Combinatorial Decomposition
JavaScript (ES6),  95 93 86  82 bytes
Expects (r)(m).

r=&gt;F=(m,x=r)=&gt;r?(g=k=&gt;!k||g(--k)*(k-x)/~k)(r)&gt;m?[...F(m-g(r--,--x)),x]:F(m,x+1):[]

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How?
Helper function
The helper function \$g\$ is used to compute:
$$\binom{x}{r}=\frac{x(x-1)\dots(x-r+1)}{r!}=\prod_{k=1}^{r}\frac{x-k+1}{k}$$
(g = k =&gt;     // g is a recursive function taking k
  !k          // if k = 0, stop the recursion and return 1
  ||          // otherwise:
    g(--k)    //   decrement k and do a recursive call with the updated value
    * (k - x) //   multiply the result by k - x
    / ~k      //   divide by -k - 1
              //   which is equivalent to g(k - 1) * (x - k + 1) / k
)(r)          // initial call to g with k = r

Main function
r =&gt;                      // r = requested number of combinations
F = (m, x = r) =&gt;         // F is a recursive function taking the target number m
                          // and a counter x, initialized to r
  r ?                     // if r is not equal to 0:
    g(r) &gt; m ?            //   if C(x, r) is greater than m:
      [ ...F(             //     append the result of a recursive call to F:
          m - g(r--, --x) //       with m - C(x - 1, r) and r - 1
        ),                //     end of recursive call
        x                 //     append x (which was decremented above)
      ]                   //
    :                     //   else:
      F(m, x + 1)         //     increment x until C(x, r) &gt; m
  :                       // else:
    []                    //   stop the recursion