Chains in Power Sets

Abstract

When studying ordered sets a natural question that arises is: How many chains are there in a typical finite, partially ordered set? Generic questions about partially ordered sets (posets) have a long history and this particular question about the number of chains in a partially ordered set has connections to questions dealing with ordered partitions and trees that may be traced back to work by Cayley [3] in 1859. Our main goal in this article is to develop recursive and closed formulas for the total number of chains and the number of chains of a particular length in the power set of an n-element set, partially ordered by set inclusion. We also discuss a canonical relationship between these chains and the preferential arrangements of an n-element set discussed in [7] and [9]. Another objective is to use this question about chains to make two pedagogical points. First, frequently in mathematics there may be several approaches to solving a problem with each approach illuminating a different aspect of the problem. Chain counting provides such an occasion for illustrating a wide variety of techniques from classical combinatorics and, as a consequence, we sketch more than one line of attack for most of our results. Second, most of the results in this article are subsumed in more advanced and modern treatments of combinatorics. (Explicit reference to the literature is contained in our concluding remarks.) However, the classical combinatorial techniques used here are elementary and accessible to those familiar with counting techniques, generating functions, distribution and occupancy problems, Stirling numbers, and elementary differential equations. In particular, the generating function techniques exploit the remarkable correspondence, discovered by Laplace, between set theoretic operations and operations on formal power series. A comprehensive discussion of these classical techniques may be found in most advanced undergraduate texts on combinatorics (e.g., [11, Chapters 2, 3, 5, and 6]). Before we begin, a word of caution is in order. As in elementary number theory, questions about partially ordered sets may be easy to pose but quite difficult to resolve. Indeed, a simply stated open question [12, p. 103], closely related to the one we discuss here, is: What is the number of antichains in the power set of an n-element set? (An antichain is a subset of a poset in which no two elements are comparable.) We start by fixing some notation. Identifying an arbitrary n-element set in the natural way with Xn = (1, 2, .. ., n}, we partially order the power set 9Y(X,T) by set inclusion. If C: No C N1 C ... C Nk is a chain in A(Xn), we say C has length k. In particular, chains consisting of a single element of 32(Xn) have length 0. Furthermore, an k and an will denote, respectively, the number of chains of length k and the