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Your data matches 134 different statistics following compositions of up to 3 maps.
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Matching statistic: St001933
St001933: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1]
=> 1
[2]
=> 1
[1,1]
=> 2
[3]
=> 1
[2,1]
=> 1
[1,1,1]
=> 3
[4]
=> 1
[3,1]
=> 1
[2,2]
=> 2
[2,1,1]
=> 2
[1,1,1,1]
=> 4
[5]
=> 1
[4,1]
=> 1
[3,2]
=> 1
[3,1,1]
=> 2
[2,2,1]
=> 2
[2,1,1,1]
=> 3
[1,1,1,1,1]
=> 5
[6]
=> 1
[5,1]
=> 1
[4,2]
=> 1
[4,1,1]
=> 2
[3,3]
=> 2
[3,2,1]
=> 1
[3,1,1,1]
=> 3
[2,2,2]
=> 3
[2,2,1,1]
=> 2
[2,1,1,1,1]
=> 4
[1,1,1,1,1,1]
=> 6
[7]
=> 1
[6,1]
=> 1
[5,2]
=> 1
[5,1,1]
=> 2
[4,3]
=> 1
[4,2,1]
=> 1
[4,1,1,1]
=> 3
[3,3,1]
=> 2
[3,2,2]
=> 2
[3,2,1,1]
=> 2
[3,1,1,1,1]
=> 4
[2,2,2,1]
=> 3
[2,2,1,1,1]
=> 3
[2,1,1,1,1,1]
=> 5
[1,1,1,1,1,1,1]
=> 7
[8]
=> 1
[7,1]
=> 1
[6,2]
=> 1
[6,1,1]
=> 2
[5,3]
=> 1
[5,2,1]
=> 1
Description
The largest multiplicity of a part in an integer partition.
Matching statistic: St001232
(load all 6 compositions to match this statistic)
(load all 6 compositions to match this statistic)
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
St001232: Dyck paths ⟶ ℤResult quality: 46% ●values known / values provided: 46%●distinct values known / distinct values provided: 56%
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
St001232: Dyck paths ⟶ ℤResult quality: 46% ●values known / values provided: 46%●distinct values known / distinct values provided: 56%
Values
[1]
=> []
=> ?
=> ?
=> ? = 1
[2]
=> []
=> ?
=> ?
=> ? ∊ {1,2}
[1,1]
=> [1]
=> []
=> []
=> ? ∊ {1,2}
[3]
=> []
=> ?
=> ?
=> ? ∊ {1,3}
[2,1]
=> [1]
=> []
=> []
=> ? ∊ {1,3}
[1,1,1]
=> [1,1]
=> [1]
=> [1,0,1,0]
=> 1
[4]
=> []
=> ?
=> ?
=> ? ∊ {1,2,4}
[3,1]
=> [1]
=> []
=> []
=> ? ∊ {1,2,4}
[2,2]
=> [2]
=> []
=> []
=> ? ∊ {1,2,4}
[2,1,1]
=> [1,1]
=> [1]
=> [1,0,1,0]
=> 1
[1,1,1,1]
=> [1,1,1]
=> [1,1]
=> [1,0,1,1,0,0]
=> 2
[5]
=> []
=> ?
=> ?
=> ? ∊ {1,2,5}
[4,1]
=> [1]
=> []
=> []
=> ? ∊ {1,2,5}
[3,2]
=> [2]
=> []
=> []
=> ? ∊ {1,2,5}
[3,1,1]
=> [1,1]
=> [1]
=> [1,0,1,0]
=> 1
[2,2,1]
=> [2,1]
=> [1]
=> [1,0,1,0]
=> 1
[2,1,1,1]
=> [1,1,1]
=> [1,1]
=> [1,0,1,1,0,0]
=> 2
[1,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> 3
[6]
=> []
=> ?
=> ?
=> ? ∊ {1,2,3,6}
[5,1]
=> [1]
=> []
=> []
=> ? ∊ {1,2,3,6}
[4,2]
=> [2]
=> []
=> []
=> ? ∊ {1,2,3,6}
[4,1,1]
=> [1,1]
=> [1]
=> [1,0,1,0]
=> 1
[3,3]
=> [3]
=> []
=> []
=> ? ∊ {1,2,3,6}
[3,2,1]
=> [2,1]
=> [1]
=> [1,0,1,0]
=> 1
[3,1,1,1]
=> [1,1,1]
=> [1,1]
=> [1,0,1,1,0,0]
=> 2
[2,2,2]
=> [2,2]
=> [2]
=> [1,1,0,0,1,0]
=> 1
[2,2,1,1]
=> [2,1,1]
=> [1,1]
=> [1,0,1,1,0,0]
=> 2
[2,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> 3
[1,1,1,1,1,1]
=> [1,1,1,1,1]
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> 4
[7]
=> []
=> ?
=> ?
=> ? ∊ {1,2,2,3,7}
[6,1]
=> [1]
=> []
=> []
=> ? ∊ {1,2,2,3,7}
[5,2]
=> [2]
=> []
=> []
=> ? ∊ {1,2,2,3,7}
[5,1,1]
=> [1,1]
=> [1]
=> [1,0,1,0]
=> 1
[4,3]
=> [3]
=> []
=> []
=> ? ∊ {1,2,2,3,7}
[4,2,1]
=> [2,1]
=> [1]
=> [1,0,1,0]
=> 1
[4,1,1,1]
=> [1,1,1]
=> [1,1]
=> [1,0,1,1,0,0]
=> 2
[3,3,1]
=> [3,1]
=> [1]
=> [1,0,1,0]
=> 1
[3,2,2]
=> [2,2]
=> [2]
=> [1,1,0,0,1,0]
=> 1
[3,2,1,1]
=> [2,1,1]
=> [1,1]
=> [1,0,1,1,0,0]
=> 2
[3,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> 3
[2,2,2,1]
=> [2,2,1]
=> [2,1]
=> [1,0,1,0,1,0]
=> ? ∊ {1,2,2,3,7}
[2,2,1,1,1]
=> [2,1,1,1]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> 3
[2,1,1,1,1,1]
=> [1,1,1,1,1]
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> 4
[1,1,1,1,1,1,1]
=> [1,1,1,1,1,1]
=> [1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> 5
[8]
=> []
=> ?
=> ?
=> ? ∊ {1,2,2,2,3,4,8}
[7,1]
=> [1]
=> []
=> []
=> ? ∊ {1,2,2,2,3,4,8}
[6,2]
=> [2]
=> []
=> []
=> ? ∊ {1,2,2,2,3,4,8}
[6,1,1]
=> [1,1]
=> [1]
=> [1,0,1,0]
=> 1
[5,3]
=> [3]
=> []
=> []
=> ? ∊ {1,2,2,2,3,4,8}
[5,2,1]
=> [2,1]
=> [1]
=> [1,0,1,0]
=> 1
[5,1,1,1]
=> [1,1,1]
=> [1,1]
=> [1,0,1,1,0,0]
=> 2
[4,4]
=> [4]
=> []
=> []
=> ? ∊ {1,2,2,2,3,4,8}
[4,3,1]
=> [3,1]
=> [1]
=> [1,0,1,0]
=> 1
[4,2,2]
=> [2,2]
=> [2]
=> [1,1,0,0,1,0]
=> 1
[4,2,1,1]
=> [2,1,1]
=> [1,1]
=> [1,0,1,1,0,0]
=> 2
[4,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> 3
[3,3,2]
=> [3,2]
=> [2]
=> [1,1,0,0,1,0]
=> 1
[3,3,1,1]
=> [3,1,1]
=> [1,1]
=> [1,0,1,1,0,0]
=> 2
[3,2,2,1]
=> [2,2,1]
=> [2,1]
=> [1,0,1,0,1,0]
=> ? ∊ {1,2,2,2,3,4,8}
[3,2,1,1,1]
=> [2,1,1,1]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> 3
[3,1,1,1,1,1]
=> [1,1,1,1,1]
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> 4
[2,2,2,2]
=> [2,2,2]
=> [2,2]
=> [1,1,0,0,1,1,0,0]
=> 2
[2,2,2,1,1]
=> [2,2,1,1]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> ? ∊ {1,2,2,2,3,4,8}
[2,2,1,1,1,1]
=> [2,1,1,1,1]
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> 4
[2,1,1,1,1,1,1]
=> [1,1,1,1,1,1]
=> [1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> 5
[1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1]
=> [1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> 6
[9]
=> []
=> ?
=> ?
=> ? ∊ {1,2,2,2,2,3,3,3,4,7,9}
[8,1]
=> [1]
=> []
=> []
=> ? ∊ {1,2,2,2,2,3,3,3,4,7,9}
[7,2]
=> [2]
=> []
=> []
=> ? ∊ {1,2,2,2,2,3,3,3,4,7,9}
[7,1,1]
=> [1,1]
=> [1]
=> [1,0,1,0]
=> 1
[6,3]
=> [3]
=> []
=> []
=> ? ∊ {1,2,2,2,2,3,3,3,4,7,9}
[6,2,1]
=> [2,1]
=> [1]
=> [1,0,1,0]
=> 1
[6,1,1,1]
=> [1,1,1]
=> [1,1]
=> [1,0,1,1,0,0]
=> 2
[5,4]
=> [4]
=> []
=> []
=> ? ∊ {1,2,2,2,2,3,3,3,4,7,9}
[5,3,1]
=> [3,1]
=> [1]
=> [1,0,1,0]
=> 1
[5,2,2]
=> [2,2]
=> [2]
=> [1,1,0,0,1,0]
=> 1
[5,2,1,1]
=> [2,1,1]
=> [1,1]
=> [1,0,1,1,0,0]
=> 2
[5,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> 3
[4,4,1]
=> [4,1]
=> [1]
=> [1,0,1,0]
=> 1
[4,3,2]
=> [3,2]
=> [2]
=> [1,1,0,0,1,0]
=> 1
[4,3,1,1]
=> [3,1,1]
=> [1,1]
=> [1,0,1,1,0,0]
=> 2
[4,2,2,1]
=> [2,2,1]
=> [2,1]
=> [1,0,1,0,1,0]
=> ? ∊ {1,2,2,2,2,3,3,3,4,7,9}
[4,2,1,1,1]
=> [2,1,1,1]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> 3
[3,3,2,1]
=> [3,2,1]
=> [2,1]
=> [1,0,1,0,1,0]
=> ? ∊ {1,2,2,2,2,3,3,3,4,7,9}
[3,2,2,1,1]
=> [2,2,1,1]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> ? ∊ {1,2,2,2,2,3,3,3,4,7,9}
[2,2,2,2,1]
=> [2,2,2,1]
=> [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> ? ∊ {1,2,2,2,2,3,3,3,4,7,9}
[2,2,2,1,1,1]
=> [2,2,1,1,1]
=> [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> ? ∊ {1,2,2,2,2,3,3,3,4,7,9}
[1,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> ? ∊ {1,2,2,2,2,3,3,3,4,7,9}
[10]
=> []
=> ?
=> ?
=> ? ∊ {1,1,2,2,2,2,2,2,3,3,3,4,4,5,7,8,10}
[9,1]
=> [1]
=> []
=> []
=> ? ∊ {1,1,2,2,2,2,2,2,3,3,3,4,4,5,7,8,10}
[8,2]
=> [2]
=> []
=> []
=> ? ∊ {1,1,2,2,2,2,2,2,3,3,3,4,4,5,7,8,10}
[7,3]
=> [3]
=> []
=> []
=> ? ∊ {1,1,2,2,2,2,2,2,3,3,3,4,4,5,7,8,10}
[6,4]
=> [4]
=> []
=> []
=> ? ∊ {1,1,2,2,2,2,2,2,3,3,3,4,4,5,7,8,10}
[5,5]
=> [5]
=> []
=> []
=> ? ∊ {1,1,2,2,2,2,2,2,3,3,3,4,4,5,7,8,10}
[5,2,2,1]
=> [2,2,1]
=> [2,1]
=> [1,0,1,0,1,0]
=> ? ∊ {1,1,2,2,2,2,2,2,3,3,3,4,4,5,7,8,10}
[4,3,2,1]
=> [3,2,1]
=> [2,1]
=> [1,0,1,0,1,0]
=> ? ∊ {1,1,2,2,2,2,2,2,3,3,3,4,4,5,7,8,10}
[4,2,2,1,1]
=> [2,2,1,1]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> ? ∊ {1,1,2,2,2,2,2,2,3,3,3,4,4,5,7,8,10}
[3,3,3,1]
=> [3,3,1]
=> [3,1]
=> [1,1,0,1,0,0,1,0]
=> ? ∊ {1,1,2,2,2,2,2,2,3,3,3,4,4,5,7,8,10}
[3,3,2,1,1]
=> [3,2,1,1]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> ? ∊ {1,1,2,2,2,2,2,2,3,3,3,4,4,5,7,8,10}
[3,2,2,2,1]
=> [2,2,2,1]
=> [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> ? ∊ {1,1,2,2,2,2,2,2,3,3,3,4,4,5,7,8,10}
Description
The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2.
Matching statistic: St000392
(load all 11 compositions to match this statistic)
(load all 11 compositions to match this statistic)
Mp00095: Integer partitions —to binary word⟶ Binary words
Mp00104: Binary words —reverse⟶ Binary words
St000392: Binary words ⟶ ℤResult quality: 37% ●values known / values provided: 37%●distinct values known / distinct values provided: 62%
Mp00104: Binary words —reverse⟶ Binary words
St000392: Binary words ⟶ ℤResult quality: 37% ●values known / values provided: 37%●distinct values known / distinct values provided: 62%
Values
[1]
=> 10 => 01 => 1
[2]
=> 100 => 001 => 1
[1,1]
=> 110 => 011 => 2
[3]
=> 1000 => 0001 => 1
[2,1]
=> 1010 => 0101 => 1
[1,1,1]
=> 1110 => 0111 => 3
[4]
=> 10000 => 00001 => 1
[3,1]
=> 10010 => 01001 => 1
[2,2]
=> 1100 => 0011 => 2
[2,1,1]
=> 10110 => 01101 => 2
[1,1,1,1]
=> 11110 => 01111 => 4
[5]
=> 100000 => 000001 => 1
[4,1]
=> 100010 => 010001 => 1
[3,2]
=> 10100 => 00101 => 1
[3,1,1]
=> 100110 => 011001 => 2
[2,2,1]
=> 11010 => 01011 => 2
[2,1,1,1]
=> 101110 => 011101 => 3
[1,1,1,1,1]
=> 111110 => 011111 => 5
[6]
=> 1000000 => 0000001 => 1
[5,1]
=> 1000010 => 0100001 => 1
[4,2]
=> 100100 => 001001 => 1
[4,1,1]
=> 1000110 => 0110001 => 2
[3,3]
=> 11000 => 00011 => 2
[3,2,1]
=> 101010 => 010101 => 1
[3,1,1,1]
=> 1001110 => 0111001 => 3
[2,2,2]
=> 11100 => 00111 => 3
[2,2,1,1]
=> 110110 => 011011 => 2
[2,1,1,1,1]
=> 1011110 => 0111101 => 4
[1,1,1,1,1,1]
=> 1111110 => 0111111 => 6
[7]
=> 10000000 => 00000001 => 1
[6,1]
=> 10000010 => 01000001 => 1
[5,2]
=> 1000100 => 0010001 => 1
[5,1,1]
=> 10000110 => 01100001 => 2
[4,3]
=> 101000 => 000101 => 1
[4,2,1]
=> 1001010 => 0101001 => 1
[4,1,1,1]
=> 10001110 => 01110001 => 3
[3,3,1]
=> 110010 => 010011 => 2
[3,2,2]
=> 101100 => 001101 => 2
[3,2,1,1]
=> 1010110 => 0110101 => 2
[3,1,1,1,1]
=> 10011110 => 01111001 => 4
[2,2,2,1]
=> 111010 => 010111 => 3
[2,2,1,1,1]
=> 1101110 => 0111011 => 3
[2,1,1,1,1,1]
=> 10111110 => 01111101 => 5
[1,1,1,1,1,1,1]
=> 11111110 => 01111111 => 7
[8]
=> 100000000 => 000000001 => 1
[7,1]
=> 100000010 => 010000001 => 1
[6,2]
=> 10000100 => 00100001 => 1
[6,1,1]
=> 100000110 => 011000001 => 2
[5,3]
=> 1001000 => 0001001 => 1
[5,2,1]
=> 10001010 => 01010001 => 1
[11]
=> 100000000000 => 000000000001 => ? ∊ {1,1,1,1,1,2,2,2,2,2,3,3,3,4,4,4,4,5,5,5,6,6,7,7,8,9,11}
[10,1]
=> 100000000010 => 010000000001 => ? ∊ {1,1,1,1,1,2,2,2,2,2,3,3,3,4,4,4,4,5,5,5,6,6,7,7,8,9,11}
[9,2]
=> 10000000100 => 00100000001 => ? ∊ {1,1,1,1,1,2,2,2,2,2,3,3,3,4,4,4,4,5,5,5,6,6,7,7,8,9,11}
[9,1,1]
=> 100000000110 => 011000000001 => ? ∊ {1,1,1,1,1,2,2,2,2,2,3,3,3,4,4,4,4,5,5,5,6,6,7,7,8,9,11}
[8,2,1]
=> 10000001010 => 01010000001 => ? ∊ {1,1,1,1,1,2,2,2,2,2,3,3,3,4,4,4,4,5,5,5,6,6,7,7,8,9,11}
[8,1,1,1]
=> 100000001110 => 011100000001 => ? ∊ {1,1,1,1,1,2,2,2,2,2,3,3,3,4,4,4,4,5,5,5,6,6,7,7,8,9,11}
[7,3,1]
=> 1000010010 => 0100100001 => ? ∊ {1,1,1,1,1,2,2,2,2,2,3,3,3,4,4,4,4,5,5,5,6,6,7,7,8,9,11}
[7,2,2]
=> 1000001100 => 0011000001 => ? ∊ {1,1,1,1,1,2,2,2,2,2,3,3,3,4,4,4,4,5,5,5,6,6,7,7,8,9,11}
[7,2,1,1]
=> 10000010110 => 01101000001 => ? ∊ {1,1,1,1,1,2,2,2,2,2,3,3,3,4,4,4,4,5,5,5,6,6,7,7,8,9,11}
[7,1,1,1,1]
=> 100000011110 => 011110000001 => ? ∊ {1,1,1,1,1,2,2,2,2,2,3,3,3,4,4,4,4,5,5,5,6,6,7,7,8,9,11}
[6,3,1,1]
=> 1000100110 => 0110010001 => ? ∊ {1,1,1,1,1,2,2,2,2,2,3,3,3,4,4,4,4,5,5,5,6,6,7,7,8,9,11}
[6,2,2,1]
=> 1000011010 => 0101100001 => ? ∊ {1,1,1,1,1,2,2,2,2,2,3,3,3,4,4,4,4,5,5,5,6,6,7,7,8,9,11}
[6,2,1,1,1]
=> 10000101110 => 01110100001 => ? ∊ {1,1,1,1,1,2,2,2,2,2,3,3,3,4,4,4,4,5,5,5,6,6,7,7,8,9,11}
[5,2,1,1,1,1]
=> 10001011110 => 01111010001 => ? ∊ {1,1,1,1,1,2,2,2,2,2,3,3,3,4,4,4,4,5,5,5,6,6,7,7,8,9,11}
[5,1,1,1,1,1,1]
=> 100001111110 => 011111100001 => ? ∊ {1,1,1,1,1,2,2,2,2,2,3,3,3,4,4,4,4,5,5,5,6,6,7,7,8,9,11}
[4,3,1,1,1,1]
=> 1010011110 => 0111100101 => ? ∊ {1,1,1,1,1,2,2,2,2,2,3,3,3,4,4,4,4,5,5,5,6,6,7,7,8,9,11}
[4,2,2,1,1,1]
=> 1001101110 => 0111011001 => ? ∊ {1,1,1,1,1,2,2,2,2,2,3,3,3,4,4,4,4,5,5,5,6,6,7,7,8,9,11}
[4,2,1,1,1,1,1]
=> 10010111110 => 01111101001 => ? ∊ {1,1,1,1,1,2,2,2,2,2,3,3,3,4,4,4,4,5,5,5,6,6,7,7,8,9,11}
[4,1,1,1,1,1,1,1]
=> 100011111110 => 011111110001 => ? ∊ {1,1,1,1,1,2,2,2,2,2,3,3,3,4,4,4,4,5,5,5,6,6,7,7,8,9,11}
[3,3,1,1,1,1,1]
=> 1100111110 => 0111110011 => ? ∊ {1,1,1,1,1,2,2,2,2,2,3,3,3,4,4,4,4,5,5,5,6,6,7,7,8,9,11}
[3,2,2,1,1,1,1]
=> 1011011110 => 0111101101 => ? ∊ {1,1,1,1,1,2,2,2,2,2,3,3,3,4,4,4,4,5,5,5,6,6,7,7,8,9,11}
[3,2,1,1,1,1,1,1]
=> 10101111110 => 01111110101 => ? ∊ {1,1,1,1,1,2,2,2,2,2,3,3,3,4,4,4,4,5,5,5,6,6,7,7,8,9,11}
[3,1,1,1,1,1,1,1,1]
=> 100111111110 => 011111111001 => ? ∊ {1,1,1,1,1,2,2,2,2,2,3,3,3,4,4,4,4,5,5,5,6,6,7,7,8,9,11}
[2,2,2,1,1,1,1,1]
=> 1110111110 => 0111110111 => ? ∊ {1,1,1,1,1,2,2,2,2,2,3,3,3,4,4,4,4,5,5,5,6,6,7,7,8,9,11}
[2,2,1,1,1,1,1,1,1]
=> 11011111110 => 01111111011 => ? ∊ {1,1,1,1,1,2,2,2,2,2,3,3,3,4,4,4,4,5,5,5,6,6,7,7,8,9,11}
[2,1,1,1,1,1,1,1,1,1]
=> 101111111110 => 011111111101 => ? ∊ {1,1,1,1,1,2,2,2,2,2,3,3,3,4,4,4,4,5,5,5,6,6,7,7,8,9,11}
[1,1,1,1,1,1,1,1,1,1,1]
=> 111111111110 => 011111111111 => ? ∊ {1,1,1,1,1,2,2,2,2,2,3,3,3,4,4,4,4,5,5,5,6,6,7,7,8,9,11}
[12]
=> 1000000000000 => 0000000000001 => ? ∊ {1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,4,4,4,4,5,5,5,5,6,6,6,6,7,7,8,8,9,10,12}
[11,1]
=> 1000000000010 => ? => ? ∊ {1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,4,4,4,4,5,5,5,5,6,6,6,6,7,7,8,8,9,10,12}
[10,2]
=> 100000000100 => 001000000001 => ? ∊ {1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,4,4,4,4,5,5,5,5,6,6,6,6,7,7,8,8,9,10,12}
[10,1,1]
=> 1000000000110 => ? => ? ∊ {1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,4,4,4,4,5,5,5,5,6,6,6,6,7,7,8,8,9,10,12}
[9,3]
=> 10000001000 => 00010000001 => ? ∊ {1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,4,4,4,4,5,5,5,5,6,6,6,6,7,7,8,8,9,10,12}
[9,2,1]
=> 100000001010 => ? => ? ∊ {1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,4,4,4,4,5,5,5,5,6,6,6,6,7,7,8,8,9,10,12}
[9,1,1,1]
=> 1000000001110 => ? => ? ∊ {1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,4,4,4,4,5,5,5,5,6,6,6,6,7,7,8,8,9,10,12}
[8,3,1]
=> 10000010010 => 01001000001 => ? ∊ {1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,4,4,4,4,5,5,5,5,6,6,6,6,7,7,8,8,9,10,12}
[8,2,2]
=> 10000001100 => 00110000001 => ? ∊ {1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,4,4,4,4,5,5,5,5,6,6,6,6,7,7,8,8,9,10,12}
[8,2,1,1]
=> 100000010110 => 011010000001 => ? ∊ {1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,4,4,4,4,5,5,5,5,6,6,6,6,7,7,8,8,9,10,12}
[8,1,1,1,1]
=> 1000000011110 => ? => ? ∊ {1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,4,4,4,4,5,5,5,5,6,6,6,6,7,7,8,8,9,10,12}
[7,4,1]
=> 1000100010 => 0100010001 => ? ∊ {1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,4,4,4,4,5,5,5,5,6,6,6,6,7,7,8,8,9,10,12}
[7,3,1,1]
=> 10000100110 => 01100100001 => ? ∊ {1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,4,4,4,4,5,5,5,5,6,6,6,6,7,7,8,8,9,10,12}
[7,2,2,1]
=> 10000011010 => 01011000001 => ? ∊ {1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,4,4,4,4,5,5,5,5,6,6,6,6,7,7,8,8,9,10,12}
[7,2,1,1,1]
=> 100000101110 => ? => ? ∊ {1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,4,4,4,4,5,5,5,5,6,6,6,6,7,7,8,8,9,10,12}
[7,1,1,1,1,1]
=> 1000000111110 => ? => ? ∊ {1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,4,4,4,4,5,5,5,5,6,6,6,6,7,7,8,8,9,10,12}
[6,4,1,1]
=> 1001000110 => 0110001001 => ? ∊ {1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,4,4,4,4,5,5,5,5,6,6,6,6,7,7,8,8,9,10,12}
[6,3,2,1]
=> 1000101010 => 0101010001 => ? ∊ {1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,4,4,4,4,5,5,5,5,6,6,6,6,7,7,8,8,9,10,12}
[6,3,1,1,1]
=> 10001001110 => 01110010001 => ? ∊ {1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,4,4,4,4,5,5,5,5,6,6,6,6,7,7,8,8,9,10,12}
[6,2,2,2]
=> 1000011100 => 0011100001 => ? ∊ {1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,4,4,4,4,5,5,5,5,6,6,6,6,7,7,8,8,9,10,12}
[6,2,2,1,1]
=> 10000110110 => 01101100001 => ? ∊ {1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,4,4,4,4,5,5,5,5,6,6,6,6,7,7,8,8,9,10,12}
[6,1,1,1,1,1,1]
=> 1000001111110 => ? => ? ∊ {1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,4,4,4,4,5,5,5,5,6,6,6,6,7,7,8,8,9,10,12}
[5,3,1,1,1,1]
=> 10010011110 => 01111001001 => ? ∊ {1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,4,4,4,4,5,5,5,5,6,6,6,6,7,7,8,8,9,10,12}
Description
The length of the longest run of ones in a binary word.
Matching statistic: St001372
(load all 10 compositions to match this statistic)
(load all 10 compositions to match this statistic)
Mp00095: Integer partitions —to binary word⟶ Binary words
Mp00104: Binary words —reverse⟶ Binary words
St001372: Binary words ⟶ ℤResult quality: 35% ●values known / values provided: 35%●distinct values known / distinct values provided: 62%
Mp00104: Binary words —reverse⟶ Binary words
St001372: Binary words ⟶ ℤResult quality: 35% ●values known / values provided: 35%●distinct values known / distinct values provided: 62%
Values
[1]
=> 10 => 01 => 1
[2]
=> 100 => 001 => 1
[1,1]
=> 110 => 011 => 2
[3]
=> 1000 => 0001 => 1
[2,1]
=> 1010 => 0101 => 1
[1,1,1]
=> 1110 => 0111 => 3
[4]
=> 10000 => 00001 => 1
[3,1]
=> 10010 => 01001 => 1
[2,2]
=> 1100 => 0011 => 2
[2,1,1]
=> 10110 => 01101 => 2
[1,1,1,1]
=> 11110 => 01111 => 4
[5]
=> 100000 => 000001 => 1
[4,1]
=> 100010 => 010001 => 1
[3,2]
=> 10100 => 00101 => 1
[3,1,1]
=> 100110 => 011001 => 2
[2,2,1]
=> 11010 => 01011 => 2
[2,1,1,1]
=> 101110 => 011101 => 3
[1,1,1,1,1]
=> 111110 => 011111 => 5
[6]
=> 1000000 => 0000001 => 1
[5,1]
=> 1000010 => 0100001 => 1
[4,2]
=> 100100 => 001001 => 1
[4,1,1]
=> 1000110 => 0110001 => 2
[3,3]
=> 11000 => 00011 => 2
[3,2,1]
=> 101010 => 010101 => 1
[3,1,1,1]
=> 1001110 => 0111001 => 3
[2,2,2]
=> 11100 => 00111 => 3
[2,2,1,1]
=> 110110 => 011011 => 2
[2,1,1,1,1]
=> 1011110 => 0111101 => 4
[1,1,1,1,1,1]
=> 1111110 => 0111111 => 6
[7]
=> 10000000 => 00000001 => 1
[6,1]
=> 10000010 => 01000001 => 1
[5,2]
=> 1000100 => 0010001 => 1
[5,1,1]
=> 10000110 => 01100001 => 2
[4,3]
=> 101000 => 000101 => 1
[4,2,1]
=> 1001010 => 0101001 => 1
[4,1,1,1]
=> 10001110 => 01110001 => 3
[3,3,1]
=> 110010 => 010011 => 2
[3,2,2]
=> 101100 => 001101 => 2
[3,2,1,1]
=> 1010110 => 0110101 => 2
[3,1,1,1,1]
=> 10011110 => 01111001 => 4
[2,2,2,1]
=> 111010 => 010111 => 3
[2,2,1,1,1]
=> 1101110 => 0111011 => 3
[2,1,1,1,1,1]
=> 10111110 => 01111101 => 5
[1,1,1,1,1,1,1]
=> 11111110 => 01111111 => 7
[8]
=> 100000000 => 000000001 => 1
[7,1]
=> 100000010 => 010000001 => 1
[6,2]
=> 10000100 => 00100001 => 1
[6,1,1]
=> 100000110 => 011000001 => 2
[5,3]
=> 1001000 => 0001001 => 1
[5,2,1]
=> 10001010 => 01010001 => 1
[11]
=> 100000000000 => 000000000001 => ? ∊ {1,1,1,1,1,1,2,2,2,2,2,3,3,3,4,4,4,4,5,5,5,6,6,7,7,8,9,11}
[10,1]
=> 100000000010 => 010000000001 => ? ∊ {1,1,1,1,1,1,2,2,2,2,2,3,3,3,4,4,4,4,5,5,5,6,6,7,7,8,9,11}
[9,2]
=> 10000000100 => 00100000001 => ? ∊ {1,1,1,1,1,1,2,2,2,2,2,3,3,3,4,4,4,4,5,5,5,6,6,7,7,8,9,11}
[9,1,1]
=> 100000000110 => 011000000001 => ? ∊ {1,1,1,1,1,1,2,2,2,2,2,3,3,3,4,4,4,4,5,5,5,6,6,7,7,8,9,11}
[8,3]
=> 1000001000 => 0001000001 => ? ∊ {1,1,1,1,1,1,2,2,2,2,2,3,3,3,4,4,4,4,5,5,5,6,6,7,7,8,9,11}
[8,2,1]
=> 10000001010 => 01010000001 => ? ∊ {1,1,1,1,1,1,2,2,2,2,2,3,3,3,4,4,4,4,5,5,5,6,6,7,7,8,9,11}
[8,1,1,1]
=> 100000001110 => 011100000001 => ? ∊ {1,1,1,1,1,1,2,2,2,2,2,3,3,3,4,4,4,4,5,5,5,6,6,7,7,8,9,11}
[7,3,1]
=> 1000010010 => 0100100001 => ? ∊ {1,1,1,1,1,1,2,2,2,2,2,3,3,3,4,4,4,4,5,5,5,6,6,7,7,8,9,11}
[7,2,2]
=> 1000001100 => 0011000001 => ? ∊ {1,1,1,1,1,1,2,2,2,2,2,3,3,3,4,4,4,4,5,5,5,6,6,7,7,8,9,11}
[7,2,1,1]
=> 10000010110 => 01101000001 => ? ∊ {1,1,1,1,1,1,2,2,2,2,2,3,3,3,4,4,4,4,5,5,5,6,6,7,7,8,9,11}
[7,1,1,1,1]
=> 100000011110 => 011110000001 => ? ∊ {1,1,1,1,1,1,2,2,2,2,2,3,3,3,4,4,4,4,5,5,5,6,6,7,7,8,9,11}
[6,3,1,1]
=> 1000100110 => 0110010001 => ? ∊ {1,1,1,1,1,1,2,2,2,2,2,3,3,3,4,4,4,4,5,5,5,6,6,7,7,8,9,11}
[6,2,2,1]
=> 1000011010 => 0101100001 => ? ∊ {1,1,1,1,1,1,2,2,2,2,2,3,3,3,4,4,4,4,5,5,5,6,6,7,7,8,9,11}
[6,2,1,1,1]
=> 10000101110 => 01110100001 => ? ∊ {1,1,1,1,1,1,2,2,2,2,2,3,3,3,4,4,4,4,5,5,5,6,6,7,7,8,9,11}
[5,2,1,1,1,1]
=> 10001011110 => 01111010001 => ? ∊ {1,1,1,1,1,1,2,2,2,2,2,3,3,3,4,4,4,4,5,5,5,6,6,7,7,8,9,11}
[5,1,1,1,1,1,1]
=> 100001111110 => 011111100001 => ? ∊ {1,1,1,1,1,1,2,2,2,2,2,3,3,3,4,4,4,4,5,5,5,6,6,7,7,8,9,11}
[4,3,1,1,1,1]
=> 1010011110 => 0111100101 => ? ∊ {1,1,1,1,1,1,2,2,2,2,2,3,3,3,4,4,4,4,5,5,5,6,6,7,7,8,9,11}
[4,2,2,1,1,1]
=> 1001101110 => 0111011001 => ? ∊ {1,1,1,1,1,1,2,2,2,2,2,3,3,3,4,4,4,4,5,5,5,6,6,7,7,8,9,11}
[4,2,1,1,1,1,1]
=> 10010111110 => 01111101001 => ? ∊ {1,1,1,1,1,1,2,2,2,2,2,3,3,3,4,4,4,4,5,5,5,6,6,7,7,8,9,11}
[4,1,1,1,1,1,1,1]
=> 100011111110 => 011111110001 => ? ∊ {1,1,1,1,1,1,2,2,2,2,2,3,3,3,4,4,4,4,5,5,5,6,6,7,7,8,9,11}
[3,3,1,1,1,1,1]
=> 1100111110 => 0111110011 => ? ∊ {1,1,1,1,1,1,2,2,2,2,2,3,3,3,4,4,4,4,5,5,5,6,6,7,7,8,9,11}
[3,2,2,1,1,1,1]
=> 1011011110 => 0111101101 => ? ∊ {1,1,1,1,1,1,2,2,2,2,2,3,3,3,4,4,4,4,5,5,5,6,6,7,7,8,9,11}
[3,2,1,1,1,1,1,1]
=> 10101111110 => 01111110101 => ? ∊ {1,1,1,1,1,1,2,2,2,2,2,3,3,3,4,4,4,4,5,5,5,6,6,7,7,8,9,11}
[3,1,1,1,1,1,1,1,1]
=> 100111111110 => 011111111001 => ? ∊ {1,1,1,1,1,1,2,2,2,2,2,3,3,3,4,4,4,4,5,5,5,6,6,7,7,8,9,11}
[2,2,2,1,1,1,1,1]
=> 1110111110 => 0111110111 => ? ∊ {1,1,1,1,1,1,2,2,2,2,2,3,3,3,4,4,4,4,5,5,5,6,6,7,7,8,9,11}
[2,2,1,1,1,1,1,1,1]
=> 11011111110 => 01111111011 => ? ∊ {1,1,1,1,1,1,2,2,2,2,2,3,3,3,4,4,4,4,5,5,5,6,6,7,7,8,9,11}
[2,1,1,1,1,1,1,1,1,1]
=> 101111111110 => 011111111101 => ? ∊ {1,1,1,1,1,1,2,2,2,2,2,3,3,3,4,4,4,4,5,5,5,6,6,7,7,8,9,11}
[1,1,1,1,1,1,1,1,1,1,1]
=> 111111111110 => 011111111111 => ? ∊ {1,1,1,1,1,1,2,2,2,2,2,3,3,3,4,4,4,4,5,5,5,6,6,7,7,8,9,11}
[12]
=> 1000000000000 => 0000000000001 => ? ∊ {1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,4,4,4,4,5,5,5,5,6,6,6,6,7,7,8,8,9,10,12}
[11,1]
=> 1000000000010 => ? => ? ∊ {1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,4,4,4,4,5,5,5,5,6,6,6,6,7,7,8,8,9,10,12}
[10,2]
=> 100000000100 => 001000000001 => ? ∊ {1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,4,4,4,4,5,5,5,5,6,6,6,6,7,7,8,8,9,10,12}
[10,1,1]
=> 1000000000110 => ? => ? ∊ {1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,4,4,4,4,5,5,5,5,6,6,6,6,7,7,8,8,9,10,12}
[9,3]
=> 10000001000 => 00010000001 => ? ∊ {1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,4,4,4,4,5,5,5,5,6,6,6,6,7,7,8,8,9,10,12}
[9,2,1]
=> 100000001010 => ? => ? ∊ {1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,4,4,4,4,5,5,5,5,6,6,6,6,7,7,8,8,9,10,12}
[9,1,1,1]
=> 1000000001110 => ? => ? ∊ {1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,4,4,4,4,5,5,5,5,6,6,6,6,7,7,8,8,9,10,12}
[8,4]
=> 1000010000 => 0000100001 => ? ∊ {1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,4,4,4,4,5,5,5,5,6,6,6,6,7,7,8,8,9,10,12}
[8,3,1]
=> 10000010010 => 01001000001 => ? ∊ {1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,4,4,4,4,5,5,5,5,6,6,6,6,7,7,8,8,9,10,12}
[8,2,2]
=> 10000001100 => 00110000001 => ? ∊ {1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,4,4,4,4,5,5,5,5,6,6,6,6,7,7,8,8,9,10,12}
[8,2,1,1]
=> 100000010110 => 011010000001 => ? ∊ {1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,4,4,4,4,5,5,5,5,6,6,6,6,7,7,8,8,9,10,12}
[8,1,1,1,1]
=> 1000000011110 => ? => ? ∊ {1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,4,4,4,4,5,5,5,5,6,6,6,6,7,7,8,8,9,10,12}
[7,4,1]
=> 1000100010 => 0100010001 => ? ∊ {1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,4,4,4,4,5,5,5,5,6,6,6,6,7,7,8,8,9,10,12}
[7,3,2]
=> 1000010100 => 0010100001 => ? ∊ {1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,4,4,4,4,5,5,5,5,6,6,6,6,7,7,8,8,9,10,12}
[7,3,1,1]
=> 10000100110 => 01100100001 => ? ∊ {1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,4,4,4,4,5,5,5,5,6,6,6,6,7,7,8,8,9,10,12}
[7,2,2,1]
=> 10000011010 => 01011000001 => ? ∊ {1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,4,4,4,4,5,5,5,5,6,6,6,6,7,7,8,8,9,10,12}
[7,2,1,1,1]
=> 100000101110 => ? => ? ∊ {1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,4,4,4,4,5,5,5,5,6,6,6,6,7,7,8,8,9,10,12}
[7,1,1,1,1,1]
=> 1000000111110 => ? => ? ∊ {1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,4,4,4,4,5,5,5,5,6,6,6,6,7,7,8,8,9,10,12}
[6,4,1,1]
=> 1001000110 => 0110001001 => ? ∊ {1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,4,4,4,4,5,5,5,5,6,6,6,6,7,7,8,8,9,10,12}
[6,3,2,1]
=> 1000101010 => 0101010001 => ? ∊ {1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,4,4,4,4,5,5,5,5,6,6,6,6,7,7,8,8,9,10,12}
[6,3,1,1,1]
=> 10001001110 => 01110010001 => ? ∊ {1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,4,4,4,4,5,5,5,5,6,6,6,6,7,7,8,8,9,10,12}
[6,2,2,2]
=> 1000011100 => 0011100001 => ? ∊ {1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,4,4,4,4,5,5,5,5,6,6,6,6,7,7,8,8,9,10,12}
Description
The length of a longest cyclic run of ones of a binary word.
Consider the binary word as a cyclic arrangement of ones and zeros. Then this statistic is the length of the longest continuous sequence of ones in this arrangement.
Matching statistic: St000326
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Mp00095: Integer partitions —to binary word⟶ Binary words
Mp00135: Binary words —rotate front-to-back⟶ Binary words
Mp00224: Binary words —runsort⟶ Binary words
St000326: Binary words ⟶ ℤResult quality: 34% ●values known / values provided: 34%●distinct values known / distinct values provided: 62%
Mp00135: Binary words —rotate front-to-back⟶ Binary words
Mp00224: Binary words —runsort⟶ Binary words
St000326: Binary words ⟶ ℤResult quality: 34% ●values known / values provided: 34%●distinct values known / distinct values provided: 62%
Values
[1]
=> 10 => 01 => 01 => 2 = 1 + 1
[2]
=> 100 => 001 => 001 => 3 = 2 + 1
[1,1]
=> 110 => 101 => 011 => 2 = 1 + 1
[3]
=> 1000 => 0001 => 0001 => 4 = 3 + 1
[2,1]
=> 1010 => 0101 => 0101 => 2 = 1 + 1
[1,1,1]
=> 1110 => 1101 => 0111 => 2 = 1 + 1
[4]
=> 10000 => 00001 => 00001 => 5 = 4 + 1
[3,1]
=> 10010 => 00101 => 00101 => 3 = 2 + 1
[2,2]
=> 1100 => 1001 => 0011 => 3 = 2 + 1
[2,1,1]
=> 10110 => 01101 => 01011 => 2 = 1 + 1
[1,1,1,1]
=> 11110 => 11101 => 01111 => 2 = 1 + 1
[5]
=> 100000 => 000001 => 000001 => 6 = 5 + 1
[4,1]
=> 100010 => 000101 => 000101 => 4 = 3 + 1
[3,2]
=> 10100 => 01001 => 00101 => 3 = 2 + 1
[3,1,1]
=> 100110 => 001101 => 001101 => 3 = 2 + 1
[2,2,1]
=> 11010 => 10101 => 01011 => 2 = 1 + 1
[2,1,1,1]
=> 101110 => 011101 => 010111 => 2 = 1 + 1
[1,1,1,1,1]
=> 111110 => 111101 => 011111 => 2 = 1 + 1
[6]
=> 1000000 => 0000001 => 0000001 => 7 = 6 + 1
[5,1]
=> 1000010 => 0000101 => 0000101 => 5 = 4 + 1
[4,2]
=> 100100 => 001001 => 001001 => 3 = 2 + 1
[4,1,1]
=> 1000110 => 0001101 => 0001101 => 4 = 3 + 1
[3,3]
=> 11000 => 10001 => 00011 => 4 = 3 + 1
[3,2,1]
=> 101010 => 010101 => 010101 => 2 = 1 + 1
[3,1,1,1]
=> 1001110 => 0011101 => 0011101 => 3 = 2 + 1
[2,2,2]
=> 11100 => 11001 => 00111 => 3 = 2 + 1
[2,2,1,1]
=> 110110 => 101101 => 010111 => 2 = 1 + 1
[2,1,1,1,1]
=> 1011110 => 0111101 => 0101111 => 2 = 1 + 1
[1,1,1,1,1,1]
=> 1111110 => 1111101 => 0111111 => 2 = 1 + 1
[7]
=> 10000000 => 00000001 => 00000001 => 8 = 7 + 1
[6,1]
=> 10000010 => 00000101 => 00000101 => 6 = 5 + 1
[5,2]
=> 1000100 => 0001001 => 0001001 => 4 = 3 + 1
[5,1,1]
=> 10000110 => 00001101 => 00001101 => 5 = 4 + 1
[4,3]
=> 101000 => 010001 => 000101 => 4 = 3 + 1
[4,2,1]
=> 1001010 => 0010101 => 0010101 => 3 = 2 + 1
[4,1,1,1]
=> 10001110 => 00011101 => 00011101 => 4 = 3 + 1
[3,3,1]
=> 110010 => 100101 => 001011 => 3 = 2 + 1
[3,2,2]
=> 101100 => 011001 => 001011 => 3 = 2 + 1
[3,2,1,1]
=> 1010110 => 0101101 => 0101011 => 2 = 1 + 1
[3,1,1,1,1]
=> 10011110 => 00111101 => 00111101 => 3 = 2 + 1
[2,2,2,1]
=> 111010 => 110101 => 010111 => 2 = 1 + 1
[2,2,1,1,1]
=> 1101110 => 1011101 => 0101111 => 2 = 1 + 1
[2,1,1,1,1,1]
=> 10111110 => 01111101 => 01011111 => 2 = 1 + 1
[1,1,1,1,1,1,1]
=> 11111110 => 11111101 => 01111111 => 2 = 1 + 1
[8]
=> 100000000 => 000000001 => 000000001 => 9 = 8 + 1
[7,1]
=> 100000010 => 000000101 => 000000101 => 7 = 6 + 1
[6,2]
=> 10000100 => 00001001 => 00001001 => 5 = 4 + 1
[6,1,1]
=> 100000110 => 000001101 => 000001101 => 6 = 5 + 1
[5,3]
=> 1001000 => 0010001 => 0001001 => 4 = 3 + 1
[5,2,1]
=> 10001010 => 00010101 => 00010101 => 4 = 3 + 1
[11]
=> 100000000000 => ? => ? => ? ∊ {1,1,1,1,1,1,2,2,2,2,2,3,3,3,4,4,4,4,5,5,5,6,6,7,7,8,9,11} + 1
[10,1]
=> 100000000010 => ? => ? => ? ∊ {1,1,1,1,1,1,2,2,2,2,2,3,3,3,4,4,4,4,5,5,5,6,6,7,7,8,9,11} + 1
[9,2]
=> 10000000100 => 00000001001 => ? => ? ∊ {1,1,1,1,1,1,2,2,2,2,2,3,3,3,4,4,4,4,5,5,5,6,6,7,7,8,9,11} + 1
[9,1,1]
=> 100000000110 => ? => ? => ? ∊ {1,1,1,1,1,1,2,2,2,2,2,3,3,3,4,4,4,4,5,5,5,6,6,7,7,8,9,11} + 1
[8,3]
=> 1000001000 => ? => ? => ? ∊ {1,1,1,1,1,1,2,2,2,2,2,3,3,3,4,4,4,4,5,5,5,6,6,7,7,8,9,11} + 1
[8,2,1]
=> 10000001010 => 00000010101 => ? => ? ∊ {1,1,1,1,1,1,2,2,2,2,2,3,3,3,4,4,4,4,5,5,5,6,6,7,7,8,9,11} + 1
[8,1,1,1]
=> 100000001110 => ? => ? => ? ∊ {1,1,1,1,1,1,2,2,2,2,2,3,3,3,4,4,4,4,5,5,5,6,6,7,7,8,9,11} + 1
[7,3,1]
=> 1000010010 => ? => ? => ? ∊ {1,1,1,1,1,1,2,2,2,2,2,3,3,3,4,4,4,4,5,5,5,6,6,7,7,8,9,11} + 1
[7,2,2]
=> 1000001100 => ? => ? => ? ∊ {1,1,1,1,1,1,2,2,2,2,2,3,3,3,4,4,4,4,5,5,5,6,6,7,7,8,9,11} + 1
[7,2,1,1]
=> 10000010110 => 00000101101 => ? => ? ∊ {1,1,1,1,1,1,2,2,2,2,2,3,3,3,4,4,4,4,5,5,5,6,6,7,7,8,9,11} + 1
[7,1,1,1,1]
=> 100000011110 => ? => ? => ? ∊ {1,1,1,1,1,1,2,2,2,2,2,3,3,3,4,4,4,4,5,5,5,6,6,7,7,8,9,11} + 1
[6,3,1,1]
=> 1000100110 => ? => ? => ? ∊ {1,1,1,1,1,1,2,2,2,2,2,3,3,3,4,4,4,4,5,5,5,6,6,7,7,8,9,11} + 1
[6,2,2,1]
=> 1000011010 => ? => ? => ? ∊ {1,1,1,1,1,1,2,2,2,2,2,3,3,3,4,4,4,4,5,5,5,6,6,7,7,8,9,11} + 1
[6,2,1,1,1]
=> 10000101110 => 00001011101 => ? => ? ∊ {1,1,1,1,1,1,2,2,2,2,2,3,3,3,4,4,4,4,5,5,5,6,6,7,7,8,9,11} + 1
[5,2,1,1,1,1]
=> 10001011110 => 00010111101 => ? => ? ∊ {1,1,1,1,1,1,2,2,2,2,2,3,3,3,4,4,4,4,5,5,5,6,6,7,7,8,9,11} + 1
[5,1,1,1,1,1,1]
=> 100001111110 => ? => ? => ? ∊ {1,1,1,1,1,1,2,2,2,2,2,3,3,3,4,4,4,4,5,5,5,6,6,7,7,8,9,11} + 1
[4,3,1,1,1,1]
=> 1010011110 => ? => ? => ? ∊ {1,1,1,1,1,1,2,2,2,2,2,3,3,3,4,4,4,4,5,5,5,6,6,7,7,8,9,11} + 1
[4,2,2,1,1,1]
=> 1001101110 => ? => ? => ? ∊ {1,1,1,1,1,1,2,2,2,2,2,3,3,3,4,4,4,4,5,5,5,6,6,7,7,8,9,11} + 1
[4,2,1,1,1,1,1]
=> 10010111110 => 00101111101 => ? => ? ∊ {1,1,1,1,1,1,2,2,2,2,2,3,3,3,4,4,4,4,5,5,5,6,6,7,7,8,9,11} + 1
[4,1,1,1,1,1,1,1]
=> 100011111110 => ? => ? => ? ∊ {1,1,1,1,1,1,2,2,2,2,2,3,3,3,4,4,4,4,5,5,5,6,6,7,7,8,9,11} + 1
[3,3,1,1,1,1,1]
=> 1100111110 => ? => ? => ? ∊ {1,1,1,1,1,1,2,2,2,2,2,3,3,3,4,4,4,4,5,5,5,6,6,7,7,8,9,11} + 1
[3,2,2,1,1,1,1]
=> 1011011110 => ? => ? => ? ∊ {1,1,1,1,1,1,2,2,2,2,2,3,3,3,4,4,4,4,5,5,5,6,6,7,7,8,9,11} + 1
[3,2,1,1,1,1,1,1]
=> 10101111110 => 01011111101 => ? => ? ∊ {1,1,1,1,1,1,2,2,2,2,2,3,3,3,4,4,4,4,5,5,5,6,6,7,7,8,9,11} + 1
[3,1,1,1,1,1,1,1,1]
=> 100111111110 => ? => ? => ? ∊ {1,1,1,1,1,1,2,2,2,2,2,3,3,3,4,4,4,4,5,5,5,6,6,7,7,8,9,11} + 1
[2,2,2,1,1,1,1,1]
=> 1110111110 => 1101111101 => ? => ? ∊ {1,1,1,1,1,1,2,2,2,2,2,3,3,3,4,4,4,4,5,5,5,6,6,7,7,8,9,11} + 1
[2,2,1,1,1,1,1,1,1]
=> 11011111110 => 10111111101 => ? => ? ∊ {1,1,1,1,1,1,2,2,2,2,2,3,3,3,4,4,4,4,5,5,5,6,6,7,7,8,9,11} + 1
[2,1,1,1,1,1,1,1,1,1]
=> 101111111110 => ? => ? => ? ∊ {1,1,1,1,1,1,2,2,2,2,2,3,3,3,4,4,4,4,5,5,5,6,6,7,7,8,9,11} + 1
[1,1,1,1,1,1,1,1,1,1,1]
=> 111111111110 => ? => ? => ? ∊ {1,1,1,1,1,1,2,2,2,2,2,3,3,3,4,4,4,4,5,5,5,6,6,7,7,8,9,11} + 1
[12]
=> 1000000000000 => ? => ? => ? ∊ {1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,3,4,4,4,4,4,4,5,5,5,5,6,6,6,6,7,7,8,8,9,10,12} + 1
[11,1]
=> 1000000000010 => ? => ? => ? ∊ {1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,3,4,4,4,4,4,4,5,5,5,5,6,6,6,6,7,7,8,8,9,10,12} + 1
[10,2]
=> 100000000100 => 000000001001 => ? => ? ∊ {1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,3,4,4,4,4,4,4,5,5,5,5,6,6,6,6,7,7,8,8,9,10,12} + 1
[10,1,1]
=> 1000000000110 => ? => ? => ? ∊ {1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,3,4,4,4,4,4,4,5,5,5,5,6,6,6,6,7,7,8,8,9,10,12} + 1
[9,3]
=> 10000001000 => ? => ? => ? ∊ {1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,3,4,4,4,4,4,4,5,5,5,5,6,6,6,6,7,7,8,8,9,10,12} + 1
[9,2,1]
=> 100000001010 => ? => ? => ? ∊ {1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,3,4,4,4,4,4,4,5,5,5,5,6,6,6,6,7,7,8,8,9,10,12} + 1
[9,1,1,1]
=> 1000000001110 => ? => ? => ? ∊ {1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,3,4,4,4,4,4,4,5,5,5,5,6,6,6,6,7,7,8,8,9,10,12} + 1
[8,4]
=> 1000010000 => 0000100001 => ? => ? ∊ {1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,3,4,4,4,4,4,4,5,5,5,5,6,6,6,6,7,7,8,8,9,10,12} + 1
[8,3,1]
=> 10000010010 => ? => ? => ? ∊ {1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,3,4,4,4,4,4,4,5,5,5,5,6,6,6,6,7,7,8,8,9,10,12} + 1
[8,2,2]
=> 10000001100 => ? => ? => ? ∊ {1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,3,4,4,4,4,4,4,5,5,5,5,6,6,6,6,7,7,8,8,9,10,12} + 1
[8,2,1,1]
=> 100000010110 => ? => ? => ? ∊ {1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,3,4,4,4,4,4,4,5,5,5,5,6,6,6,6,7,7,8,8,9,10,12} + 1
[8,1,1,1,1]
=> 1000000011110 => ? => ? => ? ∊ {1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,3,4,4,4,4,4,4,5,5,5,5,6,6,6,6,7,7,8,8,9,10,12} + 1
[7,4,1]
=> 1000100010 => ? => ? => ? ∊ {1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,3,4,4,4,4,4,4,5,5,5,5,6,6,6,6,7,7,8,8,9,10,12} + 1
[7,3,2]
=> 1000010100 => ? => ? => ? ∊ {1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,3,4,4,4,4,4,4,5,5,5,5,6,6,6,6,7,7,8,8,9,10,12} + 1
[7,3,1,1]
=> 10000100110 => ? => ? => ? ∊ {1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,3,4,4,4,4,4,4,5,5,5,5,6,6,6,6,7,7,8,8,9,10,12} + 1
[7,2,2,1]
=> 10000011010 => ? => ? => ? ∊ {1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,3,4,4,4,4,4,4,5,5,5,5,6,6,6,6,7,7,8,8,9,10,12} + 1
[7,2,1,1,1]
=> 100000101110 => ? => ? => ? ∊ {1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,3,4,4,4,4,4,4,5,5,5,5,6,6,6,6,7,7,8,8,9,10,12} + 1
[7,1,1,1,1,1]
=> 1000000111110 => ? => ? => ? ∊ {1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,3,4,4,4,4,4,4,5,5,5,5,6,6,6,6,7,7,8,8,9,10,12} + 1
[6,4,1,1]
=> 1001000110 => 0010001101 => ? => ? ∊ {1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,3,4,4,4,4,4,4,5,5,5,5,6,6,6,6,7,7,8,8,9,10,12} + 1
[6,3,2,1]
=> 1000101010 => 0001010101 => ? => ? ∊ {1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,3,4,4,4,4,4,4,5,5,5,5,6,6,6,6,7,7,8,8,9,10,12} + 1
[6,3,1,1,1]
=> 10001001110 => ? => ? => ? ∊ {1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,3,4,4,4,4,4,4,5,5,5,5,6,6,6,6,7,7,8,8,9,10,12} + 1
[6,2,2,2]
=> 1000011100 => 0000111001 => ? => ? ∊ {1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,3,4,4,4,4,4,4,5,5,5,5,6,6,6,6,7,7,8,8,9,10,12} + 1
Description
The position of the first one in a binary word after appending a 1 at the end.
Regarding the binary word as a subset of $\{1,\dots,n,n+1\}$ that contains $n+1$, this is the minimal element of the set.
Matching statistic: St000147
Mp00095: Integer partitions —to binary word⟶ Binary words
Mp00178: Binary words —to composition⟶ Integer compositions
Mp00040: Integer compositions —to partition⟶ Integer partitions
St000147: Integer partitions ⟶ ℤResult quality: 33% ●values known / values provided: 33%●distinct values known / distinct values provided: 62%
Mp00178: Binary words —to composition⟶ Integer compositions
Mp00040: Integer compositions —to partition⟶ Integer partitions
St000147: Integer partitions ⟶ ℤResult quality: 33% ●values known / values provided: 33%●distinct values known / distinct values provided: 62%
Values
[1]
=> 10 => [1,2] => [2,1]
=> 2 = 1 + 1
[2]
=> 100 => [1,3] => [3,1]
=> 3 = 2 + 1
[1,1]
=> 110 => [1,1,2] => [2,1,1]
=> 2 = 1 + 1
[3]
=> 1000 => [1,4] => [4,1]
=> 4 = 3 + 1
[2,1]
=> 1010 => [1,2,2] => [2,2,1]
=> 2 = 1 + 1
[1,1,1]
=> 1110 => [1,1,1,2] => [2,1,1,1]
=> 2 = 1 + 1
[4]
=> 10000 => [1,5] => [5,1]
=> 5 = 4 + 1
[3,1]
=> 10010 => [1,3,2] => [3,2,1]
=> 3 = 2 + 1
[2,2]
=> 1100 => [1,1,3] => [3,1,1]
=> 3 = 2 + 1
[2,1,1]
=> 10110 => [1,2,1,2] => [2,2,1,1]
=> 2 = 1 + 1
[1,1,1,1]
=> 11110 => [1,1,1,1,2] => [2,1,1,1,1]
=> 2 = 1 + 1
[5]
=> 100000 => [1,6] => [6,1]
=> 6 = 5 + 1
[4,1]
=> 100010 => [1,4,2] => [4,2,1]
=> 4 = 3 + 1
[3,2]
=> 10100 => [1,2,3] => [3,2,1]
=> 3 = 2 + 1
[3,1,1]
=> 100110 => [1,3,1,2] => [3,2,1,1]
=> 3 = 2 + 1
[2,2,1]
=> 11010 => [1,1,2,2] => [2,2,1,1]
=> 2 = 1 + 1
[2,1,1,1]
=> 101110 => [1,2,1,1,2] => [2,2,1,1,1]
=> 2 = 1 + 1
[1,1,1,1,1]
=> 111110 => [1,1,1,1,1,2] => [2,1,1,1,1,1]
=> 2 = 1 + 1
[6]
=> 1000000 => [1,7] => [7,1]
=> 7 = 6 + 1
[5,1]
=> 1000010 => [1,5,2] => [5,2,1]
=> 5 = 4 + 1
[4,2]
=> 100100 => [1,3,3] => [3,3,1]
=> 3 = 2 + 1
[4,1,1]
=> 1000110 => [1,4,1,2] => [4,2,1,1]
=> 4 = 3 + 1
[3,3]
=> 11000 => [1,1,4] => [4,1,1]
=> 4 = 3 + 1
[3,2,1]
=> 101010 => [1,2,2,2] => [2,2,2,1]
=> 2 = 1 + 1
[3,1,1,1]
=> 1001110 => [1,3,1,1,2] => [3,2,1,1,1]
=> 3 = 2 + 1
[2,2,2]
=> 11100 => [1,1,1,3] => [3,1,1,1]
=> 3 = 2 + 1
[2,2,1,1]
=> 110110 => [1,1,2,1,2] => [2,2,1,1,1]
=> 2 = 1 + 1
[2,1,1,1,1]
=> 1011110 => [1,2,1,1,1,2] => [2,2,1,1,1,1]
=> 2 = 1 + 1
[1,1,1,1,1,1]
=> 1111110 => [1,1,1,1,1,1,2] => [2,1,1,1,1,1,1]
=> 2 = 1 + 1
[7]
=> 10000000 => [1,8] => [8,1]
=> 8 = 7 + 1
[6,1]
=> 10000010 => [1,6,2] => [6,2,1]
=> 6 = 5 + 1
[5,2]
=> 1000100 => [1,4,3] => [4,3,1]
=> 4 = 3 + 1
[5,1,1]
=> 10000110 => [1,5,1,2] => [5,2,1,1]
=> 5 = 4 + 1
[4,3]
=> 101000 => [1,2,4] => [4,2,1]
=> 4 = 3 + 1
[4,2,1]
=> 1001010 => [1,3,2,2] => [3,2,2,1]
=> 3 = 2 + 1
[4,1,1,1]
=> 10001110 => [1,4,1,1,2] => [4,2,1,1,1]
=> 4 = 3 + 1
[3,3,1]
=> 110010 => [1,1,3,2] => [3,2,1,1]
=> 3 = 2 + 1
[3,2,2]
=> 101100 => [1,2,1,3] => [3,2,1,1]
=> 3 = 2 + 1
[3,2,1,1]
=> 1010110 => [1,2,2,1,2] => [2,2,2,1,1]
=> 2 = 1 + 1
[3,1,1,1,1]
=> 10011110 => [1,3,1,1,1,2] => [3,2,1,1,1,1]
=> 3 = 2 + 1
[2,2,2,1]
=> 111010 => [1,1,1,2,2] => [2,2,1,1,1]
=> 2 = 1 + 1
[2,2,1,1,1]
=> 1101110 => [1,1,2,1,1,2] => [2,2,1,1,1,1]
=> 2 = 1 + 1
[2,1,1,1,1,1]
=> 10111110 => [1,2,1,1,1,1,2] => [2,2,1,1,1,1,1]
=> 2 = 1 + 1
[1,1,1,1,1,1,1]
=> 11111110 => [1,1,1,1,1,1,1,2] => [2,1,1,1,1,1,1,1]
=> 2 = 1 + 1
[8]
=> 100000000 => [1,9] => [9,1]
=> 9 = 8 + 1
[7,1]
=> 100000010 => [1,7,2] => [7,2,1]
=> 7 = 6 + 1
[6,2]
=> 10000100 => [1,5,3] => [5,3,1]
=> 5 = 4 + 1
[6,1,1]
=> 100000110 => [1,6,1,2] => [6,2,1,1]
=> 6 = 5 + 1
[5,3]
=> 1001000 => [1,3,4] => [4,3,1]
=> 4 = 3 + 1
[5,2,1]
=> 10001010 => [1,4,2,2] => [4,2,2,1]
=> 4 = 3 + 1
[11]
=> 100000000000 => ? => ?
=> ? ∊ {1,1,1,1,1,2,2,2,2,2,2,3,3,3,3,4,4,4,4,5,5,5,5,6,6,7,8,9,11} + 1
[10,1]
=> 100000000010 => ? => ?
=> ? ∊ {1,1,1,1,1,2,2,2,2,2,2,3,3,3,3,4,4,4,4,5,5,5,5,6,6,7,8,9,11} + 1
[9,1,1]
=> 100000000110 => ? => ?
=> ? ∊ {1,1,1,1,1,2,2,2,2,2,2,3,3,3,3,4,4,4,4,5,5,5,5,6,6,7,8,9,11} + 1
[8,3]
=> 1000001000 => ? => ?
=> ? ∊ {1,1,1,1,1,2,2,2,2,2,2,3,3,3,3,4,4,4,4,5,5,5,5,6,6,7,8,9,11} + 1
[8,2,1]
=> 10000001010 => [1,7,2,2] => ?
=> ? ∊ {1,1,1,1,1,2,2,2,2,2,2,3,3,3,3,4,4,4,4,5,5,5,5,6,6,7,8,9,11} + 1
[8,1,1,1]
=> 100000001110 => ? => ?
=> ? ∊ {1,1,1,1,1,2,2,2,2,2,2,3,3,3,3,4,4,4,4,5,5,5,5,6,6,7,8,9,11} + 1
[7,3,1]
=> 1000010010 => ? => ?
=> ? ∊ {1,1,1,1,1,2,2,2,2,2,2,3,3,3,3,4,4,4,4,5,5,5,5,6,6,7,8,9,11} + 1
[7,2,2]
=> 1000001100 => ? => ?
=> ? ∊ {1,1,1,1,1,2,2,2,2,2,2,3,3,3,3,4,4,4,4,5,5,5,5,6,6,7,8,9,11} + 1
[7,2,1,1]
=> 10000010110 => [1,6,2,1,2] => ?
=> ? ∊ {1,1,1,1,1,2,2,2,2,2,2,3,3,3,3,4,4,4,4,5,5,5,5,6,6,7,8,9,11} + 1
[7,1,1,1,1]
=> 100000011110 => ? => ?
=> ? ∊ {1,1,1,1,1,2,2,2,2,2,2,3,3,3,3,4,4,4,4,5,5,5,5,6,6,7,8,9,11} + 1
[6,3,1,1]
=> 1000100110 => ? => ?
=> ? ∊ {1,1,1,1,1,2,2,2,2,2,2,3,3,3,3,4,4,4,4,5,5,5,5,6,6,7,8,9,11} + 1
[6,2,2,1]
=> 1000011010 => ? => ?
=> ? ∊ {1,1,1,1,1,2,2,2,2,2,2,3,3,3,3,4,4,4,4,5,5,5,5,6,6,7,8,9,11} + 1
[6,2,1,1,1]
=> 10000101110 => [1,5,2,1,1,2] => ?
=> ? ∊ {1,1,1,1,1,2,2,2,2,2,2,3,3,3,3,4,4,4,4,5,5,5,5,6,6,7,8,9,11} + 1
[6,1,1,1,1,1]
=> 100000111110 => [1,6,1,1,1,1,2] => ?
=> ? ∊ {1,1,1,1,1,2,2,2,2,2,2,3,3,3,3,4,4,4,4,5,5,5,5,6,6,7,8,9,11} + 1
[5,3,1,1,1]
=> 1001001110 => [1,3,3,1,1,2] => ?
=> ? ∊ {1,1,1,1,1,2,2,2,2,2,2,3,3,3,3,4,4,4,4,5,5,5,5,6,6,7,8,9,11} + 1
[5,2,2,1,1]
=> 1000110110 => [1,4,1,2,1,2] => ?
=> ? ∊ {1,1,1,1,1,2,2,2,2,2,2,3,3,3,3,4,4,4,4,5,5,5,5,6,6,7,8,9,11} + 1
[5,2,1,1,1,1]
=> 10001011110 => [1,4,2,1,1,1,2] => ?
=> ? ∊ {1,1,1,1,1,2,2,2,2,2,2,3,3,3,3,4,4,4,4,5,5,5,5,6,6,7,8,9,11} + 1
[5,1,1,1,1,1,1]
=> 100001111110 => ? => ?
=> ? ∊ {1,1,1,1,1,2,2,2,2,2,2,3,3,3,3,4,4,4,4,5,5,5,5,6,6,7,8,9,11} + 1
[4,3,1,1,1,1]
=> 1010011110 => ? => ?
=> ? ∊ {1,1,1,1,1,2,2,2,2,2,2,3,3,3,3,4,4,4,4,5,5,5,5,6,6,7,8,9,11} + 1
[4,2,2,1,1,1]
=> 1001101110 => ? => ?
=> ? ∊ {1,1,1,1,1,2,2,2,2,2,2,3,3,3,3,4,4,4,4,5,5,5,5,6,6,7,8,9,11} + 1
[4,2,1,1,1,1,1]
=> 10010111110 => [1,3,2,1,1,1,1,2] => ?
=> ? ∊ {1,1,1,1,1,2,2,2,2,2,2,3,3,3,3,4,4,4,4,5,5,5,5,6,6,7,8,9,11} + 1
[4,1,1,1,1,1,1,1]
=> 100011111110 => ? => ?
=> ? ∊ {1,1,1,1,1,2,2,2,2,2,2,3,3,3,3,4,4,4,4,5,5,5,5,6,6,7,8,9,11} + 1
[3,3,1,1,1,1,1]
=> 1100111110 => ? => ?
=> ? ∊ {1,1,1,1,1,2,2,2,2,2,2,3,3,3,3,4,4,4,4,5,5,5,5,6,6,7,8,9,11} + 1
[3,2,2,1,1,1,1]
=> 1011011110 => ? => ?
=> ? ∊ {1,1,1,1,1,2,2,2,2,2,2,3,3,3,3,4,4,4,4,5,5,5,5,6,6,7,8,9,11} + 1
[3,2,1,1,1,1,1,1]
=> 10101111110 => [1,2,2,1,1,1,1,1,2] => ?
=> ? ∊ {1,1,1,1,1,2,2,2,2,2,2,3,3,3,3,4,4,4,4,5,5,5,5,6,6,7,8,9,11} + 1
[3,1,1,1,1,1,1,1,1]
=> 100111111110 => ? => ?
=> ? ∊ {1,1,1,1,1,2,2,2,2,2,2,3,3,3,3,4,4,4,4,5,5,5,5,6,6,7,8,9,11} + 1
[2,2,2,1,1,1,1,1]
=> 1110111110 => [1,1,1,2,1,1,1,1,2] => ?
=> ? ∊ {1,1,1,1,1,2,2,2,2,2,2,3,3,3,3,4,4,4,4,5,5,5,5,6,6,7,8,9,11} + 1
[2,1,1,1,1,1,1,1,1,1]
=> 101111111110 => ? => ?
=> ? ∊ {1,1,1,1,1,2,2,2,2,2,2,3,3,3,3,4,4,4,4,5,5,5,5,6,6,7,8,9,11} + 1
[1,1,1,1,1,1,1,1,1,1,1]
=> 111111111110 => ? => ?
=> ? ∊ {1,1,1,1,1,2,2,2,2,2,2,3,3,3,3,4,4,4,4,5,5,5,5,6,6,7,8,9,11} + 1
[12]
=> 1000000000000 => ? => ?
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,5,5,5,5,6,6,6,6,7,7,8,8,9,10,12} + 1
[11,1]
=> 1000000000010 => ? => ?
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,5,5,5,5,6,6,6,6,7,7,8,8,9,10,12} + 1
[10,2]
=> 100000000100 => [1,9,3] => ?
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,5,5,5,5,6,6,6,6,7,7,8,8,9,10,12} + 1
[10,1,1]
=> 1000000000110 => ? => ?
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,5,5,5,5,6,6,6,6,7,7,8,8,9,10,12} + 1
[9,3]
=> 10000001000 => ? => ?
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,5,5,5,5,6,6,6,6,7,7,8,8,9,10,12} + 1
[9,2,1]
=> 100000001010 => ? => ?
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,5,5,5,5,6,6,6,6,7,7,8,8,9,10,12} + 1
[9,1,1,1]
=> 1000000001110 => ? => ?
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,5,5,5,5,6,6,6,6,7,7,8,8,9,10,12} + 1
[8,3,1]
=> 10000010010 => ? => ?
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,5,5,5,5,6,6,6,6,7,7,8,8,9,10,12} + 1
[8,2,2]
=> 10000001100 => ? => ?
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,5,5,5,5,6,6,6,6,7,7,8,8,9,10,12} + 1
[8,2,1,1]
=> 100000010110 => ? => ?
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,5,5,5,5,6,6,6,6,7,7,8,8,9,10,12} + 1
[8,1,1,1,1]
=> 1000000011110 => ? => ?
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,5,5,5,5,6,6,6,6,7,7,8,8,9,10,12} + 1
[7,4,1]
=> 1000100010 => ? => ?
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,5,5,5,5,6,6,6,6,7,7,8,8,9,10,12} + 1
[7,3,2]
=> 1000010100 => ? => ?
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,5,5,5,5,6,6,6,6,7,7,8,8,9,10,12} + 1
[7,3,1,1]
=> 10000100110 => ? => ?
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,5,5,5,5,6,6,6,6,7,7,8,8,9,10,12} + 1
[7,2,2,1]
=> 10000011010 => ? => ?
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,5,5,5,5,6,6,6,6,7,7,8,8,9,10,12} + 1
[7,2,1,1,1]
=> 100000101110 => ? => ?
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,5,5,5,5,6,6,6,6,7,7,8,8,9,10,12} + 1
[7,1,1,1,1,1]
=> 1000000111110 => ? => ?
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,5,5,5,5,6,6,6,6,7,7,8,8,9,10,12} + 1
[6,4,1,1]
=> 1001000110 => [1,3,4,1,2] => ?
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,5,5,5,5,6,6,6,6,7,7,8,8,9,10,12} + 1
[6,3,2,1]
=> 1000101010 => [1,4,2,2,2] => ?
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,5,5,5,5,6,6,6,6,7,7,8,8,9,10,12} + 1
[6,3,1,1,1]
=> 10001001110 => ? => ?
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,5,5,5,5,6,6,6,6,7,7,8,8,9,10,12} + 1
[6,2,2,2]
=> 1000011100 => [1,5,1,1,3] => ?
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,5,5,5,5,6,6,6,6,7,7,8,8,9,10,12} + 1
Description
The largest part of an integer partition.
Matching statistic: St000845
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00023: Dyck paths —to non-crossing permutation⟶ Permutations
Mp00065: Permutations —permutation poset⟶ Posets
St000845: Posets ⟶ ℤResult quality: 31% ●values known / values provided: 31%●distinct values known / distinct values provided: 38%
Mp00023: Dyck paths —to non-crossing permutation⟶ Permutations
Mp00065: Permutations —permutation poset⟶ Posets
St000845: Posets ⟶ ℤResult quality: 31% ●values known / values provided: 31%●distinct values known / distinct values provided: 38%
Values
[1]
=> [1,0,1,0]
=> [1,2] => ([(0,1)],2)
=> 1
[2]
=> [1,1,0,0,1,0]
=> [2,1,3] => ([(0,2),(1,2)],3)
=> 1
[1,1]
=> [1,0,1,1,0,0]
=> [1,3,2] => ([(0,1),(0,2)],3)
=> 2
[3]
=> [1,1,1,0,0,0,1,0]
=> [3,2,1,4] => ([(0,3),(1,3),(2,3)],4)
=> 1
[2,1]
=> [1,0,1,0,1,0]
=> [1,2,3] => ([(0,2),(2,1)],3)
=> 1
[1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [1,4,3,2] => ([(0,1),(0,2),(0,3)],4)
=> 3
[4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [4,3,2,1,5] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 1
[3,1]
=> [1,1,0,1,0,0,1,0]
=> [2,3,1,4] => ([(0,3),(1,2),(2,3)],4)
=> 1
[2,2]
=> [1,1,0,0,1,1,0,0]
=> [2,1,4,3] => ([(0,2),(0,3),(1,2),(1,3)],4)
=> 2
[2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,3,4,2] => ([(0,2),(0,3),(3,1)],4)
=> 2
[1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,5,4,3,2] => ([(0,1),(0,2),(0,3),(0,4)],5)
=> 4
[5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> [5,4,3,2,1,6] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> 1
[4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [4,2,3,1,5] => ([(0,4),(1,4),(2,3),(3,4)],5)
=> 1
[3,2]
=> [1,1,0,0,1,0,1,0]
=> [2,1,3,4] => ([(0,3),(1,3),(3,2)],4)
=> 1
[3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [1,3,2,4] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
[2,2,1]
=> [1,0,1,0,1,1,0,0]
=> [1,2,4,3] => ([(0,3),(3,1),(3,2)],4)
=> 2
[2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,5,3,4,2] => ([(0,2),(0,3),(0,4),(4,1)],5)
=> 3
[1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,6,5,4,3,2] => ([(0,1),(0,2),(0,3),(0,4),(0,5)],6)
=> 5
[6]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> [6,5,4,3,2,1,7] => ([(0,6),(1,6),(2,6),(3,6),(4,6),(5,6)],7)
=> 1
[5,1]
=> [1,1,1,1,0,1,0,0,0,0,1,0]
=> [5,3,4,2,1,6] => ([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
=> 1
[4,2]
=> [1,1,1,0,0,1,0,0,1,0]
=> [3,2,4,1,5] => ([(0,4),(1,3),(2,3),(3,4)],5)
=> 1
[4,1,1]
=> [1,1,0,1,1,0,0,0,1,0]
=> [2,4,3,1,5] => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> 2
[3,3]
=> [1,1,1,0,0,0,1,1,0,0]
=> [3,2,1,5,4] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> 2
[3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> 1
[3,1,1,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,4,3,5,2] => ([(0,1),(0,2),(0,3),(2,4),(3,4)],5)
=> 3
[2,2,2]
=> [1,1,0,0,1,1,1,0,0,0]
=> [2,1,5,4,3] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4)],5)
=> 3
[2,2,1,1]
=> [1,0,1,1,0,1,1,0,0,0]
=> [1,3,5,4,2] => ([(0,3),(0,4),(4,1),(4,2)],5)
=> 2
[2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> [1,6,4,5,3,2] => ([(0,2),(0,3),(0,4),(0,5),(5,1)],6)
=> 4
[1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,7,6,5,4,3,2] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6)],7)
=> 6
[7]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> [7,6,5,4,3,2,1,8] => ([(0,7),(1,7),(2,7),(3,7),(4,7),(5,7),(6,7)],8)
=> ? ∊ {1,7}
[6,1]
=> [1,1,1,1,1,0,1,0,0,0,0,0,1,0]
=> [6,5,3,4,2,1,7] => ([(0,6),(1,6),(2,6),(3,6),(4,5),(5,6)],7)
=> 1
[5,2]
=> [1,1,1,1,0,0,1,0,0,0,1,0]
=> [5,3,2,4,1,6] => ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> 1
[5,1,1]
=> [1,1,1,0,1,1,0,0,0,0,1,0]
=> [5,2,4,3,1,6] => ([(0,5),(1,5),(2,3),(2,4),(3,5),(4,5)],6)
=> 2
[4,3]
=> [1,1,1,0,0,0,1,0,1,0]
=> [3,2,1,4,5] => ([(0,4),(1,4),(2,4),(4,3)],5)
=> 1
[4,2,1]
=> [1,1,0,1,0,1,0,0,1,0]
=> [2,3,4,1,5] => ([(0,4),(1,2),(2,3),(3,4)],5)
=> 1
[4,1,1,1]
=> [1,0,1,1,1,0,0,0,1,0]
=> [1,4,3,2,5] => ([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
=> 3
[3,3,1]
=> [1,1,0,1,0,0,1,1,0,0]
=> [2,3,1,5,4] => ([(0,3),(0,4),(1,2),(2,3),(2,4)],5)
=> 2
[3,2,2]
=> [1,1,0,0,1,1,0,1,0,0]
=> [2,1,4,5,3] => ([(0,3),(0,4),(1,3),(1,4),(4,2)],5)
=> 2
[3,2,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,3,4,5,2] => ([(0,2),(0,4),(3,1),(4,3)],5)
=> 2
[3,1,1,1,1]
=> [1,0,1,1,1,1,0,0,1,0,0,0]
=> [1,6,4,3,5,2] => ([(0,1),(0,2),(0,3),(0,4),(3,5),(4,5)],6)
=> 4
[2,2,2,1]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,2,5,4,3] => ([(0,4),(4,1),(4,2),(4,3)],5)
=> 3
[2,2,1,1,1]
=> [1,0,1,1,1,0,1,1,0,0,0,0]
=> [1,6,3,5,4,2] => ([(0,3),(0,4),(0,5),(5,1),(5,2)],6)
=> 3
[2,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,1,0,0,0,0,0]
=> [1,7,6,4,5,3,2] => ([(0,2),(0,3),(0,4),(0,5),(0,6),(6,1)],7)
=> 5
[1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [1,8,7,6,5,4,3,2] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7)],8)
=> ? ∊ {1,7}
[8]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0]
=> [8,7,6,5,4,3,2,1,9] => ([(0,8),(1,8),(2,8),(3,8),(4,8),(5,8),(6,8),(7,8)],9)
=> ? ∊ {1,1,6,8}
[7,1]
=> [1,1,1,1,1,1,0,1,0,0,0,0,0,0,1,0]
=> [7,6,4,5,3,2,1,8] => ([(0,7),(1,7),(2,7),(3,7),(4,7),(5,6),(6,7)],8)
=> ? ∊ {1,1,6,8}
[6,2]
=> [1,1,1,1,1,0,0,1,0,0,0,0,1,0]
=> [6,4,3,5,2,1,7] => ([(0,6),(1,6),(2,6),(3,5),(4,5),(5,6)],7)
=> 1
[6,1,1]
=> [1,1,1,1,0,1,1,0,0,0,0,0,1,0]
=> [6,3,5,4,2,1,7] => ([(0,6),(1,6),(2,6),(3,4),(3,5),(4,6),(5,6)],7)
=> 2
[5,3]
=> [1,1,1,1,0,0,0,1,0,0,1,0]
=> [4,3,2,5,1,6] => ([(0,5),(1,5),(2,5),(3,4),(5,4)],6)
=> 1
[5,2,1]
=> [1,1,1,0,1,0,1,0,0,0,1,0]
=> [5,2,3,4,1,6] => ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
=> 1
[5,1,1,1]
=> [1,1,0,1,1,1,0,0,0,0,1,0]
=> [2,5,4,3,1,6] => ([(0,5),(1,2),(1,3),(1,4),(2,5),(3,5),(4,5)],6)
=> 3
[4,4]
=> [1,1,1,1,0,0,0,0,1,1,0,0]
=> [4,3,2,1,6,5] => ([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> 2
[4,3,1]
=> [1,1,0,1,0,0,1,0,1,0]
=> [2,3,1,4,5] => ([(0,4),(1,2),(2,4),(4,3)],5)
=> 1
[4,2,2]
=> [1,1,0,0,1,1,0,0,1,0]
=> [2,1,4,3,5] => ([(0,3),(0,4),(1,3),(1,4),(3,2),(4,2)],5)
=> 2
[2,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,1,0,0,0,0,0,0]
=> [1,8,7,5,6,4,3,2] => ([(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(7,1)],8)
=> ? ∊ {1,1,6,8}
[1,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [1,9,8,7,6,5,4,3,2] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(0,8)],9)
=> ? ∊ {1,1,6,8}
[9]
=> [1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,1,0]
=> [9,8,7,6,5,4,3,2,1,10] => ([(0,9),(1,9),(2,9),(3,9),(4,9),(5,9),(6,9),(7,9),(8,9)],10)
=> ? ∊ {1,1,1,2,5,6,7,9}
[8,1]
=> [1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,1,0]
=> [8,7,6,4,5,3,2,1,9] => ([(0,8),(1,8),(2,8),(3,8),(4,8),(5,8),(6,7),(7,8)],9)
=> ? ∊ {1,1,1,2,5,6,7,9}
[7,2]
=> [1,1,1,1,1,1,0,0,1,0,0,0,0,0,1,0]
=> [7,6,4,3,5,2,1,8] => ([(0,7),(1,7),(2,7),(3,7),(4,6),(5,6),(6,7)],8)
=> ? ∊ {1,1,1,2,5,6,7,9}
[7,1,1]
=> [1,1,1,1,1,0,1,1,0,0,0,0,0,0,1,0]
=> [7,6,3,5,4,2,1,8] => ([(0,7),(1,7),(2,7),(3,7),(4,5),(4,6),(5,7),(6,7)],8)
=> ? ∊ {1,1,1,2,5,6,7,9}
[3,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,1,0,0,0,0,0]
=> [1,8,7,5,4,6,3,2] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(5,7),(6,7)],8)
=> ? ∊ {1,1,1,2,5,6,7,9}
[2,2,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,1,1,0,0,0,0,0,0]
=> [1,8,7,4,6,5,3,2] => ([(0,3),(0,4),(0,5),(0,6),(0,7),(7,1),(7,2)],8)
=> ? ∊ {1,1,1,2,5,6,7,9}
[2,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0]
=> [1,9,8,7,5,6,4,3,2] => ([(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(0,8),(8,1)],9)
=> ? ∊ {1,1,1,2,5,6,7,9}
[1,1,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0]
=> [1,10,9,8,7,6,5,4,3,2] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(0,8),(0,9)],10)
=> ? ∊ {1,1,1,2,5,6,7,9}
[10]
=> [1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,1,0]
=> [10,9,8,7,6,5,4,3,2,1,11] => ([(0,10),(1,10),(2,10),(3,10),(4,10),(5,10),(6,10),(7,10),(8,10),(9,10)],11)
=> ? ∊ {1,1,1,1,1,2,3,4,5,5,6,6,7,8,10}
[9,1]
=> [1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0,1,0]
=> [9,8,7,5,6,4,3,2,1,10] => ([(0,9),(1,9),(2,9),(3,9),(4,9),(5,9),(6,9),(7,8),(8,9)],10)
=> ? ∊ {1,1,1,1,1,2,3,4,5,5,6,6,7,8,10}
[8,2]
=> [1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0,1,0]
=> [8,7,5,4,6,3,2,1,9] => ([(0,8),(1,8),(2,8),(3,8),(4,8),(5,7),(6,7),(7,8)],9)
=> ? ∊ {1,1,1,1,1,2,3,4,5,5,6,6,7,8,10}
[8,1,1]
=> [1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0,1,0]
=> [8,7,4,6,5,3,2,1,9] => ([(0,8),(1,8),(2,8),(3,8),(4,8),(5,6),(5,7),(6,8),(7,8)],9)
=> ? ∊ {1,1,1,1,1,2,3,4,5,5,6,6,7,8,10}
[7,3]
=> [1,1,1,1,1,1,0,0,0,1,0,0,0,0,1,0]
=> [7,5,4,3,6,2,1,8] => ([(0,7),(1,7),(2,7),(3,6),(4,6),(5,6),(6,7)],8)
=> ? ∊ {1,1,1,1,1,2,3,4,5,5,6,6,7,8,10}
[7,2,1]
=> [1,1,1,1,1,0,1,0,1,0,0,0,0,0,1,0]
=> [7,6,3,4,5,2,1,8] => ([(0,7),(1,7),(2,7),(3,7),(4,5),(5,6),(6,7)],8)
=> ? ∊ {1,1,1,1,1,2,3,4,5,5,6,6,7,8,10}
[7,1,1,1]
=> [1,1,1,1,0,1,1,1,0,0,0,0,0,0,1,0]
=> [7,3,6,5,4,2,1,8] => ([(0,7),(1,7),(2,7),(3,4),(3,5),(3,6),(4,7),(5,7),(6,7)],8)
=> ? ∊ {1,1,1,1,1,2,3,4,5,5,6,6,7,8,10}
[4,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,0,1,0,0,0,0]
=> [1,8,6,5,4,7,3,2] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(4,7),(5,7),(6,7)],8)
=> ? ∊ {1,1,1,1,1,2,3,4,5,5,6,6,7,8,10}
[3,2,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,1,0,1,0,0,0,0,0]
=> [1,8,7,4,5,6,3,2] => ([(0,2),(0,3),(0,4),(0,5),(0,7),(6,1),(7,6)],8)
=> ? ∊ {1,1,1,1,1,2,3,4,5,5,6,6,7,8,10}
[3,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0]
=> [1,9,8,6,5,7,4,3,2] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(6,8),(7,8)],9)
=> ? ∊ {1,1,1,1,1,2,3,4,5,5,6,6,7,8,10}
[2,2,2,2,2]
=> [1,1,0,0,1,1,1,1,1,0,0,0,0,0]
=> [2,1,7,6,5,4,3] => ([(0,2),(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6)],7)
=> ? ∊ {1,1,1,1,1,2,3,4,5,5,6,6,7,8,10}
[2,2,2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,1,1,0,0,0,0,0,0]
=> [1,8,4,7,6,5,3,2] => ([(0,4),(0,5),(0,6),(0,7),(7,1),(7,2),(7,3)],8)
=> ? ∊ {1,1,1,1,1,2,3,4,5,5,6,6,7,8,10}
[2,2,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0]
=> [1,9,8,5,7,6,4,3,2] => ([(0,3),(0,4),(0,5),(0,6),(0,7),(0,8),(8,1),(8,2)],9)
=> ? ∊ {1,1,1,1,1,2,3,4,5,5,6,6,7,8,10}
[2,1,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0]
=> [1,10,9,8,6,7,5,4,3,2] => ([(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(0,8),(0,9),(9,1)],10)
=> ? ∊ {1,1,1,1,1,2,3,4,5,5,6,6,7,8,10}
[1,1,1,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0]
=> [1,11,10,9,8,7,6,5,4,3,2] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(0,8),(0,9),(0,10)],11)
=> ? ∊ {1,1,1,1,1,2,3,4,5,5,6,6,7,8,10}
[11]
=> [1,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,0,1,0]
=> [11,10,9,8,7,6,5,4,3,2,1,12] => ([(0,11),(1,11),(2,11),(3,11),(4,11),(5,11),(6,11),(7,11),(8,11),(9,11),(10,11)],12)
=> ? ∊ {1,1,1,1,1,1,1,2,2,2,3,4,4,4,4,5,5,5,6,6,7,7,8,9,11}
[10,1]
=> [1,1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0,0,1,0]
=> [10,9,8,7,5,6,4,3,2,1,11] => ?
=> ? ∊ {1,1,1,1,1,1,1,2,2,2,3,4,4,4,4,5,5,5,6,6,7,7,8,9,11}
[9,2]
=> [1,1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0,0,1,0]
=> [9,8,7,5,4,6,3,2,1,10] => ?
=> ? ∊ {1,1,1,1,1,1,1,2,2,2,3,4,4,4,4,5,5,5,6,6,7,7,8,9,11}
[9,1,1]
=> [1,1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0,0,1,0]
=> [9,8,7,4,6,5,3,2,1,10] => ?
=> ? ∊ {1,1,1,1,1,1,1,2,2,2,3,4,4,4,4,5,5,5,6,6,7,7,8,9,11}
[8,3]
=> [1,1,1,1,1,1,1,0,0,0,1,0,0,0,0,0,1,0]
=> [8,7,5,4,3,6,2,1,9] => ?
=> ? ∊ {1,1,1,1,1,1,1,2,2,2,3,4,4,4,4,5,5,5,6,6,7,7,8,9,11}
[8,2,1]
=> [1,1,1,1,1,1,0,1,0,1,0,0,0,0,0,0,1,0]
=> [8,7,4,5,6,3,2,1,9] => ?
=> ? ∊ {1,1,1,1,1,1,1,2,2,2,3,4,4,4,4,5,5,5,6,6,7,7,8,9,11}
[8,1,1,1]
=> [1,1,1,1,1,0,1,1,1,0,0,0,0,0,0,0,1,0]
=> [8,7,3,6,5,4,2,1,9] => ?
=> ? ∊ {1,1,1,1,1,1,1,2,2,2,3,4,4,4,4,5,5,5,6,6,7,7,8,9,11}
[7,4]
=> [1,1,1,1,1,1,0,0,0,0,1,0,0,0,1,0]
=> [7,5,4,3,2,6,1,8] => ([(0,7),(1,7),(2,7),(3,7),(4,6),(5,6),(7,6)],8)
=> ? ∊ {1,1,1,1,1,1,1,2,2,2,3,4,4,4,4,5,5,5,6,6,7,7,8,9,11}
[7,3,1]
=> [1,1,1,1,1,0,1,0,0,1,0,0,0,0,1,0]
=> [7,5,3,4,6,2,1,8] => ?
=> ? ∊ {1,1,1,1,1,1,1,2,2,2,3,4,4,4,4,5,5,5,6,6,7,7,8,9,11}
[7,2,2]
=> [1,1,1,1,1,0,0,1,1,0,0,0,0,0,1,0]
=> [7,4,3,6,5,2,1,8] => ([(0,7),(1,7),(2,7),(3,5),(3,6),(4,5),(4,6),(5,7),(6,7)],8)
=> ? ∊ {1,1,1,1,1,1,1,2,2,2,3,4,4,4,4,5,5,5,6,6,7,7,8,9,11}
[7,2,1,1]
=> [1,1,1,1,0,1,1,0,1,0,0,0,0,0,1,0]
=> [7,3,6,4,5,2,1,8] => ([(0,7),(1,7),(2,7),(3,4),(3,5),(4,6),(5,7),(6,7)],8)
=> ? ∊ {1,1,1,1,1,1,1,2,2,2,3,4,4,4,4,5,5,5,6,6,7,7,8,9,11}
[7,1,1,1,1]
=> [1,1,1,0,1,1,1,1,0,0,0,0,0,0,1,0]
=> [7,2,6,5,4,3,1,8] => ([(0,7),(1,7),(2,3),(2,4),(2,5),(2,6),(3,7),(4,7),(5,7),(6,7)],8)
=> ? ∊ {1,1,1,1,1,1,1,2,2,2,3,4,4,4,4,5,5,5,6,6,7,7,8,9,11}
[5,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,0,0,1,0,0,0]
=> [1,8,6,5,4,3,7,2] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(3,7),(4,7),(5,7),(6,7)],8)
=> ? ∊ {1,1,1,1,1,1,1,2,2,2,3,4,4,4,4,5,5,5,6,6,7,7,8,9,11}
[4,2,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,1,0,0,1,0,0,0,0]
=> [1,8,6,4,5,7,3,2] => ([(0,2),(0,3),(0,4),(0,5),(0,6),(1,7),(5,7),(6,1)],8)
=> ? ∊ {1,1,1,1,1,1,1,2,2,2,3,4,4,4,4,5,5,5,6,6,7,7,8,9,11}
[4,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,0,0,0,1,0,0,0,0,0]
=> [1,9,8,6,5,4,7,3,2] => ?
=> ? ∊ {1,1,1,1,1,1,1,2,2,2,3,4,4,4,4,5,5,5,6,6,7,7,8,9,11}
[3,3,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,1,1,0,0,0,0,0]
=> [1,8,5,4,7,6,3,2] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(4,6),(4,7),(5,6),(5,7)],8)
=> ? ∊ {1,1,1,1,1,1,1,2,2,2,3,4,4,4,4,5,5,5,6,6,7,7,8,9,11}
[3,2,2,2,2]
=> [1,1,0,0,1,1,1,1,0,1,0,0,0,0]
=> [2,1,7,5,6,4,3] => ([(0,3),(0,4),(0,5),(0,6),(1,3),(1,4),(1,5),(1,6),(6,2)],7)
=> ? ∊ {1,1,1,1,1,1,1,2,2,2,3,4,4,4,4,5,5,5,6,6,7,7,8,9,11}
[3,2,2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,1,0,1,0,0,0,0,0]
=> [1,8,4,7,5,6,3,2] => ?
=> ? ∊ {1,1,1,1,1,1,1,2,2,2,3,4,4,4,4,5,5,5,6,6,7,7,8,9,11}
[3,2,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,1,0,1,0,0,0,0,0,0]
=> [1,9,8,5,6,7,4,3,2] => ([(0,2),(0,3),(0,4),(0,5),(0,6),(0,8),(7,1),(8,7)],9)
=> ? ∊ {1,1,1,1,1,1,1,2,2,2,3,4,4,4,4,5,5,5,6,6,7,7,8,9,11}
[3,1,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0,0]
=> [1,10,9,8,6,5,7,4,3,2] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(0,8),(7,9),(8,9)],10)
=> ? ∊ {1,1,1,1,1,1,1,2,2,2,3,4,4,4,4,5,5,5,6,6,7,7,8,9,11}
[2,2,2,2,1,1,1]
=> [1,0,1,1,1,0,1,1,1,1,0,0,0,0,0,0]
=> [1,8,3,7,6,5,4,2] => ([(0,5),(0,6),(0,7),(7,1),(7,2),(7,3),(7,4)],8)
=> ? ∊ {1,1,1,1,1,1,1,2,2,2,3,4,4,4,4,5,5,5,6,6,7,7,8,9,11}
Description
The maximal number of elements covered by an element in a poset.
Matching statistic: St000846
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00023: Dyck paths —to non-crossing permutation⟶ Permutations
Mp00065: Permutations —permutation poset⟶ Posets
St000846: Posets ⟶ ℤResult quality: 31% ●values known / values provided: 31%●distinct values known / distinct values provided: 38%
Mp00023: Dyck paths —to non-crossing permutation⟶ Permutations
Mp00065: Permutations —permutation poset⟶ Posets
St000846: Posets ⟶ ℤResult quality: 31% ●values known / values provided: 31%●distinct values known / distinct values provided: 38%
Values
[1]
=> [1,0,1,0]
=> [1,2] => ([(0,1)],2)
=> 1
[2]
=> [1,1,0,0,1,0]
=> [2,1,3] => ([(0,2),(1,2)],3)
=> 2
[1,1]
=> [1,0,1,1,0,0]
=> [1,3,2] => ([(0,1),(0,2)],3)
=> 1
[3]
=> [1,1,1,0,0,0,1,0]
=> [3,2,1,4] => ([(0,3),(1,3),(2,3)],4)
=> 3
[2,1]
=> [1,0,1,0,1,0]
=> [1,2,3] => ([(0,2),(2,1)],3)
=> 1
[1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [1,4,3,2] => ([(0,1),(0,2),(0,3)],4)
=> 1
[4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [4,3,2,1,5] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 4
[3,1]
=> [1,1,0,1,0,0,1,0]
=> [2,3,1,4] => ([(0,3),(1,2),(2,3)],4)
=> 2
[2,2]
=> [1,1,0,0,1,1,0,0]
=> [2,1,4,3] => ([(0,2),(0,3),(1,2),(1,3)],4)
=> 2
[2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,3,4,2] => ([(0,2),(0,3),(3,1)],4)
=> 1
[1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,5,4,3,2] => ([(0,1),(0,2),(0,3),(0,4)],5)
=> 1
[5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> [5,4,3,2,1,6] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> 5
[4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [4,2,3,1,5] => ([(0,4),(1,4),(2,3),(3,4)],5)
=> 3
[3,2]
=> [1,1,0,0,1,0,1,0]
=> [2,1,3,4] => ([(0,3),(1,3),(3,2)],4)
=> 2
[3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [1,3,2,4] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
[2,2,1]
=> [1,0,1,0,1,1,0,0]
=> [1,2,4,3] => ([(0,3),(3,1),(3,2)],4)
=> 1
[2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,5,3,4,2] => ([(0,2),(0,3),(0,4),(4,1)],5)
=> 1
[1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,6,5,4,3,2] => ([(0,1),(0,2),(0,3),(0,4),(0,5)],6)
=> 1
[6]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> [6,5,4,3,2,1,7] => ([(0,6),(1,6),(2,6),(3,6),(4,6),(5,6)],7)
=> 6
[5,1]
=> [1,1,1,1,0,1,0,0,0,0,1,0]
=> [5,3,4,2,1,6] => ([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
=> 4
[4,2]
=> [1,1,1,0,0,1,0,0,1,0]
=> [3,2,4,1,5] => ([(0,4),(1,3),(2,3),(3,4)],5)
=> 2
[4,1,1]
=> [1,1,0,1,1,0,0,0,1,0]
=> [2,4,3,1,5] => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> 3
[3,3]
=> [1,1,1,0,0,0,1,1,0,0]
=> [3,2,1,5,4] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> 3
[3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> 1
[3,1,1,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,4,3,5,2] => ([(0,1),(0,2),(0,3),(2,4),(3,4)],5)
=> 2
[2,2,2]
=> [1,1,0,0,1,1,1,0,0,0]
=> [2,1,5,4,3] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4)],5)
=> 2
[2,2,1,1]
=> [1,0,1,1,0,1,1,0,0,0]
=> [1,3,5,4,2] => ([(0,3),(0,4),(4,1),(4,2)],5)
=> 1
[2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> [1,6,4,5,3,2] => ([(0,2),(0,3),(0,4),(0,5),(5,1)],6)
=> 1
[1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,7,6,5,4,3,2] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6)],7)
=> 1
[7]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> [7,6,5,4,3,2,1,8] => ([(0,7),(1,7),(2,7),(3,7),(4,7),(5,7),(6,7)],8)
=> ? ∊ {1,7}
[6,1]
=> [1,1,1,1,1,0,1,0,0,0,0,0,1,0]
=> [6,5,3,4,2,1,7] => ([(0,6),(1,6),(2,6),(3,6),(4,5),(5,6)],7)
=> 5
[5,2]
=> [1,1,1,1,0,0,1,0,0,0,1,0]
=> [5,3,2,4,1,6] => ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> 3
[5,1,1]
=> [1,1,1,0,1,1,0,0,0,0,1,0]
=> [5,2,4,3,1,6] => ([(0,5),(1,5),(2,3),(2,4),(3,5),(4,5)],6)
=> 4
[4,3]
=> [1,1,1,0,0,0,1,0,1,0]
=> [3,2,1,4,5] => ([(0,4),(1,4),(2,4),(4,3)],5)
=> 3
[4,2,1]
=> [1,1,0,1,0,1,0,0,1,0]
=> [2,3,4,1,5] => ([(0,4),(1,2),(2,3),(3,4)],5)
=> 2
[4,1,1,1]
=> [1,0,1,1,1,0,0,0,1,0]
=> [1,4,3,2,5] => ([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
=> 3
[3,3,1]
=> [1,1,0,1,0,0,1,1,0,0]
=> [2,3,1,5,4] => ([(0,3),(0,4),(1,2),(2,3),(2,4)],5)
=> 2
[3,2,2]
=> [1,1,0,0,1,1,0,1,0,0]
=> [2,1,4,5,3] => ([(0,3),(0,4),(1,3),(1,4),(4,2)],5)
=> 2
[3,2,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,3,4,5,2] => ([(0,2),(0,4),(3,1),(4,3)],5)
=> 1
[3,1,1,1,1]
=> [1,0,1,1,1,1,0,0,1,0,0,0]
=> [1,6,4,3,5,2] => ([(0,1),(0,2),(0,3),(0,4),(3,5),(4,5)],6)
=> 2
[2,2,2,1]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,2,5,4,3] => ([(0,4),(4,1),(4,2),(4,3)],5)
=> 1
[2,2,1,1,1]
=> [1,0,1,1,1,0,1,1,0,0,0,0]
=> [1,6,3,5,4,2] => ([(0,3),(0,4),(0,5),(5,1),(5,2)],6)
=> 1
[2,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,1,0,0,0,0,0]
=> [1,7,6,4,5,3,2] => ([(0,2),(0,3),(0,4),(0,5),(0,6),(6,1)],7)
=> 1
[1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [1,8,7,6,5,4,3,2] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7)],8)
=> ? ∊ {1,7}
[8]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0]
=> [8,7,6,5,4,3,2,1,9] => ([(0,8),(1,8),(2,8),(3,8),(4,8),(5,8),(6,8),(7,8)],9)
=> ? ∊ {1,1,6,8}
[7,1]
=> [1,1,1,1,1,1,0,1,0,0,0,0,0,0,1,0]
=> [7,6,4,5,3,2,1,8] => ([(0,7),(1,7),(2,7),(3,7),(4,7),(5,6),(6,7)],8)
=> ? ∊ {1,1,6,8}
[6,2]
=> [1,1,1,1,1,0,0,1,0,0,0,0,1,0]
=> [6,4,3,5,2,1,7] => ([(0,6),(1,6),(2,6),(3,5),(4,5),(5,6)],7)
=> 4
[6,1,1]
=> [1,1,1,1,0,1,1,0,0,0,0,0,1,0]
=> [6,3,5,4,2,1,7] => ([(0,6),(1,6),(2,6),(3,4),(3,5),(4,6),(5,6)],7)
=> 5
[5,3]
=> [1,1,1,1,0,0,0,1,0,0,1,0]
=> [4,3,2,5,1,6] => ([(0,5),(1,5),(2,5),(3,4),(5,4)],6)
=> 3
[5,2,1]
=> [1,1,1,0,1,0,1,0,0,0,1,0]
=> [5,2,3,4,1,6] => ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
=> 3
[5,1,1,1]
=> [1,1,0,1,1,1,0,0,0,0,1,0]
=> [2,5,4,3,1,6] => ([(0,5),(1,2),(1,3),(1,4),(2,5),(3,5),(4,5)],6)
=> 4
[4,4]
=> [1,1,1,1,0,0,0,0,1,1,0,0]
=> [4,3,2,1,6,5] => ([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> 4
[4,3,1]
=> [1,1,0,1,0,0,1,0,1,0]
=> [2,3,1,4,5] => ([(0,4),(1,2),(2,4),(4,3)],5)
=> 2
[4,2,2]
=> [1,1,0,0,1,1,0,0,1,0]
=> [2,1,4,3,5] => ([(0,3),(0,4),(1,3),(1,4),(3,2),(4,2)],5)
=> 2
[2,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,1,0,0,0,0,0,0]
=> [1,8,7,5,6,4,3,2] => ([(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(7,1)],8)
=> ? ∊ {1,1,6,8}
[1,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [1,9,8,7,6,5,4,3,2] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(0,8)],9)
=> ? ∊ {1,1,6,8}
[9]
=> [1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,1,0]
=> [9,8,7,6,5,4,3,2,1,10] => ([(0,9),(1,9),(2,9),(3,9),(4,9),(5,9),(6,9),(7,9),(8,9)],10)
=> ? ∊ {1,1,1,2,5,6,7,9}
[8,1]
=> [1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,1,0]
=> [8,7,6,4,5,3,2,1,9] => ([(0,8),(1,8),(2,8),(3,8),(4,8),(5,8),(6,7),(7,8)],9)
=> ? ∊ {1,1,1,2,5,6,7,9}
[7,2]
=> [1,1,1,1,1,1,0,0,1,0,0,0,0,0,1,0]
=> [7,6,4,3,5,2,1,8] => ([(0,7),(1,7),(2,7),(3,7),(4,6),(5,6),(6,7)],8)
=> ? ∊ {1,1,1,2,5,6,7,9}
[7,1,1]
=> [1,1,1,1,1,0,1,1,0,0,0,0,0,0,1,0]
=> [7,6,3,5,4,2,1,8] => ([(0,7),(1,7),(2,7),(3,7),(4,5),(4,6),(5,7),(6,7)],8)
=> ? ∊ {1,1,1,2,5,6,7,9}
[3,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,1,0,0,0,0,0]
=> [1,8,7,5,4,6,3,2] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(5,7),(6,7)],8)
=> ? ∊ {1,1,1,2,5,6,7,9}
[2,2,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,1,1,0,0,0,0,0,0]
=> [1,8,7,4,6,5,3,2] => ([(0,3),(0,4),(0,5),(0,6),(0,7),(7,1),(7,2)],8)
=> ? ∊ {1,1,1,2,5,6,7,9}
[2,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0]
=> [1,9,8,7,5,6,4,3,2] => ([(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(0,8),(8,1)],9)
=> ? ∊ {1,1,1,2,5,6,7,9}
[1,1,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0]
=> [1,10,9,8,7,6,5,4,3,2] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(0,8),(0,9)],10)
=> ? ∊ {1,1,1,2,5,6,7,9}
[10]
=> [1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,1,0]
=> [10,9,8,7,6,5,4,3,2,1,11] => ([(0,10),(1,10),(2,10),(3,10),(4,10),(5,10),(6,10),(7,10),(8,10),(9,10)],11)
=> ? ∊ {1,1,1,1,1,2,2,3,4,5,6,6,7,8,10}
[9,1]
=> [1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0,1,0]
=> [9,8,7,5,6,4,3,2,1,10] => ([(0,9),(1,9),(2,9),(3,9),(4,9),(5,9),(6,9),(7,8),(8,9)],10)
=> ? ∊ {1,1,1,1,1,2,2,3,4,5,6,6,7,8,10}
[8,2]
=> [1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0,1,0]
=> [8,7,5,4,6,3,2,1,9] => ([(0,8),(1,8),(2,8),(3,8),(4,8),(5,7),(6,7),(7,8)],9)
=> ? ∊ {1,1,1,1,1,2,2,3,4,5,6,6,7,8,10}
[8,1,1]
=> [1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0,1,0]
=> [8,7,4,6,5,3,2,1,9] => ([(0,8),(1,8),(2,8),(3,8),(4,8),(5,6),(5,7),(6,8),(7,8)],9)
=> ? ∊ {1,1,1,1,1,2,2,3,4,5,6,6,7,8,10}
[7,3]
=> [1,1,1,1,1,1,0,0,0,1,0,0,0,0,1,0]
=> [7,5,4,3,6,2,1,8] => ([(0,7),(1,7),(2,7),(3,6),(4,6),(5,6),(6,7)],8)
=> ? ∊ {1,1,1,1,1,2,2,3,4,5,6,6,7,8,10}
[7,2,1]
=> [1,1,1,1,1,0,1,0,1,0,0,0,0,0,1,0]
=> [7,6,3,4,5,2,1,8] => ([(0,7),(1,7),(2,7),(3,7),(4,5),(5,6),(6,7)],8)
=> ? ∊ {1,1,1,1,1,2,2,3,4,5,6,6,7,8,10}
[7,1,1,1]
=> [1,1,1,1,0,1,1,1,0,0,0,0,0,0,1,0]
=> [7,3,6,5,4,2,1,8] => ([(0,7),(1,7),(2,7),(3,4),(3,5),(3,6),(4,7),(5,7),(6,7)],8)
=> ? ∊ {1,1,1,1,1,2,2,3,4,5,6,6,7,8,10}
[4,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,0,1,0,0,0,0]
=> [1,8,6,5,4,7,3,2] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(4,7),(5,7),(6,7)],8)
=> ? ∊ {1,1,1,1,1,2,2,3,4,5,6,6,7,8,10}
[3,2,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,1,0,1,0,0,0,0,0]
=> [1,8,7,4,5,6,3,2] => ([(0,2),(0,3),(0,4),(0,5),(0,7),(6,1),(7,6)],8)
=> ? ∊ {1,1,1,1,1,2,2,3,4,5,6,6,7,8,10}
[3,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0]
=> [1,9,8,6,5,7,4,3,2] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(6,8),(7,8)],9)
=> ? ∊ {1,1,1,1,1,2,2,3,4,5,6,6,7,8,10}
[2,2,2,2,2]
=> [1,1,0,0,1,1,1,1,1,0,0,0,0,0]
=> [2,1,7,6,5,4,3] => ([(0,2),(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6)],7)
=> ? ∊ {1,1,1,1,1,2,2,3,4,5,6,6,7,8,10}
[2,2,2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,1,1,0,0,0,0,0,0]
=> [1,8,4,7,6,5,3,2] => ([(0,4),(0,5),(0,6),(0,7),(7,1),(7,2),(7,3)],8)
=> ? ∊ {1,1,1,1,1,2,2,3,4,5,6,6,7,8,10}
[2,2,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0]
=> [1,9,8,5,7,6,4,3,2] => ([(0,3),(0,4),(0,5),(0,6),(0,7),(0,8),(8,1),(8,2)],9)
=> ? ∊ {1,1,1,1,1,2,2,3,4,5,6,6,7,8,10}
[2,1,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0]
=> [1,10,9,8,6,7,5,4,3,2] => ([(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(0,8),(0,9),(9,1)],10)
=> ? ∊ {1,1,1,1,1,2,2,3,4,5,6,6,7,8,10}
[1,1,1,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0]
=> [1,11,10,9,8,7,6,5,4,3,2] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(0,8),(0,9),(0,10)],11)
=> ? ∊ {1,1,1,1,1,2,2,3,4,5,6,6,7,8,10}
[11]
=> [1,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,0,1,0]
=> [11,10,9,8,7,6,5,4,3,2,1,12] => ([(0,11),(1,11),(2,11),(3,11),(4,11),(5,11),(6,11),(7,11),(8,11),(9,11),(10,11)],12)
=> ? ∊ {1,1,1,1,1,1,1,2,2,2,2,3,4,4,4,5,5,5,6,6,7,7,8,9,11}
[10,1]
=> [1,1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0,0,1,0]
=> [10,9,8,7,5,6,4,3,2,1,11] => ?
=> ? ∊ {1,1,1,1,1,1,1,2,2,2,2,3,4,4,4,5,5,5,6,6,7,7,8,9,11}
[9,2]
=> [1,1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0,0,1,0]
=> [9,8,7,5,4,6,3,2,1,10] => ?
=> ? ∊ {1,1,1,1,1,1,1,2,2,2,2,3,4,4,4,5,5,5,6,6,7,7,8,9,11}
[9,1,1]
=> [1,1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0,0,1,0]
=> [9,8,7,4,6,5,3,2,1,10] => ?
=> ? ∊ {1,1,1,1,1,1,1,2,2,2,2,3,4,4,4,5,5,5,6,6,7,7,8,9,11}
[8,3]
=> [1,1,1,1,1,1,1,0,0,0,1,0,0,0,0,0,1,0]
=> [8,7,5,4,3,6,2,1,9] => ?
=> ? ∊ {1,1,1,1,1,1,1,2,2,2,2,3,4,4,4,5,5,5,6,6,7,7,8,9,11}
[8,2,1]
=> [1,1,1,1,1,1,0,1,0,1,0,0,0,0,0,0,1,0]
=> [8,7,4,5,6,3,2,1,9] => ?
=> ? ∊ {1,1,1,1,1,1,1,2,2,2,2,3,4,4,4,5,5,5,6,6,7,7,8,9,11}
[8,1,1,1]
=> [1,1,1,1,1,0,1,1,1,0,0,0,0,0,0,0,1,0]
=> [8,7,3,6,5,4,2,1,9] => ?
=> ? ∊ {1,1,1,1,1,1,1,2,2,2,2,3,4,4,4,5,5,5,6,6,7,7,8,9,11}
[7,4]
=> [1,1,1,1,1,1,0,0,0,0,1,0,0,0,1,0]
=> [7,5,4,3,2,6,1,8] => ([(0,7),(1,7),(2,7),(3,7),(4,6),(5,6),(7,6)],8)
=> ? ∊ {1,1,1,1,1,1,1,2,2,2,2,3,4,4,4,5,5,5,6,6,7,7,8,9,11}
[7,3,1]
=> [1,1,1,1,1,0,1,0,0,1,0,0,0,0,1,0]
=> [7,5,3,4,6,2,1,8] => ?
=> ? ∊ {1,1,1,1,1,1,1,2,2,2,2,3,4,4,4,5,5,5,6,6,7,7,8,9,11}
[7,2,2]
=> [1,1,1,1,1,0,0,1,1,0,0,0,0,0,1,0]
=> [7,4,3,6,5,2,1,8] => ([(0,7),(1,7),(2,7),(3,5),(3,6),(4,5),(4,6),(5,7),(6,7)],8)
=> ? ∊ {1,1,1,1,1,1,1,2,2,2,2,3,4,4,4,5,5,5,6,6,7,7,8,9,11}
[7,2,1,1]
=> [1,1,1,1,0,1,1,0,1,0,0,0,0,0,1,0]
=> [7,3,6,4,5,2,1,8] => ([(0,7),(1,7),(2,7),(3,4),(3,5),(4,6),(5,7),(6,7)],8)
=> ? ∊ {1,1,1,1,1,1,1,2,2,2,2,3,4,4,4,5,5,5,6,6,7,7,8,9,11}
[7,1,1,1,1]
=> [1,1,1,0,1,1,1,1,0,0,0,0,0,0,1,0]
=> [7,2,6,5,4,3,1,8] => ([(0,7),(1,7),(2,3),(2,4),(2,5),(2,6),(3,7),(4,7),(5,7),(6,7)],8)
=> ? ∊ {1,1,1,1,1,1,1,2,2,2,2,3,4,4,4,5,5,5,6,6,7,7,8,9,11}
[5,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,0,0,1,0,0,0]
=> [1,8,6,5,4,3,7,2] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(3,7),(4,7),(5,7),(6,7)],8)
=> ? ∊ {1,1,1,1,1,1,1,2,2,2,2,3,4,4,4,5,5,5,6,6,7,7,8,9,11}
[4,2,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,1,0,0,1,0,0,0,0]
=> [1,8,6,4,5,7,3,2] => ([(0,2),(0,3),(0,4),(0,5),(0,6),(1,7),(5,7),(6,1)],8)
=> ? ∊ {1,1,1,1,1,1,1,2,2,2,2,3,4,4,4,5,5,5,6,6,7,7,8,9,11}
[4,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,0,0,0,1,0,0,0,0,0]
=> [1,9,8,6,5,4,7,3,2] => ?
=> ? ∊ {1,1,1,1,1,1,1,2,2,2,2,3,4,4,4,5,5,5,6,6,7,7,8,9,11}
[3,3,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,1,1,0,0,0,0,0]
=> [1,8,5,4,7,6,3,2] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(4,6),(4,7),(5,6),(5,7)],8)
=> ? ∊ {1,1,1,1,1,1,1,2,2,2,2,3,4,4,4,5,5,5,6,6,7,7,8,9,11}
[3,2,2,2,2]
=> [1,1,0,0,1,1,1,1,0,1,0,0,0,0]
=> [2,1,7,5,6,4,3] => ([(0,3),(0,4),(0,5),(0,6),(1,3),(1,4),(1,5),(1,6),(6,2)],7)
=> ? ∊ {1,1,1,1,1,1,1,2,2,2,2,3,4,4,4,5,5,5,6,6,7,7,8,9,11}
[3,2,2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,1,0,1,0,0,0,0,0]
=> [1,8,4,7,5,6,3,2] => ?
=> ? ∊ {1,1,1,1,1,1,1,2,2,2,2,3,4,4,4,5,5,5,6,6,7,7,8,9,11}
[3,2,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,1,0,1,0,0,0,0,0,0]
=> [1,9,8,5,6,7,4,3,2] => ([(0,2),(0,3),(0,4),(0,5),(0,6),(0,8),(7,1),(8,7)],9)
=> ? ∊ {1,1,1,1,1,1,1,2,2,2,2,3,4,4,4,5,5,5,6,6,7,7,8,9,11}
[3,1,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0,0]
=> [1,10,9,8,6,5,7,4,3,2] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(0,8),(7,9),(8,9)],10)
=> ? ∊ {1,1,1,1,1,1,1,2,2,2,2,3,4,4,4,5,5,5,6,6,7,7,8,9,11}
[2,2,2,2,1,1,1]
=> [1,0,1,1,1,0,1,1,1,1,0,0,0,0,0,0]
=> [1,8,3,7,6,5,4,2] => ([(0,5),(0,6),(0,7),(7,1),(7,2),(7,3),(7,4)],8)
=> ? ∊ {1,1,1,1,1,1,1,2,2,2,2,3,4,4,4,5,5,5,6,6,7,7,8,9,11}
Description
The maximal number of elements covering an element of a poset.
Matching statistic: St000451
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00123: Dyck paths —Barnabei-Castronuovo involution⟶ Dyck paths
Mp00024: Dyck paths —to 321-avoiding permutation⟶ Permutations
St000451: Permutations ⟶ ℤResult quality: 30% ●values known / values provided: 30%●distinct values known / distinct values provided: 62%
Mp00123: Dyck paths —Barnabei-Castronuovo involution⟶ Dyck paths
Mp00024: Dyck paths —to 321-avoiding permutation⟶ Permutations
St000451: Permutations ⟶ ℤResult quality: 30% ●values known / values provided: 30%●distinct values known / distinct values provided: 62%
Values
[1]
=> [1,0,1,0]
=> [1,0,1,0]
=> [2,1] => 2 = 1 + 1
[2]
=> [1,1,0,0,1,0]
=> [1,0,1,1,0,0]
=> [2,3,1] => 2 = 1 + 1
[1,1]
=> [1,0,1,1,0,0]
=> [1,1,0,0,1,0]
=> [3,1,2] => 3 = 2 + 1
[3]
=> [1,1,1,0,0,0,1,0]
=> [1,0,1,1,1,0,0,0]
=> [2,3,4,1] => 2 = 1 + 1
[2,1]
=> [1,0,1,0,1,0]
=> [1,1,0,1,0,0]
=> [1,3,2] => 2 = 1 + 1
[1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [1,1,1,0,0,0,1,0]
=> [4,1,2,3] => 4 = 3 + 1
[4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => 2 = 1 + 1
[3,1]
=> [1,1,0,1,0,0,1,0]
=> [1,1,0,1,1,0,0,0]
=> [1,3,4,2] => 2 = 1 + 1
[2,2]
=> [1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> [3,4,1,2] => 3 = 2 + 1
[2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,1,1,0,0,1,0,0]
=> [1,4,2,3] => 3 = 2 + 1
[1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> [5,1,2,3,4] => 5 = 4 + 1
[5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> [2,3,4,5,6,1] => 2 = 1 + 1
[4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> [1,3,4,5,2] => 2 = 1 + 1
[3,2]
=> [1,1,0,0,1,0,1,0]
=> [1,1,0,0,1,0,1,0]
=> [3,1,4,2] => 3 = 2 + 1
[3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [1,1,1,0,1,0,0,0]
=> [1,2,4,3] => 2 = 1 + 1
[2,2,1]
=> [1,0,1,0,1,1,0,0]
=> [1,0,1,0,1,1,0,0]
=> [2,4,1,3] => 3 = 2 + 1
[2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> [1,5,2,3,4] => 4 = 3 + 1
[1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> [6,1,2,3,4,5] => 6 = 5 + 1
[6]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [2,3,4,5,6,7,1] => 2 = 1 + 1
[5,1]
=> [1,1,1,1,0,1,0,0,0,0,1,0]
=> [1,1,0,1,1,1,1,0,0,0,0,0]
=> [1,3,4,5,6,2] => 2 = 1 + 1
[4,2]
=> [1,1,1,0,0,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> [3,1,4,5,2] => 3 = 2 + 1
[4,1,1]
=> [1,1,0,1,1,0,0,0,1,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> [1,2,4,5,3] => 2 = 1 + 1
[3,3]
=> [1,1,1,0,0,0,1,1,0,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> [3,4,5,1,2] => 3 = 2 + 1
[3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0]
=> [2,1,4,3] => 2 = 1 + 1
[3,1,1,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> [1,2,5,3,4] => 3 = 2 + 1
[2,2,2]
=> [1,1,0,0,1,1,1,0,0,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> [4,5,1,2,3] => 4 = 3 + 1
[2,2,1,1]
=> [1,0,1,1,0,1,1,0,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> [2,5,1,3,4] => 4 = 3 + 1
[2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,1,0,0]
=> [1,6,2,3,4,5] => 5 = 4 + 1
[1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> [7,1,2,3,4,5,6] => 7 = 6 + 1
[7]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [2,3,4,5,6,7,8,1] => 2 = 1 + 1
[6,1]
=> [1,1,1,1,1,0,1,0,0,0,0,0,1,0]
=> [1,1,0,1,1,1,1,1,0,0,0,0,0,0]
=> [1,3,4,5,6,7,2] => 2 = 1 + 1
[5,2]
=> [1,1,1,1,0,0,1,0,0,0,1,0]
=> [1,1,0,0,1,1,1,0,0,0,1,0]
=> [3,1,4,5,6,2] => 3 = 2 + 1
[5,1,1]
=> [1,1,1,0,1,1,0,0,0,0,1,0]
=> [1,1,1,0,1,1,1,0,0,0,0,0]
=> [1,2,4,5,6,3] => 2 = 1 + 1
[4,3]
=> [1,1,1,0,0,0,1,0,1,0]
=> [1,1,0,0,1,1,0,1,0,0]
=> [3,4,1,5,2] => 3 = 2 + 1
[4,2,1]
=> [1,1,0,1,0,1,0,0,1,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> [2,1,4,5,3] => 2 = 1 + 1
[4,1,1,1]
=> [1,0,1,1,1,0,0,0,1,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> [1,2,3,5,4] => 2 = 1 + 1
[3,3,1]
=> [1,1,0,1,0,0,1,1,0,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> [2,4,5,1,3] => 3 = 2 + 1
[3,2,2]
=> [1,1,0,0,1,1,0,1,0,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> [4,1,5,2,3] => 4 = 3 + 1
[3,2,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> [2,1,5,3,4] => 3 = 2 + 1
[3,1,1,1,1]
=> [1,0,1,1,1,1,0,0,1,0,0,0]
=> [1,1,1,1,1,0,0,0,1,0,0,0]
=> [1,2,6,3,4,5] => 4 = 3 + 1
[2,2,2,1]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,0,1,1,0,0]
=> [3,5,1,2,4] => 4 = 3 + 1
[2,2,1,1,1]
=> [1,0,1,1,1,0,1,1,0,0,0,0]
=> [1,0,1,1,1,0,0,0,1,1,0,0]
=> [2,6,1,3,4,5] => 5 = 4 + 1
[2,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,1,0,0,0,0,0]
=> [1,1,1,1,1,1,0,0,0,0,0,1,0,0]
=> [1,7,2,3,4,5,6] => 6 = 5 + 1
[1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> [8,1,2,3,4,5,6,7] => 8 = 7 + 1
[8]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0]
=> [1,0,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [2,3,4,5,6,7,8,9,1] => 2 = 1 + 1
[7,1]
=> [1,1,1,1,1,1,0,1,0,0,0,0,0,0,1,0]
=> [1,1,0,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [1,3,4,5,6,7,8,2] => ? ∊ {1,3} + 1
[6,2]
=> [1,1,1,1,1,0,0,1,0,0,0,0,1,0]
=> [1,1,0,0,1,1,1,1,0,0,0,0,1,0]
=> [3,1,4,5,6,7,2] => ? ∊ {1,3} + 1
[6,1,1]
=> [1,1,1,1,0,1,1,0,0,0,0,0,1,0]
=> [1,1,1,0,1,1,1,1,0,0,0,0,0,0]
=> [1,2,4,5,6,7,3] => 2 = 1 + 1
[5,3]
=> [1,1,1,1,0,0,0,1,0,0,1,0]
=> [1,1,0,0,1,1,1,0,0,1,0,0]
=> [3,4,1,5,6,2] => 3 = 2 + 1
[5,2,1]
=> [1,1,1,0,1,0,1,0,0,0,1,0]
=> [1,0,1,0,1,1,1,0,0,0,1,0]
=> [2,1,4,5,6,3] => 2 = 1 + 1
[5,1,1,1]
=> [1,1,0,1,1,1,0,0,0,0,1,0]
=> [1,1,1,1,0,1,1,0,0,0,0,0]
=> [1,2,3,5,6,4] => 2 = 1 + 1
[4,4]
=> [1,1,1,1,0,0,0,0,1,1,0,0]
=> [1,1,0,0,1,1,1,1,0,0,0,0]
=> [3,4,5,6,1,2] => 3 = 2 + 1
[8,1]
=> [1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,1,0]
=> [1,1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [1,3,4,5,6,7,8,9,2] => ? ∊ {1,1,1,2,3,5,5} + 1
[7,2]
=> [1,1,1,1,1,1,0,0,1,0,0,0,0,0,1,0]
=> [1,1,0,0,1,1,1,1,1,0,0,0,0,0,1,0]
=> [3,1,4,5,6,7,8,2] => ? ∊ {1,1,1,2,3,5,5} + 1
[7,1,1]
=> [1,1,1,1,1,0,1,1,0,0,0,0,0,0,1,0]
=> [1,1,1,0,1,1,1,1,1,0,0,0,0,0,0,0]
=> [1,2,4,5,6,7,8,3] => ? ∊ {1,1,1,2,3,5,5} + 1
[6,3]
=> [1,1,1,1,1,0,0,0,1,0,0,0,1,0]
=> [1,1,0,0,1,1,1,1,0,0,0,1,0,0]
=> [3,4,1,5,6,7,2] => ? ∊ {1,1,1,2,3,5,5} + 1
[6,2,1]
=> [1,1,1,1,0,1,0,1,0,0,0,0,1,0]
=> [1,0,1,0,1,1,1,1,0,0,0,0,1,0]
=> [2,1,4,5,6,7,3] => ? ∊ {1,1,1,2,3,5,5} + 1
[3,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,1,0,0,0,0,0]
=> [1,1,1,1,1,1,1,0,0,0,0,0,1,0,0,0]
=> [1,2,8,3,4,5,6,7] => ? ∊ {1,1,1,2,3,5,5} + 1
[2,2,2,1,1,1]
=> [1,0,1,1,1,0,1,1,1,0,0,0,0,0]
=> [1,1,0,1,1,1,0,0,0,0,1,1,0,0]
=> [3,7,1,2,4,5,6] => ? ∊ {1,1,1,2,3,5,5} + 1
[9,1]
=> [1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0,1,0]
=> [1,1,0,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0]
=> [1,3,4,5,6,7,8,9,10,2] => ? ∊ {1,1,1,1,2,2,2,3,4,4,4,5,6,6} + 1
[8,2]
=> [1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0,1,0]
=> [1,1,0,0,1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> [3,1,4,5,6,7,8,9,2] => ? ∊ {1,1,1,1,2,2,2,3,4,4,4,5,6,6} + 1
[8,1,1]
=> [1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0,1,0]
=> [1,1,1,0,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [1,2,4,5,6,7,8,9,3] => ? ∊ {1,1,1,1,2,2,2,3,4,4,4,5,6,6} + 1
[7,3]
=> [1,1,1,1,1,1,0,0,0,1,0,0,0,0,1,0]
=> [1,1,0,0,1,1,1,1,1,0,0,0,0,1,0,0]
=> [3,4,1,5,6,7,8,2] => ? ∊ {1,1,1,1,2,2,2,3,4,4,4,5,6,6} + 1
[7,2,1]
=> [1,1,1,1,1,0,1,0,1,0,0,0,0,0,1,0]
=> [1,0,1,0,1,1,1,1,1,0,0,0,0,0,1,0]
=> [2,1,4,5,6,7,8,3] => ? ∊ {1,1,1,1,2,2,2,3,4,4,4,5,6,6} + 1
[6,4]
=> [1,1,1,1,1,0,0,0,0,1,0,0,1,0]
=> [1,1,0,0,1,1,1,1,0,0,1,0,0,0]
=> [3,4,5,1,6,7,2] => ? ∊ {1,1,1,1,2,2,2,3,4,4,4,5,6,6} + 1
[6,3,1]
=> [1,1,1,1,0,1,0,0,1,0,0,0,1,0]
=> [1,0,1,0,1,1,1,1,0,0,0,1,0,0]
=> [2,4,1,5,6,7,3] => ? ∊ {1,1,1,1,2,2,2,3,4,4,4,5,6,6} + 1
[6,2,2]
=> [1,1,1,1,0,0,1,1,0,0,0,0,1,0]
=> [1,1,1,0,0,1,1,1,0,0,0,0,1,0]
=> [4,1,2,5,6,7,3] => ? ∊ {1,1,1,1,2,2,2,3,4,4,4,5,6,6} + 1
[6,2,1,1]
=> [1,1,1,0,1,1,0,1,0,0,0,0,1,0]
=> [1,0,1,1,0,1,1,1,0,0,0,0,1,0]
=> [2,1,3,5,6,7,4] => ? ∊ {1,1,1,1,2,2,2,3,4,4,4,5,6,6} + 1
[3,3,1,1,1,1]
=> [1,0,1,1,1,1,0,0,1,1,0,0,0,0]
=> [1,0,1,1,1,1,0,0,0,1,1,0,0,0]
=> [2,3,7,1,4,5,6] => ? ∊ {1,1,1,1,2,2,2,3,4,4,4,5,6,6} + 1
[3,2,2,1,1,1]
=> [1,0,1,1,1,0,1,1,0,1,0,0,0,0]
=> [1,1,0,1,1,1,0,0,0,0,1,0,1,0]
=> [3,1,7,2,4,5,6] => ? ∊ {1,1,1,1,2,2,2,3,4,4,4,5,6,6} + 1
[3,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,1,0,0,0]
=> [1,2,9,3,4,5,6,7,8] => ? ∊ {1,1,1,1,2,2,2,3,4,4,4,5,6,6} + 1
[2,2,2,2,1,1]
=> [1,0,1,1,0,1,1,1,1,0,0,0,0,0]
=> [1,1,1,0,1,1,0,0,0,0,1,1,0,0]
=> [4,7,1,2,3,5,6] => ? ∊ {1,1,1,1,2,2,2,3,4,4,4,5,6,6} + 1
[2,2,2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,1,1,0,0,0,0,0,0]
=> [1,1,0,1,1,1,1,0,0,0,0,0,1,1,0,0]
=> [3,8,1,2,4,5,6,7] => ? ∊ {1,1,1,1,2,2,2,3,4,4,4,5,6,6} + 1
[11]
=> [1,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,0,1,0]
=> [1,0,1,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,0]
=> ? => ? ∊ {1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,4,4,4,5,5,5,5,5,6,7,7,11} + 1
[10,1]
=> [1,1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0,0,1,0]
=> [1,1,0,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0]
=> [1,3,4,5,6,7,8,9,10,11,2] => ? ∊ {1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,4,4,4,5,5,5,5,5,6,7,7,11} + 1
[9,2]
=> [1,1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0,0,1,0]
=> [1,1,0,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> [3,1,4,5,6,7,8,9,10,2] => ? ∊ {1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,4,4,4,5,5,5,5,5,6,7,7,11} + 1
[9,1,1]
=> [1,1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0,0,1,0]
=> [1,1,1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0]
=> [1,2,4,5,6,7,8,9,10,3] => ? ∊ {1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,4,4,4,5,5,5,5,5,6,7,7,11} + 1
[8,3]
=> [1,1,1,1,1,1,1,0,0,0,1,0,0,0,0,0,1,0]
=> [1,1,0,0,1,1,1,1,1,1,0,0,0,0,0,1,0,0]
=> [3,4,1,5,6,7,8,9,2] => ? ∊ {1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,4,4,4,5,5,5,5,5,6,7,7,11} + 1
[8,2,1]
=> [1,1,1,1,1,1,0,1,0,1,0,0,0,0,0,0,1,0]
=> [1,0,1,0,1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> [2,1,4,5,6,7,8,9,3] => ? ∊ {1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,4,4,4,5,5,5,5,5,6,7,7,11} + 1
[8,1,1,1]
=> [1,1,1,1,1,0,1,1,1,0,0,0,0,0,0,0,1,0]
=> [1,1,1,1,0,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [1,2,3,5,6,7,8,9,4] => ? ∊ {1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,4,4,4,5,5,5,5,5,6,7,7,11} + 1
[7,4]
=> [1,1,1,1,1,1,0,0,0,0,1,0,0,0,1,0]
=> [1,1,0,0,1,1,1,1,1,0,0,0,1,0,0,0]
=> [3,4,5,1,6,7,8,2] => ? ∊ {1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,4,4,4,5,5,5,5,5,6,7,7,11} + 1
[7,3,1]
=> [1,1,1,1,1,0,1,0,0,1,0,0,0,0,1,0]
=> [1,0,1,0,1,1,1,1,1,0,0,0,0,1,0,0]
=> [2,4,1,5,6,7,8,3] => ? ∊ {1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,4,4,4,5,5,5,5,5,6,7,7,11} + 1
[7,2,2]
=> [1,1,1,1,1,0,0,1,1,0,0,0,0,0,1,0]
=> [1,1,1,0,0,1,1,1,1,0,0,0,0,0,1,0]
=> [4,1,2,5,6,7,8,3] => ? ∊ {1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,4,4,4,5,5,5,5,5,6,7,7,11} + 1
[7,2,1,1]
=> [1,1,1,1,0,1,1,0,1,0,0,0,0,0,1,0]
=> [1,0,1,1,0,1,1,1,1,0,0,0,0,0,1,0]
=> [2,1,3,5,6,7,8,4] => ? ∊ {1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,4,4,4,5,5,5,5,5,6,7,7,11} + 1
[7,1,1,1,1]
=> [1,1,1,0,1,1,1,1,0,0,0,0,0,0,1,0]
=> [1,1,1,1,1,0,1,1,1,0,0,0,0,0,0,0]
=> [1,2,3,4,6,7,8,5] => ? ∊ {1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,4,4,4,5,5,5,5,5,6,7,7,11} + 1
[6,5]
=> [1,1,1,1,1,0,0,0,0,0,1,0,1,0]
=> [1,1,0,0,1,1,1,1,0,1,0,0,0,0]
=> [3,4,5,6,1,7,2] => ? ∊ {1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,4,4,4,5,5,5,5,5,6,7,7,11} + 1
[6,4,1]
=> [1,1,1,1,0,1,0,0,0,1,0,0,1,0]
=> [1,0,1,0,1,1,1,1,0,0,1,0,0,0]
=> [2,4,5,1,6,7,3] => ? ∊ {1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,4,4,4,5,5,5,5,5,6,7,7,11} + 1
[6,3,1,1]
=> [1,1,1,0,1,1,0,0,1,0,0,0,1,0]
=> [1,0,1,1,0,1,1,1,0,0,0,1,0,0]
=> [2,3,1,5,6,7,4] => ? ∊ {1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,4,4,4,5,5,5,5,5,6,7,7,11} + 1
[6,2,2,1]
=> [1,1,1,0,1,0,1,1,0,0,0,0,1,0]
=> [1,1,0,1,0,1,1,1,0,0,0,0,1,0]
=> [3,1,2,5,6,7,4] => ? ∊ {1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,4,4,4,5,5,5,5,5,6,7,7,11} + 1
[6,2,1,1,1]
=> [1,1,0,1,1,1,0,1,0,0,0,0,1,0]
=> [1,0,1,1,1,0,1,1,0,0,0,0,1,0]
=> [2,1,3,4,6,7,5] => ? ∊ {1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,4,4,4,5,5,5,5,5,6,7,7,11} + 1
[5,5,1]
=> [1,1,1,1,0,1,0,0,0,0,1,1,0,0]
=> [1,0,1,0,1,1,1,1,1,0,0,0,0,0]
=> [2,4,5,6,7,1,3] => ? ∊ {1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,4,4,4,5,5,5,5,5,6,7,7,11} + 1
[5,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,0,0,1,0,0,0]
=> [1,1,1,1,1,1,1,0,0,0,1,0,0,0,0,0]
=> [1,2,3,4,8,5,6,7] => ? ∊ {1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,4,4,4,5,5,5,5,5,6,7,7,11} + 1
[4,3,1,1,1,1]
=> [1,0,1,1,1,1,0,0,1,0,1,0,0,0]
=> [1,0,1,1,1,1,0,0,0,1,0,1,0,0]
=> [2,3,1,7,4,5,6] => ? ∊ {1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,4,4,4,5,5,5,5,5,6,7,7,11} + 1
[4,2,2,1,1,1]
=> [1,0,1,1,1,0,1,1,0,0,1,0,0,0]
=> [1,1,0,1,1,1,0,0,0,1,0,0,1,0]
=> [3,1,2,7,4,5,6] => ? ∊ {1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,4,4,4,5,5,5,5,5,6,7,7,11} + 1
[4,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,0,0,0,1,0,0,0,0,0]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,1,0,0,0,0]
=> [1,2,3,9,4,5,6,7,8] => ? ∊ {1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,4,4,4,5,5,5,5,5,6,7,7,11} + 1
[3,3,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,1,1,0,0,0,0,0]
=> [1,0,1,1,1,1,1,0,0,0,0,1,1,0,0,0]
=> [2,3,8,1,4,5,6,7] => ? ∊ {1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,4,4,4,5,5,5,5,5,6,7,7,11} + 1
[3,2,2,2,2]
=> [1,1,0,0,1,1,1,1,0,1,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0,1,0,1,0]
=> [6,1,7,2,3,4,5] => ? ∊ {1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,4,4,4,5,5,5,5,5,6,7,7,11} + 1
[3,2,2,2,1,1]
=> [1,0,1,1,0,1,1,1,0,1,0,0,0,0]
=> [1,1,1,0,1,1,0,0,0,0,1,0,1,0]
=> [4,1,7,2,3,5,6] => ? ∊ {1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,4,4,4,5,5,5,5,5,6,7,7,11} + 1
[3,2,2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,1,0,1,0,0,0,0,0]
=> [1,1,0,1,1,1,1,0,0,0,0,0,1,0,1,0]
=> [3,1,8,2,4,5,6,7] => ? ∊ {1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,4,4,4,5,5,5,5,5,6,7,7,11} + 1
[3,1,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0,0]
=> [1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0,0,0]
=> [1,2,10,3,4,5,6,7,8,9] => ? ∊ {1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,4,4,4,5,5,5,5,5,6,7,7,11} + 1
Description
The length of the longest pattern of the form k 1 2...(k-1).
Matching statistic: St000982
(load all 4 compositions to match this statistic)
(load all 4 compositions to match this statistic)
Mp00095: Integer partitions —to binary word⟶ Binary words
Mp00136: Binary words —rotate back-to-front⟶ Binary words
Mp00268: Binary words —zeros to flag zeros⟶ Binary words
St000982: Binary words ⟶ ℤResult quality: 29% ●values known / values provided: 29%●distinct values known / distinct values provided: 56%
Mp00136: Binary words —rotate back-to-front⟶ Binary words
Mp00268: Binary words —zeros to flag zeros⟶ Binary words
St000982: Binary words ⟶ ℤResult quality: 29% ●values known / values provided: 29%●distinct values known / distinct values provided: 56%
Values
[1]
=> 10 => 01 => 00 => 2 = 1 + 1
[2]
=> 100 => 010 => 100 => 2 = 1 + 1
[1,1]
=> 110 => 011 => 000 => 3 = 2 + 1
[3]
=> 1000 => 0100 => 0100 => 2 = 1 + 1
[2,1]
=> 1010 => 0101 => 1100 => 2 = 1 + 1
[1,1,1]
=> 1110 => 0111 => 0000 => 4 = 3 + 1
[4]
=> 10000 => 01000 => 10100 => 2 = 1 + 1
[3,1]
=> 10010 => 01001 => 00100 => 2 = 1 + 1
[2,2]
=> 1100 => 0110 => 1000 => 3 = 2 + 1
[2,1,1]
=> 10110 => 01011 => 11100 => 3 = 2 + 1
[1,1,1,1]
=> 11110 => 01111 => 00000 => 5 = 4 + 1
[5]
=> 100000 => 010000 => 010100 => 2 = 1 + 1
[4,1]
=> 100010 => 010001 => 110100 => 2 = 1 + 1
[3,2]
=> 10100 => 01010 => 01100 => 2 = 1 + 1
[3,1,1]
=> 100110 => 010011 => 000100 => 3 = 2 + 1
[2,2,1]
=> 11010 => 01101 => 11000 => 3 = 2 + 1
[2,1,1,1]
=> 101110 => 010111 => 111100 => 4 = 3 + 1
[1,1,1,1,1]
=> 111110 => 011111 => 000000 => 6 = 5 + 1
[6]
=> 1000000 => 0100000 => 1010100 => 2 = 1 + 1
[5,1]
=> 1000010 => 0100001 => 0010100 => 2 = 1 + 1
[4,2]
=> 100100 => 010010 => 100100 => 2 = 1 + 1
[4,1,1]
=> 1000110 => 0100011 => 1110100 => 3 = 2 + 1
[3,3]
=> 11000 => 01100 => 01000 => 3 = 2 + 1
[3,2,1]
=> 101010 => 010101 => 001100 => 2 = 1 + 1
[3,1,1,1]
=> 1001110 => 0100111 => 0000100 => 4 = 3 + 1
[2,2,2]
=> 11100 => 01110 => 10000 => 4 = 3 + 1
[2,2,1,1]
=> 110110 => 011011 => 111000 => 3 = 2 + 1
[2,1,1,1,1]
=> 1011110 => 0101111 => 1111100 => 5 = 4 + 1
[1,1,1,1,1,1]
=> 1111110 => 0111111 => 0000000 => 7 = 6 + 1
[7]
=> 10000000 => 01000000 => 01010100 => 2 = 1 + 1
[6,1]
=> 10000010 => 01000001 => 11010100 => 2 = 1 + 1
[5,2]
=> 1000100 => 0100010 => 0110100 => 2 = 1 + 1
[5,1,1]
=> 10000110 => 01000011 => 00010100 => 3 = 2 + 1
[4,3]
=> 101000 => 010100 => 101100 => 2 = 1 + 1
[4,2,1]
=> 1001010 => 0100101 => 1100100 => 2 = 1 + 1
[4,1,1,1]
=> 10001110 => 01000111 => 11110100 => 4 = 3 + 1
[3,3,1]
=> 110010 => 011001 => 001000 => 3 = 2 + 1
[3,2,2]
=> 101100 => 010110 => 011100 => 3 = 2 + 1
[3,2,1,1]
=> 1010110 => 0101011 => 0001100 => 3 = 2 + 1
[3,1,1,1,1]
=> 10011110 => 01001111 => 00000100 => 5 = 4 + 1
[2,2,2,1]
=> 111010 => 011101 => 110000 => 4 = 3 + 1
[2,2,1,1,1]
=> 1101110 => 0110111 => 1111000 => 4 = 3 + 1
[2,1,1,1,1,1]
=> 10111110 => 01011111 => 11111100 => 6 = 5 + 1
[1,1,1,1,1,1,1]
=> 11111110 => 01111111 => 00000000 => 8 = 7 + 1
[8]
=> 100000000 => 010000000 => 101010100 => 2 = 1 + 1
[7,1]
=> 100000010 => 010000001 => 001010100 => 2 = 1 + 1
[6,2]
=> 10000100 => 01000010 => 10010100 => 2 = 1 + 1
[6,1,1]
=> 100000110 => 010000011 => 111010100 => 3 = 2 + 1
[5,3]
=> 1001000 => 0100100 => 0100100 => 2 = 1 + 1
[5,2,1]
=> 10001010 => 01000101 => 00110100 => 2 = 1 + 1
[9]
=> 1000000000 => 0100000000 => 0101010100 => ? ∊ {1,3,5,7} + 1
[6,1,1,1]
=> 1000001110 => 0100000111 => 1111010100 => ? ∊ {1,3,5,7} + 1
[4,1,1,1,1,1]
=> 1000111110 => 0100011111 => 1111110100 => ? ∊ {1,3,5,7} + 1
[2,1,1,1,1,1,1,1]
=> 1011111110 => 0101111111 => 1111111100 => ? ∊ {1,3,5,7} + 1
[10]
=> 10000000000 => 01000000000 => 10101010100 => ? ∊ {1,1,1,2,3,4,4,5,6,6,7,8,10} + 1
[8,2]
=> 1000000100 => 0100000010 => 1001010100 => ? ∊ {1,1,1,2,3,4,4,5,6,6,7,8,10} + 1
[8,1,1]
=> 10000000110 => 01000000011 => 11101010100 => ? ∊ {1,1,1,2,3,4,4,5,6,6,7,8,10} + 1
[7,2,1]
=> 1000001010 => 0100000101 => 0011010100 => ? ∊ {1,1,1,2,3,4,4,5,6,6,7,8,10} + 1
[7,1,1,1]
=> 10000001110 => 01000000111 => 00001010100 => ? ∊ {1,1,1,2,3,4,4,5,6,6,7,8,10} + 1
[6,1,1,1,1]
=> 10000011110 => 01000001111 => 11111010100 => ? ∊ {1,1,1,2,3,4,4,5,6,6,7,8,10} + 1
[5,1,1,1,1,1]
=> 10000111110 => 01000011111 => 00000010100 => ? ∊ {1,1,1,2,3,4,4,5,6,6,7,8,10} + 1
[4,2,1,1,1,1]
=> 1001011110 => 0100101111 => 1111100100 => ? ∊ {1,1,1,2,3,4,4,5,6,6,7,8,10} + 1
[4,1,1,1,1,1,1]
=> 10001111110 => 01000111111 => 11111110100 => ? ∊ {1,1,1,2,3,4,4,5,6,6,7,8,10} + 1
[3,1,1,1,1,1,1,1]
=> 10011111110 => 01001111111 => 00000000100 => ? ∊ {1,1,1,2,3,4,4,5,6,6,7,8,10} + 1
[2,2,1,1,1,1,1,1]
=> 1101111110 => 0110111111 => 1111111000 => ? ∊ {1,1,1,2,3,4,4,5,6,6,7,8,10} + 1
[2,1,1,1,1,1,1,1,1]
=> 10111111110 => 01011111111 => 11111111100 => ? ∊ {1,1,1,2,3,4,4,5,6,6,7,8,10} + 1
[1,1,1,1,1,1,1,1,1,1]
=> 11111111110 => 01111111111 => 00000000000 => ? ∊ {1,1,1,2,3,4,4,5,6,6,7,8,10} + 1
[11]
=> 100000000000 => ? => ? => ? ∊ {1,1,1,1,1,2,2,2,2,2,3,3,3,4,4,4,4,5,5,5,5,6,6,7,7,8,9,11} + 1
[10,1]
=> 100000000010 => ? => ? => ? ∊ {1,1,1,1,1,2,2,2,2,2,3,3,3,4,4,4,4,5,5,5,5,6,6,7,7,8,9,11} + 1
[9,2]
=> 10000000100 => 01000000010 => ? => ? ∊ {1,1,1,1,1,2,2,2,2,2,3,3,3,4,4,4,4,5,5,5,5,6,6,7,7,8,9,11} + 1
[9,1,1]
=> 100000000110 => ? => ? => ? ∊ {1,1,1,1,1,2,2,2,2,2,3,3,3,4,4,4,4,5,5,5,5,6,6,7,7,8,9,11} + 1
[8,2,1]
=> 10000001010 => 01000000101 => ? => ? ∊ {1,1,1,1,1,2,2,2,2,2,3,3,3,4,4,4,4,5,5,5,5,6,6,7,7,8,9,11} + 1
[8,1,1,1]
=> 100000001110 => ? => ? => ? ∊ {1,1,1,1,1,2,2,2,2,2,3,3,3,4,4,4,4,5,5,5,5,6,6,7,7,8,9,11} + 1
[7,3,1]
=> 1000010010 => ? => ? => ? ∊ {1,1,1,1,1,2,2,2,2,2,3,3,3,4,4,4,4,5,5,5,5,6,6,7,7,8,9,11} + 1
[7,2,2]
=> 1000001100 => ? => ? => ? ∊ {1,1,1,1,1,2,2,2,2,2,3,3,3,4,4,4,4,5,5,5,5,6,6,7,7,8,9,11} + 1
[7,2,1,1]
=> 10000010110 => 01000001011 => ? => ? ∊ {1,1,1,1,1,2,2,2,2,2,3,3,3,4,4,4,4,5,5,5,5,6,6,7,7,8,9,11} + 1
[7,1,1,1,1]
=> 100000011110 => ? => ? => ? ∊ {1,1,1,1,1,2,2,2,2,2,3,3,3,4,4,4,4,5,5,5,5,6,6,7,7,8,9,11} + 1
[6,3,1,1]
=> 1000100110 => ? => ? => ? ∊ {1,1,1,1,1,2,2,2,2,2,3,3,3,4,4,4,4,5,5,5,5,6,6,7,7,8,9,11} + 1
[6,2,2,1]
=> 1000011010 => ? => ? => ? ∊ {1,1,1,1,1,2,2,2,2,2,3,3,3,4,4,4,4,5,5,5,5,6,6,7,7,8,9,11} + 1
[6,2,1,1,1]
=> 10000101110 => 01000010111 => ? => ? ∊ {1,1,1,1,1,2,2,2,2,2,3,3,3,4,4,4,4,5,5,5,5,6,6,7,7,8,9,11} + 1
[6,1,1,1,1,1]
=> 100000111110 => 010000011111 => 111111010100 => ? ∊ {1,1,1,1,1,2,2,2,2,2,3,3,3,4,4,4,4,5,5,5,5,6,6,7,7,8,9,11} + 1
[5,2,1,1,1,1]
=> 10001011110 => 01000101111 => ? => ? ∊ {1,1,1,1,1,2,2,2,2,2,3,3,3,4,4,4,4,5,5,5,5,6,6,7,7,8,9,11} + 1
[5,1,1,1,1,1,1]
=> 100001111110 => ? => ? => ? ∊ {1,1,1,1,1,2,2,2,2,2,3,3,3,4,4,4,4,5,5,5,5,6,6,7,7,8,9,11} + 1
[4,3,1,1,1,1]
=> 1010011110 => ? => ? => ? ∊ {1,1,1,1,1,2,2,2,2,2,3,3,3,4,4,4,4,5,5,5,5,6,6,7,7,8,9,11} + 1
[4,2,2,1,1,1]
=> 1001101110 => ? => ? => ? ∊ {1,1,1,1,1,2,2,2,2,2,3,3,3,4,4,4,4,5,5,5,5,6,6,7,7,8,9,11} + 1
[4,2,1,1,1,1,1]
=> 10010111110 => 01001011111 => ? => ? ∊ {1,1,1,1,1,2,2,2,2,2,3,3,3,4,4,4,4,5,5,5,5,6,6,7,7,8,9,11} + 1
[4,1,1,1,1,1,1,1]
=> 100011111110 => ? => ? => ? ∊ {1,1,1,1,1,2,2,2,2,2,3,3,3,4,4,4,4,5,5,5,5,6,6,7,7,8,9,11} + 1
[3,3,1,1,1,1,1]
=> 1100111110 => ? => ? => ? ∊ {1,1,1,1,1,2,2,2,2,2,3,3,3,4,4,4,4,5,5,5,5,6,6,7,7,8,9,11} + 1
[3,2,2,1,1,1,1]
=> 1011011110 => ? => ? => ? ∊ {1,1,1,1,1,2,2,2,2,2,3,3,3,4,4,4,4,5,5,5,5,6,6,7,7,8,9,11} + 1
[3,2,1,1,1,1,1,1]
=> 10101111110 => 01010111111 => ? => ? ∊ {1,1,1,1,1,2,2,2,2,2,3,3,3,4,4,4,4,5,5,5,5,6,6,7,7,8,9,11} + 1
[3,1,1,1,1,1,1,1,1]
=> 100111111110 => ? => ? => ? ∊ {1,1,1,1,1,2,2,2,2,2,3,3,3,4,4,4,4,5,5,5,5,6,6,7,7,8,9,11} + 1
[2,2,2,1,1,1,1,1]
=> 1110111110 => 0111011111 => ? => ? ∊ {1,1,1,1,1,2,2,2,2,2,3,3,3,4,4,4,4,5,5,5,5,6,6,7,7,8,9,11} + 1
[2,2,1,1,1,1,1,1,1]
=> 11011111110 => 01101111111 => ? => ? ∊ {1,1,1,1,1,2,2,2,2,2,3,3,3,4,4,4,4,5,5,5,5,6,6,7,7,8,9,11} + 1
[2,1,1,1,1,1,1,1,1,1]
=> 101111111110 => ? => ? => ? ∊ {1,1,1,1,1,2,2,2,2,2,3,3,3,4,4,4,4,5,5,5,5,6,6,7,7,8,9,11} + 1
[1,1,1,1,1,1,1,1,1,1,1]
=> 111111111110 => ? => ? => ? ∊ {1,1,1,1,1,2,2,2,2,2,3,3,3,4,4,4,4,5,5,5,5,6,6,7,7,8,9,11} + 1
[12]
=> 1000000000000 => ? => ? => ? ∊ {1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,5,5,5,5,6,6,6,6,7,7,8,8,9,10,12} + 1
[11,1]
=> 1000000000010 => ? => ? => ? ∊ {1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,5,5,5,5,6,6,6,6,7,7,8,8,9,10,12} + 1
[10,2]
=> 100000000100 => 010000000010 => 100101010100 => ? ∊ {1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,5,5,5,5,6,6,6,6,7,7,8,8,9,10,12} + 1
[10,1,1]
=> 1000000000110 => ? => ? => ? ∊ {1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,5,5,5,5,6,6,6,6,7,7,8,8,9,10,12} + 1
[9,3]
=> 10000001000 => ? => ? => ? ∊ {1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,5,5,5,5,6,6,6,6,7,7,8,8,9,10,12} + 1
Description
The length of the longest constant subword.
The following 124 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St001060The distinguishing index of a graph. St000225Difference between largest and smallest parts in a partition. St000983The length of the longest alternating subword. St000141The maximum drop size of a permutation. St000366The number of double descents of a permutation. St000710The number of big deficiencies of a permutation. St000711The number of big exceedences of a permutation. St000373The number of weak exceedences of a permutation that are also mid-points of a decreasing subsequence of length $3$. St000028The number of stack-sorts needed to sort a permutation. St000031The number of cycles in the cycle decomposition of a permutation. St000371The number of mid points of decreasing subsequences of length 3 in a permutation. St000381The largest part of an integer composition. St000617The number of global maxima of a Dyck path. St001210Gives the maximal vector space dimension of the first Ext-group between an indecomposable module X and the regular module A, when A is the Nakayama algebra corresponding to the Dyck path. St001231The number of simple modules that are non-projective and non-injective with the property that they have projective dimension equal to one and that also the Auslander-Reiten translates of the module and the inverse Auslander-Reiten translate of the module have the same projective dimension. St001234The number of indecomposable three dimensional modules with projective dimension one. St000155The number of exceedances (also excedences) of a permutation. St000864The number of circled entries of the shifted recording tableau of a permutation. St001089Number of indecomposable projective non-injective modules minus the number of indecomposable projective non-injective modules with dominant dimension equal to the injective dimension in the corresponding Nakayama algebra. St001219Number of simple modules S in the corresponding Nakayama algebra such that the Auslander-Reiten sequence ending at S has the property that all modules in the exact sequence are reflexive. St001226The number of integers i such that the radical of the i-th indecomposable projective module has vanishing first extension group with the Jacobson radical J in the corresponding Nakayama algebra. St001683The number of distinct positions of the pattern letter 3 in occurrences of 132 in a permutation. St001687The number of distinct positions of the pattern letter 2 in occurrences of 213 in a permutation. St001744The number of occurrences of the arrow pattern 1-2 with an arrow from 1 to 2 in a permutation. St000444The length of the maximal rise of a Dyck path. St001290The first natural number n such that the tensor product of n copies of D(A) is zero for the corresponding Nakayama algebra A. St000969We make a CNakayama algebra out of the LNakayama algebra (corresponding to the Dyck path) $[c_0,c_1,...,c_{n-1}]$ by adding $c_0$ to $c_{n-1}$. St000358The number of occurrences of the pattern 31-2. St000654The first descent of a permutation. St001727The number of invisible inversions of a permutation. St000757The length of the longest weakly inreasing subsequence of parts of an integer composition. St000899The maximal number of repetitions of an integer composition. St000904The maximal number of repetitions of an integer composition. St000260The radius of a connected graph. St000456The monochromatic index of a connected graph. St000765The number of weak records in an integer composition. St000771The largest multiplicity of a distance Laplacian eigenvalue in a connected graph. St000772The multiplicity of the largest distance Laplacian eigenvalue in a connected graph. St000013The height of a Dyck path. St001133The smallest label in the subtree rooted at the sister of 1 in the decreasing labelled binary unordered tree associated with the perfect matching. St001134The largest label in the subtree rooted at the sister of 1 in the leaf labelled binary unordered tree associated with the perfect matching. St001199The dominant dimension of $eAe$ for the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St001652The length of a longest interval of consecutive numbers. St001662The length of the longest factor of consecutive numbers in a permutation. St000954Number of times the corresponding LNakayama algebra has $Ext^i(D(A),A)=0$ for $i>0$. St000442The maximal area to the right of an up step of a Dyck path. St000996The number of exclusive left-to-right maxima of a permutation. St001024Maximum of dominant dimensions of the simple modules in the Nakayama algebra corresponding to the Dyck path. St000223The number of nestings in the permutation. St000007The number of saliances of the permutation. St000054The first entry of the permutation. St000214The number of adjacencies of a permutation. St000215The number of adjacencies of a permutation, zero appended. St000740The last entry of a permutation. St001491The number of indecomposable projective-injective modules in the algebra corresponding to a subset. St000782The indicator function of whether a given perfect matching is an L & P matching. St000308The height of the tree associated to a permutation. St000723The maximal cardinality of a set of vertices with the same neighbourhood in a graph. St001880The number of 2-Gorenstein indecomposable injective modules in the incidence algebra of the lattice. St000193The row of the unique '1' in the first column of the alternating sign matrix. St001431Half of the Loewy length minus one of a modified stable Auslander algebra of the Nakayama algebra corresponding to the Dyck path. St001553The number of indecomposable summands of the square of the Jacobson radical as a bimodule in the Nakayama algebra corresponding to the Dyck path. St001184Number of indecomposable injective modules with grade at least 1 in the corresponding Nakayama algebra. St000051The size of the left subtree of a binary tree. St000199The column of the unique '1' in the last row of the alternating sign matrix. St000200The row of the unique '1' in the last column of the alternating sign matrix. St000314The number of left-to-right-maxima of a permutation. St000335The difference of lower and upper interactions. St000338The number of pixed points of a permutation. St000461The rix statistic of a permutation. St000541The number of indices greater than or equal to 2 of a permutation such that all smaller indices appear to its right. St000989The number of final rises of a permutation. St000990The first ascent of a permutation. St001185The number of indecomposable injective modules of grade at least 2 in the corresponding Nakayama algebra. St001194The injective dimension of $A/AfA$ in the corresponding Nakayama algebra $A$ when $Af$ is the minimal faithful projective-injective left $A$-module St001201The grade of the simple module $S_0$ in the special CNakayama algebra corresponding to the Dyck path. St001233The number of indecomposable 2-dimensional modules with projective dimension one. St001264The smallest index i such that the i-th simple module has projective dimension equal to the global dimension of the corresponding Nakayama algebra. St001266The largest vector space dimension of an indecomposable non-projective module that is reflexive in the corresponding Nakayama algebra. St001640The number of ascent tops in the permutation such that all smaller elements appear before. St000061The number of nodes on the left branch of a binary tree. St000066The column of the unique '1' in the first row of the alternating sign matrix. St000297The number of leading ones in a binary word. St000365The number of double ascents of a permutation. St000542The number of left-to-right-minima of a permutation. St000838The number of terminal right-hand endpoints when the vertices are written in order. St000991The number of right-to-left minima of a permutation. St001513The number of nested exceedences of a permutation. St001557The number of inversions of the second entry of a permutation. St001811The Castelnuovo-Mumford regularity of a permutation. St001948The number of augmented double ascents of a permutation. St001960The number of descents of a permutation minus one if its first entry is not one. St000025The number of initial rises of a Dyck path. St000738The first entry in the last row of a standard tableau. St001239The largest vector space dimension of the double dual of a simple module in the corresponding Nakayama algebra. St001240The number of indecomposable modules e_i J^2 that have injective dimension at most one in the corresponding Nakayama algebra St001473The absolute value of the sum of all entries of the Coxeter matrix of the corresponding LNakayama algebra. St000684The global dimension of the LNakayama algebra associated to a Dyck path. St001630The global dimension of the incidence algebra of the lattice over the rational numbers. St001875The number of simple modules with projective dimension at most 1. St001878The projective dimension of the simple modules corresponding to the minimum of L in the incidence algebra of the lattice L. St000688The global dimension minus the dominant dimension of the LNakayama algebra associated to a Dyck path. St000986The multiplicity of the eigenvalue zero of the adjacency matrix of the graph. St001026The maximum of the projective dimensions of the indecomposable non-projective injective modules minus the minimum in the Nakayama algebra corresponding to the Dyck path. St001066The number of simple reflexive modules in the corresponding Nakayama algebra. St001067The number of simple modules of dominant dimension at least two in the corresponding Nakayama algebra. St001126Number of simple module that are 1-regular in the corresponding Nakayama algebra. St001192The maximal dimension of $Ext_A^2(S,A)$ for a simple module $S$ over the corresponding Nakayama algebra $A$. St001238The number of simple modules S such that the Auslander-Reiten translate of S is isomorphic to the Nakayama functor applied to the second syzygy of S. St001483The number of simple module modules that appear in the socle of the regular module but have no nontrivial selfextensions with the regular module. St000686The finitistic dominant dimension of a Dyck path. St001167The number of simple modules that appear as the top of an indecomposable non-projective modules that is reflexive in the corresponding Nakayama algebra. St001223Number of indecomposable projective non-injective modules P such that the modules X and Y in a an Auslander-Reiten sequence ending at P are torsionless. St001235The global dimension of the corresponding Comp-Nakayama algebra. St001253The number of non-projective indecomposable reflexive modules in the corresponding Nakayama algebra. St001438The number of missing boxes of a skew partition. St001487The number of inner corners of a skew partition. St001435The number of missing boxes in the first row. St001868The number of alignments of type NE of a signed permutation. St000454The largest eigenvalue of a graph if it is integral. St000259The diameter of a connected graph. St001198The number of simple modules in the algebra $eAe$ with projective dimension at most 1 in the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St001206The maximal dimension of an indecomposable projective $eAe$-module (that is the height of the corresponding Dyck path) of the corresponding Nakayama algebra with minimal faithful projective-injective module $eA$. St001867The number of alignments of type EN of a signed permutation.
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