Identifier
Values
[[3,2,1],[2,1]] => ([(0,2),(2,1)],3) => ([(0,1),(0,2),(1,3),(2,3)],4) => 3
[[4,2],[2]] => ([(0,2),(2,1)],3) => ([(0,1),(0,2),(1,3),(2,3)],4) => 3
[[3,2,1],[1,1]] => ([(0,2),(2,1)],3) => ([(0,1),(0,2),(1,3),(2,3)],4) => 3
[[3,3,1],[2,1]] => ([(0,2),(2,1)],3) => ([(0,1),(0,2),(1,3),(2,3)],4) => 3
[[3,2,1],[2]] => ([(0,2),(2,1)],3) => ([(0,1),(0,2),(1,3),(2,3)],4) => 3
[[3,2,2],[2,1]] => ([(0,2),(2,1)],3) => ([(0,1),(0,2),(1,3),(2,3)],4) => 3
[[2,2,1,1],[1,1]] => ([(0,2),(2,1)],3) => ([(0,1),(0,2),(1,3),(2,3)],4) => 3
[[5,2],[2]] => ([(0,2),(2,1)],3) => ([(0,1),(0,2),(1,3),(2,3)],4) => 3
[[4,2,1],[1,1]] => ([(0,2),(2,1)],3) => ([(0,1),(0,2),(1,3),(2,3)],4) => 3
[[5,3],[3]] => ([(0,2),(2,1)],3) => ([(0,1),(0,2),(1,3),(2,3)],4) => 3
[[3,2,1],[1]] => ([(0,2),(2,1)],3) => ([(0,1),(0,2),(1,3),(2,3)],4) => 3
[[3,2,2,1],[1,1,1]] => ([(0,2),(2,1)],3) => ([(0,1),(0,2),(1,3),(2,3)],4) => 3
[[4,4,1],[3,1]] => ([(0,2),(2,1)],3) => ([(0,1),(0,2),(1,3),(2,3)],4) => 3
[[4,3,1],[3]] => ([(0,2),(2,1)],3) => ([(0,1),(0,2),(1,3),(2,3)],4) => 3
[[3,3,2],[2,1]] => ([(0,2),(2,1)],3) => ([(0,1),(0,2),(1,3),(2,3)],4) => 3
[[3,2,1,1],[2]] => ([(0,2),(2,1)],3) => ([(0,1),(0,2),(1,3),(2,3)],4) => 3
[[4,3,3],[3,2]] => ([(0,2),(2,1)],3) => ([(0,1),(0,2),(1,3),(2,3)],4) => 3
[[3,3,3,1],[2,2,1]] => ([(0,2),(2,1)],3) => ([(0,1),(0,2),(1,3),(2,3)],4) => 3
[[3,2,2,2],[2,1,1]] => ([(0,2),(2,1)],3) => ([(0,1),(0,2),(1,3),(2,3)],4) => 3
[[2,2,2,1,1],[1,1,1]] => ([(0,2),(2,1)],3) => ([(0,1),(0,2),(1,3),(2,3)],4) => 3
[[2,2,1,1,1],[1,1]] => ([(0,2),(2,1)],3) => ([(0,1),(0,2),(1,3),(2,3)],4) => 3
[[6,2],[2]] => ([(0,2),(2,1)],3) => ([(0,1),(0,2),(1,3),(2,3)],4) => 3
[[5,2,1],[1,1]] => ([(0,2),(2,1)],3) => ([(0,1),(0,2),(1,3),(2,3)],4) => 3
[[5,3],[2]] => ([(0,2),(2,1)],3) => ([(0,1),(0,2),(1,3),(2,3)],4) => 3
[[4,3,1],[1,1]] => ([(0,2),(2,1)],3) => ([(0,1),(0,2),(1,3),(2,3)],4) => 3
[[4,2,1],[1]] => ([(0,2),(2,1)],3) => ([(0,1),(0,2),(1,3),(2,3)],4) => 3
[[4,2,2,1],[1,1,1]] => ([(0,2),(2,1)],3) => ([(0,1),(0,2),(1,3),(2,3)],4) => 3
[[6,4],[4]] => ([(0,2),(2,1)],3) => ([(0,1),(0,2),(1,3),(2,3)],4) => 3
[[4,4,1],[2,1]] => ([(0,2),(2,1)],3) => ([(0,1),(0,2),(1,3),(2,3)],4) => 3
[[4,3,1],[2]] => ([(0,1),(0,2),(1,3),(2,3)],4) => ([(0,1),(0,2),(1,3),(2,3)],4) => 3
[[4,4,2],[2,2]] => ([(0,2),(2,1)],3) => ([(0,1),(0,2),(1,3),(2,3)],4) => 3
[[4,2,2],[2]] => ([(0,2),(2,1)],3) => ([(0,1),(0,2),(1,3),(2,3)],4) => 3
[[3,3,2,1],[1,1,1]] => ([(0,2),(2,1)],3) => ([(0,1),(0,2),(1,3),(2,3)],4) => 3
[[3,3,1,1],[1,1]] => ([(0,2),(2,1)],3) => ([(0,1),(0,2),(1,3),(2,3)],4) => 3
[[3,2,1,1],[1]] => ([(0,2),(2,1)],3) => ([(0,1),(0,2),(1,3),(2,3)],4) => 3
[[3,2,2,1],[1,1]] => ([(0,1),(0,2),(1,3),(2,3)],4) => ([(0,1),(0,2),(1,3),(2,3)],4) => 3
[[3,2,2,2,1],[1,1,1,1]] => ([(0,2),(2,1)],3) => ([(0,1),(0,2),(1,3),(2,3)],4) => 3
[[5,5,1],[4,1]] => ([(0,2),(2,1)],3) => ([(0,1),(0,2),(1,3),(2,3)],4) => 3
[[5,4,1],[4]] => ([(0,2),(2,1)],3) => ([(0,1),(0,2),(1,3),(2,3)],4) => 3
[[4,4,2],[3,1]] => ([(0,1),(0,2),(1,3),(2,3)],4) => ([(0,1),(0,2),(1,3),(2,3)],4) => 3
[[4,3,2],[3]] => ([(0,2),(2,1)],3) => ([(0,1),(0,2),(1,3),(2,3)],4) => 3
[[4,3,1,1],[3]] => ([(0,2),(2,1)],3) => ([(0,1),(0,2),(1,3),(2,3)],4) => 3
[[4,4,3],[3,2]] => ([(0,2),(2,1)],3) => ([(0,1),(0,2),(1,3),(2,3)],4) => 3
[[4,3,3],[3,1]] => ([(0,2),(2,1)],3) => ([(0,1),(0,2),(1,3),(2,3)],4) => 3
[[3,3,3,1],[2,1,1]] => ([(0,2),(2,1)],3) => ([(0,1),(0,2),(1,3),(2,3)],4) => 3
[[3,2,2,1],[2]] => ([(0,2),(2,1)],3) => ([(0,1),(0,2),(1,3),(2,3)],4) => 3
[[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) => 3
[[3,2,1,1,1],[2]] => ([(0,2),(2,1)],3) => ([(0,1),(0,2),(1,3),(2,3)],4) => 3
[[5,4,4],[4,3]] => ([(0,2),(2,1)],3) => ([(0,1),(0,2),(1,3),(2,3)],4) => 3
[[4,4,4,1],[3,3,1]] => ([(0,2),(2,1)],3) => ([(0,1),(0,2),(1,3),(2,3)],4) => 3
[[3,3,3,2],[2,2,1]] => ([(0,2),(2,1)],3) => ([(0,1),(0,2),(1,3),(2,3)],4) => 3
[[3,3,2,2],[2,2]] => ([(0,2),(2,1)],3) => ([(0,1),(0,2),(1,3),(2,3)],4) => 3
[[3,2,2,2],[2,1]] => ([(0,2),(2,1)],3) => ([(0,1),(0,2),(1,3),(2,3)],4) => 3
[[2,2,2,1,1],[1,1]] => ([(0,2),(2,1)],3) => ([(0,1),(0,2),(1,3),(2,3)],4) => 3
[[4,3,3,3],[3,2,2]] => ([(0,2),(2,1)],3) => ([(0,1),(0,2),(1,3),(2,3)],4) => 3
[[3,3,3,3,1],[2,2,2,1]] => ([(0,2),(2,1)],3) => ([(0,1),(0,2),(1,3),(2,3)],4) => 3
[[3,2,2,2,2],[2,1,1,1]] => ([(0,2),(2,1)],3) => ([(0,1),(0,2),(1,3),(2,3)],4) => 3
[[2,2,2,2,1,1],[1,1,1,1]] => ([(0,2),(2,1)],3) => ([(0,1),(0,2),(1,3),(2,3)],4) => 3
[[2,2,1,1,1,1],[1,1]] => ([(0,2),(2,1)],3) => ([(0,1),(0,2),(1,3),(2,3)],4) => 3
[[7,2],[2]] => ([(0,2),(2,1)],3) => ([(0,1),(0,2),(1,3),(2,3)],4) => 3
[[6,2,1],[1,1]] => ([(0,2),(2,1)],3) => ([(0,1),(0,2),(1,3),(2,3)],4) => 3
[[6,3],[2]] => ([(0,2),(2,1)],3) => ([(0,1),(0,2),(1,3),(2,3)],4) => 3
[[5,3,1],[1,1]] => ([(0,2),(2,1)],3) => ([(0,1),(0,2),(1,3),(2,3)],4) => 3
[[5,2,1],[1]] => ([(0,2),(2,1)],3) => ([(0,1),(0,2),(1,3),(2,3)],4) => 3
[[5,2,2,1],[1,1,1]] => ([(0,2),(2,1)],3) => ([(0,1),(0,2),(1,3),(2,3)],4) => 3
[[6,4],[3]] => ([(0,2),(2,1)],3) => ([(0,1),(0,2),(1,3),(2,3)],4) => 3
[[4,3,1],[1]] => ([(0,2),(2,1)],3) => ([(0,1),(0,2),(1,3),(2,3)],4) => 3
[[4,3,2],[1,1]] => ([(0,2),(2,1)],3) => ([(0,1),(0,2),(1,3),(2,3)],4) => 3
[[5,2,2],[2]] => ([(0,2),(2,1)],3) => ([(0,1),(0,2),(1,3),(2,3)],4) => 3
[[4,2,1,1],[1]] => ([(0,2),(2,1)],3) => ([(0,1),(0,2),(1,3),(2,3)],4) => 3
[[5,3,3],[2,2]] => ([(0,2),(2,1)],3) => ([(0,1),(0,2),(1,3),(2,3)],4) => 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) => 3
[[4,2,2,2,1],[1,1,1,1]] => ([(0,2),(2,1)],3) => ([(0,1),(0,2),(1,3),(2,3)],4) => 3
[[7,5],[5]] => ([(0,2),(2,1)],3) => ([(0,1),(0,2),(1,3),(2,3)],4) => 3
[[5,5,1],[3,1]] => ([(0,2),(2,1)],3) => ([(0,1),(0,2),(1,3),(2,3)],4) => 3
[[5,4,1],[3]] => ([(0,1),(0,2),(1,3),(2,3)],4) => ([(0,1),(0,2),(1,3),(2,3)],4) => 3
[[4,3,1,1],[2]] => ([(0,1),(0,2),(1,3),(2,3)],4) => ([(0,1),(0,2),(1,3),(2,3)],4) => 3
[[5,5,3],[3,3]] => ([(0,2),(2,1)],3) => ([(0,1),(0,2),(1,3),(2,3)],4) => 3
[[4,3,3],[2,1]] => ([(0,2),(2,1)],3) => ([(0,1),(0,2),(1,3),(2,3)],4) => 3
[[3,2,2,1],[1]] => ([(0,2),(2,1)],3) => ([(0,1),(0,2),(1,3),(2,3)],4) => 3
[[3,3,2,2,1],[1,1,1,1]] => ([(0,2),(2,1)],3) => ([(0,1),(0,2),(1,3),(2,3)],4) => 3
[[3,3,1,1,1],[1,1]] => ([(0,2),(2,1)],3) => ([(0,1),(0,2),(1,3),(2,3)],4) => 3
[[3,2,1,1,1],[1]] => ([(0,2),(2,1)],3) => ([(0,1),(0,2),(1,3),(2,3)],4) => 3
[[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) => 4
[[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) => 3
[[3,2,2,2,2,1],[1,1,1,1,1]] => ([(0,2),(2,1)],3) => ([(0,1),(0,2),(1,3),(2,3)],4) => 3
[[6,6,1],[5,1]] => ([(0,2),(2,1)],3) => ([(0,1),(0,2),(1,3),(2,3)],4) => 3
[[6,5,1],[5]] => ([(0,2),(2,1)],3) => ([(0,1),(0,2),(1,3),(2,3)],4) => 3
[[5,5,2],[4,1]] => ([(0,1),(0,2),(1,3),(2,3)],4) => ([(0,1),(0,2),(1,3),(2,3)],4) => 3
[[5,4,2],[4]] => ([(0,2),(2,1)],3) => ([(0,1),(0,2),(1,3),(2,3)],4) => 3
[[5,4,1,1],[4]] => ([(0,2),(2,1)],3) => ([(0,1),(0,2),(1,3),(2,3)],4) => 3
[[4,4,3],[3,1]] => ([(0,2),(2,1)],3) => ([(0,1),(0,2),(1,3),(2,3)],4) => 3
[[3,3,2,1],[2]] => ([(0,2),(2,1)],3) => ([(0,1),(0,2),(1,3),(2,3)],4) => 3
[[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) => 4
[[4,3,1,1,1],[3]] => ([(0,2),(2,1)],3) => ([(0,1),(0,2),(1,3),(2,3)],4) => 3
[[5,5,4],[4,3]] => ([(0,2),(2,1)],3) => ([(0,1),(0,2),(1,3),(2,3)],4) => 3
[[5,4,4],[4,2]] => ([(0,2),(2,1)],3) => ([(0,1),(0,2),(1,3),(2,3)],4) => 3
[[3,3,3,1],[2,1]] => ([(0,2),(2,1)],3) => ([(0,1),(0,2),(1,3),(2,3)],4) => 3
[[3,3,3,2],[2,1,1]] => ([(0,2),(2,1)],3) => ([(0,1),(0,2),(1,3),(2,3)],4) => 3
[[3,2,2,1,1],[2]] => ([(0,2),(2,1)],3) => ([(0,1),(0,2),(1,3),(2,3)],4) => 3
[[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) => 3
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Description
The number of simple modules with projective dimension at most 1.
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.
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.
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.
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.
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