Identifier
Values
[2] => [1,0,1,0] => [1,2] => ([(0,1)],2) => 1
[3] => [1,0,1,0,1,0] => [1,2,3] => ([(0,2),(2,1)],3) => 1
[2,1] => [1,0,1,1,0,0] => [1,3,2] => ([(0,1),(0,2)],3) => 2
[4] => [1,0,1,0,1,0,1,0] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4) => 1
[3,1] => [1,0,1,0,1,1,0,0] => [1,2,4,3] => ([(0,3),(3,1),(3,2)],4) => 1
[2,1,1] => [1,0,1,1,0,1,0,0] => [1,3,4,2] => ([(0,2),(0,3),(3,1)],4) => 2
[5] => [1,0,1,0,1,0,1,0,1,0] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5) => 1
[4,1] => [1,0,1,0,1,0,1,1,0,0] => [1,2,3,5,4] => ([(0,3),(3,4),(4,1),(4,2)],5) => 1
[3,2] => [1,0,1,1,1,0,0,0] => [1,4,2,3] => ([(0,2),(0,3),(3,1)],4) => 2
[3,1,1] => [1,0,1,0,1,1,0,1,0,0] => [1,2,4,5,3] => ([(0,4),(3,2),(4,1),(4,3)],5) => 1
[2,2,1] => [1,1,1,0,0,1,0,0] => [3,1,4,2] => ([(0,3),(1,2),(1,3)],4) => 0
[2,1,1,1] => [1,0,1,1,0,1,0,1,0,0] => [1,3,4,5,2] => ([(0,2),(0,4),(3,1),(4,3)],5) => 2
[6] => [1,0,1,0,1,0,1,0,1,0,1,0] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6) => 1
[5,1] => [1,0,1,0,1,0,1,0,1,1,0,0] => [1,2,3,4,6,5] => ([(0,4),(3,5),(4,3),(5,1),(5,2)],6) => 1
[4,2] => [1,0,1,0,1,1,1,0,0,0] => [1,2,5,3,4] => ([(0,4),(3,2),(4,1),(4,3)],5) => 1
[4,1,1] => [1,0,1,0,1,0,1,1,0,1,0,0] => [1,2,3,5,6,4] => ([(0,4),(3,2),(4,5),(5,1),(5,3)],6) => 1
[3,2,1] => [1,0,1,1,1,0,0,1,0,0] => [1,4,2,5,3] => ([(0,2),(0,3),(2,4),(3,1),(3,4)],5) => 2
[3,1,1,1] => [1,0,1,0,1,1,0,1,0,1,0,0] => [1,2,4,5,6,3] => ([(0,5),(3,4),(4,2),(5,1),(5,3)],6) => 1
[2,2,1,1] => [1,1,1,0,0,1,0,1,0,0] => [3,1,4,5,2] => ([(0,4),(1,2),(1,4),(4,3)],5) => 0
[2,1,1,1,1] => [1,0,1,1,0,1,0,1,0,1,0,0] => [1,3,4,5,6,2] => ([(0,2),(0,5),(3,4),(4,1),(5,3)],6) => 2
[7] => [1,0,1,0,1,0,1,0,1,0,1,0,1,0] => [1,2,3,4,5,6,7] => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7) => 1
[5,2] => [1,0,1,0,1,0,1,1,1,0,0,0] => [1,2,3,6,4,5] => ([(0,4),(3,2),(4,5),(5,1),(5,3)],6) => 1
[4,3] => [1,0,1,1,1,0,1,0,0,0] => [1,4,5,2,3] => ([(0,3),(0,4),(3,2),(4,1)],5) => 2
[4,2,1] => [1,0,1,0,1,1,1,0,0,1,0,0] => [1,2,5,3,6,4] => ([(0,4),(2,5),(3,1),(3,5),(4,2),(4,3)],6) => 2
[3,3,1] => [1,1,1,0,1,0,0,1,0,0] => [3,4,1,5,2] => ([(0,3),(1,2),(1,4),(3,4)],5) => 1
[3,2,2] => [1,0,1,1,1,1,0,0,0,0] => [1,5,2,3,4] => ([(0,2),(0,4),(3,1),(4,3)],5) => 2
[3,2,1,1] => [1,0,1,1,1,0,0,1,0,1,0,0] => [1,4,2,5,6,3] => ([(0,3),(0,4),(3,5),(4,1),(4,5),(5,2)],6) => 2
[2,2,2,1] => [1,1,1,1,0,0,0,1,0,0] => [4,1,2,5,3] => ([(0,4),(1,2),(2,3),(2,4)],5) => 1
[2,2,1,1,1] => [1,1,1,0,0,1,0,1,0,1,0,0] => [3,1,4,5,6,2] => ([(0,5),(1,3),(1,5),(4,2),(5,4)],6) => 0
[5,3] => [1,0,1,0,1,1,1,0,1,0,0,0] => [1,2,5,6,3,4] => ([(0,5),(3,2),(4,1),(5,3),(5,4)],6) => 1
[4,3,1] => [1,0,1,1,1,0,1,0,0,1,0,0] => [1,4,5,2,6,3] => ([(0,3),(0,4),(2,5),(3,2),(4,1),(4,5)],6) => 2
[4,2,2] => [1,0,1,0,1,1,1,1,0,0,0,0] => [1,2,6,3,4,5] => ([(0,5),(3,4),(4,2),(5,1),(5,3)],6) => 1
[3,3,2] => [1,1,1,0,1,1,0,0,0,0] => [3,5,1,2,4] => ([(0,3),(1,2),(1,4),(3,4)],5) => 1
[3,3,1,1] => [1,1,1,0,1,0,0,1,0,1,0,0] => [3,4,1,5,6,2] => ([(0,4),(1,3),(1,5),(4,5),(5,2)],6) => 1
[3,2,2,1] => [1,0,1,1,1,1,0,0,0,1,0,0] => [1,5,2,3,6,4] => ([(0,2),(0,4),(2,5),(3,1),(3,5),(4,3)],6) => 2
[2,2,2,1,1] => [1,1,1,1,0,0,0,1,0,1,0,0] => [4,1,2,5,6,3] => ([(0,5),(1,4),(4,2),(4,5),(5,3)],6) => 1
[5,4] => [1,0,1,1,1,0,1,0,1,0,0,0] => [1,4,5,6,2,3] => ([(0,4),(0,5),(3,2),(4,3),(5,1)],6) => 2
[4,4,1] => [1,1,1,0,1,0,1,0,0,1,0,0] => [3,4,5,1,6,2] => ([(0,4),(1,3),(1,5),(2,5),(4,2)],6) => 1
[4,3,2] => [1,0,1,1,1,0,1,1,0,0,0,0] => [1,4,6,2,3,5] => ([(0,3),(0,4),(2,5),(3,2),(4,1),(4,5)],6) => 2
[3,3,2,1] => [1,1,1,0,1,1,0,0,0,1,0,0] => [3,5,1,2,6,4] => ([(0,3),(1,2),(1,4),(2,5),(3,4),(3,5)],6) => 1
[3,2,2,2] => [1,0,1,1,1,1,0,1,0,0,0,0] => [1,5,6,2,3,4] => ([(0,4),(0,5),(3,2),(4,3),(5,1)],6) => 2
[2,2,2,2,1] => [1,1,1,1,0,1,0,0,0,1,0,0] => [4,5,1,2,6,3] => ([(0,3),(1,4),(3,5),(4,2),(4,5)],6) => 2
[4,4,2] => [1,1,1,0,1,0,1,1,0,0,0,0] => [3,4,6,1,2,5] => ([(0,3),(1,4),(3,5),(4,2),(4,5)],6) => 2
[4,3,3] => [1,0,1,1,1,1,1,0,0,0,0,0] => [1,6,2,3,4,5] => ([(0,2),(0,5),(3,4),(4,1),(5,3)],6) => 2
[3,3,3,1] => [1,1,1,1,1,0,0,0,0,1,0,0] => [5,1,2,3,6,4] => ([(0,5),(1,3),(3,4),(4,2),(4,5)],6) => 1
[3,3,2,2] => [1,1,1,0,1,1,0,1,0,0,0,0] => [3,5,6,1,2,4] => ([(0,3),(1,4),(1,5),(3,5),(4,2)],6) => 1
[4,4,3] => [1,1,1,0,1,1,1,0,0,0,0,0] => [3,6,1,2,4,5] => ([(0,4),(1,3),(1,5),(4,5),(5,2)],6) => 1
[3,3,3,2] => [1,1,1,1,1,0,0,1,0,0,0,0] => [5,1,6,2,3,4] => ([(0,5),(1,4),(1,5),(3,2),(4,3)],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
parallelogram polyomino
Description
Return the Dyck path corresponding to the partition interpreted as a parallogram polyomino.
The Ferrers diagram of an integer partition can be interpreted as a parallogram polyomino, such that each part corresponds to a column.
This map returns the corresponding Dyck path.
Map
to 321-avoiding permutation (Krattenthaler)
Description
Krattenthaler's bijection to 321-avoiding permutations.
Draw the path of semilength $n$ in an $n\times n$ square matrix, starting at the upper left corner, with right and down steps, and staying below the diagonal. Then the permutation matrix is obtained by placing ones into the cells corresponding to the peaks of the path and placing ones into the remaining columns from left to right, such that the row indices of the cells increase.
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
$$\{ (2, 1) \prec (5, 2),\ (2, 1) \prec (4, 4),\ (2, 1) \prec (3, 5),\ (1, 3) \prec (4, 4),\ (1, 3) \prec (3, 5) \}.$$