There is a well-known Hopf structure on permutations, due to Malvenuto and Reutenauer. Here, the F basis elements are permutations in one-line notation, the product of two permutations is their shifted shuffle, and the coproduct of a permutation is deconcatenate and standardise.

Now there are several "pictorial" ways to represent permutations, such as permutation tableaux, alternative tableaux, tree-like tableaux, ... . What makes them "pictorial" to me is that they do not involve the labels 1,2,...,n.

I would like to know if descriptions of the product and coproduct of the Malvenuto-Reutenauer Hopf algebra in terms of such "pictorial" objects are known.

This description can be in terms of any basis. I only know of one case, which I will post as an answer, so you can see the sort of description that I am after.

A self paired Hopf algebra on double posets and a Littlewood–Richardson rule.) The Malvenuto-Reutenauer Hopf algebra is a subquotient of it (it is actually a quotient of the Hopf algebra of special double posets), and it is defined in a way which does not involve the labels 1,2,...,n. $\endgroup$