What is the count of the strict partitions of n in k parts not exceeding m?

Question

Lets say we had a $k,m,n \in \mathbb{N}$ where $k < m \le n$. How many different sets $X_1,..,X_m$ with $|X_i|=k$ for $i=1,..,m$, where the sets do not include duplicates, for which the sum of their elements is equal to n?

$$n = X_{sum}^i := \sum_{j=1}^k x_j^i, \; x_j^i \in X_i$$

It holds, that all elements $x_j^i \le m$ and for each pair $|X_i \cap X_j| < k$, where $i=1,..,m$ and $j=1,..,m$ but $i\neq j$.

As example, we take $k=3$, $n=15$ and show all possibilities for m:

m = 6 : 6+5+4                (1)
m = 7 : 7+5+3, 7+6+2         (2)
m = 8 : 8+4+3, 8+5+2, 8+6+1  (3)
m = 9 : 9+4+2, 9+5+1         (2)
m = 10: 10+3+2, 10+4+1       (2)
m = 11: 11+3+1               (1)
m = 12: 12+2+1               (1)

Is there a general formula to calculate this values for $n,k,m$ or at least for $k=3$?

Finally my goal was if $C_{n,k,m}$ is the count of the values, the calculation of

$$C_{n,k} = \sum_{i=1}^n (C_{n,k,i} - 1).$$

I found out a formula for $C_{n,2}$ which was:

$$C_{n,2} = {n - 2 \choose 2},$$

but I was not able to generalize it for k, since it does not hold, that

$$C_{n,k} = {n - 2 \choose k},$$

for all k.

Answer 1

I'm assuming from your list of examples that what you are looking for is the number of partitions of $n$ into $k$ distinct parts with largest part equal to $m$ (this is different from the question in the title, but seems to fit your examples best).

The number of partitions of $n$ with at most $k$ parts, each of length at most $m$, is the coefficient of $q^n$ in the Gaussian binomial coefficient $\binom{m+k}{k}_q$. To have only those with exactly $k$ parts, one of which is exactly of length $m$, we first get rid of all partitions in $k-1$ parts or less, and then of all those whose largest part is at most $m-1$, and add back those in at most $k-1$ parts of length at most $m-1$, which we have deleted twice. This number is then the coefficient of $q^n$ in $$\binom{m+k}{k}_q -\binom{m+k-1}{k}_q - \binom{m+k-1}{k-1}_q + \binom{m+k-2}{k-1}_q.$$

Now, we want partitions of $n$ having these conditions and having distinct parts. The trick here is to see that such partitions are in bijection with partitions of $n-\binom{k}{2}$ with exactly $k$ (not necessarily distinct) parts and largest part of length $m-k+1$. This trick is explained for instance in this answer.

Therefore, the number you are looking for is the coefficient of $q^{n-\binom{k}{2}}$ in $$\binom{m+1}{k}_q -\binom{m}{k}_q - \binom{m}{k-1}_q + \binom{m-1}{k-1}_q.$$

In the particular example where $m=12$ and $k=3$, this expression is the polynomial $$q^{30} + q^{29} + 2q^{28} + 2q^{27} + 3q^{26} + 3q^{25} + 4q^{24} + 4q^{23} + 5q^{22} + 5q^{21} + 5q^{20} + 4q^{19} + 4q^{18} + 3q^{17} + 3q^{16} + 2q^{15} + 2q^{14} + q^{13} + q^{12},$$

so for $n=15$, you look at the coefficient in front of $q^{15-\binom{k}{2}} = q^{12}$, which is $1$ as expected. For $n=23$, you would take the coefficient in front of $q^{20}$, which is $5$; this corresponds to the five partitions $12+10+1 = 12+9+2 = 12+8+3 = 12+7+4 = 12+6+5$.