A Triangulation of the Flow Polytope of the Zigzag Graph
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866913407939117056 |
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| author | Brunner, Rachel Hanusa, Christopher R. H. |
| author_facet | Brunner, Rachel Hanusa, Christopher R. H. |
| contents | We show that the dual graph of the triangulation of the flow polytope of the zigzag graph adorned with the length-reverse-length framing is a subgraph of a grid graph. Through Mészáros, Morales, and Striker's bijection between simplices of the triangulation, integer flows of a different, supplemental flow polytope, we provide a simple numerical characterization of the adjacency between the triangulation's simplices in terms of their corresponding integer flows. The proofs result from the development of Postnikov and Stanley's sequences of noncrossing bipartite trees as combinatorial objects we call groves. We propose two new statistics derived from this construction that we conjecture recover the $h^*$-polynomial of the flow polytope of the zigzag graph. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2406_20048 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | A Triangulation of the Flow Polytope of the Zigzag Graph Brunner, Rachel Hanusa, Christopher R. H. Combinatorics Primary: 05A19, 05C21, 52B12, 52B05, 52C07, Secondary: 05C05, 05C20, 05C30, 52A38, 52B11 We show that the dual graph of the triangulation of the flow polytope of the zigzag graph adorned with the length-reverse-length framing is a subgraph of a grid graph. Through Mészáros, Morales, and Striker's bijection between simplices of the triangulation, integer flows of a different, supplemental flow polytope, we provide a simple numerical characterization of the adjacency between the triangulation's simplices in terms of their corresponding integer flows. The proofs result from the development of Postnikov and Stanley's sequences of noncrossing bipartite trees as combinatorial objects we call groves. We propose two new statistics derived from this construction that we conjecture recover the $h^*$-polynomial of the flow polytope of the zigzag graph. |
| title | A Triangulation of the Flow Polytope of the Zigzag Graph |
| topic | Combinatorics Primary: 05A19, 05C21, 52B12, 52B05, 52C07, Secondary: 05C05, 05C20, 05C30, 52A38, 52B11 |
| url | https://arxiv.org/abs/2406.20048 |