For the Malvenuto-Reutenauer Hopf algebra of permutations, we provide a cancellation-free antipode formula for permutations starting with a decreasing sequence consisting of two positive integers and ending with an increasing one. As a consequence, we confirm two conjectures posed by Carolina Benedetti and Bruce E. Sagan.
Matroid-minor Hopf algebra: a cancellation-free antipode formula and other applications of sign-reversing involutions
Remarks on the conjectures of Capparelli, Meurman, Primc and Primc
A cancellation-free antipode formula (for uniform matroids) for the restriction-contraction matroid Hopf algebra
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