Identifier
Values
00 => [2] => 10 => 11 => 2
11 => [2] => 10 => 11 => 2
000 => [3] => 100 => 011 => 1
001 => [2,1] => 101 => 110 => 1
011 => [1,2] => 110 => 111 => 3
100 => [1,2] => 110 => 111 => 3
110 => [2,1] => 101 => 110 => 1
111 => [3] => 100 => 011 => 1
0000 => [4] => 1000 => 0011 => 1
0001 => [3,1] => 1001 => 0110 => 2
0010 => [2,1,1] => 1011 => 1100 => 1
0011 => [2,2] => 1010 => 1101 => 2
0100 => [1,1,2] => 1110 => 1111 => 4
0110 => [1,2,1] => 1101 => 1110 => 2
0111 => [1,3] => 1100 => 0111 => 2
1000 => [1,3] => 1100 => 0111 => 2
1001 => [1,2,1] => 1101 => 1110 => 2
1011 => [1,1,2] => 1110 => 1111 => 4
1100 => [2,2] => 1010 => 1101 => 2
1101 => [2,1,1] => 1011 => 1100 => 1
1110 => [3,1] => 1001 => 0110 => 2
1111 => [4] => 1000 => 0011 => 1
search for individual values
searching the database for the individual values of this statistic
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
to binary word
Description
Return the composition as a binary word, treating ones as separators.
Encoding a positive integer $i$ as the word $10\dots 0$ consisting of a one followed by $i-1$ zeros, the binary word of a composition $(i_1,\dots,i_k)$ is the concatenation of of words for $i_1,\dots,i_k$.
The image of this map contains precisely the words which do not begin with a $0$.
Map
delta morphism
Description
Applies the delta morphism to a binary word.
The delta morphism of a finite word $w$ is the integer compositions composed of the lengths of consecutive runs of the same letter in $w$.
Map
path rowmotion
Description
Return the rowmotion of the binary word, regarded as a lattice path.
Consider the binary word of length $n$ as a lattice path with $n$ steps, where a 1 corresponds to an up step and a 0 corresponds to a down step.
This map returns the path whose peaks are the valleys of the original path with an up step appended.