Restricted partitions including zero, without repeated numbers

Question

I'd like to try to develop some formal maths for listing the degeneracies of spinless fermion states in a harmonic oscillator. For those who don't know much quantum physics, I'm essentially trying to count the number distinct k-tuples whose entries sum to some number n (up to commutativity, ie. (123) = (213) = (312) = (321)), as well as adding the restriction that no two numbers in this k-tuple can be repeated.

Number of ways to write n as a sum of k nonnegative integers

The post in the link above helped me with the case of bosons (the same deal, but no repetition restrictions). I'm hoping someone could help me out, as I started to develop a flawed formalism and I'm too motivated to stop now. If you test the case j=3, k=3, you obtain 2 ways to write the j-tuple: (210) and (012), but these are just the same in my terms. Thanks a lot!

Please use MathJax to format your posts and to transcribe images of text. — gen-ℤ ready to perish, Sep 10 '17 at 06:39
Note that the ways to write an increasing $j$-tuple of distinct nonnegative entries that sum to $k$ is, by incrementing each entry by one, equivalent to ways to write an increasing $j$-tuple of distinct positive entries that sum to $j+k$. This latter is the set of integer partitions of $j+k$ into exactly $j$ distinct parts. — hardmath, Sep 10 '17 at 15:50
Sometimes integer partitions with distinct parts are called strict partitions. See What is the count of the strict partitions of n in k parts not exceeding m? and Partition an integer n into exactly k distinct parts — hardmath, Sep 10 '17 at 16:35
Will you be satisfied with counting these partitions of $k$ into $j$ distinct parts, or will you need to list (construct) all of them? — hardmath, Sep 10 '17 at 16:42
Just counting works. The solution posted by Lord Shark worked well, but this is a very nice way to look at it as well, given one knows something about strict partitions. Thanks very much! — rsek1996, Sep 10 '17 at 18:08

score 2 · Answer 1 · answered Sep 10 '17 at 06:17

I presume zero cannot be repeated. Denote the number of partitions $(p_1,\ldots,p_k)$ with $p_1>p_2>\cdots>p_k\ge0$ and $\sum_i p_i=n$ by $a_{n,k}$. We can express the $a_{n,k}$ as the coefficients of a generating function $$\sum_{n,k}a_{n,k}x^nt^k=\prod_{m=0}^\infty(1+x^m t).$$ To see this, note that the terms in the product involving $t^k$ are $x^{p_1+p_2+\cdots+p_k}t^k$ with $p_1>\cdots>p_k$. I'm not sure what one can do with this. You could rewrite it like $$\sum_{n,k}a_{n,k}x^nt^k=\prod_{m=0}^\infty\frac{1-x^{2m}t^2}{1-x^m t}.$$ (This kind of manipulation is occasionally useful in partition theory.)

Restricted partitions including zero, without repeated numbers

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