A cancellation-free antipode formula (for uniform matroids) for the   restriction-contraction matroid Hopf algebra
  In this paper, we give a cancellation-free antipode formula (for uniform matroids) for the restriction-contraction matroid Hopf algebra, using the technique of splitting and merging via a sign-reversing involution. The cancellation-free formula expresses the antipode of uniform matroids as a sum over certain ordered set partitions. 
Eric Bucher, Jacob P. Matherne
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