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()