Arithmetic Properties of Non-Squashing Partitions into Distinct Parts

A partition \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$n = p_1 + p_2 + \cdots + p_k $$\end{document} with \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$1 \leq p_1 \leq p_2 \leq \cdots \leq p_k $$\end{document} is non-squashing if \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$p_1 + \cdots + p_j \leqslant p_{j + 1} \;{\text{for}}\;1 \leqslant j \leqslant k - 1.$$\end{document} On their way towards the solution of a certain box-stacking problem, Sloane and Sellers were led to consider the number b(n) of non-squashing partitions of n into distinct parts. Sloane and Sellers did briefly consider congruences for b(n) modulo 2. In this paper we show that 2r-2 is the exact power of 2 dividing the difference b(2r+1n)−b(2r-1n) for n odd and r ≥ 2.