Abstract. We present a complete analytic solution to the problem of counting planar graphs. We prove an estimate gn ∼ g ·n −7/2 γ n n! for the number gn of labelled planar graphs on n vertices, where γ and g are explicit computable constants. We show that the number of edges in random planar graphs is asymptotically normal with linear mean and variance and, as a consequence, the number of edges is sharply concentrated around its expected value. Moreover we prove an estimate g(q) · n −4 γ(q) n n! for the number of planar graphs with n vertices and ⌊qn⌋ edges, where γ(q) is an analytic function of q. We also show that the number of connected components in a random planar graph is distributed asymptotically as a shifted Poisson law 1 + P (ν), where ν is an explicit constant. Additional Gaussian and Poisson limit laws for random planar graphs are derived. The proofs are based on singularity analysis of generating functions and on perturbation of singularities.

In their study of cyclic pattern containment, Domagalski et al. [4] conjecture differential equations for the generating functions of circular permutations avoiding consecutive patterns of length 3. In this note, we prove and significantly generalize these conjectures. We show that, for every consecutive pattern σ beginning with 1, the bivariate generating function counting occurrences of σ in circular permutations can be obtained from the generating function counting occurrences of σ in (linear) permutations. This includes all the patterns for which the latter generating function is known.

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Fix two lattice paths P and Q from (0, 0) to (m, r) that use East and North steps with P never going above Q. We show that the lattice paths that go from (0, 0) to (m, r) and that remain in the region bounded by P and Q can be identified with the bases of a particular type of transversal matroid, which we call a lattice path matroid. We consider a variety of enumerative aspects of these matroids and we study three important matroid invariants, namely the Tutte polynomial and, for special types of lattice path matroids, the characteristic polynomial and the β invariant. In particular, we show that the Tutte polynomial is the generating function for two basic lattice path statistics and we show that certain sequences of lattice path matroids give rise to sequences of Tutte polynomials for which there are relatively simple generating functions. We show that Tutte polynomials of lattice path matroids can be computed in polynomial time. Also, we obtain a new result about lattice paths from an analysis of the β invariant of certain lattice path matroids.

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