Abstract
It follows from a theorem of Derksen [J. Algebraic Combin., 30 (2009) 43–86] that the Tutte polynomial of a rank-r matroid on an n-set is “naturally” a linear combination of Tutte polynomials of rank-r size-n freedom matroids. However, the Tutte polynomials of rank-r size-n freedom matroids are not linearly independent. We construct two natural bases for these polynomials and as a corollary, we prove that the Tutte polynomials of rank-r matroids of size-n span a subspace of dimension \({r(n-r)+1}\). We also find a generating set for the linear relations between Tutte polynomials of freedom matroids. This generating set is indexed by a pair of intervals, one of size 2 and one of size 4, in the weak order of freedom matroids. This weak order is a distributive lattice and a sublattice of Young’s partition lattice.
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Kung, J.P. Syzygies on Tutte Polynomials of Freedom Matroids. Ann. Comb. 21, 605–628 (2017). https://doi.org/10.1007/s00026-017-0370-0
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DOI: https://doi.org/10.1007/s00026-017-0370-0