- FindStat

Identifier

Mp00224: Binary words —runsort⟶ Binary words
Mp00273: Binary words —Gray previous⟶ Binary words
St001524: Binary words ⟶ ℤ

Values

0 => 0 => 1 => 0

1 => 1 => 0 => 0

00 => 00 => 01 => 0

01 => 01 => 11 => 1

10 => 01 => 11 => 1

11 => 11 => 10 => 0

000 => 000 => 001 => 0

001 => 001 => 101 => 1

010 => 001 => 101 => 1

011 => 011 => 010 => 1

100 => 001 => 101 => 1

101 => 011 => 010 => 1

110 => 011 => 010 => 1

111 => 111 => 011 => 0

0000 => 0000 => 0001 => 1

0001 => 0001 => 1001 => 2

0010 => 0001 => 1001 => 2

0011 => 0011 => 0010 => 1

0100 => 0001 => 1001 => 2

0101 => 0101 => 0111 => 1

0110 => 0011 => 0010 => 1

0111 => 0111 => 1111 => 2

1000 => 0001 => 1001 => 2

1001 => 0011 => 0010 => 1

1010 => 0011 => 0010 => 1

1011 => 0111 => 1111 => 2

1100 => 0011 => 0010 => 1

1101 => 0111 => 1111 => 2

1110 => 0111 => 1111 => 2

1111 => 1111 => 1011 => 1

00000 => 00000 => 00001 => 1

00001 => 00001 => 10001 => 2

00010 => 00001 => 10001 => 2

00011 => 00011 => 00010 => 1

00100 => 00001 => 10001 => 2

00101 => 00101 => 00111 => 0

00110 => 00011 => 00010 => 1

00111 => 00111 => 10111 => 1

01000 => 00001 => 10001 => 2

01001 => 00101 => 00111 => 0

01010 => 00101 => 00111 => 0

01011 => 01011 => 11011 => 2

01100 => 00011 => 00010 => 1

01101 => 01011 => 11011 => 2

01110 => 00111 => 10111 => 1

01111 => 01111 => 01011 => 1

10000 => 00001 => 10001 => 2

10001 => 00011 => 00010 => 1

10010 => 00011 => 00010 => 1

10011 => 00111 => 10111 => 1

10100 => 00011 => 00010 => 1

10101 => 01011 => 11011 => 2

10110 => 00111 => 10111 => 1

10111 => 01111 => 01011 => 1

11000 => 00011 => 00010 => 1

11001 => 00111 => 10111 => 1

11010 => 00111 => 10111 => 1

11011 => 01111 => 01011 => 1

11100 => 00111 => 10111 => 1

11101 => 01111 => 01011 => 1

11110 => 01111 => 01011 => 1

11111 => 11111 => 01111 => 1

000000 => 000000 => 000001 => 2

000001 => 000001 => 100001 => 3

000010 => 000001 => 100001 => 3

000011 => 000011 => 000010 => 2

000100 => 000001 => 100001 => 3

000101 => 000101 => 000111 => 0

000110 => 000011 => 000010 => 2

000111 => 000111 => 100111 => 1

001000 => 000001 => 100001 => 3

001001 => 001001 => 001101 => 2

001010 => 000101 => 000111 => 0

001011 => 001011 => 101011 => 1

001100 => 000011 => 000010 => 2

001101 => 001101 => 101101 => 3

001110 => 000111 => 100111 => 1

001111 => 001111 => 001011 => 0

010000 => 000001 => 100001 => 3

010001 => 000101 => 000111 => 0

010010 => 000101 => 000111 => 0

010011 => 001101 => 101101 => 3

010100 => 000101 => 000111 => 0

010101 => 010101 => 110101 => 1

010110 => 001011 => 101011 => 1

010111 => 010111 => 011111 => 2

011000 => 000011 => 000010 => 2

011001 => 001011 => 101011 => 1

011010 => 001011 => 101011 => 1

011011 => 011011 => 010011 => 2

011100 => 000111 => 100111 => 1

011101 => 010111 => 011111 => 2

011110 => 001111 => 001011 => 0

011111 => 011111 => 111111 => 3

100000 => 000001 => 100001 => 3

100001 => 000011 => 000010 => 2

100010 => 000011 => 000010 => 2

100011 => 000111 => 100111 => 1

100100 => 000011 => 000010 => 2

100101 => 001011 => 101011 => 1

100110 => 000111 => 100111 => 1

>>> Load all 1022 entries. <<<

search for individual values

searching the database for the individual values of this statistic

/ search for generating function

searching the database for statistics with the same generating function

click to show known generating functions

$F_{1} = 2$

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

$F_{3} = 2 + 6\ q$

$F_{4} = 8\ q + 8\ q^{2}$

$F_{5} = 3 + 21\ q + 8\ q^{2}$

$F_{6} = 14 + 18\ q + 18\ q^{2} + 14\ q^{3}$

$F_{7} = 12 + 54\ q + 48\ q^{2} + 14\ q^{3}$

$F_{8} = 100\ q + 102\ q^{2} + 28\ q^{3} + 26\ q^{4}$

$F_{9} = 38 + 130\ q + 197\ q^{2} + 130\ q^{3} + 17\ q^{4}$

Description

The degree of symmetry of a binary word.
For a binary word

$w$ of length

$n$ , this is the number of positions

$i\leq n/2$ such that

$w_i = w_{n+1-i}$ .

Map

Gray previous

Description

The previous element in the Gray code for binary words.
Let

$w$ be a binary word. If the number of ones in

$w$ is odd, return the binary word obtained by flipping the first bit of

$w$ . If

$w$ has only zeros, return the word with the final bit flipped. Otherwise, return the word obtained by flipping the bit after the first one.

Map

runsort

Description

The word obtained by sorting the weakly increasing runs lexicographically.