- FindStat

Identifier

Mp00224: Binary words —runsort⟶ Binary words
Mp00178: Binary words —to composition⟶ Integer compositions
St001777: Integer compositions ⟶ ℤ

Values

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

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

00 => 00 => [3] => 0

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

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

11 => 11 => [1,1,1] => 2

000 => 000 => [4] => 0

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

010 => 001 => [3,1] => 1

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

100 => 001 => [3,1] => 1

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

110 => 011 => [2,1,1] => 2

111 => 111 => [1,1,1,1] => 3

0000 => 0000 => [5] => 0

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

0010 => 0001 => [4,1] => 1

0011 => 0011 => [3,1,1] => 2

0100 => 0001 => [4,1] => 1

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

0110 => 0011 => [3,1,1] => 2

0111 => 0111 => [2,1,1,1] => 3

1000 => 0001 => [4,1] => 1

1001 => 0011 => [3,1,1] => 2

1010 => 0011 => [3,1,1] => 2

1011 => 0111 => [2,1,1,1] => 3

1100 => 0011 => [3,1,1] => 2

1101 => 0111 => [2,1,1,1] => 3

1110 => 0111 => [2,1,1,1] => 3

1111 => 1111 => [1,1,1,1,1] => 4

00000 => 00000 => [6] => 0

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

00010 => 00001 => [5,1] => 1

00011 => 00011 => [4,1,1] => 2

00100 => 00001 => [5,1] => 1

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

00110 => 00011 => [4,1,1] => 2

00111 => 00111 => [3,1,1,1] => 3

01000 => 00001 => [5,1] => 1

01001 => 00101 => [3,2,1] => 2

01010 => 00101 => [3,2,1] => 2

01011 => 01011 => [2,2,1,1] => 3

01100 => 00011 => [4,1,1] => 2

01101 => 01011 => [2,2,1,1] => 3

01110 => 00111 => [3,1,1,1] => 3

01111 => 01111 => [2,1,1,1,1] => 4

10000 => 00001 => [5,1] => 1

10001 => 00011 => [4,1,1] => 2

10010 => 00011 => [4,1,1] => 2

10011 => 00111 => [3,1,1,1] => 3

10100 => 00011 => [4,1,1] => 2

10101 => 01011 => [2,2,1,1] => 3

10110 => 00111 => [3,1,1,1] => 3

10111 => 01111 => [2,1,1,1,1] => 4

11000 => 00011 => [4,1,1] => 2

11001 => 00111 => [3,1,1,1] => 3

11010 => 00111 => [3,1,1,1] => 3

11011 => 01111 => [2,1,1,1,1] => 4

11100 => 00111 => [3,1,1,1] => 3

11101 => 01111 => [2,1,1,1,1] => 4

11110 => 01111 => [2,1,1,1,1] => 4

11111 => 11111 => [1,1,1,1,1,1] => 5

000000 => 000000 => [7] => 0

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

000010 => 000001 => [6,1] => 1

000011 => 000011 => [5,1,1] => 2

000100 => 000001 => [6,1] => 1

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

000110 => 000011 => [5,1,1] => 2

000111 => 000111 => [4,1,1,1] => 3

001000 => 000001 => [6,1] => 1

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

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

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

001100 => 000011 => [5,1,1] => 2

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

001110 => 000111 => [4,1,1,1] => 3

001111 => 001111 => [3,1,1,1,1] => 4

010000 => 000001 => [6,1] => 1

010001 => 000101 => [4,2,1] => 2

010010 => 000101 => [4,2,1] => 2

010011 => 001101 => [3,1,2,1] => 2

010100 => 000101 => [4,2,1] => 2

010101 => 010101 => [2,2,2,1] => 3

010110 => 001011 => [3,2,1,1] => 3

010111 => 010111 => [2,2,1,1,1] => 4

011000 => 000011 => [5,1,1] => 2

011001 => 001011 => [3,2,1,1] => 3

011010 => 001011 => [3,2,1,1] => 3

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

011100 => 000111 => [4,1,1,1] => 3

011101 => 010111 => [2,2,1,1,1] => 4

011110 => 001111 => [3,1,1,1,1] => 4

011111 => 011111 => [2,1,1,1,1,1] => 5

100000 => 000001 => [6,1] => 1

100001 => 000011 => [5,1,1] => 2

100010 => 000011 => [5,1,1] => 2

100011 => 000111 => [4,1,1,1] => 3

100100 => 000011 => [5,1,1] => 2

100101 => 001011 => [3,2,1,1] => 3

100110 => 000111 => [4,1,1,1] => 3

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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

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.

Map

runsort

Description

The word obtained by sorting the weakly increasing runs lexicographically.