Number of solutions to $a_1 + a_2 + \dots + a_k = n$ where $n \gt 0$ and $0 \lt a_1 \leq a_2 \leq \dots \leq a_k$ are integers.

Question

I know how to find the number of solutions to the equation:

$$a_1 + a_2 + \dots + a_k = n$$

where $n$ is a given positive integer and $a_1$, $a_2$, $\dots$, $a_n$ are positive integers. The number of solutions to this equation is:

$$\binom{n - 1}{k - 1}$$

This can be imagined as $n$ balls arranged on a straight line and selecting $k - 1$ gaps from a total of $n - 1$ gaps between them as partition boundaries. The $k - 1$ partition boundaries divide the $n$ balls into $k$ partitions. The number of balls in the $i$th partition is $a_i$.

Now, I don't know how to find the number of solutions to the same equation when we have an additional constraint: $$0 < a_1 \leq a_2 \leq \dots \leq a_k.$$ Could you please help me?

The number of solutions is called the number of partitions of $n$ into $k$ parts. There is no known closed form. — André Nicolas, Mar 20 '12 at 14:23
Related: http://math.stackexchange.com/questions/1867469/number-of-integer-triplets-a-b-c-such-that-abc-and-abc-n?noredirect=1&lq=1. — StubbornAtom, Dec 23 '16 at 06:46

Marc van Leeuwen · Answer 1 · 2016-06-01T05:34:34.900

While there is no closed form for the number of partitions of $n$ into $k$ parts, these numbers are not hard to compute. First let's get rid of the awkward initial strict inequality: by setting $a'_i=a_i-1$ we get $0\leq a'_1\leq\cdots\leq a'_k$ and $a'_1+\cdots+a'_k=n-k$. This means we want to count weak partitions of $m=n-k$ into $k$ parts, where weak means the parts can be $0$. Now forming a Young diagram by putting $a'_i$ squares in row $k+1-i$, we are counting Young diagrams with $m$ squares that fit in the first $k$ rows. This is the same as counting Young diagrams with $m$ squares and columns of length at most $k$, or counting partitions of $m$ into parts (arbitrarily many) of size at most $k$.

Now such partition problems have easy generating series. In this case the number we want is the coefficient of $X^m$ in the product of formal power series: $$ \frac1{(1-X)(1-X^2)\cdots(1-X^k)} = \prod_{i=1}^k\frac1{1-X^i}. $$ Multiplying a power series by $\frac1{1-X^i}$ is obtained by adding, in an increasing order, to each coefficient of $X^j$ with $j\geq i$, the coefficient of $X^{j-i}$ (the latter is in general already modified during this run). Thus the following little program (in C++) computes you number in the final entry of the array $c$:

int m = n-k;
vector<int> c(m+1,0); c[0] = 1; // series with m+1 coefs, set to unity
for (int i=1; i<=k; ++i)
  for (int j=i; j<=m; ++j)
    c[j] += c[j-i];
int count = c[m];

@draks: Then $a_i\geq i$ for all $i$, so one sets $a'_i=a_i-i$ instead, leading to the same reduced problem, but with $m=n-\binom{k+1}2$ instead of $m=n-k$ (assuming of course this number is non-negative, for otherwise there are no solutions). — Marc van Leeuwen, Apr 08 '12 at 07:38
@draks: For additional information on counting partitions with all distinct parts, see this previous Question and Answer. — hardmath, May 31 '16 at 20:03

score 1 · Answer 2 · answered Nov 12 '21 at 14:32

Number of solutions is $S(n, k)$.

If $a_1=1$, we need the number of solutions of: $a_2+⋯+a_k=n-1$, which $S(n-1, k-1)$

If $a_1>1$, minus all $k$ numbers by 1:
$$(a_1-1)+(a_2-1)⋯+(a_k-1)=n-k$$ The number of solutions of this is $S(n-k, k)$.

So $S(n, k)=S(n-1, k-1)+S(n-k, k)$

Number of solutions to $a_1 + a_2 + \dots + a_k = n$ where $n \gt 0$ and $0 \lt a_1 \leq a_2 \leq \dots \leq a_k$ are integers.

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