- FindStat

Identifier

Mp00234: Binary words —valleys-to-peaks⟶ Binary words
St001372: Binary words ⟶ ℤ

Values

0 => 1 => 1

1 => 1 => 1

00 => 01 => 1

01 => 10 => 1

10 => 11 => 2

11 => 11 => 2

000 => 001 => 1

001 => 010 => 1

010 => 101 => 2

011 => 101 => 2

100 => 101 => 2

101 => 110 => 2

110 => 111 => 3

111 => 111 => 3

0000 => 0001 => 1

0001 => 0010 => 1

0010 => 0101 => 1

0011 => 0101 => 1

0100 => 1001 => 2

0101 => 1010 => 1

0110 => 1011 => 3

0111 => 1011 => 3

1000 => 1001 => 2

1001 => 1010 => 1

1010 => 1101 => 3

1011 => 1101 => 3

1100 => 1101 => 3

1101 => 1110 => 3

1110 => 1111 => 4

1111 => 1111 => 4

00000 => 00001 => 1

00001 => 00010 => 1

00010 => 00101 => 1

00011 => 00101 => 1

00100 => 01001 => 1

00101 => 01010 => 2

00110 => 01011 => 2

00111 => 01011 => 2

01000 => 10001 => 2

01001 => 10010 => 1

01010 => 10101 => 2

01011 => 10101 => 2

01100 => 10101 => 2

01101 => 10110 => 2

01110 => 10111 => 4

01111 => 10111 => 4

10000 => 10001 => 2

10001 => 10010 => 1

10010 => 10101 => 2

10011 => 10101 => 2

10100 => 11001 => 3

10101 => 11010 => 2

10110 => 11011 => 4

10111 => 11011 => 4

11000 => 11001 => 3

11001 => 11010 => 2

11010 => 11101 => 4

11011 => 11101 => 4

11100 => 11101 => 4

11101 => 11110 => 4

11110 => 11111 => 5

11111 => 11111 => 5

000000 => 000001 => 1

000001 => 000010 => 1

000010 => 000101 => 1

000011 => 000101 => 1

000100 => 001001 => 1

000101 => 001010 => 2

000110 => 001011 => 2

000111 => 001011 => 2

001000 => 010001 => 1

001001 => 010010 => 2

001010 => 010101 => 1

001011 => 010101 => 1

001100 => 010101 => 1

001101 => 010110 => 3

001110 => 010111 => 3

001111 => 010111 => 3

010000 => 100001 => 2

010001 => 100010 => 1

010010 => 100101 => 2

010011 => 100101 => 2

010100 => 101001 => 2

010101 => 101010 => 1

010110 => 101011 => 3

010111 => 101011 => 3

011000 => 101001 => 2

011001 => 101010 => 1

011010 => 101101 => 2

011011 => 101101 => 2

011100 => 101101 => 2

011101 => 101110 => 3

011110 => 101111 => 5

011111 => 101111 => 5

100000 => 100001 => 2

100001 => 100010 => 1

100010 => 100101 => 2

100011 => 100101 => 2

100100 => 101001 => 2

100101 => 101010 => 1

100110 => 101011 => 3

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$F_{1} = 2\ q$

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

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

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

$F_{5} = 7\ q + 13\ q^{2} + 2\ q^{3} + 8\ q^{4} + 2\ q^{5}$

$F_{6} = 14\ q + 19\ q^{2} + 17\ q^{3} + 2\ q^{4} + 10\ q^{5} + 2\ q^{6}$

$F_{7} = 21\ q + 42\ q^{2} + 28\ q^{3} + 21\ q^{4} + 2\ q^{5} + 12\ q^{6} + 2\ q^{7}$

$F_{8} = 39\ q + 83\ q^{2} + 57\ q^{3} + 34\ q^{4} + 25\ q^{5} + 2\ q^{6} + 14\ q^{7} + 2\ q^{8}$

$F_{9} = 65\ q + 161\ q^{2} + 122\ q^{3} + 75\ q^{4} + 40\ q^{5} + 29\ q^{6} + 2\ q^{7} + 16\ q^{8} + 2\ q^{9}$

Description

The length of a longest cyclic run of ones of a binary word.
Consider the binary word as a cyclic arrangement of ones and zeros. Then this statistic is the length of the longest continuous sequence of ones in this arrangement.

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.