Antipodes and involutions

@article{Benedetti2017AntipodesAI,
  title={Antipodes and involutions},
  author={Carolina Benedetti and Bruce E. Sagan},
  journal={J. Comb. Theory, Ser. A},
  year={2017},
  volume={148},
  pages={275-315}
}
If H is a connected, graded Hopf algebra, then Takeuchi’s formula can be used to compute its antipode. However, there is usually massive cancellation in the result. We show how sign-reversing involutions can sometimes be used to obtain cancellationfree formulas. We apply this idea to the Hopf algebras of polynomials, graphs, and noncommutative symmetric functions.