Abstract:

For a fixed integer $m\geq2$, we say that a partition $n=p_1+p_2+\cdots+p_k$ of a natural number $n$ is $m$-non-squashing if $p_1\geq1$ and $(m-1)(p_1+\cdots+p_{j-1})\leq p_j$ for $2\leq j\leq k$. In this paper we give a new bijective proof that the number of $m$-non-squashing partitions of $n$ is equal to the number of $m$-ary partitions of $n$. Moreover, we prove a similar result for a certain restricted $m$-non-squashing partition function $c(n)$ which is a natural generalization of the function which enumerates non-squashing partitions into distinct parts (originally introduced by Sloane and the second author). Finally, we prove that for each integer $r\geq2$,

\begin{displaymath}c(m^{r+1}n)-c(m^r n)\equiv0\pmod{m^{r-1}/d^{r-2}}, \end{displaymath}

where $d=\gcd(2,m)$.