Total variation asymptotics for independent process approximations of logarithmic multisets and selections

Abstract

Many natural unlabeled combinatorial structures, such as random partitions of the integer n, or random monic polynomials over a finite field of degree n, or unlabeled mapping patterns on n points may be described as multisets. In the usual statistical language, a multiset is an unordered sample in which number of items is variable, but the sum is a fixed value n. For these structures, the process counting the number of components of various sizes is equal in distribution to a process of independent, but not identically distributed random variables, conditioned on the value of a weighted sum. By restricting to the first b coordinates, it is possible to compare the combinatorial process directly to the independent process, and to estimate the total variation distance db(n) between these distributions. For a broad class of examples similar to the Ewens sampling formula we give asymptotics for db(n) which are valid for b=o(n/log n). The polynomial and random mapping pattern examples are covered by this result, but not the example of partitions. Similar results for selections, which are multisets with no repeated parts, such as square free polynomials, are also derived. The proofs of these results use large deviations bounds and singularity analysis of generating functions. © 1997 John Wiley & Sons, Inc. Random Struct. Alg., 11, 51–80, 1997