The Hopf algebra of diagonal rectangulations
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arXiv
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| Format: | Preprint |
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2010
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| _version_ | 1866911674469974016 |
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| author | Law, Shirley Reading, Nathan |
| author_facet | Law, Shirley Reading, Nathan |
| contents | We define and study a combinatorial Hopf algebra dRec with basis elements indexed by diagonal rectangulations of a square. This Hopf algebra provides an intrinsic combinatorial realization of the Hopf algebra tBax of twisted Baxter permutations, which previously had only been described extrinsically as a sub Hopf algebra of the Malvenuto-Reutenauer Hopf algebra of permutations. We describe the natural lattice structure on diagonal rectangulations, analogous to the Tamari lattice on triangulations, and observe that diagonal rectangulations index the vertices of a polytope analogous to the associahedron. We give an explicit bijection between twisted Baxter permutations and the better-known Baxter permutations, and describe the resulting Hopf algebra structure on Baxter permutations. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_1011_2118 |
| institution | arXiv |
| publishDate | 2010 |
| record_format | arxiv |
| spellingShingle | The Hopf algebra of diagonal rectangulations Law, Shirley Reading, Nathan Combinatorics Primary 05E15, 16T30, Secondary 05A19, 05A05 We define and study a combinatorial Hopf algebra dRec with basis elements indexed by diagonal rectangulations of a square. This Hopf algebra provides an intrinsic combinatorial realization of the Hopf algebra tBax of twisted Baxter permutations, which previously had only been described extrinsically as a sub Hopf algebra of the Malvenuto-Reutenauer Hopf algebra of permutations. We describe the natural lattice structure on diagonal rectangulations, analogous to the Tamari lattice on triangulations, and observe that diagonal rectangulations index the vertices of a polytope analogous to the associahedron. We give an explicit bijection between twisted Baxter permutations and the better-known Baxter permutations, and describe the resulting Hopf algebra structure on Baxter permutations. |
| title | The Hopf algebra of diagonal rectangulations |
| topic | Combinatorics Primary 05E15, 16T30, Secondary 05A19, 05A05 |
| url | https://arxiv.org/abs/1011.2118 |