We call progressive paths and rushed paths two families of Dyck paths studied by Asinowski and Jelínek, which have the same enumerating sequence (OEIS entry A287709).We present a bijection proving this fact.Rushed paths turn out to be in bijection with one-sided trees, introduced by Durhuus and Ünel, which have an asymptotic enumeration involving a stretched exponential.We conclude by presenting several other classes of related lattice paths and directed animals that may have similar asymptotic properties.
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Axel Bacher (2024) studied this question.
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