Sample  Papers

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 for q on the order of n log n. In a series of papers written around 1980, E. M. Wright enumerated connected (n,q) graphs for q up to n+o(n^{1/3}). This paper gives an asymptotic count for connected (n,q) graphs which holds uniformly in q as n becomes large.
(Electronic version of this paper unavailable.)
2.   A simplified guide to large antichains in the partition lattice, (with L. H. Harper).
Description.   A partition of the set [n]={1,2, ..., n} is a collection of pairwise disjoint subsets, called blocks, of [n] whose union equals [n]. One partition is a refinement of another if it is obtained by further partitioning one or more blocks. A collection of partitions is an antichain if no two are related by refinement. All the partitions with a fixed number, k, of blocks forms an antichain whose size is S(n,k), a Stirling number of the second kind. We tell how to construct antichains whose size is greater than n^{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 if X_n is a real nonnegative log-concave sequence, then the coefficients A_n determined by \sum A_n u^n = exp(\sum X_j u^j/j) satisfy the inequalities A_{n-1}A_{n+1} \le A_n^2 \le ((n+1)/n)A_{n-1}A_{n+1}.
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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.
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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 than e+o(1) times as large as the maximum rank size p(n,k). ( e = 2.71828...)
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