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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.

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  • $\begingroup$ 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. $\endgroup$ – darij grinberg Dec 19 '14 at 21:02
  • $\begingroup$ 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! $\endgroup$ – Amy Pang Dec 19 '14 at 22:25
  • $\begingroup$ Oh! Sorry for this; I read "plane" and thought this had to be something different. $\endgroup$ – darij grinberg Dec 19 '14 at 22:55
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(This answer came from Yannic Vargas.)

Foissy defines a bijection between special plane posets and permutations (p10). The set of special plane posets inherit a Hopf structure from the set of double posets: the product is disjoint union and the coproduct is "removing an ideal" (p6). The isomorphism between this Hopf structure and the Malvenuto-Reutenauer algebra is given in Theorem 18, p13.

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