Identifier
Values
[1,1] => [1] => [1] => 10 => 1
[2,1] => [1] => [1] => 10 => 1
[1,1,1] => [1,1] => [2] => 100 => 1
[3,1] => [1] => [1] => 10 => 1
[2,2] => [2] => [1,1] => 110 => 1
[2,1,1] => [1,1] => [2] => 100 => 1
[1,1,1,1] => [1,1,1] => [2,1] => 1010 => 0
[4,1] => [1] => [1] => 10 => 1
[3,2] => [2] => [1,1] => 110 => 1
[3,1,1] => [1,1] => [2] => 100 => 1
[2,2,1] => [2,1] => [3] => 1000 => 1
[2,1,1,1] => [1,1,1] => [2,1] => 1010 => 0
[5,1] => [1] => [1] => 10 => 1
[4,2] => [2] => [1,1] => 110 => 1
[4,1,1] => [1,1] => [2] => 100 => 1
[3,3] => [3] => [1,1,1] => 1110 => 2
[3,2,1] => [2,1] => [3] => 1000 => 1
[3,1,1,1] => [1,1,1] => [2,1] => 1010 => 0
[2,2,1,1] => [2,1,1] => [2,2] => 1100 => 1
[6,1] => [1] => [1] => 10 => 1
[5,2] => [2] => [1,1] => 110 => 1
[5,1,1] => [1,1] => [2] => 100 => 1
[4,3] => [3] => [1,1,1] => 1110 => 2
[4,2,1] => [2,1] => [3] => 1000 => 1
[4,1,1,1] => [1,1,1] => [2,1] => 1010 => 0
[3,2,1,1] => [2,1,1] => [2,2] => 1100 => 1
[7,1] => [1] => [1] => 10 => 1
[6,2] => [2] => [1,1] => 110 => 1
[6,1,1] => [1,1] => [2] => 100 => 1
[5,3] => [3] => [1,1,1] => 1110 => 2
[5,2,1] => [2,1] => [3] => 1000 => 1
[5,1,1,1] => [1,1,1] => [2,1] => 1010 => 0
[4,2,1,1] => [2,1,1] => [2,2] => 1100 => 1
[8,1] => [1] => [1] => 10 => 1
[7,2] => [2] => [1,1] => 110 => 1
[7,1,1] => [1,1] => [2] => 100 => 1
[6,3] => [3] => [1,1,1] => 1110 => 2
[6,2,1] => [2,1] => [3] => 1000 => 1
[6,1,1,1] => [1,1,1] => [2,1] => 1010 => 0
[5,2,1,1] => [2,1,1] => [2,2] => 1100 => 1
[9,1] => [1] => [1] => 10 => 1
[8,2] => [2] => [1,1] => 110 => 1
[8,1,1] => [1,1] => [2] => 100 => 1
[7,3] => [3] => [1,1,1] => 1110 => 2
[7,2,1] => [2,1] => [3] => 1000 => 1
[7,1,1,1] => [1,1,1] => [2,1] => 1010 => 0
[6,2,1,1] => [2,1,1] => [2,2] => 1100 => 1
[10,1] => [1] => [1] => 10 => 1
[9,2] => [2] => [1,1] => 110 => 1
[9,1,1] => [1,1] => [2] => 100 => 1
[8,3] => [3] => [1,1,1] => 1110 => 2
[8,2,1] => [2,1] => [3] => 1000 => 1
[8,1,1,1] => [1,1,1] => [2,1] => 1010 => 0
[7,2,1,1] => [2,1,1] => [2,2] => 1100 => 1
[11,1] => [1] => [1] => 10 => 1
[10,2] => [2] => [1,1] => 110 => 1
[10,1,1] => [1,1] => [2] => 100 => 1
[9,3] => [3] => [1,1,1] => 1110 => 2
[9,2,1] => [2,1] => [3] => 1000 => 1
[9,1,1,1] => [1,1,1] => [2,1] => 1010 => 0
[8,2,1,1] => [2,1,1] => [2,2] => 1100 => 1
[12,1] => [1] => [1] => 10 => 1
[11,2] => [2] => [1,1] => 110 => 1
[11,1,1] => [1,1] => [2] => 100 => 1
[10,3] => [3] => [1,1,1] => 1110 => 2
[10,2,1] => [2,1] => [3] => 1000 => 1
[10,1,1,1] => [1,1,1] => [2,1] => 1010 => 0
[9,2,1,1] => [2,1,1] => [2,2] => 1100 => 1
[13,1] => [1] => [1] => 10 => 1
[12,2] => [2] => [1,1] => 110 => 1
[12,1,1] => [1,1] => [2] => 100 => 1
[11,3] => [3] => [1,1,1] => 1110 => 2
[11,2,1] => [2,1] => [3] => 1000 => 1
[11,1,1,1] => [1,1,1] => [2,1] => 1010 => 0
[10,2,1,1] => [2,1,1] => [2,2] => 1100 => 1
[14,1] => [1] => [1] => 10 => 1
[13,2] => [2] => [1,1] => 110 => 1
[13,1,1] => [1,1] => [2] => 100 => 1
[12,3] => [3] => [1,1,1] => 1110 => 2
[12,2,1] => [2,1] => [3] => 1000 => 1
[12,1,1,1] => [1,1,1] => [2,1] => 1010 => 0
[11,2,1,1] => [2,1,1] => [2,2] => 1100 => 1
[15,1] => [1] => [1] => 10 => 1
[14,2] => [2] => [1,1] => 110 => 1
[14,1,1] => [1,1] => [2] => 100 => 1
[13,3] => [3] => [1,1,1] => 1110 => 2
[13,2,1] => [2,1] => [3] => 1000 => 1
[13,1,1,1] => [1,1,1] => [2,1] => 1010 => 0
[12,2,1,1] => [2,1,1] => [2,2] => 1100 => 1
[16,1] => [1] => [1] => 10 => 1
[15,2] => [2] => [1,1] => 110 => 1
[15,1,1] => [1,1] => [2] => 100 => 1
[14,3] => [3] => [1,1,1] => 1110 => 2
[14,2,1] => [2,1] => [3] => 1000 => 1
[14,1,1,1] => [1,1,1] => [2,1] => 1010 => 0
[13,2,1,1] => [2,1,1] => [2,2] => 1100 => 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
first row removal
Description
Removes the first entry of an integer partition
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
Loehr-Warrington
Description
Return a partition whose diagonal inversion number is the length of the preimage.