# Hopf structures on “pictorial” descriptions of permutations

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.

• Are you aware of the Hopf algebra of double posets? (See Malvenuto & Reutenauer, 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. – darij grinberg Dec 19 '14 at 21:02
• Darij, this is the answer that Yannic gave, that I posted below. He didn't tell me about this particular paper though, so thanks for the reference! – Amy Pang Dec 19 '14 at 22:25
• Oh! Sorry for this; I read "plane" and thought this had to be something different. – darij grinberg Dec 19 '14 at 22:55