Identifier
Values
([(0,1)],2) => [1] => 10 => 11 => 2
([(1,2)],3) => [1] => 10 => 11 => 2
([(0,2),(1,2)],3) => [1,1] => 110 => 100 => 1
([(0,1),(0,2),(1,2)],3) => [3] => 1000 => 1101 => 2
([(2,3)],4) => [1] => 10 => 11 => 2
([(1,3),(2,3)],4) => [1,1] => 110 => 100 => 1
([(0,3),(1,3),(2,3)],4) => [1,1,1] => 1110 => 1011 => 2
([(0,3),(1,2)],4) => [1,1] => 110 => 100 => 1
([(0,3),(1,2),(2,3)],4) => [1,1,1] => 1110 => 1011 => 2
([(1,2),(1,3),(2,3)],4) => [3] => 1000 => 1101 => 2
([(3,4)],5) => [1] => 10 => 11 => 2
([(2,4),(3,4)],5) => [1,1] => 110 => 100 => 1
([(1,4),(2,4),(3,4)],5) => [1,1,1] => 1110 => 1011 => 2
([(1,4),(2,3)],5) => [1,1] => 110 => 100 => 1
([(1,4),(2,3),(3,4)],5) => [1,1,1] => 1110 => 1011 => 2
([(0,1),(2,4),(3,4)],5) => [1,1,1] => 1110 => 1011 => 2
([(2,3),(2,4),(3,4)],5) => [3] => 1000 => 1101 => 2
([(4,5)],6) => [1] => 10 => 11 => 2
([(3,5),(4,5)],6) => [1,1] => 110 => 100 => 1
([(2,5),(3,5),(4,5)],6) => [1,1,1] => 1110 => 1011 => 2
([(2,5),(3,4)],6) => [1,1] => 110 => 100 => 1
([(2,5),(3,4),(4,5)],6) => [1,1,1] => 1110 => 1011 => 2
([(1,2),(3,5),(4,5)],6) => [1,1,1] => 1110 => 1011 => 2
([(3,4),(3,5),(4,5)],6) => [3] => 1000 => 1101 => 2
([(0,5),(1,4),(2,3)],6) => [1,1,1] => 1110 => 1011 => 2
([(5,6)],7) => [1] => 10 => 11 => 2
([(4,6),(5,6)],7) => [1,1] => 110 => 100 => 1
([(3,6),(4,6),(5,6)],7) => [1,1,1] => 1110 => 1011 => 2
([(3,6),(4,5)],7) => [1,1] => 110 => 100 => 1
([(3,6),(4,5),(5,6)],7) => [1,1,1] => 1110 => 1011 => 2
([(2,3),(4,6),(5,6)],7) => [1,1,1] => 1110 => 1011 => 2
([(4,5),(4,6),(5,6)],7) => [3] => 1000 => 1101 => 2
([(1,6),(2,5),(3,4)],7) => [1,1,1] => 1110 => 1011 => 2
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 edge-partition of biconnected components
Description
Sends a graph to the partition recording the number of edges in its biconnected components.
The biconnected components are also known as blocks of a graph.
Map
alternating inverse
Description
Sends a binary word $w_1\cdots w_m$ to the binary word $v_1 \cdots v_m$ with $v_i = w_i$ if $i$ is odd and $v_i = 1 - w_i$ if $i$ is even.
This map is used in [1], see Definitions 3.2 and 5.1.
Map
to binary word
Description
Return the partition as binary word, by traversing its shape from the first row to the last row, down steps as 1 and left steps as 0.