Identifier
-
Mp00095:
Integer partitions
—to binary word⟶
Binary words
Mp00096: Binary words —Foata bijection⟶ Binary words
Mp00224: Binary words —runsort⟶ Binary words
St000543: Binary words ⟶ ℤ
Values
[1] => 10 => 10 => 01 => 2
[2] => 100 => 010 => 001 => 3
[1,1] => 110 => 110 => 011 => 3
[3] => 1000 => 0010 => 0001 => 4
[2,1] => 1010 => 1100 => 0011 => 4
[1,1,1] => 1110 => 1110 => 0111 => 4
[4] => 10000 => 00010 => 00001 => 5
[3,1] => 10010 => 10100 => 00011 => 5
[2,2] => 1100 => 0110 => 0011 => 4
[2,1,1] => 10110 => 11010 => 00111 => 5
[1,1,1,1] => 11110 => 11110 => 01111 => 5
[5] => 100000 => 000010 => 000001 => 6
[4,1] => 100010 => 100100 => 000011 => 6
[3,2] => 10100 => 01100 => 00011 => 5
[3,1,1] => 100110 => 101010 => 001011 => 6
[2,2,1] => 11010 => 11100 => 00111 => 5
[2,1,1,1] => 101110 => 110110 => 001111 => 6
[1,1,1,1,1] => 111110 => 111110 => 011111 => 6
[6] => 1000000 => 0000010 => 0000001 => 7
[5,1] => 1000010 => 1000100 => 0000011 => 7
[4,2] => 100100 => 010100 => 000101 => 6
[4,1,1] => 1000110 => 1001010 => 0001011 => 7
[3,3] => 11000 => 00110 => 00011 => 5
[3,2,1] => 101010 => 111000 => 000111 => 6
[3,1,1,1] => 1001110 => 1010110 => 0010111 => 7
[2,2,2] => 11100 => 01110 => 00111 => 5
[2,2,1,1] => 110110 => 111010 => 001111 => 6
[2,1,1,1,1] => 1011110 => 1101110 => 0011111 => 7
[1,1,1,1,1,1] => 1111110 => 1111110 => 0111111 => 7
[7] => 10000000 => 00000010 => 00000001 => 8
[6,1] => 10000010 => 10000100 => 00000011 => 8
[5,2] => 1000100 => 0100100 => 0000101 => 7
[5,1,1] => 10000110 => 10001010 => 00001011 => 8
[4,3] => 101000 => 001100 => 000011 => 6
[4,2,1] => 1001010 => 1101000 => 0000111 => 7
[4,1,1,1] => 10001110 => 10010110 => 00010111 => 8
[3,3,1] => 110010 => 101100 => 000111 => 6
[3,2,2] => 101100 => 011010 => 001011 => 6
[3,2,1,1] => 1010110 => 1110010 => 0001111 => 7
[3,1,1,1,1] => 10011110 => 10101110 => 00101111 => 8
[2,2,2,1] => 111010 => 111100 => 001111 => 6
[2,2,1,1,1] => 1101110 => 1110110 => 0011111 => 7
[2,1,1,1,1,1] => 10111110 => 11011110 => 00111111 => 8
[1,1,1,1,1,1,1] => 11111110 => 11111110 => 01111111 => 8
[8] => 100000000 => 000000010 => 000000001 => 9
[7,1] => 100000010 => 100000100 => 000000011 => 9
[6,2] => 10000100 => 01000100 => 00000101 => 8
[6,1,1] => 100000110 => 100001010 => 000001011 => 9
[5,3] => 1001000 => 0010100 => 0000101 => 7
[5,2,1] => 10001010 => 11001000 => 00000111 => 8
[5,1,1,1] => 100001110 => 100010110 => 000010111 => 9
[4,4] => 110000 => 000110 => 000011 => 6
[4,3,1] => 1010010 => 1011000 => 0000111 => 7
[4,2,2] => 1001100 => 0101010 => 0010101 => 7
[4,2,1,1] => 10010110 => 11010010 => 00010111 => 8
[4,1,1,1,1] => 100011110 => 100101110 => 000101111 => 9
[3,3,2] => 110100 => 011100 => 000111 => 6
[3,3,1,1] => 1100110 => 1011010 => 0010111 => 7
[3,2,2,1] => 1011010 => 1110100 => 0001111 => 7
[3,2,1,1,1] => 10101110 => 11100110 => 00011111 => 8
[3,1,1,1,1,1] => 100111110 => 101011110 => 001011111 => 9
[2,2,2,2] => 111100 => 011110 => 001111 => 6
[2,2,2,1,1] => 1110110 => 1111010 => 0011111 => 7
[2,2,1,1,1,1] => 11011110 => 11101110 => 00111111 => 8
[2,1,1,1,1,1,1] => 101111110 => 110111110 => 001111111 => 9
[1,1,1,1,1,1,1,1] => 111111110 => 111111110 => 011111111 => 9
[9] => 1000000000 => 0000000010 => 0000000001 => 10
[8,1] => 1000000010 => 1000000100 => 0000000011 => 10
[7,2] => 100000100 => 010000100 => 000000101 => 9
[7,1,1] => 1000000110 => 1000001010 => 0000001011 => 10
[6,3] => 10001000 => 00100100 => 00001001 => 8
[6,2,1] => 100001010 => 110001000 => 000000111 => 9
[6,1,1,1] => 1000001110 => 1000010110 => 0000010111 => 10
[5,4] => 1010000 => 0001100 => 0000011 => 7
[5,3,1] => 10010010 => 10101000 => 00001011 => 8
[5,2,2] => 10001100 => 01001010 => 00010101 => 8
[5,2,1,1] => 100010110 => 110010010 => 000100111 => 9
[5,1,1,1,1] => 1000011110 => 1000101110 => 0000101111 => 10
[4,4,1] => 1100010 => 1001100 => 0000111 => 7
[4,3,2] => 1010100 => 0111000 => 0000111 => 7
[4,3,1,1] => 10100110 => 10110010 => 00010111 => 8
[4,2,2,1] => 10011010 => 11010100 => 00010111 => 8
[4,2,1,1,1] => 100101110 => 110100110 => 000110111 => 9
[4,1,1,1,1,1] => 1000111110 => 1001011110 => 0001011111 => 10
[3,3,3] => 111000 => 001110 => 000111 => 6
[3,3,2,1] => 1101010 => 1111000 => 0001111 => 7
[3,3,1,1,1] => 11001110 => 10110110 => 00110111 => 8
[3,2,2,2] => 1011100 => 0110110 => 0011011 => 7
[3,2,2,1,1] => 10110110 => 11101010 => 00101111 => 8
[3,2,1,1,1,1] => 101011110 => 111001110 => 000111111 => 9
[2,2,2,2,1] => 1111010 => 1111100 => 0011111 => 7
[2,2,2,1,1,1] => 11101110 => 11110110 => 00111111 => 8
[2,2,1,1,1,1,1] => 110111110 => 111011110 => 001111111 => 9
[8,2] => 1000000100 => 0100000100 => 0000000101 => 10
[7,3] => 100001000 => 001000100 => 000001001 => 9
[7,2,1] => 1000001010 => 1100001000 => 0000000111 => 10
[6,4] => 10010000 => 00010100 => 00000101 => 8
[6,3,1] => 100010010 => 101001000 => 000001011 => 9
[6,2,2] => 100001100 => 010001010 => 000010101 => 9
[6,2,1,1] => 1000010110 => 1100010010 => 0000100111 => 10
[5,5] => 1100000 => 0000110 => 0000011 => 7
>>> Load all 319 entries. <<<
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Description
The size of the conjugacy class of a binary word.
Two words $u$ and $v$ are conjugate, if $u=w_1 w_2$ and $v=w_2 w_1$, see Section 1.3 of [1].
Two words $u$ and $v$ are conjugate, if $u=w_1 w_2$ and $v=w_2 w_1$, see Section 1.3 of [1].
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.
For instance, to compute $\phi(4154223)$, the sequence of words is
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 $w_{i+1} < v_k$.
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.
Map
runsort
Description
The word obtained by sorting the weakly increasing runs lexicographically.
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.
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