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\begin{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 $r\geq2$,
  \[ c(m^{r+1}n)-c(m^r n)\equiv0\pmod{m^{r-1}/d^{r-2}}, \] where
  $d=\gcd(2,m)$.

\end{abstract}

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