flagGeomTuttePolynomial -- computes the flag-geometric Tutte polynomial of flag matroids

Synopsis

Usage:

flagGeomTuttePolynomial(FM)
Inputs:
- FM, a flag matroid,
Outputs:
- a ring element, a polynomial in variables $x,y$

Description

This method computes the flag-geometric Tutte polynomial of a FlagMatroid, defined via a push-pull of the KClass of the flag matroid. See Definition 6.1 of [DES20]. The following is the example 8.24 in [CDMS18].

i1 : FM = flagMatroid {uniformMatroid(1,3),uniformMatroid(2,3)}

o1 = a "flag matroid" with rank sequence {1, 2} on 3 elements 

o1 : FlagMatroid

i2 : flagGeomTuttePolynomial FM

      2 2    2       2    2           2
o2 = x y  + x y + x*y  + x  + 2x*y + y

o2 : ZZ[x, y]

The following example negatively answers Conjecture 9.2 of [CDMS18], which had conjectured that all coefficients of the flag-geometric Tutte polynomial of a flag matroid are nonnegative.

i1 : FM = flagMatroid {uniformMatroid(1,5),uniformMatroid(3,5)}

o1 = a flag matroid with rank sequence {1, 3} on 5 elements

o1 : FlagMatroid

i2 : flagGeomTuttePolynomial FM

      3 4    3 3     2 4    3 2    2 3       4    3      2 2       3     4    3     2        2
o2 = x y  + x y  + 2x y  + x y  - x y  + 3x*y  + x y + 6x y  + 9x*y  + 4y  + x  + 3x y + 3x*y  +
      3
     y

o2 : ZZ[x, y]

Here is another counterexample but one where no constituent matroids have rank 1 or corank 1.

i1 : FM = flagMatroid {uniformMatroid(2,6),uniformMatroid(4,6)}

o1 = a flag matroid with rank sequence {2, 4} on 6 elements

o1 : FlagMatroid

i2 : time flagGeomTuttePolynomial FM -- used 691.322 seconds

       4 4     4 3     3 4     4 2     3 3     2 4     4       3 2      2 3       4     4      3
o2 = x y  + 2x y  + 2x y  + 3x y  - 6x y  + 3x y  + 4x y + 18x y  + 18x y  + 4x*y  + 5x  + 14x y
           2 2        3     4     3     2        2     3
      + 18x y  + 14x*y  + 5y  + 2x  + 6x y + 6x*y  + 2y

o2 : ZZ[x, y]

When the flag matroid has a single constituent (i.e. is a matroid), it agrees with the usual Tutte polynomial.

i3 : M = matroid graph{{a,b},{b,c},{c,a},{a,d}}

o3 = a "matroid" of rank 3 on 4 elements

o3 : Matroid

i4 : flagGeomTuttePolynomial flagMatroid {M}, tuttePolynomial M

       3    2         3    2
o4 = (x  + x  + x*y, x  + x  + x*y)

o4 : Sequence

Caveat

The computation often does not finish within a reasonable time (< 10 min) if the ground set is bigger than 5.

Ways to use flagGeomTuttePolynomial :

flagGeomTuttePolynomial(FlagMatroid)

For the programmer