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Your data matches 35 different statistics following compositions of up to 3 maps.
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Matching statistic: St000617
Mp00027: Dyck paths —to partition⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00227: Dyck paths —Delest-Viennot-inverse⟶ Dyck paths
St000617: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00227: Dyck paths —Delest-Viennot-inverse⟶ Dyck paths
St000617: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> []
=> []
=> []
=> 1
[1,0,1,0]
=> [1]
=> [1,0]
=> [1,0]
=> 1
[1,1,0,0]
=> []
=> []
=> []
=> 1
[1,0,1,0,1,0]
=> [2,1]
=> [1,0,1,1,0,0]
=> [1,1,0,0,1,0]
=> 1
[1,0,1,1,0,0]
=> [1,1]
=> [1,1,0,0]
=> [1,0,1,0]
=> 2
[1,1,0,0,1,0]
=> [2]
=> [1,0,1,0]
=> [1,1,0,0]
=> 1
[1,1,0,1,0,0]
=> [1]
=> [1,0]
=> [1,0]
=> 1
[1,1,1,0,0,0]
=> []
=> []
=> []
=> 1
[1,0,1,0,1,0,1,0]
=> [3,2,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,1,1,0,0,1,0,0,1,0]
=> 1
[1,0,1,0,1,1,0,0]
=> [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> 2
[1,0,1,1,0,0,1,0]
=> [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> 1
[1,0,1,1,0,1,0,0]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0]
=> 1
[1,0,1,1,1,0,0,0]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> [1,0,1,0,1,0]
=> 3
[1,1,0,0,1,0,1,0]
=> [3,2]
=> [1,0,1,1,1,0,0,0]
=> [1,1,1,0,0,1,0,0]
=> 1
[1,1,0,0,1,1,0,0]
=> [2,2]
=> [1,1,1,0,0,0]
=> [1,1,0,1,0,0]
=> 2
[1,1,0,1,0,0,1,0]
=> [3,1]
=> [1,0,1,0,1,1,0,0]
=> [1,1,1,0,0,0,1,0]
=> 1
[1,1,0,1,0,1,0,0]
=> [2,1]
=> [1,0,1,1,0,0]
=> [1,1,0,0,1,0]
=> 1
[1,1,0,1,1,0,0,0]
=> [1,1]
=> [1,1,0,0]
=> [1,0,1,0]
=> 2
[1,1,1,0,0,0,1,0]
=> [3]
=> [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> 1
[1,1,1,0,0,1,0,0]
=> [2]
=> [1,0,1,0]
=> [1,1,0,0]
=> 1
[1,1,1,0,1,0,0,0]
=> [1]
=> [1,0]
=> [1,0]
=> 1
[1,1,1,1,0,0,0,0]
=> []
=> []
=> []
=> 1
[1,0,1,0,1,0,1,0,1,0]
=> [4,3,2,1]
=> [1,0,1,1,1,0,1,1,0,0,0,1,0,0]
=> [1,1,1,1,0,0,1,0,0,1,0,0,1,0]
=> 1
[1,0,1,0,1,0,1,1,0,0]
=> [3,3,2,1]
=> [1,1,1,0,1,1,0,0,0,1,0,0]
=> [1,1,1,0,1,0,0,1,0,0,1,0]
=> 2
[1,0,1,0,1,1,0,1,0,0]
=> [3,2,2,1]
=> [1,0,1,1,1,1,0,0,0,1,0,0]
=> [1,1,1,0,0,1,0,1,0,0,1,0]
=> 1
[1,0,1,0,1,1,1,0,0,0]
=> [2,2,2,1]
=> [1,1,1,1,0,0,0,1,0,0]
=> [1,1,0,1,0,1,0,0,1,0]
=> 3
[1,0,1,1,0,0,1,1,0,0]
=> [3,3,1,1]
=> [1,1,1,0,1,0,0,1,0,1,0,0]
=> [1,1,1,0,1,0,0,0,1,0,1,0]
=> 2
[1,0,1,1,0,1,0,1,0,0]
=> [3,2,1,1]
=> [1,0,1,1,1,0,0,1,0,1,0,0]
=> [1,1,1,0,0,1,0,0,1,0,1,0]
=> 1
[1,0,1,1,0,1,1,0,0,0]
=> [2,2,1,1]
=> [1,1,1,0,0,1,0,1,0,0]
=> [1,1,0,1,0,0,1,0,1,0]
=> 2
[1,0,1,1,1,0,0,1,0,0]
=> [3,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,1,0,0,0,1,0,1,0,1,0]
=> 1
[1,0,1,1,1,0,1,0,0,0]
=> [2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> 1
[1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0]
=> 4
[1,1,0,0,1,0,1,0,1,0]
=> [4,3,2]
=> [1,0,1,1,1,0,1,1,0,0,0,0]
=> [1,1,1,1,0,0,1,0,0,1,0,0]
=> 1
[1,1,0,0,1,0,1,1,0,0]
=> [3,3,2]
=> [1,1,1,0,1,1,0,0,0,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> 2
[1,1,0,0,1,1,0,0,1,0]
=> [4,2,2]
=> [1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,1,0,1,0,0]
=> 1
[1,1,0,0,1,1,0,1,0,0]
=> [3,2,2]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> 1
[1,1,0,0,1,1,1,0,0,0]
=> [2,2,2]
=> [1,1,1,1,0,0,0,0]
=> [1,1,0,1,0,1,0,0]
=> 3
[1,1,0,1,0,0,1,0,1,0]
=> [4,3,1]
=> [1,0,1,1,1,0,1,0,0,1,0,0]
=> [1,1,1,1,0,0,1,0,0,0,1,0]
=> 1
[1,1,0,1,0,0,1,1,0,0]
=> [3,3,1]
=> [1,1,1,0,1,0,0,1,0,0]
=> [1,1,1,0,1,0,0,0,1,0]
=> 2
[1,1,0,1,0,1,0,0,1,0]
=> [4,2,1]
=> [1,0,1,0,1,1,1,0,0,1,0,0]
=> [1,1,1,1,0,0,0,1,0,0,1,0]
=> 1
[1,1,0,1,0,1,0,1,0,0]
=> [3,2,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,1,1,0,0,1,0,0,1,0]
=> 1
[1,1,0,1,0,1,1,0,0,0]
=> [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> 2
[1,1,0,1,1,0,0,0,1,0]
=> [4,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,1,0,0,0,0,1,0,1,0]
=> 1
[1,1,0,1,1,0,0,1,0,0]
=> [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> 1
[1,1,0,1,1,0,1,0,0,0]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0]
=> 1
[1,1,0,1,1,1,0,0,0,0]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> [1,0,1,0,1,0]
=> 3
[1,1,1,0,0,0,1,0,1,0]
=> [4,3]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> 1
[1,1,1,0,0,0,1,1,0,0]
=> [3,3]
=> [1,1,1,0,1,0,0,0]
=> [1,1,1,0,1,0,0,0]
=> 2
[1,1,1,0,0,1,0,0,1,0]
=> [4,2]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> 1
[1,1,1,0,0,1,0,1,0,0]
=> [3,2]
=> [1,0,1,1,1,0,0,0]
=> [1,1,1,0,0,1,0,0]
=> 1
Description
The number of global maxima of a Dyck path.
Matching statistic: St000326
Mp00027: Dyck paths —to partition⟶ Integer partitions
Mp00095: Integer partitions —to binary word⟶ Binary words
Mp00105: Binary words —complement⟶ Binary words
St000326: Binary words ⟶ ℤResult quality: 89% ●values known / values provided: 95%●distinct values known / distinct values provided: 89%
Mp00095: Integer partitions —to binary word⟶ Binary words
Mp00105: Binary words —complement⟶ Binary words
St000326: Binary words ⟶ ℤResult quality: 89% ●values known / values provided: 95%●distinct values known / distinct values provided: 89%
Values
[1,0]
=> []
=> => => ? = 1 + 1
[1,0,1,0]
=> [1]
=> 10 => 01 => 2 = 1 + 1
[1,1,0,0]
=> []
=> => => ? = 1 + 1
[1,0,1,0,1,0]
=> [2,1]
=> 1010 => 0101 => 2 = 1 + 1
[1,0,1,1,0,0]
=> [1,1]
=> 110 => 001 => 3 = 2 + 1
[1,1,0,0,1,0]
=> [2]
=> 100 => 011 => 2 = 1 + 1
[1,1,0,1,0,0]
=> [1]
=> 10 => 01 => 2 = 1 + 1
[1,1,1,0,0,0]
=> []
=> => => ? = 1 + 1
[1,0,1,0,1,0,1,0]
=> [3,2,1]
=> 101010 => 010101 => 2 = 1 + 1
[1,0,1,0,1,1,0,0]
=> [2,2,1]
=> 11010 => 00101 => 3 = 2 + 1
[1,0,1,1,0,0,1,0]
=> [3,1,1]
=> 100110 => 011001 => 2 = 1 + 1
[1,0,1,1,0,1,0,0]
=> [2,1,1]
=> 10110 => 01001 => 2 = 1 + 1
[1,0,1,1,1,0,0,0]
=> [1,1,1]
=> 1110 => 0001 => 4 = 3 + 1
[1,1,0,0,1,0,1,0]
=> [3,2]
=> 10100 => 01011 => 2 = 1 + 1
[1,1,0,0,1,1,0,0]
=> [2,2]
=> 1100 => 0011 => 3 = 2 + 1
[1,1,0,1,0,0,1,0]
=> [3,1]
=> 10010 => 01101 => 2 = 1 + 1
[1,1,0,1,0,1,0,0]
=> [2,1]
=> 1010 => 0101 => 2 = 1 + 1
[1,1,0,1,1,0,0,0]
=> [1,1]
=> 110 => 001 => 3 = 2 + 1
[1,1,1,0,0,0,1,0]
=> [3]
=> 1000 => 0111 => 2 = 1 + 1
[1,1,1,0,0,1,0,0]
=> [2]
=> 100 => 011 => 2 = 1 + 1
[1,1,1,0,1,0,0,0]
=> [1]
=> 10 => 01 => 2 = 1 + 1
[1,1,1,1,0,0,0,0]
=> []
=> => => ? = 1 + 1
[1,0,1,0,1,0,1,0,1,0]
=> [4,3,2,1]
=> 10101010 => 01010101 => 2 = 1 + 1
[1,0,1,0,1,0,1,1,0,0]
=> [3,3,2,1]
=> 1101010 => 0010101 => 3 = 2 + 1
[1,0,1,0,1,1,0,1,0,0]
=> [3,2,2,1]
=> 1011010 => 0100101 => 2 = 1 + 1
[1,0,1,0,1,1,1,0,0,0]
=> [2,2,2,1]
=> 111010 => 000101 => 4 = 3 + 1
[1,0,1,1,0,0,1,1,0,0]
=> [3,3,1,1]
=> 1100110 => 0011001 => 3 = 2 + 1
[1,0,1,1,0,1,0,1,0,0]
=> [3,2,1,1]
=> 1010110 => 0101001 => 2 = 1 + 1
[1,0,1,1,0,1,1,0,0,0]
=> [2,2,1,1]
=> 110110 => 001001 => 3 = 2 + 1
[1,0,1,1,1,0,0,1,0,0]
=> [3,1,1,1]
=> 1001110 => 0110001 => 2 = 1 + 1
[1,0,1,1,1,0,1,0,0,0]
=> [2,1,1,1]
=> 101110 => 010001 => 2 = 1 + 1
[1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1]
=> 11110 => 00001 => 5 = 4 + 1
[1,1,0,0,1,0,1,0,1,0]
=> [4,3,2]
=> 1010100 => 0101011 => 2 = 1 + 1
[1,1,0,0,1,0,1,1,0,0]
=> [3,3,2]
=> 110100 => 001011 => 3 = 2 + 1
[1,1,0,0,1,1,0,0,1,0]
=> [4,2,2]
=> 1001100 => 0110011 => 2 = 1 + 1
[1,1,0,0,1,1,0,1,0,0]
=> [3,2,2]
=> 101100 => 010011 => 2 = 1 + 1
[1,1,0,0,1,1,1,0,0,0]
=> [2,2,2]
=> 11100 => 00011 => 4 = 3 + 1
[1,1,0,1,0,0,1,0,1,0]
=> [4,3,1]
=> 1010010 => 0101101 => 2 = 1 + 1
[1,1,0,1,0,0,1,1,0,0]
=> [3,3,1]
=> 110010 => 001101 => 3 = 2 + 1
[1,1,0,1,0,1,0,0,1,0]
=> [4,2,1]
=> 1001010 => 0110101 => 2 = 1 + 1
[1,1,0,1,0,1,0,1,0,0]
=> [3,2,1]
=> 101010 => 010101 => 2 = 1 + 1
[1,1,0,1,0,1,1,0,0,0]
=> [2,2,1]
=> 11010 => 00101 => 3 = 2 + 1
[1,1,0,1,1,0,0,0,1,0]
=> [4,1,1]
=> 1000110 => 0111001 => 2 = 1 + 1
[1,1,0,1,1,0,0,1,0,0]
=> [3,1,1]
=> 100110 => 011001 => 2 = 1 + 1
[1,1,0,1,1,0,1,0,0,0]
=> [2,1,1]
=> 10110 => 01001 => 2 = 1 + 1
[1,1,0,1,1,1,0,0,0,0]
=> [1,1,1]
=> 1110 => 0001 => 4 = 3 + 1
[1,1,1,0,0,0,1,0,1,0]
=> [4,3]
=> 101000 => 010111 => 2 = 1 + 1
[1,1,1,0,0,0,1,1,0,0]
=> [3,3]
=> 11000 => 00111 => 3 = 2 + 1
[1,1,1,0,0,1,0,0,1,0]
=> [4,2]
=> 100100 => 011011 => 2 = 1 + 1
[1,1,1,0,0,1,0,1,0,0]
=> [3,2]
=> 10100 => 01011 => 2 = 1 + 1
[1,1,1,0,0,1,1,0,0,0]
=> [2,2]
=> 1100 => 0011 => 3 = 2 + 1
[1,1,1,0,1,0,0,0,1,0]
=> [4,1]
=> 100010 => 011101 => 2 = 1 + 1
[1,1,1,0,1,0,0,1,0,0]
=> [3,1]
=> 10010 => 01101 => 2 = 1 + 1
[1,1,1,0,1,0,1,0,0,0]
=> [2,1]
=> 1010 => 0101 => 2 = 1 + 1
[1,1,1,1,1,0,0,0,0,0]
=> []
=> => => ? = 1 + 1
[1,1,1,1,1,1,0,0,0,0,0,0]
=> []
=> => => ? = 1 + 1
[1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [5,5,4,3,2,1]
=> 11010101010 => 00101010101 => ? = 2 + 1
[1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> []
=> => => ? = 1 + 1
[1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [6,6,5,4,3,2,1]
=> 1101010101010 => 0010101010101 => ? = 2 + 1
[1,0,1,1,0,0,1,1,1,1,1,0,0,0,0,0]
=> [3,3,3,3,3,1,1]
=> 1111100110 => ? => ? = 5 + 1
[1,1,0,1,0,1,0,1,0,1,0,1,1,0,0,0]
=> [5,5,4,3,2,1]
=> 11010101010 => 00101010101 => ? = 2 + 1
[1,1,1,1,1,0,0,1,0,0,0,0,1,0,1,0]
=> [7,6,2]
=> 1010000100 => 0101111011 => ? = 1 + 1
[1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> []
=> => => ? = 1 + 1
[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]
=> 100000000000 => 011111111111 => ? = 1 + 1
[1,1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0,0,1,0]
=> [10,1]
=> 100000000010 => 011111111101 => ? = 1 + 1
[1,1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0,0,1,0]
=> [9,2]
=> 10000000100 => 01111111011 => ? = 1 + 1
[1,1,1,1,1,1,1,0,0,0,1,0,0,0,0,0,1,0]
=> [8,3]
=> 1000001000 => 0111110111 => ? = 1 + 1
[1,0,1,1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0,0]
=> [2,1,1,1,1,1,1,1,1,1]
=> 101111111110 => 010000000001 => ? = 1 + 1
[1,1,1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0,0,0,1,0]
=> [10,2]
=> 100000000100 => 011111111011 => ? = 1 + 1
[1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,1,0,0]
=> [8,8]
=> 1100000000 => 0011111111 => ? = 2 + 1
[1,1,1,1,1,1,0,0,1,0,0,0,0,0,1,1,0,0]
=> [7,7,2]
=> 1100000100 => 0011111011 => ? = 2 + 1
[]
=> []
=> => => ? = 1 + 1
[1,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0,0]
=> [6,6,5,4,3,2,1]
=> 1101010101010 => 0010101010101 => ? = 2 + 1
[1,1,0,1,1,0,0,1,1,1,1,1,0,0,0,0,0,0]
=> [3,3,3,3,3,1,1]
=> 1111100110 => ? => ? = 5 + 1
[1,1,1,0,1,0,1,0,1,0,1,0,1,1,0,0,0,0]
=> [5,5,4,3,2,1]
=> 11010101010 => 00101010101 => ? = 2 + 1
[1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0]
=> []
=> => => ? = 1 + 1
[1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0]
=> []
=> => => ? = 1 + 1
[1,1,1,1,1,0,1,0,1,0,1,0,1,0,1,1,0,0,0,0,0,0]
=> [5,5,4,3,2,1]
=> 11010101010 => 00101010101 => ? = 2 + 1
[1,1,1,1,0,1,0,1,0,1,0,1,0,1,1,0,0,0,0,0]
=> [5,5,4,3,2,1]
=> 11010101010 => 00101010101 => ? = 2 + 1
[1,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,0]
=> []
=> => => ? = 1 + 1
[1,1,0,0,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0]
=> [2,2,2,2,2,2,2,2,2]
=> 11111111100 => 00000000011 => ? = 9 + 1
[1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,1,1,0,0]
=> [9,9]
=> 11000000000 => 00111111111 => ? = 2 + 1
[1,1,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0,0,0]
=> [6,6,5,4,3,2,1]
=> 1101010101010 => 0010101010101 => ? = 2 + 1
[1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,1,1,0,0]
=> [10,10]
=> 110000000000 => 001111111111 => ? = 2 + 1
[1,1,1,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0,0,0,0]
=> [6,6,5,4,3,2,1]
=> 1101010101010 => 0010101010101 => ? = 2 + 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: St000382
Mp00027: Dyck paths —to partition⟶ Integer partitions
Mp00095: Integer partitions —to binary word⟶ Binary words
Mp00097: Binary words —delta morphism⟶ Integer compositions
St000382: Integer compositions ⟶ ℤResult quality: 89% ●values known / values provided: 93%●distinct values known / distinct values provided: 89%
Mp00095: Integer partitions —to binary word⟶ Binary words
Mp00097: Binary words —delta morphism⟶ Integer compositions
St000382: Integer compositions ⟶ ℤResult quality: 89% ●values known / values provided: 93%●distinct values known / distinct values provided: 89%
Values
[1,0]
=> []
=> => [] => ? = 1
[1,0,1,0]
=> [1]
=> 10 => [1,1] => 1
[1,1,0,0]
=> []
=> => [] => ? = 1
[1,0,1,0,1,0]
=> [2,1]
=> 1010 => [1,1,1,1] => 1
[1,0,1,1,0,0]
=> [1,1]
=> 110 => [2,1] => 2
[1,1,0,0,1,0]
=> [2]
=> 100 => [1,2] => 1
[1,1,0,1,0,0]
=> [1]
=> 10 => [1,1] => 1
[1,1,1,0,0,0]
=> []
=> => [] => ? = 1
[1,0,1,0,1,0,1,0]
=> [3,2,1]
=> 101010 => [1,1,1,1,1,1] => 1
[1,0,1,0,1,1,0,0]
=> [2,2,1]
=> 11010 => [2,1,1,1] => 2
[1,0,1,1,0,0,1,0]
=> [3,1,1]
=> 100110 => [1,2,2,1] => 1
[1,0,1,1,0,1,0,0]
=> [2,1,1]
=> 10110 => [1,1,2,1] => 1
[1,0,1,1,1,0,0,0]
=> [1,1,1]
=> 1110 => [3,1] => 3
[1,1,0,0,1,0,1,0]
=> [3,2]
=> 10100 => [1,1,1,2] => 1
[1,1,0,0,1,1,0,0]
=> [2,2]
=> 1100 => [2,2] => 2
[1,1,0,1,0,0,1,0]
=> [3,1]
=> 10010 => [1,2,1,1] => 1
[1,1,0,1,0,1,0,0]
=> [2,1]
=> 1010 => [1,1,1,1] => 1
[1,1,0,1,1,0,0,0]
=> [1,1]
=> 110 => [2,1] => 2
[1,1,1,0,0,0,1,0]
=> [3]
=> 1000 => [1,3] => 1
[1,1,1,0,0,1,0,0]
=> [2]
=> 100 => [1,2] => 1
[1,1,1,0,1,0,0,0]
=> [1]
=> 10 => [1,1] => 1
[1,1,1,1,0,0,0,0]
=> []
=> => [] => ? = 1
[1,0,1,0,1,0,1,0,1,0]
=> [4,3,2,1]
=> 10101010 => [1,1,1,1,1,1,1,1] => 1
[1,0,1,0,1,0,1,1,0,0]
=> [3,3,2,1]
=> 1101010 => [2,1,1,1,1,1] => 2
[1,0,1,0,1,1,0,1,0,0]
=> [3,2,2,1]
=> 1011010 => [1,1,2,1,1,1] => 1
[1,0,1,0,1,1,1,0,0,0]
=> [2,2,2,1]
=> 111010 => [3,1,1,1] => 3
[1,0,1,1,0,0,1,1,0,0]
=> [3,3,1,1]
=> 1100110 => [2,2,2,1] => 2
[1,0,1,1,0,1,0,1,0,0]
=> [3,2,1,1]
=> 1010110 => [1,1,1,1,2,1] => 1
[1,0,1,1,0,1,1,0,0,0]
=> [2,2,1,1]
=> 110110 => [2,1,2,1] => 2
[1,0,1,1,1,0,0,1,0,0]
=> [3,1,1,1]
=> 1001110 => [1,2,3,1] => 1
[1,0,1,1,1,0,1,0,0,0]
=> [2,1,1,1]
=> 101110 => [1,1,3,1] => 1
[1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1]
=> 11110 => [4,1] => 4
[1,1,0,0,1,0,1,0,1,0]
=> [4,3,2]
=> 1010100 => [1,1,1,1,1,2] => 1
[1,1,0,0,1,0,1,1,0,0]
=> [3,3,2]
=> 110100 => [2,1,1,2] => 2
[1,1,0,0,1,1,0,0,1,0]
=> [4,2,2]
=> 1001100 => [1,2,2,2] => 1
[1,1,0,0,1,1,0,1,0,0]
=> [3,2,2]
=> 101100 => [1,1,2,2] => 1
[1,1,0,0,1,1,1,0,0,0]
=> [2,2,2]
=> 11100 => [3,2] => 3
[1,1,0,1,0,0,1,0,1,0]
=> [4,3,1]
=> 1010010 => [1,1,1,2,1,1] => 1
[1,1,0,1,0,0,1,1,0,0]
=> [3,3,1]
=> 110010 => [2,2,1,1] => 2
[1,1,0,1,0,1,0,0,1,0]
=> [4,2,1]
=> 1001010 => [1,2,1,1,1,1] => 1
[1,1,0,1,0,1,0,1,0,0]
=> [3,2,1]
=> 101010 => [1,1,1,1,1,1] => 1
[1,1,0,1,0,1,1,0,0,0]
=> [2,2,1]
=> 11010 => [2,1,1,1] => 2
[1,1,0,1,1,0,0,0,1,0]
=> [4,1,1]
=> 1000110 => [1,3,2,1] => 1
[1,1,0,1,1,0,0,1,0,0]
=> [3,1,1]
=> 100110 => [1,2,2,1] => 1
[1,1,0,1,1,0,1,0,0,0]
=> [2,1,1]
=> 10110 => [1,1,2,1] => 1
[1,1,0,1,1,1,0,0,0,0]
=> [1,1,1]
=> 1110 => [3,1] => 3
[1,1,1,0,0,0,1,0,1,0]
=> [4,3]
=> 101000 => [1,1,1,3] => 1
[1,1,1,0,0,0,1,1,0,0]
=> [3,3]
=> 11000 => [2,3] => 2
[1,1,1,0,0,1,0,0,1,0]
=> [4,2]
=> 100100 => [1,2,1,2] => 1
[1,1,1,0,0,1,0,1,0,0]
=> [3,2]
=> 10100 => [1,1,1,2] => 1
[1,1,1,0,0,1,1,0,0,0]
=> [2,2]
=> 1100 => [2,2] => 2
[1,1,1,0,1,0,0,0,1,0]
=> [4,1]
=> 100010 => [1,3,1,1] => 1
[1,1,1,0,1,0,0,1,0,0]
=> [3,1]
=> 10010 => [1,2,1,1] => 1
[1,1,1,0,1,0,1,0,0,0]
=> [2,1]
=> 1010 => [1,1,1,1] => 1
[1,1,1,1,1,0,0,0,0,0]
=> []
=> => [] => ? = 1
[1,1,1,1,1,1,0,0,0,0,0,0]
=> []
=> => [] => ? = 1
[1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [5,5,4,3,2,1]
=> 11010101010 => ? => ? = 2
[1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> []
=> => [] => ? = 1
[1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [6,6,5,4,3,2,1]
=> 1101010101010 => ? => ? = 2
[1,0,1,0,1,1,1,1,1,0,1,0,0,0,0,0]
=> [3,2,2,2,2,2,1]
=> 1011111010 => [1,1,5,1,1,1] => ? = 1
[1,0,1,1,0,0,1,1,1,1,1,0,0,0,0,0]
=> [3,3,3,3,3,1,1]
=> 1111100110 => ? => ? = 5
[1,1,0,1,0,1,0,1,0,1,0,1,1,0,0,0]
=> [5,5,4,3,2,1]
=> 11010101010 => ? => ? = 2
[1,1,1,1,1,0,0,1,0,0,0,0,1,0,1,0]
=> [7,6,2]
=> 1010000100 => ? => ? = 1
[1,1,1,1,1,0,1,0,0,0,0,0,1,0,1,0]
=> [7,6,1]
=> 1010000010 => [1,1,1,5,1,1] => ? = 1
[1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> []
=> => [] => ? = 1
[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]
=> 100000000000 => ? => ? = 1
[1,1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0,0,1,0]
=> [10,1]
=> 100000000010 => ? => ? = 1
[1,1,1,1,1,1,1,0,0,0,1,0,0,0,0,0,1,0]
=> [8,3]
=> 1000001000 => ? => ? = 1
[1,0,1,1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0,0]
=> [2,1,1,1,1,1,1,1,1,1]
=> 101111111110 => ? => ? = 1
[1,1,1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0,0,0,1,0]
=> [10,2]
=> 100000000100 => ? => ? = 1
[1,0,1,1,0,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [2,2,2,2,2,2,1,1]
=> 1111110110 => [6,1,2,1] => ? = 6
[1,1,1,1,1,1,0,1,0,0,0,0,0,0,1,1,0,0]
=> [7,7,1]
=> 1100000010 => [2,6,1,1] => ? = 2
[1,1,0,0,1,1,1,1,1,1,0,1,0,0,0,0,0,0]
=> [3,2,2,2,2,2,2]
=> 1011111100 => [1,1,6,2] => ? = 1
[1,1,1,1,1,1,0,0,1,0,0,0,0,0,1,1,0,0]
=> [7,7,2]
=> 1100000100 => ? => ? = 2
[1,1,0,0,1,1,1,1,1,0,1,1,0,0,0,0,0,0]
=> [3,3,2,2,2,2,2]
=> 1101111100 => ? => ? = 2
[1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0,1,0]
=> [9,8]
=> 10100000000 => [1,1,1,8] => ? = 1
[1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,1,1,0,0]
=> [8,8,1]
=> 11000000010 => [2,7,1,1] => ? = 2
[1,1,0,0,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0]
=> [3,2,2,2,2,2,2,2]
=> 10111111100 => [1,1,7,2] => ? = 1
[1,0,1,0,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [2,2,2,2,2,2,2,2,1]
=> 11111111010 => [8,1,1,1] => ? = 8
[]
=> []
=> => [] => ? = 1
[1,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0,0]
=> [6,6,5,4,3,2,1]
=> 1101010101010 => ? => ? = 2
[1,1,0,1,0,1,1,1,1,1,0,1,0,0,0,0,0,0]
=> [3,2,2,2,2,2,1]
=> 1011111010 => [1,1,5,1,1,1] => ? = 1
[1,1,0,1,1,0,0,1,1,1,1,1,0,0,0,0,0,0]
=> [3,3,3,3,3,1,1]
=> 1111100110 => ? => ? = 5
[1,1,1,0,1,0,1,0,1,0,1,0,1,1,0,0,0,0]
=> [5,5,4,3,2,1]
=> 11010101010 => ? => ? = 2
[1,1,1,1,1,1,0,1,0,0,0,0,0,1,0,1,0,0]
=> [7,6,1]
=> 1010000010 => [1,1,1,5,1,1] => ? = 1
[1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0]
=> []
=> => [] => ? = 1
[1,1,1,0,0,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [3,3,3,3,3,3,3]
=> 1111111000 => ? => ? = 7
[1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0]
=> []
=> => [] => ? = 1
[1,1,1,1,1,0,1,0,1,0,1,0,1,0,1,1,0,0,0,0,0,0]
=> [5,5,4,3,2,1]
=> 11010101010 => ? => ? = 2
[1,1,1,1,0,1,0,1,0,1,0,1,0,1,1,0,0,0,0,0]
=> [5,5,4,3,2,1]
=> 11010101010 => ? => ? = 2
[1,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,0]
=> []
=> => [] => ? = 1
[1,1,0,0,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0]
=> [2,2,2,2,2,2,2,2,2]
=> 11111111100 => ? => ? = 9
[1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,1,1,0,0]
=> [9,9]
=> 11000000000 => [2,9] => ? = 2
[1,1,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0,0,0]
=> [6,6,5,4,3,2,1]
=> 1101010101010 => ? => ? = 2
[1,1,1,1,1,1,1,0,1,0,0,0,0,0,1,0,1,0,0,0]
=> [7,6,1]
=> 1010000010 => [1,1,1,5,1,1] => ? = 1
[1,1,1,0,0,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0]
=> [3,2,2,2,2,2,2]
=> 1011111100 => [1,1,6,2] => ? = 1
[1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,1,1,0,0]
=> [10,10]
=> 110000000000 => [2,10] => ? = 2
[1,1,1,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0,0,0,0]
=> [6,6,5,4,3,2,1]
=> 1101010101010 => ? => ? = 2
Description
The first part of an integer composition.
Matching statistic: St000297
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00027: Dyck paths —to partition⟶ Integer partitions
Mp00095: Integer partitions —to binary word⟶ Binary words
Mp00135: Binary words —rotate front-to-back⟶ Binary words
St000297: Binary words ⟶ ℤResult quality: 78% ●values known / values provided: 93%●distinct values known / distinct values provided: 78%
Mp00095: Integer partitions —to binary word⟶ Binary words
Mp00135: Binary words —rotate front-to-back⟶ Binary words
St000297: Binary words ⟶ ℤResult quality: 78% ●values known / values provided: 93%●distinct values known / distinct values provided: 78%
Values
[1,0]
=> []
=> => ? => ? = 1 - 1
[1,0,1,0]
=> [1]
=> 10 => 01 => 0 = 1 - 1
[1,1,0,0]
=> []
=> => ? => ? = 1 - 1
[1,0,1,0,1,0]
=> [2,1]
=> 1010 => 0101 => 0 = 1 - 1
[1,0,1,1,0,0]
=> [1,1]
=> 110 => 101 => 1 = 2 - 1
[1,1,0,0,1,0]
=> [2]
=> 100 => 001 => 0 = 1 - 1
[1,1,0,1,0,0]
=> [1]
=> 10 => 01 => 0 = 1 - 1
[1,1,1,0,0,0]
=> []
=> => ? => ? = 1 - 1
[1,0,1,0,1,0,1,0]
=> [3,2,1]
=> 101010 => 010101 => 0 = 1 - 1
[1,0,1,0,1,1,0,0]
=> [2,2,1]
=> 11010 => 10101 => 1 = 2 - 1
[1,0,1,1,0,0,1,0]
=> [3,1,1]
=> 100110 => 001101 => 0 = 1 - 1
[1,0,1,1,0,1,0,0]
=> [2,1,1]
=> 10110 => 01101 => 0 = 1 - 1
[1,0,1,1,1,0,0,0]
=> [1,1,1]
=> 1110 => 1101 => 2 = 3 - 1
[1,1,0,0,1,0,1,0]
=> [3,2]
=> 10100 => 01001 => 0 = 1 - 1
[1,1,0,0,1,1,0,0]
=> [2,2]
=> 1100 => 1001 => 1 = 2 - 1
[1,1,0,1,0,0,1,0]
=> [3,1]
=> 10010 => 00101 => 0 = 1 - 1
[1,1,0,1,0,1,0,0]
=> [2,1]
=> 1010 => 0101 => 0 = 1 - 1
[1,1,0,1,1,0,0,0]
=> [1,1]
=> 110 => 101 => 1 = 2 - 1
[1,1,1,0,0,0,1,0]
=> [3]
=> 1000 => 0001 => 0 = 1 - 1
[1,1,1,0,0,1,0,0]
=> [2]
=> 100 => 001 => 0 = 1 - 1
[1,1,1,0,1,0,0,0]
=> [1]
=> 10 => 01 => 0 = 1 - 1
[1,1,1,1,0,0,0,0]
=> []
=> => ? => ? = 1 - 1
[1,0,1,0,1,0,1,0,1,0]
=> [4,3,2,1]
=> 10101010 => 01010101 => 0 = 1 - 1
[1,0,1,0,1,0,1,1,0,0]
=> [3,3,2,1]
=> 1101010 => 1010101 => 1 = 2 - 1
[1,0,1,0,1,1,0,1,0,0]
=> [3,2,2,1]
=> 1011010 => 0110101 => 0 = 1 - 1
[1,0,1,0,1,1,1,0,0,0]
=> [2,2,2,1]
=> 111010 => 110101 => 2 = 3 - 1
[1,0,1,1,0,0,1,1,0,0]
=> [3,3,1,1]
=> 1100110 => 1001101 => 1 = 2 - 1
[1,0,1,1,0,1,0,1,0,0]
=> [3,2,1,1]
=> 1010110 => 0101101 => 0 = 1 - 1
[1,0,1,1,0,1,1,0,0,0]
=> [2,2,1,1]
=> 110110 => 101101 => 1 = 2 - 1
[1,0,1,1,1,0,0,1,0,0]
=> [3,1,1,1]
=> 1001110 => 0011101 => 0 = 1 - 1
[1,0,1,1,1,0,1,0,0,0]
=> [2,1,1,1]
=> 101110 => 011101 => 0 = 1 - 1
[1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1]
=> 11110 => 11101 => 3 = 4 - 1
[1,1,0,0,1,0,1,0,1,0]
=> [4,3,2]
=> 1010100 => 0101001 => 0 = 1 - 1
[1,1,0,0,1,0,1,1,0,0]
=> [3,3,2]
=> 110100 => 101001 => 1 = 2 - 1
[1,1,0,0,1,1,0,0,1,0]
=> [4,2,2]
=> 1001100 => 0011001 => 0 = 1 - 1
[1,1,0,0,1,1,0,1,0,0]
=> [3,2,2]
=> 101100 => 011001 => 0 = 1 - 1
[1,1,0,0,1,1,1,0,0,0]
=> [2,2,2]
=> 11100 => 11001 => 2 = 3 - 1
[1,1,0,1,0,0,1,0,1,0]
=> [4,3,1]
=> 1010010 => 0100101 => 0 = 1 - 1
[1,1,0,1,0,0,1,1,0,0]
=> [3,3,1]
=> 110010 => 100101 => 1 = 2 - 1
[1,1,0,1,0,1,0,0,1,0]
=> [4,2,1]
=> 1001010 => 0010101 => 0 = 1 - 1
[1,1,0,1,0,1,0,1,0,0]
=> [3,2,1]
=> 101010 => 010101 => 0 = 1 - 1
[1,1,0,1,0,1,1,0,0,0]
=> [2,2,1]
=> 11010 => 10101 => 1 = 2 - 1
[1,1,0,1,1,0,0,0,1,0]
=> [4,1,1]
=> 1000110 => 0001101 => 0 = 1 - 1
[1,1,0,1,1,0,0,1,0,0]
=> [3,1,1]
=> 100110 => 001101 => 0 = 1 - 1
[1,1,0,1,1,0,1,0,0,0]
=> [2,1,1]
=> 10110 => 01101 => 0 = 1 - 1
[1,1,0,1,1,1,0,0,0,0]
=> [1,1,1]
=> 1110 => 1101 => 2 = 3 - 1
[1,1,1,0,0,0,1,0,1,0]
=> [4,3]
=> 101000 => 010001 => 0 = 1 - 1
[1,1,1,0,0,0,1,1,0,0]
=> [3,3]
=> 11000 => 10001 => 1 = 2 - 1
[1,1,1,0,0,1,0,0,1,0]
=> [4,2]
=> 100100 => 001001 => 0 = 1 - 1
[1,1,1,0,0,1,0,1,0,0]
=> [3,2]
=> 10100 => 01001 => 0 = 1 - 1
[1,1,1,0,0,1,1,0,0,0]
=> [2,2]
=> 1100 => 1001 => 1 = 2 - 1
[1,1,1,0,1,0,0,0,1,0]
=> [4,1]
=> 100010 => 000101 => 0 = 1 - 1
[1,1,1,0,1,0,0,1,0,0]
=> [3,1]
=> 10010 => 00101 => 0 = 1 - 1
[1,1,1,0,1,0,1,0,0,0]
=> [2,1]
=> 1010 => 0101 => 0 = 1 - 1
[1,1,1,1,1,0,0,0,0,0]
=> []
=> => ? => ? = 1 - 1
[1,1,1,1,1,1,0,0,0,0,0,0]
=> []
=> => ? => ? = 1 - 1
[1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> []
=> => ? => ? = 1 - 1
[1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [6,6,5,4,3,2,1]
=> 1101010101010 => ? => ? = 2 - 1
[1,0,1,0,1,1,1,1,1,0,1,0,0,0,0,0]
=> [3,2,2,2,2,2,1]
=> 1011111010 => ? => ? = 1 - 1
[1,0,1,1,0,0,1,1,1,1,1,0,0,0,0,0]
=> [3,3,3,3,3,1,1]
=> 1111100110 => ? => ? = 5 - 1
[1,1,1,1,1,0,0,1,0,0,0,0,1,0,1,0]
=> [7,6,2]
=> 1010000100 => ? => ? = 1 - 1
[1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> []
=> => ? => ? = 1 - 1
[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]
=> 100000000000 => ? => ? = 1 - 1
[1,1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0,0,1,0]
=> [10,1]
=> 100000000010 => ? => ? = 1 - 1
[1,1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0,0,1,0]
=> [9,2]
=> 10000000100 => 00000001001 => ? = 1 - 1
[1,1,1,1,1,1,1,0,0,0,1,0,0,0,0,0,1,0]
=> [8,3]
=> 1000001000 => ? => ? = 1 - 1
[1,0,1,1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0,0]
=> [2,1,1,1,1,1,1,1,1,1]
=> 101111111110 => ? => ? = 1 - 1
[1,1,1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0,0,0,1,0]
=> [10,2]
=> 100000000100 => 000000001001 => ? = 1 - 1
[1,0,1,1,0,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [2,2,2,2,2,2,1,1]
=> 1111110110 => 1111101101 => ? = 6 - 1
[1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0,1,0]
=> [8,7]
=> 1010000000 => ? => ? = 1 - 1
[1,1,1,1,1,1,0,1,0,0,0,0,0,0,1,1,0,0]
=> [7,7,1]
=> 1100000010 => 1000000101 => ? = 2 - 1
[1,1,0,0,1,1,1,1,1,1,0,1,0,0,0,0,0,0]
=> [3,2,2,2,2,2,2]
=> 1011111100 => ? => ? = 1 - 1
[1,0,1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [2,2,2,2,2,2,2,1]
=> 1111111010 => 1111110101 => ? = 7 - 1
[1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,1,0,0]
=> [8,8]
=> 1100000000 => 1000000001 => ? = 2 - 1
[1,1,1,1,1,1,0,0,1,0,0,0,0,0,1,1,0,0]
=> [7,7,2]
=> 1100000100 => ? => ? = 2 - 1
[1,1,0,0,1,1,1,1,1,0,1,1,0,0,0,0,0,0]
=> [3,3,2,2,2,2,2]
=> 1101111100 => ? => ? = 2 - 1
[1,1,0,0,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [2,2,2,2,2,2,2,2]
=> 1111111100 => 1111111001 => ? = 8 - 1
[1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0,1,0]
=> [9,8]
=> 10100000000 => ? => ? = 1 - 1
[1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,1,1,0,0]
=> [8,8,1]
=> 11000000010 => 10000000101 => ? = 2 - 1
[1,1,0,0,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0]
=> [3,2,2,2,2,2,2,2]
=> 10111111100 => ? => ? = 1 - 1
[1,0,1,0,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [2,2,2,2,2,2,2,2,1]
=> 11111111010 => 11111110101 => ? = 8 - 1
[]
=> []
=> => ? => ? = 1 - 1
[1,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0,0]
=> [6,6,5,4,3,2,1]
=> 1101010101010 => ? => ? = 2 - 1
[1,1,0,1,0,1,1,1,1,1,0,1,0,0,0,0,0,0]
=> [3,2,2,2,2,2,1]
=> 1011111010 => ? => ? = 1 - 1
[1,1,0,1,1,0,0,1,1,1,1,1,0,0,0,0,0,0]
=> [3,3,3,3,3,1,1]
=> 1111100110 => ? => ? = 5 - 1
[1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0]
=> []
=> => ? => ? = 1 - 1
[1,1,1,0,0,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [3,3,3,3,3,3,3]
=> 1111111000 => ? => ? = 7 - 1
[1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0]
=> []
=> => ? => ? = 1 - 1
[1,1,0,1,0,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [3,3,3,3,3,3,1]
=> 1111110010 => 1111100101 => ? = 6 - 1
[1,1,0,1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [2,2,2,2,2,2,2,1]
=> 1111111010 => 1111110101 => ? = 7 - 1
[1,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,0]
=> []
=> => ? => ? = 1 - 1
[1,1,0,0,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0]
=> [2,2,2,2,2,2,2,2,2]
=> 11111111100 => ? => ? = 9 - 1
[1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,1,1,0,0]
=> [9,9]
=> 11000000000 => 10000000001 => ? = 2 - 1
[1,1,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0,0,0]
=> [6,6,5,4,3,2,1]
=> 1101010101010 => ? => ? = 2 - 1
[1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0,1,0,0]
=> [8,7]
=> 1010000000 => ? => ? = 1 - 1
[1,1,1,0,0,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0]
=> [3,2,2,2,2,2,2]
=> 1011111100 => ? => ? = 1 - 1
[1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,1,1,0,0]
=> [10,10]
=> 110000000000 => 100000000001 => ? = 2 - 1
[1,1,1,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0,0,0,0]
=> [6,6,5,4,3,2,1]
=> 1101010101010 => ? => ? = 2 - 1
Description
The number of leading ones in a binary word.
Matching statistic: St000383
Mp00027: Dyck paths —to partition⟶ Integer partitions
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00102: Dyck paths —rise composition⟶ Integer compositions
St000383: Integer compositions ⟶ ℤResult quality: 67% ●values known / values provided: 87%●distinct values known / distinct values provided: 67%
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00102: Dyck paths —rise composition⟶ Integer compositions
St000383: Integer compositions ⟶ ℤResult quality: 67% ●values known / values provided: 87%●distinct values known / distinct values provided: 67%
Values
[1,0]
=> []
=> []
=> [] => ? = 1
[1,0,1,0]
=> [1]
=> [1,0,1,0]
=> [1,1] => 1
[1,1,0,0]
=> []
=> []
=> [] => ? = 1
[1,0,1,0,1,0]
=> [2,1]
=> [1,0,1,0,1,0]
=> [1,1,1] => 1
[1,0,1,1,0,0]
=> [1,1]
=> [1,0,1,1,0,0]
=> [1,2] => 2
[1,1,0,0,1,0]
=> [2]
=> [1,1,0,0,1,0]
=> [2,1] => 1
[1,1,0,1,0,0]
=> [1]
=> [1,0,1,0]
=> [1,1] => 1
[1,1,1,0,0,0]
=> []
=> []
=> [] => ? = 1
[1,0,1,0,1,0,1,0]
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1] => 1
[1,0,1,0,1,1,0,0]
=> [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> [1,1,2] => 2
[1,0,1,1,0,0,1,0]
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [1,2,1] => 1
[1,0,1,1,0,1,0,0]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,2,1] => 1
[1,0,1,1,1,0,0,0]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [1,3] => 3
[1,1,0,0,1,0,1,0]
=> [3,2]
=> [1,1,0,0,1,0,1,0]
=> [2,1,1] => 1
[1,1,0,0,1,1,0,0]
=> [2,2]
=> [1,1,0,0,1,1,0,0]
=> [2,2] => 2
[1,1,0,1,0,0,1,0]
=> [3,1]
=> [1,1,0,1,0,0,1,0]
=> [2,1,1] => 1
[1,1,0,1,0,1,0,0]
=> [2,1]
=> [1,0,1,0,1,0]
=> [1,1,1] => 1
[1,1,0,1,1,0,0,0]
=> [1,1]
=> [1,0,1,1,0,0]
=> [1,2] => 2
[1,1,1,0,0,0,1,0]
=> [3]
=> [1,1,1,0,0,0,1,0]
=> [3,1] => 1
[1,1,1,0,0,1,0,0]
=> [2]
=> [1,1,0,0,1,0]
=> [2,1] => 1
[1,1,1,0,1,0,0,0]
=> [1]
=> [1,0,1,0]
=> [1,1] => 1
[1,1,1,1,0,0,0,0]
=> []
=> []
=> [] => ? = 1
[1,0,1,0,1,0,1,0,1,0]
=> [4,3,2,1]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1] => 1
[1,0,1,0,1,0,1,1,0,0]
=> [3,3,2,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,2] => 2
[1,0,1,0,1,1,0,1,0,0]
=> [3,2,2,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,1,2,1] => 1
[1,0,1,0,1,1,1,0,0,0]
=> [2,2,2,1]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,1,3] => 3
[1,0,1,1,0,0,1,1,0,0]
=> [3,3,1,1]
=> [1,0,1,1,0,0,1,1,0,0]
=> [1,2,2] => 2
[1,0,1,1,0,1,0,1,0,0]
=> [3,2,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,2,1,1] => 1
[1,0,1,1,0,1,1,0,0,0]
=> [2,2,1,1]
=> [1,0,1,1,0,1,1,0,0,0]
=> [1,2,2] => 2
[1,0,1,1,1,0,0,1,0,0]
=> [3,1,1,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,3,1] => 1
[1,0,1,1,1,0,1,0,0,0]
=> [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,3,1] => 1
[1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,4] => 4
[1,1,0,0,1,0,1,0,1,0]
=> [4,3,2]
=> [1,1,0,0,1,0,1,0,1,0]
=> [2,1,1,1] => 1
[1,1,0,0,1,0,1,1,0,0]
=> [3,3,2]
=> [1,1,0,0,1,0,1,1,0,0]
=> [2,1,2] => 2
[1,1,0,0,1,1,0,0,1,0]
=> [4,2,2]
=> [1,1,0,0,1,1,0,0,1,0]
=> [2,2,1] => 1
[1,1,0,0,1,1,0,1,0,0]
=> [3,2,2]
=> [1,1,0,0,1,1,0,1,0,0]
=> [2,2,1] => 1
[1,1,0,0,1,1,1,0,0,0]
=> [2,2,2]
=> [1,1,0,0,1,1,1,0,0,0]
=> [2,3] => 3
[1,1,0,1,0,0,1,0,1,0]
=> [4,3,1]
=> [1,1,0,1,0,0,1,0,1,0]
=> [2,1,1,1] => 1
[1,1,0,1,0,0,1,1,0,0]
=> [3,3,1]
=> [1,1,0,1,0,0,1,1,0,0]
=> [2,1,2] => 2
[1,1,0,1,0,1,0,0,1,0]
=> [4,2,1]
=> [1,1,0,1,0,1,0,0,1,0]
=> [2,1,1,1] => 1
[1,1,0,1,0,1,0,1,0,0]
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1] => 1
[1,1,0,1,0,1,1,0,0,0]
=> [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> [1,1,2] => 2
[1,1,0,1,1,0,0,0,1,0]
=> [4,1,1]
=> [1,1,0,1,1,0,0,0,1,0]
=> [2,2,1] => 1
[1,1,0,1,1,0,0,1,0,0]
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [1,2,1] => 1
[1,1,0,1,1,0,1,0,0,0]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,2,1] => 1
[1,1,0,1,1,1,0,0,0,0]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [1,3] => 3
[1,1,1,0,0,0,1,0,1,0]
=> [4,3]
=> [1,1,1,0,0,0,1,0,1,0]
=> [3,1,1] => 1
[1,1,1,0,0,0,1,1,0,0]
=> [3,3]
=> [1,1,1,0,0,0,1,1,0,0]
=> [3,2] => 2
[1,1,1,0,0,1,0,0,1,0]
=> [4,2]
=> [1,1,1,0,0,1,0,0,1,0]
=> [3,1,1] => 1
[1,1,1,0,0,1,0,1,0,0]
=> [3,2]
=> [1,1,0,0,1,0,1,0]
=> [2,1,1] => 1
[1,1,1,0,0,1,1,0,0,0]
=> [2,2]
=> [1,1,0,0,1,1,0,0]
=> [2,2] => 2
[1,1,1,0,1,0,0,0,1,0]
=> [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [3,1,1] => 1
[1,1,1,0,1,0,0,1,0,0]
=> [3,1]
=> [1,1,0,1,0,0,1,0]
=> [2,1,1] => 1
[1,1,1,0,1,0,1,0,0,0]
=> [2,1]
=> [1,0,1,0,1,0]
=> [1,1,1] => 1
[1,1,1,1,1,0,0,0,0,0]
=> []
=> []
=> [] => ? = 1
[1,1,1,1,1,1,0,0,0,0,0,0]
=> []
=> []
=> [] => ? = 1
[1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> []
=> []
=> [] => ? = 1
[1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [6,6,5,4,3,2,1]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,1,1,2] => ? = 2
[1,0,1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [2,2,2,2,2,2,1]
=> [1,0,1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,1,6] => ? = 6
[1,0,1,1,0,0,1,1,1,1,1,0,0,0,0,0]
=> [3,3,3,3,3,1,1]
=> [1,0,1,1,0,0,1,1,1,1,1,0,0,0,0,0]
=> [1,2,5] => ? = 5
[1,0,1,1,0,1,1,1,1,1,0,0,0,0,0,0]
=> [2,2,2,2,2,1,1]
=> [1,0,1,1,0,1,1,1,1,1,0,0,0,0,0,0]
=> [1,2,5] => ? = 5
[1,0,1,1,1,0,1,1,1,1,0,0,0,0,0,0]
=> [2,2,2,2,1,1,1]
=> [1,0,1,1,1,0,1,1,1,1,0,0,0,0,0,0]
=> [1,3,4] => ? = 4
[1,1,0,0,1,0,1,1,1,1,1,0,0,0,0,0]
=> [3,3,3,3,3,2]
=> [1,1,0,0,1,0,1,1,1,1,1,0,0,0,0,0]
=> [2,1,5] => ? = 5
[1,1,0,0,1,1,1,1,1,0,1,0,0,0,0,0]
=> [3,2,2,2,2,2]
=> [1,1,0,0,1,1,1,1,1,0,1,0,0,0,0,0]
=> [2,5,1] => ? = 1
[1,1,0,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [2,2,2,2,2,2]
=> [1,1,0,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [2,6] => ? = 6
[1,1,0,1,0,0,1,1,1,1,1,0,0,0,0,0]
=> [3,3,3,3,3,1]
=> [1,1,0,1,0,0,1,1,1,1,1,0,0,0,0,0]
=> [2,1,5] => ? = 5
[1,1,1,0,0,0,1,1,1,1,1,0,0,0,0,0]
=> [3,3,3,3,3]
=> [1,1,1,0,0,0,1,1,1,1,1,0,0,0,0,0]
=> [3,5] => ? = 5
[1,1,1,1,0,0,1,0,0,0,1,1,1,0,0,0]
=> [5,5,5,2]
=> [1,1,1,1,0,0,1,0,0,0,1,1,1,0,0,0]
=> [4,1,3] => ? = 3
[1,1,1,1,0,1,0,0,0,0,1,1,1,0,0,0]
=> [5,5,5,1]
=> [1,1,1,1,0,1,0,0,0,0,1,1,1,0,0,0]
=> [4,1,3] => ? = 3
[1,1,1,1,1,0,0,0,0,0,1,0,1,1,0,0]
=> [6,6,5]
=> [1,1,1,1,1,0,0,0,0,0,1,0,1,1,0,0]
=> [5,1,2] => ? = 2
[1,1,1,1,1,0,0,0,0,0,1,1,1,0,0,0]
=> [5,5,5]
=> [1,1,1,1,1,0,0,0,0,0,1,1,1,0,0,0]
=> [5,3] => ? = 3
[1,1,1,1,1,0,0,0,0,1,0,0,1,1,0,0]
=> [6,6,4]
=> [1,1,1,1,1,0,0,0,0,1,0,0,1,1,0,0]
=> [5,1,2] => ? = 2
[1,1,1,1,1,0,0,1,0,0,0,0,1,1,0,0]
=> [6,6,2]
=> [1,1,1,1,1,0,0,1,0,0,0,0,1,1,0,0]
=> [5,1,2] => ? = 2
[1,1,1,1,1,0,1,0,0,0,0,0,1,1,0,0]
=> [6,6,1]
=> [1,1,1,1,1,0,1,0,0,0,0,0,1,1,0,0]
=> [5,1,2] => ? = 2
[1,1,1,1,1,1,0,0,0,0,0,0,1,1,0,0]
=> [6,6]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,1,0,0]
=> [6,2] => ? = 2
[1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> []
=> []
=> [] => ? = 1
[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]
=> [1,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,0,1,0]
=> [11,1] => ? = 1
[1,1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0,0,1,0]
=> [10,1]
=> [1,1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0,0,1,0]
=> [9,1,1] => ? = 1
[1,0,1,1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0,0]
=> [2,1,1,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0,0]
=> [1,9,1] => ? = 1
[1,1,1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0,0,0,1,0]
=> [10,2]
=> [1,1,1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0,0,0,1,0]
=> [9,1,1] => ? = 1
[1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,1,0,0]
=> [7,7]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,1,0,0]
=> [7,2] => ? = 2
[1,1,0,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [2,2,2,2,2,2,2]
=> [1,1,0,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [2,7] => ? = 7
[1,0,1,1,0,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [2,2,2,2,2,2,1,1]
=> [1,0,1,1,0,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [1,2,6] => ? = 6
[1,1,1,1,1,1,0,1,0,0,0,0,0,0,1,1,0,0]
=> [7,7,1]
=> [1,1,1,1,1,1,0,1,0,0,0,0,0,0,1,1,0,0]
=> [6,1,2] => ? = 2
[1,1,0,0,1,1,1,1,1,1,0,1,0,0,0,0,0,0]
=> [3,2,2,2,2,2,2]
=> [1,1,0,0,1,1,1,1,1,1,0,1,0,0,0,0,0,0]
=> [2,6,1] => ? = 1
[1,0,1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [2,2,2,2,2,2,2,1]
=> [1,0,1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [1,1,7] => ? = 7
[1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,1,0,0]
=> [8,8]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,1,0,0]
=> [8,2] => ? = 2
[1,1,1,1,1,1,0,0,1,0,0,0,0,0,1,1,0,0]
=> [7,7,2]
=> [1,1,1,1,1,1,0,0,1,0,0,0,0,0,1,1,0,0]
=> [6,1,2] => ? = 2
[1,1,0,0,1,1,1,1,1,0,1,1,0,0,0,0,0,0]
=> [3,3,2,2,2,2,2]
=> [1,1,0,0,1,1,1,1,1,0,1,1,0,0,0,0,0,0]
=> [2,5,2] => ? = 2
[1,1,0,0,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [2,2,2,2,2,2,2,2]
=> [1,1,0,0,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [2,8] => ? = 8
[1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,1,1,0,0]
=> [8,8,1]
=> [1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,1,1,0,0]
=> [7,1,2] => ? = 2
[1,1,0,0,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0]
=> [3,2,2,2,2,2,2,2]
=> [1,1,0,0,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0]
=> [2,7,1] => ? = 1
[1,0,1,0,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [2,2,2,2,2,2,2,2,1]
=> [1,0,1,0,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [1,1,8] => ? = 8
[]
=> []
=> []
=> [] => ? = 1
[1,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0,0]
=> [6,6,5,4,3,2,1]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,1,1,2] => ? = 2
[1,1,0,1,0,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [2,2,2,2,2,2,1]
=> [1,0,1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,1,6] => ? = 6
[1,1,0,1,1,0,0,1,1,1,1,1,0,0,0,0,0,0]
=> [3,3,3,3,3,1,1]
=> [1,0,1,1,0,0,1,1,1,1,1,0,0,0,0,0]
=> [1,2,5] => ? = 5
[1,1,0,1,1,0,1,1,1,1,1,0,0,0,0,0,0,0]
=> [2,2,2,2,2,1,1]
=> [1,0,1,1,0,1,1,1,1,1,0,0,0,0,0,0]
=> [1,2,5] => ? = 5
[1,1,1,0,0,1,0,1,1,1,1,1,0,0,0,0,0,0]
=> [3,3,3,3,3,2]
=> [1,1,0,0,1,0,1,1,1,1,1,0,0,0,0,0]
=> [2,1,5] => ? = 5
[1,1,1,0,0,1,1,1,1,1,0,1,0,0,0,0,0,0]
=> [3,2,2,2,2,2]
=> [1,1,0,0,1,1,1,1,1,0,1,0,0,0,0,0]
=> [2,5,1] => ? = 1
Description
The last part of an integer composition.
Matching statistic: St001107
Mp00027: Dyck paths —to partition⟶ Integer partitions
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00132: Dyck paths —switch returns and last double rise⟶ Dyck paths
St001107: Dyck paths ⟶ ℤResult quality: 67% ●values known / values provided: 81%●distinct values known / distinct values provided: 67%
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00132: Dyck paths —switch returns and last double rise⟶ Dyck paths
St001107: Dyck paths ⟶ ℤResult quality: 67% ●values known / values provided: 81%●distinct values known / distinct values provided: 67%
Values
[1,0]
=> []
=> []
=> []
=> ? = 1 - 1
[1,0,1,0]
=> [1]
=> [1,0,1,0]
=> [1,0,1,0]
=> 0 = 1 - 1
[1,1,0,0]
=> []
=> []
=> []
=> ? = 1 - 1
[1,0,1,0,1,0]
=> [2,1]
=> [1,0,1,0,1,0]
=> [1,0,1,0,1,0]
=> 0 = 1 - 1
[1,0,1,1,0,0]
=> [1,1]
=> [1,0,1,1,0,0]
=> [1,1,0,1,0,0]
=> 1 = 2 - 1
[1,1,0,0,1,0]
=> [2]
=> [1,1,0,0,1,0]
=> [1,1,0,0,1,0]
=> 0 = 1 - 1
[1,1,0,1,0,0]
=> [1]
=> [1,0,1,0]
=> [1,0,1,0]
=> 0 = 1 - 1
[1,1,1,0,0,0]
=> []
=> []
=> []
=> ? = 1 - 1
[1,0,1,0,1,0,1,0]
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0]
=> 0 = 1 - 1
[1,0,1,0,1,1,0,0]
=> [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,0]
=> 1 = 2 - 1
[1,0,1,1,0,0,1,0]
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,1,0,0,1,0]
=> 0 = 1 - 1
[1,0,1,1,0,1,0,0]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,0,1,1,0,1,0,0]
=> 0 = 1 - 1
[1,0,1,1,1,0,0,0]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [1,1,1,0,1,0,0,0]
=> 2 = 3 - 1
[1,1,0,0,1,0,1,0]
=> [3,2]
=> [1,1,0,0,1,0,1,0]
=> [1,1,0,0,1,0,1,0]
=> 0 = 1 - 1
[1,1,0,0,1,1,0,0]
=> [2,2]
=> [1,1,0,0,1,1,0,0]
=> [1,1,1,0,0,1,0,0]
=> 1 = 2 - 1
[1,1,0,1,0,0,1,0]
=> [3,1]
=> [1,1,0,1,0,0,1,0]
=> [1,0,1,1,0,0,1,0]
=> 0 = 1 - 1
[1,1,0,1,0,1,0,0]
=> [2,1]
=> [1,0,1,0,1,0]
=> [1,0,1,0,1,0]
=> 0 = 1 - 1
[1,1,0,1,1,0,0,0]
=> [1,1]
=> [1,0,1,1,0,0]
=> [1,1,0,1,0,0]
=> 1 = 2 - 1
[1,1,1,0,0,0,1,0]
=> [3]
=> [1,1,1,0,0,0,1,0]
=> [1,1,1,0,0,0,1,0]
=> 0 = 1 - 1
[1,1,1,0,0,1,0,0]
=> [2]
=> [1,1,0,0,1,0]
=> [1,1,0,0,1,0]
=> 0 = 1 - 1
[1,1,1,0,1,0,0,0]
=> [1]
=> [1,0,1,0]
=> [1,0,1,0]
=> 0 = 1 - 1
[1,1,1,1,0,0,0,0]
=> []
=> []
=> []
=> ? = 1 - 1
[1,0,1,0,1,0,1,0,1,0]
=> [4,3,2,1]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> 0 = 1 - 1
[1,0,1,0,1,0,1,1,0,0]
=> [3,3,2,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> 1 = 2 - 1
[1,0,1,0,1,1,0,1,0,0]
=> [3,2,2,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,0,1,1,0,1,0,1,0,0]
=> 0 = 1 - 1
[1,0,1,0,1,1,1,0,0,0]
=> [2,2,2,1]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> 2 = 3 - 1
[1,0,1,1,0,0,1,1,0,0]
=> [3,3,1,1]
=> [1,0,1,1,0,0,1,1,0,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> 1 = 2 - 1
[1,0,1,1,0,1,0,1,0,0]
=> [3,2,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,0,1,0,1,1,0,1,0,0]
=> 0 = 1 - 1
[1,0,1,1,0,1,1,0,0,0]
=> [2,2,1,1]
=> [1,0,1,1,0,1,1,0,0,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> 1 = 2 - 1
[1,0,1,1,1,0,0,1,0,0]
=> [3,1,1,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,0,1,1,1,0,0,1,0,0]
=> 0 = 1 - 1
[1,0,1,1,1,0,1,0,0,0]
=> [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,0,1,1,1,0,1,0,0,0]
=> 0 = 1 - 1
[1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> 3 = 4 - 1
[1,1,0,0,1,0,1,0,1,0]
=> [4,3,2]
=> [1,1,0,0,1,0,1,0,1,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> 0 = 1 - 1
[1,1,0,0,1,0,1,1,0,0]
=> [3,3,2]
=> [1,1,0,0,1,0,1,1,0,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> 1 = 2 - 1
[1,1,0,0,1,1,0,0,1,0]
=> [4,2,2]
=> [1,1,0,0,1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0,1,0]
=> 0 = 1 - 1
[1,1,0,0,1,1,0,1,0,0]
=> [3,2,2]
=> [1,1,0,0,1,1,0,1,0,0]
=> [1,1,0,0,1,1,0,1,0,0]
=> 0 = 1 - 1
[1,1,0,0,1,1,1,0,0,0]
=> [2,2,2]
=> [1,1,0,0,1,1,1,0,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> 2 = 3 - 1
[1,1,0,1,0,0,1,0,1,0]
=> [4,3,1]
=> [1,1,0,1,0,0,1,0,1,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> 0 = 1 - 1
[1,1,0,1,0,0,1,1,0,0]
=> [3,3,1]
=> [1,1,0,1,0,0,1,1,0,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> 1 = 2 - 1
[1,1,0,1,0,1,0,0,1,0]
=> [4,2,1]
=> [1,1,0,1,0,1,0,0,1,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> 0 = 1 - 1
[1,1,0,1,0,1,0,1,0,0]
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0]
=> 0 = 1 - 1
[1,1,0,1,0,1,1,0,0,0]
=> [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,0]
=> 1 = 2 - 1
[1,1,0,1,1,0,0,0,1,0]
=> [4,1,1]
=> [1,1,0,1,1,0,0,0,1,0]
=> [1,1,0,1,1,0,0,0,1,0]
=> 0 = 1 - 1
[1,1,0,1,1,0,0,1,0,0]
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,1,0,0,1,0]
=> 0 = 1 - 1
[1,1,0,1,1,0,1,0,0,0]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,0,1,1,0,1,0,0]
=> 0 = 1 - 1
[1,1,0,1,1,1,0,0,0,0]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [1,1,1,0,1,0,0,0]
=> 2 = 3 - 1
[1,1,1,0,0,0,1,0,1,0]
=> [4,3]
=> [1,1,1,0,0,0,1,0,1,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> 0 = 1 - 1
[1,1,1,0,0,0,1,1,0,0]
=> [3,3]
=> [1,1,1,0,0,0,1,1,0,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> 1 = 2 - 1
[1,1,1,0,0,1,0,0,1,0]
=> [4,2]
=> [1,1,1,0,0,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> 0 = 1 - 1
[1,1,1,0,0,1,0,1,0,0]
=> [3,2]
=> [1,1,0,0,1,0,1,0]
=> [1,1,0,0,1,0,1,0]
=> 0 = 1 - 1
[1,1,1,0,0,1,1,0,0,0]
=> [2,2]
=> [1,1,0,0,1,1,0,0]
=> [1,1,1,0,0,1,0,0]
=> 1 = 2 - 1
[1,1,1,0,1,0,0,0,1,0]
=> [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> 0 = 1 - 1
[1,1,1,0,1,0,0,1,0,0]
=> [3,1]
=> [1,1,0,1,0,0,1,0]
=> [1,0,1,1,0,0,1,0]
=> 0 = 1 - 1
[1,1,1,0,1,0,1,0,0,0]
=> [2,1]
=> [1,0,1,0,1,0]
=> [1,0,1,0,1,0]
=> 0 = 1 - 1
[1,1,1,1,1,0,0,0,0,0]
=> []
=> []
=> []
=> ? = 1 - 1
[1,1,1,1,1,1,0,0,0,0,0,0]
=> []
=> []
=> []
=> ? = 1 - 1
[1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> []
=> []
=> []
=> ? = 1 - 1
[1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [6,6,5,4,3,2,1]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 2 - 1
[1,0,1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [2,2,2,2,2,2,1]
=> [1,0,1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,1,1,1,1,1,0,1,0,1,0,0,0,0,0,0]
=> ? = 6 - 1
[1,0,1,1,0,0,1,1,1,1,1,0,0,0,0,0]
=> [3,3,3,3,3,1,1]
=> [1,0,1,1,0,0,1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1,1,0,1,0,0,1,0,0,0,0,0]
=> ? = 5 - 1
[1,0,1,1,0,1,1,1,1,1,0,0,0,0,0,0]
=> [2,2,2,2,2,1,1]
=> [1,0,1,1,0,1,1,1,1,1,0,0,0,0,0,0]
=> [1,1,1,1,1,0,1,1,0,1,0,0,0,0,0,0]
=> ? = 5 - 1
[1,0,1,1,1,0,1,1,1,1,0,0,0,0,0,0]
=> [2,2,2,2,1,1,1]
=> [1,0,1,1,1,0,1,1,1,1,0,0,0,0,0,0]
=> [1,1,1,1,0,1,1,1,0,1,0,0,0,0,0,0]
=> ? = 4 - 1
[1,1,0,0,1,0,1,1,1,1,1,0,0,0,0,0]
=> [3,3,3,3,3,2]
=> [1,1,0,0,1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1,1,0,0,1,0,1,0,0,0,0,0]
=> ? = 5 - 1
[1,1,0,0,1,1,1,0,1,1,1,0,0,0,0,0]
=> [3,3,3,2,2,2]
=> [1,1,0,0,1,1,1,0,1,1,1,0,0,0,0,0]
=> [1,1,1,1,0,0,1,1,1,0,1,0,0,0,0,0]
=> ? = 3 - 1
[1,1,0,0,1,1,1,1,0,1,1,0,0,0,0,0]
=> [3,3,2,2,2,2]
=> [1,1,0,0,1,1,1,1,0,1,1,0,0,0,0,0]
=> [1,1,1,0,0,1,1,1,1,0,1,0,0,0,0,0]
=> ? = 2 - 1
[1,1,0,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [2,2,2,2,2,2]
=> [1,1,0,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0]
=> ? = 6 - 1
[1,1,0,1,0,0,1,1,1,1,1,0,0,0,0,0]
=> [3,3,3,3,3,1]
=> [1,1,0,1,0,0,1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1,0,1,1,0,0,1,0,0,0,0,0]
=> ? = 5 - 1
[1,1,1,0,0,0,1,1,1,1,0,1,0,0,0,0]
=> [4,3,3,3,3]
=> [1,1,1,0,0,0,1,1,1,1,0,1,0,0,0,0]
=> [1,1,1,0,0,0,1,1,1,1,0,1,0,0,0,0]
=> ? = 1 - 1
[1,1,1,0,0,0,1,1,1,1,1,0,0,0,0,0]
=> [3,3,3,3,3]
=> [1,1,1,0,0,0,1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1,1,1,0,0,0,1,0,0,0,0,0]
=> ? = 5 - 1
[1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0]
=> [4,4,4,4]
=> [1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,1,1,1,0,0,0,0,1,0,0,0,0]
=> ? = 4 - 1
[1,1,1,1,0,0,1,0,0,0,1,1,1,0,0,0]
=> [5,5,5,2]
=> [1,1,1,1,0,0,1,0,0,0,1,1,1,0,0,0]
=> [1,1,1,1,0,0,1,1,1,0,0,0,1,0,0,0]
=> ? = 3 - 1
[1,1,1,1,0,1,0,0,0,0,1,1,1,0,0,0]
=> [5,5,5,1]
=> [1,1,1,1,0,1,0,0,0,0,1,1,1,0,0,0]
=> [1,1,1,0,1,1,1,1,0,0,0,0,1,0,0,0]
=> ? = 3 - 1
[1,1,1,1,1,0,0,0,0,0,1,0,1,1,0,0]
=> [6,6,5]
=> [1,1,1,1,1,0,0,0,0,0,1,0,1,1,0,0]
=> [1,1,1,1,1,1,0,0,0,0,0,1,0,1,0,0]
=> ? = 2 - 1
[1,1,1,1,1,0,0,0,0,0,1,1,0,1,0,0]
=> [6,5,5]
=> [1,1,1,1,1,0,0,0,0,0,1,1,0,1,0,0]
=> [1,1,1,1,1,0,0,0,0,0,1,1,0,1,0,0]
=> ? = 1 - 1
[1,1,1,1,1,0,0,0,0,0,1,1,1,0,0,0]
=> [5,5,5]
=> [1,1,1,1,1,0,0,0,0,0,1,1,1,0,0,0]
=> [1,1,1,1,1,1,1,0,0,0,0,0,1,0,0,0]
=> ? = 3 - 1
[1,1,1,1,1,0,0,0,0,1,0,0,1,1,0,0]
=> [6,6,4]
=> [1,1,1,1,1,0,0,0,0,1,0,0,1,1,0,0]
=> [1,1,1,1,1,0,0,0,0,1,1,0,0,1,0,0]
=> ? = 2 - 1
[1,1,1,1,1,0,0,1,0,0,0,0,1,1,0,0]
=> [6,6,2]
=> [1,1,1,1,1,0,0,1,0,0,0,0,1,1,0,0]
=> [1,1,1,0,0,1,1,1,1,0,0,0,0,1,0,0]
=> ? = 2 - 1
[1,1,1,1,1,0,1,0,0,0,0,0,1,1,0,0]
=> [6,6,1]
=> [1,1,1,1,1,0,1,0,0,0,0,0,1,1,0,0]
=> [1,1,0,1,1,1,1,1,0,0,0,0,0,1,0,0]
=> ? = 2 - 1
[1,1,1,1,1,1,0,0,0,0,0,0,1,0,1,0]
=> [7,6]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0,1,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0,1,0]
=> ? = 1 - 1
[1,1,1,1,1,1,0,0,0,0,0,0,1,1,0,0]
=> [6,6]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,1,0,0]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,1,0,0]
=> ? = 2 - 1
[1,1,1,1,1,1,0,0,0,0,0,1,0,0,1,0]
=> [7,5]
=> [1,1,1,1,1,1,0,0,0,0,0,1,0,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0,1,1,0,0,1,0]
=> ? = 1 - 1
[1,1,1,1,1,1,0,0,0,0,1,0,0,0,1,0]
=> [7,4]
=> [1,1,1,1,1,1,0,0,0,0,1,0,0,0,1,0]
=> [1,1,1,1,0,0,0,0,1,1,1,0,0,0,1,0]
=> ? = 1 - 1
[1,1,1,1,1,1,0,0,0,1,0,0,0,0,1,0]
=> [7,3]
=> [1,1,1,1,1,1,0,0,0,1,0,0,0,0,1,0]
=> [1,1,1,0,0,0,1,1,1,1,0,0,0,0,1,0]
=> ? = 1 - 1
[1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> [7]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> ? = 1 - 1
[1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> []
=> []
=> []
=> ? = 1 - 1
[1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0]
=> [8]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0]
=> ? = 1 - 1
[1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,1,0]
=> [9]
=> [1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,1,0]
=> [1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,1,0]
=> ? = 1 - 1
[1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,1,0]
=> [8,1]
=> [1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,1,0]
=> [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> ? = 1 - 1
[1,0,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0]
=> [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]
=> [1,0,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0]
=> ? = 1 - 1
[1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,1,0]
=> [10]
=> [1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,1,0]
=> [1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,1,0]
=> ? = 1 - 1
[1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0,1,0]
=> [9,1]
=> [1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0,1,0]
=> [1,0,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0]
=> ? = 1 - 1
[1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0,1,0]
=> [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]
=> ? = 1 - 1
[1,0,1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0]
=> [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]
=> [1,0,1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0]
=> ? = 1 - 1
[1,0,1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0]
=> [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]
=> [1,0,1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0]
=> ? = 1 - 1
[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]
=> [1,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,0,1,0]
=> [1,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,0,1,0]
=> ? = 1 - 1
[1,1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0,0,1,0]
=> [10,1]
=> [1,1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0,0,1,0]
=> [1,0,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,1,0]
=> ? = 1 - 1
[1,1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0,0,1,0]
=> [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]
=> ? = 1 - 1
[1,1,1,1,1,1,1,0,0,0,1,0,0,0,0,0,1,0]
=> [8,3]
=> [1,1,1,1,1,1,1,0,0,0,1,0,0,0,0,0,1,0]
=> [1,1,1,0,0,0,1,1,1,1,1,0,0,0,0,0,1,0]
=> ? = 1 - 1
[1,0,1,1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0,0]
=> [2,1,1,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0,0]
=> [1,0,1,1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0,0]
=> ? = 1 - 1
[1,1,1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0,0,0,1,0]
=> [10,2]
=> [1,1,1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0,0,0,1,0]
=> [1,1,0,0,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0]
=> ? = 1 - 1
Description
The number of times one can erase the first up and the last down step in a Dyck path and still remain a Dyck path.
In other words, this is the lowest height of a valley of a Dyck path, or its semilength in case of the unique path without valleys.
Matching statistic: St001038
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00027: Dyck paths —to partition⟶ Integer partitions
Mp00044: Integer partitions —conjugate⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
St001038: Dyck paths ⟶ ℤResult quality: 67% ●values known / values provided: 80%●distinct values known / distinct values provided: 67%
Mp00044: Integer partitions —conjugate⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
St001038: Dyck paths ⟶ ℤResult quality: 67% ●values known / values provided: 80%●distinct values known / distinct values provided: 67%
Values
[1,0]
=> []
=> []
=> []
=> ? = 1
[1,0,1,0]
=> [1]
=> [1]
=> [1,0]
=> ? = 1
[1,1,0,0]
=> []
=> []
=> []
=> ? = 1
[1,0,1,0,1,0]
=> [2,1]
=> [2,1]
=> [1,0,1,1,0,0]
=> 1
[1,0,1,1,0,0]
=> [1,1]
=> [2]
=> [1,0,1,0]
=> 2
[1,1,0,0,1,0]
=> [2]
=> [1,1]
=> [1,1,0,0]
=> 1
[1,1,0,1,0,0]
=> [1]
=> [1]
=> [1,0]
=> ? = 1
[1,1,1,0,0,0]
=> []
=> []
=> []
=> ? = 1
[1,0,1,0,1,0,1,0]
=> [3,2,1]
=> [3,2,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> 1
[1,0,1,0,1,1,0,0]
=> [2,2,1]
=> [3,2]
=> [1,0,1,1,1,0,0,0]
=> 2
[1,0,1,1,0,0,1,0]
=> [3,1,1]
=> [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> 1
[1,0,1,1,0,1,0,0]
=> [2,1,1]
=> [3,1]
=> [1,0,1,0,1,1,0,0]
=> 1
[1,0,1,1,1,0,0,0]
=> [1,1,1]
=> [3]
=> [1,0,1,0,1,0]
=> 3
[1,1,0,0,1,0,1,0]
=> [3,2]
=> [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> 1
[1,1,0,0,1,1,0,0]
=> [2,2]
=> [2,2]
=> [1,1,1,0,0,0]
=> 2
[1,1,0,1,0,0,1,0]
=> [3,1]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 1
[1,1,0,1,0,1,0,0]
=> [2,1]
=> [2,1]
=> [1,0,1,1,0,0]
=> 1
[1,1,0,1,1,0,0,0]
=> [1,1]
=> [2]
=> [1,0,1,0]
=> 2
[1,1,1,0,0,0,1,0]
=> [3]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> 1
[1,1,1,0,0,1,0,0]
=> [2]
=> [1,1]
=> [1,1,0,0]
=> 1
[1,1,1,0,1,0,0,0]
=> [1]
=> [1]
=> [1,0]
=> ? = 1
[1,1,1,1,0,0,0,0]
=> []
=> []
=> []
=> ? = 1
[1,0,1,0,1,0,1,0,1,0]
=> [4,3,2,1]
=> [4,3,2,1]
=> [1,0,1,1,1,0,1,1,0,0,0,1,0,0]
=> 1
[1,0,1,0,1,0,1,1,0,0]
=> [3,3,2,1]
=> [4,3,2]
=> [1,0,1,1,1,0,1,1,0,0,0,0]
=> 2
[1,0,1,0,1,1,0,1,0,0]
=> [3,2,2,1]
=> [4,3,1]
=> [1,0,1,1,1,0,1,0,0,1,0,0]
=> 1
[1,0,1,0,1,1,1,0,0,0]
=> [2,2,2,1]
=> [4,3]
=> [1,0,1,1,1,0,1,0,0,0]
=> 3
[1,0,1,1,0,0,1,1,0,0]
=> [3,3,1,1]
=> [4,2,2]
=> [1,0,1,0,1,1,1,1,0,0,0,0]
=> 2
[1,0,1,1,0,1,0,1,0,0]
=> [3,2,1,1]
=> [4,2,1]
=> [1,0,1,0,1,1,1,0,0,1,0,0]
=> 1
[1,0,1,1,0,1,1,0,0,0]
=> [2,2,1,1]
=> [4,2]
=> [1,0,1,0,1,1,1,0,0,0]
=> 2
[1,0,1,1,1,0,0,1,0,0]
=> [3,1,1,1]
=> [4,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,0]
=> 1
[1,0,1,1,1,0,1,0,0,0]
=> [2,1,1,1]
=> [4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> 1
[1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1]
=> [4]
=> [1,0,1,0,1,0,1,0]
=> 4
[1,1,0,0,1,0,1,0,1,0]
=> [4,3,2]
=> [3,3,2,1]
=> [1,1,1,0,1,1,0,0,0,1,0,0]
=> 1
[1,1,0,0,1,0,1,1,0,0]
=> [3,3,2]
=> [3,3,2]
=> [1,1,1,0,1,1,0,0,0,0]
=> 2
[1,1,0,0,1,1,0,0,1,0]
=> [4,2,2]
=> [3,3,1,1]
=> [1,1,1,0,1,0,0,1,0,1,0,0]
=> 1
[1,1,0,0,1,1,0,1,0,0]
=> [3,2,2]
=> [3,3,1]
=> [1,1,1,0,1,0,0,1,0,0]
=> 1
[1,1,0,0,1,1,1,0,0,0]
=> [2,2,2]
=> [3,3]
=> [1,1,1,0,1,0,0,0]
=> 3
[1,1,0,1,0,0,1,0,1,0]
=> [4,3,1]
=> [3,2,2,1]
=> [1,0,1,1,1,1,0,0,0,1,0,0]
=> 1
[1,1,0,1,0,0,1,1,0,0]
=> [3,3,1]
=> [3,2,2]
=> [1,0,1,1,1,1,0,0,0,0]
=> 2
[1,1,0,1,0,1,0,0,1,0]
=> [4,2,1]
=> [3,2,1,1]
=> [1,0,1,1,1,0,0,1,0,1,0,0]
=> 1
[1,1,0,1,0,1,0,1,0,0]
=> [3,2,1]
=> [3,2,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> 1
[1,1,0,1,0,1,1,0,0,0]
=> [2,2,1]
=> [3,2]
=> [1,0,1,1,1,0,0,0]
=> 2
[1,1,0,1,1,0,0,0,1,0]
=> [4,1,1]
=> [3,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> 1
[1,1,0,1,1,0,0,1,0,0]
=> [3,1,1]
=> [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> 1
[1,1,0,1,1,0,1,0,0,0]
=> [2,1,1]
=> [3,1]
=> [1,0,1,0,1,1,0,0]
=> 1
[1,1,0,1,1,1,0,0,0,0]
=> [1,1,1]
=> [3]
=> [1,0,1,0,1,0]
=> 3
[1,1,1,0,0,0,1,0,1,0]
=> [4,3]
=> [2,2,2,1]
=> [1,1,1,1,0,0,0,1,0,0]
=> 1
[1,1,1,0,0,0,1,1,0,0]
=> [3,3]
=> [2,2,2]
=> [1,1,1,1,0,0,0,0]
=> 2
[1,1,1,0,0,1,0,0,1,0]
=> [4,2]
=> [2,2,1,1]
=> [1,1,1,0,0,1,0,1,0,0]
=> 1
[1,1,1,0,0,1,0,1,0,0]
=> [3,2]
=> [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> 1
[1,1,1,0,0,1,1,0,0,0]
=> [2,2]
=> [2,2]
=> [1,1,1,0,0,0]
=> 2
[1,1,1,0,1,0,0,0,1,0]
=> [4,1]
=> [2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> 1
[1,1,1,0,1,0,0,1,0,0]
=> [3,1]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 1
[1,1,1,0,1,0,1,0,0,0]
=> [2,1]
=> [2,1]
=> [1,0,1,1,0,0]
=> 1
[1,1,1,0,1,1,0,0,0,0]
=> [1,1]
=> [2]
=> [1,0,1,0]
=> 2
[1,1,1,1,0,0,0,0,1,0]
=> [4]
=> [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> 1
[1,1,1,1,0,0,0,1,0,0]
=> [3]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> 1
[1,1,1,1,0,1,0,0,0,0]
=> [1]
=> [1]
=> [1,0]
=> ? = 1
[1,1,1,1,1,0,0,0,0,0]
=> []
=> []
=> []
=> ? = 1
[1,1,1,1,1,0,1,0,0,0,0,0]
=> [1]
=> [1]
=> [1,0]
=> ? = 1
[1,1,1,1,1,1,0,0,0,0,0,0]
=> []
=> []
=> []
=> ? = 1
[1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [5,5,4,3,2,1]
=> [6,5,4,3,2]
=> [1,0,1,1,1,0,1,1,1,0,1,1,0,0,0,1,0,0,0,0]
=> ? = 2
[1,1,1,1,0,0,0,0,1,0,1,0,1,0]
=> [6,5,4]
=> [3,3,3,3,2,1]
=> [1,1,1,1,1,1,0,0,0,1,0,0,0,1,0,0]
=> ? = 1
[1,1,1,1,0,0,0,0,1,1,0,0,1,0]
=> [6,4,4]
=> [3,3,3,3,1,1]
=> [1,1,1,1,1,1,0,0,0,0,0,1,0,1,0,0]
=> ? = 1
[1,1,1,1,0,0,1,0,0,0,1,0,1,0]
=> [6,5,2]
=> [3,3,2,2,2,1]
=> [1,1,1,0,1,1,0,1,0,1,0,0,0,1,0,0]
=> ? = 1
[1,1,1,1,0,0,1,0,0,1,0,0,1,0]
=> [6,4,2]
=> [3,3,2,2,1,1]
=> [1,1,1,0,1,1,0,1,0,0,0,1,0,1,0,0]
=> ? = 1
[1,1,1,1,1,1,0,1,0,0,0,0,0,0]
=> [1]
=> [1]
=> [1,0]
=> ? = 1
[1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> []
=> []
=> []
=> ? = 1
[1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [6,6,5,4,3,2,1]
=> [7,6,5,4,3,2]
=> [1,0,1,1,1,0,1,1,1,0,1,1,1,0,0,1,0,0,0,1,0,0,0,0]
=> ? = 2
[1,0,1,0,1,1,1,1,1,0,1,0,0,0,0,0]
=> [3,2,2,2,2,2,1]
=> [7,6,1]
=> [1,0,1,1,1,0,1,0,1,0,1,0,1,0,0,1,0,0]
=> ? = 1
[1,0,1,1,0,0,1,1,1,1,1,0,0,0,0,0]
=> [3,3,3,3,3,1,1]
=> [7,5,5]
=> [1,0,1,0,1,1,1,1,1,0,1,0,1,0,0,0,0,0]
=> ? = 5
[1,0,1,1,1,1,1,1,0,0,1,0,0,0,0,0]
=> [3,1,1,1,1,1,1]
=> [7,1,1]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> ? = 1
[1,1,0,0,1,0,1,1,1,1,1,0,0,0,0,0]
=> [3,3,3,3,3,2]
=> [6,6,5]
=> [1,1,1,0,1,1,1,0,1,0,1,0,0,0,0,0]
=> ? = 5
[1,1,0,0,1,1,1,0,1,1,1,0,0,0,0,0]
=> [3,3,3,2,2,2]
=> [6,6,3]
=> [1,1,1,0,1,0,1,0,1,1,1,0,0,0,0,0]
=> ? = 3
[1,1,0,0,1,1,1,1,0,1,1,0,0,0,0,0]
=> [3,3,2,2,2,2]
=> [6,6,2]
=> [1,1,1,0,1,0,1,0,1,0,1,1,0,0,0,0]
=> ? = 2
[1,1,0,0,1,1,1,1,1,0,1,0,0,0,0,0]
=> [3,2,2,2,2,2]
=> [6,6,1]
=> [1,1,1,0,1,0,1,0,1,0,1,0,0,1,0,0]
=> ? = 1
[1,1,0,1,0,1,0,1,0,1,0,1,1,0,0,0]
=> [5,5,4,3,2,1]
=> [6,5,4,3,2]
=> [1,0,1,1,1,0,1,1,1,0,1,1,0,0,0,1,0,0,0,0]
=> ? = 2
[1,1,1,0,0,0,1,1,1,1,0,1,0,0,0,0]
=> [4,3,3,3,3]
=> [5,5,5,1]
=> [1,1,1,1,1,0,1,0,1,0,0,0,0,1,0,0]
=> ? = 1
[1,1,1,1,0,0,1,0,0,0,1,1,1,0,0,0]
=> [5,5,5,2]
=> [4,4,3,3,3]
=> [1,1,1,0,1,1,1,1,0,1,0,0,0,0,0,0]
=> ? = 3
[1,1,1,1,1,0,0,0,0,0,1,0,1,1,0,0]
=> [6,6,5]
=> [3,3,3,3,3,2]
=> [1,1,1,1,1,1,0,1,0,0,0,1,0,0,0,0]
=> ? = 2
[1,1,1,1,1,0,0,0,0,0,1,1,0,1,0,0]
=> [6,5,5]
=> [3,3,3,3,3,1]
=> [1,1,1,1,1,1,0,1,0,0,0,0,0,1,0,0]
=> ? = 1
[1,1,1,1,1,0,0,0,0,1,0,0,1,1,0,0]
=> [6,6,4]
=> [3,3,3,3,2,2]
=> [1,1,1,1,1,1,0,0,0,1,0,1,0,0,0,0]
=> ? = 2
[1,1,1,1,1,0,0,0,0,1,0,1,0,1,0,0]
=> [6,5,4]
=> [3,3,3,3,2,1]
=> [1,1,1,1,1,1,0,0,0,1,0,0,0,1,0,0]
=> ? = 1
[1,1,1,1,1,0,0,0,0,1,1,0,0,1,0,0]
=> [6,4,4]
=> [3,3,3,3,1,1]
=> [1,1,1,1,1,1,0,0,0,0,0,1,0,1,0,0]
=> ? = 1
[1,1,1,1,1,0,0,1,0,0,0,0,1,0,1,0]
=> [7,6,2]
=> [3,3,2,2,2,2,1]
=> [1,1,1,0,1,1,0,1,0,1,0,1,0,0,0,1,0,0]
=> ? = 1
[1,1,1,1,1,0,0,1,0,0,0,0,1,1,0,0]
=> [6,6,2]
=> [3,3,2,2,2,2]
=> [1,1,1,0,1,1,0,1,0,1,0,1,0,0,0,0]
=> ? = 2
[1,1,1,1,1,0,0,1,0,0,0,1,0,1,0,0]
=> [6,5,2]
=> [3,3,2,2,2,1]
=> [1,1,1,0,1,1,0,1,0,1,0,0,0,1,0,0]
=> ? = 1
[1,1,1,1,1,0,0,1,0,0,1,0,0,1,0,0]
=> [6,4,2]
=> [3,3,2,2,1,1]
=> [1,1,1,0,1,1,0,1,0,0,0,1,0,1,0,0]
=> ? = 1
[1,1,1,1,1,0,1,0,0,0,0,0,1,0,1,0]
=> [7,6,1]
=> [3,2,2,2,2,2,1]
=> [1,0,1,1,1,1,0,1,0,1,0,1,0,0,0,1,0,0]
=> ? = 1
[1,1,1,1,1,1,0,0,0,0,0,0,1,0,1,0]
=> [7,6]
=> [2,2,2,2,2,2,1]
=> [1,1,1,1,0,1,0,1,0,1,0,0,0,1,0,0]
=> ? = 1
[1,1,1,1,1,1,0,0,0,0,0,1,0,0,1,0]
=> [7,5]
=> [2,2,2,2,2,1,1]
=> [1,1,1,1,0,1,0,1,0,0,0,1,0,1,0,0]
=> ? = 1
[1,1,1,1,1,1,0,0,0,0,1,0,0,0,1,0]
=> [7,4]
=> [2,2,2,2,1,1,1]
=> [1,1,1,1,0,1,0,0,0,1,0,1,0,1,0,0]
=> ? = 1
[1,1,1,1,1,1,0,0,0,1,0,0,0,0,1,0]
=> [7,3]
=> [2,2,2,1,1,1,1]
=> [1,1,1,1,0,0,0,1,0,1,0,1,0,1,0,0]
=> ? = 1
[1,1,1,1,1,1,0,0,1,0,0,0,0,0,1,0]
=> [7,2]
=> [2,2,1,1,1,1,1]
=> [1,1,1,0,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 1
[1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0]
=> [1]
=> [1]
=> [1,0]
=> ? = 1
[1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> []
=> []
=> []
=> ? = 1
[1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0]
=> [8]
=> [1,1,1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 1
[1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,1,0]
=> [9]
=> [1,1,1,1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 1
[1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,1,0]
=> [8,1]
=> [2,1,1,1,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 1
[1,0,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0]
=> [2,1,1,1,1,1,1,1]
=> [8,1]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> ? = 1
Description
The minimal height of a column in the parallelogram polyomino associated with the Dyck path.
Matching statistic: St000993
(load all 9 compositions to match this statistic)
(load all 9 compositions to match this statistic)
Mp00027: Dyck paths —to partition⟶ Integer partitions
St000993: Integer partitions ⟶ ℤResult quality: 67% ●values known / values provided: 78%●distinct values known / distinct values provided: 67%
St000993: Integer partitions ⟶ ℤResult quality: 67% ●values known / values provided: 78%●distinct values known / distinct values provided: 67%
Values
[1,0]
=> []
=> ? = 1
[1,0,1,0]
=> [1]
=> ? = 1
[1,1,0,0]
=> []
=> ? = 1
[1,0,1,0,1,0]
=> [2,1]
=> 1
[1,0,1,1,0,0]
=> [1,1]
=> 2
[1,1,0,0,1,0]
=> [2]
=> 1
[1,1,0,1,0,0]
=> [1]
=> ? = 1
[1,1,1,0,0,0]
=> []
=> ? = 1
[1,0,1,0,1,0,1,0]
=> [3,2,1]
=> 1
[1,0,1,0,1,1,0,0]
=> [2,2,1]
=> 2
[1,0,1,1,0,0,1,0]
=> [3,1,1]
=> 1
[1,0,1,1,0,1,0,0]
=> [2,1,1]
=> 1
[1,0,1,1,1,0,0,0]
=> [1,1,1]
=> 3
[1,1,0,0,1,0,1,0]
=> [3,2]
=> 1
[1,1,0,0,1,1,0,0]
=> [2,2]
=> 2
[1,1,0,1,0,0,1,0]
=> [3,1]
=> 1
[1,1,0,1,0,1,0,0]
=> [2,1]
=> 1
[1,1,0,1,1,0,0,0]
=> [1,1]
=> 2
[1,1,1,0,0,0,1,0]
=> [3]
=> 1
[1,1,1,0,0,1,0,0]
=> [2]
=> 1
[1,1,1,0,1,0,0,0]
=> [1]
=> ? = 1
[1,1,1,1,0,0,0,0]
=> []
=> ? = 1
[1,0,1,0,1,0,1,0,1,0]
=> [4,3,2,1]
=> 1
[1,0,1,0,1,0,1,1,0,0]
=> [3,3,2,1]
=> 2
[1,0,1,0,1,1,0,1,0,0]
=> [3,2,2,1]
=> 1
[1,0,1,0,1,1,1,0,0,0]
=> [2,2,2,1]
=> 3
[1,0,1,1,0,0,1,1,0,0]
=> [3,3,1,1]
=> 2
[1,0,1,1,0,1,0,1,0,0]
=> [3,2,1,1]
=> 1
[1,0,1,1,0,1,1,0,0,0]
=> [2,2,1,1]
=> 2
[1,0,1,1,1,0,0,1,0,0]
=> [3,1,1,1]
=> 1
[1,0,1,1,1,0,1,0,0,0]
=> [2,1,1,1]
=> 1
[1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1]
=> 4
[1,1,0,0,1,0,1,0,1,0]
=> [4,3,2]
=> 1
[1,1,0,0,1,0,1,1,0,0]
=> [3,3,2]
=> 2
[1,1,0,0,1,1,0,0,1,0]
=> [4,2,2]
=> 1
[1,1,0,0,1,1,0,1,0,0]
=> [3,2,2]
=> 1
[1,1,0,0,1,1,1,0,0,0]
=> [2,2,2]
=> 3
[1,1,0,1,0,0,1,0,1,0]
=> [4,3,1]
=> 1
[1,1,0,1,0,0,1,1,0,0]
=> [3,3,1]
=> 2
[1,1,0,1,0,1,0,0,1,0]
=> [4,2,1]
=> 1
[1,1,0,1,0,1,0,1,0,0]
=> [3,2,1]
=> 1
[1,1,0,1,0,1,1,0,0,0]
=> [2,2,1]
=> 2
[1,1,0,1,1,0,0,0,1,0]
=> [4,1,1]
=> 1
[1,1,0,1,1,0,0,1,0,0]
=> [3,1,1]
=> 1
[1,1,0,1,1,0,1,0,0,0]
=> [2,1,1]
=> 1
[1,1,0,1,1,1,0,0,0,0]
=> [1,1,1]
=> 3
[1,1,1,0,0,0,1,0,1,0]
=> [4,3]
=> 1
[1,1,1,0,0,0,1,1,0,0]
=> [3,3]
=> 2
[1,1,1,0,0,1,0,0,1,0]
=> [4,2]
=> 1
[1,1,1,0,0,1,0,1,0,0]
=> [3,2]
=> 1
[1,1,1,0,0,1,1,0,0,0]
=> [2,2]
=> 2
[1,1,1,0,1,0,0,0,1,0]
=> [4,1]
=> 1
[1,1,1,0,1,0,0,1,0,0]
=> [3,1]
=> 1
[1,1,1,0,1,0,1,0,0,0]
=> [2,1]
=> 1
[1,1,1,0,1,1,0,0,0,0]
=> [1,1]
=> 2
[1,1,1,1,0,0,0,0,1,0]
=> [4]
=> 1
[1,1,1,1,0,0,0,1,0,0]
=> [3]
=> 1
[1,1,1,1,0,1,0,0,0,0]
=> [1]
=> ? = 1
[1,1,1,1,1,0,0,0,0,0]
=> []
=> ? = 1
[1,0,1,0,1,0,1,0,1,1,0,0]
=> [4,4,3,2,1]
=> ? = 2
[1,1,0,0,1,0,1,0,1,1,0,0]
=> [4,4,3,2]
=> ? = 2
[1,1,1,1,1,0,1,0,0,0,0,0]
=> [1]
=> ? = 1
[1,1,1,1,1,1,0,0,0,0,0,0]
=> []
=> ? = 1
[1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [5,5,4,3,2,1]
=> ? = 2
[1,0,1,1,0,0,1,1,1,1,0,0,0,0]
=> [3,3,3,3,1,1]
=> ? = 4
[1,1,0,0,1,0,1,1,1,1,0,0,0,0]
=> [3,3,3,3,2]
=> ? = 4
[1,1,0,0,1,1,0,1,1,1,0,0,0,0]
=> [3,3,3,2,2]
=> ? = 3
[1,1,0,1,0,0,1,1,1,1,0,0,0,0]
=> [3,3,3,3,1]
=> ? = 4
[1,1,0,1,0,1,0,1,0,1,1,0,0,0]
=> [4,4,3,2,1]
=> ? = 2
[1,1,1,0,0,0,1,1,1,0,1,0,0,0]
=> [4,3,3,3]
=> ? = 1
[1,1,1,0,0,1,0,0,1,1,1,0,0,0]
=> [4,4,4,2]
=> ? = 3
[1,1,1,0,0,1,0,1,0,1,1,0,0,0]
=> [4,4,3,2]
=> ? = 2
[1,1,1,0,1,0,0,0,1,1,1,0,0,0]
=> [4,4,4,1]
=> ? = 3
[1,1,1,1,0,0,0,0,1,0,1,0,1,0]
=> [6,5,4]
=> ? = 1
[1,1,1,1,0,0,0,0,1,0,1,1,0,0]
=> [5,5,4]
=> ? = 2
[1,1,1,1,0,0,0,0,1,1,0,0,1,0]
=> [6,4,4]
=> ? = 1
[1,1,1,1,0,0,0,1,0,0,1,1,0,0]
=> [5,5,3]
=> ? = 2
[1,1,1,1,0,0,1,0,0,0,1,0,1,0]
=> [6,5,2]
=> ? = 1
[1,1,1,1,1,1,0,1,0,0,0,0,0,0]
=> [1]
=> ? = 1
[1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> []
=> ? = 1
[1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [6,6,5,4,3,2,1]
=> ? = 2
[1,0,1,0,1,1,1,1,1,0,1,0,0,0,0,0]
=> [3,2,2,2,2,2,1]
=> ? = 1
[1,0,1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [2,2,2,2,2,2,1]
=> ? = 6
[1,0,1,1,0,0,1,1,1,1,1,0,0,0,0,0]
=> [3,3,3,3,3,1,1]
=> ? = 5
[1,1,0,0,1,0,1,1,1,1,1,0,0,0,0,0]
=> [3,3,3,3,3,2]
=> ? = 5
[1,1,0,0,1,1,1,0,1,1,1,0,0,0,0,0]
=> [3,3,3,2,2,2]
=> ? = 3
[1,1,0,0,1,1,1,1,0,1,1,0,0,0,0,0]
=> [3,3,2,2,2,2]
=> ? = 2
[1,1,0,0,1,1,1,1,1,0,1,0,0,0,0,0]
=> [3,2,2,2,2,2]
=> ? = 1
[1,1,0,1,0,0,1,1,1,1,1,0,0,0,0,0]
=> [3,3,3,3,3,1]
=> ? = 5
[1,1,0,1,0,1,0,1,0,1,0,1,1,0,0,0]
=> [5,5,4,3,2,1]
=> ? = 2
[1,1,0,1,1,0,0,1,1,1,1,0,0,0,0,0]
=> [3,3,3,3,1,1]
=> ? = 4
[1,1,1,0,0,0,1,1,1,1,0,1,0,0,0,0]
=> [4,3,3,3,3]
=> ? = 1
[1,1,1,0,0,0,1,1,1,1,1,0,0,0,0,0]
=> [3,3,3,3,3]
=> ? = 5
[1,1,1,0,0,1,0,1,1,1,1,0,0,0,0,0]
=> [3,3,3,3,2]
=> ? = 4
[1,1,1,0,0,1,1,0,1,1,1,0,0,0,0,0]
=> [3,3,3,2,2]
=> ? = 3
[1,1,1,0,1,0,0,1,1,1,1,0,0,0,0,0]
=> [3,3,3,3,1]
=> ? = 4
[1,1,1,0,1,0,1,0,1,0,1,1,0,0,0,0]
=> [4,4,3,2,1]
=> ? = 2
[1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0]
=> [4,4,4,4]
=> ? = 4
[1,1,1,1,0,0,0,1,1,1,0,1,0,0,0,0]
=> [4,3,3,3]
=> ? = 1
[1,1,1,1,0,0,1,0,0,0,1,1,1,0,0,0]
=> [5,5,5,2]
=> ? = 3
Description
The multiplicity of the largest part of an integer partition.
Matching statistic: St001733
Mp00027: Dyck paths —to partition⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00227: Dyck paths —Delest-Viennot-inverse⟶ Dyck paths
St001733: Dyck paths ⟶ ℤResult quality: 67% ●values known / values provided: 76%●distinct values known / distinct values provided: 67%
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00227: Dyck paths —Delest-Viennot-inverse⟶ Dyck paths
St001733: Dyck paths ⟶ ℤResult quality: 67% ●values known / values provided: 76%●distinct values known / distinct values provided: 67%
Values
[1,0]
=> []
=> []
=> []
=> ? = 1
[1,0,1,0]
=> [1]
=> [1,0]
=> [1,0]
=> 1
[1,1,0,0]
=> []
=> []
=> []
=> ? = 1
[1,0,1,0,1,0]
=> [2,1]
=> [1,0,1,1,0,0]
=> [1,1,0,0,1,0]
=> 1
[1,0,1,1,0,0]
=> [1,1]
=> [1,1,0,0]
=> [1,0,1,0]
=> 2
[1,1,0,0,1,0]
=> [2]
=> [1,0,1,0]
=> [1,1,0,0]
=> 1
[1,1,0,1,0,0]
=> [1]
=> [1,0]
=> [1,0]
=> 1
[1,1,1,0,0,0]
=> []
=> []
=> []
=> ? = 1
[1,0,1,0,1,0,1,0]
=> [3,2,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,1,1,0,0,1,0,0,1,0]
=> 1
[1,0,1,0,1,1,0,0]
=> [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> 2
[1,0,1,1,0,0,1,0]
=> [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> 1
[1,0,1,1,0,1,0,0]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0]
=> 1
[1,0,1,1,1,0,0,0]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> [1,0,1,0,1,0]
=> 3
[1,1,0,0,1,0,1,0]
=> [3,2]
=> [1,0,1,1,1,0,0,0]
=> [1,1,1,0,0,1,0,0]
=> 1
[1,1,0,0,1,1,0,0]
=> [2,2]
=> [1,1,1,0,0,0]
=> [1,1,0,1,0,0]
=> 2
[1,1,0,1,0,0,1,0]
=> [3,1]
=> [1,0,1,0,1,1,0,0]
=> [1,1,1,0,0,0,1,0]
=> 1
[1,1,0,1,0,1,0,0]
=> [2,1]
=> [1,0,1,1,0,0]
=> [1,1,0,0,1,0]
=> 1
[1,1,0,1,1,0,0,0]
=> [1,1]
=> [1,1,0,0]
=> [1,0,1,0]
=> 2
[1,1,1,0,0,0,1,0]
=> [3]
=> [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> 1
[1,1,1,0,0,1,0,0]
=> [2]
=> [1,0,1,0]
=> [1,1,0,0]
=> 1
[1,1,1,0,1,0,0,0]
=> [1]
=> [1,0]
=> [1,0]
=> 1
[1,1,1,1,0,0,0,0]
=> []
=> []
=> []
=> ? = 1
[1,0,1,0,1,0,1,0,1,0]
=> [4,3,2,1]
=> [1,0,1,1,1,0,1,1,0,0,0,1,0,0]
=> [1,1,1,1,0,0,1,0,0,1,0,0,1,0]
=> 1
[1,0,1,0,1,0,1,1,0,0]
=> [3,3,2,1]
=> [1,1,1,0,1,1,0,0,0,1,0,0]
=> [1,1,1,0,1,0,0,1,0,0,1,0]
=> 2
[1,0,1,0,1,1,0,1,0,0]
=> [3,2,2,1]
=> [1,0,1,1,1,1,0,0,0,1,0,0]
=> [1,1,1,0,0,1,0,1,0,0,1,0]
=> 1
[1,0,1,0,1,1,1,0,0,0]
=> [2,2,2,1]
=> [1,1,1,1,0,0,0,1,0,0]
=> [1,1,0,1,0,1,0,0,1,0]
=> 3
[1,0,1,1,0,0,1,1,0,0]
=> [3,3,1,1]
=> [1,1,1,0,1,0,0,1,0,1,0,0]
=> [1,1,1,0,1,0,0,0,1,0,1,0]
=> 2
[1,0,1,1,0,1,0,1,0,0]
=> [3,2,1,1]
=> [1,0,1,1,1,0,0,1,0,1,0,0]
=> [1,1,1,0,0,1,0,0,1,0,1,0]
=> 1
[1,0,1,1,0,1,1,0,0,0]
=> [2,2,1,1]
=> [1,1,1,0,0,1,0,1,0,0]
=> [1,1,0,1,0,0,1,0,1,0]
=> 2
[1,0,1,1,1,0,0,1,0,0]
=> [3,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,1,0,0,0,1,0,1,0,1,0]
=> 1
[1,0,1,1,1,0,1,0,0,0]
=> [2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> 1
[1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0]
=> 4
[1,1,0,0,1,0,1,0,1,0]
=> [4,3,2]
=> [1,0,1,1,1,0,1,1,0,0,0,0]
=> [1,1,1,1,0,0,1,0,0,1,0,0]
=> 1
[1,1,0,0,1,0,1,1,0,0]
=> [3,3,2]
=> [1,1,1,0,1,1,0,0,0,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> 2
[1,1,0,0,1,1,0,0,1,0]
=> [4,2,2]
=> [1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,1,0,1,0,0]
=> 1
[1,1,0,0,1,1,0,1,0,0]
=> [3,2,2]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> 1
[1,1,0,0,1,1,1,0,0,0]
=> [2,2,2]
=> [1,1,1,1,0,0,0,0]
=> [1,1,0,1,0,1,0,0]
=> 3
[1,1,0,1,0,0,1,0,1,0]
=> [4,3,1]
=> [1,0,1,1,1,0,1,0,0,1,0,0]
=> [1,1,1,1,0,0,1,0,0,0,1,0]
=> 1
[1,1,0,1,0,0,1,1,0,0]
=> [3,3,1]
=> [1,1,1,0,1,0,0,1,0,0]
=> [1,1,1,0,1,0,0,0,1,0]
=> 2
[1,1,0,1,0,1,0,0,1,0]
=> [4,2,1]
=> [1,0,1,0,1,1,1,0,0,1,0,0]
=> [1,1,1,1,0,0,0,1,0,0,1,0]
=> 1
[1,1,0,1,0,1,0,1,0,0]
=> [3,2,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,1,1,0,0,1,0,0,1,0]
=> 1
[1,1,0,1,0,1,1,0,0,0]
=> [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> 2
[1,1,0,1,1,0,0,0,1,0]
=> [4,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,1,0,0,0,0,1,0,1,0]
=> 1
[1,1,0,1,1,0,0,1,0,0]
=> [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> 1
[1,1,0,1,1,0,1,0,0,0]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0]
=> 1
[1,1,0,1,1,1,0,0,0,0]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> [1,0,1,0,1,0]
=> 3
[1,1,1,0,0,0,1,0,1,0]
=> [4,3]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> 1
[1,1,1,0,0,0,1,1,0,0]
=> [3,3]
=> [1,1,1,0,1,0,0,0]
=> [1,1,1,0,1,0,0,0]
=> 2
[1,1,1,0,0,1,0,0,1,0]
=> [4,2]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> 1
[1,1,1,0,0,1,0,1,0,0]
=> [3,2]
=> [1,0,1,1,1,0,0,0]
=> [1,1,1,0,0,1,0,0]
=> 1
[1,1,1,0,0,1,1,0,0,0]
=> [2,2]
=> [1,1,1,0,0,0]
=> [1,1,0,1,0,0]
=> 2
[1,1,1,0,1,0,0,0,1,0]
=> [4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> 1
[1,1,1,0,1,0,0,1,0,0]
=> [3,1]
=> [1,0,1,0,1,1,0,0]
=> [1,1,1,0,0,0,1,0]
=> 1
[1,1,1,0,1,0,1,0,0,0]
=> [2,1]
=> [1,0,1,1,0,0]
=> [1,1,0,0,1,0]
=> 1
[1,1,1,1,1,0,0,0,0,0]
=> []
=> []
=> []
=> ? = 1
[1,0,1,0,1,0,1,0,1,1,0,0]
=> [4,4,3,2,1]
=> [1,1,1,0,1,1,1,0,0,1,0,0,0,1,0,0]
=> [1,1,1,1,0,1,0,0,1,0,0,1,0,0,1,0]
=> ? = 2
[1,1,1,1,1,1,0,0,0,0,0,0]
=> []
=> []
=> []
=> ? = 1
[1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [5,5,4,3,2,1]
=> [1,1,1,0,1,1,1,0,1,1,0,0,0,1,0,0,0,1,0,0]
=> [1,1,1,1,1,0,1,0,0,1,0,0,1,0,0,1,0,0,1,0]
=> ? = 2
[1,0,1,0,1,1,1,1,0,1,0,0,0,0]
=> [3,2,2,2,2,1]
=> [1,0,1,1,1,1,0,1,0,1,0,0,0,1,0,0]
=> [1,1,1,0,0,1,0,1,0,1,0,1,0,0,1,0]
=> ? = 1
[1,0,1,1,0,0,1,1,1,1,0,0,0,0]
=> [3,3,3,3,1,1]
=> [1,1,1,1,1,1,0,0,0,0,0,1,0,1,0,0]
=> [1,1,1,0,1,0,1,0,1,0,0,0,1,0,1,0]
=> ? = 4
[1,0,1,1,0,1,1,1,0,1,0,0,0,0]
=> [3,2,2,2,1,1]
=> [1,0,1,1,1,1,0,1,0,0,0,1,0,1,0,0]
=> [1,1,1,0,0,1,0,1,0,1,0,0,1,0,1,0]
=> ? = 1
[1,0,1,1,1,0,0,1,1,1,0,0,0,0]
=> [3,3,3,1,1,1]
=> [1,1,1,1,1,0,0,0,0,1,0,1,0,1,0,0]
=> [1,1,1,0,1,0,1,0,0,0,1,0,1,0,1,0]
=> ? = 3
[1,0,1,1,1,1,1,0,0,1,0,0,0,0]
=> [3,1,1,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,1,0,0,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 1
[1,1,0,1,0,1,0,1,0,1,1,0,0,0]
=> [4,4,3,2,1]
=> [1,1,1,0,1,1,1,0,0,1,0,0,0,1,0,0]
=> [1,1,1,1,0,1,0,0,1,0,0,1,0,0,1,0]
=> ? = 2
[1,1,1,1,0,0,0,0,1,0,1,0,1,0]
=> [6,5,4]
=> [1,0,1,1,1,0,1,1,1,0,1,0,0,0,0,0]
=> [1,1,1,1,1,1,0,0,1,0,0,1,0,0,0,0]
=> ? = 1
[1,1,1,1,0,0,0,0,1,1,0,0,1,0]
=> [6,4,4]
=> [1,0,1,0,1,1,1,1,1,0,1,0,0,0,0,0]
=> [1,1,1,1,1,1,0,0,0,1,0,1,0,0,0,0]
=> ? = 1
[1,1,1,1,0,0,1,0,0,0,1,0,1,0]
=> [6,5,2]
=> [1,0,1,1,1,0,1,0,1,0,1,1,0,0,0,0]
=> [1,1,1,1,1,1,0,0,1,0,0,0,0,1,0,0]
=> ? = 1
[1,1,1,1,0,0,1,0,0,1,0,0,1,0]
=> [6,4,2]
=> [1,0,1,0,1,1,1,0,1,0,1,1,0,0,0,0]
=> [1,1,1,1,1,1,0,0,0,1,0,0,0,1,0,0]
=> ? = 1
[1,1,1,1,0,1,0,0,0,0,1,0,1,0]
=> [6,5,1]
=> [1,0,1,1,1,0,1,0,1,0,1,0,0,1,0,0]
=> [1,1,1,1,1,1,0,0,1,0,0,0,0,0,1,0]
=> ? = 1
[1,1,1,1,0,1,0,0,0,1,0,0,1,0]
=> [6,4,1]
=> [1,0,1,0,1,1,1,0,1,0,1,0,0,1,0,0]
=> [1,1,1,1,1,1,0,0,0,1,0,0,0,0,1,0]
=> ? = 1
[1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> []
=> []
=> []
=> ? = 1
[1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [6,6,5,4,3,2,1]
=> [1,1,1,0,1,1,1,0,1,1,1,0,0,1,0,0,0,1,0,0,0,1,0,0]
=> [1,1,1,1,1,1,0,1,0,0,1,0,0,1,0,0,1,0,0,1,0,0,1,0]
=> ? = 2
[1,0,1,0,1,1,1,1,1,0,1,0,0,0,0,0]
=> [3,2,2,2,2,2,1]
=> [1,0,1,1,1,1,0,1,0,1,0,1,0,0,0,1,0,0]
=> [1,1,1,0,0,1,0,1,0,1,0,1,0,1,0,0,1,0]
=> ? = 1
[1,0,1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [2,2,2,2,2,2,1]
=> [1,1,1,1,0,1,0,1,0,1,0,0,0,1,0,0]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,0,1,0]
=> ? = 6
[1,0,1,1,0,0,1,1,1,1,1,0,0,0,0,0]
=> [3,3,3,3,3,1,1]
=> [1,1,1,1,1,1,0,1,0,0,0,0,0,1,0,1,0,0]
=> [1,1,1,0,1,0,1,0,1,0,1,0,0,0,1,0,1,0]
=> ? = 5
[1,0,1,1,0,1,1,1,1,1,0,0,0,0,0,0]
=> [2,2,2,2,2,1,1]
=> [1,1,1,1,0,1,0,1,0,0,0,1,0,1,0,0]
=> [1,1,0,1,0,1,0,1,0,1,0,0,1,0,1,0]
=> ? = 5
[1,0,1,1,1,0,1,1,1,1,0,0,0,0,0,0]
=> [2,2,2,2,1,1,1]
=> [1,1,1,1,0,1,0,0,0,1,0,1,0,1,0,0]
=> [1,1,0,1,0,1,0,1,0,0,1,0,1,0,1,0]
=> ? = 4
[1,0,1,1,1,1,1,1,0,0,1,0,0,0,0,0]
=> [3,1,1,1,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,1,0,0,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 1
[1,1,0,0,1,0,1,1,1,1,1,0,0,0,0,0]
=> [3,3,3,3,3,2]
=> [1,1,1,1,1,1,0,1,0,0,0,1,0,0,0,0]
=> [1,1,1,0,1,0,1,0,1,0,1,0,0,1,0,0]
=> ? = 5
[1,1,0,0,1,1,1,0,1,1,1,0,0,0,0,0]
=> [3,3,3,2,2,2]
=> [1,1,1,1,1,0,0,1,0,1,0,1,0,0,0,0]
=> [1,1,1,0,1,0,1,0,0,1,0,1,0,1,0,0]
=> ? = 3
[1,1,0,0,1,1,1,1,0,1,1,0,0,0,0,0]
=> [3,3,2,2,2,2]
=> [1,1,1,0,1,1,0,1,0,1,0,1,0,0,0,0]
=> [1,1,1,0,1,0,0,1,0,1,0,1,0,1,0,0]
=> ? = 2
[1,1,0,0,1,1,1,1,1,0,1,0,0,0,0,0]
=> [3,2,2,2,2,2]
=> [1,0,1,1,1,1,0,1,0,1,0,1,0,0,0,0]
=> [1,1,1,0,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 1
[1,1,0,1,0,0,1,1,1,1,1,0,0,0,0,0]
=> [3,3,3,3,3,1]
=> [1,1,1,1,1,1,0,1,0,0,0,0,0,1,0,0]
=> [1,1,1,0,1,0,1,0,1,0,1,0,0,0,1,0]
=> ? = 5
[1,1,0,1,0,1,0,1,0,1,0,1,1,0,0,0]
=> [5,5,4,3,2,1]
=> [1,1,1,0,1,1,1,0,1,1,0,0,0,1,0,0,0,1,0,0]
=> [1,1,1,1,1,0,1,0,0,1,0,0,1,0,0,1,0,0,1,0]
=> ? = 2
[1,1,0,1,0,1,1,1,1,0,1,0,0,0,0,0]
=> [3,2,2,2,2,1]
=> [1,0,1,1,1,1,0,1,0,1,0,0,0,1,0,0]
=> [1,1,1,0,0,1,0,1,0,1,0,1,0,0,1,0]
=> ? = 1
[1,1,0,1,1,0,0,1,1,1,1,0,0,0,0,0]
=> [3,3,3,3,1,1]
=> [1,1,1,1,1,1,0,0,0,0,0,1,0,1,0,0]
=> [1,1,1,0,1,0,1,0,1,0,0,0,1,0,1,0]
=> ? = 4
[1,1,0,1,1,0,1,1,1,0,1,0,0,0,0,0]
=> [3,2,2,2,1,1]
=> [1,0,1,1,1,1,0,1,0,0,0,1,0,1,0,0]
=> [1,1,1,0,0,1,0,1,0,1,0,0,1,0,1,0]
=> ? = 1
[1,1,0,1,1,1,0,0,1,1,1,0,0,0,0,0]
=> [3,3,3,1,1,1]
=> [1,1,1,1,1,0,0,0,0,1,0,1,0,1,0,0]
=> [1,1,1,0,1,0,1,0,0,0,1,0,1,0,1,0]
=> ? = 3
[1,1,0,1,1,1,1,1,0,0,1,0,0,0,0,0]
=> [3,1,1,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,1,0,0,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 1
[1,1,1,0,0,0,1,1,1,1,0,1,0,0,0,0]
=> [4,3,3,3,3]
=> [1,0,1,1,1,1,1,1,0,1,0,0,0,0,0,0]
=> [1,1,1,1,0,0,1,0,1,0,1,0,1,0,0,0]
=> ? = 1
[1,1,1,0,1,0,1,0,1,0,1,1,0,0,0,0]
=> [4,4,3,2,1]
=> [1,1,1,0,1,1,1,0,0,1,0,0,0,1,0,0]
=> [1,1,1,1,0,1,0,0,1,0,0,1,0,0,1,0]
=> ? = 2
[1,1,1,1,0,0,1,0,0,0,1,1,1,0,0,0]
=> [5,5,5,2]
=> [1,1,1,1,1,0,1,0,1,0,0,1,0,0,0,0]
=> [1,1,1,1,1,0,1,0,1,0,0,0,0,1,0,0]
=> ? = 3
[1,1,1,1,0,1,0,0,0,0,1,1,1,0,0,0]
=> [5,5,5,1]
=> [1,1,1,1,1,0,1,0,1,0,0,0,0,1,0,0]
=> [1,1,1,1,1,0,1,0,1,0,0,0,0,0,1,0]
=> ? = 3
[1,1,1,1,1,0,0,0,0,0,1,0,1,1,0,0]
=> [6,6,5]
=> [1,1,1,0,1,1,1,0,1,0,1,0,0,0,0,0]
=> [1,1,1,1,1,1,0,1,0,0,1,0,0,0,0,0]
=> ? = 2
[1,1,1,1,1,0,0,0,0,0,1,1,0,1,0,0]
=> [6,5,5]
=> [1,0,1,1,1,1,1,0,1,0,1,0,0,0,0,0]
=> [1,1,1,1,1,1,0,0,1,0,1,0,0,0,0,0]
=> ? = 1
[1,1,1,1,1,0,0,0,0,1,0,0,1,1,0,0]
=> [6,6,4]
=> [1,1,1,0,1,0,1,1,1,0,1,0,0,0,0,0]
=> [1,1,1,1,1,1,0,1,0,0,0,1,0,0,0,0]
=> ? = 2
[1,1,1,1,1,0,0,0,0,1,0,1,0,1,0,0]
=> [6,5,4]
=> [1,0,1,1,1,0,1,1,1,0,1,0,0,0,0,0]
=> [1,1,1,1,1,1,0,0,1,0,0,1,0,0,0,0]
=> ? = 1
[1,1,1,1,1,0,0,0,0,1,1,0,0,1,0,0]
=> [6,4,4]
=> [1,0,1,0,1,1,1,1,1,0,1,0,0,0,0,0]
=> [1,1,1,1,1,1,0,0,0,1,0,1,0,0,0,0]
=> ? = 1
[1,1,1,1,1,0,0,1,0,0,0,0,1,0,1,0]
=> [7,6,2]
=> [1,0,1,1,1,0,1,0,1,0,1,0,1,1,0,0,0,0]
=> [1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,1,0,0]
=> ? = 1
[1,1,1,1,1,0,0,1,0,0,0,0,1,1,0,0]
=> [6,6,2]
=> [1,1,1,0,1,0,1,0,1,0,1,1,0,0,0,0]
=> [1,1,1,1,1,1,0,1,0,0,0,0,0,1,0,0]
=> ? = 2
Description
The number of weak left to right maxima of a Dyck path.
A weak left to right maximum is a peak whose height is larger than or equal to the height of all peaks to its
left.
Matching statistic: St000733
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00027: Dyck paths —to partition⟶ Integer partitions
Mp00045: Integer partitions —reading tableau⟶ Standard tableaux
St000733: Standard tableaux ⟶ ℤResult quality: 67% ●values known / values provided: 74%●distinct values known / distinct values provided: 67%
Mp00045: Integer partitions —reading tableau⟶ Standard tableaux
St000733: Standard tableaux ⟶ ℤResult quality: 67% ●values known / values provided: 74%●distinct values known / distinct values provided: 67%
Values
[1,0]
=> []
=> []
=> ? = 1
[1,0,1,0]
=> [1]
=> [[1]]
=> 1
[1,1,0,0]
=> []
=> []
=> ? = 1
[1,0,1,0,1,0]
=> [2,1]
=> [[1,3],[2]]
=> 1
[1,0,1,1,0,0]
=> [1,1]
=> [[1],[2]]
=> 2
[1,1,0,0,1,0]
=> [2]
=> [[1,2]]
=> 1
[1,1,0,1,0,0]
=> [1]
=> [[1]]
=> 1
[1,1,1,0,0,0]
=> []
=> []
=> ? = 1
[1,0,1,0,1,0,1,0]
=> [3,2,1]
=> [[1,3,6],[2,5],[4]]
=> 1
[1,0,1,0,1,1,0,0]
=> [2,2,1]
=> [[1,3],[2,5],[4]]
=> 2
[1,0,1,1,0,0,1,0]
=> [3,1,1]
=> [[1,4,5],[2],[3]]
=> 1
[1,0,1,1,0,1,0,0]
=> [2,1,1]
=> [[1,4],[2],[3]]
=> 1
[1,0,1,1,1,0,0,0]
=> [1,1,1]
=> [[1],[2],[3]]
=> 3
[1,1,0,0,1,0,1,0]
=> [3,2]
=> [[1,2,5],[3,4]]
=> 1
[1,1,0,0,1,1,0,0]
=> [2,2]
=> [[1,2],[3,4]]
=> 2
[1,1,0,1,0,0,1,0]
=> [3,1]
=> [[1,3,4],[2]]
=> 1
[1,1,0,1,0,1,0,0]
=> [2,1]
=> [[1,3],[2]]
=> 1
[1,1,0,1,1,0,0,0]
=> [1,1]
=> [[1],[2]]
=> 2
[1,1,1,0,0,0,1,0]
=> [3]
=> [[1,2,3]]
=> 1
[1,1,1,0,0,1,0,0]
=> [2]
=> [[1,2]]
=> 1
[1,1,1,0,1,0,0,0]
=> [1]
=> [[1]]
=> 1
[1,1,1,1,0,0,0,0]
=> []
=> []
=> ? = 1
[1,0,1,0,1,0,1,0,1,0]
=> [4,3,2,1]
=> [[1,3,6,10],[2,5,9],[4,8],[7]]
=> 1
[1,0,1,0,1,0,1,1,0,0]
=> [3,3,2,1]
=> [[1,3,6],[2,5,9],[4,8],[7]]
=> 2
[1,0,1,0,1,1,0,1,0,0]
=> [3,2,2,1]
=> [[1,3,8],[2,5],[4,7],[6]]
=> 1
[1,0,1,0,1,1,1,0,0,0]
=> [2,2,2,1]
=> [[1,3],[2,5],[4,7],[6]]
=> 3
[1,0,1,1,0,0,1,1,0,0]
=> [3,3,1,1]
=> [[1,4,5],[2,7,8],[3],[6]]
=> 2
[1,0,1,1,0,1,0,1,0,0]
=> [3,2,1,1]
=> [[1,4,7],[2,6],[3],[5]]
=> 1
[1,0,1,1,0,1,1,0,0,0]
=> [2,2,1,1]
=> [[1,4],[2,6],[3],[5]]
=> 2
[1,0,1,1,1,0,0,1,0,0]
=> [3,1,1,1]
=> [[1,5,6],[2],[3],[4]]
=> 1
[1,0,1,1,1,0,1,0,0,0]
=> [2,1,1,1]
=> [[1,5],[2],[3],[4]]
=> 1
[1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1]
=> [[1],[2],[3],[4]]
=> 4
[1,1,0,0,1,0,1,0,1,0]
=> [4,3,2]
=> [[1,2,5,9],[3,4,8],[6,7]]
=> 1
[1,1,0,0,1,0,1,1,0,0]
=> [3,3,2]
=> [[1,2,5],[3,4,8],[6,7]]
=> 2
[1,1,0,0,1,1,0,0,1,0]
=> [4,2,2]
=> [[1,2,7,8],[3,4],[5,6]]
=> 1
[1,1,0,0,1,1,0,1,0,0]
=> [3,2,2]
=> [[1,2,7],[3,4],[5,6]]
=> 1
[1,1,0,0,1,1,1,0,0,0]
=> [2,2,2]
=> [[1,2],[3,4],[5,6]]
=> 3
[1,1,0,1,0,0,1,0,1,0]
=> [4,3,1]
=> [[1,3,4,8],[2,6,7],[5]]
=> 1
[1,1,0,1,0,0,1,1,0,0]
=> [3,3,1]
=> [[1,3,4],[2,6,7],[5]]
=> 2
[1,1,0,1,0,1,0,0,1,0]
=> [4,2,1]
=> [[1,3,6,7],[2,5],[4]]
=> 1
[1,1,0,1,0,1,0,1,0,0]
=> [3,2,1]
=> [[1,3,6],[2,5],[4]]
=> 1
[1,1,0,1,0,1,1,0,0,0]
=> [2,2,1]
=> [[1,3],[2,5],[4]]
=> 2
[1,1,0,1,1,0,0,0,1,0]
=> [4,1,1]
=> [[1,4,5,6],[2],[3]]
=> 1
[1,1,0,1,1,0,0,1,0,0]
=> [3,1,1]
=> [[1,4,5],[2],[3]]
=> 1
[1,1,0,1,1,0,1,0,0,0]
=> [2,1,1]
=> [[1,4],[2],[3]]
=> 1
[1,1,0,1,1,1,0,0,0,0]
=> [1,1,1]
=> [[1],[2],[3]]
=> 3
[1,1,1,0,0,0,1,0,1,0]
=> [4,3]
=> [[1,2,3,7],[4,5,6]]
=> 1
[1,1,1,0,0,0,1,1,0,0]
=> [3,3]
=> [[1,2,3],[4,5,6]]
=> 2
[1,1,1,0,0,1,0,0,1,0]
=> [4,2]
=> [[1,2,5,6],[3,4]]
=> 1
[1,1,1,0,0,1,0,1,0,0]
=> [3,2]
=> [[1,2,5],[3,4]]
=> 1
[1,1,1,0,0,1,1,0,0,0]
=> [2,2]
=> [[1,2],[3,4]]
=> 2
[1,1,1,0,1,0,0,0,1,0]
=> [4,1]
=> [[1,3,4,5],[2]]
=> 1
[1,1,1,0,1,0,0,1,0,0]
=> [3,1]
=> [[1,3,4],[2]]
=> 1
[1,1,1,0,1,0,1,0,0,0]
=> [2,1]
=> [[1,3],[2]]
=> 1
[1,1,1,1,1,0,0,0,0,0]
=> []
=> []
=> ? = 1
[1,1,1,1,1,1,0,0,0,0,0,0]
=> []
=> []
=> ? = 1
[1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [5,5,4,3,2,1]
=> [[1,3,6,10,15],[2,5,9,14,20],[4,8,13,19],[7,12,18],[11,17],[16]]
=> ? = 2
[1,0,1,0,1,1,1,1,0,1,0,0,0,0]
=> [3,2,2,2,2,1]
=> [[1,3,12],[2,5],[4,7],[6,9],[8,11],[10]]
=> ? = 1
[1,0,1,0,1,1,1,1,1,0,0,0,0,0]
=> [2,2,2,2,2,1]
=> [[1,3],[2,5],[4,7],[6,9],[8,11],[10]]
=> ? = 5
[1,0,1,1,0,0,1,1,1,1,0,0,0,0]
=> [3,3,3,3,1,1]
=> [[1,4,5],[2,7,8],[3,10,11],[6,13,14],[9],[12]]
=> ? = 4
[1,0,1,1,0,1,1,1,0,1,0,0,0,0]
=> [3,2,2,2,1,1]
=> [[1,4,11],[2,6],[3,8],[5,10],[7],[9]]
=> ? = 1
[1,0,1,1,1,0,0,1,1,1,0,0,0,0]
=> [3,3,3,1,1,1]
=> [[1,5,6],[2,8,9],[3,11,12],[4],[7],[10]]
=> ? = 3
[1,1,0,0,1,0,1,1,1,1,0,0,0,0]
=> [3,3,3,3,2]
=> [[1,2,5],[3,4,8],[6,7,11],[9,10,14],[12,13]]
=> ? = 4
[1,1,0,0,1,1,0,1,1,1,0,0,0,0]
=> [3,3,3,2,2]
=> [[1,2,7],[3,4,10],[5,6,13],[8,9],[11,12]]
=> ? = 3
[1,1,0,0,1,1,1,0,1,1,0,0,0,0]
=> [3,3,2,2,2]
=> [[1,2,9],[3,4,12],[5,6],[7,8],[10,11]]
=> ? = 2
[1,1,0,0,1,1,1,1,0,1,0,0,0,0]
=> [3,2,2,2,2]
=> [[1,2,11],[3,4],[5,6],[7,8],[9,10]]
=> ? = 1
[1,1,0,1,0,0,1,1,1,1,0,0,0,0]
=> [3,3,3,3,1]
=> [[1,3,4],[2,6,7],[5,9,10],[8,12,13],[11]]
=> ? = 4
[1,1,1,0,0,0,1,1,1,0,1,0,0,0]
=> [4,3,3,3]
=> [[1,2,3,13],[4,5,6],[7,8,9],[10,11,12]]
=> ? = 1
[1,1,1,0,0,0,1,1,1,1,0,0,0,0]
=> [3,3,3,3]
=> [[1,2,3],[4,5,6],[7,8,9],[10,11,12]]
=> ? = 4
[1,1,1,0,0,1,0,0,1,1,1,0,0,0]
=> [4,4,4,2]
=> [[1,2,5,6],[3,4,9,10],[7,8,13,14],[11,12]]
=> ? = 3
[1,1,1,0,1,0,0,0,1,1,1,0,0,0]
=> [4,4,4,1]
=> [[1,3,4,5],[2,7,8,9],[6,11,12,13],[10]]
=> ? = 3
[1,1,1,1,0,0,0,0,1,0,1,0,1,0]
=> [6,5,4]
=> [[1,2,3,4,9,15],[5,6,7,8,14],[10,11,12,13]]
=> ? = 1
[1,1,1,1,0,0,0,0,1,0,1,1,0,0]
=> [5,5,4]
=> [[1,2,3,4,9],[5,6,7,8,14],[10,11,12,13]]
=> ? = 2
[1,1,1,1,0,0,0,0,1,1,0,0,1,0]
=> [6,4,4]
=> [[1,2,3,4,13,14],[5,6,7,8],[9,10,11,12]]
=> ? = 1
[1,1,1,1,0,0,0,0,1,1,1,0,0,0]
=> [4,4,4]
=> [[1,2,3,4],[5,6,7,8],[9,10,11,12]]
=> ? = 3
[1,1,1,1,0,0,0,1,0,0,1,1,0,0]
=> [5,5,3]
=> [[1,2,3,7,8],[4,5,6,12,13],[9,10,11]]
=> ? = 2
[1,1,1,1,0,0,1,0,0,0,1,0,1,0]
=> [6,5,2]
=> [[1,2,5,6,7,13],[3,4,10,11,12],[8,9]]
=> ? = 1
[1,1,1,1,0,0,1,0,0,0,1,1,0,0]
=> [5,5,2]
=> [[1,2,5,6,7],[3,4,10,11,12],[8,9]]
=> ? = 2
[1,1,1,1,0,1,0,0,0,0,1,0,1,0]
=> [6,5,1]
=> [[1,3,4,5,6,12],[2,8,9,10,11],[7]]
=> ? = 1
[1,1,1,1,0,1,0,0,0,0,1,1,0,0]
=> [5,5,1]
=> [[1,3,4,5,6],[2,8,9,10,11],[7]]
=> ? = 2
[1,1,1,1,0,1,0,0,0,1,0,0,1,0]
=> [6,4,1]
=> [[1,3,4,5,10,11],[2,7,8,9],[6]]
=> ? = 1
[1,1,1,1,1,0,0,0,0,0,1,0,1,0]
=> [6,5]
=> [[1,2,3,4,5,11],[6,7,8,9,10]]
=> ? = 1
[1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> []
=> []
=> ? = 1
[1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [6,6,5,4,3,2,1]
=> [[1,3,6,10,15,21],[2,5,9,14,20,27],[4,8,13,19,26],[7,12,18,25],[11,17,24],[16,23],[22]]
=> ? = 2
[1,0,1,0,1,1,1,1,1,0,1,0,0,0,0,0]
=> [3,2,2,2,2,2,1]
=> [[1,3,14],[2,5],[4,7],[6,9],[8,11],[10,13],[12]]
=> ? = 1
[1,0,1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [2,2,2,2,2,2,1]
=> [[1,3],[2,5],[4,7],[6,9],[8,11],[10,13],[12]]
=> ? = 6
[1,0,1,1,0,0,1,1,1,1,1,0,0,0,0,0]
=> [3,3,3,3,3,1,1]
=> [[1,4,5],[2,7,8],[3,10,11],[6,13,14],[9,16,17],[12],[15]]
=> ? = 5
[1,0,1,1,0,1,1,1,1,1,0,0,0,0,0,0]
=> [2,2,2,2,2,1,1]
=> [[1,4],[2,6],[3,8],[5,10],[7,12],[9],[11]]
=> ? = 5
[1,0,1,1,1,0,1,1,1,1,0,0,0,0,0,0]
=> [2,2,2,2,1,1,1]
=> [[1,5],[2,7],[3,9],[4,11],[6],[8],[10]]
=> ? = 4
[1,1,0,0,1,0,1,1,1,1,1,0,0,0,0,0]
=> [3,3,3,3,3,2]
=> [[1,2,5],[3,4,8],[6,7,11],[9,10,14],[12,13,17],[15,16]]
=> ? = 5
[1,1,0,0,1,1,1,0,1,1,1,0,0,0,0,0]
=> [3,3,3,2,2,2]
=> [[1,2,9],[3,4,12],[5,6,15],[7,8],[10,11],[13,14]]
=> ? = 3
[1,1,0,0,1,1,1,1,0,1,1,0,0,0,0,0]
=> [3,3,2,2,2,2]
=> [[1,2,11],[3,4,14],[5,6],[7,8],[9,10],[12,13]]
=> ? = 2
[1,1,0,0,1,1,1,1,1,0,1,0,0,0,0,0]
=> [3,2,2,2,2,2]
=> [[1,2,13],[3,4],[5,6],[7,8],[9,10],[11,12]]
=> ? = 1
[1,1,0,1,0,0,1,1,1,1,1,0,0,0,0,0]
=> [3,3,3,3,3,1]
=> [[1,3,4],[2,6,7],[5,9,10],[8,12,13],[11,15,16],[14]]
=> ? = 5
[1,1,0,1,0,1,0,1,0,1,0,1,1,0,0,0]
=> [5,5,4,3,2,1]
=> [[1,3,6,10,15],[2,5,9,14,20],[4,8,13,19],[7,12,18],[11,17],[16]]
=> ? = 2
[1,1,0,1,0,1,1,1,1,0,1,0,0,0,0,0]
=> [3,2,2,2,2,1]
=> [[1,3,12],[2,5],[4,7],[6,9],[8,11],[10]]
=> ? = 1
[1,1,0,1,0,1,1,1,1,1,0,0,0,0,0,0]
=> [2,2,2,2,2,1]
=> [[1,3],[2,5],[4,7],[6,9],[8,11],[10]]
=> ? = 5
[1,1,0,1,1,0,0,1,1,1,1,0,0,0,0,0]
=> [3,3,3,3,1,1]
=> [[1,4,5],[2,7,8],[3,10,11],[6,13,14],[9],[12]]
=> ? = 4
[1,1,0,1,1,0,1,1,1,0,1,0,0,0,0,0]
=> [3,2,2,2,1,1]
=> [[1,4,11],[2,6],[3,8],[5,10],[7],[9]]
=> ? = 1
[1,1,0,1,1,1,0,0,1,1,1,0,0,0,0,0]
=> [3,3,3,1,1,1]
=> [[1,5,6],[2,8,9],[3,11,12],[4],[7],[10]]
=> ? = 3
Description
The row containing the largest entry of a standard tableau.
The following 25 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000883The number of longest increasing subsequences of a permutation. St001264The smallest index i such that the i-th simple module has projective dimension equal to the global dimension of the corresponding Nakayama algebra. St000654The first descent of a permutation. St000765The number of weak records in an integer composition. St000766The number of inversions of an integer composition. St001803The maximal overlap of the cylindrical tableau associated with a tableau. St000278The size of the preimage of the map 'to partition' from Integer compositions to Integer partitions. St000175Degree of the polynomial counting the number of semistandard Young tableaux when stretching the shape. St001810The number of fixed points of a permutation smaller than its largest moved point. St001199The dominant dimension of $eAe$ for the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St000234The number of global ascents of a permutation. St000771The largest multiplicity of a distance Laplacian eigenvalue in a connected graph. St000772The multiplicity of the largest distance Laplacian eigenvalue in 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 $eA$. St000264The girth of a graph, which is not a tree. St001545The second Elser number of a connected graph. St000056The decomposition (or block) number of a permutation. St001292The injective dimension of the tensor product of two copies of the dual of the Nakayama algebra associated to a Dyck path. St001875The number of simple modules with projective dimension at most 1. St001846The number of elements which do not have a complement in the lattice. St001549The number of restricted non-inversions between exceedances. St001200The number of simple modules in $eAe$ with projective dimension at most 2 in the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St000455The second largest eigenvalue of a graph if it is integral. St001194The injective dimension of $A/AfA$ in the corresponding Nakayama algebra $A$ when $Af$ is the minimal faithful projective-injective left $A$-module
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