Identifier
Values
[1,2,2] => [[3,2,1],[1]] => ([(0,2),(2,1)],3) => ([(0,1),(0,2),(1,3),(2,3)],4) => 1
[2,2,1] => [[3,3,2],[2,1]] => ([(0,2),(2,1)],3) => ([(0,1),(0,2),(1,3),(2,3)],4) => 1
[1,1,2,1,1] => [[2,2,2,1,1],[1,1]] => ([(0,2),(2,1)],3) => ([(0,1),(0,2),(1,3),(2,3)],4) => 1
[1,1,2,2] => [[3,2,1,1],[1]] => ([(0,2),(2,1)],3) => ([(0,1),(0,2),(1,3),(2,3)],4) => 1
[1,2,1,2] => [[3,2,2,1],[1,1]] => ([(0,1),(0,2),(1,3),(2,3)],4) => ([(0,1),(0,2),(1,3),(2,3)],4) => 1
[1,2,3] => [[4,2,1],[1]] => ([(0,2),(2,1)],3) => ([(0,1),(0,2),(1,3),(2,3)],4) => 1
[1,3,2] => [[4,3,1],[2]] => ([(0,1),(0,2),(1,3),(2,3)],4) => ([(0,1),(0,2),(1,3),(2,3)],4) => 1
[2,1,2,1] => [[3,3,2,2],[2,1,1]] => ([(0,1),(0,2),(1,3),(2,3)],4) => ([(0,1),(0,2),(1,3),(2,3)],4) => 1
[2,2,1,1] => [[3,3,3,2],[2,2,1]] => ([(0,2),(2,1)],3) => ([(0,1),(0,2),(1,3),(2,3)],4) => 1
[2,3,1] => [[4,4,2],[3,1]] => ([(0,1),(0,2),(1,3),(2,3)],4) => ([(0,1),(0,2),(1,3),(2,3)],4) => 1
[3,2,1] => [[4,4,3],[3,2]] => ([(0,2),(2,1)],3) => ([(0,1),(0,2),(1,3),(2,3)],4) => 1
[3,3] => [[5,3],[2]] => ([(0,2),(2,1)],3) => ([(0,1),(0,2),(1,3),(2,3)],4) => 1
[1,1,1,2,1,1] => [[2,2,2,1,1,1],[1,1]] => ([(0,2),(2,1)],3) => ([(0,1),(0,2),(1,3),(2,3)],4) => 1
[1,1,1,2,2] => [[3,2,1,1,1],[1]] => ([(0,2),(2,1)],3) => ([(0,1),(0,2),(1,3),(2,3)],4) => 1
[1,1,2,1,1,1] => [[2,2,2,2,1,1],[1,1,1]] => ([(0,2),(2,1)],3) => ([(0,1),(0,2),(1,3),(2,3)],4) => 1
[1,1,2,3] => [[4,2,1,1],[1]] => ([(0,2),(2,1)],3) => ([(0,1),(0,2),(1,3),(2,3)],4) => 1
[1,1,3,1,1] => [[3,3,3,1,1],[2,2]] => ([(0,2),(2,1)],3) => ([(0,1),(0,2),(1,3),(2,3)],4) => 1
[1,1,3,2] => [[4,3,1,1],[2]] => ([(0,1),(0,2),(1,3),(2,3)],4) => ([(0,1),(0,2),(1,3),(2,3)],4) => 1
[1,2,1,1,2] => [[3,2,2,2,1],[1,1,1]] => ([(0,1),(0,2),(1,3),(2,3)],4) => ([(0,1),(0,2),(1,3),(2,3)],4) => 1
[1,2,1,3] => [[4,2,2,1],[1,1]] => ([(0,1),(0,2),(1,3),(2,3)],4) => ([(0,1),(0,2),(1,3),(2,3)],4) => 1
[1,2,4] => [[5,2,1],[1]] => ([(0,2),(2,1)],3) => ([(0,1),(0,2),(1,3),(2,3)],4) => 1
[1,3,1,2] => [[4,3,3,1],[2,2]] => ([(0,2),(0,3),(1,4),(2,4),(3,1)],5) => ([(0,3),(1,4),(2,4),(3,1),(3,2)],5) => 1
[1,4,2] => [[5,4,1],[3]] => ([(0,1),(0,2),(1,3),(2,3)],4) => ([(0,1),(0,2),(1,3),(2,3)],4) => 1
[2,1,1,2,1] => [[3,3,2,2,2],[2,1,1,1]] => ([(0,1),(0,2),(1,3),(2,3)],4) => ([(0,1),(0,2),(1,3),(2,3)],4) => 1
[2,1,3,1] => [[4,4,2,2],[3,1,1]] => ([(0,2),(0,3),(1,4),(2,4),(3,1)],5) => ([(0,3),(1,4),(2,4),(3,1),(3,2)],5) => 1
[2,2,1,1,1] => [[3,3,3,3,2],[2,2,2,1]] => ([(0,2),(2,1)],3) => ([(0,1),(0,2),(1,3),(2,3)],4) => 1
[2,3,1,1] => [[4,4,4,2],[3,3,1]] => ([(0,1),(0,2),(1,3),(2,3)],4) => ([(0,1),(0,2),(1,3),(2,3)],4) => 1
[2,4,1] => [[5,5,2],[4,1]] => ([(0,1),(0,2),(1,3),(2,3)],4) => ([(0,1),(0,2),(1,3),(2,3)],4) => 1
[3,1,2,1] => [[4,4,3,3],[3,2,2]] => ([(0,1),(0,2),(1,3),(2,3)],4) => ([(0,1),(0,2),(1,3),(2,3)],4) => 1
[3,1,3] => [[5,3,3],[2,2]] => ([(0,2),(2,1)],3) => ([(0,1),(0,2),(1,3),(2,3)],4) => 1
[3,2,1,1] => [[4,4,4,3],[3,3,2]] => ([(0,2),(2,1)],3) => ([(0,1),(0,2),(1,3),(2,3)],4) => 1
[3,4] => [[6,3],[2]] => ([(0,2),(2,1)],3) => ([(0,1),(0,2),(1,3),(2,3)],4) => 1
[4,2,1] => [[5,5,4],[4,3]] => ([(0,2),(2,1)],3) => ([(0,1),(0,2),(1,3),(2,3)],4) => 1
[4,3] => [[6,4],[3]] => ([(0,2),(2,1)],3) => ([(0,1),(0,2),(1,3),(2,3)],4) => 1
[1,1,1,1,2,1,1] => [[2,2,2,1,1,1,1],[1,1]] => ([(0,2),(2,1)],3) => ([(0,1),(0,2),(1,3),(2,3)],4) => 1
[1,1,1,1,2,2] => [[3,2,1,1,1,1],[1]] => ([(0,2),(2,1)],3) => ([(0,1),(0,2),(1,3),(2,3)],4) => 1
[1,1,1,3,2] => [[4,3,1,1,1],[2]] => ([(0,1),(0,2),(1,3),(2,3)],4) => ([(0,1),(0,2),(1,3),(2,3)],4) => 1
[1,1,2,1,1,1,1] => [[2,2,2,2,2,1,1],[1,1,1,1]] => ([(0,2),(2,1)],3) => ([(0,1),(0,2),(1,3),(2,3)],4) => 1
[1,2,1,1,1,2] => [[3,2,2,2,2,1],[1,1,1,1]] => ([(0,1),(0,2),(1,3),(2,3)],4) => ([(0,1),(0,2),(1,3),(2,3)],4) => 1
[1,3,1,1,2] => [[4,3,3,3,1],[2,2,2]] => ([(0,2),(0,3),(1,4),(2,4),(3,1)],5) => ([(0,3),(1,4),(2,4),(3,1),(3,2)],5) => 1
[1,4,1,2] => [[5,4,4,1],[3,3]] => ([(0,2),(0,3),(1,4),(2,4),(3,1)],5) => ([(0,3),(1,4),(2,4),(3,1),(3,2)],5) => 1
[2,1,1,1,2,1] => [[3,3,2,2,2,2],[2,1,1,1,1]] => ([(0,1),(0,2),(1,3),(2,3)],4) => ([(0,1),(0,2),(1,3),(2,3)],4) => 1
[2,1,1,3,1] => [[4,4,2,2,2],[3,1,1,1]] => ([(0,2),(0,3),(1,4),(2,4),(3,1)],5) => ([(0,3),(1,4),(2,4),(3,1),(3,2)],5) => 1
[2,2,1,1,1,1] => [[3,3,3,3,3,2],[2,2,2,2,1]] => ([(0,2),(2,1)],3) => ([(0,1),(0,2),(1,3),(2,3)],4) => 1
[2,3,1,1,1] => [[4,4,4,4,2],[3,3,3,1]] => ([(0,1),(0,2),(1,3),(2,3)],4) => ([(0,1),(0,2),(1,3),(2,3)],4) => 1
[2,4,1,1] => [[5,5,5,2],[4,4,1]] => ([(0,1),(0,2),(1,3),(2,3)],4) => ([(0,1),(0,2),(1,3),(2,3)],4) => 1
[3,1,1,2,1] => [[4,4,3,3,3],[3,2,2,2]] => ([(0,1),(0,2),(1,3),(2,3)],4) => ([(0,1),(0,2),(1,3),(2,3)],4) => 1
[3,1,1,3] => [[5,3,3,3],[2,2,2]] => ([(0,2),(2,1)],3) => ([(0,1),(0,2),(1,3),(2,3)],4) => 1
[3,2,1,1,1] => [[4,4,4,4,3],[3,3,3,2]] => ([(0,2),(2,1)],3) => ([(0,1),(0,2),(1,3),(2,3)],4) => 1
[4,1,2,1] => [[5,5,4,4],[4,3,3]] => ([(0,1),(0,2),(1,3),(2,3)],4) => ([(0,1),(0,2),(1,3),(2,3)],4) => 1
[4,1,3] => [[6,4,4],[3,3]] => ([(0,2),(2,1)],3) => ([(0,1),(0,2),(1,3),(2,3)],4) => 1
[4,2,1,1] => [[5,5,5,4],[4,4,3]] => ([(0,2),(2,1)],3) => ([(0,1),(0,2),(1,3),(2,3)],4) => 1
search for individual values
searching the database for the individual values of this statistic
Description
Number of indecomposable injective modules with projective dimension 2.
Map
to ribbon
Description
The ribbon shape corresponding to an integer composition.
For an integer composition $(a_1, \dots, a_n)$, this is the ribbon shape whose $i$th row from the bottom has $a_i$ cells.
Map
dominating sublattice
Description
Return the sublattice of the dominance order induced by the support of the expansion of the skew Schur function into Schur functions.
Consider the expansion of the skew Schur function $s_{\lambda/\mu}=\sum_\nu c^\lambda_{\mu, \nu} s_\nu$ as a linear combination of straight Schur functions.
It is shown in [1] that the subposet of the dominance order whose elements are the partitions $\nu$ with $c^\lambda_{\mu, \nu} > 0$ form a lattice.
The example $\lambda = (5^2,4^2,1)$ and $\mu=(3,2)$ shows that this lattice is not a sublattice of the dominance order.
Map
lattice of congruences
Description
The lattice of congruences of a lattice.
A congruence of a lattice is an equivalence relation such that $a_1 \cong a_2$ and $b_1 \cong b_2$ implies $a_1 \vee b_1 \cong a_2 \vee b_2$ and $a_1 \wedge b_1 \cong a_2 \wedge b_2$.
The set of congruences ordered by refinement forms a lattice.