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

## Synopsis

• Usage:
flagGeomTuttePolynomial(FM)
• Inputs:
• FM, ,
• Outputs:
• , 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.