- FindStat

Identifier

Mp00104: Binary words —reverse⟶ Binary words
Mp00280: Binary words —path rowmotion⟶ Binary words
Mp00104: Binary words —reverse⟶ Binary words
St000326: Binary words ⟶ ℤ

Values

0 => 0 => 1 => 1 => 1

1 => 1 => 0 => 0 => 2

00 => 00 => 01 => 10 => 1

01 => 10 => 11 => 11 => 1

10 => 01 => 10 => 01 => 2

11 => 11 => 00 => 00 => 3

000 => 000 => 001 => 100 => 1

001 => 100 => 011 => 110 => 1

010 => 010 => 101 => 101 => 1

011 => 110 => 111 => 111 => 1

100 => 001 => 010 => 010 => 2

101 => 101 => 110 => 011 => 2

110 => 011 => 100 => 001 => 3

111 => 111 => 000 => 000 => 4

0000 => 0000 => 0001 => 1000 => 1

0001 => 1000 => 0011 => 1100 => 1

0010 => 0100 => 1001 => 1001 => 1

0011 => 1100 => 0111 => 1110 => 1

0100 => 0010 => 0101 => 1010 => 1

0101 => 1010 => 1101 => 1011 => 1

0110 => 0110 => 1011 => 1101 => 1

0111 => 1110 => 1111 => 1111 => 1

1000 => 0001 => 0010 => 0100 => 2

1001 => 1001 => 0110 => 0110 => 2

1010 => 0101 => 1010 => 0101 => 2

1011 => 1101 => 1110 => 0111 => 2

1100 => 0011 => 0100 => 0010 => 3

1101 => 1011 => 1100 => 0011 => 3

1110 => 0111 => 1000 => 0001 => 4

1111 => 1111 => 0000 => 0000 => 5

00000 => 00000 => 00001 => 10000 => 1

00001 => 10000 => 00011 => 11000 => 1

00010 => 01000 => 10001 => 10001 => 1

00011 => 11000 => 00111 => 11100 => 1

00100 => 00100 => 01001 => 10010 => 1

00101 => 10100 => 11001 => 10011 => 1

00110 => 01100 => 10011 => 11001 => 1

00111 => 11100 => 01111 => 11110 => 1

01000 => 00010 => 00101 => 10100 => 1

01001 => 10010 => 01101 => 10110 => 1

01010 => 01010 => 10101 => 10101 => 1

01011 => 11010 => 11101 => 10111 => 1

01100 => 00110 => 01011 => 11010 => 1

01101 => 10110 => 11011 => 11011 => 1

01110 => 01110 => 10111 => 11101 => 1

01111 => 11110 => 11111 => 11111 => 1

10000 => 00001 => 00010 => 01000 => 2

10001 => 10001 => 00110 => 01100 => 2

10010 => 01001 => 10010 => 01001 => 2

10011 => 11001 => 01110 => 01110 => 2

10100 => 00101 => 01010 => 01010 => 2

10101 => 10101 => 11010 => 01011 => 2

10110 => 01101 => 10110 => 01101 => 2

10111 => 11101 => 11110 => 01111 => 2

11000 => 00011 => 00100 => 00100 => 3

11001 => 10011 => 01100 => 00110 => 3

11010 => 01011 => 10100 => 00101 => 3

11011 => 11011 => 11100 => 00111 => 3

11100 => 00111 => 01000 => 00010 => 4

11101 => 10111 => 11000 => 00011 => 4

11110 => 01111 => 10000 => 00001 => 5

11111 => 11111 => 00000 => 00000 => 6

000000 => 000000 => 000001 => 100000 => 1

000001 => 100000 => 000011 => 110000 => 1

000010 => 010000 => 100001 => 100001 => 1

000011 => 110000 => 000111 => 111000 => 1

000100 => 001000 => 010001 => 100010 => 1

000101 => 101000 => 110001 => 100011 => 1

000110 => 011000 => 100011 => 110001 => 1

000111 => 111000 => 001111 => 111100 => 1

001000 => 000100 => 001001 => 100100 => 1

001001 => 100100 => 011001 => 100110 => 1

001010 => 010100 => 101001 => 100101 => 1

001011 => 110100 => 111001 => 100111 => 1

001100 => 001100 => 010011 => 110010 => 1

001101 => 101100 => 110011 => 110011 => 1

001110 => 011100 => 100111 => 111001 => 1

001111 => 111100 => 011111 => 111110 => 1

010000 => 000010 => 000101 => 101000 => 1

010001 => 100010 => 001101 => 101100 => 1

010010 => 010010 => 100101 => 101001 => 1

010011 => 110010 => 011101 => 101110 => 1

010100 => 001010 => 010101 => 101010 => 1

010101 => 101010 => 110101 => 101011 => 1

010110 => 011010 => 101101 => 101101 => 1

010111 => 111010 => 111101 => 101111 => 1

011000 => 000110 => 001011 => 110100 => 1

011001 => 100110 => 011011 => 110110 => 1

011010 => 010110 => 101011 => 110101 => 1

011011 => 110110 => 111011 => 110111 => 1

011100 => 001110 => 010111 => 111010 => 1

011101 => 101110 => 110111 => 111011 => 1

011110 => 011110 => 101111 => 111101 => 1

011111 => 111110 => 111111 => 111111 => 1

100000 => 000001 => 000010 => 010000 => 2

100001 => 100001 => 000110 => 011000 => 2

100010 => 010001 => 100010 => 010001 => 2

100011 => 110001 => 001110 => 011100 => 2

100100 => 001001 => 010010 => 010010 => 2

100101 => 101001 => 110010 => 010011 => 2

100110 => 011001 => 100110 => 011001 => 2

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$F_{1} = q + q^{2}$

$F_{2} = 2\ q + q^{2} + q^{3}$

$F_{3} = 4\ q + 2\ q^{2} + q^{3} + q^{4}$

$F_{4} = 8\ q + 4\ q^{2} + 2\ q^{3} + q^{4} + q^{5}$

$F_{5} = 16\ q + 8\ q^{2} + 4\ q^{3} + 2\ q^{4} + q^{5} + q^{6}$

$F_{6} = 32\ q + 16\ q^{2} + 8\ q^{3} + 4\ q^{4} + 2\ q^{5} + q^{6} + q^{7}$

$F_{7} = 64\ q + 32\ q^{2} + 16\ q^{3} + 8\ q^{4} + 4\ q^{5} + 2\ q^{6} + q^{7} + q^{8}$

$F_{8} = 128\ q + 64\ q^{2} + 32\ q^{3} + 16\ q^{4} + 8\ q^{5} + 4\ q^{6} + 2\ q^{7} + q^{8} + q^{9}$

$F_{9} = 256\ q + 128\ q^{2} + 64\ q^{3} + 32\ q^{4} + 16\ q^{5} + 8\ q^{6} + 4\ q^{7} + 2\ q^{8} + q^{9} + q^{10}$

Description

The position of the first one in a binary word after appending a 1 at the end.
Regarding the binary word as a subset of

$\{1,\dots,n,n+1\}$ that contains

$n+1$ , this is the minimal element of the set.

Map

reverse

Description

Return the reversal of a binary word.

Map

path rowmotion

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.