Identifier
Values
([(0,1)],2) => [1] => [1] => 10 => 1
([(1,2)],3) => [1] => [1] => 10 => 1
([(0,2),(1,2)],3) => [1,1] => [2] => 100 => 1
([(0,1),(0,2),(1,2)],3) => [3] => [1,1,1] => 1110 => 2
([(2,3)],4) => [1] => [1] => 10 => 1
([(1,3),(2,3)],4) => [1,1] => [2] => 100 => 1
([(0,3),(1,3),(2,3)],4) => [1,1,1] => [3] => 1000 => 1
([(0,3),(1,2)],4) => [1,1] => [2] => 100 => 1
([(0,3),(1,2),(2,3)],4) => [1,1,1] => [3] => 1000 => 1
([(1,2),(1,3),(2,3)],4) => [3] => [1,1,1] => 1110 => 2
([(3,4)],5) => [1] => [1] => 10 => 1
([(2,4),(3,4)],5) => [1,1] => [2] => 100 => 1
([(1,4),(2,4),(3,4)],5) => [1,1,1] => [3] => 1000 => 1
([(1,4),(2,3)],5) => [1,1] => [2] => 100 => 1
([(1,4),(2,3),(3,4)],5) => [1,1,1] => [3] => 1000 => 1
([(0,1),(2,4),(3,4)],5) => [1,1,1] => [3] => 1000 => 1
([(2,3),(2,4),(3,4)],5) => [3] => [1,1,1] => 1110 => 2
([(4,5)],6) => [1] => [1] => 10 => 1
([(3,5),(4,5)],6) => [1,1] => [2] => 100 => 1
([(2,5),(3,5),(4,5)],6) => [1,1,1] => [3] => 1000 => 1
([(2,5),(3,4)],6) => [1,1] => [2] => 100 => 1
([(2,5),(3,4),(4,5)],6) => [1,1,1] => [3] => 1000 => 1
([(1,2),(3,5),(4,5)],6) => [1,1,1] => [3] => 1000 => 1
([(3,4),(3,5),(4,5)],6) => [3] => [1,1,1] => 1110 => 2
([(0,5),(1,4),(2,3)],6) => [1,1,1] => [3] => 1000 => 1
([(5,6)],7) => [1] => [1] => 10 => 1
([(4,6),(5,6)],7) => [1,1] => [2] => 100 => 1
([(3,6),(4,6),(5,6)],7) => [1,1,1] => [3] => 1000 => 1
([(3,6),(4,5)],7) => [1,1] => [2] => 100 => 1
([(3,6),(4,5),(5,6)],7) => [1,1,1] => [3] => 1000 => 1
([(2,3),(4,6),(5,6)],7) => [1,1,1] => [3] => 1000 => 1
([(4,5),(4,6),(5,6)],7) => [3] => [1,1,1] => 1110 => 2
([(1,6),(2,5),(3,4)],7) => [1,1,1] => [3] => 1000 => 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 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.
Map
conjugate
Description
Return the conjugate partition of the partition.
The conjugate partition of the partition $\lambda$ of $n$ is the partition $\lambda^*$ whose Ferrers diagram is obtained from the diagram of $\lambda$ by interchanging rows with columns.
This is also called the associated partition or the transpose in the literature.
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.