- FindStat

Identifier

Mp00317: Integer partitions —odd parts⟶ Binary words
Mp00105: Binary words —complement⟶ Binary words
Mp00096: Binary words —Foata bijection⟶ Binary words
St000291: Binary words ⟶ ℤ

Values

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

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

[1,1] => 11 => 00 => 00 => 0

[3] => 1 => 0 => 0 => 0

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

[1,1,1] => 111 => 000 => 000 => 0

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

[3,1] => 11 => 00 => 00 => 0

[2,2] => 00 => 11 => 11 => 0

[2,1,1] => 011 => 100 => 010 => 1

[1,1,1,1] => 1111 => 0000 => 0000 => 0

[5] => 1 => 0 => 0 => 0

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

[3,2] => 10 => 01 => 01 => 0

[3,1,1] => 111 => 000 => 000 => 0

[2,2,1] => 001 => 110 => 110 => 1

[2,1,1,1] => 0111 => 1000 => 0010 => 1

[1,1,1,1,1] => 11111 => 00000 => 00000 => 0

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

[5,1] => 11 => 00 => 00 => 0

[4,2] => 00 => 11 => 11 => 0

[4,1,1] => 011 => 100 => 010 => 1

[3,3] => 11 => 00 => 00 => 0

[3,2,1] => 101 => 010 => 100 => 1

[3,1,1,1] => 1111 => 0000 => 0000 => 0

[2,2,2] => 000 => 111 => 111 => 0

[2,2,1,1] => 0011 => 1100 => 0110 => 1

[2,1,1,1,1] => 01111 => 10000 => 00010 => 1

[1,1,1,1,1,1] => 111111 => 000000 => 000000 => 0

[7] => 1 => 0 => 0 => 0

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

[5,2] => 10 => 01 => 01 => 0

[5,1,1] => 111 => 000 => 000 => 0

[4,3] => 01 => 10 => 10 => 1

[4,2,1] => 001 => 110 => 110 => 1

[4,1,1,1] => 0111 => 1000 => 0010 => 1

[3,3,1] => 111 => 000 => 000 => 0

[3,2,2] => 100 => 011 => 011 => 0

[3,2,1,1] => 1011 => 0100 => 0100 => 1

[3,1,1,1,1] => 11111 => 00000 => 00000 => 0

[2,2,2,1] => 0001 => 1110 => 1110 => 1

[2,2,1,1,1] => 00111 => 11000 => 00110 => 1

[2,1,1,1,1,1] => 011111 => 100000 => 000010 => 1

[1,1,1,1,1,1,1] => 1111111 => 0000000 => 0000000 => 0

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

[7,1] => 11 => 00 => 00 => 0

[6,2] => 00 => 11 => 11 => 0

[6,1,1] => 011 => 100 => 010 => 1

[5,3] => 11 => 00 => 00 => 0

[5,2,1] => 101 => 010 => 100 => 1

[5,1,1,1] => 1111 => 0000 => 0000 => 0

[4,4] => 00 => 11 => 11 => 0

[4,3,1] => 011 => 100 => 010 => 1

[4,2,2] => 000 => 111 => 111 => 0

[4,2,1,1] => 0011 => 1100 => 0110 => 1

[4,1,1,1,1] => 01111 => 10000 => 00010 => 1

[3,3,2] => 110 => 001 => 001 => 0

[3,3,1,1] => 1111 => 0000 => 0000 => 0

[3,2,2,1] => 1001 => 0110 => 1010 => 2

[3,2,1,1,1] => 10111 => 01000 => 00100 => 1

[3,1,1,1,1,1] => 111111 => 000000 => 000000 => 0

[2,2,2,2] => 0000 => 1111 => 1111 => 0

[2,2,2,1,1] => 00011 => 11100 => 01110 => 1

[2,2,1,1,1,1] => 001111 => 110000 => 000110 => 1

[2,1,1,1,1,1,1] => 0111111 => 1000000 => 0000010 => 1

[1,1,1,1,1,1,1,1] => 11111111 => 00000000 => 00000000 => 0

[9] => 1 => 0 => 0 => 0

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

[7,2] => 10 => 01 => 01 => 0

[7,1,1] => 111 => 000 => 000 => 0

[6,3] => 01 => 10 => 10 => 1

[6,2,1] => 001 => 110 => 110 => 1

[6,1,1,1] => 0111 => 1000 => 0010 => 1

[5,4] => 10 => 01 => 01 => 0

[5,3,1] => 111 => 000 => 000 => 0

[5,2,2] => 100 => 011 => 011 => 0

[5,2,1,1] => 1011 => 0100 => 0100 => 1

[5,1,1,1,1] => 11111 => 00000 => 00000 => 0

[4,4,1] => 001 => 110 => 110 => 1

[4,3,2] => 010 => 101 => 101 => 1

[4,3,1,1] => 0111 => 1000 => 0010 => 1

[4,2,2,1] => 0001 => 1110 => 1110 => 1

[4,2,1,1,1] => 00111 => 11000 => 00110 => 1

[4,1,1,1,1,1] => 011111 => 100000 => 000010 => 1

[3,3,3] => 111 => 000 => 000 => 0

[3,3,2,1] => 1101 => 0010 => 1000 => 1

[3,3,1,1,1] => 11111 => 00000 => 00000 => 0

[3,2,2,2] => 1000 => 0111 => 0111 => 0

[3,2,2,1,1] => 10011 => 01100 => 01010 => 2

[3,2,1,1,1,1] => 101111 => 010000 => 000100 => 1

[3,1,1,1,1,1,1] => 1111111 => 0000000 => 0000000 => 0

[2,2,2,2,1] => 00001 => 11110 => 11110 => 1

[2,2,2,1,1,1] => 000111 => 111000 => 001110 => 1

[2,2,1,1,1,1,1] => 0011111 => 1100000 => 0000110 => 1

[2,1,1,1,1,1,1,1] => 01111111 => 10000000 => 00000010 => 1

[1,1,1,1,1,1,1,1,1] => 111111111 => 000000000 => 000000000 => 0

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

[9,1] => 11 => 00 => 00 => 0

[8,2] => 00 => 11 => 11 => 0

[8,1,1] => 011 => 100 => 010 => 1

[7,3] => 11 => 00 => 00 => 0

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$F_{1} = 1$

$F_{2} = 2$

$F_{3} = 2 + q$

$F_{4} = 4 + q$

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

$F_{6} = 7 + 4\ q$

$F_{7} = 7 + 8\ q$

$F_{8} = 12 + 9\ q + q^{2}$

$F_{9} = 12 + 17\ q + q^{2}$

$F_{10} = 19 + 20\ q + 3\ q^{2}$

Description

The number of descents of a binary word.

Map

complement

Description

Send a binary word to the word obtained by interchanging the two letters.

Map

odd parts

Description

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

Map

Foata bijection

Description

The Foata bijection

$\phi$ is a bijection on the set of words of given content (by a slight generalization of Section 2 in [1]).
Given a word

$w_1 w_2 ... w_n$ , compute the image inductively by starting with

$\phi(w_1) = w_1$ . At the

$i$ -th step, if

$\phi(w_1 w_2 ... w_i) = v_1 v_2 ... v_i$ , define

$\phi(w_1 w_2 ... w_i w_{i+1})$ by placing

$w_{i+1}$ on the end of the word

$v_1 v_2 ... v_i$ and breaking the word up into blocks as follows.

If $w_{i+1} \geq v_i$ , place a vertical line to the right of each $v_k$ for which $w_{i+1} \geq v_k$ .
If $w_{i+1} < v_i$ , place a vertical line to the right of each $v_k$ for which $w_{i+1} < v_k$ .

In either case, place a vertical line at the start of the word as well. Now, within each block between vertical lines, cyclically shift the entries one place to the right.
For instance, to compute

$\phi(4154223)$ , the sequence of words is

4,
|4|1 -- > 41,
|4|1|5 -- > 415,
|415|4 -- > 5414,
|5|4|14|2 -- > 54412,
|5441|2|2 -- > 154422,
|1|5442|2|3 -- > 1254423.

$\phi(4154223) = 1254423$ .