- FindStat

Identifier

Mp00224: Binary words —runsort⟶ Binary words
Mp00104: Binary words —reverse⟶ Binary words
Mp00269: Binary words —flag zeros to zeros⟶ Binary words
St001491: Binary words ⟶ ℤ

Values

=> 1 => 1 => 1 => 1
=> 00 => 00 => 01 => 1
=> 11 => 11 => 11 => 2
=> 000 => 000 => 011 => 1
=> 001 => 100 => 010 => 1
=> 001 => 100 => 010 => 1
=> 011 => 110 => 001 => 1
=> 001 => 100 => 010 => 1
=> 011 => 110 => 001 => 1
=> 011 => 110 => 001 => 1
=> 111 => 111 => 111 => 3
=> 0000 => 0000 => 0111 => 2
=> 0001 => 1000 => 0110 => 2
=> 0001 => 1000 => 0110 => 2
=> 0011 => 1100 => 0101 => 0
=> 0001 => 1000 => 0110 => 2
=> 0011 => 1100 => 0101 => 0
=> 0111 => 1110 => 0011 => 1
=> 0001 => 1000 => 0110 => 2
=> 0011 => 1100 => 0101 => 0
=> 0011 => 1100 => 0101 => 0
=> 0111 => 1110 => 0011 => 1
=> 0011 => 1100 => 0101 => 0
=> 0111 => 1110 => 0011 => 1
=> 0111 => 1110 => 0011 => 1
=> 1111 => 1111 => 1111 => 4

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$F_{3} = 7\ q + q^{3}$

Description

The number of indecomposable projective-injective modules in the algebra corresponding to a subset.
Let

$A_n=K[x]/(x^n)$ .
We associate to a nonempty subset S of an (n-1)-set the module

$M_S$ , which is the direct sum of

$A_n$ -modules with indecomposable non-projective direct summands of dimension

$i$ when

$i$ is in

$S$ (note that such modules have vector space dimension at most n-1). Then the corresponding algebra associated to S is the stable endomorphism ring of

$M_S$ . We decode the subset as a binary word so that for example the subset

$S=\{1,3 \}$ of

$\{1,2,3 \}$ is decoded as 101.

Map

reverse

Description

Return the reversal of a binary word.

Map

runsort

Description

The word obtained by sorting the weakly increasing runs lexicographically.

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

$0\dots 0$ .