Identifier
Values
[1,1] => [1] => 10 => 10 => 1
[2,1] => [1] => 10 => 10 => 1
[1,1,1] => [1,1] => 110 => 110 => 1
[3,1] => [1] => 10 => 10 => 1
[2,2] => [2] => 100 => 010 => 1
[2,1,1] => [1,1] => 110 => 110 => 1
[1,1,1,1] => [1,1,1] => 1110 => 1110 => 2
[4,1] => [1] => 10 => 10 => 1
[3,2] => [2] => 100 => 010 => 1
[3,1,1] => [1,1] => 110 => 110 => 1
[2,2,1] => [2,1] => 1010 => 1100 => 1
[2,1,1,1] => [1,1,1] => 1110 => 1110 => 2
[5,1] => [1] => 10 => 10 => 1
[4,2] => [2] => 100 => 010 => 1
[4,1,1] => [1,1] => 110 => 110 => 1
[3,3] => [3] => 1000 => 0010 => 1
[3,2,1] => [2,1] => 1010 => 1100 => 1
[3,1,1,1] => [1,1,1] => 1110 => 1110 => 2
[2,2,2] => [2,2] => 1100 => 0110 => 2
[6,1] => [1] => 10 => 10 => 1
[5,2] => [2] => 100 => 010 => 1
[5,1,1] => [1,1] => 110 => 110 => 1
[4,3] => [3] => 1000 => 0010 => 1
[4,2,1] => [2,1] => 1010 => 1100 => 1
[4,1,1,1] => [1,1,1] => 1110 => 1110 => 2
[3,2,2] => [2,2] => 1100 => 0110 => 2
[7,1] => [1] => 10 => 10 => 1
[6,2] => [2] => 100 => 010 => 1
[6,1,1] => [1,1] => 110 => 110 => 1
[5,3] => [3] => 1000 => 0010 => 1
[5,2,1] => [2,1] => 1010 => 1100 => 1
[5,1,1,1] => [1,1,1] => 1110 => 1110 => 2
[4,2,2] => [2,2] => 1100 => 0110 => 2
[8,1] => [1] => 10 => 10 => 1
[7,2] => [2] => 100 => 010 => 1
[7,1,1] => [1,1] => 110 => 110 => 1
[6,3] => [3] => 1000 => 0010 => 1
[6,2,1] => [2,1] => 1010 => 1100 => 1
[6,1,1,1] => [1,1,1] => 1110 => 1110 => 2
[5,2,2] => [2,2] => 1100 => 0110 => 2
[9,1] => [1] => 10 => 10 => 1
[8,2] => [2] => 100 => 010 => 1
[8,1,1] => [1,1] => 110 => 110 => 1
[7,3] => [3] => 1000 => 0010 => 1
[7,2,1] => [2,1] => 1010 => 1100 => 1
[7,1,1,1] => [1,1,1] => 1110 => 1110 => 2
[6,2,2] => [2,2] => 1100 => 0110 => 2
[10,1] => [1] => 10 => 10 => 1
[9,2] => [2] => 100 => 010 => 1
[9,1,1] => [1,1] => 110 => 110 => 1
[8,3] => [3] => 1000 => 0010 => 1
[8,2,1] => [2,1] => 1010 => 1100 => 1
[8,1,1,1] => [1,1,1] => 1110 => 1110 => 2
[7,2,2] => [2,2] => 1100 => 0110 => 2
[11,1] => [1] => 10 => 10 => 1
[10,2] => [2] => 100 => 010 => 1
[10,1,1] => [1,1] => 110 => 110 => 1
[9,3] => [3] => 1000 => 0010 => 1
[9,2,1] => [2,1] => 1010 => 1100 => 1
[9,1,1,1] => [1,1,1] => 1110 => 1110 => 2
[8,2,2] => [2,2] => 1100 => 0110 => 2
[12,1] => [1] => 10 => 10 => 1
[11,2] => [2] => 100 => 010 => 1
[11,1,1] => [1,1] => 110 => 110 => 1
[10,3] => [3] => 1000 => 0010 => 1
[10,2,1] => [2,1] => 1010 => 1100 => 1
[10,1,1,1] => [1,1,1] => 1110 => 1110 => 2
[9,2,2] => [2,2] => 1100 => 0110 => 2
[13,1] => [1] => 10 => 10 => 1
[12,2] => [2] => 100 => 010 => 1
[12,1,1] => [1,1] => 110 => 110 => 1
[11,3] => [3] => 1000 => 0010 => 1
[11,2,1] => [2,1] => 1010 => 1100 => 1
[11,1,1,1] => [1,1,1] => 1110 => 1110 => 2
[10,2,2] => [2,2] => 1100 => 0110 => 2
[14,1] => [1] => 10 => 10 => 1
[13,2] => [2] => 100 => 010 => 1
[13,1,1] => [1,1] => 110 => 110 => 1
[12,3] => [3] => 1000 => 0010 => 1
[12,2,1] => [2,1] => 1010 => 1100 => 1
[12,1,1,1] => [1,1,1] => 1110 => 1110 => 2
[11,2,2] => [2,2] => 1100 => 0110 => 2
[15,1] => [1] => 10 => 10 => 1
[14,2] => [2] => 100 => 010 => 1
[14,1,1] => [1,1] => 110 => 110 => 1
[13,3] => [3] => 1000 => 0010 => 1
[13,2,1] => [2,1] => 1010 => 1100 => 1
[13,1,1,1] => [1,1,1] => 1110 => 1110 => 2
[12,2,2] => [2,2] => 1100 => 0110 => 2
[16,1] => [1] => 10 => 10 => 1
[15,2] => [2] => 100 => 010 => 1
[15,1,1] => [1,1] => 110 => 110 => 1
[14,3] => [3] => 1000 => 0010 => 1
[14,2,1] => [2,1] => 1010 => 1100 => 1
[14,1,1,1] => [1,1,1] => 1110 => 1110 => 2
[13,2,2] => [2,2] => 1100 => 0110 => 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
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
Foata bijection
Description
The Foata bijection $\phi$ is a bijection on the set of words of given content (by a slight generalization of Section 2 in [1]).
Given a word $w_1 w_2 ... w_n$, compute the image inductively by starting with $\phi(w_1) = w_1$. At the $i$-th step, if $\phi(w_1 w_2 ... w_i) = v_1 v_2 ... v_i$, define $\phi(w_1 w_2 ... w_i w_{i+1})$ by placing $w_{i+1}$ on the end of the word $v_1 v_2 ... v_i$ and breaking the word up into blocks as follows.
  • If $w_{i+1} \geq v_i$, place a vertical line to the right of each $v_k$ for which $w_{i+1} \geq v_k$.
  • If $w_{i+1} < v_i$, place a vertical line to the right of each $v_k$ for which $w_{i+1} < v_k$.
In either case, place a vertical line at the start of the word as well. Now, within each block between vertical lines, cyclically shift the entries one place to the right.
For instance, to compute $\phi(4154223)$, the sequence of words is
  • 4,
  • |4|1 -- > 41,
  • |4|1|5 -- > 415,
  • |415|4 -- > 5414,
  • |5|4|14|2 -- > 54412,
  • |5441|2|2 -- > 154422,
  • |1|5442|2|3 -- > 1254423.
So $\phi(4154223) = 1254423$.