A catalog of posets and lattices.¶
Some common posets can be accessed through the posets.<tab>
object:
sage: posets.PentagonPoset()
Finite lattice containing 5 elements
Moreover, the set of all posets of order \(n\) is represented by Posets(n)
:
sage: Posets(5)
Posets containing 5 elements
The infinite set of all posets can be used to find minimal examples:
sage: for P in Posets():
....: if not P.is_series_parallel():
....: break
sage: P
Finite poset containing 4 elements
Catalog of common posets:
AntichainPoset() |
Return an antichain on \(n\) elements. |
BooleanLattice() |
Return the Boolean lattice on \(2^n\) elements. |
ChainPoset() |
Return a chain on \(n\) elements. |
Crown() |
Return the crown poset on \(2n\) elements. |
DexterSemilattice() |
Return the Dexter semilattice. |
DiamondPoset() |
Return the lattice of rank two on \(n\) elements. |
DivisorLattice() |
Return the divisor lattice of an integer. |
IntegerCompositions() |
Return the poset of integer compositions of \(n\). |
IntegerPartitions() |
Return the poset of integer partitions of n . |
IntegerPartitionsDominanceOrder() |
Return the lattice of integer partitions on the integer \(n\) ordered by dominance. |
NoncrossingPartitions() |
Return the poset of noncrossing partitions of a finite Coxeter group W . |
PentagonPoset() |
Return the Pentagon poset. |
PermutationPattern() |
Return the Permutation pattern poset. |
PermutationPatternInterval() |
Return an interval in the Permutation pattern poset. |
PermutationPatternOccurrenceInterval() |
Return the occurrence poset for a pair of comparable elements in the Permutation pattern poset. |
PowerPoset() |
Return a power poset. |
ProductOfChains() |
Return a product of chain posets. |
RandomLattice() |
Return a random lattice on \(n\) elements. |
RandomPoset() |
Return a random poset on \(n\) elements. |
RestrictedIntegerPartitions() |
Return the poset of integer partitions of \(n\), ordered by restricted refinement. |
SetPartitions() |
Return the poset of set partitions of the set \(\{1,\dots,n\}\). |
ShardPoset() |
Return the shard intersection order. |
SSTPoset() |
Return the poset on semistandard tableaux of shape \(s\) and largest entry \(f\) that is ordered by componentwise comparison. |
StandardExample() |
Return the standard example of a poset with dimension \(n\). |
SymmetricGroupAbsoluteOrderPoset() |
The poset of permutations with respect to absolute order. |
SymmetricGroupBruhatIntervalPoset() |
The poset of permutations with respect to Bruhat order. |
SymmetricGroupBruhatOrderPoset() |
The poset of permutations with respect to Bruhat order. |
SymmetricGroupWeakOrderPoset() |
The poset of permutations of \(\{ 1, 2, \ldots, n \}\) with respect to the weak order. |
TamariLattice() |
Return the Tamari lattice. |
TetrahedralPoset() |
Return the Tetrahedral poset with \(n-1\) layers based on the input colors. |
UpDownPoset() |
Return the up-down poset on \(n\) elements. |
YoungDiagramPoset() |
Return the poset of cells in the Young diagram of a partition. |
YoungsLattice() |
Return Young’s Lattice up to rank \(n\). |
YoungsLatticePrincipalOrderIdeal() |
Return the principal order ideal of the partition \(lam\) in Young’s Lattice. |
YoungFibonacci() |
Return the Young-Fibonacci lattice up to rank \(n\). |
Constructions¶
-
sage.combinat.posets.poset_examples.
Posets
¶ A collection of posets and lattices.
EXAMPLES:
sage: posets.BooleanLattice(3) Finite lattice containing 8 elements sage: posets.ChainPoset(3) Finite lattice containing 3 elements sage: posets.RandomPoset(17,.15) Finite poset containing 17 elements
The category of all posets:
sage: Posets() Category of posets
The enumerated set of all posets on \(3\) elements, up to an isomorphism:
sage: Posets(3) Posets containing 3 elements
See also
Posets
,FinitePosets
,Poset()
-
sage.combinat.posets.poset_examples.
posets
¶ A collection of posets and lattices.
EXAMPLES:
sage: posets.BooleanLattice(3) Finite lattice containing 8 elements sage: posets.ChainPoset(3) Finite lattice containing 3 elements sage: posets.RandomPoset(17,.15) Finite poset containing 17 elements
The category of all posets:
sage: Posets() Category of posets
The enumerated set of all posets on \(3\) elements, up to an isomorphism:
sage: Posets(3) Posets containing 3 elements
See also
Posets
,FinitePosets
,Poset()