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pPartitionRing -- produces the p-partition ring of a poset

Synopsis

Description

Recall that a $P$-partition for a naturally labeled poset $P$ on vertices $1, \ldots, n$ is a function $f: P \rightarrow \mathbb{NN}$ which is order-reversing, i.e., if $i < j$ in $P$ then $f(i) \geq f(j)$ in $\mathbb{NN}$. To a $P$-partition $f$ we can assign the monomial $t_1^{f(1)} \ldots t_n^{f(n)}$. The $P$-partition ring is the ring spanned by the monomials corresponding to $P$-partitions.

The $P$-partition ring is more simply generated by the monomials corresponding to the connected order ideals of $P$. This method returns the toric quotient algebra, whose toric ideal is minimally generated, isomorphic to the $P$-partition ring.

i1 : P = poset {{1,2}, {2,4}, {3,4}, {3,5}};
 
i2 : pPartitionRing P

     QQ[t   , t      , t   , t      , t            , t               ]
         {3}   {3, 4}   {0}   {0, 1}   {0, 1, 2, 3}   {0, 1, 2, 3, 4}
o2 = -----------------------------------------------------------------
                t      t             - t   t
                 {3, 4} {0, 1, 2, 3}    {3} {0, 1, 2, 3, 4}

o2 : QuotientRing

In some cases, it may be faster to use the FourTiTwo method toricGroebner to generate the toric relations. Using the Strategy "4ti2" tells the method to use this approach.

i3 : pPartitionRing(divisorPoset 6, Strategy => "4ti2")

     QQ[t   , t      , t      , t         , t            ]
         {0}   {0, 1}   {0, 2}   {0, 1, 2}   {0, 1, 2, 3}
o3 = -----------------------------------------------------
                t      t       - t   t
                 {0, 1} {0, 2}    {0} {0, 1, 2}

o3 : QuotientRing

See also

Ways to use pPartitionRing :

For the programmer

The object pPartitionRing is a method function with options.