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Counting isomorphism classes of $β$-normal linear lambda terms

Unanticipated connections between different fragments of lambda calculus and different families of embedded graphs (a.k.a. "maps") motivate the problem of enumerating $β$-normal linear lambda terms. In this brief note, it is shown (by appeal to a theorem of Arquès and Beraud) that the sequence counting isomorphism classes of $β$-normal linear lambda terms up to free exchange of adjacent lambda abstractions coincides with the sequence counting isomorphism classes of rooted maps on oriented surfaces (A000698).