Is there are exactly one partition, given length of partition and maximum number?
Python 2, 38 bytes



lambda n,k,m:m&gt;n-k&lt;2or-1&lt;k*m-n&lt;2+2/m*k


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Characterizes the cases where there's only one legal partition.

Consider the partitions of \$n\$ into exactly \$k\$ parts each between \$1\$ and \$m\$ inclusive. There's only two ways to for there to be only one such partition.


The number \$n\$ is close to as small as possible (\$n=k\$ or \$n=k+1)\$ or as large as possible (\$n=mk-1\$ or \$n=mk)\$ for such a partitions. This leave only enough "wiggle room" for only one partition in each case:


\$n=1+1+\cdots +1+1\$.
\$n=2+1+\cdots+1+1\$.
\$n=m+m+\cdots+m+(m-1)\$.
\$n=m+m+\cdots+m+m\$. 

We allow only \$m=2\$ maximum, forcing a unique partition made of 1's and 2's. Such a partition requires \$k \leq n \leq 2k\$.


The challenge lets us assume that \$n&gt;k\$, so we can omit the \$n=k\$ case and \$k \leq n\$ check, and, as feersum observed, can let us check \$n=k+1\$ as \$n&lt;k+2\$.  The \$m=1\$ case makes it impossible to make a partition with \$n&gt;k\$, so we make sure it's always rejected. This gives the condition


  \$n \in \{ k+1, mk-1, mk\}\$and \$ m&gt;1\$, or \$n \leq 2k \$ and \$m=2\$