- FindStat

Identifier

Mp00316: Binary words —inverse Foata bijection⟶ Binary words
Mp00280: Binary words —path rowmotion⟶ Binary words
St000393: Binary words ⟶ ℤ

Values

0 => 0 => 1 => 1

1 => 1 => 0 => 1

00 => 00 => 01 => 1

01 => 01 => 10 => 2

10 => 10 => 11 => 2

11 => 11 => 00 => 2

000 => 000 => 001 => 2

001 => 001 => 010 => 2

010 => 100 => 011 => 2

011 => 011 => 100 => 3

100 => 010 => 101 => 2

101 => 101 => 110 => 3

110 => 110 => 111 => 3

111 => 111 => 000 => 3

0000 => 0000 => 0001 => 3

0001 => 0001 => 0010 => 3

0010 => 1000 => 0011 => 3

0011 => 0011 => 0100 => 3

0100 => 0100 => 1001 => 3

0101 => 1001 => 0110 => 3

0110 => 1100 => 0111 => 3

0111 => 0111 => 1000 => 4

1000 => 0010 => 0101 => 2

1001 => 0101 => 1010 => 3

1010 => 0110 => 1011 => 3

1011 => 1011 => 1100 => 4

1100 => 1010 => 1101 => 3

1101 => 1101 => 1110 => 4

1110 => 1110 => 1111 => 4

1111 => 1111 => 0000 => 4

00000 => 00000 => 00001 => 4

00001 => 00001 => 00010 => 4

00010 => 10000 => 00011 => 4

00011 => 00011 => 00100 => 4

00100 => 01000 => 10001 => 4

00101 => 10001 => 00110 => 4

00110 => 11000 => 00111 => 4

00111 => 00111 => 01000 => 4

01000 => 00100 => 01001 => 3

01001 => 01001 => 10010 => 4

01010 => 01100 => 10011 => 4

01011 => 10011 => 01100 => 4

01100 => 10100 => 11001 => 4

01101 => 11001 => 01110 => 4

01110 => 11100 => 01111 => 4

01111 => 01111 => 10000 => 5

10000 => 00010 => 00101 => 3

10001 => 00101 => 01010 => 3

10010 => 00110 => 01011 => 3

10011 => 01011 => 10100 => 4

10100 => 10010 => 01101 => 3

10101 => 01101 => 10110 => 4

10110 => 01110 => 10111 => 4

10111 => 10111 => 11000 => 5

11000 => 01010 => 10101 => 3

11001 => 10101 => 11010 => 4

11010 => 10110 => 11011 => 4

11011 => 11011 => 11100 => 5

11100 => 11010 => 11101 => 4

11101 => 11101 => 11110 => 5

11110 => 11110 => 11111 => 5

11111 => 11111 => 00000 => 5

000000 => 000000 => 000001 => 5

000001 => 000001 => 000010 => 5

000010 => 100000 => 000011 => 5

000011 => 000011 => 000100 => 5

000100 => 010000 => 100001 => 5

000101 => 100001 => 000110 => 5

000110 => 110000 => 000111 => 5

000111 => 000111 => 001000 => 5

001000 => 001000 => 010001 => 4

001001 => 010001 => 100010 => 5

001010 => 011000 => 100011 => 5

001011 => 100011 => 001100 => 5

001100 => 101000 => 110001 => 5

001101 => 110001 => 001110 => 5

001110 => 111000 => 001111 => 5

001111 => 001111 => 010000 => 5

010000 => 000100 => 001001 => 4

010001 => 001001 => 010010 => 4

010010 => 001100 => 010011 => 4

010011 => 010011 => 100100 => 5

010100 => 100100 => 011001 => 4

010101 => 011001 => 100110 => 5

010110 => 011100 => 100111 => 5

010111 => 100111 => 011000 => 5

011000 => 010100 => 101001 => 4

011001 => 101001 => 110010 => 5

011010 => 101100 => 110011 => 5

011011 => 110011 => 011100 => 5

011100 => 110100 => 111001 => 5

011101 => 111001 => 011110 => 5

011110 => 111100 => 011111 => 5

011111 => 011111 => 100000 => 6

100000 => 000010 => 000101 => 4

100001 => 000101 => 001010 => 4

100010 => 000110 => 001011 => 4

100011 => 001011 => 010100 => 4

100100 => 100010 => 001101 => 4

100101 => 001101 => 010110 => 4

100110 => 001110 => 010111 => 4

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$F_{1} = 2\ q$

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

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

$F_{4} = q^{2} + 10\ q^{3} + 5\ q^{4}$

$F_{5} = 6\ q^{3} + 20\ q^{4} + 6\ q^{5}$

$F_{6} = q^{3} + 21\ q^{4} + 35\ q^{5} + 7\ q^{6}$

$F_{7} = 8\ q^{4} + 56\ q^{5} + 56\ q^{6} + 8\ q^{7}$

$F_{8} = q^{4} + 36\ q^{5} + 126\ q^{6} + 84\ q^{7} + 9\ q^{8}$

$F_{9} = 10\ q^{5} + 120\ q^{6} + 252\ q^{7} + 120\ q^{8} + 10\ q^{9}$

Description

The number of strictly increasing runs in a binary word.

Map

inverse Foata bijection

Description

The inverse of Foata's bijection.
See Mp00096Foata bijection.

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.