In this extended abstract we focus on enumerative properties of
fighting fish: in particular we provide a new decomposition and we
show that the number of fighting fish with *i* left lower free edges
and *j* right lower free edges is equal to

These numbers are known to count rooted planar non-separable maps with
*i*+1 vertices and *j*+1 faces, or two-stack-sortable permutations
with respect to ascending and descending runs, or left ternary trees
with respect to vertices with even and odd abscissa. However we have
been unable until now to provide any explicit bijection between our
fish and such structures. Instead we provide new refined generating
functions for left ternary trees to prove further equidistribution
results.

Received: November 14, 2016. Accepted: February 17, 2017. Final version: April 1, 2017.

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