- FindStat

Identifier

Mp00224: Binary words —runsort⟶ Binary words
Mp00178: Binary words —to composition⟶ Integer compositions
St000763: Integer compositions ⟶ ℤ

Values

0 => 0 => [2] => 1

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

00 => 00 => [3] => 1

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

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

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

000 => 000 => [4] => 1

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

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

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

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

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

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

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

0000 => 0000 => [5] => 1

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

00000 => 00000 => [6] => 1

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

000000 => 000000 => [7] => 1

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

>>> Load all 511 entries. <<<

search for individual values

searching the database for the individual values of this statistic

/ search for generating function

searching the database for statistics with the same generating function

click to show known generating functions

$F_{1} = 2\ q$

$F_{2} = 4\ q$

$F_{3} = 8\ q$

$F_{4} = 16\ q$

$F_{5} = 32\ q$

$F_{6} = 64\ q$

$F_{7} = 128\ q$

$F_{8} = 256\ q$

Description

The sum of the positions of the strong records of an integer composition.
A strong record is an element

$a_i$ such that

$a_i > a_j$ for all

$j < i$ . This statistic is the sum of the positions of the strong records.

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.