E. Rodney Canfield

The asymptotic number of labeled graphs with given degree sequences

Abstract Asymptotics are obtained for the number of n × n symmetric non-negative integer matrices subject to the following constraints: (i) each row sum is specified and bounded, (ii) the entries are bounded, and (iii) a specified “sparse” set of entries must be zero. The result can be interpreted in terms of incidence matrices for labeled graphs.

The asymptotic number of tree-rooted maps on a surface

Let S be a surface. We asymptotically enumerate two classes of n-edged maps on S as N → ∞: tree-rooted and tree-rooted smooth. These results are based on a system of equations enumerating single vertex maps and on a relation found by Walsh and Lehman for the case of orientable surfaces between tree-rooted and single vertex maps.

The asymptotic number of labeled connected graphs with a given number of vertices and edges

Let c(n, q) be the number of connected labeled graphs with n vertices and q ≤ N = (2n) edges. Let x = q/n and k = q − n. We determine functions wk ˜ 1. a(x) and φ(x) such that c(n, q) ˜ wk(qN)enφ(x)+a(x) uniformly for all n and q ≥ n. If ϵ > 0 is fixed, n ∞ and 4q > (1 + ϵ)n log n, this formula simplifies to c(n, q) ˜ (Nq) exp(–ne−2q/n). on the other hand, if k = o(n1/2), this formula simplifies to c(n, n + k) ˜ 1/2 wk (3/π)1/2 (e/12k)k/2nn−(3k−1)/2.

Central and local limit theorems for the coefficients of polynomials of binomial type

Abstract We introduce the problem of establishing a central limit theorem for the coefficients of a sequence of polynomials Pn(x) of binomial type; that is, a sequence Pn(x) satisfying exp (xg(u)) = ∑ n=0 ∞ P n (x)( u n n! ) for some (formal) power series g(u) lacking constant term. We give a complete answer in the case when g(u) is a polynomial, and point out the widest known class of nonpolynomial power series g(u) for which the corresponding central limit theorem is known true. We also give the least restrictive conditions known for the coefficients of Pn(x) which permit passage from a central to a local limit theorem, as well as a simple criterion for the generating function g(u) which assures these conditions on the coefficients of Pn(x). The latter criterion is a new and general result concerning log concavity of doubly indexed sequences of numbers with combinatorial significance. Asymptotic formulas for the coefficients of Pn(x) are developed.

The number of rooted maps on an orientable surface

Let mg(n) be the number of rooted n edged maps on an orientable surface of genus g > 0. The generating function Mg(x) = Σ mg(n) xn is a rational function of ϱ = (1 − 12x)12 whose denominator factors completely into powers of ϱ, ϱ + 2, and ϱ + 5. We calculate M2(x) and M3(x). Unfortunately, we have not been able to discern a pattern in the sequence Mg(x) from the values for g ≤ 3.

On a problem of rota

Abstract Let S ( n , k ) denote Stirling numbers of the second kind; for each n , let K n be such that S ( n , K n ) ⩾ S ( n , k ) for all k . Also, let P ( n ) denote the lattice of partitions of an n -element set. Say that a collection of partitions from P ( n ) is incomparable if no two are related by refinement. Rota has asked if for all n , the largest possible incomparable collection in P ( n ) contains S ( n , K n ) partitions. In this paper, we construct for all n sufficiently large an incomparable collection in P ( n ) containing strictly more than S ( n , K n ) partitions. We also estimate how large n must be for this construction to work.

The Number of Degree-Restricted Rooted Maps on the Sphere

Let

The asymptotic number of rooted maps on a surface II: enumeration by vertices and faces

D

A loop-free algorithm for generating the linear extensions of a poset

be a set of positive integers. Let

Remarks on an asymptotic method in combinatorics

m(n)