Identifier
-
Mp00090:
Permutations
—cycle-as-one-line notation⟶
Permutations
Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
Mp00109: Permutations —descent word⟶ Binary words
St001491: Binary words ⟶ ℤ
Values
[4,1,2,3] => [1,4,3,2] => [1,2,4,3] => 001 => 1
[4,1,3,2] => [1,4,2,3] => [1,2,4,3] => 001 => 1
[4,2,1,3] => [1,4,3,2] => [1,2,4,3] => 001 => 1
[4,2,3,1] => [1,4,2,3] => [1,2,4,3] => 001 => 1
[4,3,1,2] => [1,4,2,3] => [1,2,4,3] => 001 => 1
[4,3,2,1] => [1,4,2,3] => [1,2,4,3] => 001 => 1
[1,5,2,3,4] => [1,2,5,4,3] => [1,2,3,5,4] => 0001 => 1
[1,5,2,4,3] => [1,2,5,3,4] => [1,2,3,5,4] => 0001 => 1
[1,5,3,2,4] => [1,2,5,4,3] => [1,2,3,5,4] => 0001 => 1
[1,5,3,4,2] => [1,2,5,3,4] => [1,2,3,5,4] => 0001 => 1
[1,5,4,2,3] => [1,2,5,3,4] => [1,2,3,5,4] => 0001 => 1
[1,5,4,3,2] => [1,2,5,3,4] => [1,2,3,5,4] => 0001 => 1
[2,5,1,3,4] => [1,2,5,4,3] => [1,2,3,5,4] => 0001 => 1
[2,5,1,4,3] => [1,2,5,3,4] => [1,2,3,5,4] => 0001 => 1
[2,5,3,1,4] => [1,2,5,4,3] => [1,2,3,5,4] => 0001 => 1
[2,5,3,4,1] => [1,2,5,3,4] => [1,2,3,5,4] => 0001 => 1
[2,5,4,1,3] => [1,2,5,3,4] => [1,2,3,5,4] => 0001 => 1
[2,5,4,3,1] => [1,2,5,3,4] => [1,2,3,5,4] => 0001 => 1
[3,1,5,2,4] => [1,3,5,4,2] => [1,2,3,5,4] => 0001 => 1
[3,1,5,4,2] => [1,3,5,2,4] => [1,2,3,5,4] => 0001 => 1
[3,2,5,1,4] => [1,3,5,4,2] => [1,2,3,5,4] => 0001 => 1
[3,2,5,4,1] => [1,3,5,2,4] => [1,2,3,5,4] => 0001 => 1
[3,4,5,1,2] => [1,3,5,2,4] => [1,2,3,5,4] => 0001 => 1
[3,4,5,2,1] => [1,3,5,2,4] => [1,2,3,5,4] => 0001 => 1
[4,1,2,3,5] => [1,4,3,2,5] => [1,2,4,3,5] => 0010 => 1
[4,1,2,5,3] => [1,4,5,3,2] => [1,2,4,3,5] => 0010 => 1
[4,1,3,2,5] => [1,4,2,3,5] => [1,2,4,3,5] => 0010 => 1
[4,1,3,5,2] => [1,4,5,2,3] => [1,2,4,3,5] => 0010 => 1
[4,1,5,2,3] => [1,4,2,3,5] => [1,2,4,3,5] => 0010 => 1
[4,1,5,3,2] => [1,4,3,5,2] => [1,2,4,5,3] => 0001 => 1
[4,2,1,3,5] => [1,4,3,2,5] => [1,2,4,3,5] => 0010 => 1
[4,2,1,5,3] => [1,4,5,3,2] => [1,2,4,3,5] => 0010 => 1
[4,2,3,1,5] => [1,4,2,3,5] => [1,2,4,3,5] => 0010 => 1
[4,2,3,5,1] => [1,4,5,2,3] => [1,2,4,3,5] => 0010 => 1
[4,2,5,1,3] => [1,4,2,3,5] => [1,2,4,3,5] => 0010 => 1
[4,2,5,3,1] => [1,4,3,5,2] => [1,2,4,5,3] => 0001 => 1
[4,3,1,2,5] => [1,4,2,3,5] => [1,2,4,3,5] => 0010 => 1
[4,3,1,5,2] => [1,4,5,2,3] => [1,2,4,3,5] => 0010 => 1
[4,3,2,1,5] => [1,4,2,3,5] => [1,2,4,3,5] => 0010 => 1
[4,3,2,5,1] => [1,4,5,2,3] => [1,2,4,3,5] => 0010 => 1
[4,3,5,1,2] => [1,4,2,3,5] => [1,2,4,3,5] => 0010 => 1
[4,3,5,2,1] => [1,4,2,3,5] => [1,2,4,3,5] => 0010 => 1
[4,5,1,2,3] => [1,4,2,5,3] => [1,2,4,5,3] => 0001 => 1
[4,5,1,3,2] => [1,4,3,2,5] => [1,2,4,3,5] => 0010 => 1
[4,5,2,1,3] => [1,4,2,5,3] => [1,2,4,5,3] => 0001 => 1
[4,5,2,3,1] => [1,4,3,2,5] => [1,2,4,3,5] => 0010 => 1
[4,5,3,1,2] => [1,4,2,5,3] => [1,2,4,5,3] => 0001 => 1
[4,5,3,2,1] => [1,4,2,5,3] => [1,2,4,5,3] => 0001 => 1
[5,1,2,3,4] => [1,5,4,3,2] => [1,2,5,3,4] => 0010 => 1
[5,1,2,4,3] => [1,5,3,2,4] => [1,2,5,4,3] => 0011 => 1
[5,1,3,2,4] => [1,5,4,2,3] => [1,2,5,3,4] => 0010 => 1
[5,1,3,4,2] => [1,5,2,3,4] => [1,2,5,4,3] => 0011 => 1
[5,1,4,2,3] => [1,5,3,4,2] => [1,2,5,3,4] => 0010 => 1
[5,1,4,3,2] => [1,5,2,3,4] => [1,2,5,4,3] => 0011 => 1
[5,2,1,3,4] => [1,5,4,3,2] => [1,2,5,3,4] => 0010 => 1
[5,2,1,4,3] => [1,5,3,2,4] => [1,2,5,4,3] => 0011 => 1
[5,2,3,1,4] => [1,5,4,2,3] => [1,2,5,3,4] => 0010 => 1
[5,2,3,4,1] => [1,5,2,3,4] => [1,2,5,4,3] => 0011 => 1
[5,2,4,1,3] => [1,5,3,4,2] => [1,2,5,3,4] => 0010 => 1
[5,2,4,3,1] => [1,5,2,3,4] => [1,2,5,4,3] => 0011 => 1
[5,3,1,2,4] => [1,5,4,2,3] => [1,2,5,3,4] => 0010 => 1
[5,3,1,4,2] => [1,5,2,3,4] => [1,2,5,4,3] => 0011 => 1
[5,3,2,1,4] => [1,5,4,2,3] => [1,2,5,3,4] => 0010 => 1
[5,3,2,4,1] => [1,5,2,3,4] => [1,2,5,4,3] => 0011 => 1
[5,3,4,1,2] => [1,5,2,3,4] => [1,2,5,4,3] => 0011 => 1
[5,3,4,2,1] => [1,5,2,3,4] => [1,2,5,4,3] => 0011 => 1
[5,4,1,2,3] => [1,5,3,2,4] => [1,2,5,4,3] => 0011 => 1
[5,4,1,3,2] => [1,5,2,4,3] => [1,2,5,3,4] => 0010 => 1
[5,4,2,1,3] => [1,5,3,2,4] => [1,2,5,4,3] => 0011 => 1
[5,4,2,3,1] => [1,5,2,4,3] => [1,2,5,3,4] => 0010 => 1
[5,4,3,1,2] => [1,5,2,4,3] => [1,2,5,3,4] => 0010 => 1
[5,4,3,2,1] => [1,5,2,4,3] => [1,2,5,3,4] => 0010 => 1
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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.
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
cycle-as-one-line notation
Description
Return the permutation obtained by concatenating the cycles of a permutation, each written with minimal element first, sorted by minimal element.
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.
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.
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