Obtaining a poset of plane partitions via Sage

asked 2021-08-29 17:12:52 +0200

klaaa
671 ●3 ●14 ●38

updated 2023-05-11 14:11:50 +0200

FrédéricC

5125 ●3 ●43 ●111

In Sage there is the command YoungsLatticePrincipalOrderIdeal(p) to obtain the lattice of partitions contained in p (via Ferrer diagrams) with maximal element p.

My question is whether there is the same command for obtaining via Sage the finite poset of all plane partitions that are contained in a given plane partition p ?

(see https://en.wikipedia.org/wiki/Plane_p... )

edit: It would also be interesting how to obtain the poset of plane partitions of $m$ for a given $m \leq n$ as one can do the same for the Young lattice in Sage via P=posets.YoungsLattice(5).

Thank you for any help.

answered 2021-08-29 17:50:45 +0200

FrédéricC

5125 ●3 ●43 ●111

updated 2021-08-30 08:48:46 +0200

Like this

sage: def PP(a,b,c): 
....:         return posets.ProductOfChains([a,b,c]).order_ideals_lattice() 
sage: PP(2,2,2)                                                                 
Finite lattice containing 20 element

for the full poset. Then you can take any principal order filter.

EDIT Another way:

sage: P = PlanePartitions([4,3,2])                                                
sage: q = P.random_element()                                                      
sage: qt = q.to_tableau()                                                         
sage: Poset(([x.to_tableau() for x in P if x.to_tableau() <= qt] ,lambda x,y: x<=y))                                                                      
Finite poset containing 459 elements

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Comments

Thank you. Sorry I do not fully understand you answer. For example how to obtain the poset of all plane partitions of $m$ with $m \leq n$ for a given $n$? Is it obtained by taking all your posets as in your answer with $a+b+c \leq n$?

klaaa ( 2021-08-29 18:33:03 +0200 )edit

Not clear what you want. See also https://doc.sagemath.org/html/en/refe...

FrédéricC ( 2021-08-30 08:37:03 +0200 )edit

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