Identifier
Values
[1,1,0,0] => [2,1] => [2,1] => 1 => 1
[1,0,1,1,0,0] => [1,3,2] => [1,3,2] => 01 => 1
[1,1,0,0,1,0] => [2,1,3] => [2,1,3] => 10 => 1
[1,1,0,1,0,0] => [2,3,1] => [3,2,1] => 11 => 2
[1,1,1,0,0,0] => [3,2,1] => [3,2,1] => 11 => 2
[1,0,1,0,1,1,0,0] => [1,2,4,3] => [1,2,4,3] => 001 => 1
[1,0,1,1,0,0,1,0] => [1,3,2,4] => [1,3,2,4] => 010 => 1
[1,0,1,1,0,1,0,0] => [1,3,4,2] => [1,4,3,2] => 011 => 1
[1,0,1,1,1,0,0,0] => [1,4,3,2] => [1,4,3,2] => 011 => 1
[1,1,0,0,1,0,1,0] => [2,1,3,4] => [2,1,3,4] => 100 => 1
[1,1,0,0,1,1,0,0] => [2,1,4,3] => [2,1,4,3] => 101 => 2
[1,1,0,1,0,0,1,0] => [2,3,1,4] => [3,2,1,4] => 110 => 1
[1,1,0,1,0,1,0,0] => [2,3,4,1] => [4,2,3,1] => 101 => 2
[1,1,0,1,1,0,0,0] => [2,4,3,1] => [4,3,2,1] => 111 => 3
[1,1,1,0,0,0,1,0] => [3,2,1,4] => [3,2,1,4] => 110 => 1
[1,1,1,0,0,1,0,0] => [3,2,4,1] => [4,2,3,1] => 101 => 2
[1,1,1,0,1,0,0,0] => [3,4,2,1] => [4,3,2,1] => 111 => 3
[1,1,1,1,0,0,0,0] => [4,3,2,1] => [4,3,2,1] => 111 => 3
[1,0,1,0,1,0,1,1,0,0] => [1,2,3,5,4] => [1,2,3,5,4] => 0001 => 1
[1,0,1,0,1,1,0,0,1,0] => [1,2,4,3,5] => [1,2,4,3,5] => 0010 => 1
[1,0,1,0,1,1,0,1,0,0] => [1,2,4,5,3] => [1,2,5,4,3] => 0011 => 1
[1,0,1,0,1,1,1,0,0,0] => [1,2,5,4,3] => [1,2,5,4,3] => 0011 => 1
[1,0,1,1,0,0,1,0,1,0] => [1,3,2,4,5] => [1,3,2,4,5] => 0100 => 1
[1,0,1,1,0,0,1,1,0,0] => [1,3,2,5,4] => [1,3,2,5,4] => 0101 => 0
[1,0,1,1,0,1,0,0,1,0] => [1,3,4,2,5] => [1,4,3,2,5] => 0110 => 2
[1,0,1,1,0,1,0,1,0,0] => [1,3,4,5,2] => [1,5,3,4,2] => 0101 => 0
[1,0,1,1,0,1,1,0,0,0] => [1,3,5,4,2] => [1,5,4,3,2] => 0111 => 2
[1,0,1,1,1,0,0,0,1,0] => [1,4,3,2,5] => [1,4,3,2,5] => 0110 => 2
[1,0,1,1,1,0,0,1,0,0] => [1,4,3,5,2] => [1,5,3,4,2] => 0101 => 0
[1,0,1,1,1,0,1,0,0,0] => [1,4,5,3,2] => [1,5,4,3,2] => 0111 => 2
[1,0,1,1,1,1,0,0,0,0] => [1,5,4,3,2] => [1,5,4,3,2] => 0111 => 2
[1,1,0,0,1,0,1,0,1,0] => [2,1,3,4,5] => [2,1,3,4,5] => 1000 => 1
[1,1,0,0,1,0,1,1,0,0] => [2,1,3,5,4] => [2,1,3,5,4] => 1001 => 2
[1,1,0,0,1,1,0,0,1,0] => [2,1,4,3,5] => [2,1,4,3,5] => 1010 => 0
[1,1,0,0,1,1,0,1,0,0] => [2,1,4,5,3] => [2,1,5,4,3] => 1011 => 2
[1,1,0,0,1,1,1,0,0,0] => [2,1,5,4,3] => [2,1,5,4,3] => 1011 => 2
[1,1,0,1,0,0,1,0,1,0] => [2,3,1,4,5] => [3,2,1,4,5] => 1100 => 1
[1,1,0,1,0,0,1,1,0,0] => [2,3,1,5,4] => [3,2,1,5,4] => 1101 => 2
[1,1,0,1,0,1,0,0,1,0] => [2,3,4,1,5] => [4,2,3,1,5] => 1010 => 0
[1,1,0,1,0,1,0,1,0,0] => [2,3,4,5,1] => [5,2,3,4,1] => 1001 => 2
[1,1,0,1,0,1,1,0,0,0] => [2,3,5,4,1] => [5,2,4,3,1] => 1011 => 2
[1,1,0,1,1,0,0,0,1,0] => [2,4,3,1,5] => [4,3,2,1,5] => 1110 => 2
[1,1,0,1,1,0,0,1,0,0] => [2,4,3,5,1] => [5,3,2,4,1] => 1101 => 2
[1,1,0,1,1,0,1,0,0,0] => [2,4,5,3,1] => [5,4,3,2,1] => 1111 => 4
[1,1,0,1,1,1,0,0,0,0] => [2,5,4,3,1] => [5,4,3,2,1] => 1111 => 4
[1,1,1,0,0,0,1,0,1,0] => [3,2,1,4,5] => [3,2,1,4,5] => 1100 => 1
[1,1,1,0,0,0,1,1,0,0] => [3,2,1,5,4] => [3,2,1,5,4] => 1101 => 2
[1,1,1,0,0,1,0,0,1,0] => [3,2,4,1,5] => [4,2,3,1,5] => 1010 => 0
[1,1,1,0,0,1,0,1,0,0] => [3,2,4,5,1] => [5,2,3,4,1] => 1001 => 2
[1,1,1,0,0,1,1,0,0,0] => [3,2,5,4,1] => [5,2,4,3,1] => 1011 => 2
[1,1,1,0,1,0,0,0,1,0] => [3,4,2,1,5] => [4,3,2,1,5] => 1110 => 2
[1,1,1,0,1,0,0,1,0,0] => [3,4,2,5,1] => [5,3,2,4,1] => 1101 => 2
[1,1,1,0,1,0,1,0,0,0] => [3,4,5,2,1] => [5,4,3,2,1] => 1111 => 4
[1,1,1,0,1,1,0,0,0,0] => [3,5,4,2,1] => [5,4,3,2,1] => 1111 => 4
[1,1,1,1,0,0,0,0,1,0] => [4,3,2,1,5] => [4,3,2,1,5] => 1110 => 2
[1,1,1,1,0,0,0,1,0,0] => [4,3,2,5,1] => [5,3,2,4,1] => 1101 => 2
[1,1,1,1,0,0,1,0,0,0] => [4,3,5,2,1] => [5,4,3,2,1] => 1111 => 4
[1,1,1,1,0,1,0,0,0,0] => [4,5,3,2,1] => [5,4,3,2,1] => 1111 => 4
[1,1,1,1,1,0,0,0,0,0] => [5,4,3,2,1] => [5,4,3,2,1] => 1111 => 4
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 312-avoiding permutation
Description
Sends a Dyck path to the 312-avoiding permutation according to Bandlow-Killpatrick.
This map is defined in [1] and sends the area (St000012The area of a Dyck path.) to the inversion number (St000018The number of inversions of a permutation.).
Map
descent word
Description
The descent positions of a permutation as a binary word.
For a permutation $\pi$ of $n$ letters and each $1\leq i\leq n-1$ such that $\pi(i) > \pi(i+1)$ we set $w_i=1$, otherwise $w_i=0$.
Thus, the length of the word is one less the size of the permutation. In particular, the descent word is undefined for the empty permutation.
Map
Demazure product with inverse
Description
This map sends a permutation $\pi$ to $\pi^{-1} \star \pi$ where $\star$ denotes the Demazure product on permutations.
This map is a surjection onto the set of involutions, i.e., the set of permutations $\pi$ for which $\pi = \pi^{-1}$.