- FindStat

Identifier

Mp00234: Binary words —valleys-to-peaks⟶ Binary words
Mp00097: Binary words —delta morphism⟶ Integer compositions
St001777: Integer compositions ⟶ ℤ

Values

0 => 1 => [1] => 0

1 => 1 => [1] => 0

00 => 01 => [1,1] => 1

01 => 10 => [1,1] => 1

10 => 11 => [2] => 0

11 => 11 => [2] => 0

000 => 001 => [2,1] => 1

001 => 010 => [1,1,1] => 2

010 => 101 => [1,1,1] => 2

011 => 101 => [1,1,1] => 2

100 => 101 => [1,1,1] => 2

101 => 110 => [2,1] => 1

110 => 111 => [3] => 0

111 => 111 => [3] => 0

0000 => 0001 => [3,1] => 1

0001 => 0010 => [2,1,1] => 2

0010 => 0101 => [1,1,1,1] => 3

0011 => 0101 => [1,1,1,1] => 3

0100 => 1001 => [1,2,1] => 1

0101 => 1010 => [1,1,1,1] => 3

0110 => 1011 => [1,1,2] => 1

0111 => 1011 => [1,1,2] => 1

1000 => 1001 => [1,2,1] => 1

1001 => 1010 => [1,1,1,1] => 3

1010 => 1101 => [2,1,1] => 2

1011 => 1101 => [2,1,1] => 2

1100 => 1101 => [2,1,1] => 2

1101 => 1110 => [3,1] => 1

1110 => 1111 => [4] => 0

1111 => 1111 => [4] => 0

00000 => 00001 => [4,1] => 1

00001 => 00010 => [3,1,1] => 2

00010 => 00101 => [2,1,1,1] => 3

00011 => 00101 => [2,1,1,1] => 3

00100 => 01001 => [1,1,2,1] => 2

00101 => 01010 => [1,1,1,1,1] => 4

00110 => 01011 => [1,1,1,2] => 2

00111 => 01011 => [1,1,1,2] => 2

01000 => 10001 => [1,3,1] => 1

01001 => 10010 => [1,2,1,1] => 2

01010 => 10101 => [1,1,1,1,1] => 4

01011 => 10101 => [1,1,1,1,1] => 4

01100 => 10101 => [1,1,1,1,1] => 4

01101 => 10110 => [1,1,2,1] => 2

01110 => 10111 => [1,1,3] => 1

01111 => 10111 => [1,1,3] => 1

10000 => 10001 => [1,3,1] => 1

10001 => 10010 => [1,2,1,1] => 2

10010 => 10101 => [1,1,1,1,1] => 4

10011 => 10101 => [1,1,1,1,1] => 4

10100 => 11001 => [2,2,1] => 2

10101 => 11010 => [2,1,1,1] => 3

10110 => 11011 => [2,1,2] => 1

10111 => 11011 => [2,1,2] => 1

11000 => 11001 => [2,2,1] => 2

11001 => 11010 => [2,1,1,1] => 3

11010 => 11101 => [3,1,1] => 2

11011 => 11101 => [3,1,1] => 2

11100 => 11101 => [3,1,1] => 2

11101 => 11110 => [4,1] => 1

11110 => 11111 => [5] => 0

11111 => 11111 => [5] => 0

000000 => 000001 => [5,1] => 1

000001 => 000010 => [4,1,1] => 2

000010 => 000101 => [3,1,1,1] => 3

000011 => 000101 => [3,1,1,1] => 3

000100 => 001001 => [2,1,2,1] => 2

000101 => 001010 => [2,1,1,1,1] => 4

000110 => 001011 => [2,1,1,2] => 2

000111 => 001011 => [2,1,1,2] => 2

001000 => 010001 => [1,1,3,1] => 2

001001 => 010010 => [1,1,2,1,1] => 3

001010 => 010101 => [1,1,1,1,1,1] => 5

001011 => 010101 => [1,1,1,1,1,1] => 5

001100 => 010101 => [1,1,1,1,1,1] => 5

001101 => 010110 => [1,1,1,2,1] => 3

001110 => 010111 => [1,1,1,3] => 2

001111 => 010111 => [1,1,1,3] => 2

010000 => 100001 => [1,4,1] => 1

010001 => 100010 => [1,3,1,1] => 2

010010 => 100101 => [1,2,1,1,1] => 3

010011 => 100101 => [1,2,1,1,1] => 3

010100 => 101001 => [1,1,1,2,1] => 3

010101 => 101010 => [1,1,1,1,1,1] => 5

010110 => 101011 => [1,1,1,1,2] => 3

010111 => 101011 => [1,1,1,1,2] => 3

011000 => 101001 => [1,1,1,2,1] => 3

011001 => 101010 => [1,1,1,1,1,1] => 5

011010 => 101101 => [1,1,2,1,1] => 3

011011 => 101101 => [1,1,2,1,1] => 3

011100 => 101101 => [1,1,2,1,1] => 3

011101 => 101110 => [1,1,3,1] => 2

011110 => 101111 => [1,1,4] => 1

011111 => 101111 => [1,1,4] => 1

100000 => 100001 => [1,4,1] => 1

100001 => 100010 => [1,3,1,1] => 2

100010 => 100101 => [1,2,1,1,1] => 3

100011 => 100101 => [1,2,1,1,1] => 3

100100 => 101001 => [1,1,1,2,1] => 3

100101 => 101010 => [1,1,1,1,1,1] => 5

100110 => 101011 => [1,1,1,1,2] => 3

>>> Load all 1357 entries. <<<

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$F_{1} = 2$

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

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

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

$F_{5} = 2 + 8\ q + 12\ q^{2} + 4\ q^{3} + 6\ q^{4}$

$F_{6} = 2 + 12\ q + 16\ q^{2} + 22\ q^{3} + 6\ q^{4} + 6\ q^{5}$

$F_{7} = 2 + 14\ q + 32\ q^{2} + 36\ q^{3} + 28\ q^{4} + 6\ q^{5} + 10\ q^{6}$

$F_{8} = 2 + 18\ q + 48\ q^{2} + 72\ q^{3} + 48\ q^{4} + 48\ q^{5} + 10\ q^{6} + 10\ q^{7}$

$F_{9} = 2 + 20\ q + 76\ q^{2} + 114\ q^{3} + 122\ q^{4} + 92\ q^{5} + 60\ q^{6} + 10\ q^{7} + 16\ q^{8}$

Description

The number of weak descents in an integer composition.
A weak descent of an integer composition

$\alpha=(a_1, \dots, a_n)$ is an index

$1\leq i < n$ such that

$a_i \geq a_{i+1}$ .

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.

Map

delta morphism

Description

Applies the delta morphism to a binary word.
The delta morphism of a finite word

$w$ is the integer compositions composed of the lengths of consecutive runs of the same letter in

$w$ .