Given a number $n \in \Bbb{N}$. In how many ways can $n$ be written as $\prod_{i=1}^{k}n_i$ such that $n1|n2|\ldots|n_k|n$?

I came to this problem in a question a few days ago. I have not found any duplicates so I assume there are none. In fact, the proposed problem is equivalent to "How many abelian groups are there with order $n$?". First of all I tried to solve the problem where $n=p^a$ with $p$ prime. Looking at some examples like $p=p^{i}p^{i.j}p^{a-i-i.j}$ I came to the conclusion that each solution corresponds to the sizes of a partition of a set with $a$ elements (in the example the partition sizes are $(i,i.j,a-i-i.j)$, and vice versa. Then I tried some solutions where $n=p^aq^b$ where $p$ and $q$ prime and coprime. For every solution we write the exponents of $p$ and $q$ in the factors of the decomposition in a matrix where the rows are these exponents per factor and the columns correspond to the exponents of $a$ resp. $b$. An example Let $n=72$ then a solution is $72=12.6$ (I reversed the order on purpose). This would give rise to a matrix of the form $\bigl( \begin{smallmatrix} 2 & 1 \\ 1 & 1 \end{smallmatrix} \bigr)$, because $12=2^2.3^1$ and $6=2^1.3^1$. In the same way the decomposition $72=6.6.2$ corresponds to the matrix $\bigl( \begin{smallmatrix} 1 & 1 \\ 1 & 1 \\ 1 & 0 \end{smallmatrix}\bigr)$. The only conditions on these matrices is that the sum of the first column is $3$, because $2$ occurs with exponent $3$ in $72$, that the sum of the elements of the second colum is $2$ and that the elements of each column are non increasing. I suspect that, if I order these partitions by decreasing size I can fill in for the first column any list of sizes of a partition of $3$ elements, and any list of sizes of a partition of $2$ elements in the second column. This would imply that there are $3.2=6$ abelian groups of order $72$. If my suspicion is right then for any number $n=p_1^{e_1}p_2^{e_2}.\ldots.p_k^{e_k}$ there are $\prod_{i=1}^{k}P(e^i)$ abelian groups of order $n$ with $P(a)$ the number of partitions of $a$ elements.

The following remarks establish an exact formula for these chains in terms of the cycle index of the symmetric group on $v$ elements, with $v$ being the maximal exponent of a prime in the prime factorization of $n.$

Suppose $$n = \prod_{q=1}^m p_q^{v_q}$$ is the prime factoriation of $n.$ Now if we consider the exponents $w_1, w_2, w_3, \ldots, w_k$ of $p_q$ dividing $n_1, n_2, n_3, \ldots, n_k$ then we must have

$$w_1\le w_2\le w_3\le\cdots\le w_k \quad\text{and}\quad \sum_{j=1}^k w_k = v_q.$$

The number of such $k$-tuples can be calculated with the Polya Enumeration Theorem (PET) and has the species equation $$\mathfrak{M}_{=k} (\epsilon + \mathcal{Z} + \mathcal{Z}^2 + \mathcal{Z}^3 + \cdots).$$

This translates to the generating function equation $$[z^{v_q}] Z(S_k)\left(\frac{1}{1-z}\right).$$

It follows that the count $P_{n,k}$ of such tuples/chains is given by (this is the promised exact formula) $$P_{n,k} = \prod_{q=1}^m [z^{v_q}] Z(S_k)\left(\frac{1}{1-z}\right).$$

This includes chains that have a prefix of a string of one factors, which we are apparently not counting here. These inadmissible chains among the chains counted by $P_{n,k}$ have the property that they correspond bijectively to the chains from $P_{n,k-1}.$ Hence the desired value $Q_{n, k}$ is given by

$$Q_{n,k} = P_{n,k} - P_{n, k-1}.$$

Here are some examples of the cycle indices that are used: $$Z(S_3) = 1/6\,{a_{{1}}}^{3}+1/2\,a_{{2}}a_{{1}}+1/3\,a_{{3}},$$ $$Z(S_4) = 1/24\,{a_{{1}}}^{4}+1/4\,a_{{2}}{a_{{1}}}^{2} \\+1/3\,a_{{3}}a_{{1}}+1/8\,{a_{{2}}}^{2}+1/4\,a_{{4}}$$ and $$Z(S_5) = {\frac {{a_{{1}}}^{5}}{120}}+1/12\,a_{{2}}{a_{{1}}}^{3}\\ +1/6\,a_{{3}}{a_{{1}}}^{2}+1/8\,a_{{1}}{a_{{2}}}^{2}\\ +1/4\,a_{{4}}a_{{1}}+1/6\,a_{{2}}a_{{3}}+1/5\,a_{{5}}.$$

It is not difficult to see that the maximum $k$ such that an admissible chain for $n$ exists is the maximal exponent call it $v$ of a prime $p$ in the prime factorization of $n.$ (This was also observed by the first responder above.) The chain is obtained by distributing $v_q$ copies of $p_q$ into the $v$ slots starting from the right, with divisibility being determined from the left. The value in a slot is the product of the primes that have been distributed into it.

It follows that the total number $T_n$ of admissible chains is given by $$T_n = \sum_{k=1}^v Q_{n,k} = \sum_{k=1}^v (P_{n,k} - P_{n,{k-1}}) = P_{n, v} = \prod_{q=1}^m [z^{v_q}] Z(S_v)\left(\frac{1}{1-z}\right).$$

This gives the sequence $$1, 1, 1, 2, 1, 1, 1, 3, 2, 1, 1, 2, 1, 1, 1, 5, 1, 2, 1, 2, 1,\\ 1, 1, 3, 2, 1, 3, 2, 1, 1, 1, 7, 1, 1, 1, 4, 1, 1, 1, 3, 1, 1, 1,\\ 2, 2, 1, 1, 5, 2, 2, 1, 2, 1, 3, 1, 3, 1, 1, 1, 2, 1, 1, 2, 11,\ldots$$ which is OEIS A000688 where we find that indeed it counts the number of Abelian groups as surmised by the OP.

The Maple code for these including a verification of the values of $Q_{n,k}$ by brute force (routine Q_ex) is as follows:

with(numtheory);
with(combinat);

pet_cycleind_symm :=
proc(n)
local p, s;
option remember;

if n=0 then return 1; fi;

end;

pet_varinto_cind :=
proc(poly, ind)
local subs1, subs2, polyvars, indvars, v, pot, res;

res := ind;

polyvars := indets(poly);
indvars := indets(ind);

for v in indvars do
pot := op(1, v);

subs1 :=
[seq(polyvars[k]=polyvars[k]^pot,
k=1..nops(polyvars))];

subs2 := [v=subs(subs1, poly)];

res := subs(subs2, res);
od;

res;
end;

P :=
proc(n, k)
option remember;
local pv, p, ind, gf;

ind := pet_cycleind_symm(k);
gf := pet_varinto_cind(1/(1-z), ind);

p := 1;

for pv in ifactors(n)[2] do
p := p*coeftayl(gf, z=0, pv[2]);
od;

p;
end;

Q :=
proc(n, k)
option remember;
P(n,k) - P(n, k-1);
end;

T :=
proc(n)
option remember;
local pv, mx;

mx := -1;
for pv in ifactors(n)[2] do
if pv[2] > mx then
mx := pv[2];
fi;
od;

P(n, mx);
end;

chooserep :=
proc(data, sofar, k, res)
if k=0 then
res[sofar] := 1;
return;
fi;

if nops(data) > 0 then
if nops(sofar) = 0 or data[1] mod sofar[-1] = 0 then
chooserep(data, [op(sofar), data[1]], k-1, res);
fi;
chooserep([op(2..nops(data), data)], sofar, k, res);
fi;
end;

Q_ex :=
proc(n, k)
option remember;
local res, tupl, divchoose;

res :=0;

divchoose := table();
chooserep(sort(convert(divisors(n) minus {1}, list)),
[], k, divchoose);

for tupl in [indices(divchoose, 'nolist')] do
if convert(tupl, *) = n then
res := res+1;
fi;
od;

res;
end;


Addendum. One possible improvement of Q_ex would be not to collect all chains and then check them for admissibility and instead check admissibility as they are generated. The code looks the way it does because it started with the Maple command choose as its paradigm.

If you write "as the product of k numbers", and you found the answer for $p^a$, then I'd say the answer for $p^a q^b r^c$ would be the product of the answers for $p^a$, $q^b$ and $r^c$.

If you exclude trivial answers where $n_1 = 1$, then the largest k would be the largest of the prime exponents in the factorisation. So all you need is f (a, k) = "number of ways to write a as the sum of k non-decreasing non-negative integers"; you factor $n>1$ into the product of powers of primes $p_i^{a_i}$, let K = largest of the $a_i$, then for each $1 ≤ k ≤ K$ calculate the product of $f (a_i, k)$ over all i in the factorisation, and add the products.

• I don't understand what you mean by "then the largest k would be the largest of the prime exponents in the factorisation". can you give an example? – Marc Bogaerts Oct 2 '14 at 16:35
• (+1) Concise answer that represents the givens of the problem. – Marko Riedel Oct 3 '14 at 1:32