- FindStat

Identifier

Mp00104: Binary words —reverse⟶ Binary words
Mp00269: Binary words —flag zeros to zeros⟶ Binary words
Mp00178: Binary words —to composition⟶ Integer compositions
St001235: Integer compositions ⟶ ℤ

Values

=> 0 => 0 => [2] => 1
=> 1 => 1 => [1,1] => 2
=> 00 => 01 => [2,1] => 2
=> 10 => 00 => [3] => 1
=> 01 => 10 => [1,2] => 2
=> 11 => 11 => [1,1,1] => 3
=> 000 => 011 => [2,1,1] => 3
=> 100 => 010 => [2,2] => 2
=> 010 => 000 => [4] => 1
=> 110 => 001 => [3,1] => 2
=> 001 => 101 => [1,2,1] => 2
=> 101 => 100 => [1,3] => 2
=> 011 => 110 => [1,1,2] => 3
=> 111 => 111 => [1,1,1,1] => 4
=> 0000 => 0111 => [2,1,1,1] => 4
=> 1000 => 0110 => [2,1,2] => 3
=> 0100 => 0100 => [2,3] => 2
=> 1100 => 0101 => [2,2,1] => 2
=> 0010 => 0001 => [4,1] => 2
=> 1010 => 0000 => [5] => 1
=> 0110 => 0010 => [3,2] => 2
=> 1110 => 0011 => [3,1,1] => 3
=> 0001 => 1011 => [1,2,1,1] => 3
=> 1001 => 1010 => [1,2,2] => 2
=> 0101 => 1000 => [1,4] => 2
=> 1101 => 1001 => [1,3,1] => 2
=> 0011 => 1101 => [1,1,2,1] => 3
=> 1011 => 1100 => [1,1,3] => 3
=> 0111 => 1110 => [1,1,1,2] => 4
=> 1111 => 1111 => [1,1,1,1,1] => 5
=> 00000 => 01111 => [2,1,1,1,1] => 5
=> 10000 => 01110 => [2,1,1,2] => 4
=> 01000 => 01100 => [2,1,3] => 3
=> 11000 => 01101 => [2,1,2,1] => 3
=> 00100 => 01001 => [2,3,1] => 2
=> 10100 => 01000 => [2,4] => 2
=> 01100 => 01010 => [2,2,2] => 2
=> 11100 => 01011 => [2,2,1,1] => 3
=> 00010 => 00011 => [4,1,1] => 3
=> 10010 => 00010 => [4,2] => 2
=> 01010 => 00000 => [6] => 1
=> 11010 => 00001 => [5,1] => 2
=> 00110 => 00101 => [3,2,1] => 2
=> 10110 => 00100 => [3,3] => 2
=> 01110 => 00110 => [3,1,2] => 3
=> 11110 => 00111 => [3,1,1,1] => 4
=> 00001 => 10111 => [1,2,1,1,1] => 4
=> 10001 => 10110 => [1,2,1,2] => 3
=> 01001 => 10100 => [1,2,3] => 2
=> 11001 => 10101 => [1,2,2,1] => 2
=> 00101 => 10001 => [1,4,1] => 2
=> 10101 => 10000 => [1,5] => 2
=> 01101 => 10010 => [1,3,2] => 2
=> 11101 => 10011 => [1,3,1,1] => 3
=> 00011 => 11011 => [1,1,2,1,1] => 3
=> 10011 => 11010 => [1,1,2,2] => 3
=> 01011 => 11000 => [1,1,4] => 3
=> 11011 => 11001 => [1,1,3,1] => 3
=> 00111 => 11101 => [1,1,1,2,1] => 4
=> 10111 => 11100 => [1,1,1,3] => 4
=> 01111 => 11110 => [1,1,1,1,2] => 5
=> 11111 => 11111 => [1,1,1,1,1,1] => 6
 =>  =>  => [1] => 1

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$F_{1} = q + q^{2}$

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

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

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

$F_{5} = q + 12\ q^{2} + 11\ q^{3} + 5\ q^{4} + 2\ q^{5} + q^{6}$

Description

The global dimension of the corresponding Comp-Nakayama algebra.
We identify the composition [n1-1,n2-1,...,nr-1] with the Nakayama algebra with Kupisch series [n1,n1-1,...,2,n2,n2-1,...,2,...,nr,nr-1,...,3,2,1]. We call such Nakayama algebras with Kupisch series corresponding to a integer composition "Comp-Nakayama algebra".

Map

reverse

Description

Return the reversal of a binary word.

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

flag zeros to zeros

Description

Return a binary word of the same length, such that the number of zeros equals the number of occurrences of

$10$ in the word obtained from the original word by prepending the reverse of the complement.
For example, the image of the word

$w=1\dots 1$ is

$1\dots 1$ , because

$0\dots 01\dots 1$ has no occurrences of

$10$ . The words

$10\dots 10$ and

$010\dots 10$ have image

$0\dots 0$ .