- FindStat

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Identifier

Mp00280: Binary words —path rowmotion⟶ Binary words

Images

0 => 1

1 => 0

00 => 01

01 => 10

10 => 11

11 => 00

000 => 001

001 => 010

010 => 101

011 => 100

100 => 011

101 => 110

110 => 111

111 => 000

0000 => 0001

0001 => 0010

0010 => 0101

0011 => 0100

0100 => 1001

0101 => 1010

0110 => 1011

0111 => 1000

1000 => 0011

1001 => 0110

1010 => 1101

1011 => 1100

1100 => 0111

1101 => 1110

1110 => 1111

1111 => 0000

00000 => 00001

00001 => 00010

00010 => 00101

00011 => 00100

00100 => 01001

00101 => 01010

00110 => 01011

00111 => 01000

01000 => 10001

01001 => 10010

01010 => 10101

01011 => 10100

01100 => 10011

01101 => 10110

01110 => 10111

01111 => 10000

10000 => 00011

10001 => 00110

10010 => 01101

10011 => 01100

10100 => 11001

10101 => 11010

10110 => 11011

10111 => 11000

11000 => 00111

11001 => 01110

11010 => 11101

11011 => 11100

11100 => 01111

11101 => 11110

11110 => 11111

11111 => 00000

000000 => 000001

000001 => 000010

000010 => 000101

000011 => 000100

000100 => 001001

000101 => 001010

000110 => 001011

000111 => 001000

001000 => 010001

001001 => 010010

001010 => 010101

001011 => 010100

001100 => 010011

001101 => 010110

001110 => 010111

001111 => 010000

010000 => 100001

010001 => 100010

010010 => 100101

010011 => 100100

010100 => 101001

010101 => 101010

010110 => 101011

010111 => 101000

011000 => 100011

011001 => 100110

011010 => 101101

011011 => 101100

011100 => 100111

011101 => 101110

011110 => 101111

011111 => 100000

100000 => 000011

100001 => 000110

100010 => 001101

100011 => 001100

100100 => 011001

100101 => 011010

>>> Load all 2761 entries. <<<

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

Description

Return the rowmotion of the binary word, regarded as a lattice path.
Consider the binary word of length $n$ as a lattice path with $n$ steps, where a 1 corresponds to an up step and a 0 corresponds to a down step.
This map returns the path whose peaks are the valleys of the original path with an up step appended.

Properties

bijective, graded

Sage code

def mapping(w):
    n = len(w)
    w1 = []
    a = 0
    i = 0
    while i < n:
        if w[i]:
            a += 1
            i += 1
        elif i+1 == n:
            w1.extend([1]*(a+1))
            i += 2
            a = 0
        elif w[i+1]:
            w1.extend([1]*(a+1) + [0])
            i += 2
            a = 0
        else:
            w1.append(0)
            i += 1
    w1.extend([0]*a)
    return Word(w1)

Weight

Created

Jan 16, 2023 at 17:43 by Martin Rubey

Updated

Jan 16, 2023 at 17:43 by Martin Rubey