Geodesic
Fighting Fish and Two-Stack Sortable Permutations
Séminaire lotharingien de combinatoire, 80B (2018)

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In 2017, Duchi, Guerrini, Rinaldi and Schaeffer proposed new combinatorial objects called "fighting fish", which are counted by the same formula as more classical objects, such as two-stack sortable permutations and non-separable planar maps. In this article, we explore the bijective aspect of fighting fish by establishing a bijection to two-stack sortable permutations, using a new recursive decomposition of these permutations. With our bijection, we give combinatorial explanations of several results on fighting fish proved previously with generating functions. Using the decomposition, we also prove the algebraicity of a generating function of two-stack sortable permutations, extending a result of Bousquet-Mélou (1998). 

@article{SLC_2018_80B_a6,
     author = {Wenjie Fang},
     title = {Fighting {Fish} and {Two-Stack} {Sortable} {Permutations}},
     journal = {S\'eminaire lotharingien de combinatoire},
     publisher = {mathdoc},
     volume = {80B},
     year = {2018},
     url = {http://geodesic.mathdoc.fr/item/SLC_2018_80B_a6/}
}
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Wenjie Fang. Fighting Fish and Two-Stack Sortable Permutations. Séminaire lotharingien de combinatoire, 80B (2018). http://geodesic.mathdoc.fr/item/SLC_2018_80B_a6/