We address the enumeration of p-valent planar maps equipped with a spanning forest, with a weight z per face and a weight u per connected component of the forest. Equivalently, we count p-valent maps equipped with a spanning tree, with a weight z per face and a weight μ := u + 1 per internally active edge, in the sense of Tutte; or the (dual) p-angulations equipped with a recurrent sandpile configuration, with a weight z per vertex and a variable μ := u + 1 that keeps track of the level of the configuration. This enumeration problem also corresponds to the limit q → 0 of the q-state Potts model on p-angulations. Our approach is purely combinatorial. The associated generating function, denoted F (z, u), is expressed in terms of a pair of series defined implicitly by a system involving doubly hypergeometric series. We derive from this system that F (z, u) is differentially algebraic in z, that is, satisfies a differential equation in z with polynomial coefficients in z and u. This has recently been proved to hold for the more general Potts model on 3-valent maps, but via a much more involved and less combinatorial proof. For u ≥ −1, we study the singularities of F (z, u) and the corresponding asymptotic behaviour of its nth coefficient. For u > 0, we find the standard behaviour of planar maps, with a subexponential term in n −5/2 . At u = 0 we witness a phase transition with a term n −3 . When u ∈ [−1, 0), we obtain an extremely unusual behaviour in n −3 (ln n) −2 . To our knowledge, this is a new "universality class" for planar maps. We analyze the phase transition at u = 0 in terms of the sandpile model on large maps, and find it to be of infinite order.

Abstract. Walks on Young's lattice of integer partitions encode many objects of algebraic and combinatorial interest. Chen et al. established connections between such walks and arc diagrams. We show that walks that start at ∅, end at a row shape, and only visit partitions of bounded height are in bijection with a new type of arc diagram -open diagrams. Remarkably two subclasses of open diagrams are equinumerous with well known objects: standard Young tableaux of bounded height, and Baxter permutations. We give an explicit combinatorial bijection in the former case.

we explain how these results have a simple application to the combinatorics of lambda calculus, verifying the conjecture that a certain natural family of lambda terms is equinumerous with bridgeless maps.