Identifier
Values
[1,0,1,1,0,0] => [3,1,4,2] => [3,1,4,2] => ([(0,3),(1,2),(1,3)],4) => 0
[1,1,0,0,1,0] => [2,4,1,3] => [2,4,1,3] => ([(0,3),(1,2),(1,3)],4) => 0
[1,0,1,0,1,1,0,0] => [4,1,2,5,3] => [4,1,5,3,2] => ([(0,4),(1,2),(1,3),(1,4)],5) => 0
[1,0,1,1,0,0,1,0] => [3,1,5,2,4] => [3,1,5,4,2] => ([(0,3),(0,4),(1,2),(1,3),(1,4)],5) => 0
[1,0,1,1,1,0,0,0] => [3,1,4,5,2] => [3,1,5,4,2] => ([(0,3),(0,4),(1,2),(1,3),(1,4)],5) => 0
[1,1,0,0,1,0,1,0] => [2,5,1,3,4] => [2,5,1,4,3] => ([(0,3),(0,4),(1,2),(1,3),(1,4)],5) => 0
[1,1,0,0,1,1,0,0] => [2,4,1,5,3] => [2,5,1,4,3] => ([(0,3),(0,4),(1,2),(1,3),(1,4)],5) => 0
[1,1,0,1,1,0,0,0] => [4,3,1,5,2] => [4,3,1,5,2] => ([(0,4),(1,4),(2,3),(2,4)],5) => 0
[1,1,1,0,0,0,1,0] => [2,3,5,1,4] => [2,5,4,1,3] => ([(0,4),(1,2),(1,3),(1,4)],5) => 0
[1,1,1,0,0,1,0,0] => [2,5,4,1,3] => [2,5,4,1,3] => ([(0,4),(1,2),(1,3),(1,4)],5) => 0
[1,0,1,0,1,0,1,1,0,0] => [5,1,2,3,6,4] => [5,1,6,4,3,2] => ([(0,5),(1,2),(1,3),(1,4),(1,5)],6) => 0
[1,0,1,0,1,1,0,0,1,0] => [4,1,2,6,3,5] => [4,1,6,5,3,2] => ([(0,4),(0,5),(1,2),(1,3),(1,4),(1,5)],6) => 0
[1,0,1,0,1,1,1,0,0,0] => [4,1,2,5,6,3] => [4,1,6,5,3,2] => ([(0,4),(0,5),(1,2),(1,3),(1,4),(1,5)],6) => 0
[1,0,1,1,0,0,1,0,1,0] => [3,1,6,2,4,5] => [3,1,6,5,4,2] => ([(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5)],6) => 0
[1,0,1,1,0,0,1,1,0,0] => [3,1,5,2,6,4] => [3,1,6,5,4,2] => ([(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5)],6) => 0
[1,0,1,1,0,1,1,0,0,0] => [5,1,4,2,6,3] => [5,1,6,4,3,2] => ([(0,5),(1,2),(1,3),(1,4),(1,5)],6) => 0
[1,0,1,1,1,0,0,0,1,0] => [3,1,4,6,2,5] => [3,1,6,5,4,2] => ([(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5)],6) => 0
[1,0,1,1,1,0,0,1,0,0] => [3,1,6,5,2,4] => [3,1,6,5,4,2] => ([(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5)],6) => 0
[1,0,1,1,1,1,0,0,0,0] => [3,1,4,5,6,2] => [3,1,6,5,4,2] => ([(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5)],6) => 0
[1,1,0,0,1,0,1,0,1,0] => [2,6,1,3,4,5] => [2,6,1,5,4,3] => ([(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5)],6) => 0
[1,1,0,0,1,0,1,1,0,0] => [2,5,1,3,6,4] => [2,6,1,5,4,3] => ([(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5)],6) => 0
[1,1,0,0,1,1,0,0,1,0] => [2,4,1,6,3,5] => [2,6,1,5,4,3] => ([(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5)],6) => 0
[1,1,0,0,1,1,0,1,0,0] => [2,6,1,5,3,4] => [2,6,1,5,4,3] => ([(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5)],6) => 0
[1,1,0,0,1,1,1,0,0,0] => [2,4,1,5,6,3] => [2,6,1,5,4,3] => ([(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5)],6) => 0
[1,1,0,1,0,0,1,1,0,0] => [5,3,1,2,6,4] => [5,3,1,6,4,2] => ([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5)],6) => 0
[1,1,0,1,0,1,1,0,0,0] => [5,4,1,2,6,3] => [5,4,1,6,3,2] => ([(0,5),(1,5),(2,3),(2,4),(2,5)],6) => 0
[1,1,0,1,1,0,0,0,1,0] => [4,3,1,6,2,5] => [4,3,1,6,5,2] => ([(0,4),(0,5),(1,4),(1,5),(2,3),(2,4),(2,5)],6) => 0
[1,1,0,1,1,1,0,0,0,0] => [4,3,1,5,6,2] => [4,3,1,6,5,2] => ([(0,4),(0,5),(1,4),(1,5),(2,3),(2,4),(2,5)],6) => 0
[1,1,1,0,0,0,1,0,1,0] => [2,3,6,1,4,5] => [2,6,5,1,4,3] => ([(0,4),(0,5),(1,2),(1,3),(1,4),(1,5)],6) => 0
[1,1,1,0,0,0,1,1,0,0] => [2,3,5,1,6,4] => [2,6,5,1,4,3] => ([(0,4),(0,5),(1,2),(1,3),(1,4),(1,5)],6) => 0
[1,1,1,0,0,1,0,0,1,0] => [2,6,4,1,3,5] => [2,6,5,1,4,3] => ([(0,4),(0,5),(1,2),(1,3),(1,4),(1,5)],6) => 0
[1,1,1,0,0,1,0,1,0,0] => [2,6,5,1,3,4] => [2,6,5,1,4,3] => ([(0,4),(0,5),(1,2),(1,3),(1,4),(1,5)],6) => 0
[1,1,1,0,0,1,1,0,0,0] => [2,5,4,1,6,3] => [2,6,5,1,4,3] => ([(0,4),(0,5),(1,2),(1,3),(1,4),(1,5)],6) => 0
[1,1,1,0,1,1,0,0,0,0] => [5,3,4,1,6,2] => [5,3,6,1,4,2] => ([(0,4),(1,4),(1,5),(2,3),(2,5)],6) => 0
[1,1,1,1,0,0,0,0,1,0] => [2,3,4,6,1,5] => [2,6,5,4,1,3] => ([(0,5),(1,2),(1,3),(1,4),(1,5)],6) => 0
[1,1,1,1,0,0,0,1,0,0] => [2,3,6,5,1,4] => [2,6,5,4,1,3] => ([(0,5),(1,2),(1,3),(1,4),(1,5)],6) => 0
[1,1,1,1,0,0,1,0,0,0] => [2,6,4,5,1,3] => [2,6,5,4,1,3] => ([(0,5),(1,2),(1,3),(1,4),(1,5)],6) => 0
search for individual values
searching the database for the individual values of this statistic
Description
The number of indecomposable injective modules $I$ with $dim Ext^1(I,A)=1$ for the incidence algebra A of a poset.
Map
Ringel
Description
The Ringel permutation of the LNakayama algebra corresponding to a Dyck path.
Map
permutation poset
Description
Sends a permutation to its permutation poset.
For a permutation $\pi$ of length $n$, this poset has vertices
$$\{ (i,\pi(i))\ :\ 1 \leq i \leq n \}$$
and the cover relation is given by $(w, x) \leq (y, z)$ if $w \leq y$ and $x \leq z$.
For example, the permutation $[3,1,5,4,2]$ is mapped to the poset with cover relations
$$\{ (2, 1) \prec (5, 2),\ (2, 1) \prec (4, 4),\ (2, 1) \prec (3, 5),\ (1, 3) \prec (4, 4),\ (1, 3) \prec (3, 5) \}.$$
Map
Simion-Schmidt map
Description
The Simion-Schmidt map sends any permutation to a $123$-avoiding permutation.
Details can be found in [1].
In particular, this is a bijection between $132$-avoiding permutations and $123$-avoiding permutations, see [1, Proposition 19].