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
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i2 : flagGeomTuttePolynomial FM
2 2 2 2 2 2
o2 = x y + x y + x*y + x + 2x*y + y
o2 : ZZ[x, y]
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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
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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]
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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
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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]
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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
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i4 : flagGeomTuttePolynomial flagMatroid {M}, tuttePolynomial M
3 2 3 2
o4 = (x + x + x*y, x + x + x*y)
o4 : Sequence
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The computation often does not finish within a reasonable time (< 10 min) if the ground set is bigger than 5.
The object flagGeomTuttePolynomial is a method function.