- FindStat

Identifier

Mp00317: Integer partitions —odd parts⟶ Binary words
Mp00224: Binary words —runsort⟶ Binary words
Mp00104: Binary words —reverse⟶ Binary words
St001437: Binary words ⟶ ℤ

Values

[1] => 1 => 1 => 1 => 1

[2] => 0 => 0 => 0 => 1

[1,1] => 11 => 11 => 11 => 2

[3] => 1 => 1 => 1 => 1

[2,1] => 01 => 01 => 10 => 2

[1,1,1] => 111 => 111 => 111 => 3

[4] => 0 => 0 => 0 => 1

[3,1] => 11 => 11 => 11 => 2

[2,2] => 00 => 00 => 00 => 2

[2,1,1] => 011 => 011 => 110 => 3

[1,1,1,1] => 1111 => 1111 => 1111 => 4

[5] => 1 => 1 => 1 => 1

[4,1] => 01 => 01 => 10 => 2

[3,2] => 10 => 01 => 10 => 2

[3,1,1] => 111 => 111 => 111 => 3

[2,2,1] => 001 => 001 => 100 => 3

[2,1,1,1] => 0111 => 0111 => 1110 => 4

[1,1,1,1,1] => 11111 => 11111 => 11111 => 5

[6] => 0 => 0 => 0 => 1

[5,1] => 11 => 11 => 11 => 2

[4,2] => 00 => 00 => 00 => 2

[4,1,1] => 011 => 011 => 110 => 3

[3,3] => 11 => 11 => 11 => 2

[3,2,1] => 101 => 011 => 110 => 3

[3,1,1,1] => 1111 => 1111 => 1111 => 4

[2,2,2] => 000 => 000 => 000 => 3

[2,2,1,1] => 0011 => 0011 => 1100 => 4

[2,1,1,1,1] => 01111 => 01111 => 11110 => 5

[1,1,1,1,1,1] => 111111 => 111111 => 111111 => 6

[7] => 1 => 1 => 1 => 1

[6,1] => 01 => 01 => 10 => 2

[5,2] => 10 => 01 => 10 => 2

[5,1,1] => 111 => 111 => 111 => 3

[4,3] => 01 => 01 => 10 => 2

[4,2,1] => 001 => 001 => 100 => 3

[4,1,1,1] => 0111 => 0111 => 1110 => 4

[3,3,1] => 111 => 111 => 111 => 3

[3,2,2] => 100 => 001 => 100 => 3

[3,2,1,1] => 1011 => 0111 => 1110 => 4

[3,1,1,1,1] => 11111 => 11111 => 11111 => 5

[2,2,2,1] => 0001 => 0001 => 1000 => 4

[2,2,1,1,1] => 00111 => 00111 => 11100 => 5

[2,1,1,1,1,1] => 011111 => 011111 => 111110 => 6

[1,1,1,1,1,1,1] => 1111111 => 1111111 => 1111111 => 7

[8] => 0 => 0 => 0 => 1

[7,1] => 11 => 11 => 11 => 2

[6,2] => 00 => 00 => 00 => 2

[6,1,1] => 011 => 011 => 110 => 3

[5,3] => 11 => 11 => 11 => 2

[5,2,1] => 101 => 011 => 110 => 3

[5,1,1,1] => 1111 => 1111 => 1111 => 4

[4,4] => 00 => 00 => 00 => 2

[4,3,1] => 011 => 011 => 110 => 3

[4,2,2] => 000 => 000 => 000 => 3

[4,2,1,1] => 0011 => 0011 => 1100 => 4

[4,1,1,1,1] => 01111 => 01111 => 11110 => 5

[3,3,2] => 110 => 011 => 110 => 3

[3,3,1,1] => 1111 => 1111 => 1111 => 4

[3,2,2,1] => 1001 => 0011 => 1100 => 4

[3,2,1,1,1] => 10111 => 01111 => 11110 => 5

[3,1,1,1,1,1] => 111111 => 111111 => 111111 => 6

[2,2,2,2] => 0000 => 0000 => 0000 => 4

[2,2,2,1,1] => 00011 => 00011 => 11000 => 5

[2,2,1,1,1,1] => 001111 => 001111 => 111100 => 6

[2,1,1,1,1,1,1] => 0111111 => 0111111 => 1111110 => 7

[1,1,1,1,1,1,1,1] => 11111111 => 11111111 => 11111111 => 8

[9] => 1 => 1 => 1 => 1

[8,1] => 01 => 01 => 10 => 2

[7,2] => 10 => 01 => 10 => 2

[7,1,1] => 111 => 111 => 111 => 3

[6,3] => 01 => 01 => 10 => 2

[6,2,1] => 001 => 001 => 100 => 3

[6,1,1,1] => 0111 => 0111 => 1110 => 4

[5,4] => 10 => 01 => 10 => 2

[5,3,1] => 111 => 111 => 111 => 3

[5,2,2] => 100 => 001 => 100 => 3

[5,2,1,1] => 1011 => 0111 => 1110 => 4

[5,1,1,1,1] => 11111 => 11111 => 11111 => 5

[4,4,1] => 001 => 001 => 100 => 3

[4,3,2] => 010 => 001 => 100 => 3

[4,3,1,1] => 0111 => 0111 => 1110 => 4

[4,2,2,1] => 0001 => 0001 => 1000 => 4

[4,2,1,1,1] => 00111 => 00111 => 11100 => 5

[4,1,1,1,1,1] => 011111 => 011111 => 111110 => 6

[3,3,3] => 111 => 111 => 111 => 3

[3,3,2,1] => 1101 => 0111 => 1110 => 4

[3,3,1,1,1] => 11111 => 11111 => 11111 => 5

[3,2,2,2] => 1000 => 0001 => 1000 => 4

[3,2,2,1,1] => 10011 => 00111 => 11100 => 5

[3,2,1,1,1,1] => 101111 => 011111 => 111110 => 6

[3,1,1,1,1,1,1] => 1111111 => 1111111 => 1111111 => 7

[2,2,2,2,1] => 00001 => 00001 => 10000 => 5

[2,2,2,1,1,1] => 000111 => 000111 => 111000 => 6

[2,2,1,1,1,1,1] => 0011111 => 0011111 => 1111100 => 7

[2,1,1,1,1,1,1,1] => 01111111 => 01111111 => 11111110 => 8

[1,1,1,1,1,1,1,1,1] => 111111111 => 111111111 => 111111111 => 9

[10] => 0 => 0 => 0 => 1

[9,1] => 11 => 11 => 11 => 2

[8,2] => 00 => 00 => 00 => 2

[8,1,1] => 011 => 011 => 110 => 3

[7,3] => 11 => 11 => 11 => 2

>>> Load all 1104 entries. <<<

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$F_{1} = q$

$F_{2} = q + q^{2}$

$F_{3} = q + q^{2} + q^{3}$

$F_{4} = q + 2\ q^{2} + q^{3} + q^{4}$

$F_{5} = q + 2\ q^{2} + 2\ q^{3} + q^{4} + q^{5}$

$F_{6} = q + 3\ q^{2} + 3\ q^{3} + 2\ q^{4} + q^{5} + q^{6}$

$F_{7} = q + 3\ q^{2} + 4\ q^{3} + 3\ q^{4} + 2\ q^{5} + q^{6} + q^{7}$

$F_{8} = q + 4\ q^{2} + 5\ q^{3} + 5\ q^{4} + 3\ q^{5} + 2\ q^{6} + q^{7} + q^{8}$

$F_{9} = q + 4\ q^{2} + 7\ q^{3} + 6\ q^{4} + 5\ q^{5} + 3\ q^{6} + 2\ q^{7} + q^{8} + q^{9}$

Description

The flex of a binary word.
This is the product of the lex statistic (St001436The index of a given binary word in the lex-order among all its cyclic shifts., augmented by 1) and its frequency (St000627The exponent of a binary word.), see [1, §8].

Map

reverse

Description

Return the reversal of a binary word.

Map

odd parts

Description

Return the binary word indicating which parts of the partition are odd.

Map

runsort

Description

The word obtained by sorting the weakly increasing runs lexicographically.