- FindStat

Identifier

Mp00234: Binary words —valleys-to-peaks⟶ Binary words
Mp00178: Binary words —to composition⟶ Integer compositions
St000758: Integer compositions ⟶ ℤ

Values

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

1 => 1 => [1,1] => 1

00 => 01 => [2,1] => 1

01 => 10 => [1,2] => 2

10 => 11 => [1,1,1] => 1

11 => 11 => [1,1,1] => 1

000 => 001 => [3,1] => 1

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

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

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

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

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

110 => 111 => [1,1,1,1] => 1

111 => 111 => [1,1,1,1] => 1

0000 => 0001 => [4,1] => 1

0001 => 0010 => [3,2] => 2

0010 => 0101 => [2,2,1] => 2

0011 => 0101 => [2,2,1] => 2

0100 => 1001 => [1,3,1] => 2

0101 => 1010 => [1,2,2] => 2

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

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

1000 => 1001 => [1,3,1] => 2

1001 => 1010 => [1,2,2] => 2

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

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

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

1101 => 1110 => [1,1,1,2] => 2

1110 => 1111 => [1,1,1,1,1] => 1

1111 => 1111 => [1,1,1,1,1] => 1

00000 => 00001 => [5,1] => 1

00001 => 00010 => [4,2] => 2

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

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

00100 => 01001 => [2,3,1] => 2

00101 => 01010 => [2,2,2] => 2

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

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

01000 => 10001 => [1,4,1] => 2

01001 => 10010 => [1,3,2] => 2

01010 => 10101 => [1,2,2,1] => 2

01011 => 10101 => [1,2,2,1] => 2

01100 => 10101 => [1,2,2,1] => 2

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

01110 => 10111 => [1,2,1,1,1] => 2

01111 => 10111 => [1,2,1,1,1] => 2

10000 => 10001 => [1,4,1] => 2

10001 => 10010 => [1,3,2] => 2

10010 => 10101 => [1,2,2,1] => 2

10011 => 10101 => [1,2,2,1] => 2

10100 => 11001 => [1,1,3,1] => 2

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

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

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

11000 => 11001 => [1,1,3,1] => 2

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

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

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

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

11101 => 11110 => [1,1,1,1,2] => 2

11110 => 11111 => [1,1,1,1,1,1] => 1

11111 => 11111 => [1,1,1,1,1,1] => 1

000000 => 000001 => [6,1] => 1

000001 => 000010 => [5,2] => 2

000010 => 000101 => [4,2,1] => 2

000011 => 000101 => [4,2,1] => 2

000100 => 001001 => [3,3,1] => 2

000101 => 001010 => [3,2,2] => 2

000110 => 001011 => [3,2,1,1] => 2

000111 => 001011 => [3,2,1,1] => 2

001000 => 010001 => [2,4,1] => 2

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

001010 => 010101 => [2,2,2,1] => 2

001011 => 010101 => [2,2,2,1] => 2

001100 => 010101 => [2,2,2,1] => 2

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

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

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

010000 => 100001 => [1,5,1] => 2

010001 => 100010 => [1,4,2] => 2

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

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

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

010101 => 101010 => [1,2,2,2] => 2

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

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

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

011001 => 101010 => [1,2,2,2] => 2

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

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

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

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

011110 => 101111 => [1,2,1,1,1,1] => 2

011111 => 101111 => [1,2,1,1,1,1] => 2

100000 => 100001 => [1,5,1] => 2

100001 => 100010 => [1,4,2] => 2

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

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

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

100101 => 101010 => [1,2,2,2] => 2

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

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Description

The length of the longest staircase fitting into an integer composition.
For a given composition $c_1,\dots,c_n$, this is the maximal number $\ell$ such that there are indices $i_1 < \dots < i_\ell$ with $c_{i_k} \geq k$, see [def.3.1, 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

to composition

Description

The composition corresponding to a binary word.
Prepending $1$ to a binary word $w$, the $i$-th part of the composition equals $1$ plus the number of zeros after the $i$-th $1$ in $w$.
This map is not surjective, since the empty composition does not have a preimage.