,180,181,185,185,188,188,189,189,193,193,199,199,202,202,205,205,207,209,209,211,211,214,214,219,219,220,220,225,225,227,227,230,233,241,241,251,251,252,252,255,255,255,255,268,268,281,295,295,297,297,303,303,314,345,345,359,359,429}
[1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [5,4,4,3,2,1]
=> 10110101010 => 01010101101 => ? ∊ {1,25,25,26,31,31,37,40,40,42,50,50,51,51,52,52,54,54,55,55,61,61,62,62,62,66,66,66,66,67,67,67,67,70,70,75,76,76,78,78,79,79,79,79,81,81,82,82,84,85,85,87,87,89,89,91,91,94,94,95,95,95,95,96,96,97,97,97,97,99,99,104,104,105,105,106,106,107,107,107,109,109,110,110,112,112,113,113,114,114,115,115,116,116,118,118,118,121,121,123,123,123,123,125,125,126,127,127,129,129,132,132,132,134,136,136,136,136,137,137,139,139,140,140,142,142,145,145,145,145,145,145,145,145,146,146,151,151,151,151,156,156,156,157,157,157,157,158,158,161,161,162,163,163,165,165,171,171,171,171,173,173,174,174,179,179,180,180,181,185,185,188,188,189,189,193,193,199,199,202,202,205,205,207,209,209,211,211,214,214,219,219,220,220,225,225,227,227,230,233,241,241,251,251,252,252,255,255,255,255,268,268,281,295,295,297,297,303,303,314,345,345,359,359,429}
[1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [4,4,4,3,2,1]
=> 1110101010 => 0101010111 => ? ∊ {1,25,25,26,31,31,37,40,40,42,50,50,51,51,52,52,54,54,55,55,61,61,62,62,62,66,66,66,66,67,67,67,67,70,70,75,76,76,78,78,79,79,79,79,81,81,82,82,84,85,85,87,87,89,89,91,91,94,94,95,95,95,95,96,96,97,97,97,97,99,99,104,104,105,105,106,106,107,107,107,109,109,110,110,112,112,113,113,114,114,115,115,116,116,118,118,118,121,121,123,123,123,123,125,125,126,127,127,129,129,132,132,132,134,136,136,136,136,137,137,139,139,140,140,142,142,145,145,145,145,145,145,145,145,146,146,151,151,151,151,156,156,156,157,157,157,157,158,158,161,161,162,163,163,165,165,171,171,171,171,173,173,174,174,179,179,180,180,181,185,185,188,188,189,189,193,193,199,199,202,202,205,205,207,209,209,211,211,214,214,219,219,220,220,225,225,227,227,230,233,241,241,251,251,252,252,255,255,255,255,268,268,281,295,295,297,297,303,303,314,345,345,359,359,429}
[1,0,1,0,1,0,1,1,0,0,1,0,1,0]
=> [6,5,3,3,2,1]
=> 101001101010 => ? => ? ∊ {1,25,25,26,31,31,37,40,40,42,50,50,51,51,52,52,54,54,55,55,61,61,62,62,62,66,66,66,66,67,67,67,67,70,70,75,76,76,78,78,79,79,79,79,81,81,82,82,84,85,85,87,87,89,89,91,91,94,94,95,95,95,95,96,96,97,97,97,97,99,99,104,104,105,105,106,106,107,107,107,109,109,110,110,112,112,113,113,114,114,115,115,116,116,118,118,118,121,121,123,123,123,123,125,125,126,127,127,129,129,132,132,132,134,136,136,136,136,137,137,139,139,140,140,142,142,145,145,145,145,145,145,145,145,146,146,151,151,151,151,156,156,156,157,157,157,157,158,158,161,161,162,163,163,165,165,171,171,171,171,173,173,174,174,179,179,180,180,181,185,185,188,188,189,189,193,193,199,199,202,202,205,205,207,209,209,211,211,214,214,219,219,220,220,225,225,227,227,230,233,241,241,251,251,252,252,255,255,255,255,268,268,281,295,295,297,297,303,303,314,345,345,359,359,429}
[1,0,1,0,1,0,1,1,0,0,1,1,0,0]
=> [5,5,3,3,2,1]
=> 11001101010 => 01010110011 => ? ∊ {1,25,25,26,31,31,37,40,40,42,50,50,51,51,52,52,54,54,55,55,61,61,62,62,62,66,66,66,66,67,67,67,67,70,70,75,76,76,78,78,79,79,79,79,81,81,82,82,84,85,85,87,87,89,89,91,91,94,94,95,95,95,95,96,96,97,97,97,97,99,99,104,104,105,105,106,106,107,107,107,109,109,110,110,112,112,113,113,114,114,115,115,116,116,118,118,118,121,121,123,123,123,123,125,125,126,127,127,129,129,132,132,132,134,136,136,136,136,137,137,139,139,140,140,142,142,145,145,145,145,145,145,145,145,146,146,151,151,151,151,156,156,156,157,157,157,157,158,158,161,161,162,163,163,165,165,171,171,171,171,173,173,174,174,179,179,180,180,181,185,185,188,188,189,189,193,193,199,199,202,202,205,205,207,209,209,211,211,214,214,219,219,220,220,225,225,227,227,230,233,241,241,251,251,252,252,255,255,255,255,268,268,281,295,295,297,297,303,303,314,345,345,359,359,429}
[1,0,1,0,1,0,1,1,0,1,0,0,1,0]
=> [6,4,3,3,2,1]
=> 100101101010 => 010101101001 => ? ∊ {1,25,25,26,31,31,37,40,40,42,50,50,51,51,52,52,54,54,55,55,61,61,62,62,62,66,66,66,66,67,67,67,67,70,70,75,76,76,78,78,79,79,79,79,81,81,82,82,84,85,85,87,87,89,89,91,91,94,94,95,95,95,95,96,96,97,97,97,97,99,99,104,104,105,105,106,106,107,107,107,109,109,110,110,112,112,113,113,114,114,115,115,116,116,118,118,118,121,121,123,123,123,123,125,125,126,127,127,129,129,132,132,132,134,136,136,136,136,137,137,139,139,140,140,142,142,145,145,145,145,145,145,145,145,146,146,151,151,151,151,156,156,156,157,157,157,157,158,158,161,161,162,163,163,165,165,171,171,171,171,173,173,174,174,179,179,180,180,181,185,185,188,188,189,189,193,193,199,199,202,202,205,205,207,209,209,211,211,214,214,219,219,220,220,225,225,227,227,230,233,241,241,251,251,252,252,255,255,255,255,268,268,281,295,295,297,297,303,303,314,345,345,359,359,429}
[1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [5,4,3,3,2,1]
=> 10101101010 => 01010110101 => ? ∊ {1,25,25,26,31,31,37,40,40,42,50,50,51,51,52,52,54,54,55,55,61,61,62,62,62,66,66,66,66,67,67,67,67,70,70,75,76,76,78,78,79,79,79,79,81,81,82,82,84,85,85,87,87,89,89,91,91,94,94,95,95,95,95,96,96,97,97,97,97,99,99,104,104,105,105,106,106,107,107,107,109,109,110,110,112,112,113,113,114,114,115,115,116,116,118,118,118,121,121,123,123,123,123,125,125,126,127,127,129,129,132,132,132,134,136,136,136,136,137,137,139,139,140,140,142,142,145,145,145,145,145,145,145,145,146,146,151,151,151,151,156,156,156,157,157,157,157,158,158,161,161,162,163,163,165,165,171,171,171,171,173,173,174,174,179,179,180,180,181,185,185,188,188,189,189,193,193,199,199,202,202,205,205,207,209,209,211,211,214,214,219,219,220,220,225,225,227,227,230,233,241,241,251,251,252,252,255,255,255,255,268,268,281,295,295,297,297,303,303,314,345,345,359,359,429}
[1,0,1,0,1,0,1,1,0,1,1,0,0,0]
=> [4,4,3,3,2,1]
=> 1101101010 => 0101011011 => ? ∊ {1,25,25,26,31,31,37,40,40,42,50,50,51,51,52,52,54,54,55,55,61,61,62,62,62,66,66,66,66,67,67,67,67,70,70,75,76,76,78,78,79,79,79,79,81,81,82,82,84,85,85,87,87,89,89,91,91,94,94,95,95,95,95,96,96,97,97,97,97,99,99,104,104,105,105,106,106,107,107,107,109,109,110,110,112,112,113,113,114,114,115,115,116,116,118,118,118,121,121,123,123,123,123,125,125,126,127,127,129,129,132,132,132,134,136,136,136,136,137,137,139,139,140,140,142,142,145,145,145,145,145,145,145,145,146,146,151,151,151,151,156,156,156,157,157,157,157,158,158,161,161,162,163,163,165,165,171,171,171,171,173,173,174,174,179,179,180,180,181,185,185,188,188,189,189,193,193,199,199,202,202,205,205,207,209,209,211,211,214,214,219,219,220,220,225,225,227,227,230,233,241,241,251,251,252,252,255,255,255,255,268,268,281,295,295,297,297,303,303,314,345,345,359,359,429}
[1,0,1,0,1,0,1,1,1,0,0,0,1,0]
=> [6,3,3,3,2,1]
=> 100011101010 => ? => ? ∊ {1,25,25,26,31,31,37,40,40,42,50,50,51,51,52,52,54,54,55,55,61,61,62,62,62,66,66,66,66,67,67,67,67,70,70,75,76,76,78,78,79,79,79,79,81,81,82,82,84,85,85,87,87,89,89,91,91,94,94,95,95,95,95,96,96,97,97,97,97,99,99,104,104,105,105,106,106,107,107,107,109,109,110,110,112,112,113,113,114,114,115,115,116,116,118,118,118,121,121,123,123,123,123,125,125,126,127,127,129,129,132,132,132,134,136,136,136,136,137,137,139,139,140,140,142,142,145,145,145,145,145,145,145,145,146,146,151,151,151,151,156,156,156,157,157,157,157,158,158,161,161,162,163,163,165,165,171,171,171,171,173,173,174,174,179,179,180,180,181,185,185,188,188,189,189,193,193,199,199,202,202,205,205,207,209,209,211,211,214,214,219,219,220,220,225,225,227,227,230,233,241,241,251,251,252,252,255,255,255,255,268,268,281,295,295,297,297,303,303,314,345,345,359,359,429}
[1,0,1,0,1,0,1,1,1,0,0,1,0,0]
=> [5,3,3,3,2,1]
=> 10011101010 => 01010111001 => ? ∊ {1,25,25,26,31,31,37,40,40,42,50,50,51,51,52,52,54,54,55,55,61,61,62,62,62,66,66,66,66,67,67,67,67,70,70,75,76,76,78,78,79,79,79,79,81,81,82,82,84,85,85,87,87,89,89,91,91,94,94,95,95,95,95,96,96,97,97,97,97,99,99,104,104,105,105,106,106,107,107,107,109,109,110,110,112,112,113,113,114,114,115,115,116,116,118,118,118,121,121,123,123,123,123,125,125,126,127,127,129,129,132,132,132,134,136,136,136,136,137,137,139,139,140,140,142,142,145,145,145,145,145,145,145,145,146,146,151,151,151,151,156,156,156,157,157,157,157,158,158,161,161,162,163,163,165,165,171,171,171,171,173,173,174,174,179,179,180,180,181,185,185,188,188,189,189,193,193,199,199,202,202,205,205,207,209,209,211,211,214,214,219,219,220,220,225,225,227,227,230,233,241,241,251,251,252,252,255,255,255,255,268,268,281,295,295,297,297,303,303,314,345,345,359,359,429}
[1,0,1,0,1,0,1,1,1,0,1,0,0,0]
=> [4,3,3,3,2,1]
=> 1011101010 => 0101011101 => ? ∊ {1,25,25,26,31,31,37,40,40,42,50,50,51,51,52,52,54,54,55,55,61,61,62,62,62,66,66,66,66,67,67,67,67,70,70,75,76,76,78,78,79,79,79,79,81,81,82,82,84,85,85,87,87,89,89,91,91,94,94,95,95,95,95,96,96,97,97,97,97,99,99,104,104,105,105,106,106,107,107,107,109,109,110,110,112,112,113,113,114,114,115,115,116,116,118,118,118,121,121,123,123,123,123,125,125,126,127,127,129,129,132,132,132,134,136,136,136,136,137,137,139,139,140,140,142,142,145,145,145,145,145,145,145,145,146,146,151,151,151,151,156,156,156,157,157,157,157,158,158,161,161,162,163,163,165,165,171,171,171,171,173,173,174,174,179,179,180,180,181,185,185,188,188,189,189,193,193,199,199,202,202,205,205,207,209,209,211,211,214,214,219,219,220,220,225,225,227,227,230,233,241,241,251,251,252,252,255,255,255,255,268,268,281,295,295,297,297,303,303,314,345,345,359,359,429}
[1,0,1,0,1,1,0,0,1,0,1,0,1,0]
=> [6,5,4,2,2,1]
=> 101010011010 => 010110010101 => ? ∊ {1,25,25,26,31,31,37,40,40,42,50,50,51,51,52,52,54,54,55,55,61,61,62,62,62,66,66,66,66,67,67,67,67,70,70,75,76,76,78,78,79,79,79,79,81,81,82,82,84,85,85,87,87,89,89,91,91,94,94,95,95,95,95,96,96,97,97,97,97,99,99,104,104,105,105,106,106,107,107,107,109,109,110,110,112,112,113,113,114,114,115,115,116,116,118,118,118,121,121,123,123,123,123,125,125,126,127,127,129,129,132,132,132,134,136,136,136,136,137,137,139,139,140,140,142,142,145,145,145,145,145,145,145,145,146,146,151,151,151,151,156,156,156,157,157,157,157,158,158,161,161,162,163,163,165,165,171,171,171,171,173,173,174,174,179,179,180,180,181,185,185,188,188,189,189,193,193,199,199,202,202,205,205,207,209,209,211,211,214,214,219,219,220,220,225,225,227,227,230,233,241,241,251,251,252,252,255,255,255,255,268,268,281,295,295,297,297,303,303,314,345,345,359,359,429}
[1,0,1,0,1,1,0,0,1,0,1,1,0,0]
=> [5,5,4,2,2,1]
=> 11010011010 => 01011001011 => ? ∊ {1,25,25,26,31,31,37,40,40,42,50,50,51,51,52,52,54,54,55,55,61,61,62,62,62,66,66,66,66,67,67,67,67,70,70,75,76,76,78,78,79,79,79,79,81,81,82,82,84,85,85,87,87,89,89,91,91,94,94,95,95,95,95,96,96,97,97,97,97,99,99,104,104,105,105,106,106,107,107,107,109,109,110,110,112,112,113,113,114,114,115,115,116,116,118,118,118,121,121,123,123,123,123,125,125,126,127,127,129,129,132,132,132,134,136,136,136,136,137,137,139,139,140,140,142,142,145,145,145,145,145,145,145,145,146,146,151,151,151,151,156,156,156,157,157,157,157,158,158,161,161,162,163,163,165,165,171,171,171,171,173,173,174,174,179,179,180,180,181,185,185,188,188,189,189,193,193,199,199,202,202,205,205,207,209,209,211,211,214,214,219,219,220,220,225,225,227,227,230,233,241,241,251,251,252,252,255,255,255,255,268,268,281,295,295,297,297,303,303,314,345,345,359,359,429}
[1,0,1,0,1,1,0,0,1,1,0,0,1,0]
=> [6,4,4,2,2,1]
=> 100110011010 => 010110011001 => ? ∊ {1,25,25,26,31,31,37,40,40,42,50,50,51,51,52,52,54,54,55,55,61,61,62,62,62,66,66,66,66,67,67,67,67,70,70,75,76,76,78,78,79,79,79,79,81,81,82,82,84,85,85,87,87,89,89,91,91,94,94,95,95,95,95,96,96,97,97,97,97,99,99,104,104,105,105,106,106,107,107,107,109,109,110,110,112,112,113,113,114,114,115,115,116,116,118,118,118,121,121,123,123,123,123,125,125,126,127,127,129,129,132,132,132,134,136,136,136,136,137,137,139,139,140,140,142,142,145,145,145,145,145,145,145,145,146,146,151,151,151,151,156,156,156,157,157,157,157,158,158,161,161,162,163,163,165,165,171,171,171,171,173,173,174,174,179,179,180,180,181,185,185,188,188,189,189,193,193,199,199,202,202,205,205,207,209,209,211,211,214,214,219,219,220,220,225,225,227,227,230,233,241,241,251,251,252,252,255,255,255,255,268,268,281,295,295,297,297,303,303,314,345,345,359,359,429}
[1,0,1,0,1,1,0,0,1,1,0,1,0,0]
=> [5,4,4,2,2,1]
=> 10110011010 => 01011001101 => ? ∊ {1,25,25,26,31,31,37,40,40,42,50,50,51,51,52,52,54,54,55,55,61,61,62,62,62,66,66,66,66,67,67,67,67,70,70,75,76,76,78,78,79,79,79,79,81,81,82,82,84,85,85,87,87,89,89,91,91,94,94,95,95,95,95,96,96,97,97,97,97,99,99,104,104,105,105,106,106,107,107,107,109,109,110,110,112,112,113,113,114,114,115,115,116,116,118,118,118,121,121,123,123,123,123,125,125,126,127,127,129,129,132,132,132,134,136,136,136,136,137,137,139,139,140,140,142,142,145,145,145,145,145,145,145,145,146,146,151,151,151,151,156,156,156,157,157,157,157,158,158,161,161,162,163,163,165,165,171,171,171,171,173,173,174,174,179,179,180,180,181,185,185,188,188,189,189,193,193,199,199,202,202,205,205,207,209,209,211,211,214,214,219,219,220,220,225,225,227,227,230,233,241,241,251,251,252,252,255,255,255,255,268,268,281,295,295,297,297,303,303,314,345,345,359,359,429}
[1,0,1,0,1,1,0,0,1,1,1,0,0,0]
=> [4,4,4,2,2,1]
=> 1110011010 => 0101100111 => ? ∊ {1,25,25,26,31,31,37,40,40,42,50,50,51,51,52,52,54,54,55,55,61,61,62,62,62,66,66,66,66,67,67,67,67,70,70,75,76,76,78,78,79,79,79,79,81,81,82,82,84,85,85,87,87,89,89,91,91,94,94,95,95,95,95,96,96,97,97,97,97,99,99,104,104,105,105,106,106,107,107,107,109,109,110,110,112,112,113,113,114,114,115,115,116,116,118,118,118,121,121,123,123,123,123,125,125,126,127,127,129,129,132,132,132,134,136,136,136,136,137,137,139,139,140,140,142,142,145,145,145,145,145,145,145,145,146,146,151,151,151,151,156,156,156,157,157,157,157,158,158,161,161,162,163,163,165,165,171,171,171,171,173,173,174,174,179,179,180,180,181,185,185,188,188,189,189,193,193,199,199,202,202,205,205,207,209,209,211,211,214,214,219,219,220,220,225,225,227,227,230,233,241,241,251,251,252,252,255,255,255,255,268,268,281,295,295,297,297,303,303,314,345,345,359,359,429}
[1,0,1,0,1,1,0,1,0,0,1,0,1,0]
=> [6,5,3,2,2,1]
=> 101001011010 => 010110100101 => ? ∊ {1,25,25,26,31,31,37,40,40,42,50,50,51,51,52,52,54,54,55,55,61,61,62,62,62,66,66,66,66,67,67,67,67,70,70,75,76,76,78,78,79,79,79,79,81,81,82,82,84,85,85,87,87,89,89,91,91,94,94,95,95,95,95,96,96,97,97,97,97,99,99,104,104,105,105,106,106,107,107,107,109,109,110,110,112,112,113,113,114,114,115,115,116,116,118,118,118,121,121,123,123,123,123,125,125,126,127,127,129,129,132,132,132,134,136,136,136,136,137,137,139,139,140,140,142,142,145,145,145,145,145,145,145,145,146,146,151,151,151,151,156,156,156,157,157,157,157,158,158,161,161,162,163,163,165,165,171,171,171,171,173,173,174,174,179,179,180,180,181,185,185,188,188,189,189,193,193,199,199,202,202,205,205,207,209,209,211,211,214,214,219,219,220,220,225,225,227,227,230,233,241,241,251,251,252,252,255,255,255,255,268,268,281,295,295,297,297,303,303,314,345,345,359,359,429}
[1,0,1,0,1,1,0,1,0,0,1,1,0,0]
=> [5,5,3,2,2,1]
=> 11001011010 => 01011010011 => ? ∊ {1,25,25,26,31,31,37,40,40,42,50,50,51,51,52,52,54,54,55,55,61,61,62,62,62,66,66,66,66,67,67,67,67,70,70,75,76,76,78,78,79,79,79,79,81,81,82,82,84,85,85,87,87,89,89,91,91,94,94,95,95,95,95,96,96,97,97,97,97,99,99,104,104,105,105,106,106,107,107,107,109,109,110,110,112,112,113,113,114,114,115,115,116,116,118,118,118,121,121,123,123,123,123,125,125,126,127,127,129,129,132,132,132,134,136,136,136,136,137,137,139,139,140,140,142,142,145,145,145,145,145,145,145,145,146,146,151,151,151,151,156,156,156,157,157,157,157,158,158,161,161,162,163,163,165,165,171,171,171,171,173,173,174,174,179,179,180,180,181,185,185,188,188,189,189,193,193,199,199,202,202,205,205,207,209,209,211,211,214,214,219,219,220,220,225,225,227,227,230,233,241,241,251,251,252,252,255,255,255,255,268,268,281,295,295,297,297,303,303,314,345,345,359,359,429}
[1,0,1,0,1,1,0,1,0,1,0,0,1,0]
=> [6,4,3,2,2,1]
=> 100101011010 => 010110101001 => ? ∊ {1,25,25,26,31,31,37,40,40,42,50,50,51,51,52,52,54,54,55,55,61,61,62,62,62,66,66,66,66,67,67,67,67,70,70,75,76,76,78,78,79,79,79,79,81,81,82,82,84,85,85,87,87,89,89,91,91,94,94,95,95,95,95,96,96,97,97,97,97,99,99,104,104,105,105,106,106,107,107,107,109,109,110,110,112,112,113,113,114,114,115,115,116,116,118,118,118,121,121,123,123,123,123,125,125,126,127,127,129,129,132,132,132,134,136,136,136,136,137,137,139,139,140,140,142,142,145,145,145,145,145,145,145,145,146,146,151,151,151,151,156,156,156,157,157,157,157,158,158,161,161,162,163,163,165,165,171,171,171,171,173,173,174,174,179,179,180,180,181,185,185,188,188,189,189,193,193,199,199,202,202,205,205,207,209,209,211,211,214,214,219,219,220,220,225,225,227,227,230,233,241,241,251,251,252,252,255,255,255,255,268,268,281,295,295,297,297,303,303,314,345,345,359,359,429}
[1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> [5,4,3,2,2,1]
=> 10101011010 => 01011010101 => ? ∊ {1,25,25,26,31,31,37,40,40,42,50,50,51,51,52,52,54,54,55,55,61,61,62,62,62,66,66,66,66,67,67,67,67,70,70,75,76,76,78,78,79,79,79,79,81,81,82,82,84,85,85,87,87,89,89,91,91,94,94,95,95,95,95,96,96,97,97,97,97,99,99,104,104,105,105,106,106,107,107,107,109,109,110,110,112,112,113,113,114,114,115,115,116,116,118,118,118,121,121,123,123,123,123,125,125,126,127,127,129,129,132,132,132,134,136,136,136,136,137,137,139,139,140,140,142,142,145,145,145,145,145,145,145,145,146,146,151,151,151,151,156,156,156,157,157,157,157,158,158,161,161,162,163,163,165,165,171,171,171,171,173,173,174,174,179,179,180,180,181,185,185,188,188,189,189,193,193,199,199,202,202,205,205,207,209,209,211,211,214,214,219,219,220,220,225,225,227,227,230,233,241,241,251,251,252,252,255,255,255,255,268,268,281,295,295,297,297,303,303,314,345,345,359,359,429}
[1,0,1,0,1,1,0,1,0,1,1,0,0,0]
=> [4,4,3,2,2,1]
=> 1101011010 => 0101101011 => ? ∊ {1,25,25,26,31,31,37,40,40,42,50,50,51,51,52,52,54,54,55,55,61,61,62,62,62,66,66,66,66,67,67,67,67,70,70,75,76,76,78,78,79,79,79,79,81,81,82,82,84,85,85,87,87,89,89,91,91,94,94,95,95,95,95,96,96,97,97,97,97,99,99,104,104,105,105,106,106,107,107,107,109,109,110,110,112,112,113,113,114,114,115,115,116,116,118,118,118,121,121,123,123,123,123,125,125,126,127,127,129,129,132,132,132,134,136,136,136,136,137,137,139,139,140,140,142,142,145,145,145,145,145,145,145,145,146,146,151,151,151,151,156,156,156,157,157,157,157,158,158,161,161,162,163,163,165,165,171,171,171,171,173,173,174,174,179,179,180,180,181,185,185,188,188,189,189,193,193,199,199,202,202,205,205,207,209,209,211,211,214,214,219,219,220,220,225,225,227,227,230,233,241,241,251,251,252,252,255,255,255,255,268,268,281,295,295,297,297,303,303,314,345,345,359,359,429}
[1,0,1,0,1,1,0,1,1,0,0,0,1,0]
=> [6,3,3,2,2,1]
=> 100011011010 => ? => ? ∊ {1,25,25,26,31,31,37,40,40,42,50,50,51,51,52,52,54,54,55,55,61,61,62,62,62,66,66,66,66,67,67,67,67,70,70,75,76,76,78,78,79,79,79,79,81,81,82,82,84,85,85,87,87,89,89,91,91,94,94,95,95,95,95,96,96,97,97,97,97,99,99,104,104,105,105,106,106,107,107,107,109,109,110,110,112,112,113,113,114,114,115,115,116,116,118,118,118,121,121,123,123,123,123,125,125,126,127,127,129,129,132,132,132,134,136,136,136,136,137,137,139,139,140,140,142,142,145,145,145,145,145,145,145,145,146,146,151,151,151,151,156,156,156,157,157,157,157,158,158,161,161,162,163,163,165,165,171,171,171,171,173,173,174,174,179,179,180,180,181,185,185,188,188,189,189,193,193,199,199,202,202,205,205,207,209,209,211,211,214,214,219,219,220,220,225,225,227,227,230,233,241,241,251,251,252,252,255,255,255,255,268,268,281,295,295,297,297,303,303,314,345,345,359,359,429}
[1,0,1,0,1,1,0,1,1,0,0,1,0,0]
=> [5,3,3,2,2,1]
=> 10011011010 => ? => ? ∊ {1,25,25,26,31,31,37,40,40,42,50,50,51,51,52,52,54,54,55,55,61,61,62,62,62,66,66,66,66,67,67,67,67,70,70,75,76,76,78,78,79,79,79,79,81,81,82,82,84,85,85,87,87,89,89,91,91,94,94,95,95,95,95,96,96,97,97,97,97,99,99,104,104,105,105,106,106,107,107,107,109,109,110,110,112,112,113,113,114,114,115,115,116,116,118,118,118,121,121,123,123,123,123,125,125,126,127,127,129,129,132,132,132,134,136,136,136,136,137,137,139,139,140,140,142,142,145,145,145,145,145,145,145,145,146,146,151,151,151,151,156,156,156,157,157,157,157,158,158,161,161,162,163,163,165,165,171,171,171,171,173,173,174,174,179,179,180,180,181,185,185,188,188,189,189,193,193,199,199,202,202,205,205,207,209,209,211,211,214,214,219,219,220,220,225,225,227,227,230,233,241,241,251,251,252,252,255,255,255,255,268,268,281,295,295,297,297,303,303,314,345,345,359,359,429}
[1,0,1,0,1,1,0,1,1,0,1,0,0,0]
=> [4,3,3,2,2,1]
=> 1011011010 => 0101101101 => ? ∊ {1,25,25,26,31,31,37,40,40,42,50,50,51,51,52,52,54,54,55,55,61,61,62,62,62,66,66,66,66,67,67,67,67,70,70,75,76,76,78,78,79,79,79,79,81,81,82,82,84,85,85,87,87,89,89,91,91,94,94,95,95,95,95,96,96,97,97,97,97,99,99,104,104,105,105,106,106,107,107,107,109,109,110,110,112,112,113,113,114,114,115,115,116,116,118,118,118,121,121,123,123,123,123,125,125,126,127,127,129,129,132,132,132,134,136,136,136,136,137,137,139,139,140,140,142,142,145,145,145,145,145,145,145,145,146,146,151,151,151,151,156,156,156,157,157,157,157,158,158,161,161,162,163,163,165,165,171,171,171,171,173,173,174,174,179,179,180,180,181,185,185,188,188,189,189,193,193,199,199,202,202,205,205,207,209,209,211,211,214,214,219,219,220,220,225,225,227,227,230,233,241,241,251,251,252,252,255,255,255,255,268,268,281,295,295,297,297,303,303,314,345,345,359,359,429}
[1,0,1,0,1,1,1,0,0,0,1,0,1,0]
=> [6,5,2,2,2,1]
=> 101000111010 => 010111000101 => ? ∊ {1,25,25,26,31,31,37,40,40,42,50,50,51,51,52,52,54,54,55,55,61,61,62,62,62,66,66,66,66,67,67,67,67,70,70,75,76,76,78,78,79,79,79,79,81,81,82,82,84,85,85,87,87,89,89,91,91,94,94,95,95,95,95,96,96,97,97,97,97,99,99,104,104,105,105,106,106,107,107,107,109,109,110,110,112,112,113,113,114,114,115,115,116,116,118,118,118,121,121,123,123,123,123,125,125,126,127,127,129,129,132,132,132,134,136,136,136,136,137,137,139,139,140,140,142,142,145,145,145,145,145,145,145,145,146,146,151,151,151,151,156,156,156,157,157,157,157,158,158,161,161,162,163,163,165,165,171,171,171,171,173,173,174,174,179,179,180,180,181,185,185,188,188,189,189,193,193,199,199,202,202,205,205,207,209,209,211,211,214,214,219,219,220,220,225,225,227,227,230,233,241,241,251,251,252,252,255,255,255,255,268,268,281,295,295,297,297,303,303,314,345,345,359,359,429}
[1,0,1,0,1,1,1,0,0,0,1,1,0,0]
=> [5,5,2,2,2,1]
=> 11000111010 => ? => ? ∊ {1,25,25,26,31,31,37,40,40,42,50,50,51,51,52,52,54,54,55,55,61,61,62,62,62,66,66,66,66,67,67,67,67,70,70,75,76,76,78,78,79,79,79,79,81,81,82,82,84,85,85,87,87,89,89,91,91,94,94,95,95,95,95,96,96,97,97,97,97,99,99,104,104,105,105,106,106,107,107,107,109,109,110,110,112,112,113,113,114,114,115,115,116,116,118,118,118,121,121,123,123,123,123,125,125,126,127,127,129,129,132,132,132,134,136,136,136,136,137,137,139,139,140,140,142,142,145,145,145,145,145,145,145,145,146,146,151,151,151,151,156,156,156,157,157,157,157,158,158,161,161,162,163,163,165,165,171,171,171,171,173,173,174,174,179,179,180,180,181,185,185,188,188,189,189,193,193,199,199,202,202,205,205,207,209,209,211,211,214,214,219,219,220,220,225,225,227,227,230,233,241,241,251,251,252,252,255,255,255,255,268,268,281,295,295,297,297,303,303,314,345,345,359,359,429}
[1,0,1,0,1,1,1,0,0,1,0,0,1,0]
=> [6,4,2,2,2,1]
=> 100100111010 => 010111001001 => ? ∊ {1,25,25,26,31,31,37,40,40,42,50,50,51,51,52,52,54,54,55,55,61,61,62,62,62,66,66,66,66,67,67,67,67,70,70,75,76,76,78,78,79,79,79,79,81,81,82,82,84,85,85,87,87,89,89,91,91,94,94,95,95,95,95,96,96,97,97,97,97,99,99,104,104,105,105,106,106,107,107,107,109,109,110,110,112,112,113,113,114,114,115,115,116,116,118,118,118,121,121,123,123,123,123,125,125,126,127,127,129,129,132,132,132,134,136,136,136,136,137,137,139,139,140,140,142,142,145,145,145,145,145,145,145,145,146,146,151,151,151,151,156,156,156,157,157,157,157,158,158,161,161,162,163,163,165,165,171,171,171,171,173,173,174,174,179,179,180,180,181,185,185,188,188,189,189,193,193,199,199,202,202,205,205,207,209,209,211,211,214,214,219,219,220,220,225,225,227,227,230,233,241,241,251,251,252,252,255,255,255,255,268,268,281,295,295,297,297,303,303,314,345,345,359,359,429}
[1,0,1,0,1,1,1,0,0,1,0,1,0,0]
=> [5,4,2,2,2,1]
=> 10100111010 => 01011100101 => ? ∊ {1,25,25,26,31,31,37,40,40,42,50,50,51,51,52,52,54,54,55,55,61,61,62,62,62,66,66,66,66,67,67,67,67,70,70,75,76,76,78,78,79,79,79,79,81,81,82,82,84,85,85,87,87,89,89,91,91,94,94,95,95,95,95,96,96,97,97,97,97,99,99,104,104,105,105,106,106,107,107,107,109,109,110,110,112,112,113,113,114,114,115,115,116,116,118,118,118,121,121,123,123,123,123,125,125,126,127,127,129,129,132,132,132,134,136,136,136,136,137,137,139,139,140,140,142,142,145,145,145,145,145,145,145,145,146,146,151,151,151,151,156,156,156,157,157,157,157,158,158,161,161,162,163,163,165,165,171,171,171,171,173,173,174,174,179,179,180,180,181,185,185,188,188,189,189,193,193,199,199,202,202,205,205,207,209,209,211,211,214,214,219,219,220,220,225,225,227,227,230,233,241,241,251,251,252,252,255,255,255,255,268,268,281,295,295,297,297,303,303,314,345,345,359,359,429}
[1,0,1,0,1,1,1,0,0,1,1,0,0,0]
=> [4,4,2,2,2,1]
=> 1100111010 => 0101110011 => ? ∊ {1,25,25,26,31,31,37,40,40,42,50,50,51,51,52,52,54,54,55,55,61,61,62,62,62,66,66,66,66,67,67,67,67,70,70,75,76,76,78,78,79,79,79,79,81,81,82,82,84,85,85,87,87,89,89,91,91,94,94,95,95,95,95,96,96,97,97,97,97,99,99,104,104,105,105,106,106,107,107,107,109,109,110,110,112,112,113,113,114,114,115,115,116,116,118,118,118,121,121,123,123,123,123,125,125,126,127,127,129,129,132,132,132,134,136,136,136,136,137,137,139,139,140,140,142,142,145,145,145,145,145,145,145,145,146,146,151,151,151,151,156,156,156,157,157,157,157,158,158,161,161,162,163,163,165,165,171,171,171,171,173,173,174,174,179,179,180,180,181,185,185,188,188,189,189,193,193,199,199,202,202,205,205,207,209,209,211,211,214,214,219,219,220,220,225,225,227,227,230,233,241,241,251,251,252,252,255,255,255,255,268,268,281,295,295,297,297,303,303,314,345,345,359,359,429}
Description
The number of Dyck paths above the lattice path given by a binary word.
One may treat a binary word as a lattice path starting at the origin and treating $1$'s as steps $(1,0)$ and $0$'s as steps $(0,1)$. Given a binary word $w$, this statistic counts the number of lattice paths from the origin to the same endpoint as $w$ that stay weakly above $w$.
See [[St001312]] for this statistic on compositions treated as bounce paths.
Matching statistic: St001389
Mp00199: Dyck paths —prime Dyck path⟶ Dyck paths
Mp00122: Dyck paths —Elizalde-Deutsch bijection⟶ Dyck paths
Mp00027: Dyck paths —to partition⟶ Integer partitions
St001389: Integer partitions ⟶ ℤResult quality: 29% ●values known / values provided: 44%●distinct values known / distinct values provided: 29%
Mp00122: Dyck paths —Elizalde-Deutsch bijection⟶ Dyck paths
Mp00027: Dyck paths —to partition⟶ Integer partitions
St001389: Integer partitions ⟶ ℤResult quality: 29% ●values known / values provided: 44%●distinct values known / distinct values provided: 29%
Values
[1,0,1,0]
=> [1,1,0,1,0,0]
=> [1,0,1,1,0,0]
=> [1,1]
=> 1
[1,1,0,0]
=> [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> [2,1]
=> 2
[1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,0]
=> [1,0,1,1,0,0,1,0]
=> [3,1,1]
=> 3
[1,0,1,1,0,0]
=> [1,1,0,1,1,0,0,0]
=> [1,0,1,1,0,1,0,0]
=> [2,1,1]
=> 2
[1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> [1,0,1,1,1,0,0,0]
=> [1,1,1]
=> 1
[1,1,0,1,0,0]
=> [1,1,1,0,1,0,0,0]
=> [1,0,1,0,1,1,0,0]
=> [2,2,1]
=> 3
[1,1,1,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> [3,2,1]
=> 5
[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]
=> [3,3,1,1]
=> 6
[1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> [1,0,1,1,0,1,1,0,0,0]
=> [2,2,1,1]
=> 3
[1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> [4,3,1,1]
=> 9
[1,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> [1,0,1,1,0,1,0,0,1,0]
=> [4,2,1,1]
=> 7
[1,0,1,1,1,0,0,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> [1,0,1,1,0,1,0,1,0,0]
=> [3,2,1,1]
=> 5
[1,1,0,0,1,0,1,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> [1,0,1,1,1,0,0,1,0,0]
=> [3,1,1,1]
=> 3
[1,1,0,0,1,1,0,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1]
=> 1
[1,1,0,1,0,0,1,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> [4,1,1,1]
=> 4
[1,1,0,1,0,1,0,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> [4,2,2,1]
=> 10
[1,1,0,1,1,0,0,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> [1,0,1,0,1,1,0,1,0,0]
=> [3,2,2,1]
=> 7
[1,1,1,0,0,0,1,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> [1,0,1,1,1,0,1,0,0,0]
=> [2,1,1,1]
=> 2
[1,1,1,0,0,1,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> [2,2,2,1]
=> 4
[1,1,1,0,1,0,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> [3,3,2,1]
=> 9
[1,1,1,1,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [4,3,2,1]
=> 14
[1,0,1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,1,0,0]
=> [1,0,1,1,0,0,1,1,0,0,1,0]
=> [5,3,3,1,1]
=> 22
[1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,1,1,0,0,0]
=> [1,0,1,1,0,1,1,0,0,0,1,0]
=> [5,2,2,1,1]
=> 13
[1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,0,1,1,0,0,1,0,0]
=> [1,0,1,1,0,0,1,1,0,1,0,0]
=> [4,3,3,1,1]
=> 16
[1,0,1,0,1,1,0,1,0,0]
=> [1,1,0,1,0,1,1,0,1,0,0,0]
=> [1,0,1,1,0,1,1,0,0,1,0,0]
=> [4,2,2,1,1]
=> 10
[1,0,1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,1,1,0,0,0,0]
=> [1,0,1,1,0,1,1,1,0,0,0,0]
=> [2,2,2,1,1]
=> 4
[1,0,1,1,0,0,1,0,1,0]
=> [1,1,0,1,1,0,0,1,0,1,0,0]
=> [1,0,1,1,0,0,1,1,1,0,0,0]
=> [3,3,3,1,1]
=> 10
[1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,1,1,0,0,1,1,0,0,0]
=> [1,0,1,1,0,1,1,0,1,0,0,0]
=> [3,2,2,1,1]
=> 7
[1,0,1,1,0,1,0,0,1,0]
=> [1,1,0,1,1,0,1,0,0,1,0,0]
=> [1,0,1,1,0,0,1,0,1,1,0,0]
=> [4,4,3,1,1]
=> 19
[1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,1,1,0,1,0,1,0,0,0]
=> [1,0,1,1,0,1,0,0,1,1,0,0]
=> [4,4,2,1,1]
=> 16
[1,0,1,1,0,1,1,0,0,0]
=> [1,1,0,1,1,0,1,1,0,0,0,0]
=> [1,0,1,1,0,1,0,1,1,0,0,0]
=> [3,3,2,1,1]
=> 9
[1,0,1,1,1,0,0,0,1,0]
=> [1,1,0,1,1,1,0,0,0,1,0,0]
=> [1,0,1,1,0,0,1,0,1,0,1,0]
=> [5,4,3,1,1]
=> 28
[1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,1,1,1,0,0,1,0,0,0]
=> [1,0,1,1,0,1,0,0,1,0,1,0]
=> [5,4,2,1,1]
=> 23
[1,0,1,1,1,0,1,0,0,0]
=> [1,1,0,1,1,1,0,1,0,0,0,0]
=> [1,0,1,1,0,1,0,1,0,0,1,0]
=> [5,3,2,1,1]
=> 19
[1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,1,1,1,0,0,0,0,0]
=> [1,0,1,1,0,1,0,1,0,1,0,0]
=> [4,3,2,1,1]
=> 14
[1,1,0,0,1,0,1,0,1,0]
=> [1,1,1,0,0,1,0,1,0,1,0,0]
=> [1,0,1,1,1,0,0,1,0,0,1,0]
=> [5,3,1,1,1]
=> 12
[1,1,0,0,1,0,1,1,0,0]
=> [1,1,1,0,0,1,0,1,1,0,0,0]
=> [1,0,1,1,1,1,0,0,0,0,1,0]
=> [5,1,1,1,1]
=> 5
[1,1,0,0,1,1,0,0,1,0]
=> [1,1,1,0,0,1,1,0,0,1,0,0]
=> [1,0,1,1,1,0,0,1,0,1,0,0]
=> [4,3,1,1,1]
=> 9
[1,1,0,0,1,1,0,1,0,0]
=> [1,1,1,0,0,1,1,0,1,0,0,0]
=> [1,0,1,1,1,1,0,0,0,1,0,0]
=> [4,1,1,1,1]
=> 4
[1,1,0,0,1,1,1,0,0,0]
=> [1,1,1,0,0,1,1,1,0,0,0,0]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> [2,1,1,1,1]
=> 2
[1,1,0,1,0,0,1,0,1,0]
=> [1,1,1,0,1,0,0,1,0,1,0,0]
=> [1,0,1,1,1,0,0,1,1,0,0,0]
=> [3,3,1,1,1]
=> 6
[1,1,0,1,0,0,1,1,0,0]
=> [1,1,1,0,1,0,0,1,1,0,0,0]
=> [1,0,1,1,1,1,0,0,1,0,0,0]
=> [3,1,1,1,1]
=> 3
[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]
=> [4,4,1,1,1]
=> 10
[1,1,0,1,0,1,0,1,0,0]
=> [1,1,1,0,1,0,1,0,1,0,0,0]
=> [1,0,1,0,1,1,0,0,1,1,0,0]
=> [4,4,2,2,1]
=> 22
[1,1,0,1,0,1,1,0,0,0]
=> [1,1,1,0,1,0,1,1,0,0,0,0]
=> [1,0,1,0,1,1,0,1,1,0,0,0]
=> [3,3,2,2,1]
=> 12
[1,1,0,1,1,0,0,0,1,0]
=> [1,1,1,0,1,1,0,0,0,1,0,0]
=> [1,0,1,1,1,0,0,0,1,0,1,0]
=> [5,4,1,1,1]
=> 14
[1,1,0,1,1,0,0,1,0,0]
=> [1,1,1,0,1,1,0,0,1,0,0,0]
=> [1,0,1,0,1,1,0,0,1,0,1,0]
=> [5,4,2,2,1]
=> 32
[1,1,0,1,1,0,1,0,0,0]
=> [1,1,1,0,1,1,0,1,0,0,0,0]
=> [1,0,1,0,1,1,0,1,0,0,1,0]
=> [5,3,2,2,1]
=> 26
[1,1,0,1,1,1,0,0,0,0]
=> [1,1,1,0,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> [4,3,2,2,1]
=> 19
[1,1,1,0,0,0,1,0,1,0]
=> [1,1,1,1,0,0,0,1,0,1,0,0]
=> [1,0,1,1,1,0,1,1,0,0,0,0]
=> [2,2,1,1,1]
=> 3
[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]
=> [5,5,3,3,1,1]
=> ? ∊ {48,48,53,53,53,62,62,62,62,62,63,63,66,66,72,72,75,75,75,76,76,84,89,89,90,90,104,104,107,132}
[1,0,1,0,1,1,0,0,1,0,1,0]
=> [1,1,0,1,0,1,1,0,0,1,0,1,0,0]
=> [1,0,1,1,0,0,1,1,0,0,1,0,1,0]
=> [6,5,3,3,1,1]
=> ? ∊ {48,48,53,53,53,62,62,62,62,62,63,63,66,66,72,72,75,75,75,76,76,84,89,89,90,90,104,104,107,132}
[1,0,1,0,1,1,0,1,0,0,1,0]
=> [1,1,0,1,0,1,1,0,1,0,0,1,0,0]
=> [1,0,1,1,0,0,1,1,0,1,0,0,1,0]
=> [6,4,3,3,1,1]
=> ? ∊ {48,48,53,53,53,62,62,62,62,62,63,63,66,66,72,72,75,75,75,76,76,84,89,89,90,90,104,104,107,132}
[1,0,1,1,0,1,0,1,0,0,1,0]
=> [1,1,0,1,1,0,1,0,1,0,0,1,0,0]
=> [1,0,1,1,0,0,1,0,1,1,0,0,1,0]
=> [6,4,4,3,1,1]
=> ? ∊ {48,48,53,53,53,62,62,62,62,62,63,63,66,66,72,72,75,75,75,76,76,84,89,89,90,90,104,104,107,132}
[1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,1,0,1,1,0,1,0,1,0,1,0,0,0]
=> [1,0,1,1,0,1,0,0,1,1,0,0,1,0]
=> [6,4,4,2,1,1]
=> ? ∊ {48,48,53,53,53,62,62,62,62,62,63,63,66,66,72,72,75,75,75,76,76,84,89,89,90,90,104,104,107,132}
[1,0,1,1,0,1,1,0,0,0,1,0]
=> [1,1,0,1,1,0,1,1,0,0,0,1,0,0]
=> [1,0,1,1,0,0,1,0,1,1,0,1,0,0]
=> [5,4,4,3,1,1]
=> ? ∊ {48,48,53,53,53,62,62,62,62,62,63,63,66,66,72,72,75,75,75,76,76,84,89,89,90,90,104,104,107,132}
[1,0,1,1,1,0,1,0,0,0,1,0]
=> [1,1,0,1,1,1,0,1,0,0,0,1,0,0]
=> [1,0,1,1,0,0,1,0,1,0,1,1,0,0]
=> [5,5,4,3,1,1]
=> ? ∊ {48,48,53,53,53,62,62,62,62,62,63,63,66,66,72,72,75,75,75,76,76,84,89,89,90,90,104,104,107,132}
[1,0,1,1,1,0,1,0,0,1,0,0]
=> [1,1,0,1,1,1,0,1,0,0,1,0,0,0]
=> [1,0,1,1,0,1,0,0,1,0,1,1,0,0]
=> [5,5,4,2,1,1]
=> ? ∊ {48,48,53,53,53,62,62,62,62,62,63,63,66,66,72,72,75,75,75,76,76,84,89,89,90,90,104,104,107,132}
[1,0,1,1,1,1,0,0,0,0,1,0]
=> [1,1,0,1,1,1,1,0,0,0,0,1,0,0]
=> [1,0,1,1,0,0,1,0,1,0,1,0,1,0]
=> [6,5,4,3,1,1]
=> ? ∊ {48,48,53,53,53,62,62,62,62,62,63,63,66,66,72,72,75,75,75,76,76,84,89,89,90,90,104,104,107,132}
[1,0,1,1,1,1,0,0,0,1,0,0]
=> [1,1,0,1,1,1,1,0,0,0,1,0,0,0]
=> [1,0,1,1,0,1,0,0,1,0,1,0,1,0]
=> [6,5,4,2,1,1]
=> ? ∊ {48,48,53,53,53,62,62,62,62,62,63,63,66,66,72,72,75,75,75,76,76,84,89,89,90,90,104,104,107,132}
[1,0,1,1,1,1,0,0,1,0,0,0]
=> [1,1,0,1,1,1,1,0,0,1,0,0,0,0]
=> [1,0,1,1,0,1,0,1,0,0,1,0,1,0]
=> [6,5,3,2,1,1]
=> ? ∊ {48,48,53,53,53,62,62,62,62,62,63,63,66,66,72,72,75,75,75,76,76,84,89,89,90,90,104,104,107,132}
[1,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,1,0,1,0,1,0,1,0,1,0,0,0]
=> [1,0,1,0,1,1,0,0,1,1,0,0,1,0]
=> [6,4,4,2,2,1]
=> ? ∊ {48,48,53,53,53,62,62,62,62,62,63,63,66,66,72,72,75,75,75,76,76,84,89,89,90,90,104,104,107,132}
[1,1,0,1,0,1,1,0,0,1,0,0]
=> [1,1,1,0,1,0,1,1,0,0,1,0,0,0]
=> [1,0,1,0,1,1,0,0,1,1,0,1,0,0]
=> [5,4,4,2,2,1]
=> ? ∊ {48,48,53,53,53,62,62,62,62,62,63,63,66,66,72,72,75,75,75,76,76,84,89,89,90,90,104,104,107,132}
[1,1,0,1,1,0,1,0,0,1,0,0]
=> [1,1,1,0,1,1,0,1,0,0,1,0,0,0]
=> [1,0,1,0,1,1,0,0,1,0,1,1,0,0]
=> [5,5,4,2,2,1]
=> ? ∊ {48,48,53,53,53,62,62,62,62,62,63,63,66,66,72,72,75,75,75,76,76,84,89,89,90,90,104,104,107,132}
[1,1,0,1,1,0,1,0,1,0,0,0]
=> [1,1,1,0,1,1,0,1,0,1,0,0,0,0]
=> [1,0,1,0,1,1,0,1,0,0,1,1,0,0]
=> [5,5,3,2,2,1]
=> ? ∊ {48,48,53,53,53,62,62,62,62,62,63,63,66,66,72,72,75,75,75,76,76,84,89,89,90,90,104,104,107,132}
[1,1,0,1,1,1,0,0,0,0,1,0]
=> [1,1,1,0,1,1,1,0,0,0,0,1,0,0]
=> [1,0,1,1,1,0,0,0,1,0,1,0,1,0]
=> [6,5,4,1,1,1]
=> ? ∊ {48,48,53,53,53,62,62,62,62,62,63,63,66,66,72,72,75,75,75,76,76,84,89,89,90,90,104,104,107,132}
[1,1,0,1,1,1,0,0,0,1,0,0]
=> [1,1,1,0,1,1,1,0,0,0,1,0,0,0]
=> [1,0,1,0,1,1,0,0,1,0,1,0,1,0]
=> [6,5,4,2,2,1]
=> ? ∊ {48,48,53,53,53,62,62,62,62,62,63,63,66,66,72,72,75,75,75,76,76,84,89,89,90,90,104,104,107,132}
[1,1,0,1,1,1,0,0,1,0,0,0]
=> [1,1,1,0,1,1,1,0,0,1,0,0,0,0]
=> [1,0,1,0,1,1,0,1,0,0,1,0,1,0]
=> [6,5,3,2,2,1]
=> ? ∊ {48,48,53,53,53,62,62,62,62,62,63,63,66,66,72,72,75,75,75,76,76,84,89,89,90,90,104,104,107,132}
[1,1,0,1,1,1,0,1,0,0,0,0]
=> [1,1,1,0,1,1,1,0,1,0,0,0,0,0]
=> [1,0,1,0,1,1,0,1,0,1,0,0,1,0]
=> [6,4,3,2,2,1]
=> ? ∊ {48,48,53,53,53,62,62,62,62,62,63,63,66,66,72,72,75,75,75,76,76,84,89,89,90,90,104,104,107,132}
[1,1,1,0,1,0,1,0,1,0,0,0]
=> [1,1,1,1,0,1,0,1,0,1,0,0,0,0]
=> [1,0,1,0,1,0,1,1,0,0,1,1,0,0]
=> [5,5,3,3,2,1]
=> ? ∊ {48,48,53,53,53,62,62,62,62,62,63,63,66,66,72,72,75,75,75,76,76,84,89,89,90,90,104,104,107,132}
[1,1,1,0,1,1,0,0,0,1,0,0]
=> [1,1,1,1,0,1,1,0,0,0,1,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0,1,0,1,0]
=> [6,5,2,2,2,1]
=> ? ∊ {48,48,53,53,53,62,62,62,62,62,63,63,66,66,72,72,75,75,75,76,76,84,89,89,90,90,104,104,107,132}
[1,1,1,0,1,1,0,0,1,0,0,0]
=> [1,1,1,1,0,1,1,0,0,1,0,0,0,0]
=> [1,0,1,0,1,0,1,1,0,0,1,0,1,0]
=> [6,5,3,3,2,1]
=> ? ∊ {48,48,53,53,53,62,62,62,62,62,63,63,66,66,72,72,75,75,75,76,76,84,89,89,90,90,104,104,107,132}
[1,1,1,0,1,1,0,1,0,0,0,0]
=> [1,1,1,1,0,1,1,0,1,0,0,0,0,0]
=> [1,0,1,0,1,0,1,1,0,1,0,0,1,0]
=> [6,4,3,3,2,1]
=> ? ∊ {48,48,53,53,53,62,62,62,62,62,63,63,66,66,72,72,75,75,75,76,76,84,89,89,90,90,104,104,107,132}
[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]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [5,4,3,3,2,1]
=> ? ∊ {48,48,53,53,53,62,62,62,62,62,63,63,66,66,72,72,75,75,75,76,76,84,89,89,90,90,104,104,107,132}
[1,1,1,1,0,1,0,0,1,0,0,0]
=> [1,1,1,1,1,0,1,0,0,1,0,0,0,0]
=> [1,0,1,0,1,0,1,1,1,0,0,0,1,0]
=> [6,3,3,3,2,1]
=> ? ∊ {48,48,53,53,53,62,62,62,62,62,63,63,66,66,72,72,75,75,75,76,76,84,89,89,90,90,104,104,107,132}
[1,1,1,1,0,1,0,1,0,0,0,0]
=> [1,1,1,1,1,0,1,0,1,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,1,0,0,1,0]
=> [6,4,4,3,2,1]
=> ? ∊ {48,48,53,53,53,62,62,62,62,62,63,63,66,66,72,72,75,75,75,76,76,84,89,89,90,90,104,104,107,132}
[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]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [5,4,4,3,2,1]
=> ? ∊ {48,48,53,53,53,62,62,62,62,62,63,63,66,66,72,72,75,75,75,76,76,84,89,89,90,90,104,104,107,132}
[1,1,1,1,1,0,0,1,0,0,0,0]
=> [1,1,1,1,1,1,0,0,1,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [4,4,4,3,2,1]
=> ? ∊ {48,48,53,53,53,62,62,62,62,62,63,63,66,66,72,72,75,75,75,76,76,84,89,89,90,90,104,104,107,132}
[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]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [5,5,4,3,2,1]
=> ? ∊ {48,48,53,53,53,62,62,62,62,62,63,63,66,66,72,72,75,75,75,76,76,84,89,89,90,90,104,104,107,132}
[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]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [6,5,4,3,2,1]
=> ? ∊ {48,48,53,53,53,62,62,62,62,62,63,63,66,66,72,72,75,75,75,76,76,84,89,89,90,90,104,104,107,132}
[1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,0,1,1,0,0,1,1,0,0,1,1,0,0,1,0]
=> [7,5,5,3,3,1,1]
=> ? ∊ {27,27,34,34,35,35,36,36,37,40,40,43,43,43,43,45,45,46,46,46,46,46,47,47,47,47,48,48,48,48,50,50,50,50,50,50,51,51,51,51,52,52,52,52,53,53,53,54,54,55,55,55,55,55,55,55,55,56,56,61,61,61,61,62,62,62,62,62,62,62,62,63,63,65,65,65,65,66,66,66,66,66,66,67,67,67,67,69,70,70,70,70,71,71,72,72,75,75,75,75,75,75,75,76,76,76,76,78,78,79,79,79,79,80,80,81,81,81,81,82,82,83,83,84,85,85,87,87,89,89,89,89,90,90,91,91,94,94,95,95,95,95,95,95,96,96,96,96,97,97,97,97,99,99,99,99,103,103,104,104,105,105,105,105,106,106,107,107,107,108,108,109,109,110,110,112,112,113,113,114,114,115,115,116,116,117,117,118,118,118,121,121,123,123,123,123,125,125,126,127,127,129,129,132,132,132,134,136,136,136,136,137,137,139,139,140,140,142,142,145,145,145,145,145,145,145,145,146,146,151,151,151,151,156,156,156,157,157,157,157,158,158,161,161,162,163,163,165,165,171,171,171,171,173,173,174,174,179,179,180,180,181,185,185,188,188,189,189,193,193,199,199,202,202,205,205,207,209,209,211,211,214,214,219,219,220,220,225,225,227,227,230,233,241,241,251,251,252,252,255,255,255,255,268,268,281,295,295,297,297,303,303,314,345,345,359,359,429}
[1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,1,0,1,0,1,1,0,0,0]
=> [1,0,1,1,0,1,1,0,0,0,1,1,0,0,1,0]
=> [7,5,5,2,2,1,1]
=> ? ∊ {27,27,34,34,35,35,36,36,37,40,40,43,43,43,43,45,45,46,46,46,46,46,47,47,47,47,48,48,48,48,50,50,50,50,50,50,51,51,51,51,52,52,52,52,53,53,53,54,54,55,55,55,55,55,55,55,55,56,56,61,61,61,61,62,62,62,62,62,62,62,62,63,63,65,65,65,65,66,66,66,66,66,66,67,67,67,67,69,70,70,70,70,71,71,72,72,75,75,75,75,75,75,75,76,76,76,76,78,78,79,79,79,79,80,80,81,81,81,81,82,82,83,83,84,85,85,87,87,89,89,89,89,90,90,91,91,94,94,95,95,95,95,95,95,96,96,96,96,97,97,97,97,99,99,99,99,103,103,104,104,105,105,105,105,106,106,107,107,107,108,108,109,109,110,110,112,112,113,113,114,114,115,115,116,116,117,117,118,118,118,121,121,123,123,123,123,125,125,126,127,127,129,129,132,132,132,134,136,136,136,136,137,137,139,139,140,140,142,142,145,145,145,145,145,145,145,145,146,146,151,151,151,151,156,156,156,157,157,157,157,158,158,161,161,162,163,163,165,165,171,171,171,171,173,173,174,174,179,179,180,180,181,185,185,188,188,189,189,193,193,199,199,202,202,205,205,207,209,209,211,211,214,214,219,219,220,220,225,225,227,227,230,233,241,241,251,251,252,252,255,255,255,255,268,268,281,295,295,297,297,303,303,314,345,345,359,359,429}
[1,0,1,0,1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,0,1,0,1,0,1,1,0,0,1,0,0]
=> [1,0,1,1,0,0,1,1,0,1,1,0,0,0,1,0]
=> [7,4,4,3,3,1,1]
=> ? ∊ {27,27,34,34,35,35,36,36,37,40,40,43,43,43,43,45,45,46,46,46,46,46,47,47,47,47,48,48,48,48,50,50,50,50,50,50,51,51,51,51,52,52,52,52,53,53,53,54,54,55,55,55,55,55,55,55,55,56,56,61,61,61,61,62,62,62,62,62,62,62,62,63,63,65,65,65,65,66,66,66,66,66,66,67,67,67,67,69,70,70,70,70,71,71,72,72,75,75,75,75,75,75,75,76,76,76,76,78,78,79,79,79,79,80,80,81,81,81,81,82,82,83,83,84,85,85,87,87,89,89,89,89,90,90,91,91,94,94,95,95,95,95,95,95,96,96,96,96,97,97,97,97,99,99,99,99,103,103,104,104,105,105,105,105,106,106,107,107,107,108,108,109,109,110,110,112,112,113,113,114,114,115,115,116,116,117,117,118,118,118,121,121,123,123,123,123,125,125,126,127,127,129,129,132,132,132,134,136,136,136,136,137,137,139,139,140,140,142,142,145,145,145,145,145,145,145,145,146,146,151,151,151,151,156,156,156,157,157,157,157,158,158,161,161,162,163,163,165,165,171,171,171,171,173,173,174,174,179,179,180,180,181,185,185,188,188,189,189,193,193,199,199,202,202,205,205,207,209,209,211,211,214,214,219,219,220,220,225,225,227,227,230,233,241,241,251,251,252,252,255,255,255,255,268,268,281,295,295,297,297,303,303,314,345,345,359,359,429}
[1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,0,1,0,1,0,1,0,1,1,0,1,0,0,0]
=> [1,0,1,1,0,1,1,0,0,1,1,0,0,0,1,0]
=> [7,4,4,2,2,1,1]
=> ? ∊ {27,27,34,34,35,35,36,36,37,40,40,43,43,43,43,45,45,46,46,46,46,46,47,47,47,47,48,48,48,48,50,50,50,50,50,50,51,51,51,51,52,52,52,52,53,53,53,54,54,55,55,55,55,55,55,55,55,56,56,61,61,61,61,62,62,62,62,62,62,62,62,63,63,65,65,65,65,66,66,66,66,66,66,67,67,67,67,69,70,70,70,70,71,71,72,72,75,75,75,75,75,75,75,76,76,76,76,78,78,79,79,79,79,80,80,81,81,81,81,82,82,83,83,84,85,85,87,87,89,89,89,89,90,90,91,91,94,94,95,95,95,95,95,95,96,96,96,96,97,97,97,97,99,99,99,99,103,103,104,104,105,105,105,105,106,106,107,107,107,108,108,109,109,110,110,112,112,113,113,114,114,115,115,116,116,117,117,118,118,118,121,121,123,123,123,123,125,125,126,127,127,129,129,132,132,132,134,136,136,136,136,137,137,139,139,140,140,142,142,145,145,145,145,145,145,145,145,146,146,151,151,151,151,156,156,156,157,157,157,157,158,158,161,161,162,163,163,165,165,171,171,171,171,173,173,174,174,179,179,180,180,181,185,185,188,188,189,189,193,193,199,199,202,202,205,205,207,209,209,211,211,214,214,219,219,220,220,225,225,227,227,230,233,241,241,251,251,252,252,255,255,255,255,268,268,281,295,295,297,297,303,303,314,345,345,359,359,429}
[1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,0,1,0,1,1,1,0,0,0,0]
=> [1,0,1,1,0,1,1,1,0,0,1,0,0,0,1,0]
=> [7,4,2,2,2,1,1]
=> ? ∊ {27,27,34,34,35,35,36,36,37,40,40,43,43,43,43,45,45,46,46,46,46,46,47,47,47,47,48,48,48,48,50,50,50,50,50,50,51,51,51,51,52,52,52,52,53,53,53,54,54,55,55,55,55,55,55,55,55,56,56,61,61,61,61,62,62,62,62,62,62,62,62,63,63,65,65,65,65,66,66,66,66,66,66,67,67,67,67,69,70,70,70,70,71,71,72,72,75,75,75,75,75,75,75,76,76,76,76,78,78,79,79,79,79,80,80,81,81,81,81,82,82,83,83,84,85,85,87,87,89,89,89,89,90,90,91,91,94,94,95,95,95,95,95,95,96,96,96,96,97,97,97,97,99,99,99,99,103,103,104,104,105,105,105,105,106,106,107,107,107,108,108,109,109,110,110,112,112,113,113,114,114,115,115,116,116,117,117,118,118,118,121,121,123,123,123,123,125,125,126,127,127,129,129,132,132,132,134,136,136,136,136,137,137,139,139,140,140,142,142,145,145,145,145,145,145,145,145,146,146,151,151,151,151,156,156,156,157,157,157,157,158,158,161,161,162,163,163,165,165,171,171,171,171,173,173,174,174,179,179,180,180,181,185,185,188,188,189,189,193,193,199,199,202,202,205,205,207,209,209,211,211,214,214,219,219,220,220,225,225,227,227,230,233,241,241,251,251,252,252,255,255,255,255,268,268,281,295,295,297,297,303,303,314,345,345,359,359,429}
[1,0,1,0,1,0,1,1,0,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,1,0,0,1,0,1,0,0]
=> [1,0,1,1,0,0,1,1,0,0,1,1,0,1,0,0]
=> [6,5,5,3,3,1,1]
=> ? ∊ {27,27,34,34,35,35,36,36,37,40,40,43,43,43,43,45,45,46,46,46,46,46,47,47,47,47,48,48,48,48,50,50,50,50,50,50,51,51,51,51,52,52,52,52,53,53,53,54,54,55,55,55,55,55,55,55,55,56,56,61,61,61,61,62,62,62,62,62,62,62,62,63,63,65,65,65,65,66,66,66,66,66,66,67,67,67,67,69,70,70,70,70,71,71,72,72,75,75,75,75,75,75,75,76,76,76,76,78,78,79,79,79,79,80,80,81,81,81,81,82,82,83,83,84,85,85,87,87,89,89,89,89,90,90,91,91,94,94,95,95,95,95,95,95,96,96,96,96,97,97,97,97,99,99,99,99,103,103,104,104,105,105,105,105,106,106,107,107,107,108,108,109,109,110,110,112,112,113,113,114,114,115,115,116,116,117,117,118,118,118,121,121,123,123,123,123,125,125,126,127,127,129,129,132,132,132,134,136,136,136,136,137,137,139,139,140,140,142,142,145,145,145,145,145,145,145,145,146,146,151,151,151,151,156,156,156,157,157,157,157,158,158,161,161,162,163,163,165,165,171,171,171,171,173,173,174,174,179,179,180,180,181,185,185,188,188,189,189,193,193,199,199,202,202,205,205,207,209,209,211,211,214,214,219,219,220,220,225,225,227,227,230,233,241,241,251,251,252,252,255,255,255,255,268,268,281,295,295,297,297,303,303,314,345,345,359,359,429}
[1,0,1,0,1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,1,0,1,0,1,1,0,0,1,1,0,0,0]
=> [1,0,1,1,0,1,1,0,0,0,1,1,0,1,0,0]
=> [6,5,5,2,2,1,1]
=> ? ∊ {27,27,34,34,35,35,36,36,37,40,40,43,43,43,43,45,45,46,46,46,46,46,47,47,47,47,48,48,48,48,50,50,50,50,50,50,51,51,51,51,52,52,52,52,53,53,53,54,54,55,55,55,55,55,55,55,55,56,56,61,61,61,61,62,62,62,62,62,62,62,62,63,63,65,65,65,65,66,66,66,66,66,66,67,67,67,67,69,70,70,70,70,71,71,72,72,75,75,75,75,75,75,75,76,76,76,76,78,78,79,79,79,79,80,80,81,81,81,81,82,82,83,83,84,85,85,87,87,89,89,89,89,90,90,91,91,94,94,95,95,95,95,95,95,96,96,96,96,97,97,97,97,99,99,99,99,103,103,104,104,105,105,105,105,106,106,107,107,107,108,108,109,109,110,110,112,112,113,113,114,114,115,115,116,116,117,117,118,118,118,121,121,123,123,123,123,125,125,126,127,127,129,129,132,132,132,134,136,136,136,136,137,137,139,139,140,140,142,142,145,145,145,145,145,145,145,145,146,146,151,151,151,151,156,156,156,157,157,157,157,158,158,161,161,162,163,163,165,165,171,171,171,171,173,173,174,174,179,179,180,180,181,185,185,188,188,189,189,193,193,199,199,202,202,205,205,207,209,209,211,211,214,214,219,219,220,220,225,225,227,227,230,233,241,241,251,251,252,252,255,255,255,255,268,268,281,295,295,297,297,303,303,314,345,345,359,359,429}
[1,0,1,0,1,0,1,1,0,1,0,0,1,0]
=> [1,1,0,1,0,1,0,1,1,0,1,0,0,1,0,0]
=> [1,0,1,1,0,0,1,1,0,1,1,0,0,1,0,0]
=> [6,4,4,3,3,1,1]
=> ? ∊ {27,27,34,34,35,35,36,36,37,40,40,43,43,43,43,45,45,46,46,46,46,46,47,47,47,47,48,48,48,48,50,50,50,50,50,50,51,51,51,51,52,52,52,52,53,53,53,54,54,55,55,55,55,55,55,55,55,56,56,61,61,61,61,62,62,62,62,62,62,62,62,63,63,65,65,65,65,66,66,66,66,66,66,67,67,67,67,69,70,70,70,70,71,71,72,72,75,75,75,75,75,75,75,76,76,76,76,78,78,79,79,79,79,80,80,81,81,81,81,82,82,83,83,84,85,85,87,87,89,89,89,89,90,90,91,91,94,94,95,95,95,95,95,95,96,96,96,96,97,97,97,97,99,99,99,99,103,103,104,104,105,105,105,105,106,106,107,107,107,108,108,109,109,110,110,112,112,113,113,114,114,115,115,116,116,117,117,118,118,118,121,121,123,123,123,123,125,125,126,127,127,129,129,132,132,132,134,136,136,136,136,137,137,139,139,140,140,142,142,145,145,145,145,145,145,145,145,146,146,151,151,151,151,156,156,156,157,157,157,157,158,158,161,161,162,163,163,165,165,171,171,171,171,173,173,174,174,179,179,180,180,181,185,185,188,188,189,189,193,193,199,199,202,202,205,205,207,209,209,211,211,214,214,219,219,220,220,225,225,227,227,230,233,241,241,251,251,252,252,255,255,255,255,268,268,281,295,295,297,297,303,303,314,345,345,359,359,429}
[1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,1,0,1,0,1,1,0,1,0,1,0,0,0]
=> [1,0,1,1,0,1,1,0,0,1,1,0,0,1,0,0]
=> [6,4,4,2,2,1,1]
=> ? ∊ {27,27,34,34,35,35,36,36,37,40,40,43,43,43,43,45,45,46,46,46,46,46,47,47,47,47,48,48,48,48,50,50,50,50,50,50,51,51,51,51,52,52,52,52,53,53,53,54,54,55,55,55,55,55,55,55,55,56,56,61,61,61,61,62,62,62,62,62,62,62,62,63,63,65,65,65,65,66,66,66,66,66,66,67,67,67,67,69,70,70,70,70,71,71,72,72,75,75,75,75,75,75,75,76,76,76,76,78,78,79,79,79,79,80,80,81,81,81,81,82,82,83,83,84,85,85,87,87,89,89,89,89,90,90,91,91,94,94,95,95,95,95,95,95,96,96,96,96,97,97,97,97,99,99,99,99,103,103,104,104,105,105,105,105,106,106,107,107,107,108,108,109,109,110,110,112,112,113,113,114,114,115,115,116,116,117,117,118,118,118,121,121,123,123,123,123,125,125,126,127,127,129,129,132,132,132,134,136,136,136,136,137,137,139,139,140,140,142,142,145,145,145,145,145,145,145,145,146,146,151,151,151,151,156,156,156,157,157,157,157,158,158,161,161,162,163,163,165,165,171,171,171,171,173,173,174,174,179,179,180,180,181,185,185,188,188,189,189,193,193,199,199,202,202,205,205,207,209,209,211,211,214,214,219,219,220,220,225,225,227,227,230,233,241,241,251,251,252,252,255,255,255,255,268,268,281,295,295,297,297,303,303,314,345,345,359,359,429}
[1,0,1,0,1,0,1,1,0,1,1,0,0,0]
=> [1,1,0,1,0,1,0,1,1,0,1,1,0,0,0,0]
=> [1,0,1,1,0,1,1,1,0,0,1,0,0,1,0,0]
=> [6,4,2,2,2,1,1]
=> ? ∊ {27,27,34,34,35,35,36,36,37,40,40,43,43,43,43,45,45,46,46,46,46,46,47,47,47,47,48,48,48,48,50,50,50,50,50,50,51,51,51,51,52,52,52,52,53,53,53,54,54,55,55,55,55,55,55,55,55,56,56,61,61,61,61,62,62,62,62,62,62,62,62,63,63,65,65,65,65,66,66,66,66,66,66,67,67,67,67,69,70,70,70,70,71,71,72,72,75,75,75,75,75,75,75,76,76,76,76,78,78,79,79,79,79,80,80,81,81,81,81,82,82,83,83,84,85,85,87,87,89,89,89,89,90,90,91,91,94,94,95,95,95,95,95,95,96,96,96,96,97,97,97,97,99,99,99,99,103,103,104,104,105,105,105,105,106,106,107,107,107,108,108,109,109,110,110,112,112,113,113,114,114,115,115,116,116,117,117,118,118,118,121,121,123,123,123,123,125,125,126,127,127,129,129,132,132,132,134,136,136,136,136,137,137,139,139,140,140,142,142,145,145,145,145,145,145,145,145,146,146,151,151,151,151,156,156,156,157,157,157,157,158,158,161,161,162,163,163,165,165,171,171,171,171,173,173,174,174,179,179,180,180,181,185,185,188,188,189,189,193,193,199,199,202,202,205,205,207,209,209,211,211,214,214,219,219,220,220,225,225,227,227,230,233,241,241,251,251,252,252,255,255,255,255,268,268,281,295,295,297,297,303,303,314,345,345,359,359,429}
[1,0,1,0,1,0,1,1,1,0,0,0,1,0]
=> [1,1,0,1,0,1,0,1,1,1,0,0,0,1,0,0]
=> [1,0,1,1,0,0,1,1,0,1,1,1,0,0,0,0]
=> [4,4,4,3,3,1,1]
=> ? ∊ {27,27,34,34,35,35,36,36,37,40,40,43,43,43,43,45,45,46,46,46,46,46,47,47,47,47,48,48,48,48,50,50,50,50,50,50,51,51,51,51,52,52,52,52,53,53,53,54,54,55,55,55,55,55,55,55,55,56,56,61,61,61,61,62,62,62,62,62,62,62,62,63,63,65,65,65,65,66,66,66,66,66,66,67,67,67,67,69,70,70,70,70,71,71,72,72,75,75,75,75,75,75,75,76,76,76,76,78,78,79,79,79,79,80,80,81,81,81,81,82,82,83,83,84,85,85,87,87,89,89,89,89,90,90,91,91,94,94,95,95,95,95,95,95,96,96,96,96,97,97,97,97,99,99,99,99,103,103,104,104,105,105,105,105,106,106,107,107,107,108,108,109,109,110,110,112,112,113,113,114,114,115,115,116,116,117,117,118,118,118,121,121,123,123,123,123,125,125,126,127,127,129,129,132,132,132,134,136,136,136,136,137,137,139,139,140,140,142,142,145,145,145,145,145,145,145,145,146,146,151,151,151,151,156,156,156,157,157,157,157,158,158,161,161,162,163,163,165,165,171,171,171,171,173,173,174,174,179,179,180,180,181,185,185,188,188,189,189,193,193,199,199,202,202,205,205,207,209,209,211,211,214,214,219,219,220,220,225,225,227,227,230,233,241,241,251,251,252,252,255,255,255,255,268,268,281,295,295,297,297,303,303,314,345,345,359,359,429}
[1,0,1,0,1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,1,0,1,0,1,1,1,0,0,1,0,0,0]
=> [1,0,1,1,0,1,1,0,0,1,1,1,0,0,0,0]
=> [4,4,4,2,2,1,1]
=> ? ∊ {27,27,34,34,35,35,36,36,37,40,40,43,43,43,43,45,45,46,46,46,46,46,47,47,47,47,48,48,48,48,50,50,50,50,50,50,51,51,51,51,52,52,52,52,53,53,53,54,54,55,55,55,55,55,55,55,55,56,56,61,61,61,61,62,62,62,62,62,62,62,62,63,63,65,65,65,65,66,66,66,66,66,66,67,67,67,67,69,70,70,70,70,71,71,72,72,75,75,75,75,75,75,75,76,76,76,76,78,78,79,79,79,79,80,80,81,81,81,81,82,82,83,83,84,85,85,87,87,89,89,89,89,90,90,91,91,94,94,95,95,95,95,95,95,96,96,96,96,97,97,97,97,99,99,99,99,103,103,104,104,105,105,105,105,106,106,107,107,107,108,108,109,109,110,110,112,112,113,113,114,114,115,115,116,116,117,117,118,118,118,121,121,123,123,123,123,125,125,126,127,127,129,129,132,132,132,134,136,136,136,136,137,137,139,139,140,140,142,142,145,145,145,145,145,145,145,145,146,146,151,151,151,151,156,156,156,157,157,157,157,158,158,161,161,162,163,163,165,165,171,171,171,171,173,173,174,174,179,179,180,180,181,185,185,188,188,189,189,193,193,199,199,202,202,205,205,207,209,209,211,211,214,214,219,219,220,220,225,225,227,227,230,233,241,241,251,251,252,252,255,255,255,255,268,268,281,295,295,297,297,303,303,314,345,345,359,359,429}
[1,0,1,0,1,1,0,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,1,0,0,1,0,1,0,1,0,0]
=> [1,0,1,1,0,0,1,1,0,0,1,1,1,0,0,0]
=> [5,5,5,3,3,1,1]
=> ? ∊ {27,27,34,34,35,35,36,36,37,40,40,43,43,43,43,45,45,46,46,46,46,46,47,47,47,47,48,48,48,48,50,50,50,50,50,50,51,51,51,51,52,52,52,52,53,53,53,54,54,55,55,55,55,55,55,55,55,56,56,61,61,61,61,62,62,62,62,62,62,62,62,63,63,65,65,65,65,66,66,66,66,66,66,67,67,67,67,69,70,70,70,70,71,71,72,72,75,75,75,75,75,75,75,76,76,76,76,78,78,79,79,79,79,80,80,81,81,81,81,82,82,83,83,84,85,85,87,87,89,89,89,89,90,90,91,91,94,94,95,95,95,95,95,95,96,96,96,96,97,97,97,97,99,99,99,99,103,103,104,104,105,105,105,105,106,106,107,107,107,108,108,109,109,110,110,112,112,113,113,114,114,115,115,116,116,117,117,118,118,118,121,121,123,123,123,123,125,125,126,127,127,129,129,132,132,132,134,136,136,136,136,137,137,139,139,140,140,142,142,145,145,145,145,145,145,145,145,146,146,151,151,151,151,156,156,156,157,157,157,157,158,158,161,161,162,163,163,165,165,171,171,171,171,173,173,174,174,179,179,180,180,181,185,185,188,188,189,189,193,193,199,199,202,202,205,205,207,209,209,211,211,214,214,219,219,220,220,225,225,227,227,230,233,241,241,251,251,252,252,255,255,255,255,268,268,281,295,295,297,297,303,303,314,345,345,359,359,429}
[1,0,1,0,1,1,0,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,1,0,0,1,0,1,1,0,0,0]
=> [1,0,1,1,0,1,1,0,0,0,1,1,1,0,0,0]
=> [5,5,5,2,2,1,1]
=> ? ∊ {27,27,34,34,35,35,36,36,37,40,40,43,43,43,43,45,45,46,46,46,46,46,47,47,47,47,48,48,48,48,50,50,50,50,50,50,51,51,51,51,52,52,52,52,53,53,53,54,54,55,55,55,55,55,55,55,55,56,56,61,61,61,61,62,62,62,62,62,62,62,62,63,63,65,65,65,65,66,66,66,66,66,66,67,67,67,67,69,70,70,70,70,71,71,72,72,75,75,75,75,75,75,75,76,76,76,76,78,78,79,79,79,79,80,80,81,81,81,81,82,82,83,83,84,85,85,87,87,89,89,89,89,90,90,91,91,94,94,95,95,95,95,95,95,96,96,96,96,97,97,97,97,99,99,99,99,103,103,104,104,105,105,105,105,106,106,107,107,107,108,108,109,109,110,110,112,112,113,113,114,114,115,115,116,116,117,117,118,118,118,121,121,123,123,123,123,125,125,126,127,127,129,129,132,132,132,134,136,136,136,136,137,137,139,139,140,140,142,142,145,145,145,145,145,145,145,145,146,146,151,151,151,151,156,156,156,157,157,157,157,158,158,161,161,162,163,163,165,165,171,171,171,171,173,173,174,174,179,179,180,180,181,185,185,188,188,189,189,193,193,199,199,202,202,205,205,207,209,209,211,211,214,214,219,219,220,220,225,225,227,227,230,233,241,241,251,251,252,252,255,255,255,255,268,268,281,295,295,297,297,303,303,314,345,345,359,359,429}
[1,0,1,0,1,1,0,0,1,1,0,0,1,0]
=> [1,1,0,1,0,1,1,0,0,1,1,0,0,1,0,0]
=> [1,0,1,1,0,0,1,1,0,1,1,0,1,0,0,0]
=> [5,4,4,3,3,1,1]
=> ? ∊ {27,27,34,34,35,35,36,36,37,40,40,43,43,43,43,45,45,46,46,46,46,46,47,47,47,47,48,48,48,48,50,50,50,50,50,50,51,51,51,51,52,52,52,52,53,53,53,54,54,55,55,55,55,55,55,55,55,56,56,61,61,61,61,62,62,62,62,62,62,62,62,63,63,65,65,65,65,66,66,66,66,66,66,67,67,67,67,69,70,70,70,70,71,71,72,72,75,75,75,75,75,75,75,76,76,76,76,78,78,79,79,79,79,80,80,81,81,81,81,82,82,83,83,84,85,85,87,87,89,89,89,89,90,90,91,91,94,94,95,95,95,95,95,95,96,96,96,96,97,97,97,97,99,99,99,99,103,103,104,104,105,105,105,105,106,106,107,107,107,108,108,109,109,110,110,112,112,113,113,114,114,115,115,116,116,117,117,118,118,118,121,121,123,123,123,123,125,125,126,127,127,129,129,132,132,132,134,136,136,136,136,137,137,139,139,140,140,142,142,145,145,145,145,145,145,145,145,146,146,151,151,151,151,156,156,156,157,157,157,157,158,158,161,161,162,163,163,165,165,171,171,171,171,173,173,174,174,179,179,180,180,181,185,185,188,188,189,189,193,193,199,199,202,202,205,205,207,209,209,211,211,214,214,219,219,220,220,225,225,227,227,230,233,241,241,251,251,252,252,255,255,255,255,268,268,281,295,295,297,297,303,303,314,345,345,359,359,429}
[1,0,1,0,1,1,0,0,1,1,0,1,0,0]
=> [1,1,0,1,0,1,1,0,0,1,1,0,1,0,0,0]
=> [1,0,1,1,0,1,1,0,0,1,1,0,1,0,0,0]
=> [5,4,4,2,2,1,1]
=> ? ∊ {27,27,34,34,35,35,36,36,37,40,40,43,43,43,43,45,45,46,46,46,46,46,47,47,47,47,48,48,48,48,50,50,50,50,50,50,51,51,51,51,52,52,52,52,53,53,53,54,54,55,55,55,55,55,55,55,55,56,56,61,61,61,61,62,62,62,62,62,62,62,62,63,63,65,65,65,65,66,66,66,66,66,66,67,67,67,67,69,70,70,70,70,71,71,72,72,75,75,75,75,75,75,75,76,76,76,76,78,78,79,79,79,79,80,80,81,81,81,81,82,82,83,83,84,85,85,87,87,89,89,89,89,90,90,91,91,94,94,95,95,95,95,95,95,96,96,96,96,97,97,97,97,99,99,99,99,103,103,104,104,105,105,105,105,106,106,107,107,107,108,108,109,109,110,110,112,112,113,113,114,114,115,115,116,116,117,117,118,118,118,121,121,123,123,123,123,125,125,126,127,127,129,129,132,132,132,134,136,136,136,136,137,137,139,139,140,140,142,142,145,145,145,145,145,145,145,145,146,146,151,151,151,151,156,156,156,157,157,157,157,158,158,161,161,162,163,163,165,165,171,171,171,171,173,173,174,174,179,179,180,180,181,185,185,188,188,189,189,193,193,199,199,202,202,205,205,207,209,209,211,211,214,214,219,219,220,220,225,225,227,227,230,233,241,241,251,251,252,252,255,255,255,255,268,268,281,295,295,297,297,303,303,314,345,345,359,359,429}
[1,0,1,0,1,1,0,1,0,0,1,0,1,0]
=> [1,1,0,1,0,1,1,0,1,0,0,1,0,1,0,0]
=> [1,0,1,1,0,0,1,1,0,0,1,0,1,1,0,0]
=> [6,6,5,3,3,1,1]
=> ? ∊ {27,27,34,34,35,35,36,36,37,40,40,43,43,43,43,45,45,46,46,46,46,46,47,47,47,47,48,48,48,48,50,50,50,50,50,50,51,51,51,51,52,52,52,52,53,53,53,54,54,55,55,55,55,55,55,55,55,56,56,61,61,61,61,62,62,62,62,62,62,62,62,63,63,65,65,65,65,66,66,66,66,66,66,67,67,67,67,69,70,70,70,70,71,71,72,72,75,75,75,75,75,75,75,76,76,76,76,78,78,79,79,79,79,80,80,81,81,81,81,82,82,83,83,84,85,85,87,87,89,89,89,89,90,90,91,91,94,94,95,95,95,95,95,95,96,96,96,96,97,97,97,97,99,99,99,99,103,103,104,104,105,105,105,105,106,106,107,107,107,108,108,109,109,110,110,112,112,113,113,114,114,115,115,116,116,117,117,118,118,118,121,121,123,123,123,123,125,125,126,127,127,129,129,132,132,132,134,136,136,136,136,137,137,139,139,140,140,142,142,145,145,145,145,145,145,145,145,146,146,151,151,151,151,156,156,156,157,157,157,157,158,158,161,161,162,163,163,165,165,171,171,171,171,173,173,174,174,179,179,180,180,181,185,185,188,188,189,189,193,193,199,199,202,202,205,205,207,209,209,211,211,214,214,219,219,220,220,225,225,227,227,230,233,241,241,251,251,252,252,255,255,255,255,268,268,281,295,295,297,297,303,303,314,345,345,359,359,429}
[1,0,1,0,1,1,0,1,0,0,1,1,0,0]
=> [1,1,0,1,0,1,1,0,1,0,0,1,1,0,0,0]
=> [1,0,1,1,0,1,1,0,0,0,1,0,1,1,0,0]
=> [6,6,5,2,2,1,1]
=> ? ∊ {27,27,34,34,35,35,36,36,37,40,40,43,43,43,43,45,45,46,46,46,46,46,47,47,47,47,48,48,48,48,50,50,50,50,50,50,51,51,51,51,52,52,52,52,53,53,53,54,54,55,55,55,55,55,55,55,55,56,56,61,61,61,61,62,62,62,62,62,62,62,62,63,63,65,65,65,65,66,66,66,66,66,66,67,67,67,67,69,70,70,70,70,71,71,72,72,75,75,75,75,75,75,75,76,76,76,76,78,78,79,79,79,79,80,80,81,81,81,81,82,82,83,83,84,85,85,87,87,89,89,89,89,90,90,91,91,94,94,95,95,95,95,95,95,96,96,96,96,97,97,97,97,99,99,99,99,103,103,104,104,105,105,105,105,106,106,107,107,107,108,108,109,109,110,110,112,112,113,113,114,114,115,115,116,116,117,117,118,118,118,121,121,123,123,123,123,125,125,126,127,127,129,129,132,132,132,134,136,136,136,136,137,137,139,139,140,140,142,142,145,145,145,145,145,145,145,145,146,146,151,151,151,151,156,156,156,157,157,157,157,158,158,161,161,162,163,163,165,165,171,171,171,171,173,173,174,174,179,179,180,180,181,185,185,188,188,189,189,193,193,199,199,202,202,205,205,207,209,209,211,211,214,214,219,219,220,220,225,225,227,227,230,233,241,241,251,251,252,252,255,255,255,255,268,268,281,295,295,297,297,303,303,314,345,345,359,359,429}
[1,0,1,0,1,1,0,1,0,1,0,0,1,0]
=> [1,1,0,1,0,1,1,0,1,0,1,0,0,1,0,0]
=> [1,0,1,1,0,0,1,1,0,1,0,0,1,1,0,0]
=> [6,6,4,3,3,1,1]
=> ? ∊ {27,27,34,34,35,35,36,36,37,40,40,43,43,43,43,45,45,46,46,46,46,46,47,47,47,47,48,48,48,48,50,50,50,50,50,50,51,51,51,51,52,52,52,52,53,53,53,54,54,55,55,55,55,55,55,55,55,56,56,61,61,61,61,62,62,62,62,62,62,62,62,63,63,65,65,65,65,66,66,66,66,66,66,67,67,67,67,69,70,70,70,70,71,71,72,72,75,75,75,75,75,75,75,76,76,76,76,78,78,79,79,79,79,80,80,81,81,81,81,82,82,83,83,84,85,85,87,87,89,89,89,89,90,90,91,91,94,94,95,95,95,95,95,95,96,96,96,96,97,97,97,97,99,99,99,99,103,103,104,104,105,105,105,105,106,106,107,107,107,108,108,109,109,110,110,112,112,113,113,114,114,115,115,116,116,117,117,118,118,118,121,121,123,123,123,123,125,125,126,127,127,129,129,132,132,132,134,136,136,136,136,137,137,139,139,140,140,142,142,145,145,145,145,145,145,145,145,146,146,151,151,151,151,156,156,156,157,157,157,157,158,158,161,161,162,163,163,165,165,171,171,171,171,173,173,174,174,179,179,180,180,181,185,185,188,188,189,189,193,193,199,199,202,202,205,205,207,209,209,211,211,214,214,219,219,220,220,225,225,227,227,230,233,241,241,251,251,252,252,255,255,255,255,268,268,281,295,295,297,297,303,303,314,345,345,359,359,429}
[1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,1,0,1,0,1,1,0,1,0,1,0,1,0,0,0]
=> [1,0,1,1,0,1,1,0,0,1,0,0,1,1,0,0]
=> [6,6,4,2,2,1,1]
=> ? ∊ {27,27,34,34,35,35,36,36,37,40,40,43,43,43,43,45,45,46,46,46,46,46,47,47,47,47,48,48,48,48,50,50,50,50,50,50,51,51,51,51,52,52,52,52,53,53,53,54,54,55,55,55,55,55,55,55,55,56,56,61,61,61,61,62,62,62,62,62,62,62,62,63,63,65,65,65,65,66,66,66,66,66,66,67,67,67,67,69,70,70,70,70,71,71,72,72,75,75,75,75,75,75,75,76,76,76,76,78,78,79,79,79,79,80,80,81,81,81,81,82,82,83,83,84,85,85,87,87,89,89,89,89,90,90,91,91,94,94,95,95,95,95,95,95,96,96,96,96,97,97,97,97,99,99,99,99,103,103,104,104,105,105,105,105,106,106,107,107,107,108,108,109,109,110,110,112,112,113,113,114,114,115,115,116,116,117,117,118,118,118,121,121,123,123,123,123,125,125,126,127,127,129,129,132,132,132,134,136,136,136,136,137,137,139,139,140,140,142,142,145,145,145,145,145,145,145,145,146,146,151,151,151,151,156,156,156,157,157,157,157,158,158,161,161,162,163,163,165,165,171,171,171,171,173,173,174,174,179,179,180,180,181,185,185,188,188,189,189,193,193,199,199,202,202,205,205,207,209,209,211,211,214,214,219,219,220,220,225,225,227,227,230,233,241,241,251,251,252,252,255,255,255,255,268,268,281,295,295,297,297,303,303,314,345,345,359,359,429}
Description
The number of partitions of the same length below the given integer partition.
For a partition $\lambda_1 \geq \dots \lambda_k > 0$, this number is
$$ \det\left( \binom{\lambda_{k+1-i}}{j-i+1} \right)_{1 \le i,j \le k}.$$
Matching statistic: St000108
Mp00027: Dyck paths —to partition⟶ Integer partitions
St000108: Integer partitions ⟶ ℤResult quality: 24% ●values known / values provided: 42%●distinct values known / distinct values provided: 24%
St000108: Integer partitions ⟶ ℤResult quality: 24% ●values known / values provided: 42%●distinct values known / distinct values provided: 24%
Values
[1,0,1,0]
=> [1]
=> 2
[1,1,0,0]
=> []
=> 1
[1,0,1,0,1,0]
=> [2,1]
=> 5
[1,0,1,1,0,0]
=> [1,1]
=> 3
[1,1,0,0,1,0]
=> [2]
=> 3
[1,1,0,1,0,0]
=> [1]
=> 2
[1,1,1,0,0,0]
=> []
=> 1
[1,0,1,0,1,0,1,0]
=> [3,2,1]
=> 14
[1,0,1,0,1,1,0,0]
=> [2,2,1]
=> 9
[1,0,1,1,0,0,1,0]
=> [3,1,1]
=> 10
[1,0,1,1,0,1,0,0]
=> [2,1,1]
=> 7
[1,0,1,1,1,0,0,0]
=> [1,1,1]
=> 4
[1,1,0,0,1,0,1,0]
=> [3,2]
=> 9
[1,1,0,0,1,1,0,0]
=> [2,2]
=> 6
[1,1,0,1,0,0,1,0]
=> [3,1]
=> 7
[1,1,0,1,0,1,0,0]
=> [2,1]
=> 5
[1,1,0,1,1,0,0,0]
=> [1,1]
=> 3
[1,1,1,0,0,0,1,0]
=> [3]
=> 4
[1,1,1,0,0,1,0,0]
=> [2]
=> 3
[1,1,1,0,1,0,0,0]
=> [1]
=> 2
[1,1,1,1,0,0,0,0]
=> []
=> 1
[1,0,1,0,1,0,1,0,1,0]
=> [4,3,2,1]
=> 42
[1,0,1,0,1,0,1,1,0,0]
=> [3,3,2,1]
=> 28
[1,0,1,0,1,1,0,0,1,0]
=> [4,2,2,1]
=> 32
[1,0,1,0,1,1,0,1,0,0]
=> [3,2,2,1]
=> 23
[1,0,1,0,1,1,1,0,0,0]
=> [2,2,2,1]
=> 14
[1,0,1,1,0,0,1,0,1,0]
=> [4,3,1,1]
=> 32
[1,0,1,1,0,0,1,1,0,0]
=> [3,3,1,1]
=> 22
[1,0,1,1,0,1,0,0,1,0]
=> [4,2,1,1]
=> 26
[1,0,1,1,0,1,0,1,0,0]
=> [3,2,1,1]
=> 19
[1,0,1,1,0,1,1,0,0,0]
=> [2,2,1,1]
=> 12
[1,0,1,1,1,0,0,0,1,0]
=> [4,1,1,1]
=> 17
[1,0,1,1,1,0,0,1,0,0]
=> [3,1,1,1]
=> 13
[1,0,1,1,1,0,1,0,0,0]
=> [2,1,1,1]
=> 9
[1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1]
=> 5
[1,1,0,0,1,0,1,0,1,0]
=> [4,3,2]
=> 28
[1,1,0,0,1,0,1,1,0,0]
=> [3,3,2]
=> 19
[1,1,0,0,1,1,0,0,1,0]
=> [4,2,2]
=> 22
[1,1,0,0,1,1,0,1,0,0]
=> [3,2,2]
=> 16
[1,1,0,0,1,1,1,0,0,0]
=> [2,2,2]
=> 10
[1,1,0,1,0,0,1,0,1,0]
=> [4,3,1]
=> 23
[1,1,0,1,0,0,1,1,0,0]
=> [3,3,1]
=> 16
[1,1,0,1,0,1,0,0,1,0]
=> [4,2,1]
=> 19
[1,1,0,1,0,1,0,1,0,0]
=> [3,2,1]
=> 14
[1,1,0,1,0,1,1,0,0,0]
=> [2,2,1]
=> 9
[1,1,0,1,1,0,0,0,1,0]
=> [4,1,1]
=> 13
[1,1,0,1,1,0,0,1,0,0]
=> [3,1,1]
=> 10
[1,1,0,1,1,0,1,0,0,0]
=> [2,1,1]
=> 7
[1,1,0,1,1,1,0,0,0,0]
=> [1,1,1]
=> 4
[1,1,1,0,0,0,1,0,1,0]
=> [4,3]
=> 14
[1,0,1,0,1,0,1,0,1,0,1,0]
=> [5,4,3,2,1]
=> ? ∊ {34,34,40,40,43,43,45,45,46,47,47,48,48,48,48,53,53,53,54,54,56,56,62,62,62,62,62,63,63,66,66,72,72,75,75,75,76,76,84,89,89,90,90,104,104,107,132}
[1,0,1,0,1,0,1,0,1,1,0,0]
=> [4,4,3,2,1]
=> ? ∊ {34,34,40,40,43,43,45,45,46,47,47,48,48,48,48,53,53,53,54,54,56,56,62,62,62,62,62,63,63,66,66,72,72,75,75,75,76,76,84,89,89,90,90,104,104,107,132}
[1,0,1,0,1,0,1,1,0,0,1,0]
=> [5,3,3,2,1]
=> ? ∊ {34,34,40,40,43,43,45,45,46,47,47,48,48,48,48,53,53,53,54,54,56,56,62,62,62,62,62,63,63,66,66,72,72,75,75,75,76,76,84,89,89,90,90,104,104,107,132}
[1,0,1,0,1,0,1,1,0,1,0,0]
=> [4,3,3,2,1]
=> ? ∊ {34,34,40,40,43,43,45,45,46,47,47,48,48,48,48,53,53,53,54,54,56,56,62,62,62,62,62,63,63,66,66,72,72,75,75,75,76,76,84,89,89,90,90,104,104,107,132}
[1,0,1,0,1,0,1,1,1,0,0,0]
=> [3,3,3,2,1]
=> ? ∊ {34,34,40,40,43,43,45,45,46,47,47,48,48,48,48,53,53,53,54,54,56,56,62,62,62,62,62,63,63,66,66,72,72,75,75,75,76,76,84,89,89,90,90,104,104,107,132}
[1,0,1,0,1,1,0,0,1,0,1,0]
=> [5,4,2,2,1]
=> ? ∊ {34,34,40,40,43,43,45,45,46,47,47,48,48,48,48,53,53,53,54,54,56,56,62,62,62,62,62,63,63,66,66,72,72,75,75,75,76,76,84,89,89,90,90,104,104,107,132}
[1,0,1,0,1,1,0,0,1,1,0,0]
=> [4,4,2,2,1]
=> ? ∊ {34,34,40,40,43,43,45,45,46,47,47,48,48,48,48,53,53,53,54,54,56,56,62,62,62,62,62,63,63,66,66,72,72,75,75,75,76,76,84,89,89,90,90,104,104,107,132}
[1,0,1,0,1,1,0,1,0,0,1,0]
=> [5,3,2,2,1]
=> ? ∊ {34,34,40,40,43,43,45,45,46,47,47,48,48,48,48,53,53,53,54,54,56,56,62,62,62,62,62,63,63,66,66,72,72,75,75,75,76,76,84,89,89,90,90,104,104,107,132}
[1,0,1,0,1,1,0,1,0,1,0,0]
=> [4,3,2,2,1]
=> ? ∊ {34,34,40,40,43,43,45,45,46,47,47,48,48,48,48,53,53,53,54,54,56,56,62,62,62,62,62,63,63,66,66,72,72,75,75,75,76,76,84,89,89,90,90,104,104,107,132}
[1,0,1,0,1,1,0,1,1,0,0,0]
=> [3,3,2,2,1]
=> ? ∊ {34,34,40,40,43,43,45,45,46,47,47,48,48,48,48,53,53,53,54,54,56,56,62,62,62,62,62,63,63,66,66,72,72,75,75,75,76,76,84,89,89,90,90,104,104,107,132}
[1,0,1,0,1,1,1,0,0,0,1,0]
=> [5,2,2,2,1]
=> ? ∊ {34,34,40,40,43,43,45,45,46,47,47,48,48,48,48,53,53,53,54,54,56,56,62,62,62,62,62,63,63,66,66,72,72,75,75,75,76,76,84,89,89,90,90,104,104,107,132}
[1,0,1,0,1,1,1,0,0,1,0,0]
=> [4,2,2,2,1]
=> ? ∊ {34,34,40,40,43,43,45,45,46,47,47,48,48,48,48,53,53,53,54,54,56,56,62,62,62,62,62,63,63,66,66,72,72,75,75,75,76,76,84,89,89,90,90,104,104,107,132}
[1,0,1,1,0,0,1,0,1,0,1,0]
=> [5,4,3,1,1]
=> ? ∊ {34,34,40,40,43,43,45,45,46,47,47,48,48,48,48,53,53,53,54,54,56,56,62,62,62,62,62,63,63,66,66,72,72,75,75,75,76,76,84,89,89,90,90,104,104,107,132}
[1,0,1,1,0,0,1,0,1,1,0,0]
=> [4,4,3,1,1]
=> ? ∊ {34,34,40,40,43,43,45,45,46,47,47,48,48,48,48,53,53,53,54,54,56,56,62,62,62,62,62,63,63,66,66,72,72,75,75,75,76,76,84,89,89,90,90,104,104,107,132}
[1,0,1,1,0,0,1,1,0,0,1,0]
=> [5,3,3,1,1]
=> ? ∊ {34,34,40,40,43,43,45,45,46,47,47,48,48,48,48,53,53,53,54,54,56,56,62,62,62,62,62,63,63,66,66,72,72,75,75,75,76,76,84,89,89,90,90,104,104,107,132}
[1,0,1,1,0,0,1,1,0,1,0,0]
=> [4,3,3,1,1]
=> ? ∊ {34,34,40,40,43,43,45,45,46,47,47,48,48,48,48,53,53,53,54,54,56,56,62,62,62,62,62,63,63,66,66,72,72,75,75,75,76,76,84,89,89,90,90,104,104,107,132}
[1,0,1,1,0,0,1,1,1,0,0,0]
=> [3,3,3,1,1]
=> ? ∊ {34,34,40,40,43,43,45,45,46,47,47,48,48,48,48,53,53,53,54,54,56,56,62,62,62,62,62,63,63,66,66,72,72,75,75,75,76,76,84,89,89,90,90,104,104,107,132}
[1,0,1,1,0,1,0,0,1,0,1,0]
=> [5,4,2,1,1]
=> ? ∊ {34,34,40,40,43,43,45,45,46,47,47,48,48,48,48,53,53,53,54,54,56,56,62,62,62,62,62,63,63,66,66,72,72,75,75,75,76,76,84,89,89,90,90,104,104,107,132}
[1,0,1,1,0,1,0,0,1,1,0,0]
=> [4,4,2,1,1]
=> ? ∊ {34,34,40,40,43,43,45,45,46,47,47,48,48,48,48,53,53,53,54,54,56,56,62,62,62,62,62,63,63,66,66,72,72,75,75,75,76,76,84,89,89,90,90,104,104,107,132}
[1,0,1,1,0,1,0,1,0,0,1,0]
=> [5,3,2,1,1]
=> ? ∊ {34,34,40,40,43,43,45,45,46,47,47,48,48,48,48,53,53,53,54,54,56,56,62,62,62,62,62,63,63,66,66,72,72,75,75,75,76,76,84,89,89,90,90,104,104,107,132}
[1,0,1,1,0,1,0,1,0,1,0,0]
=> [4,3,2,1,1]
=> ? ∊ {34,34,40,40,43,43,45,45,46,47,47,48,48,48,48,53,53,53,54,54,56,56,62,62,62,62,62,63,63,66,66,72,72,75,75,75,76,76,84,89,89,90,90,104,104,107,132}
[1,0,1,1,0,1,1,0,0,0,1,0]
=> [5,2,2,1,1]
=> ? ∊ {34,34,40,40,43,43,45,45,46,47,47,48,48,48,48,53,53,53,54,54,56,56,62,62,62,62,62,63,63,66,66,72,72,75,75,75,76,76,84,89,89,90,90,104,104,107,132}
[1,0,1,1,1,0,0,0,1,0,1,0]
=> [5,4,1,1,1]
=> ? ∊ {34,34,40,40,43,43,45,45,46,47,47,48,48,48,48,53,53,53,54,54,56,56,62,62,62,62,62,63,63,66,66,72,72,75,75,75,76,76,84,89,89,90,90,104,104,107,132}
[1,0,1,1,1,0,0,0,1,1,0,0]
=> [4,4,1,1,1]
=> ? ∊ {34,34,40,40,43,43,45,45,46,47,47,48,48,48,48,53,53,53,54,54,56,56,62,62,62,62,62,63,63,66,66,72,72,75,75,75,76,76,84,89,89,90,90,104,104,107,132}
[1,0,1,1,1,0,0,1,0,0,1,0]
=> [5,3,1,1,1]
=> ? ∊ {34,34,40,40,43,43,45,45,46,47,47,48,48,48,48,53,53,53,54,54,56,56,62,62,62,62,62,63,63,66,66,72,72,75,75,75,76,76,84,89,89,90,90,104,104,107,132}
[1,1,0,0,1,0,1,0,1,0,1,0]
=> [5,4,3,2]
=> ? ∊ {34,34,40,40,43,43,45,45,46,47,47,48,48,48,48,53,53,53,54,54,56,56,62,62,62,62,62,63,63,66,66,72,72,75,75,75,76,76,84,89,89,90,90,104,104,107,132}
[1,1,0,0,1,0,1,0,1,1,0,0]
=> [4,4,3,2]
=> ? ∊ {34,34,40,40,43,43,45,45,46,47,47,48,48,48,48,53,53,53,54,54,56,56,62,62,62,62,62,63,63,66,66,72,72,75,75,75,76,76,84,89,89,90,90,104,104,107,132}
[1,1,0,0,1,0,1,1,0,0,1,0]
=> [5,3,3,2]
=> ? ∊ {34,34,40,40,43,43,45,45,46,47,47,48,48,48,48,53,53,53,54,54,56,56,62,62,62,62,62,63,63,66,66,72,72,75,75,75,76,76,84,89,89,90,90,104,104,107,132}
[1,1,0,0,1,0,1,1,0,1,0,0]
=> [4,3,3,2]
=> ? ∊ {34,34,40,40,43,43,45,45,46,47,47,48,48,48,48,53,53,53,54,54,56,56,62,62,62,62,62,63,63,66,66,72,72,75,75,75,76,76,84,89,89,90,90,104,104,107,132}
[1,1,0,0,1,0,1,1,1,0,0,0]
=> [3,3,3,2]
=> ? ∊ {34,34,40,40,43,43,45,45,46,47,47,48,48,48,48,53,53,53,54,54,56,56,62,62,62,62,62,63,63,66,66,72,72,75,75,75,76,76,84,89,89,90,90,104,104,107,132}
[1,1,0,0,1,1,0,0,1,0,1,0]
=> [5,4,2,2]
=> ? ∊ {34,34,40,40,43,43,45,45,46,47,47,48,48,48,48,53,53,53,54,54,56,56,62,62,62,62,62,63,63,66,66,72,72,75,75,75,76,76,84,89,89,90,90,104,104,107,132}
[1,1,0,0,1,1,0,0,1,1,0,0]
=> [4,4,2,2]
=> ? ∊ {34,34,40,40,43,43,45,45,46,47,47,48,48,48,48,53,53,53,54,54,56,56,62,62,62,62,62,63,63,66,66,72,72,75,75,75,76,76,84,89,89,90,90,104,104,107,132}
[1,1,0,0,1,1,0,1,0,0,1,0]
=> [5,3,2,2]
=> ? ∊ {34,34,40,40,43,43,45,45,46,47,47,48,48,48,48,53,53,53,54,54,56,56,62,62,62,62,62,63,63,66,66,72,72,75,75,75,76,76,84,89,89,90,90,104,104,107,132}
[1,1,0,0,1,1,0,1,0,1,0,0]
=> [4,3,2,2]
=> ? ∊ {34,34,40,40,43,43,45,45,46,47,47,48,48,48,48,53,53,53,54,54,56,56,62,62,62,62,62,63,63,66,66,72,72,75,75,75,76,76,84,89,89,90,90,104,104,107,132}
[1,1,0,0,1,1,1,0,0,0,1,0]
=> [5,2,2,2]
=> ? ∊ {34,34,40,40,43,43,45,45,46,47,47,48,48,48,48,53,53,53,54,54,56,56,62,62,62,62,62,63,63,66,66,72,72,75,75,75,76,76,84,89,89,90,90,104,104,107,132}
[1,1,0,1,0,0,1,0,1,0,1,0]
=> [5,4,3,1]
=> ? ∊ {34,34,40,40,43,43,45,45,46,47,47,48,48,48,48,53,53,53,54,54,56,56,62,62,62,62,62,63,63,66,66,72,72,75,75,75,76,76,84,89,89,90,90,104,104,107,132}
[1,1,0,1,0,0,1,0,1,1,0,0]
=> [4,4,3,1]
=> ? ∊ {34,34,40,40,43,43,45,45,46,47,47,48,48,48,48,53,53,53,54,54,56,56,62,62,62,62,62,63,63,66,66,72,72,75,75,75,76,76,84,89,89,90,90,104,104,107,132}
[1,1,0,1,0,0,1,1,0,0,1,0]
=> [5,3,3,1]
=> ? ∊ {34,34,40,40,43,43,45,45,46,47,47,48,48,48,48,53,53,53,54,54,56,56,62,62,62,62,62,63,63,66,66,72,72,75,75,75,76,76,84,89,89,90,90,104,104,107,132}
[1,1,0,1,0,0,1,1,0,1,0,0]
=> [4,3,3,1]
=> ? ∊ {34,34,40,40,43,43,45,45,46,47,47,48,48,48,48,53,53,53,54,54,56,56,62,62,62,62,62,63,63,66,66,72,72,75,75,75,76,76,84,89,89,90,90,104,104,107,132}
[1,1,0,1,0,1,0,0,1,0,1,0]
=> [5,4,2,1]
=> ? ∊ {34,34,40,40,43,43,45,45,46,47,47,48,48,48,48,53,53,53,54,54,56,56,62,62,62,62,62,63,63,66,66,72,72,75,75,75,76,76,84,89,89,90,90,104,104,107,132}
[1,1,0,1,0,1,0,0,1,1,0,0]
=> [4,4,2,1]
=> ? ∊ {34,34,40,40,43,43,45,45,46,47,47,48,48,48,48,53,53,53,54,54,56,56,62,62,62,62,62,63,63,66,66,72,72,75,75,75,76,76,84,89,89,90,90,104,104,107,132}
[1,1,0,1,0,1,0,1,0,0,1,0]
=> [5,3,2,1]
=> ? ∊ {34,34,40,40,43,43,45,45,46,47,47,48,48,48,48,53,53,53,54,54,56,56,62,62,62,62,62,63,63,66,66,72,72,75,75,75,76,76,84,89,89,90,90,104,104,107,132}
[1,1,0,1,1,0,0,0,1,0,1,0]
=> [5,4,1,1]
=> ? ∊ {34,34,40,40,43,43,45,45,46,47,47,48,48,48,48,53,53,53,54,54,56,56,62,62,62,62,62,63,63,66,66,72,72,75,75,75,76,76,84,89,89,90,90,104,104,107,132}
[1,1,1,0,0,0,1,0,1,0,1,0]
=> [5,4,3]
=> ? ∊ {34,34,40,40,43,43,45,45,46,47,47,48,48,48,48,53,53,53,54,54,56,56,62,62,62,62,62,63,63,66,66,72,72,75,75,75,76,76,84,89,89,90,90,104,104,107,132}
[1,1,1,0,0,0,1,0,1,1,0,0]
=> [4,4,3]
=> ? ∊ {34,34,40,40,43,43,45,45,46,47,47,48,48,48,48,53,53,53,54,54,56,56,62,62,62,62,62,63,63,66,66,72,72,75,75,75,76,76,84,89,89,90,90,104,104,107,132}
[1,1,1,0,0,0,1,1,0,0,1,0]
=> [5,3,3]
=> ? ∊ {34,34,40,40,43,43,45,45,46,47,47,48,48,48,48,53,53,53,54,54,56,56,62,62,62,62,62,63,63,66,66,72,72,75,75,75,76,76,84,89,89,90,90,104,104,107,132}
[1,1,1,0,0,1,0,0,1,0,1,0]
=> [5,4,2]
=> ? ∊ {34,34,40,40,43,43,45,45,46,47,47,48,48,48,48,53,53,53,54,54,56,56,62,62,62,62,62,63,63,66,66,72,72,75,75,75,76,76,84,89,89,90,90,104,104,107,132}
[1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [6,5,4,3,2,1]
=> ? ∊ {27,27,34,34,35,35,36,36,37,40,40,43,43,43,43,45,45,46,46,46,46,46,47,47,47,47,48,48,48,48,50,50,50,50,50,50,51,51,51,51,52,52,52,52,53,53,53,54,54,55,55,55,55,55,55,55,55,56,56,61,61,61,61,62,62,62,62,62,62,62,62,63,63,65,65,65,65,66,66,66,66,66,66,67,67,67,67,69,70,70,70,70,71,71,72,72,75,75,75,75,75,75,75,76,76,76,76,78,78,79,79,79,79,80,80,81,81,81,81,82,82,83,83,84,85,85,87,87,89,89,89,89,90,90,91,91,94,94,95,95,95,95,95,95,96,96,96,96,97,97,97,97,99,99,99,99,103,103,104,104,105,105,105,105,106,106,107,107,107,108,108,109,109,110,110,112,112,113,113,114,114,115,115,116,116,117,117,118,118,118,121,121,123,123,123,123,125,125,126,127,127,129,129,132,132,132,134,136,136,136,136,137,137,139,139,140,140,142,142,145,145,145,145,145,145,145,145,146,146,151,151,151,151,156,156,156,157,157,157,157,158,158,161,161,162,163,163,165,165,171,171,171,171,173,173,174,174,179,179,180,180,181,185,185,188,188,189,189,193,193,199,199,202,202,205,205,207,209,209,211,211,214,214,219,219,220,220,225,225,227,227,230,233,241,241,251,251,252,252,255,255,255,255,268,268,281,295,295,297,297,303,303,314,345,345,359,359,429}
[1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [5,5,4,3,2,1]
=> ? ∊ {27,27,34,34,35,35,36,36,37,40,40,43,43,43,43,45,45,46,46,46,46,46,47,47,47,47,48,48,48,48,50,50,50,50,50,50,51,51,51,51,52,52,52,52,53,53,53,54,54,55,55,55,55,55,55,55,55,56,56,61,61,61,61,62,62,62,62,62,62,62,62,63,63,65,65,65,65,66,66,66,66,66,66,67,67,67,67,69,70,70,70,70,71,71,72,72,75,75,75,75,75,75,75,76,76,76,76,78,78,79,79,79,79,80,80,81,81,81,81,82,82,83,83,84,85,85,87,87,89,89,89,89,90,90,91,91,94,94,95,95,95,95,95,95,96,96,96,96,97,97,97,97,99,99,99,99,103,103,104,104,105,105,105,105,106,106,107,107,107,108,108,109,109,110,110,112,112,113,113,114,114,115,115,116,116,117,117,118,118,118,121,121,123,123,123,123,125,125,126,127,127,129,129,132,132,132,134,136,136,136,136,137,137,139,139,140,140,142,142,145,145,145,145,145,145,145,145,146,146,151,151,151,151,156,156,156,157,157,157,157,158,158,161,161,162,163,163,165,165,171,171,171,171,173,173,174,174,179,179,180,180,181,185,185,188,188,189,189,193,193,199,199,202,202,205,205,207,209,209,211,211,214,214,219,219,220,220,225,225,227,227,230,233,241,241,251,251,252,252,255,255,255,255,268,268,281,295,295,297,297,303,303,314,345,345,359,359,429}
[1,0,1,0,1,0,1,0,1,1,0,0,1,0]
=> [6,4,4,3,2,1]
=> ? ∊ {27,27,34,34,35,35,36,36,37,40,40,43,43,43,43,45,45,46,46,46,46,46,47,47,47,47,48,48,48,48,50,50,50,50,50,50,51,51,51,51,52,52,52,52,53,53,53,54,54,55,55,55,55,55,55,55,55,56,56,61,61,61,61,62,62,62,62,62,62,62,62,63,63,65,65,65,65,66,66,66,66,66,66,67,67,67,67,69,70,70,70,70,71,71,72,72,75,75,75,75,75,75,75,76,76,76,76,78,78,79,79,79,79,80,80,81,81,81,81,82,82,83,83,84,85,85,87,87,89,89,89,89,90,90,91,91,94,94,95,95,95,95,95,95,96,96,96,96,97,97,97,97,99,99,99,99,103,103,104,104,105,105,105,105,106,106,107,107,107,108,108,109,109,110,110,112,112,113,113,114,114,115,115,116,116,117,117,118,118,118,121,121,123,123,123,123,125,125,126,127,127,129,129,132,132,132,134,136,136,136,136,137,137,139,139,140,140,142,142,145,145,145,145,145,145,145,145,146,146,151,151,151,151,156,156,156,157,157,157,157,158,158,161,161,162,163,163,165,165,171,171,171,171,173,173,174,174,179,179,180,180,181,185,185,188,188,189,189,193,193,199,199,202,202,205,205,207,209,209,211,211,214,214,219,219,220,220,225,225,227,227,230,233,241,241,251,251,252,252,255,255,255,255,268,268,281,295,295,297,297,303,303,314,345,345,359,359,429}
Description
The number of partitions contained in the given partition.
Matching statistic: St000070
Mp00027: Dyck paths —to partition⟶ Integer partitions
Mp00179: Integer partitions —to skew partition⟶ Skew partitions
Mp00185: Skew partitions —cell poset⟶ Posets
St000070: Posets ⟶ ℤResult quality: 19% ●values known / values provided: 33%●distinct values known / distinct values provided: 19%
Mp00179: Integer partitions —to skew partition⟶ Skew partitions
Mp00185: Skew partitions —cell poset⟶ Posets
St000070: Posets ⟶ ℤResult quality: 19% ●values known / values provided: 33%●distinct values known / distinct values provided: 19%
Values
[1,0,1,0]
=> [1]
=> [[1],[]]
=> ([],1)
=> 2
[1,1,0,0]
=> []
=> [[],[]]
=> ([],0)
=> ? = 1
[1,0,1,0,1,0]
=> [2,1]
=> [[2,1],[]]
=> ([(0,1),(0,2)],3)
=> 5
[1,0,1,1,0,0]
=> [1,1]
=> [[1,1],[]]
=> ([(0,1)],2)
=> 3
[1,1,0,0,1,0]
=> [2]
=> [[2],[]]
=> ([(0,1)],2)
=> 3
[1,1,0,1,0,0]
=> [1]
=> [[1],[]]
=> ([],1)
=> 2
[1,1,1,0,0,0]
=> []
=> [[],[]]
=> ([],0)
=> ? = 1
[1,0,1,0,1,0,1,0]
=> [3,2,1]
=> [[3,2,1],[]]
=> ([(0,3),(0,4),(3,2),(3,5),(4,1),(4,5)],6)
=> 14
[1,0,1,0,1,1,0,0]
=> [2,2,1]
=> [[2,2,1],[]]
=> ([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
=> 9
[1,0,1,1,0,0,1,0]
=> [3,1,1]
=> [[3,1,1],[]]
=> ([(0,3),(0,4),(3,2),(4,1)],5)
=> 10
[1,0,1,1,0,1,0,0]
=> [2,1,1]
=> [[2,1,1],[]]
=> ([(0,2),(0,3),(3,1)],4)
=> 7
[1,0,1,1,1,0,0,0]
=> [1,1,1]
=> [[1,1,1],[]]
=> ([(0,2),(2,1)],3)
=> 4
[1,1,0,0,1,0,1,0]
=> [3,2]
=> [[3,2],[]]
=> ([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
=> 9
[1,1,0,0,1,1,0,0]
=> [2,2]
=> [[2,2],[]]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 6
[1,1,0,1,0,0,1,0]
=> [3,1]
=> [[3,1],[]]
=> ([(0,2),(0,3),(3,1)],4)
=> 7
[1,1,0,1,0,1,0,0]
=> [2,1]
=> [[2,1],[]]
=> ([(0,1),(0,2)],3)
=> 5
[1,1,0,1,1,0,0,0]
=> [1,1]
=> [[1,1],[]]
=> ([(0,1)],2)
=> 3
[1,1,1,0,0,0,1,0]
=> [3]
=> [[3],[]]
=> ([(0,2),(2,1)],3)
=> 4
[1,1,1,0,0,1,0,0]
=> [2]
=> [[2],[]]
=> ([(0,1)],2)
=> 3
[1,1,1,0,1,0,0,0]
=> [1]
=> [[1],[]]
=> ([],1)
=> 2
[1,1,1,1,0,0,0,0]
=> []
=> [[],[]]
=> ([],0)
=> ? = 1
[1,0,1,0,1,0,1,0,1,0]
=> [4,3,2,1]
=> [[4,3,2,1],[]]
=> ([(0,5),(0,6),(3,2),(3,8),(4,1),(4,9),(5,3),(5,7),(6,4),(6,7),(7,8),(7,9)],10)
=> ? ∊ {1,42}
[1,0,1,0,1,0,1,1,0,0]
=> [3,3,2,1]
=> [[3,3,2,1],[]]
=> ([(0,4),(0,5),(2,7),(3,1),(3,8),(4,2),(4,6),(5,3),(5,6),(6,7),(6,8)],9)
=> 28
[1,0,1,0,1,1,0,0,1,0]
=> [4,2,2,1]
=> [[4,2,2,1],[]]
=> ([(0,5),(0,6),(3,1),(4,2),(4,8),(5,3),(5,7),(6,4),(6,7),(7,8)],9)
=> 32
[1,0,1,0,1,1,0,1,0,0]
=> [3,2,2,1]
=> [[3,2,2,1],[]]
=> ([(0,4),(0,5),(3,2),(3,7),(4,3),(4,6),(5,1),(5,6),(6,7)],8)
=> 23
[1,0,1,0,1,1,1,0,0,0]
=> [2,2,2,1]
=> [[2,2,2,1],[]]
=> ([(0,2),(0,4),(2,5),(3,1),(3,6),(4,3),(4,5),(5,6)],7)
=> 14
[1,0,1,1,0,0,1,0,1,0]
=> [4,3,1,1]
=> [[4,3,1,1],[]]
=> ([(0,5),(0,6),(3,1),(4,2),(4,8),(5,3),(5,7),(6,4),(6,7),(7,8)],9)
=> 32
[1,0,1,1,0,0,1,1,0,0]
=> [3,3,1,1]
=> [[3,3,1,1],[]]
=> ([(0,4),(0,5),(1,7),(3,2),(4,3),(4,6),(5,1),(5,6),(6,7)],8)
=> 22
[1,0,1,1,0,1,0,0,1,0]
=> [4,2,1,1]
=> [[4,2,1,1],[]]
=> ([(0,5),(0,6),(3,2),(4,1),(5,3),(5,7),(6,4),(6,7)],8)
=> 26
[1,0,1,1,0,1,0,1,0,0]
=> [3,2,1,1]
=> [[3,2,1,1],[]]
=> ([(0,4),(0,5),(3,2),(4,3),(4,6),(5,1),(5,6)],7)
=> 19
[1,0,1,1,0,1,1,0,0,0]
=> [2,2,1,1]
=> [[2,2,1,1],[]]
=> ([(0,2),(0,4),(2,5),(3,1),(4,3),(4,5)],6)
=> 12
[1,0,1,1,1,0,0,0,1,0]
=> [4,1,1,1]
=> [[4,1,1,1],[]]
=> ([(0,5),(0,6),(3,2),(4,1),(5,3),(6,4)],7)
=> 17
[1,0,1,1,1,0,0,1,0,0]
=> [3,1,1,1]
=> [[3,1,1,1],[]]
=> ([(0,4),(0,5),(3,2),(4,3),(5,1)],6)
=> 13
[1,0,1,1,1,0,1,0,0,0]
=> [2,1,1,1]
=> [[2,1,1,1],[]]
=> ([(0,2),(0,4),(3,1),(4,3)],5)
=> 9
[1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1]
=> [[1,1,1,1],[]]
=> ([(0,3),(2,1),(3,2)],4)
=> 5
[1,1,0,0,1,0,1,0,1,0]
=> [4,3,2]
=> [[4,3,2],[]]
=> ([(0,4),(0,5),(2,7),(3,1),(3,8),(4,2),(4,6),(5,3),(5,6),(6,7),(6,8)],9)
=> 28
[1,1,0,0,1,0,1,1,0,0]
=> [3,3,2]
=> [[3,3,2],[]]
=> ([(0,3),(0,4),(1,7),(2,6),(3,2),(3,5),(4,1),(4,5),(5,6),(5,7)],8)
=> 19
[1,1,0,0,1,1,0,0,1,0]
=> [4,2,2]
=> [[4,2,2],[]]
=> ([(0,4),(0,5),(1,7),(3,2),(4,3),(4,6),(5,1),(5,6),(6,7)],8)
=> 22
[1,1,0,0,1,1,0,1,0,0]
=> [3,2,2]
=> [[3,2,2],[]]
=> ([(0,3),(0,4),(2,6),(3,1),(3,5),(4,2),(4,5),(5,6)],7)
=> 16
[1,1,0,0,1,1,1,0,0,0]
=> [2,2,2]
=> [[2,2,2],[]]
=> ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 10
[1,1,0,1,0,0,1,0,1,0]
=> [4,3,1]
=> [[4,3,1],[]]
=> ([(0,4),(0,5),(3,2),(3,7),(4,3),(4,6),(5,1),(5,6),(6,7)],8)
=> 23
[1,1,0,1,0,0,1,1,0,0]
=> [3,3,1]
=> [[3,3,1],[]]
=> ([(0,3),(0,4),(2,6),(3,1),(3,5),(4,2),(4,5),(5,6)],7)
=> 16
[1,1,0,1,0,1,0,0,1,0]
=> [4,2,1]
=> [[4,2,1],[]]
=> ([(0,4),(0,5),(3,2),(4,3),(4,6),(5,1),(5,6)],7)
=> 19
[1,1,0,1,0,1,0,1,0,0]
=> [3,2,1]
=> [[3,2,1],[]]
=> ([(0,3),(0,4),(3,2),(3,5),(4,1),(4,5)],6)
=> 14
[1,1,0,1,0,1,1,0,0,0]
=> [2,2,1]
=> [[2,2,1],[]]
=> ([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
=> 9
[1,1,0,1,1,0,0,0,1,0]
=> [4,1,1]
=> [[4,1,1],[]]
=> ([(0,4),(0,5),(3,2),(4,3),(5,1)],6)
=> 13
[1,1,0,1,1,0,0,1,0,0]
=> [3,1,1]
=> [[3,1,1],[]]
=> ([(0,3),(0,4),(3,2),(4,1)],5)
=> 10
[1,1,0,1,1,0,1,0,0,0]
=> [2,1,1]
=> [[2,1,1],[]]
=> ([(0,2),(0,3),(3,1)],4)
=> 7
[1,1,0,1,1,1,0,0,0,0]
=> [1,1,1]
=> [[1,1,1],[]]
=> ([(0,2),(2,1)],3)
=> 4
[1,1,1,0,0,0,1,0,1,0]
=> [4,3]
=> [[4,3],[]]
=> ([(0,2),(0,4),(2,5),(3,1),(3,6),(4,3),(4,5),(5,6)],7)
=> 14
[1,1,1,0,0,0,1,1,0,0]
=> [3,3]
=> [[3,3],[]]
=> ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 10
[1,1,1,0,0,1,0,0,1,0]
=> [4,2]
=> [[4,2],[]]
=> ([(0,2),(0,4),(2,5),(3,1),(4,3),(4,5)],6)
=> 12
[1,1,1,0,0,1,0,1,0,0]
=> [3,2]
=> [[3,2],[]]
=> ([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
=> 9
[1,1,1,0,0,1,1,0,0,0]
=> [2,2]
=> [[2,2],[]]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 6
[1,1,1,1,1,0,0,0,0,0]
=> []
=> [[],[]]
=> ([],0)
=> ? ∊ {1,42}
[1,0,1,0,1,0,1,0,1,0,1,0]
=> [5,4,3,2,1]
=> [[5,4,3,2,1],[]]
=> ([(0,7),(0,8),(3,5),(3,12),(4,6),(4,13),(5,2),(5,10),(6,1),(6,11),(7,3),(7,9),(8,4),(8,9),(9,12),(9,13),(12,10),(12,14),(13,11),(13,14)],15)
=> ? ∊ {1,30,30,31,31,34,34,34,34,35,35,37,37,40,40,41,41,42,42,42,42,43,43,45,45,46,47,47,48,48,48,48,53,53,53,54,54,56,56,62,62,62,62,62,63,63,66,66,72,72,75,75,75,76,76,84,89,89,90,90,104,104,107,132}
[1,0,1,0,1,0,1,0,1,1,0,0]
=> [4,4,3,2,1]
=> [[4,4,3,2,1],[]]
=> ([(0,6),(0,7),(2,10),(3,5),(3,11),(4,2),(4,12),(5,1),(5,9),(6,3),(6,8),(7,4),(7,8),(8,11),(8,12),(11,9),(11,13),(12,10),(12,13)],14)
=> ? ∊ {1,30,30,31,31,34,34,34,34,35,35,37,37,40,40,41,41,42,42,42,42,43,43,45,45,46,47,47,48,48,48,48,53,53,53,54,54,56,56,62,62,62,62,62,63,63,66,66,72,72,75,75,75,76,76,84,89,89,90,90,104,104,107,132}
[1,0,1,0,1,0,1,1,0,0,1,0]
=> [5,3,3,2,1]
=> [[5,3,3,2,1],[]]
=> ([(0,7),(0,8),(3,2),(4,6),(4,12),(5,3),(5,11),(6,1),(6,10),(7,4),(7,9),(8,5),(8,9),(9,11),(9,12),(11,13),(12,10),(12,13)],14)
=> ? ∊ {1,30,30,31,31,34,34,34,34,35,35,37,37,40,40,41,41,42,42,42,42,43,43,45,45,46,47,47,48,48,48,48,53,53,53,54,54,56,56,62,62,62,62,62,63,63,66,66,72,72,75,75,75,76,76,84,89,89,90,90,104,104,107,132}
[1,0,1,0,1,0,1,1,0,1,0,0]
=> [4,3,3,2,1]
=> [[4,3,3,2,1],[]]
=> ([(0,6),(0,7),(3,4),(3,11),(4,1),(4,9),(5,2),(5,10),(6,5),(6,8),(7,3),(7,8),(8,10),(8,11),(10,12),(11,9),(11,12)],13)
=> ? ∊ {1,30,30,31,31,34,34,34,34,35,35,37,37,40,40,41,41,42,42,42,42,43,43,45,45,46,47,47,48,48,48,48,53,53,53,54,54,56,56,62,62,62,62,62,63,63,66,66,72,72,75,75,75,76,76,84,89,89,90,90,104,104,107,132}
[1,0,1,0,1,0,1,1,1,0,0,0]
=> [3,3,3,2,1]
=> [[3,3,3,2,1],[]]
=> ([(0,5),(0,6),(2,9),(3,4),(3,10),(4,1),(4,8),(5,3),(5,7),(6,2),(6,7),(7,9),(7,10),(9,11),(10,8),(10,11)],12)
=> ? ∊ {1,30,30,31,31,34,34,34,34,35,35,37,37,40,40,41,41,42,42,42,42,43,43,45,45,46,47,47,48,48,48,48,53,53,53,54,54,56,56,62,62,62,62,62,63,63,66,66,72,72,75,75,75,76,76,84,89,89,90,90,104,104,107,132}
[1,0,1,0,1,1,0,0,1,0,1,0]
=> [5,4,2,2,1]
=> [[5,4,2,2,1],[]]
=> ([(0,7),(0,8),(3,5),(3,12),(4,6),(4,13),(5,2),(5,10),(6,1),(6,11),(7,3),(7,9),(8,4),(8,9),(9,12),(9,13),(12,10),(13,11)],14)
=> ? ∊ {1,30,30,31,31,34,34,34,34,35,35,37,37,40,40,41,41,42,42,42,42,43,43,45,45,46,47,47,48,48,48,48,53,53,53,54,54,56,56,62,62,62,62,62,63,63,66,66,72,72,75,75,75,76,76,84,89,89,90,90,104,104,107,132}
[1,0,1,0,1,1,0,0,1,1,0,0]
=> [4,4,2,2,1]
=> [[4,4,2,2,1],[]]
=> ([(0,6),(0,7),(2,10),(3,5),(3,11),(4,2),(4,12),(5,1),(5,9),(6,3),(6,8),(7,4),(7,8),(8,11),(8,12),(11,9),(12,10)],13)
=> ? ∊ {1,30,30,31,31,34,34,34,34,35,35,37,37,40,40,41,41,42,42,42,42,43,43,45,45,46,47,47,48,48,48,48,53,53,53,54,54,56,56,62,62,62,62,62,63,63,66,66,72,72,75,75,75,76,76,84,89,89,90,90,104,104,107,132}
[1,0,1,0,1,1,0,1,0,0,1,0]
=> [5,3,2,2,1]
=> [[5,3,2,2,1],[]]
=> ([(0,7),(0,8),(3,2),(4,6),(4,12),(5,3),(5,11),(6,1),(6,10),(7,4),(7,9),(8,5),(8,9),(9,11),(9,12),(12,10)],13)
=> ? ∊ {1,30,30,31,31,34,34,34,34,35,35,37,37,40,40,41,41,42,42,42,42,43,43,45,45,46,47,47,48,48,48,48,53,53,53,54,54,56,56,62,62,62,62,62,63,63,66,66,72,72,75,75,75,76,76,84,89,89,90,90,104,104,107,132}
[1,0,1,0,1,1,0,1,0,1,0,0]
=> [4,3,2,2,1]
=> [[4,3,2,2,1],[]]
=> ([(0,6),(0,7),(3,4),(3,11),(4,1),(4,9),(5,2),(5,10),(6,5),(6,8),(7,3),(7,8),(8,10),(8,11),(11,9)],12)
=> ? ∊ {1,30,30,31,31,34,34,34,34,35,35,37,37,40,40,41,41,42,42,42,42,43,43,45,45,46,47,47,48,48,48,48,53,53,53,54,54,56,56,62,62,62,62,62,63,63,66,66,72,72,75,75,75,76,76,84,89,89,90,90,104,104,107,132}
[1,0,1,0,1,1,0,1,1,0,0,0]
=> [3,3,2,2,1]
=> [[3,3,2,2,1],[]]
=> ([(0,5),(0,6),(2,9),(3,4),(3,10),(4,1),(4,8),(5,3),(5,7),(6,2),(6,7),(7,9),(7,10),(10,8)],11)
=> ? ∊ {1,30,30,31,31,34,34,34,34,35,35,37,37,40,40,41,41,42,42,42,42,43,43,45,45,46,47,47,48,48,48,48,53,53,53,54,54,56,56,62,62,62,62,62,63,63,66,66,72,72,75,75,75,76,76,84,89,89,90,90,104,104,107,132}
[1,0,1,0,1,1,1,0,0,0,1,0]
=> [5,2,2,2,1]
=> [[5,2,2,2,1],[]]
=> ([(0,7),(0,8),(3,4),(4,2),(5,6),(5,11),(6,1),(6,10),(7,3),(7,9),(8,5),(8,9),(9,11),(11,10)],12)
=> ? ∊ {1,30,30,31,31,34,34,34,34,35,35,37,37,40,40,41,41,42,42,42,42,43,43,45,45,46,47,47,48,48,48,48,53,53,53,54,54,56,56,62,62,62,62,62,63,63,66,66,72,72,75,75,75,76,76,84,89,89,90,90,104,104,107,132}
[1,0,1,0,1,1,1,0,0,1,0,0]
=> [4,2,2,2,1]
=> [[4,2,2,2,1],[]]
=> ([(0,6),(0,7),(3,2),(4,5),(4,10),(5,1),(5,9),(6,4),(6,8),(7,3),(7,8),(8,10),(10,9)],11)
=> ? ∊ {1,30,30,31,31,34,34,34,34,35,35,37,37,40,40,41,41,42,42,42,42,43,43,45,45,46,47,47,48,48,48,48,53,53,53,54,54,56,56,62,62,62,62,62,63,63,66,66,72,72,75,75,75,76,76,84,89,89,90,90,104,104,107,132}
[1,0,1,0,1,1,1,0,1,0,0,0]
=> [3,2,2,2,1]
=> [[3,2,2,2,1],[]]
=> ([(0,5),(0,6),(3,4),(3,9),(4,2),(4,8),(5,3),(5,7),(6,1),(6,7),(7,9),(9,8)],10)
=> ? ∊ {1,30,30,31,31,34,34,34,34,35,35,37,37,40,40,41,41,42,42,42,42,43,43,45,45,46,47,47,48,48,48,48,53,53,53,54,54,56,56,62,62,62,62,62,63,63,66,66,72,72,75,75,75,76,76,84,89,89,90,90,104,104,107,132}
[1,0,1,1,0,0,1,0,1,0,1,0]
=> [5,4,3,1,1]
=> [[5,4,3,1,1],[]]
=> ([(0,7),(0,8),(3,2),(4,6),(4,12),(5,3),(5,11),(6,1),(6,10),(7,4),(7,9),(8,5),(8,9),(9,11),(9,12),(11,13),(12,10),(12,13)],14)
=> ? ∊ {1,30,30,31,31,34,34,34,34,35,35,37,37,40,40,41,41,42,42,42,42,43,43,45,45,46,47,47,48,48,48,48,53,53,53,54,54,56,56,62,62,62,62,62,63,63,66,66,72,72,75,75,75,76,76,84,89,89,90,90,104,104,107,132}
[1,0,1,1,0,0,1,0,1,1,0,0]
=> [4,4,3,1,1]
=> [[4,4,3,1,1],[]]
=> ([(0,6),(0,7),(2,9),(3,1),(4,3),(4,10),(5,2),(5,11),(6,4),(6,8),(7,5),(7,8),(8,10),(8,11),(10,12),(11,9),(11,12)],13)
=> ? ∊ {1,30,30,31,31,34,34,34,34,35,35,37,37,40,40,41,41,42,42,42,42,43,43,45,45,46,47,47,48,48,48,48,53,53,53,54,54,56,56,62,62,62,62,62,63,63,66,66,72,72,75,75,75,76,76,84,89,89,90,90,104,104,107,132}
[1,0,1,1,0,0,1,1,0,0,1,0]
=> [5,3,3,1,1]
=> [[5,3,3,1,1],[]]
=> ([(0,7),(0,8),(3,2),(4,1),(5,3),(5,10),(6,4),(6,11),(7,5),(7,9),(8,6),(8,9),(9,10),(9,11),(10,12),(11,12)],13)
=> ? ∊ {1,30,30,31,31,34,34,34,34,35,35,37,37,40,40,41,41,42,42,42,42,43,43,45,45,46,47,47,48,48,48,48,53,53,53,54,54,56,56,62,62,62,62,62,63,63,66,66,72,72,75,75,75,76,76,84,89,89,90,90,104,104,107,132}
[1,0,1,1,0,0,1,1,0,1,0,0]
=> [4,3,3,1,1]
=> [[4,3,3,1,1],[]]
=> ([(0,6),(0,7),(3,2),(4,3),(4,9),(5,1),(5,10),(6,4),(6,8),(7,5),(7,8),(8,9),(8,10),(9,11),(10,11)],12)
=> ? ∊ {1,30,30,31,31,34,34,34,34,35,35,37,37,40,40,41,41,42,42,42,42,43,43,45,45,46,47,47,48,48,48,48,53,53,53,54,54,56,56,62,62,62,62,62,63,63,66,66,72,72,75,75,75,76,76,84,89,89,90,90,104,104,107,132}
[1,0,1,1,0,0,1,1,1,0,0,0]
=> [3,3,3,1,1]
=> [[3,3,3,1,1],[]]
=> ([(0,5),(0,6),(2,9),(3,1),(4,3),(4,8),(5,4),(5,7),(6,2),(6,7),(7,8),(7,9),(8,10),(9,10)],11)
=> ? ∊ {1,30,30,31,31,34,34,34,34,35,35,37,37,40,40,41,41,42,42,42,42,43,43,45,45,46,47,47,48,48,48,48,53,53,53,54,54,56,56,62,62,62,62,62,63,63,66,66,72,72,75,75,75,76,76,84,89,89,90,90,104,104,107,132}
[1,0,1,1,0,1,0,0,1,0,1,0]
=> [5,4,2,1,1]
=> [[5,4,2,1,1],[]]
=> ([(0,7),(0,8),(3,2),(4,6),(4,12),(5,3),(5,11),(6,1),(6,10),(7,4),(7,9),(8,5),(8,9),(9,11),(9,12),(12,10)],13)
=> ? ∊ {1,30,30,31,31,34,34,34,34,35,35,37,37,40,40,41,41,42,42,42,42,43,43,45,45,46,47,47,48,48,48,48,53,53,53,54,54,56,56,62,62,62,62,62,63,63,66,66,72,72,75,75,75,76,76,84,89,89,90,90,104,104,107,132}
[1,0,1,1,0,1,0,0,1,1,0,0]
=> [4,4,2,1,1]
=> [[4,4,2,1,1],[]]
=> ([(0,6),(0,7),(2,9),(3,1),(4,3),(4,10),(5,2),(5,11),(6,4),(6,8),(7,5),(7,8),(8,10),(8,11),(11,9)],12)
=> ? ∊ {1,30,30,31,31,34,34,34,34,35,35,37,37,40,40,41,41,42,42,42,42,43,43,45,45,46,47,47,48,48,48,48,53,53,53,54,54,56,56,62,62,62,62,62,63,63,66,66,72,72,75,75,75,76,76,84,89,89,90,90,104,104,107,132}
[1,0,1,1,0,1,0,1,0,0,1,0]
=> [5,3,2,1,1]
=> [[5,3,2,1,1],[]]
=> ([(0,7),(0,8),(3,2),(4,1),(5,3),(5,10),(6,4),(6,11),(7,5),(7,9),(8,6),(8,9),(9,10),(9,11)],12)
=> ? ∊ {1,30,30,31,31,34,34,34,34,35,35,37,37,40,40,41,41,42,42,42,42,43,43,45,45,46,47,47,48,48,48,48,53,53,53,54,54,56,56,62,62,62,62,62,63,63,66,66,72,72,75,75,75,76,76,84,89,89,90,90,104,104,107,132}
[1,0,1,1,0,1,0,1,0,1,0,0]
=> [4,3,2,1,1]
=> [[4,3,2,1,1],[]]
=> ([(0,6),(0,7),(3,2),(4,3),(4,9),(5,1),(5,10),(6,4),(6,8),(7,5),(7,8),(8,9),(8,10)],11)
=> ? ∊ {1,30,30,31,31,34,34,34,34,35,35,37,37,40,40,41,41,42,42,42,42,43,43,45,45,46,47,47,48,48,48,48,53,53,53,54,54,56,56,62,62,62,62,62,63,63,66,66,72,72,75,75,75,76,76,84,89,89,90,90,104,104,107,132}
[1,0,1,1,0,1,0,1,1,0,0,0]
=> [3,3,2,1,1]
=> [[3,3,2,1,1],[]]
=> ([(0,5),(0,6),(2,9),(3,1),(4,3),(4,8),(5,4),(5,7),(6,2),(6,7),(7,8),(7,9)],10)
=> ? ∊ {1,30,30,31,31,34,34,34,34,35,35,37,37,40,40,41,41,42,42,42,42,43,43,45,45,46,47,47,48,48,48,48,53,53,53,54,54,56,56,62,62,62,62,62,63,63,66,66,72,72,75,75,75,76,76,84,89,89,90,90,104,104,107,132}
[1,0,1,1,0,1,1,0,0,0,1,0]
=> [5,2,2,1,1]
=> [[5,2,2,1,1],[]]
=> ([(0,7),(0,8),(3,5),(4,6),(4,10),(5,1),(6,2),(7,3),(7,9),(8,4),(8,9),(9,10)],11)
=> ? ∊ {1,30,30,31,31,34,34,34,34,35,35,37,37,40,40,41,41,42,42,42,42,43,43,45,45,46,47,47,48,48,48,48,53,53,53,54,54,56,56,62,62,62,62,62,63,63,66,66,72,72,75,75,75,76,76,84,89,89,90,90,104,104,107,132}
[1,0,1,1,0,1,1,0,0,1,0,0]
=> [4,2,2,1,1]
=> [[4,2,2,1,1],[]]
=> ([(0,6),(0,7),(3,4),(3,9),(4,1),(5,2),(6,5),(6,8),(7,3),(7,8),(8,9)],10)
=> ? ∊ {1,30,30,31,31,34,34,34,34,35,35,37,37,40,40,41,41,42,42,42,42,43,43,45,45,46,47,47,48,48,48,48,53,53,53,54,54,56,56,62,62,62,62,62,63,63,66,66,72,72,75,75,75,76,76,84,89,89,90,90,104,104,107,132}
[1,0,1,1,1,0,0,0,1,0,1,0]
=> [5,4,1,1,1]
=> [[5,4,1,1,1],[]]
=> ([(0,7),(0,8),(3,4),(4,2),(5,6),(5,11),(6,1),(6,10),(7,3),(7,9),(8,5),(8,9),(9,11),(11,10)],12)
=> ? ∊ {1,30,30,31,31,34,34,34,34,35,35,37,37,40,40,41,41,42,42,42,42,43,43,45,45,46,47,47,48,48,48,48,53,53,53,54,54,56,56,62,62,62,62,62,63,63,66,66,72,72,75,75,75,76,76,84,89,89,90,90,104,104,107,132}
[1,0,1,1,1,0,0,0,1,1,0,0]
=> [4,4,1,1,1]
=> [[4,4,1,1,1],[]]
=> ([(0,6),(0,7),(2,9),(3,4),(4,1),(5,2),(5,10),(6,3),(6,8),(7,5),(7,8),(8,10),(10,9)],11)
=> ? ∊ {1,30,30,31,31,34,34,34,34,35,35,37,37,40,40,41,41,42,42,42,42,43,43,45,45,46,47,47,48,48,48,48,53,53,53,54,54,56,56,62,62,62,62,62,63,63,66,66,72,72,75,75,75,76,76,84,89,89,90,90,104,104,107,132}
[1,0,1,1,1,0,0,1,0,0,1,0]
=> [5,3,1,1,1]
=> [[5,3,1,1,1],[]]
=> ([(0,7),(0,8),(3,5),(4,6),(4,10),(5,1),(6,2),(7,3),(7,9),(8,4),(8,9),(9,10)],11)
=> ? ∊ {1,30,30,31,31,34,34,34,34,35,35,37,37,40,40,41,41,42,42,42,42,43,43,45,45,46,47,47,48,48,48,48,53,53,53,54,54,56,56,62,62,62,62,62,63,63,66,66,72,72,75,75,75,76,76,84,89,89,90,90,104,104,107,132}
[1,0,1,1,1,0,0,1,0,1,0,0]
=> [4,3,1,1,1]
=> [[4,3,1,1,1],[]]
=> ([(0,6),(0,7),(3,4),(4,1),(5,2),(5,9),(6,5),(6,8),(7,3),(7,8),(8,9)],10)
=> ? ∊ {1,30,30,31,31,34,34,34,34,35,35,37,37,40,40,41,41,42,42,42,42,43,43,45,45,46,47,47,48,48,48,48,53,53,53,54,54,56,56,62,62,62,62,62,63,63,66,66,72,72,75,75,75,76,76,84,89,89,90,90,104,104,107,132}
[1,0,1,1,1,0,1,0,0,0,1,0]
=> [5,2,1,1,1]
=> [[5,2,1,1,1],[]]
=> ([(0,7),(0,8),(3,5),(4,6),(5,2),(6,1),(7,3),(7,9),(8,4),(8,9)],10)
=> ? ∊ {1,30,30,31,31,34,34,34,34,35,35,37,37,40,40,41,41,42,42,42,42,43,43,45,45,46,47,47,48,48,48,48,53,53,53,54,54,56,56,62,62,62,62,62,63,63,66,66,72,72,75,75,75,76,76,84,89,89,90,90,104,104,107,132}
[1,1,0,0,1,0,1,0,1,0,1,0]
=> [5,4,3,2]
=> [[5,4,3,2],[]]
=> ([(0,6),(0,7),(2,10),(3,5),(3,11),(4,2),(4,12),(5,1),(5,9),(6,3),(6,8),(7,4),(7,8),(8,11),(8,12),(11,9),(11,13),(12,10),(12,13)],14)
=> ? ∊ {1,30,30,31,31,34,34,34,34,35,35,37,37,40,40,41,41,42,42,42,42,43,43,45,45,46,47,47,48,48,48,48,53,53,53,54,54,56,56,62,62,62,62,62,63,63,66,66,72,72,75,75,75,76,76,84,89,89,90,90,104,104,107,132}
[1,1,0,0,1,0,1,0,1,1,0,0]
=> [4,4,3,2]
=> [[4,4,3,2],[]]
=> ([(0,5),(0,6),(1,9),(2,8),(3,2),(3,10),(4,1),(4,11),(5,3),(5,7),(6,4),(6,7),(7,10),(7,11),(10,8),(10,12),(11,9),(11,12)],13)
=> ? ∊ {1,30,30,31,31,34,34,34,34,35,35,37,37,40,40,41,41,42,42,42,42,43,43,45,45,46,47,47,48,48,48,48,53,53,53,54,54,56,56,62,62,62,62,62,63,63,66,66,72,72,75,75,75,76,76,84,89,89,90,90,104,104,107,132}
[1,1,0,0,1,0,1,1,0,0,1,0]
=> [5,3,3,2]
=> [[5,3,3,2],[]]
=> ([(0,6),(0,7),(2,9),(3,1),(4,3),(4,10),(5,2),(5,11),(6,4),(6,8),(7,5),(7,8),(8,10),(8,11),(10,12),(11,9),(11,12)],13)
=> ? ∊ {1,30,30,31,31,34,34,34,34,35,35,37,37,40,40,41,41,42,42,42,42,43,43,45,45,46,47,47,48,48,48,48,53,53,53,54,54,56,56,62,62,62,62,62,63,63,66,66,72,72,75,75,75,76,76,84,89,89,90,90,104,104,107,132}
[1,1,0,0,1,0,1,1,0,1,0,0]
=> [4,3,3,2]
=> [[4,3,3,2],[]]
=> ([(0,5),(0,6),(2,8),(3,2),(3,10),(4,1),(4,9),(5,3),(5,7),(6,4),(6,7),(7,9),(7,10),(9,11),(10,8),(10,11)],12)
=> ? ∊ {1,30,30,31,31,34,34,34,34,35,35,37,37,40,40,41,41,42,42,42,42,43,43,45,45,46,47,47,48,48,48,48,53,53,53,54,54,56,56,62,62,62,62,62,63,63,66,66,72,72,75,75,75,76,76,84,89,89,90,90,104,104,107,132}
[1,1,0,0,1,0,1,1,1,0,0,0]
=> [3,3,3,2]
=> [[3,3,3,2],[]]
=> ([(0,4),(0,5),(1,8),(2,7),(3,2),(3,9),(4,3),(4,6),(5,1),(5,6),(6,8),(6,9),(8,10),(9,7),(9,10)],11)
=> ? ∊ {1,30,30,31,31,34,34,34,34,35,35,37,37,40,40,41,41,42,42,42,42,43,43,45,45,46,47,47,48,48,48,48,53,53,53,54,54,56,56,62,62,62,62,62,63,63,66,66,72,72,75,75,75,76,76,84,89,89,90,90,104,104,107,132}
[1,1,0,0,1,1,0,0,1,0,1,0]
=> [5,4,2,2]
=> [[5,4,2,2],[]]
=> ([(0,6),(0,7),(2,10),(3,5),(3,11),(4,2),(4,12),(5,1),(5,9),(6,3),(6,8),(7,4),(7,8),(8,11),(8,12),(11,9),(12,10)],13)
=> ? ∊ {1,30,30,31,31,34,34,34,34,35,35,37,37,40,40,41,41,42,42,42,42,43,43,45,45,46,47,47,48,48,48,48,53,53,53,54,54,56,56,62,62,62,62,62,63,63,66,66,72,72,75,75,75,76,76,84,89,89,90,90,104,104,107,132}
[1,1,0,0,1,1,0,0,1,1,0,0]
=> [4,4,2,2]
=> [[4,4,2,2],[]]
=> ([(0,5),(0,6),(1,9),(2,8),(3,2),(3,10),(4,1),(4,11),(5,3),(5,7),(6,4),(6,7),(7,10),(7,11),(10,8),(11,9)],12)
=> ? ∊ {1,30,30,31,31,34,34,34,34,35,35,37,37,40,40,41,41,42,42,42,42,43,43,45,45,46,47,47,48,48,48,48,53,53,53,54,54,56,56,62,62,62,62,62,63,63,66,66,72,72,75,75,75,76,76,84,89,89,90,90,104,104,107,132}
[1,1,0,0,1,1,0,1,0,0,1,0]
=> [5,3,2,2]
=> [[5,3,2,2],[]]
=> ([(0,6),(0,7),(2,9),(3,1),(4,3),(4,10),(5,2),(5,11),(6,4),(6,8),(7,5),(7,8),(8,10),(8,11),(11,9)],12)
=> ? ∊ {1,30,30,31,31,34,34,34,34,35,35,37,37,40,40,41,41,42,42,42,42,43,43,45,45,46,47,47,48,48,48,48,53,53,53,54,54,56,56,62,62,62,62,62,63,63,66,66,72,72,75,75,75,76,76,84,89,89,90,90,104,104,107,132}
[1,1,0,0,1,1,0,1,0,1,0,0]
=> [4,3,2,2]
=> [[4,3,2,2],[]]
=> ([(0,5),(0,6),(2,8),(3,2),(3,10),(4,1),(4,9),(5,3),(5,7),(6,4),(6,7),(7,9),(7,10),(10,8)],11)
=> ? ∊ {1,30,30,31,31,34,34,34,34,35,35,37,37,40,40,41,41,42,42,42,42,43,43,45,45,46,47,47,48,48,48,48,53,53,53,54,54,56,56,62,62,62,62,62,63,63,66,66,72,72,75,75,75,76,76,84,89,89,90,90,104,104,107,132}
[1,1,0,0,1,1,0,1,1,0,0,0]
=> [3,3,2,2]
=> [[3,3,2,2],[]]
=> ([(0,4),(0,5),(1,8),(2,7),(3,2),(3,9),(4,3),(4,6),(5,1),(5,6),(6,8),(6,9),(9,7)],10)
=> ? ∊ {1,30,30,31,31,34,34,34,34,35,35,37,37,40,40,41,41,42,42,42,42,43,43,45,45,46,47,47,48,48,48,48,53,53,53,54,54,56,56,62,62,62,62,62,63,63,66,66,72,72,75,75,75,76,76,84,89,89,90,90,104,104,107,132}
[1,1,0,0,1,1,1,0,0,0,1,0]
=> [5,2,2,2]
=> [[5,2,2,2],[]]
=> ([(0,6),(0,7),(2,9),(3,4),(4,1),(5,2),(5,10),(6,3),(6,8),(7,5),(7,8),(8,10),(10,9)],11)
=> ? ∊ {1,30,30,31,31,34,34,34,34,35,35,37,37,40,40,41,41,42,42,42,42,43,43,45,45,46,47,47,48,48,48,48,53,53,53,54,54,56,56,62,62,62,62,62,63,63,66,66,72,72,75,75,75,76,76,84,89,89,90,90,104,104,107,132}
[1,1,0,0,1,1,1,0,0,1,0,0]
=> [4,2,2,2]
=> [[4,2,2,2],[]]
=> ([(0,5),(0,6),(2,8),(3,1),(4,2),(4,9),(5,3),(5,7),(6,4),(6,7),(7,9),(9,8)],10)
=> ? ∊ {1,30,30,31,31,34,34,34,34,35,35,37,37,40,40,41,41,42,42,42,42,43,43,45,45,46,47,47,48,48,48,48,53,53,53,54,54,56,56,62,62,62,62,62,63,63,66,66,72,72,75,75,75,76,76,84,89,89,90,90,104,104,107,132}
[1,1,0,1,0,0,1,0,1,0,1,0]
=> [5,4,3,1]
=> [[5,4,3,1],[]]
=> ([(0,6),(0,7),(3,4),(3,11),(4,1),(4,9),(5,2),(5,10),(6,5),(6,8),(7,3),(7,8),(8,10),(8,11),(10,12),(11,9),(11,12)],13)
=> ? ∊ {1,30,30,31,31,34,34,34,34,35,35,37,37,40,40,41,41,42,42,42,42,43,43,45,45,46,47,47,48,48,48,48,53,53,53,54,54,56,56,62,62,62,62,62,63,63,66,66,72,72,75,75,75,76,76,84,89,89,90,90,104,104,107,132}
[1,1,0,1,0,0,1,0,1,1,0,0]
=> [4,4,3,1]
=> [[4,4,3,1],[]]
=> ([(0,5),(0,6),(2,8),(3,2),(3,10),(4,1),(4,9),(5,3),(5,7),(6,4),(6,7),(7,9),(7,10),(9,11),(10,8),(10,11)],12)
=> ? ∊ {1,30,30,31,31,34,34,34,34,35,35,37,37,40,40,41,41,42,42,42,42,43,43,45,45,46,47,47,48,48,48,48,53,53,53,54,54,56,56,62,62,62,62,62,63,63,66,66,72,72,75,75,75,76,76,84,89,89,90,90,104,104,107,132}
[1,1,0,1,0,0,1,1,0,0,1,0]
=> [5,3,3,1]
=> [[5,3,3,1],[]]
=> ([(0,6),(0,7),(3,2),(4,3),(4,9),(5,1),(5,10),(6,4),(6,8),(7,5),(7,8),(8,9),(8,10),(9,11),(10,11)],12)
=> ? ∊ {1,30,30,31,31,34,34,34,34,35,35,37,37,40,40,41,41,42,42,42,42,43,43,45,45,46,47,47,48,48,48,48,53,53,53,54,54,56,56,62,62,62,62,62,63,63,66,66,72,72,75,75,75,76,76,84,89,89,90,90,104,104,107,132}
Description
The number of antichains in a poset.
An antichain in a poset $P$ is a subset of elements of $P$ which are pairwise incomparable.
An order ideal is a subset $I$ of $P$ such that $a\in I$ and $b \leq_P a$ implies $b \in I$. Since there is a one-to-one correspondence between antichains and order ideals, this statistic is also the number of order ideals in a poset.
Matching statistic: St001232
(load all 6 compositions to match this statistic)
(load all 6 compositions to match this statistic)
Mp00099: Dyck paths —bounce path⟶ Dyck paths
Mp00101: Dyck paths —decomposition reverse⟶ Dyck paths
St001232: Dyck paths ⟶ ℤResult quality: 5% ●values known / values provided: 7%●distinct values known / distinct values provided: 5%
Mp00101: Dyck paths —decomposition reverse⟶ Dyck paths
St001232: Dyck paths ⟶ ℤResult quality: 5% ●values known / values provided: 7%●distinct values known / distinct values provided: 5%
Values
[1,0,1,0]
=> [1,0,1,0]
=> [1,1,0,0]
=> 0 = 1 - 1
[1,1,0,0]
=> [1,1,0,0]
=> [1,0,1,0]
=> 1 = 2 - 1
[1,0,1,0,1,0]
=> [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> 0 = 1 - 1
[1,0,1,1,0,0]
=> [1,0,1,1,0,0]
=> [1,1,0,1,0,0]
=> 2 = 3 - 1
[1,1,0,0,1,0]
=> [1,1,0,0,1,0]
=> [1,1,0,0,1,0]
=> 1 = 2 - 1
[1,1,0,1,0,0]
=> [1,0,1,1,0,0]
=> [1,1,0,1,0,0]
=> 2 = 3 - 1
[1,1,1,0,0,0]
=> [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> ? = 5 - 1
[1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> 0 = 1 - 1
[1,0,1,0,1,1,0,0]
=> [1,0,1,0,1,1,0,0]
=> [1,1,1,0,1,0,0,0]
=> 3 = 4 - 1
[1,0,1,1,0,0,1,0]
=> [1,0,1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> 2 = 3 - 1
[1,0,1,1,0,1,0,0]
=> [1,0,1,0,1,1,0,0]
=> [1,1,1,0,1,0,0,0]
=> 3 = 4 - 1
[1,0,1,1,1,0,0,0]
=> [1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,0,0]
=> ? ∊ {5,6,7,7,9,9,10,14} - 1
[1,1,0,0,1,0,1,0]
=> [1,1,0,0,1,0,1,0]
=> [1,1,1,0,0,0,1,0]
=> 1 = 2 - 1
[1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> ? ∊ {5,6,7,7,9,9,10,14} - 1
[1,1,0,1,0,0,1,0]
=> [1,0,1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> 2 = 3 - 1
[1,1,0,1,0,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> ? ∊ {5,6,7,7,9,9,10,14} - 1
[1,1,0,1,1,0,0,0]
=> [1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,0,0]
=> ? ∊ {5,6,7,7,9,9,10,14} - 1
[1,1,1,0,0,0,1,0]
=> [1,1,1,0,0,0,1,0]
=> [1,1,0,0,1,0,1,0]
=> ? ∊ {5,6,7,7,9,9,10,14} - 1
[1,1,1,0,0,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> ? ∊ {5,6,7,7,9,9,10,14} - 1
[1,1,1,0,1,0,0,0]
=> [1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,0,0]
=> ? ∊ {5,6,7,7,9,9,10,14} - 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]
=> ? ∊ {5,6,7,7,9,9,10,14} - 1
[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]
=> 0 = 1 - 1
[1,0,1,0,1,0,1,1,0,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> 4 = 5 - 1
[1,0,1,0,1,1,0,0,1,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> 3 = 4 - 1
[1,0,1,0,1,1,0,1,0,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> 4 = 5 - 1
[1,0,1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> ? ∊ {5,6,7,7,9,9,9,9,10,10,10,12,12,13,13,14,14,14,16,16,17,19,19,19,22,22,23,23,26,28,28,32,32,42} - 1
[1,0,1,1,0,0,1,0,1,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> 2 = 3 - 1
[1,0,1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> ? ∊ {5,6,7,7,9,9,9,9,10,10,10,12,12,13,13,14,14,14,16,16,17,19,19,19,22,22,23,23,26,28,28,32,32,42} - 1
[1,0,1,1,0,1,0,0,1,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> 3 = 4 - 1
[1,0,1,1,0,1,0,1,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> ? ∊ {5,6,7,7,9,9,9,9,10,10,10,12,12,13,13,14,14,14,16,16,17,19,19,19,22,22,23,23,26,28,28,32,32,42} - 1
[1,0,1,1,0,1,1,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> ? ∊ {5,6,7,7,9,9,9,9,10,10,10,12,12,13,13,14,14,14,16,16,17,19,19,19,22,22,23,23,26,28,28,32,32,42} - 1
[1,0,1,1,1,0,0,0,1,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> ? ∊ {5,6,7,7,9,9,9,9,10,10,10,12,12,13,13,14,14,14,16,16,17,19,19,19,22,22,23,23,26,28,28,32,32,42} - 1
[1,0,1,1,1,0,0,1,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> ? ∊ {5,6,7,7,9,9,9,9,10,10,10,12,12,13,13,14,14,14,16,16,17,19,19,19,22,22,23,23,26,28,28,32,32,42} - 1
[1,0,1,1,1,0,1,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> ? ∊ {5,6,7,7,9,9,9,9,10,10,10,12,12,13,13,14,14,14,16,16,17,19,19,19,22,22,23,23,26,28,28,32,32,42} - 1
[1,0,1,1,1,1,0,0,0,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> ? ∊ {5,6,7,7,9,9,9,9,10,10,10,12,12,13,13,14,14,14,16,16,17,19,19,19,22,22,23,23,26,28,28,32,32,42} - 1
[1,1,0,0,1,0,1,0,1,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> 1 = 2 - 1
[1,1,0,0,1,0,1,1,0,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> [1,1,1,0,1,0,0,0,1,0]
=> ? ∊ {5,6,7,7,9,9,9,9,10,10,10,12,12,13,13,14,14,14,16,16,17,19,19,19,22,22,23,23,26,28,28,32,32,42} - 1
[1,1,0,0,1,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0,1,0]
=> ? ∊ {5,6,7,7,9,9,9,9,10,10,10,12,12,13,13,14,14,14,16,16,17,19,19,19,22,22,23,23,26,28,28,32,32,42} - 1
[1,1,0,0,1,1,0,1,0,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> [1,1,1,0,1,0,0,0,1,0]
=> ? ∊ {5,6,7,7,9,9,9,9,10,10,10,12,12,13,13,14,14,14,16,16,17,19,19,19,22,22,23,23,26,28,28,32,32,42} - 1
[1,1,0,0,1,1,1,0,0,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,0,0,1,0]
=> ? ∊ {5,6,7,7,9,9,9,9,10,10,10,12,12,13,13,14,14,14,16,16,17,19,19,19,22,22,23,23,26,28,28,32,32,42} - 1
[1,1,0,1,0,0,1,0,1,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> 2 = 3 - 1
[1,1,0,1,0,0,1,1,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> ? ∊ {5,6,7,7,9,9,9,9,10,10,10,12,12,13,13,14,14,14,16,16,17,19,19,19,22,22,23,23,26,28,28,32,32,42} - 1
[1,1,0,1,0,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0,1,0]
=> ? ∊ {5,6,7,7,9,9,9,9,10,10,10,12,12,13,13,14,14,14,16,16,17,19,19,19,22,22,23,23,26,28,28,32,32,42} - 1
[1,1,0,1,0,1,0,1,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> ? ∊ {5,6,7,7,9,9,9,9,10,10,10,12,12,13,13,14,14,14,16,16,17,19,19,19,22,22,23,23,26,28,28,32,32,42} - 1
[1,1,0,1,0,1,1,0,0,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,0,0,1,0]
=> ? ∊ {5,6,7,7,9,9,9,9,10,10,10,12,12,13,13,14,14,14,16,16,17,19,19,19,22,22,23,23,26,28,28,32,32,42} - 1
[1,1,0,1,1,0,0,0,1,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> ? ∊ {5,6,7,7,9,9,9,9,10,10,10,12,12,13,13,14,14,14,16,16,17,19,19,19,22,22,23,23,26,28,28,32,32,42} - 1
[1,1,0,1,1,0,0,1,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> ? ∊ {5,6,7,7,9,9,9,9,10,10,10,12,12,13,13,14,14,14,16,16,17,19,19,19,22,22,23,23,26,28,28,32,32,42} - 1
[1,1,0,1,1,0,1,0,0,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,0,0,1,0]
=> ? ∊ {5,6,7,7,9,9,9,9,10,10,10,12,12,13,13,14,14,14,16,16,17,19,19,19,22,22,23,23,26,28,28,32,32,42} - 1
[1,1,0,1,1,1,0,0,0,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> ? ∊ {5,6,7,7,9,9,9,9,10,10,10,12,12,13,13,14,14,14,16,16,17,19,19,19,22,22,23,23,26,28,28,32,32,42} - 1
[1,1,1,0,0,0,1,0,1,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> ? ∊ {5,6,7,7,9,9,9,9,10,10,10,12,12,13,13,14,14,14,16,16,17,19,19,19,22,22,23,23,26,28,28,32,32,42} - 1
[1,1,1,0,0,0,1,1,0,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> [1,1,0,1,0,0,1,0,1,0]
=> ? ∊ {5,6,7,7,9,9,9,9,10,10,10,12,12,13,13,14,14,14,16,16,17,19,19,19,22,22,23,23,26,28,28,32,32,42} - 1
[1,1,1,0,0,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0,1,0]
=> ? ∊ {5,6,7,7,9,9,9,9,10,10,10,12,12,13,13,14,14,14,16,16,17,19,19,19,22,22,23,23,26,28,28,32,32,42} - 1
[1,1,1,0,0,1,0,1,0,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> [1,1,0,1,0,0,1,0,1,0]
=> ? ∊ {5,6,7,7,9,9,9,9,10,10,10,12,12,13,13,14,14,14,16,16,17,19,19,19,22,22,23,23,26,28,28,32,32,42} - 1
[1,1,1,0,0,1,1,0,0,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,0,0,1,0]
=> ? ∊ {5,6,7,7,9,9,9,9,10,10,10,12,12,13,13,14,14,14,16,16,17,19,19,19,22,22,23,23,26,28,28,32,32,42} - 1
[1,1,1,0,1,0,0,0,1,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> ? ∊ {5,6,7,7,9,9,9,9,10,10,10,12,12,13,13,14,14,14,16,16,17,19,19,19,22,22,23,23,26,28,28,32,32,42} - 1
[1,1,1,0,1,0,0,1,0,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> [1,1,0,1,0,0,1,0,1,0]
=> ? ∊ {5,6,7,7,9,9,9,9,10,10,10,12,12,13,13,14,14,14,16,16,17,19,19,19,22,22,23,23,26,28,28,32,32,42} - 1
[1,1,1,0,1,0,1,0,0,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,0,0,1,0]
=> ? ∊ {5,6,7,7,9,9,9,9,10,10,10,12,12,13,13,14,14,14,16,16,17,19,19,19,22,22,23,23,26,28,28,32,32,42} - 1
[1,1,1,0,1,1,0,0,0,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> ? ∊ {5,6,7,7,9,9,9,9,10,10,10,12,12,13,13,14,14,14,16,16,17,19,19,19,22,22,23,23,26,28,28,32,32,42} - 1
[1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> ? ∊ {5,6,7,7,9,9,9,9,10,10,10,12,12,13,13,14,14,14,16,16,17,19,19,19,22,22,23,23,26,28,28,32,32,42} - 1
[1,1,1,1,0,0,0,1,0,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> [1,1,0,1,0,0,1,0,1,0]
=> ? ∊ {5,6,7,7,9,9,9,9,10,10,10,12,12,13,13,14,14,14,16,16,17,19,19,19,22,22,23,23,26,28,28,32,32,42} - 1
[1,1,1,1,0,0,1,0,0,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,0,0,1,0]
=> ? ∊ {5,6,7,7,9,9,9,9,10,10,10,12,12,13,13,14,14,14,16,16,17,19,19,19,22,22,23,23,26,28,28,32,32,42} - 1
[1,1,1,1,0,1,0,0,0,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> ? ∊ {5,6,7,7,9,9,9,9,10,10,10,12,12,13,13,14,14,14,16,16,17,19,19,19,22,22,23,23,26,28,28,32,32,42} - 1
[1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> ? ∊ {5,6,7,7,9,9,9,9,10,10,10,12,12,13,13,14,14,14,16,16,17,19,19,19,22,22,23,23,26,28,28,32,32,42} - 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]
=> [1,1,1,1,1,1,0,0,0,0,0,0]
=> 0 = 1 - 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]
=> [1,1,1,1,1,0,1,0,0,0,0,0]
=> 5 = 6 - 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]
=> [1,1,1,1,1,0,0,1,0,0,0,0]
=> 4 = 5 - 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]
=> [1,1,1,1,1,0,1,0,0,0,0,0]
=> 5 = 6 - 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]
=> [1,1,1,1,0,1,0,1,0,0,0,0]
=> ? ∊ {5,6,7,7,9,9,9,9,10,10,10,11,11,12,12,13,13,14,14,14,15,15,15,15,16,16,16,16,17,18,18,19,19,19,20,20,20,21,21,22,22,23,23,24,24,25,25,26,26,28,28,28,28,30,30,30,30,31,31,32,32,33,33,34,34,34,34,35,35,37,37,40,40,41,41,42,42,42,42,43,43,45,45,46,47,47,48,48,48,48,53,53,53,54,54,56,56,62,62,62,62,62,63,63,66,66,72,72,75,75,75,76,76,84,89,89,90,90,104,104,107,132} - 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]
=> [1,1,1,1,1,0,0,0,1,0,0,0]
=> 3 = 4 - 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]
=> [1,1,1,1,0,1,0,0,1,0,0,0]
=> ? ∊ {5,6,7,7,9,9,9,9,10,10,10,11,11,12,12,13,13,14,14,14,15,15,15,15,16,16,16,16,17,18,18,19,19,19,20,20,20,21,21,22,22,23,23,24,24,25,25,26,26,28,28,28,28,30,30,30,30,31,31,32,32,33,33,34,34,34,34,35,35,37,37,40,40,41,41,42,42,42,42,43,43,45,45,46,47,47,48,48,48,48,53,53,53,54,54,56,56,62,62,62,62,62,63,63,66,66,72,72,75,75,75,76,76,84,89,89,90,90,104,104,107,132} - 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]
=> [1,1,1,1,1,0,0,1,0,0,0,0]
=> 4 = 5 - 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]
=> [1,1,1,1,0,1,0,0,1,0,0,0]
=> ? ∊ {5,6,7,7,9,9,9,9,10,10,10,11,11,12,12,13,13,14,14,14,15,15,15,15,16,16,16,16,17,18,18,19,19,19,20,20,20,21,21,22,22,23,23,24,24,25,25,26,26,28,28,28,28,30,30,30,30,31,31,32,32,33,33,34,34,34,34,35,35,37,37,40,40,41,41,42,42,42,42,43,43,45,45,46,47,47,48,48,48,48,53,53,53,54,54,56,56,62,62,62,62,62,63,63,66,66,72,72,75,75,75,76,76,84,89,89,90,90,104,104,107,132} - 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]
=> [1,1,1,1,0,1,0,1,0,0,0,0]
=> ? ∊ {5,6,7,7,9,9,9,9,10,10,10,11,11,12,12,13,13,14,14,14,15,15,15,15,16,16,16,16,17,18,18,19,19,19,20,20,20,21,21,22,22,23,23,24,24,25,25,26,26,28,28,28,28,30,30,30,30,31,31,32,32,33,33,34,34,34,34,35,35,37,37,40,40,41,41,42,42,42,42,43,43,45,45,46,47,47,48,48,48,48,53,53,53,54,54,56,56,62,62,62,62,62,63,63,66,66,72,72,75,75,75,76,76,84,89,89,90,90,104,104,107,132} - 1
[1,0,1,0,1,1,1,0,0,0,1,0]
=> [1,0,1,0,1,1,1,0,0,0,1,0]
=> [1,1,1,1,0,0,1,0,1,0,0,0]
=> ? ∊ {5,6,7,7,9,9,9,9,10,10,10,11,11,12,12,13,13,14,14,14,15,15,15,15,16,16,16,16,17,18,18,19,19,19,20,20,20,21,21,22,22,23,23,24,24,25,25,26,26,28,28,28,28,30,30,30,30,31,31,32,32,33,33,34,34,34,34,35,35,37,37,40,40,41,41,42,42,42,42,43,43,45,45,46,47,47,48,48,48,48,53,53,53,54,54,56,56,62,62,62,62,62,63,63,66,66,72,72,75,75,75,76,76,84,89,89,90,90,104,104,107,132} - 1
[1,0,1,0,1,1,1,0,0,1,0,0]
=> [1,0,1,0,1,1,0,0,1,1,0,0]
=> [1,1,1,1,0,1,0,0,1,0,0,0]
=> ? ∊ {5,6,7,7,9,9,9,9,10,10,10,11,11,12,12,13,13,14,14,14,15,15,15,15,16,16,16,16,17,18,18,19,19,19,20,20,20,21,21,22,22,23,23,24,24,25,25,26,26,28,28,28,28,30,30,30,30,31,31,32,32,33,33,34,34,34,34,35,35,37,37,40,40,41,41,42,42,42,42,43,43,45,45,46,47,47,48,48,48,48,53,53,53,54,54,56,56,62,62,62,62,62,63,63,66,66,72,72,75,75,75,76,76,84,89,89,90,90,104,104,107,132} - 1
[1,0,1,0,1,1,1,0,1,0,0,0]
=> [1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,1,0,1,0,1,0,0,0,0]
=> ? ∊ {5,6,7,7,9,9,9,9,10,10,10,11,11,12,12,13,13,14,14,14,15,15,15,15,16,16,16,16,17,18,18,19,19,19,20,20,20,21,21,22,22,23,23,24,24,25,25,26,26,28,28,28,28,30,30,30,30,31,31,32,32,33,33,34,34,34,34,35,35,37,37,40,40,41,41,42,42,42,42,43,43,45,45,46,47,47,48,48,48,48,53,53,53,54,54,56,56,62,62,62,62,62,63,63,66,66,72,72,75,75,75,76,76,84,89,89,90,90,104,104,107,132} - 1
[1,0,1,1,0,0,1,0,1,0,1,0]
=> [1,0,1,1,0,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,1,0,0]
=> 2 = 3 - 1
[1,0,1,1,0,1,0,0,1,0,1,0]
=> [1,0,1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,1,0,0,0]
=> 3 = 4 - 1
[1,1,0,0,1,0,1,0,1,0,1,0]
=> [1,1,0,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> 1 = 2 - 1
[1,1,0,1,0,0,1,0,1,0,1,0]
=> [1,0,1,1,0,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,1,0,0]
=> 2 = 3 - 1
[1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> 0 = 1 - 1
[1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,1,1,0,1,0,0,0,0,0,0]
=> 6 = 7 - 1
[1,0,1,0,1,0,1,0,1,1,0,0,1,0]
=> [1,0,1,0,1,0,1,0,1,1,0,0,1,0]
=> [1,1,1,1,1,1,0,0,1,0,0,0,0,0]
=> 5 = 6 - 1
[1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,1,1,0,1,0,0,0,0,0,0]
=> 6 = 7 - 1
[1,0,1,0,1,0,1,1,0,0,1,0,1,0]
=> [1,0,1,0,1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,1,1,1,0,0,0,1,0,0,0,0]
=> 4 = 5 - 1
[1,0,1,0,1,0,1,1,0,1,0,0,1,0]
=> [1,0,1,0,1,0,1,0,1,1,0,0,1,0]
=> [1,1,1,1,1,1,0,0,1,0,0,0,0,0]
=> 5 = 6 - 1
[1,0,1,0,1,1,0,0,1,0,1,0,1,0]
=> [1,0,1,0,1,1,0,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,0,0,0,0,1,0,0,0]
=> 3 = 4 - 1
[1,0,1,0,1,1,0,1,0,0,1,0,1,0]
=> [1,0,1,0,1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,1,1,1,0,0,0,1,0,0,0,0]
=> 4 = 5 - 1
[1,0,1,1,0,0,1,0,1,0,1,0,1,0]
=> [1,0,1,1,0,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,0,0,0,0,0,1,0,0]
=> 2 = 3 - 1
[1,0,1,1,0,1,0,0,1,0,1,0,1,0]
=> [1,0,1,0,1,1,0,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,0,0,0,0,1,0,0,0]
=> 3 = 4 - 1
[1,1,0,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,0,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> 1 = 2 - 1
[1,1,0,1,0,0,1,0,1,0,1,0,1,0]
=> [1,0,1,1,0,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,0,0,0,0,0,1,0,0]
=> 2 = 3 - 1
Description
The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2.
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