- FindStat

Identifier

Mp00234: Binary words —valleys-to-peaks⟶ Binary words
St001424: Binary words ⟶ ℤ

Values

0 => 1 => 0

1 => 1 => 0

00 => 01 => 0

01 => 10 => 0

10 => 11 => 1

11 => 11 => 1

000 => 001 => 1

001 => 010 => 0

010 => 101 => 0

011 => 101 => 0

100 => 101 => 0

101 => 110 => 1

110 => 111 => 1

111 => 111 => 1

0000 => 0001 => 1

0001 => 0010 => 1

0010 => 0101 => 1

0011 => 0101 => 1

0100 => 1001 => 1

0101 => 1010 => 1

0110 => 1011 => 1

0111 => 1011 => 1

1000 => 1001 => 1

1001 => 1010 => 1

1010 => 1101 => 1

1011 => 1101 => 1

1100 => 1101 => 1

1101 => 1110 => 1

1110 => 1111 => 2

1111 => 1111 => 2

00000 => 00001 => 2

00001 => 00010 => 1

00010 => 00101 => 2

00011 => 00101 => 2

00100 => 01001 => 1

00101 => 01010 => 2

00110 => 01011 => 2

00111 => 01011 => 2

01000 => 10001 => 1

01001 => 10010 => 1

01010 => 10101 => 2

01011 => 10101 => 2

01100 => 10101 => 2

01101 => 10110 => 1

01110 => 10111 => 1

01111 => 10111 => 1

10000 => 10001 => 1

10001 => 10010 => 1

10010 => 10101 => 2

10011 => 10101 => 2

10100 => 11001 => 2

10101 => 11010 => 2

10110 => 11011 => 1

10111 => 11011 => 1

11000 => 11001 => 2

11001 => 11010 => 2

11010 => 11101 => 1

11011 => 11101 => 1

11100 => 11101 => 1

11101 => 11110 => 2

11110 => 11111 => 2

11111 => 11111 => 2

000000 => 000001 => 2

000001 => 000010 => 2

000010 => 000101 => 2

000011 => 000101 => 2

000100 => 001001 => 2

000101 => 001010 => 3

000110 => 001011 => 3

000111 => 001011 => 3

001000 => 010001 => 1

001001 => 010010 => 2

001010 => 010101 => 2

001011 => 010101 => 2

001100 => 010101 => 2

001101 => 010110 => 2

001110 => 010111 => 2

001111 => 010111 => 2

010000 => 100001 => 2

010001 => 100010 => 1

010010 => 100101 => 2

010011 => 100101 => 2

010100 => 101001 => 2

010101 => 101010 => 2

010110 => 101011 => 3

010111 => 101011 => 3

011000 => 101001 => 2

011001 => 101010 => 2

011010 => 101101 => 2

011011 => 101101 => 2

011100 => 101101 => 2

011101 => 101110 => 1

011110 => 101111 => 2

011111 => 101111 => 2

100000 => 100001 => 2

100001 => 100010 => 1

100010 => 100101 => 2

100011 => 100101 => 2

100100 => 101001 => 2

100101 => 101010 => 2

100110 => 101011 => 3

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Description

The number of distinct squares in a binary word.
A factor of a word is a sequence of consecutive letters. This statistic records the number of distinct non-empty words $u$ such that $uu$ is a factor of the word.
Note that every word of length at least four contains a square.

Map

valleys-to-peaks

Description

Return the binary word with every valley replaced by a peak.
A valley in a binary word is a subsequence $01$, or a trailing $0$. A peak is a subsequence $10$ or a trailing $1$. This map replaces every valley with a peak.