- FindStat

Identifier

Mp00095: Integer partitions —to binary word⟶ Binary words
Mp00096: Binary words —Foata bijection⟶ Binary words
St000293: Binary words ⟶ ℤ

Values

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

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

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

[3] => 1000 => 0010 => 1

[2,1] => 1010 => 1100 => 4

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

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

[3,1] => 10010 => 10100 => 5

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

[2,1,1] => 10110 => 11010 => 5

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

[5] => 100000 => 000010 => 1

[4,1] => 100010 => 100100 => 6

[3,2] => 10100 => 01100 => 4

[3,1,1] => 100110 => 101010 => 6

[2,2,1] => 11010 => 11100 => 6

[2,1,1,1] => 101110 => 110110 => 6

[1,1,1,1,1] => 111110 => 111110 => 5

[6] => 1000000 => 0000010 => 1

[5,1] => 1000010 => 1000100 => 7

[4,2] => 100100 => 010100 => 5

[4,1,1] => 1000110 => 1001010 => 7

[3,3] => 11000 => 00110 => 2

[3,2,1] => 101010 => 111000 => 9

[3,1,1,1] => 1001110 => 1010110 => 7

[2,2,2] => 11100 => 01110 => 3

[2,2,1,1] => 110110 => 111010 => 7

[2,1,1,1,1] => 1011110 => 1101110 => 7

[1,1,1,1,1,1] => 1111110 => 1111110 => 6

[7] => 10000000 => 00000010 => 1

[6,1] => 10000010 => 10000100 => 8

[5,2] => 1000100 => 0100100 => 6

[5,1,1] => 10000110 => 10001010 => 8

[4,3] => 101000 => 001100 => 4

[4,2,1] => 1001010 => 1101000 => 11

[4,1,1,1] => 10001110 => 10010110 => 8

[3,3,1] => 110010 => 101100 => 7

[3,2,2] => 101100 => 011010 => 5

[3,2,1,1] => 1010110 => 1110010 => 10

[3,1,1,1,1] => 10011110 => 10101110 => 8

[2,2,2,1] => 111010 => 111100 => 8

[2,2,1,1,1] => 1101110 => 1110110 => 8

[2,1,1,1,1,1] => 10111110 => 11011110 => 8

[1,1,1,1,1,1,1] => 11111110 => 11111110 => 7

[8] => 100000000 => 000000010 => 1

[7,1] => 100000010 => 100000100 => 9

[6,2] => 10000100 => 01000100 => 7

[6,1,1] => 100000110 => 100001010 => 9

[5,3] => 1001000 => 0010100 => 5

[5,2,1] => 10001010 => 11001000 => 13

[5,1,1,1] => 100001110 => 100010110 => 9

[4,4] => 110000 => 000110 => 2

[4,3,1] => 1010010 => 1011000 => 10

[4,2,2] => 1001100 => 0101010 => 6

[4,2,1,1] => 10010110 => 11010010 => 12

[4,1,1,1,1] => 100011110 => 100101110 => 9

[3,3,2] => 110100 => 011100 => 6

[3,3,1,1] => 1100110 => 1011010 => 8

[3,2,2,1] => 1011010 => 1110100 => 11

[3,2,1,1,1] => 10101110 => 11100110 => 11

[3,1,1,1,1,1] => 100111110 => 101011110 => 9

[2,2,2,2] => 111100 => 011110 => 4

[2,2,2,1,1] => 1110110 => 1111010 => 9

[2,2,1,1,1,1] => 11011110 => 11101110 => 9

[2,1,1,1,1,1,1] => 101111110 => 110111110 => 9

[1,1,1,1,1,1,1,1] => 111111110 => 111111110 => 8

[9] => 1000000000 => 0000000010 => 1

[8,1] => 1000000010 => 1000000100 => 10

[7,2] => 100000100 => 010000100 => 8

[7,1,1] => 1000000110 => 1000001010 => 10

[6,3] => 10001000 => 00100100 => 6

[6,2,1] => 100001010 => 110001000 => 15

[6,1,1,1] => 1000001110 => 1000010110 => 10

[5,4] => 1010000 => 0001100 => 4

[5,3,1] => 10010010 => 10101000 => 12

[5,2,2] => 10001100 => 01001010 => 7

[5,2,1,1] => 100010110 => 110010010 => 14

[5,1,1,1,1] => 1000011110 => 1000101110 => 10

[4,4,1] => 1100010 => 1001100 => 8

[4,3,2] => 1010100 => 0111000 => 9

[4,3,1,1] => 10100110 => 10110010 => 11

[4,2,2,1] => 10011010 => 11010100 => 13

[4,2,1,1,1] => 100101110 => 110100110 => 13

[4,1,1,1,1,1] => 1000111110 => 1001011110 => 10

[3,3,3] => 111000 => 001110 => 3

[3,3,2,1] => 1101010 => 1111000 => 12

[3,3,1,1,1] => 11001110 => 10110110 => 9

[3,2,2,2] => 1011100 => 0110110 => 6

[3,2,2,1,1] => 10110110 => 11101010 => 12

[3,2,1,1,1,1] => 101011110 => 111001110 => 12

[3,1,1,1,1,1,1] => 1001111110 => 1010111110 => 10

[2,2,2,2,1] => 1111010 => 1111100 => 10

[2,2,2,1,1,1] => 11101110 => 11110110 => 10

[2,2,1,1,1,1,1] => 110111110 => 111011110 => 10

[2,1,1,1,1,1,1,1] => 1011111110 => 1101111110 => 10

[1,1,1,1,1,1,1,1,1] => 1111111110 => 1111111110 => 9

[10] => 10000000000 => 00000000010 => 1

[9,1] => 10000000010 => 10000000100 => 11

[8,2] => 1000000100 => 0100000100 => 9

[8,1,1] => 10000000110 => 10000001010 => 11

[7,3] => 100001000 => 001000100 => 7

>>> Load all 374 entries. <<<

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$F_{1} = q$

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

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

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

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

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

$F_{7} = q + q^{4} + q^{5} + q^{6} + 2\ q^{7} + 7\ q^{8} + q^{10} + q^{11}$

$F_{8} = q + q^{2} + q^{4} + q^{5} + 2\ q^{6} + q^{7} + 2\ q^{8} + 8\ q^{9} + q^{10} + 2\ q^{11} + q^{12} + q^{13}$

$F_{9} = q + q^{3} + q^{4} + 2\ q^{6} + q^{7} + 2\ q^{8} + 3\ q^{9} + 10\ q^{10} + q^{11} + 4\ q^{12} + 2\ q^{13} + q^{14} + q^{15}$

$F_{10} = q + q^{2} + 3\ q^{5} + 4\ q^{7} + q^{8} + 3\ q^{9} + 2\ q^{10} + 13\ q^{11} + q^{12} + 5\ q^{13} + 3\ q^{14} + 2\ q^{15} + 2\ q^{16} + q^{17}$

Description

The number of inversions of a binary word.

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.

$\phi(4154223) = 1254423$ .

Map

to binary word

Description

Return the partition as binary word, by traversing its shape from the first row to the last row, down steps as 1 and left steps as 0.