1. The asymptotic number of labeled connected graphs with a given number of vertices and edges, (with E. A. Bender and B. D. McKay).

Description. In the 1960's, Erd\H{o}s and R\'{e}nyi launched the theory of random graphs with papers that included (among other things) asymptotic counts of connected(n, q)graphs forqon the order ofnlogn. In a series of papers written around 1980, E. M. Wright enumerated connected(n,q)graphs forqup ton+o(n^{1/3}). This paper gives an asymptotic count for connected(n,q)graphs which holds uniformly inqasnbecomes large.

(Electronic version of this paper unavailable.)

2. A simplified guide to large antichains in the partition lattice, (with L. H. Harper).

Description. Apartitionof the set[n]={1,2, ..., n}is a collection of pairwise disjoint subsets, calledblocks, of[n]whose union equals[n]. One partition is arefinementof another if it is obtained by further partitioning one or more blocks. A collection of partitions is anantichainif no two are related by refinement. All the partitions with a fixed number,k, of blocks forms an antichain whose size isS(n,k), a Stirling number of the second kind. We tell how to construct antichains whose size is greater thann^{1/35}times the largest Stirling number.

3. The size of the largest antichain in the partition lattice.

Description. It is shown that the largest antichain in the partition lattice can be at most a constant factor times larger than the one constructed in the previous paper.

4. Log concavity and a related property of the cycle index polynomials, (with E. A. Bender).

Description. We show that ifX_nis a real nonnegative log-concave sequence, then the coefficientsA_ndetermined by\sum A_n u^n = exp(\sum X_j u^j/j)satisfy the inequalitiesA_{n-1}A_{n+1} \le A_n^2 \le ((n+1)/n)A_{n-1}A_{n+1}.

5. Durfee polynomials, (with S. Corteel and C. D. Savage).

Description. We consider various classes of integer partitions, counted according to the size of their Durfee square. We identify the mode of the Durfee square.

6. An upper bound for the size of the largest antichain in the poset of partitions of an integer, (with K. Engel).

Description. This paper undertakes the study of the size of the largest antichain in the poset of integer partitions, ordered by refinement. We show that the largest antichain among partitions of the integer $n$, is no larger thane+o(1)times as large as the maximum rank sizep(n,k). ( e = 2.71828...)