Question

You can map whole numbers to combinations when taking them in order. For example, 13 choose 3 would look like:

0 --> (0, 1, 2)
1 --> (0, 1, 3)
2 --> (0, 1, 4)
etc...

Given a particular combination, such as (0, 3, 9), is there a way to determine which whole number maps to it (26, in this case), short of writing out all the combinations in order until I hit upon the proper one? Furthermore, is there a way of doing this when counting combinations with repetitions?

If anyone is wondering, this isn't homework, but for a personal programming project.

Answer 1

Let $N(n;a_1,\dots,a_k)$ where $0\leq a_1 < a_2 < \dots < a_k < n$ be the order number of $(a_1,\dots,a_k)$ as a combination from ($n$ choose $k$).

Since there are exactly $\binom{n-1}{k-1}$ combinations with $a_1 = 0$, we have a recurrence:

if $a_1 = 0$, then $$ N(n;a_1,\dots,a_k) = N(n-1;a_2-1,\dots,a_k-1)$$

if $a_1 > 0$, then $$N(n;a_1,\dots,a_k) = \binom{n-1}{k-1} + N(n-1;a_1-1,a_2-1,\dots,a_k-1).$$

with initial condition $N(n;)=0$ (i.e., when $k=0$) for any $n$.

For example, $$N(13;0,1,4) = N(12;0,3) = N(11;2) = \binom{10}{0} + N(10;1)$$ $$= \binom{10}{0} + N(10;1) = \binom{10}{0} + \binom{9}{0} + N(9;0)$$ $$=\binom{10}{0} + \binom{9}{0} + N(8;) = 1 + 1 + 0 = 2$$ as required.

Answer 2

Well, yes, in the sense that dividing 26 by 12 choose 2 immediately tells you that 0 is the first digit. Dividing 26 by 11 choose 1 indicates that the second digit is 3. Subtracting 12 + 11 from 26 indicates that you should take the third of the (0, 3, ?) numbers, which is (0, 3, 6). This all looks quite algorithmic: you need to divide and take remainders by certain binomial coefficients or their sums.