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Identifier

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

Images

0 => 1

1 => 1

00 => 01

01 => 10

10 => 11

11 => 11

000 => 001

001 => 010

010 => 101

011 => 101

100 => 101

101 => 110

110 => 111

111 => 111

0000 => 0001

0001 => 0010

0010 => 0101

0011 => 0101

0100 => 1001

0101 => 1010

0110 => 1011

0111 => 1011

1000 => 1001

1001 => 1010

1010 => 1101

1011 => 1101

1100 => 1101

1101 => 1110

1110 => 1111

1111 => 1111

00000 => 00001

00001 => 00010

00010 => 00101

00011 => 00101

00100 => 01001

00101 => 01010

00110 => 01011

00111 => 01011

01000 => 10001

01001 => 10010

01010 => 10101

01011 => 10101

01100 => 10101

01101 => 10110

01110 => 10111

01111 => 10111

10000 => 10001

10001 => 10010

10010 => 10101

10011 => 10101

10100 => 11001

10101 => 11010

10110 => 11011

10111 => 11011

11000 => 11001

11001 => 11010

11010 => 11101

11011 => 11101

11100 => 11101

11101 => 11110

11110 => 11111

11111 => 11111

000000 => 000001

000001 => 000010

000010 => 000101

000011 => 000101

000100 => 001001

000101 => 001010

000110 => 001011

000111 => 001011

001000 => 010001

001001 => 010010

001010 => 010101

001011 => 010101

001100 => 010101

001101 => 010110

001110 => 010111

001111 => 010111

010000 => 100001

010001 => 100010

010010 => 100101

010011 => 100101

010100 => 101001

010101 => 101010

010110 => 101011

010111 => 101011

011000 => 101001

011001 => 101010

011010 => 101101

011011 => 101101

011100 => 101101

011101 => 101110

011110 => 101111

011111 => 101111

100000 => 100001

100001 => 100010

100010 => 100101

100011 => 100101

100100 => 101001

100101 => 101010

100110 => 101011

>>> Load all 2760 entries. <<<

click to show experimental identities (only identities of compositions of up to three maps are shown)

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.

References

[1] Tasoulas, I., Manes, K., Sapounakis, A., Tsikouras, P. Chains with small intervals in the lattice of binary paths MathSciNet:4054761

Properties

graded

Sage code

def mapping(w):
    w = list(w)
    f = list(w)
    for i in range(len(w)-1):
        if w[i:i+2] == [0, 1]:
            f[i:i+2] = [1, 0]
    if w[len(w)-1] == 0:
        f[len(w)-1] = 1
    return Words([0,1])(f)

Weight

Created

May 19, 2021 at 17:35 by Martin Rubey

Updated

May 19, 2021 at 17:35 by Martin Rubey