  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 $1\leq b<a\leq n$. As a
consequence, we confirm two conjectures posed by Carolina Benedetti and Bruce
E. Sagan.
