- FindStat

Identifier

Mp00158: Binary words —alternating inverse⟶ Binary words
Mp00135: Binary words —rotate front-to-back⟶ Binary words
St000630: Binary words ⟶ ℤ

Values

0 => 0 => 0 => 1

1 => 1 => 1 => 1

00 => 01 => 10 => 2

01 => 00 => 00 => 1

10 => 11 => 11 => 1

11 => 10 => 01 => 2

000 => 010 => 100 => 2

001 => 011 => 110 => 2

010 => 000 => 000 => 1

011 => 001 => 010 => 1

100 => 110 => 101 => 1

101 => 111 => 111 => 1

110 => 100 => 001 => 2

111 => 101 => 011 => 2

0000 => 0101 => 1010 => 2

0001 => 0100 => 1000 => 2

0010 => 0111 => 1110 => 2

0011 => 0110 => 1100 => 2

0100 => 0001 => 0010 => 2

0101 => 0000 => 0000 => 1

0110 => 0011 => 0110 => 1

0111 => 0010 => 0100 => 2

1000 => 1101 => 1011 => 2

1001 => 1100 => 1001 => 1

1010 => 1111 => 1111 => 1

1011 => 1110 => 1101 => 2

1100 => 1001 => 0011 => 2

1101 => 1000 => 0001 => 2

1110 => 1011 => 0111 => 2

1111 => 1010 => 0101 => 2

00000 => 01010 => 10100 => 2

00001 => 01011 => 10110 => 2

00010 => 01000 => 10000 => 2

00011 => 01001 => 10010 => 2

00100 => 01110 => 11100 => 2

00101 => 01111 => 11110 => 2

00110 => 01100 => 11000 => 2

00111 => 01101 => 11010 => 2

01000 => 00010 => 00100 => 1

01001 => 00011 => 00110 => 2

01010 => 00000 => 00000 => 1

01011 => 00001 => 00010 => 2

01100 => 00110 => 01100 => 2

01101 => 00111 => 01110 => 1

01110 => 00100 => 01000 => 2

01111 => 00101 => 01010 => 1

10000 => 11010 => 10101 => 1

10001 => 11011 => 10111 => 2

10010 => 11000 => 10001 => 1

10011 => 11001 => 10011 => 2

10100 => 11110 => 11101 => 2

10101 => 11111 => 11111 => 1

10110 => 11100 => 11001 => 2

10111 => 11101 => 11011 => 1

11000 => 10010 => 00101 => 2

11001 => 10011 => 00111 => 2

11010 => 10000 => 00001 => 2

11011 => 10001 => 00011 => 2

11100 => 10110 => 01101 => 2

11101 => 10111 => 01111 => 2

11110 => 10100 => 01001 => 2

11111 => 10101 => 01011 => 2

000000 => 010101 => 101010 => 2

000001 => 010100 => 101000 => 2

000010 => 010111 => 101110 => 2

000011 => 010110 => 101100 => 3

000100 => 010001 => 100010 => 2

000101 => 010000 => 100000 => 2

000110 => 010011 => 100110 => 3

000111 => 010010 => 100100 => 2

001000 => 011101 => 111010 => 2

001001 => 011100 => 111000 => 2

001010 => 011111 => 111110 => 2

001011 => 011110 => 111100 => 2

001100 => 011001 => 110010 => 3

001101 => 011000 => 110000 => 2

001110 => 011011 => 110110 => 2

001111 => 011010 => 110100 => 3

010000 => 000101 => 001010 => 2

010001 => 000100 => 001000 => 2

010010 => 000111 => 001110 => 2

010011 => 000110 => 001100 => 1

010100 => 000001 => 000010 => 2

010101 => 000000 => 000000 => 1

010110 => 000011 => 000110 => 2

010111 => 000010 => 000100 => 2

011000 => 001101 => 011010 => 3

011001 => 001100 => 011000 => 2

011010 => 001111 => 011110 => 1

011011 => 001110 => 011100 => 2

011100 => 001001 => 010010 => 1

011101 => 001000 => 010000 => 2

011110 => 001011 => 010110 => 3

011111 => 001010 => 010100 => 2

100000 => 110101 => 101011 => 2

100001 => 110100 => 101001 => 3

100010 => 110111 => 101111 => 2

100011 => 110110 => 101101 => 1

100100 => 110001 => 100011 => 2

100101 => 110000 => 100001 => 1

100110 => 110011 => 100111 => 2

>>> Load all 1363 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

Description

The length of the shortest palindromic decomposition of a binary word.
A palindromic decomposition (paldec for short) of a word $w=a_1,\dots,a_n$ is any list of factors $p_1,\dots,p_k$ such that $w=p_1\dots p_k$ and each $p_i$ is a palindrome, i.e. coincides with itself read backwards.

Map

rotate front-to-back

Description

The rotation of a binary word, first letter last.
This is the word obtained by moving the first letter to the end.

Map

alternating inverse

Description

Sends a binary word $w_1\cdots w_m$ to the binary word $v_1 \cdots v_m$ with $v_i = w_i$ if $i$ is odd and $v_i = 1 - w_i$ if $i$ is even.
This map is used in [1], see Definitions 3.2 and 5.1.