On the cancellation-free antipode formula for the Malvenuto-Reutenauer Hopf Algebra

For the Malvenuto-Reutenauer Hopf algebra of permutations, we provide a cancellation-free antipode formula for any permutation of the form $ab1\cdots (b-1)(b+1)\cdots (a-1)(a+1)\cdots n$, which starts with the decreasing sequence $ab$ and ends with the increasing sequence $1\cdots (b-1)(b+1)\cdots (a-1)(a+1)\cdots n$, where $1\le b<a\le n$. As a consequence, we confirm two conjectures posed by Carolina Benedetti and Bruce E. Sagan.