A Hopf algebra on permutations with a coupling product

We define a coupling product and a draw coproduct on permutations and show that they define a graded, connected, cocommutative, free Hopf algebra $\mathbb{K}\mathbb{S}$. In characteristic zero, this implies that $\mathbb{K}\mathbb{S}$ is isomorphic to a certain cocommutative Hopf algebra associated with the dual of the coradical filtration of the Malvenuto–Reutenauer Hopf algebra on permutations. The coupling product and draw coproduct are permutation analogues of the product and coproduct on the monomial basis of symmetric functions; therefore, we can say that our presentation is a monomial basis for $\mathbb{K}\mathbb{S}$.