  Chen's lemma on iterated integrals implies that certain identities involving
multiple integrals, such as the de Bruijn and Wick formulas, amount to
combinatorial identities for Pfaffians and hafnians in shuffle algebras. We
provide direct algebraic proofs of such shuffle identities, and obtain various
generalizations. We also discuss some Pfaffian identities due to Sundquist and
Ishikawa-Wakayama, and a Cauchy formula for anticommutative symmetric
functions. Finally, we extend some of the previous considerations to
hyperpfaffians and hyperhafnians.
