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Your data matches 23 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: St000392
(load all 7 compositions to match this statistic)
(load all 7 compositions to match this statistic)
Mp00095: Integer partitions —to binary word⟶ Binary words
Mp00104: Binary words —reverse⟶ Binary words
St000392: Binary words ⟶ ℤResult quality: 31% ●values known / values provided: 31%●distinct values known / distinct values provided: 62%
Mp00104: Binary words —reverse⟶ Binary words
St000392: Binary words ⟶ ℤResult quality: 31% ●values known / values provided: 31%●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
[10,1]
=> 100000000010 => 010000000001 => ? = 1
[9,2]
=> 10000000100 => 00100000001 => ? = 1
[9,1,1]
=> 100000000110 => 011000000001 => ? = 2
[8,2,1]
=> 10000001010 => 01010000001 => ? = 1
[8,1,1,1]
=> 100000001110 => 011100000001 => ? = 3
[7,3,1]
=> 1000010010 => 0100100001 => ? = 1
[7,2,2]
=> 1000001100 => 0011000001 => ? = 2
[7,2,1,1]
=> 10000010110 => 01101000001 => ? = 2
[7,1,1,1,1]
=> 100000011110 => 011110000001 => ? = 4
[6,3,1,1]
=> 1000100110 => 0110010001 => ? = 2
[6,2,2,1]
=> 1000011010 => 0101100001 => ? = 2
[6,2,1,1,1]
=> 10000101110 => 01110100001 => ? = 3
[5,2,1,1,1,1]
=> 10001011110 => 01111010001 => ? = 4
[5,1,1,1,1,1,1]
=> 100001111110 => 011111100001 => ? = 6
[4,3,1,1,1,1]
=> 1010011110 => 0111100101 => ? = 4
[4,2,2,1,1,1]
=> 1001101110 => 0111011001 => ? = 3
[4,2,1,1,1,1,1]
=> 10010111110 => 01111101001 => ? = 5
[4,1,1,1,1,1,1,1]
=> 100011111110 => 011111110001 => ? = 7
[3,3,1,1,1,1,1]
=> 1100111110 => 0111110011 => ? = 5
[3,2,2,1,1,1,1]
=> 1011011110 => 0111101101 => ? = 4
[3,2,1,1,1,1,1,1]
=> 10101111110 => 01111110101 => ? = 6
[3,1,1,1,1,1,1,1,1]
=> 100111111110 => 011111111001 => ? = 8
[2,2,2,1,1,1,1,1]
=> 1110111110 => 0111110111 => ? = 5
[2,2,1,1,1,1,1,1,1]
=> 11011111110 => 01111111011 => ? = 7
[2,1,1,1,1,1,1,1,1,1]
=> 101111111110 => 011111111101 => ? = 9
[1,1,1,1,1,1,1,1,1,1,1]
=> 111111111110 => 011111111111 => ? = 11
[12]
=> 1000000000000 => 0000000000001 => ? = 1
[11,1]
=> 1000000000010 => ? => ? = 1
[10,2]
=> 100000000100 => 001000000001 => ? = 1
[10,1,1]
=> 1000000000110 => ? => ? = 2
[9,3]
=> 10000001000 => 00010000001 => ? = 1
[9,2,1]
=> 100000001010 => ? => ? = 1
[9,1,1,1]
=> 1000000001110 => ? => ? = 3
[8,3,1]
=> 10000010010 => 01001000001 => ? = 1
[8,2,2]
=> 10000001100 => 00110000001 => ? = 2
[8,2,1,1]
=> 100000010110 => 011010000001 => ? = 2
[8,1,1,1,1]
=> 1000000011110 => ? => ? = 4
[7,4,1]
=> 1000100010 => 0100010001 => ? = 1
[7,3,1,1]
=> 10000100110 => 01100100001 => ? = 2
[7,2,2,1]
=> 10000011010 => 01011000001 => ? = 2
[7,2,1,1,1]
=> 100000101110 => ? => ? = 3
[7,1,1,1,1,1]
=> 1000000111110 => ? => ? = 5
[6,4,1,1]
=> 1001000110 => 0110001001 => ? = 2
[6,3,2,1]
=> 1000101010 => 0101010001 => ? = 1
[6,3,1,1,1]
=> 10001001110 => 01110010001 => ? = 3
[6,2,2,2]
=> 1000011100 => 0011100001 => ? = 3
[6,2,2,1,1]
=> 10000110110 => 01101100001 => ? = 2
[6,1,1,1,1,1,1]
=> 1000001111110 => ? => ? = 6
[5,3,1,1,1,1]
=> 10010011110 => 01111001001 => ? = 4
Description
The length of the longest run of ones in a binary word.
Matching statistic: St001372
(load all 7 compositions to match this statistic)
(load all 7 compositions to match this statistic)
Mp00095: Integer partitions —to binary word⟶ Binary words
Mp00104: Binary words —reverse⟶ Binary words
St001372: Binary words ⟶ ℤResult quality: 29% ●values known / values provided: 29%●distinct values known / distinct values provided: 62%
Mp00104: Binary words —reverse⟶ Binary words
St001372: Binary words ⟶ ℤResult quality: 29% ●values known / values provided: 29%●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
[10,1]
=> 100000000010 => 010000000001 => ? = 1
[9,2]
=> 10000000100 => 00100000001 => ? = 1
[9,1,1]
=> 100000000110 => 011000000001 => ? = 2
[8,3]
=> 1000001000 => 0001000001 => ? = 1
[8,2,1]
=> 10000001010 => 01010000001 => ? = 1
[8,1,1,1]
=> 100000001110 => 011100000001 => ? = 3
[7,3,1]
=> 1000010010 => 0100100001 => ? = 1
[7,2,2]
=> 1000001100 => 0011000001 => ? = 2
[7,2,1,1]
=> 10000010110 => 01101000001 => ? = 2
[7,1,1,1,1]
=> 100000011110 => 011110000001 => ? = 4
[6,3,1,1]
=> 1000100110 => 0110010001 => ? = 2
[6,2,2,1]
=> 1000011010 => 0101100001 => ? = 2
[6,2,1,1,1]
=> 10000101110 => 01110100001 => ? = 3
[5,2,1,1,1,1]
=> 10001011110 => 01111010001 => ? = 4
[5,1,1,1,1,1,1]
=> 100001111110 => 011111100001 => ? = 6
[4,3,1,1,1,1]
=> 1010011110 => 0111100101 => ? = 4
[4,2,2,1,1,1]
=> 1001101110 => 0111011001 => ? = 3
[4,2,1,1,1,1,1]
=> 10010111110 => 01111101001 => ? = 5
[4,1,1,1,1,1,1,1]
=> 100011111110 => 011111110001 => ? = 7
[3,3,1,1,1,1,1]
=> 1100111110 => 0111110011 => ? = 5
[3,2,2,1,1,1,1]
=> 1011011110 => 0111101101 => ? = 4
[3,2,1,1,1,1,1,1]
=> 10101111110 => 01111110101 => ? = 6
[3,1,1,1,1,1,1,1,1]
=> 100111111110 => 011111111001 => ? = 8
[2,2,2,1,1,1,1,1]
=> 1110111110 => 0111110111 => ? = 5
[2,2,1,1,1,1,1,1,1]
=> 11011111110 => 01111111011 => ? = 7
[2,1,1,1,1,1,1,1,1,1]
=> 101111111110 => 011111111101 => ? = 9
[1,1,1,1,1,1,1,1,1,1,1]
=> 111111111110 => 011111111111 => ? = 11
[12]
=> 1000000000000 => 0000000000001 => ? = 1
[11,1]
=> 1000000000010 => ? => ? = 1
[10,2]
=> 100000000100 => 001000000001 => ? = 1
[10,1,1]
=> 1000000000110 => ? => ? = 2
[9,3]
=> 10000001000 => 00010000001 => ? = 1
[9,2,1]
=> 100000001010 => ? => ? = 1
[9,1,1,1]
=> 1000000001110 => ? => ? = 3
[8,4]
=> 1000010000 => 0000100001 => ? = 1
[8,3,1]
=> 10000010010 => 01001000001 => ? = 1
[8,2,2]
=> 10000001100 => 00110000001 => ? = 2
[8,2,1,1]
=> 100000010110 => 011010000001 => ? = 2
[8,1,1,1,1]
=> 1000000011110 => ? => ? = 4
[7,4,1]
=> 1000100010 => 0100010001 => ? = 1
[7,3,2]
=> 1000010100 => 0010100001 => ? = 1
[7,3,1,1]
=> 10000100110 => 01100100001 => ? = 2
[7,2,2,1]
=> 10000011010 => 01011000001 => ? = 2
[7,2,1,1,1]
=> 100000101110 => ? => ? = 3
[7,1,1,1,1,1]
=> 1000000111110 => ? => ? = 5
[6,4,1,1]
=> 1001000110 => 0110001001 => ? = 2
[6,3,2,1]
=> 1000101010 => 0101010001 => ? = 1
[6,3,1,1,1]
=> 10001001110 => 01110010001 => ? = 3
[6,2,2,2]
=> 1000011100 => 0011100001 => ? = 3
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
Mp00095: Integer partitions —to binary word⟶ Binary words
Mp00105: Binary words —complement⟶ Binary words
Mp00224: Binary words —runsort⟶ Binary words
St000326: Binary words ⟶ ℤResult quality: 27% ●values known / values provided: 27%●distinct values known / distinct values provided: 62%
Mp00105: Binary words —complement⟶ Binary words
Mp00224: Binary words —runsort⟶ Binary words
St000326: Binary words ⟶ ℤResult quality: 27% ●values known / values provided: 27%●distinct values known / distinct values provided: 62%
Values
[1]
=> 10 => 01 => 01 => 2 = 1 + 1
[2]
=> 100 => 011 => 011 => 2 = 1 + 1
[1,1]
=> 110 => 001 => 001 => 3 = 2 + 1
[3]
=> 1000 => 0111 => 0111 => 2 = 1 + 1
[2,1]
=> 1010 => 0101 => 0101 => 2 = 1 + 1
[1,1,1]
=> 1110 => 0001 => 0001 => 4 = 3 + 1
[4]
=> 10000 => 01111 => 01111 => 2 = 1 + 1
[3,1]
=> 10010 => 01101 => 01011 => 2 = 1 + 1
[2,2]
=> 1100 => 0011 => 0011 => 3 = 2 + 1
[2,1,1]
=> 10110 => 01001 => 00101 => 3 = 2 + 1
[1,1,1,1]
=> 11110 => 00001 => 00001 => 5 = 4 + 1
[5]
=> 100000 => 011111 => 011111 => 2 = 1 + 1
[4,1]
=> 100010 => 011101 => 010111 => 2 = 1 + 1
[3,2]
=> 10100 => 01011 => 01011 => 2 = 1 + 1
[3,1,1]
=> 100110 => 011001 => 001011 => 3 = 2 + 1
[2,2,1]
=> 11010 => 00101 => 00101 => 3 = 2 + 1
[2,1,1,1]
=> 101110 => 010001 => 000101 => 4 = 3 + 1
[1,1,1,1,1]
=> 111110 => 000001 => 000001 => 6 = 5 + 1
[6]
=> 1000000 => 0111111 => 0111111 => 2 = 1 + 1
[5,1]
=> 1000010 => 0111101 => 0101111 => 2 = 1 + 1
[4,2]
=> 100100 => 011011 => 011011 => 2 = 1 + 1
[4,1,1]
=> 1000110 => 0111001 => 0010111 => 3 = 2 + 1
[3,3]
=> 11000 => 00111 => 00111 => 3 = 2 + 1
[3,2,1]
=> 101010 => 010101 => 010101 => 2 = 1 + 1
[3,1,1,1]
=> 1001110 => 0110001 => 0001011 => 4 = 3 + 1
[2,2,2]
=> 11100 => 00011 => 00011 => 4 = 3 + 1
[2,2,1,1]
=> 110110 => 001001 => 001001 => 3 = 2 + 1
[2,1,1,1,1]
=> 1011110 => 0100001 => 0000101 => 5 = 4 + 1
[1,1,1,1,1,1]
=> 1111110 => 0000001 => 0000001 => 7 = 6 + 1
[7]
=> 10000000 => 01111111 => 01111111 => 2 = 1 + 1
[6,1]
=> 10000010 => 01111101 => 01011111 => 2 = 1 + 1
[5,2]
=> 1000100 => 0111011 => 0110111 => 2 = 1 + 1
[5,1,1]
=> 10000110 => 01111001 => 00101111 => 3 = 2 + 1
[4,3]
=> 101000 => 010111 => 010111 => 2 = 1 + 1
[4,2,1]
=> 1001010 => 0110101 => 0101011 => 2 = 1 + 1
[4,1,1,1]
=> 10001110 => 01110001 => 00010111 => 4 = 3 + 1
[3,3,1]
=> 110010 => 001101 => 001101 => 3 = 2 + 1
[3,2,2]
=> 101100 => 010011 => 001101 => 3 = 2 + 1
[3,2,1,1]
=> 1010110 => 0101001 => 0010101 => 3 = 2 + 1
[3,1,1,1,1]
=> 10011110 => 01100001 => 00001011 => 5 = 4 + 1
[2,2,2,1]
=> 111010 => 000101 => 000101 => 4 = 3 + 1
[2,2,1,1,1]
=> 1101110 => 0010001 => 0001001 => 4 = 3 + 1
[2,1,1,1,1,1]
=> 10111110 => 01000001 => 00000101 => 6 = 5 + 1
[1,1,1,1,1,1,1]
=> 11111110 => 00000001 => 00000001 => 8 = 7 + 1
[8]
=> 100000000 => 011111111 => 011111111 => 2 = 1 + 1
[7,1]
=> 100000010 => 011111101 => 010111111 => 2 = 1 + 1
[6,2]
=> 10000100 => 01111011 => 01101111 => 2 = 1 + 1
[6,1,1]
=> 100000110 => 011111001 => 001011111 => 3 = 2 + 1
[5,3]
=> 1001000 => 0110111 => 0110111 => 2 = 1 + 1
[5,2,1]
=> 10001010 => 01110101 => 01010111 => 2 = 1 + 1
[7,1,1]
=> 1000000110 => 0111111001 => 0010111111 => ? = 2 + 1
[8,1,1]
=> 10000000110 => 01111111001 => 00101111111 => ? = 2 + 1
[7,1,1,1]
=> 10000001110 => 01111110001 => 00010111111 => ? = 3 + 1
[6,1,1,1,1]
=> 10000011110 => 01111100001 => 00001011111 => ? = 4 + 1
[5,1,1,1,1,1]
=> 10000111110 => 01111000001 => 00000101111 => ? = 5 + 1
[4,1,1,1,1,1,1]
=> 10001111110 => 01110000001 => 00000010111 => ? = 6 + 1
[3,1,1,1,1,1,1,1]
=> 10011111110 => 01100000001 => 00000001011 => ? = 7 + 1
[11]
=> 100000000000 => 011111111111 => ? => ? = 1 + 1
[10,1]
=> 100000000010 => 011111111101 => ? => ? = 1 + 1
[9,2]
=> 10000000100 => 01111111011 => ? => ? = 1 + 1
[9,1,1]
=> 100000000110 => ? => ? => ? = 2 + 1
[8,3]
=> 1000001000 => 0111110111 => ? => ? = 1 + 1
[8,2,1]
=> 10000001010 => 01111110101 => ? => ? = 1 + 1
[8,1,1,1]
=> 100000001110 => 011111110001 => ? => ? = 3 + 1
[7,3,1]
=> 1000010010 => 0111101101 => ? => ? = 1 + 1
[7,2,2]
=> 1000001100 => 0111110011 => ? => ? = 2 + 1
[7,2,1,1]
=> 10000010110 => 01111101001 => ? => ? = 2 + 1
[7,1,1,1,1]
=> 100000011110 => 011111100001 => ? => ? = 4 + 1
[6,3,1,1]
=> 1000100110 => 0111011001 => ? => ? = 2 + 1
[6,2,2,1]
=> 1000011010 => 0111100101 => ? => ? = 2 + 1
[6,2,1,1,1]
=> 10000101110 => 01111010001 => ? => ? = 3 + 1
[5,2,1,1,1,1]
=> 10001011110 => 01110100001 => ? => ? = 4 + 1
[5,1,1,1,1,1,1]
=> 100001111110 => 011110000001 => ? => ? = 6 + 1
[4,3,1,1,1,1]
=> 1010011110 => 0101100001 => ? => ? = 4 + 1
[4,2,2,1,1,1]
=> 1001101110 => 0110010001 => ? => ? = 3 + 1
[4,2,1,1,1,1,1]
=> 10010111110 => 01101000001 => ? => ? = 5 + 1
[4,1,1,1,1,1,1,1]
=> 100011111110 => 011100000001 => ? => ? = 7 + 1
[3,3,1,1,1,1,1]
=> 1100111110 => 0011000001 => ? => ? = 5 + 1
[3,2,2,1,1,1,1]
=> 1011011110 => 0100100001 => ? => ? = 4 + 1
[3,2,1,1,1,1,1,1]
=> 10101111110 => 01010000001 => ? => ? = 6 + 1
[3,1,1,1,1,1,1,1,1]
=> 100111111110 => ? => ? => ? = 8 + 1
[2,2,2,1,1,1,1,1]
=> 1110111110 => 0001000001 => ? => ? = 5 + 1
[2,2,1,1,1,1,1,1,1]
=> 11011111110 => 00100000001 => ? => ? = 7 + 1
[2,1,1,1,1,1,1,1,1,1]
=> 101111111110 => 010000000001 => ? => ? = 9 + 1
[1,1,1,1,1,1,1,1,1,1,1]
=> 111111111110 => 000000000001 => ? => ? = 11 + 1
[12]
=> 1000000000000 => 0111111111111 => ? => ? = 1 + 1
[11,1]
=> 1000000000010 => ? => ? => ? = 1 + 1
[10,2]
=> 100000000100 => 011111111011 => ? => ? = 1 + 1
[10,1,1]
=> 1000000000110 => ? => ? => ? = 2 + 1
[9,3]
=> 10000001000 => 01111110111 => ? => ? = 1 + 1
[9,2,1]
=> 100000001010 => ? => ? => ? = 1 + 1
[9,1,1,1]
=> 1000000001110 => ? => ? => ? = 3 + 1
[8,4]
=> 1000010000 => 0111101111 => ? => ? = 1 + 1
[8,3,1]
=> 10000010010 => 01111101101 => ? => ? = 1 + 1
[8,2,2]
=> 10000001100 => 01111110011 => ? => ? = 2 + 1
[8,2,1,1]
=> 100000010110 => 011111101001 => ? => ? = 2 + 1
[8,1,1,1,1]
=> 1000000011110 => ? => ? => ? = 4 + 1
[7,4,1]
=> 1000100010 => 0111011101 => ? => ? = 1 + 1
[7,3,2]
=> 1000010100 => 0111101011 => ? => ? = 1 + 1
[7,3,1,1]
=> 10000100110 => 01111011001 => ? => ? = 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: 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: 26% ●values known / values provided: 26%●distinct values known / distinct values provided: 38%
Mp00023: Dyck paths —to non-crossing permutation⟶ Permutations
Mp00065: Permutations —permutation poset⟶ Posets
St000845: Posets ⟶ ℤResult quality: 26% ●values known / values provided: 26%●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
[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)
=> ? = 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
[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
[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)
=> ? = 6
[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)
=> ? = 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
[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
[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
[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)
=> ? = 2
[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)
=> ? = 6
[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)
=> ? = 5
[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)
=> ? = 7
[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)
=> ? = 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
[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
[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
[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)
=> ? = 2
[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
[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
[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)
=> ? = 3
[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)
=> ? = 6
[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)
=> ? = 5
[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)
=> ? = 7
[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)
=> ? = 5
[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)
=> ? = 4
[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)
=> ? = 6
[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)
=> ? = 8
[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)
=> ? = 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
[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
[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
[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] => ?
=> ? = 2
[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
[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
[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] => ?
=> ? = 3
[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
[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
[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)
=> ? = 2
[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)
=> ? = 2
[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)
=> ? = 4
[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)
=> ? = 6
[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)
=> ? = 5
[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] => ?
=> ? = 7
[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)
=> ? = 5
[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)
=> ? = 4
[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] => ?
=> ? = 4
[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)
=> ? = 6
[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)
=> ? = 8
[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)
=> ? = 4
Description
The maximal number of elements covered by an element in a poset.
Matching statistic: St000982
(load all 2 compositions to match this statistic)
(load all 2 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: 24% ●values known / values provided: 24%●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: 24% ●values known / values provided: 24%●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 + 1
[6,1,1,1]
=> 1000001110 => 0100000111 => 1111010100 => ? = 3 + 1
[4,1,1,1,1,1]
=> 1000111110 => 0100011111 => 1111110100 => ? = 5 + 1
[2,1,1,1,1,1,1,1]
=> 1011111110 => 0101111111 => 1111111100 => ? = 7 + 1
[10]
=> 10000000000 => 01000000000 => 10101010100 => ? = 1 + 1
[8,2]
=> 1000000100 => 0100000010 => 1001010100 => ? = 1 + 1
[8,1,1]
=> 10000000110 => 01000000011 => 11101010100 => ? = 2 + 1
[7,2,1]
=> 1000001010 => 0100000101 => 0011010100 => ? = 1 + 1
[7,1,1,1]
=> 10000001110 => 01000000111 => 00001010100 => ? = 3 + 1
[6,1,1,1,1]
=> 10000011110 => 01000001111 => 11111010100 => ? = 4 + 1
[5,1,1,1,1,1]
=> 10000111110 => 01000011111 => 00000010100 => ? = 5 + 1
[4,2,1,1,1,1]
=> 1001011110 => 0100101111 => 1111100100 => ? = 4 + 1
[4,1,1,1,1,1,1]
=> 10001111110 => 01000111111 => 11111110100 => ? = 6 + 1
[3,1,1,1,1,1,1,1]
=> 10011111110 => 01001111111 => 00000000100 => ? = 7 + 1
[2,2,1,1,1,1,1,1]
=> 1101111110 => 0110111111 => 1111111000 => ? = 6 + 1
[2,1,1,1,1,1,1,1,1]
=> 10111111110 => 01011111111 => 11111111100 => ? = 8 + 1
[1,1,1,1,1,1,1,1,1,1]
=> 11111111110 => 01111111111 => 00000000000 => ? = 10 + 1
[11]
=> 100000000000 => ? => ? => ? = 1 + 1
[10,1]
=> 100000000010 => ? => ? => ? = 1 + 1
[9,2]
=> 10000000100 => 01000000010 => ? => ? = 1 + 1
[9,1,1]
=> 100000000110 => ? => ? => ? = 2 + 1
[8,2,1]
=> 10000001010 => 01000000101 => ? => ? = 1 + 1
[8,1,1,1]
=> 100000001110 => ? => ? => ? = 3 + 1
[7,3,1]
=> 1000010010 => ? => ? => ? = 1 + 1
[7,2,2]
=> 1000001100 => ? => ? => ? = 2 + 1
[7,2,1,1]
=> 10000010110 => 01000001011 => ? => ? = 2 + 1
[7,1,1,1,1]
=> 100000011110 => ? => ? => ? = 4 + 1
[6,3,1,1]
=> 1000100110 => ? => ? => ? = 2 + 1
[6,2,2,1]
=> 1000011010 => ? => ? => ? = 2 + 1
[6,2,1,1,1]
=> 10000101110 => 01000010111 => ? => ? = 3 + 1
[6,1,1,1,1,1]
=> 100000111110 => 010000011111 => 111111010100 => ? = 5 + 1
[5,2,1,1,1,1]
=> 10001011110 => 01000101111 => ? => ? = 4 + 1
[5,1,1,1,1,1,1]
=> 100001111110 => ? => ? => ? = 6 + 1
[4,3,1,1,1,1]
=> 1010011110 => ? => ? => ? = 4 + 1
[4,2,2,1,1,1]
=> 1001101110 => ? => ? => ? = 3 + 1
[4,2,1,1,1,1,1]
=> 10010111110 => 01001011111 => ? => ? = 5 + 1
[4,1,1,1,1,1,1,1]
=> 100011111110 => ? => ? => ? = 7 + 1
[3,3,1,1,1,1,1]
=> 1100111110 => ? => ? => ? = 5 + 1
[3,2,2,1,1,1,1]
=> 1011011110 => ? => ? => ? = 4 + 1
[3,2,1,1,1,1,1,1]
=> 10101111110 => 01010111111 => ? => ? = 6 + 1
[3,1,1,1,1,1,1,1,1]
=> 100111111110 => ? => ? => ? = 8 + 1
[2,2,2,1,1,1,1,1]
=> 1110111110 => 0111011111 => ? => ? = 5 + 1
[2,2,1,1,1,1,1,1,1]
=> 11011111110 => 01101111111 => ? => ? = 7 + 1
[2,1,1,1,1,1,1,1,1,1]
=> 101111111110 => ? => ? => ? = 9 + 1
[1,1,1,1,1,1,1,1,1,1,1]
=> 111111111110 => ? => ? => ? = 11 + 1
[12]
=> 1000000000000 => ? => ? => ? = 1 + 1
[11,1]
=> 1000000000010 => ? => ? => ? = 1 + 1
[10,2]
=> 100000000100 => 010000000010 => 100101010100 => ? = 1 + 1
[10,1,1]
=> 1000000000110 => ? => ? => ? = 2 + 1
[9,3]
=> 10000001000 => ? => ? => ? = 1 + 1
Description
The length of the longest constant subword.
Matching statistic: St000983
Mp00044: Integer partitions —conjugate⟶ Integer partitions
Mp00095: Integer partitions —to binary word⟶ Binary words
Mp00268: Binary words —zeros to flag zeros⟶ Binary words
St000983: Binary words ⟶ ℤResult quality: 20% ●values known / values provided: 20%●distinct values known / distinct values provided: 50%
Mp00095: Integer partitions —to binary word⟶ Binary words
Mp00268: Binary words —zeros to flag zeros⟶ Binary words
St000983: Binary words ⟶ ℤResult quality: 20% ●values known / values provided: 20%●distinct values known / distinct values provided: 50%
Values
[1]
=> [1]
=> 10 => 01 => 2 = 1 + 1
[2]
=> [1,1]
=> 110 => 011 => 2 = 1 + 1
[1,1]
=> [2]
=> 100 => 101 => 3 = 2 + 1
[3]
=> [1,1,1]
=> 1110 => 0111 => 2 = 1 + 1
[2,1]
=> [2,1]
=> 1010 => 1001 => 2 = 1 + 1
[1,1,1]
=> [3]
=> 1000 => 0101 => 4 = 3 + 1
[4]
=> [1,1,1,1]
=> 11110 => 01111 => 2 = 1 + 1
[3,1]
=> [2,1,1]
=> 10110 => 10001 => 2 = 1 + 1
[2,2]
=> [2,2]
=> 1100 => 1011 => 3 = 2 + 1
[2,1,1]
=> [3,1]
=> 10010 => 01101 => 3 = 2 + 1
[1,1,1,1]
=> [4]
=> 10000 => 10101 => 5 = 4 + 1
[5]
=> [1,1,1,1,1]
=> 111110 => 011111 => 2 = 1 + 1
[4,1]
=> [2,1,1,1]
=> 101110 => 100001 => 2 = 1 + 1
[3,2]
=> [2,2,1]
=> 11010 => 10011 => 2 = 1 + 1
[3,1,1]
=> [3,1,1]
=> 100110 => 011101 => 3 = 2 + 1
[2,2,1]
=> [3,2]
=> 10100 => 01001 => 3 = 2 + 1
[2,1,1,1]
=> [4,1]
=> 100010 => 100101 => 4 = 3 + 1
[1,1,1,1,1]
=> [5]
=> 100000 => 010101 => 6 = 5 + 1
[6]
=> [1,1,1,1,1,1]
=> 1111110 => 0111111 => 2 = 1 + 1
[5,1]
=> [2,1,1,1,1]
=> 1011110 => 1000001 => 2 = 1 + 1
[4,2]
=> [2,2,1,1]
=> 110110 => 100011 => 2 = 1 + 1
[4,1,1]
=> [3,1,1,1]
=> 1001110 => 0111101 => 3 = 2 + 1
[3,3]
=> [2,2,2]
=> 11100 => 10111 => 3 = 2 + 1
[3,2,1]
=> [3,2,1]
=> 101010 => 011001 => 2 = 1 + 1
[3,1,1,1]
=> [4,1,1]
=> 1000110 => 1000101 => 4 = 3 + 1
[2,2,2]
=> [3,3]
=> 11000 => 01011 => 4 = 3 + 1
[2,2,1,1]
=> [4,2]
=> 100100 => 101101 => 3 = 2 + 1
[2,1,1,1,1]
=> [5,1]
=> 1000010 => 0110101 => 5 = 4 + 1
[1,1,1,1,1,1]
=> [6]
=> 1000000 => 1010101 => 7 = 6 + 1
[7]
=> [1,1,1,1,1,1,1]
=> 11111110 => 01111111 => 2 = 1 + 1
[6,1]
=> [2,1,1,1,1,1]
=> 10111110 => 10000001 => 2 = 1 + 1
[5,2]
=> [2,2,1,1,1]
=> 1101110 => 1000011 => 2 = 1 + 1
[5,1,1]
=> [3,1,1,1,1]
=> 10011110 => 01111101 => 3 = 2 + 1
[4,3]
=> [2,2,2,1]
=> 111010 => 100111 => 2 = 1 + 1
[4,2,1]
=> [3,2,1,1]
=> 1010110 => 0111001 => 2 = 1 + 1
[4,1,1,1]
=> [4,1,1,1]
=> 10001110 => 10000101 => 4 = 3 + 1
[3,3,1]
=> [3,2,2]
=> 101100 => 010001 => 3 = 2 + 1
[3,2,2]
=> [3,3,1]
=> 110010 => 011011 => 3 = 2 + 1
[3,2,1,1]
=> [4,2,1]
=> 1001010 => 1001101 => 3 = 2 + 1
[3,1,1,1,1]
=> [5,1,1]
=> 10000110 => 01110101 => 5 = 4 + 1
[2,2,2,1]
=> [4,3]
=> 101000 => 101001 => 4 = 3 + 1
[2,2,1,1,1]
=> [5,2]
=> 1000100 => 0100101 => 4 = 3 + 1
[2,1,1,1,1,1]
=> [6,1]
=> 10000010 => 10010101 => 6 = 5 + 1
[1,1,1,1,1,1,1]
=> [7]
=> 10000000 => 01010101 => 8 = 7 + 1
[8]
=> [1,1,1,1,1,1,1,1]
=> 111111110 => 011111111 => 2 = 1 + 1
[7,1]
=> [2,1,1,1,1,1,1]
=> 101111110 => 100000001 => 2 = 1 + 1
[6,2]
=> [2,2,1,1,1,1]
=> 11011110 => 10000011 => 2 = 1 + 1
[6,1,1]
=> [3,1,1,1,1,1]
=> 100111110 => 011111101 => 3 = 2 + 1
[5,3]
=> [2,2,2,1,1]
=> 1110110 => 1000111 => 2 = 1 + 1
[5,2,1]
=> [3,2,1,1,1]
=> 10101110 => 01111001 => 2 = 1 + 1
[9]
=> [1,1,1,1,1,1,1,1,1]
=> 1111111110 => 0111111111 => ? = 1 + 1
[8,1]
=> [2,1,1,1,1,1,1,1]
=> 1011111110 => 1000000001 => ? = 1 + 1
[7,1,1]
=> [3,1,1,1,1,1,1]
=> 1001111110 => 0111111101 => ? = 2 + 1
[6,1,1,1]
=> [4,1,1,1,1,1]
=> 1000111110 => 1000000101 => ? = 3 + 1
[5,1,1,1,1]
=> [5,1,1,1,1]
=> 1000011110 => 0111110101 => ? = 4 + 1
[4,1,1,1,1,1]
=> [6,1,1,1]
=> 1000001110 => 1000010101 => ? = 5 + 1
[3,1,1,1,1,1,1]
=> [7,1,1]
=> 1000000110 => 0111010101 => ? = 6 + 1
[2,1,1,1,1,1,1,1]
=> [8,1]
=> 1000000010 => 1001010101 => ? = 7 + 1
[1,1,1,1,1,1,1,1,1]
=> [9]
=> 1000000000 => 0101010101 => ? = 9 + 1
[10]
=> [1,1,1,1,1,1,1,1,1,1]
=> 11111111110 => 01111111111 => ? = 1 + 1
[9,1]
=> [2,1,1,1,1,1,1,1,1]
=> 10111111110 => 10000000001 => ? = 1 + 1
[8,2]
=> [2,2,1,1,1,1,1,1]
=> 1101111110 => 1000000011 => ? = 1 + 1
[8,1,1]
=> [3,1,1,1,1,1,1,1]
=> 10011111110 => 01111111101 => ? = 2 + 1
[7,2,1]
=> [3,2,1,1,1,1,1]
=> 1010111110 => 0111111001 => ? = 1 + 1
[7,1,1,1]
=> [4,1,1,1,1,1,1]
=> 10001111110 => 10000000101 => ? = 3 + 1
[6,2,1,1]
=> [4,2,1,1,1,1]
=> 1001011110 => 1000001101 => ? = 2 + 1
[6,1,1,1,1]
=> [5,1,1,1,1,1]
=> 10000111110 => 01111110101 => ? = 4 + 1
[5,2,1,1,1]
=> [5,2,1,1,1]
=> 1000101110 => 0111100101 => ? = 3 + 1
[5,1,1,1,1,1]
=> [6,1,1,1,1]
=> 10000011110 => 10000010101 => ? = 5 + 1
[4,2,1,1,1,1]
=> [6,2,1,1]
=> 1000010110 => 1000110101 => ? = 4 + 1
[4,1,1,1,1,1,1]
=> [7,1,1,1]
=> 10000001110 => 01111010101 => ? = 6 + 1
[3,2,1,1,1,1,1]
=> [7,2,1]
=> 1000001010 => 0110010101 => ? = 5 + 1
[3,1,1,1,1,1,1,1]
=> [8,1,1]
=> 10000000110 => 10001010101 => ? = 7 + 1
[2,2,1,1,1,1,1,1]
=> [8,2]
=> 1000000100 => 1011010101 => ? = 6 + 1
[2,1,1,1,1,1,1,1,1]
=> [9,1]
=> 10000000010 => 01101010101 => ? = 8 + 1
[1,1,1,1,1,1,1,1,1,1]
=> [10]
=> 10000000000 => 10101010101 => ? = 10 + 1
[11]
=> [1,1,1,1,1,1,1,1,1,1,1]
=> 111111111110 => ? => ? = 1 + 1
[10,1]
=> [2,1,1,1,1,1,1,1,1,1]
=> 101111111110 => ? => ? = 1 + 1
[9,2]
=> [2,2,1,1,1,1,1,1,1]
=> 11011111110 => 10000000011 => ? = 1 + 1
[9,1,1]
=> [3,1,1,1,1,1,1,1,1]
=> 100111111110 => ? => ? = 2 + 1
[8,3]
=> [2,2,2,1,1,1,1,1]
=> 1110111110 => 1000000111 => ? = 1 + 1
[8,2,1]
=> [3,2,1,1,1,1,1,1]
=> 10101111110 => 01111111001 => ? = 1 + 1
[8,1,1,1]
=> [4,1,1,1,1,1,1,1]
=> 100011111110 => ? => ? = 3 + 1
[7,3,1]
=> [3,2,2,1,1,1,1]
=> 1011011110 => ? => ? = 1 + 1
[7,2,2]
=> [3,3,1,1,1,1,1]
=> 1100111110 => ? => ? = 2 + 1
[7,2,1,1]
=> [4,2,1,1,1,1,1]
=> 10010111110 => 10000001101 => ? = 2 + 1
[7,1,1,1,1]
=> [5,1,1,1,1,1,1]
=> 100001111110 => ? => ? = 4 + 1
[6,3,1,1]
=> [4,2,2,1,1,1]
=> 1001101110 => ? => ? = 2 + 1
[6,2,2,1]
=> [4,3,1,1,1,1]
=> 1010011110 => ? => ? = 2 + 1
[6,2,1,1,1]
=> [5,2,1,1,1,1]
=> 10001011110 => 01111100101 => ? = 3 + 1
[6,1,1,1,1,1]
=> [6,1,1,1,1,1]
=> 100000111110 => ? => ? = 5 + 1
[5,3,1,1,1]
=> [5,2,2,1,1]
=> 1000110110 => 0111000101 => ? = 3 + 1
[5,2,2,1,1]
=> [5,3,1,1,1]
=> 1001001110 => 0111101101 => ? = 2 + 1
[5,2,1,1,1,1]
=> [6,2,1,1,1]
=> 10000101110 => 10000110101 => ? = 4 + 1
[5,1,1,1,1,1,1]
=> [7,1,1,1,1]
=> 100000011110 => ? => ? = 6 + 1
[4,3,1,1,1,1]
=> [6,2,2,1]
=> 1000011010 => ? => ? = 4 + 1
[4,2,2,1,1,1]
=> [6,3,1,1]
=> 1000100110 => ? => ? = 3 + 1
[4,2,1,1,1,1,1]
=> [7,2,1,1]
=> 10000010110 => 01110010101 => ? = 5 + 1
[4,1,1,1,1,1,1,1]
=> [8,1,1,1]
=> 100000001110 => ? => ? = 7 + 1
[3,3,1,1,1,1,1]
=> [7,2,2]
=> 1000001100 => ? => ? = 5 + 1
Description
The length of the longest alternating subword.
This is the length of the longest consecutive subword of the form $010...$ or of the form $101...$.
Matching statistic: St000381
(load all 4 compositions to match this statistic)
(load all 4 compositions to match this statistic)
Mp00042: Integer partitions —initial tableau⟶ Standard tableaux
Mp00207: Standard tableaux —horizontal strip sizes⟶ Integer compositions
Mp00133: Integer compositions —delta morphism⟶ Integer compositions
St000381: Integer compositions ⟶ ℤResult quality: 13% ●values known / values provided: 13%●distinct values known / distinct values provided: 56%
Mp00207: Standard tableaux —horizontal strip sizes⟶ Integer compositions
Mp00133: Integer compositions —delta morphism⟶ Integer compositions
St000381: Integer compositions ⟶ ℤResult quality: 13% ●values known / values provided: 13%●distinct values known / distinct values provided: 56%
Values
[1]
=> [[1]]
=> [1] => [1] => 1
[2]
=> [[1,2]]
=> [2] => [1] => 1
[1,1]
=> [[1],[2]]
=> [1,1] => [2] => 2
[3]
=> [[1,2,3]]
=> [3] => [1] => 1
[2,1]
=> [[1,2],[3]]
=> [2,1] => [1,1] => 1
[1,1,1]
=> [[1],[2],[3]]
=> [1,1,1] => [3] => 3
[4]
=> [[1,2,3,4]]
=> [4] => [1] => 1
[3,1]
=> [[1,2,3],[4]]
=> [3,1] => [1,1] => 1
[2,2]
=> [[1,2],[3,4]]
=> [2,2] => [2] => 2
[2,1,1]
=> [[1,2],[3],[4]]
=> [2,1,1] => [1,2] => 2
[1,1,1,1]
=> [[1],[2],[3],[4]]
=> [1,1,1,1] => [4] => 4
[5]
=> [[1,2,3,4,5]]
=> [5] => [1] => 1
[4,1]
=> [[1,2,3,4],[5]]
=> [4,1] => [1,1] => 1
[3,2]
=> [[1,2,3],[4,5]]
=> [3,2] => [1,1] => 1
[3,1,1]
=> [[1,2,3],[4],[5]]
=> [3,1,1] => [1,2] => 2
[2,2,1]
=> [[1,2],[3,4],[5]]
=> [2,2,1] => [2,1] => 2
[2,1,1,1]
=> [[1,2],[3],[4],[5]]
=> [2,1,1,1] => [1,3] => 3
[1,1,1,1,1]
=> [[1],[2],[3],[4],[5]]
=> [1,1,1,1,1] => [5] => 5
[6]
=> [[1,2,3,4,5,6]]
=> [6] => [1] => 1
[5,1]
=> [[1,2,3,4,5],[6]]
=> [5,1] => [1,1] => 1
[4,2]
=> [[1,2,3,4],[5,6]]
=> [4,2] => [1,1] => 1
[4,1,1]
=> [[1,2,3,4],[5],[6]]
=> [4,1,1] => [1,2] => 2
[3,3]
=> [[1,2,3],[4,5,6]]
=> [3,3] => [2] => 2
[3,2,1]
=> [[1,2,3],[4,5],[6]]
=> [3,2,1] => [1,1,1] => 1
[3,1,1,1]
=> [[1,2,3],[4],[5],[6]]
=> [3,1,1,1] => [1,3] => 3
[2,2,2]
=> [[1,2],[3,4],[5,6]]
=> [2,2,2] => [3] => 3
[2,2,1,1]
=> [[1,2],[3,4],[5],[6]]
=> [2,2,1,1] => [2,2] => 2
[2,1,1,1,1]
=> [[1,2],[3],[4],[5],[6]]
=> [2,1,1,1,1] => [1,4] => 4
[1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6]]
=> [1,1,1,1,1,1] => [6] => 6
[7]
=> [[1,2,3,4,5,6,7]]
=> [7] => [1] => 1
[6,1]
=> [[1,2,3,4,5,6],[7]]
=> [6,1] => [1,1] => 1
[5,2]
=> [[1,2,3,4,5],[6,7]]
=> [5,2] => [1,1] => 1
[5,1,1]
=> [[1,2,3,4,5],[6],[7]]
=> [5,1,1] => [1,2] => 2
[4,3]
=> [[1,2,3,4],[5,6,7]]
=> [4,3] => [1,1] => 1
[4,2,1]
=> [[1,2,3,4],[5,6],[7]]
=> [4,2,1] => [1,1,1] => 1
[4,1,1,1]
=> [[1,2,3,4],[5],[6],[7]]
=> [4,1,1,1] => [1,3] => 3
[3,3,1]
=> [[1,2,3],[4,5,6],[7]]
=> [3,3,1] => [2,1] => 2
[3,2,2]
=> [[1,2,3],[4,5],[6,7]]
=> [3,2,2] => [1,2] => 2
[3,2,1,1]
=> [[1,2,3],[4,5],[6],[7]]
=> [3,2,1,1] => [1,1,2] => 2
[3,1,1,1,1]
=> [[1,2,3],[4],[5],[6],[7]]
=> [3,1,1,1,1] => [1,4] => 4
[2,2,2,1]
=> [[1,2],[3,4],[5,6],[7]]
=> [2,2,2,1] => [3,1] => 3
[2,2,1,1,1]
=> [[1,2],[3,4],[5],[6],[7]]
=> [2,2,1,1,1] => [2,3] => 3
[2,1,1,1,1,1]
=> [[1,2],[3],[4],[5],[6],[7]]
=> [2,1,1,1,1,1] => [1,5] => 5
[1,1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6],[7]]
=> [1,1,1,1,1,1,1] => [7] => 7
[8]
=> [[1,2,3,4,5,6,7,8]]
=> [8] => [1] => 1
[7,1]
=> [[1,2,3,4,5,6,7],[8]]
=> [7,1] => [1,1] => 1
[6,2]
=> [[1,2,3,4,5,6],[7,8]]
=> [6,2] => [1,1] => 1
[6,1,1]
=> [[1,2,3,4,5,6],[7],[8]]
=> [6,1,1] => [1,2] => 2
[5,3]
=> [[1,2,3,4,5],[6,7,8]]
=> [5,3] => [1,1] => 1
[5,2,1]
=> [[1,2,3,4,5],[6,7],[8]]
=> [5,2,1] => [1,1,1] => 1
[1,1,1,1,1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6],[7],[8],[9],[10]]
=> [1,1,1,1,1,1,1,1,1,1] => [10] => ? = 10
[11]
=> [[1,2,3,4,5,6,7,8,9,10,11]]
=> ? => ? => ? = 1
[10,1]
=> [[1,2,3,4,5,6,7,8,9,10],[11]]
=> ? => ? => ? = 1
[9,2]
=> [[1,2,3,4,5,6,7,8,9],[10,11]]
=> ? => ? => ? = 1
[9,1,1]
=> [[1,2,3,4,5,6,7,8,9],[10],[11]]
=> ? => ? => ? = 2
[8,3]
=> [[1,2,3,4,5,6,7,8],[9,10,11]]
=> ? => ? => ? = 1
[8,2,1]
=> [[1,2,3,4,5,6,7,8],[9,10],[11]]
=> ? => ? => ? = 1
[8,1,1,1]
=> [[1,2,3,4,5,6,7,8],[9],[10],[11]]
=> ? => ? => ? = 3
[7,4]
=> [[1,2,3,4,5,6,7],[8,9,10,11]]
=> ? => ? => ? = 1
[7,3,1]
=> [[1,2,3,4,5,6,7],[8,9,10],[11]]
=> ? => ? => ? = 1
[7,2,2]
=> [[1,2,3,4,5,6,7],[8,9],[10,11]]
=> ? => ? => ? = 2
[7,2,1,1]
=> [[1,2,3,4,5,6,7],[8,9],[10],[11]]
=> ? => ? => ? = 2
[7,1,1,1,1]
=> [[1,2,3,4,5,6,7],[8],[9],[10],[11]]
=> ? => ? => ? = 4
[6,5]
=> [[1,2,3,4,5,6],[7,8,9,10,11]]
=> ? => ? => ? = 1
[6,4,1]
=> [[1,2,3,4,5,6],[7,8,9,10],[11]]
=> ? => ? => ? = 1
[6,3,2]
=> [[1,2,3,4,5,6],[7,8,9],[10,11]]
=> ? => ? => ? = 1
[6,3,1,1]
=> [[1,2,3,4,5,6],[7,8,9],[10],[11]]
=> ? => ? => ? = 2
[6,2,2,1]
=> [[1,2,3,4,5,6],[7,8],[9,10],[11]]
=> ? => ? => ? = 2
[6,2,1,1,1]
=> [[1,2,3,4,5,6],[7,8],[9],[10],[11]]
=> ? => ? => ? = 3
[6,1,1,1,1,1]
=> [[1,2,3,4,5,6],[7],[8],[9],[10],[11]]
=> ? => ? => ? = 5
[5,5,1]
=> [[1,2,3,4,5],[6,7,8,9,10],[11]]
=> ? => ? => ? = 2
[5,4,2]
=> [[1,2,3,4,5],[6,7,8,9],[10,11]]
=> [5,4,2] => ? => ? = 1
[5,4,1,1]
=> [[1,2,3,4,5],[6,7,8,9],[10],[11]]
=> [5,4,1,1] => ? => ? = 2
[5,3,3]
=> [[1,2,3,4,5],[6,7,8],[9,10,11]]
=> [5,3,3] => ? => ? = 2
[5,3,2,1]
=> [[1,2,3,4,5],[6,7,8],[9,10],[11]]
=> [5,3,2,1] => ? => ? = 1
[5,3,1,1,1]
=> [[1,2,3,4,5],[6,7,8],[9],[10],[11]]
=> [5,3,1,1,1] => ? => ? = 3
[5,2,2,2]
=> [[1,2,3,4,5],[6,7],[8,9],[10,11]]
=> [5,2,2,2] => ? => ? = 3
[5,2,2,1,1]
=> [[1,2,3,4,5],[6,7],[8,9],[10],[11]]
=> [5,2,2,1,1] => ? => ? = 2
[5,2,1,1,1,1]
=> [[1,2,3,4,5],[6,7],[8],[9],[10],[11]]
=> ? => ? => ? = 4
[5,1,1,1,1,1,1]
=> [[1,2,3,4,5],[6],[7],[8],[9],[10],[11]]
=> ? => ? => ? = 6
[4,4,2,1]
=> [[1,2,3,4],[5,6,7,8],[9,10],[11]]
=> [4,4,2,1] => ? => ? = 2
[4,4,1,1,1]
=> [[1,2,3,4],[5,6,7,8],[9],[10],[11]]
=> [4,4,1,1,1] => ? => ? = 3
[4,3,3,1]
=> [[1,2,3,4],[5,6,7],[8,9,10],[11]]
=> [4,3,3,1] => ? => ? = 2
[4,3,2,2]
=> [[1,2,3,4],[5,6,7],[8,9],[10,11]]
=> [4,3,2,2] => ? => ? = 2
[4,3,2,1,1]
=> [[1,2,3,4],[5,6,7],[8,9],[10],[11]]
=> [4,3,2,1,1] => ? => ? = 2
[4,3,1,1,1,1]
=> [[1,2,3,4],[5,6,7],[8],[9],[10],[11]]
=> ? => ? => ? = 4
[4,2,2,2,1]
=> [[1,2,3,4],[5,6],[7,8],[9,10],[11]]
=> [4,2,2,2,1] => ? => ? = 3
[4,2,2,1,1,1]
=> [[1,2,3,4],[5,6],[7,8],[9],[10],[11]]
=> ? => ? => ? = 3
[4,2,1,1,1,1,1]
=> [[1,2,3,4],[5,6],[7],[8],[9],[10],[11]]
=> ? => ? => ? = 5
[4,1,1,1,1,1,1,1]
=> [[1,2,3,4],[5],[6],[7],[8],[9],[10],[11]]
=> ? => ? => ? = 7
[3,3,3,1,1]
=> [[1,2,3],[4,5,6],[7,8,9],[10],[11]]
=> [3,3,3,1,1] => ? => ? = 3
[3,3,2,2,1]
=> [[1,2,3],[4,5,6],[7,8],[9,10],[11]]
=> [3,3,2,2,1] => ? => ? = 2
[3,3,2,1,1,1]
=> [[1,2,3],[4,5,6],[7,8],[9],[10],[11]]
=> ? => ? => ? = 3
[3,3,1,1,1,1,1]
=> [[1,2,3],[4,5,6],[7],[8],[9],[10],[11]]
=> ? => ? => ? = 5
[3,2,2,2,2]
=> [[1,2,3],[4,5],[6,7],[8,9],[10,11]]
=> ? => ? => ? = 4
[3,2,2,2,1,1]
=> [[1,2,3],[4,5],[6,7],[8,9],[10],[11]]
=> ? => ? => ? = 3
[3,2,2,1,1,1,1]
=> [[1,2,3],[4,5],[6,7],[8],[9],[10],[11]]
=> ? => ? => ? = 4
[3,2,1,1,1,1,1,1]
=> [[1,2,3],[4,5],[6],[7],[8],[9],[10],[11]]
=> ? => ? => ? = 6
[3,1,1,1,1,1,1,1,1]
=> [[1,2,3],[4],[5],[6],[7],[8],[9],[10],[11]]
=> ? => ? => ? = 8
[2,2,2,2,2,1]
=> [[1,2],[3,4],[5,6],[7,8],[9,10],[11]]
=> ? => ? => ? = 5
Description
The largest part of an integer composition.
Matching statistic: St000969
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00227: Dyck paths —Delest-Viennot-inverse⟶ Dyck paths
Mp00132: Dyck paths —switch returns and last double rise⟶ Dyck paths
St000969: Dyck paths ⟶ ℤResult quality: 11% ●values known / values provided: 11%●distinct values known / distinct values provided: 44%
Mp00227: Dyck paths —Delest-Viennot-inverse⟶ Dyck paths
Mp00132: Dyck paths —switch returns and last double rise⟶ Dyck paths
St000969: Dyck paths ⟶ ℤResult quality: 11% ●values known / values provided: 11%●distinct values known / distinct values provided: 44%
Values
[1]
=> [1,0]
=> [1,0]
=> [1,0]
=> 2 = 1 + 1
[2]
=> [1,0,1,0]
=> [1,1,0,0]
=> [1,1,0,0]
=> 2 = 1 + 1
[1,1]
=> [1,1,0,0]
=> [1,0,1,0]
=> [1,0,1,0]
=> 3 = 2 + 1
[3]
=> [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> [1,1,1,0,0,0]
=> 2 = 1 + 1
[2,1]
=> [1,0,1,1,0,0]
=> [1,1,0,0,1,0]
=> [1,1,0,0,1,0]
=> 2 = 1 + 1
[1,1,1]
=> [1,1,0,1,0,0]
=> [1,0,1,0,1,0]
=> [1,0,1,0,1,0]
=> 4 = 3 + 1
[4]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> 2 = 1 + 1
[3,1]
=> [1,0,1,0,1,1,0,0]
=> [1,1,1,0,0,0,1,0]
=> [1,1,1,0,0,0,1,0]
=> 2 = 1 + 1
[2,2]
=> [1,1,1,0,0,0]
=> [1,1,0,1,0,0]
=> [1,0,1,1,0,0]
=> 3 = 2 + 1
[2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0]
=> [1,1,0,0,1,0,1,0]
=> 3 = 2 + 1
[1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0]
=> 5 = 4 + 1
[5]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> 2 = 1 + 1
[4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> 2 = 1 + 1
[3,2]
=> [1,0,1,1,1,0,0,0]
=> [1,1,1,0,0,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> 2 = 1 + 1
[3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> 3 = 2 + 1
[2,2,1]
=> [1,1,1,0,0,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> [1,0,1,1,0,0,1,0]
=> 3 = 2 + 1
[2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> 4 = 3 + 1
[1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> 6 = 5 + 1
[6]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0]
=> 2 = 1 + 1
[5,1]
=> [1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> 2 = 1 + 1
[4,2]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> 2 = 1 + 1
[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]
=> [1,1,1,1,0,0,0,0,1,0,1,0]
=> 3 = 2 + 1
[3,3]
=> [1,1,1,0,1,0,0,0]
=> [1,1,1,0,1,0,0,0]
=> [1,0,1,1,1,0,0,0]
=> 3 = 2 + 1
[3,2,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,1,1,0,0,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> 2 = 1 + 1
[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]
=> [1,1,1,0,0,0,1,0,1,0,1,0]
=> 4 = 3 + 1
[2,2,2]
=> [1,1,1,1,0,0,0,0]
=> [1,1,0,1,0,1,0,0]
=> [1,0,1,0,1,1,0,0]
=> 4 = 3 + 1
[2,2,1,1]
=> [1,1,1,0,0,1,0,1,0,0]
=> [1,1,0,1,0,0,1,0,1,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> 3 = 2 + 1
[2,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0,1,0,1,0]
=> [1,1,0,0,1,0,1,0,1,0,1,0]
=> 5 = 4 + 1
[1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> 7 = 6 + 1
[7]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> 2 = 1 + 1
[6,1]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> 2 = 1 + 1
[5,2]
=> [1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,1,1,0,0,0,0,1,0,0]
=> [1,1,1,1,0,0,0,0,1,1,0,0]
=> 2 = 1 + 1
[5,1,1]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,1,1,0,0,0,0,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0,1,0,1,0]
=> 3 = 2 + 1
[4,3]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> 2 = 1 + 1
[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]
=> [1,1,1,0,0,0,1,1,0,0,1,0]
=> 2 = 1 + 1
[4,1,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,1,1,0,0,0,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0,1,0,1,0,1,0]
=> 4 = 3 + 1
[3,3,1]
=> [1,1,1,0,1,0,0,1,0,0]
=> [1,1,1,0,1,0,0,0,1,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> 3 = 2 + 1
[3,2,2]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> 3 = 2 + 1
[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]
=> [1,1,0,0,1,1,0,0,1,0,1,0]
=> 3 = 2 + 1
[3,1,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,1,1,0,0,0,1,0,1,0,1,0,1,0]
=> [1,1,1,0,0,0,1,0,1,0,1,0,1,0]
=> 5 = 4 + 1
[2,2,2,1]
=> [1,1,1,1,0,0,0,1,0,0]
=> [1,1,0,1,0,1,0,0,1,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> 4 = 3 + 1
[2,2,1,1,1]
=> [1,1,1,0,0,1,0,1,0,1,0,0]
=> [1,1,0,1,0,0,1,0,1,0,1,0]
=> [1,0,1,1,0,0,1,0,1,0,1,0]
=> 4 = 3 + 1
[2,1,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,0,0,1,0,1,0,1,0,1,0,1,0]
=> 6 = 5 + 1
[1,1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> 8 = 7 + 1
[8]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 1 + 1
[7,1]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> ? = 1 + 1
[6,2]
=> [1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,1,1,1,0,0,0,0,0,1,0,0]
=> [1,1,1,1,1,0,0,0,0,0,1,1,0,0]
=> 2 = 1 + 1
[6,1,1]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0,1,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0,1,0]
=> ? = 2 + 1
[5,3]
=> [1,0,1,0,1,1,1,0,1,0,0,0]
=> [1,1,1,1,1,0,0,0,1,0,0,0]
=> [1,1,1,0,0,0,1,1,1,0,0,0]
=> 2 = 1 + 1
[5,2,1]
=> [1,0,1,0,1,0,1,1,1,0,0,1,0,0]
=> [1,1,1,1,1,0,0,0,0,1,0,0,1,0]
=> [1,1,1,1,0,0,0,0,1,1,0,0,1,0]
=> 2 = 1 + 1
[5,1,1,1]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,1,1,1,0,0,0,0,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0,1,0,1,0,1,0]
=> ? = 3 + 1
[4,4]
=> [1,1,1,0,1,0,1,0,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> 3 = 2 + 1
[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]
=> [1,1,0,0,1,1,1,0,0,0,1,0]
=> 2 = 1 + 1
[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]
=> [1,1,1,0,0,0,1,0,1,1,0,0]
=> 3 = 2 + 1
[4,1,1,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,1,1,1,0,0,0,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0,1,0,1,0,1,0,1,0]
=> ? = 4 + 1
[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]
=> [1,1,1,0,0,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 5 + 1
[2,1,1,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 6 + 1
[1,1,1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 8 + 1
[9]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0]
=> [1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0]
=> ? = 1 + 1
[8,1]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0]
=> ? = 1 + 1
[7,2]
=> [1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,1,0,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,1,0,0]
=> ? = 1 + 1
[7,1,1]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0,1,0]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0,1,0]
=> ? = 2 + 1
[6,2,1]
=> [1,0,1,0,1,0,1,0,1,1,1,0,0,1,0,0]
=> [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
[6,1,1,1]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0,1,0,1,0]
=> ? = 3 + 1
[5,2,1,1]
=> [1,0,1,0,1,0,1,1,1,0,0,1,0,1,0,0]
=> [1,1,1,1,1,0,0,0,0,1,0,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0,1,1,0,0,1,0,1,0]
=> ? = 2 + 1
[5,1,1,1,1]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,1,1,1,1,0,0,0,0,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0,1,0,1,0,1,0,1,0]
=> ? = 4 + 1
[4,2,1,1,1]
=> [1,0,1,0,1,1,1,0,0,1,0,1,0,1,0,0]
=> [1,1,1,1,0,0,0,1,0,0,1,0,1,0,1,0]
=> [1,1,1,0,0,0,1,1,0,0,1,0,1,0,1,0]
=> ? = 3 + 1
[4,1,1,1,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,1,1,0,0,0,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 5 + 1
[3,2,1,1,1,1]
=> [1,0,1,1,1,0,0,1,0,1,0,1,0,1,0,0]
=> [1,1,1,0,0,1,0,0,1,0,1,0,1,0,1,0]
=> [1,1,0,0,1,1,0,0,1,0,1,0,1,0,1,0]
=> ? = 4 + 1
[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]
=> [1,1,1,0,0,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 6 + 1
[2,2,1,1,1,1,1]
=> [1,1,1,0,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,0,1,0,0,1,0,1,0,1,0,1,0,1,0]
=> [1,0,1,1,0,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 5 + 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]
=> [1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 7 + 1
[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]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 9 + 1
[10]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0]
=> [1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0]
=> ? = 1 + 1
[9,1]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,1,0]
=> [1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,1,0]
=> ? = 1 + 1
[8,2]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0,0]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,1,0,0]
=> ? = 1 + 1
[8,1,1]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0,1,0]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0,1,0]
=> ? = 2 + 1
[7,3]
=> [1,0,1,0,1,0,1,0,1,1,1,0,1,0,0,0]
=> [1,1,1,1,1,1,1,0,0,0,0,0,1,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0,1,1,1,0,0,0]
=> ? = 1 + 1
[7,2,1]
=> [1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,1,0,0]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,1,0,0,1,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,1,0,0,1,0]
=> ? = 1 + 1
[7,1,1,1]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0,1,0,1,0]
=> ? = 3 + 1
[6,3,1]
=> [1,0,1,0,1,0,1,1,1,0,1,0,0,1,0,0]
=> [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
[6,2,2]
=> [1,0,1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,1,1,0,0,0,0,0,1,0,1,0,0]
=> [1,1,1,1,1,0,0,0,0,0,1,0,1,1,0,0]
=> ? = 2 + 1
[6,2,1,1]
=> [1,0,1,0,1,0,1,0,1,1,1,0,0,1,0,1,0,0]
=> [1,1,1,1,1,1,0,0,0,0,0,1,0,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0,1,1,0,0,1,0,1,0]
=> ? = 2 + 1
[6,1,1,1,1]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0,1,0,1,0,1,0]
=> ? = 4 + 1
[5,3,1,1]
=> [1,0,1,0,1,1,1,0,1,0,0,1,0,1,0,0]
=> [1,1,1,1,1,0,0,0,1,0,0,0,1,0,1,0]
=> [1,1,1,0,0,0,1,1,1,0,0,0,1,0,1,0]
=> ? = 2 + 1
[5,2,2,1]
=> [1,0,1,0,1,0,1,1,1,1,0,0,0,1,0,0]
=> [1,1,1,1,1,0,0,0,0,1,0,1,0,0,1,0]
=> [1,1,1,1,0,0,0,0,1,0,1,1,0,0,1,0]
=> ? = 2 + 1
[5,2,1,1,1]
=> [1,0,1,0,1,0,1,1,1,0,0,1,0,1,0,1,0,0]
=> [1,1,1,1,1,0,0,0,0,1,0,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0,1,1,0,0,1,0,1,0,1,0]
=> ? = 3 + 1
[5,1,1,1,1,1]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,1,1,1,0,0,0,0,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 5 + 1
[4,3,1,1,1]
=> [1,0,1,1,1,0,1,0,0,1,0,1,0,1,0,0]
=> [1,1,1,1,0,0,1,0,0,0,1,0,1,0,1,0]
=> [1,1,0,0,1,1,1,0,0,0,1,0,1,0,1,0]
=> ? = 3 + 1
[4,2,2,1,1]
=> [1,0,1,0,1,1,1,1,0,0,0,1,0,1,0,0]
=> [1,1,1,1,0,0,0,1,0,1,0,0,1,0,1,0]
=> [1,1,1,0,0,0,1,0,1,1,0,0,1,0,1,0]
=> ? = 2 + 1
[4,2,1,1,1,1]
=> [1,0,1,0,1,1,1,0,0,1,0,1,0,1,0,1,0,0]
=> [1,1,1,1,0,0,0,1,0,0,1,0,1,0,1,0,1,0]
=> [1,1,1,0,0,0,1,1,0,0,1,0,1,0,1,0,1,0]
=> ? = 4 + 1
[4,1,1,1,1,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,1,1,0,0,0,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 6 + 1
[3,3,1,1,1,1]
=> [1,1,1,0,1,0,0,1,0,1,0,1,0,1,0,0]
=> [1,1,1,0,1,0,0,0,1,0,1,0,1,0,1,0]
=> [1,0,1,1,1,0,0,0,1,0,1,0,1,0,1,0]
=> ? = 4 + 1
[3,2,2,1,1,1]
=> [1,0,1,1,1,1,0,0,0,1,0,1,0,1,0,0]
=> [1,1,1,0,0,1,0,1,0,0,1,0,1,0,1,0]
=> [1,1,0,0,1,0,1,1,0,0,1,0,1,0,1,0]
=> ? = 3 + 1
[3,2,1,1,1,1,1]
=> [1,0,1,1,1,0,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,1,0,0,1,0,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,0,0,1,1,0,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 5 + 1
[3,1,1,1,1,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,1,0,0,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,0,0,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 7 + 1
[2,2,2,1,1,1,1]
=> [1,1,1,1,0,0,0,1,0,1,0,1,0,1,0,0]
=> [1,1,0,1,0,1,0,0,1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,1,0,0,1,0,1,0,1,0,1,0]
=> ? = 4 + 1
[2,2,1,1,1,1,1,1]
=> [1,1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,0,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,0,1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 6 + 1
[2,1,1,1,1,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 8 + 1
[1,1,1,1,1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 10 + 1
Description
We 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 $c_{n-1}$. Then we calculate the global dimension of that CNakayama algebra.
Matching statistic: St000757
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Mp00042: Integer partitions —initial tableau⟶ Standard tableaux
Mp00207: Standard tableaux —horizontal strip sizes⟶ Integer compositions
St000757: Integer compositions ⟶ ℤResult quality: 8% ●values known / values provided: 8%●distinct values known / distinct values provided: 56%
Mp00207: Standard tableaux —horizontal strip sizes⟶ Integer compositions
St000757: Integer compositions ⟶ ℤResult quality: 8% ●values known / values provided: 8%●distinct values known / distinct values provided: 56%
Values
[1]
=> [[1]]
=> [1] => 1
[2]
=> [[1,2]]
=> [2] => 1
[1,1]
=> [[1],[2]]
=> [1,1] => 2
[3]
=> [[1,2,3]]
=> [3] => 1
[2,1]
=> [[1,2],[3]]
=> [2,1] => 1
[1,1,1]
=> [[1],[2],[3]]
=> [1,1,1] => 3
[4]
=> [[1,2,3,4]]
=> [4] => 1
[3,1]
=> [[1,2,3],[4]]
=> [3,1] => 1
[2,2]
=> [[1,2],[3,4]]
=> [2,2] => 2
[2,1,1]
=> [[1,2],[3],[4]]
=> [2,1,1] => 2
[1,1,1,1]
=> [[1],[2],[3],[4]]
=> [1,1,1,1] => 4
[5]
=> [[1,2,3,4,5]]
=> [5] => 1
[4,1]
=> [[1,2,3,4],[5]]
=> [4,1] => 1
[3,2]
=> [[1,2,3],[4,5]]
=> [3,2] => 1
[3,1,1]
=> [[1,2,3],[4],[5]]
=> [3,1,1] => 2
[2,2,1]
=> [[1,2],[3,4],[5]]
=> [2,2,1] => 2
[2,1,1,1]
=> [[1,2],[3],[4],[5]]
=> [2,1,1,1] => 3
[1,1,1,1,1]
=> [[1],[2],[3],[4],[5]]
=> [1,1,1,1,1] => 5
[6]
=> [[1,2,3,4,5,6]]
=> [6] => 1
[5,1]
=> [[1,2,3,4,5],[6]]
=> [5,1] => 1
[4,2]
=> [[1,2,3,4],[5,6]]
=> [4,2] => 1
[4,1,1]
=> [[1,2,3,4],[5],[6]]
=> [4,1,1] => 2
[3,3]
=> [[1,2,3],[4,5,6]]
=> [3,3] => 2
[3,2,1]
=> [[1,2,3],[4,5],[6]]
=> [3,2,1] => 1
[3,1,1,1]
=> [[1,2,3],[4],[5],[6]]
=> [3,1,1,1] => 3
[2,2,2]
=> [[1,2],[3,4],[5,6]]
=> [2,2,2] => 3
[2,2,1,1]
=> [[1,2],[3,4],[5],[6]]
=> [2,2,1,1] => 2
[2,1,1,1,1]
=> [[1,2],[3],[4],[5],[6]]
=> [2,1,1,1,1] => 4
[1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6]]
=> [1,1,1,1,1,1] => 6
[7]
=> [[1,2,3,4,5,6,7]]
=> [7] => 1
[6,1]
=> [[1,2,3,4,5,6],[7]]
=> [6,1] => 1
[5,2]
=> [[1,2,3,4,5],[6,7]]
=> [5,2] => 1
[5,1,1]
=> [[1,2,3,4,5],[6],[7]]
=> [5,1,1] => 2
[4,3]
=> [[1,2,3,4],[5,6,7]]
=> [4,3] => 1
[4,2,1]
=> [[1,2,3,4],[5,6],[7]]
=> [4,2,1] => 1
[4,1,1,1]
=> [[1,2,3,4],[5],[6],[7]]
=> [4,1,1,1] => 3
[3,3,1]
=> [[1,2,3],[4,5,6],[7]]
=> [3,3,1] => 2
[3,2,2]
=> [[1,2,3],[4,5],[6,7]]
=> [3,2,2] => 2
[3,2,1,1]
=> [[1,2,3],[4,5],[6],[7]]
=> [3,2,1,1] => 2
[3,1,1,1,1]
=> [[1,2,3],[4],[5],[6],[7]]
=> [3,1,1,1,1] => 4
[2,2,2,1]
=> [[1,2],[3,4],[5,6],[7]]
=> [2,2,2,1] => 3
[2,2,1,1,1]
=> [[1,2],[3,4],[5],[6],[7]]
=> [2,2,1,1,1] => 3
[2,1,1,1,1,1]
=> [[1,2],[3],[4],[5],[6],[7]]
=> [2,1,1,1,1,1] => 5
[1,1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6],[7]]
=> [1,1,1,1,1,1,1] => 7
[8]
=> [[1,2,3,4,5,6,7,8]]
=> [8] => 1
[7,1]
=> [[1,2,3,4,5,6,7],[8]]
=> [7,1] => 1
[6,2]
=> [[1,2,3,4,5,6],[7,8]]
=> [6,2] => 1
[6,1,1]
=> [[1,2,3,4,5,6],[7],[8]]
=> [6,1,1] => 2
[5,3]
=> [[1,2,3,4,5],[6,7,8]]
=> [5,3] => 1
[5,2,1]
=> [[1,2,3,4,5],[6,7],[8]]
=> [5,2,1] => 1
[10]
=> [[1,2,3,4,5,6,7,8,9,10]]
=> [10] => ? = 1
[9,1]
=> [[1,2,3,4,5,6,7,8,9],[10]]
=> [9,1] => ? = 1
[8,2]
=> [[1,2,3,4,5,6,7,8],[9,10]]
=> [8,2] => ? = 1
[8,1,1]
=> [[1,2,3,4,5,6,7,8],[9],[10]]
=> [8,1,1] => ? = 2
[7,3]
=> [[1,2,3,4,5,6,7],[8,9,10]]
=> [7,3] => ? = 1
[7,2,1]
=> [[1,2,3,4,5,6,7],[8,9],[10]]
=> [7,2,1] => ? = 1
[7,1,1,1]
=> [[1,2,3,4,5,6,7],[8],[9],[10]]
=> [7,1,1,1] => ? = 3
[6,4]
=> [[1,2,3,4,5,6],[7,8,9,10]]
=> [6,4] => ? = 1
[6,3,1]
=> [[1,2,3,4,5,6],[7,8,9],[10]]
=> [6,3,1] => ? = 1
[6,2,2]
=> [[1,2,3,4,5,6],[7,8],[9,10]]
=> [6,2,2] => ? = 2
[6,2,1,1]
=> [[1,2,3,4,5,6],[7,8],[9],[10]]
=> [6,2,1,1] => ? = 2
[6,1,1,1,1]
=> [[1,2,3,4,5,6],[7],[8],[9],[10]]
=> [6,1,1,1,1] => ? = 4
[5,5]
=> [[1,2,3,4,5],[6,7,8,9,10]]
=> [5,5] => ? = 2
[5,4,1]
=> [[1,2,3,4,5],[6,7,8,9],[10]]
=> [5,4,1] => ? = 1
[5,3,2]
=> [[1,2,3,4,5],[6,7,8],[9,10]]
=> [5,3,2] => ? = 1
[5,3,1,1]
=> [[1,2,3,4,5],[6,7,8],[9],[10]]
=> [5,3,1,1] => ? = 2
[5,2,2,1]
=> [[1,2,3,4,5],[6,7],[8,9],[10]]
=> [5,2,2,1] => ? = 2
[5,2,1,1,1]
=> [[1,2,3,4,5],[6,7],[8],[9],[10]]
=> [5,2,1,1,1] => ? = 3
[5,1,1,1,1,1]
=> [[1,2,3,4,5],[6],[7],[8],[9],[10]]
=> [5,1,1,1,1,1] => ? = 5
[4,4,2]
=> [[1,2,3,4],[5,6,7,8],[9,10]]
=> [4,4,2] => ? = 2
[4,4,1,1]
=> [[1,2,3,4],[5,6,7,8],[9],[10]]
=> [4,4,1,1] => ? = 2
[4,3,3]
=> [[1,2,3,4],[5,6,7],[8,9,10]]
=> [4,3,3] => ? = 2
[4,3,2,1]
=> [[1,2,3,4],[5,6,7],[8,9],[10]]
=> [4,3,2,1] => ? = 1
[4,3,1,1,1]
=> [[1,2,3,4],[5,6,7],[8],[9],[10]]
=> [4,3,1,1,1] => ? = 3
[4,2,2,2]
=> [[1,2,3,4],[5,6],[7,8],[9,10]]
=> [4,2,2,2] => ? = 3
[4,2,2,1,1]
=> [[1,2,3,4],[5,6],[7,8],[9],[10]]
=> [4,2,2,1,1] => ? = 2
[4,2,1,1,1,1]
=> [[1,2,3,4],[5,6],[7],[8],[9],[10]]
=> [4,2,1,1,1,1] => ? = 4
[4,1,1,1,1,1,1]
=> [[1,2,3,4],[5],[6],[7],[8],[9],[10]]
=> [4,1,1,1,1,1,1] => ? = 6
[3,3,3,1]
=> [[1,2,3],[4,5,6],[7,8,9],[10]]
=> [3,3,3,1] => ? = 3
[3,3,2,2]
=> [[1,2,3],[4,5,6],[7,8],[9,10]]
=> [3,3,2,2] => ? = 2
[3,3,2,1,1]
=> [[1,2,3],[4,5,6],[7,8],[9],[10]]
=> [3,3,2,1,1] => ? = 2
[3,3,1,1,1,1]
=> [[1,2,3],[4,5,6],[7],[8],[9],[10]]
=> [3,3,1,1,1,1] => ? = 4
[3,2,2,2,1]
=> [[1,2,3],[4,5],[6,7],[8,9],[10]]
=> [3,2,2,2,1] => ? = 3
[3,2,2,1,1,1]
=> [[1,2,3],[4,5],[6,7],[8],[9],[10]]
=> [3,2,2,1,1,1] => ? = 3
[3,2,1,1,1,1,1]
=> [[1,2,3],[4,5],[6],[7],[8],[9],[10]]
=> [3,2,1,1,1,1,1] => ? = 5
[3,1,1,1,1,1,1,1]
=> [[1,2,3],[4],[5],[6],[7],[8],[9],[10]]
=> [3,1,1,1,1,1,1,1] => ? = 7
[2,2,2,2,2]
=> [[1,2],[3,4],[5,6],[7,8],[9,10]]
=> [2,2,2,2,2] => ? = 5
[2,2,2,2,1,1]
=> [[1,2],[3,4],[5,6],[7,8],[9],[10]]
=> [2,2,2,2,1,1] => ? = 4
[2,2,2,1,1,1,1]
=> [[1,2],[3,4],[5,6],[7],[8],[9],[10]]
=> [2,2,2,1,1,1,1] => ? = 4
[2,2,1,1,1,1,1,1]
=> [[1,2],[3,4],[5],[6],[7],[8],[9],[10]]
=> [2,2,1,1,1,1,1,1] => ? = 6
[2,1,1,1,1,1,1,1,1]
=> [[1,2],[3],[4],[5],[6],[7],[8],[9],[10]]
=> [2,1,1,1,1,1,1,1,1] => ? = 8
[1,1,1,1,1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6],[7],[8],[9],[10]]
=> [1,1,1,1,1,1,1,1,1,1] => ? = 10
[11]
=> [[1,2,3,4,5,6,7,8,9,10,11]]
=> ? => ? = 1
[10,1]
=> [[1,2,3,4,5,6,7,8,9,10],[11]]
=> ? => ? = 1
[9,2]
=> [[1,2,3,4,5,6,7,8,9],[10,11]]
=> ? => ? = 1
[9,1,1]
=> [[1,2,3,4,5,6,7,8,9],[10],[11]]
=> ? => ? = 2
[8,3]
=> [[1,2,3,4,5,6,7,8],[9,10,11]]
=> ? => ? = 1
[8,2,1]
=> [[1,2,3,4,5,6,7,8],[9,10],[11]]
=> ? => ? = 1
[8,1,1,1]
=> [[1,2,3,4,5,6,7,8],[9],[10],[11]]
=> ? => ? = 3
[7,4]
=> [[1,2,3,4,5,6,7],[8,9,10,11]]
=> ? => ? = 1
Description
The length of the longest weakly inreasing subsequence of parts of an integer composition.
The following 13 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000899The maximal number of repetitions of an integer composition. St000904The maximal number of repetitions of an integer composition. St000765The number of weak records in an integer composition. St001652The length of a longest interval of consecutive numbers. St001024Maximum of dominant dimensions of the simple modules in the Nakayama algebra corresponding to the 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. St000684The global dimension of the LNakayama algebra associated to a Dyck path. St000688The global dimension minus the dominant dimension of the LNakayama algebra associated to a Dyck path. 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. St001192The maximal dimension of $Ext_A^2(S,A)$ for a simple module $S$ over the corresponding Nakayama algebra $A$. St000686The finitistic dominant dimension of a Dyck path. St001662The length of the longest factor of consecutive numbers in a permutation. St001232The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2.
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