Variations on a result of Bressoud

The well-known Rogers-Ramanujan identities have been a rich source of mathematical study over the last fifty years. In particular, Gordon’s generalization in the early 1960s led to additional work by Andrews and Bressoud in subsequent years. Unfortunately, these results lacked a certain amount of uniformity in terms of combinatorial interpretation. In this work, we provide a single combinatorial interpretation of the series sides of these generating function results by using the concept of cluster parities. This unifies the aforementioned results of Andrews and Bressoud and also allows for a strikingly broader family of q–series results to be obtained. We close the paper by proving congruences for a “degenerate case” of Bressoud’s theorem

On Non-Squashing Partitions

A partition n = p_1 + p_2 + ... + p_k with 1 <= p_1 <= p_2 <= ... <= p_k is called non-squashing if p_1 + ... + p_j <= p_{j+1} for 1 <= j <= k-1. Hirschhorn and Sellers showed that the number of non-squashing partitions of n is equal to the number of binary partitions of n. Here we exhibit an explicit bijection between the two families, and determine the number of non-squashing partitions with distinct parts, with a specified number of parts, or with a specified maximal part. We use the results to solve a certain box-stacking problem.Comment: 15 pages, 2 fig

Labor's Capital, Business Confidence, And the Market for Loanable Funds

The market for loanable funds provides a useful framework for determining changes in investment and interest rates. In the United States, a significant source of supply originates from labor in the form of pension assets. However, despite the increased contribution by labor to the supply curve over the past several decades, levels of investment have remained less than robust. Here, we highlight the changes in the demand curve for loanable funds in order to explain the empirical trends. Data series provided by the Conference Board capture the confidence of U. S. business and thus provide a gauge of Keynes’ “animal spirits”—an essential factor in the demand curve shifts. Correlation of the data series with both quarterly changes in real interest rates and quarterly changes in payroll employment offers documentation for these macroeconomic claims.