Number of integer partitions modulo 3

Question

Motivated by Parity of number of partitions of $n!/6$ and $n!/2$, I asked my computer (and my FriCAS package for guessing) for an algebraic differential equation for the number of integer partitions modulo 3. This is what it answered:

(1) -> s := [partition n for n in 1..]

   (1)  [1, 2, 3, 5, 7, 11, 15, 22, 30, 42, ...]
                                                        Type: Stream(Integer)
(32) -> guessADE([s.i for i in 1..400], safety==290, maxDerivative==2)$GUESSF PF 3

   (32)
   [
     [
         n
       [x ]f(x):
               3    2      2          ,,       4 ,   3       3          2  ,   2
             (x f(x)  + 2 x f(x) + x)f  (x) + x f (x)  + (2 x f(x) + 2 x )f (x)

           +
                            ,            2
             (2 x f(x) + 2)f (x) + 2 f(x)

         =
           0
       ,
                          3      4
      f(x) = 1 + 2 x + 2 x  + O(x )]
     ]

Of course, this is only a guess, but it seems fairly well tested. Only 110 terms were needed to guess the recurrence, all the other 290 were used to check it.

My question is: is this known, and if not so, is this interesting?

Answer 1

Well, $f''f^2 +xf' ^3+2ff'^2 =0$ modulo 3 for $f=\prod(1-x^m)=\sum_{n\in \mathbb{Z} } (-1)^nx^{n(3n+1)/2}$ may be quickly seen as follows.

Differentiating the power series for $f$ and expanding the brackets we see that we should prove that $$ \sum_{a(3a+1)/2+b(3b+1)/2+c(3c+1)/2=n} (-1)^{a+b+c} \left(a(a-2)-abc+2ab\right) $$ is divisible by 3. Multiplying the sum by 3 and using cycling shift of variables we reduce it to proving that the sum $$ \sum_{a(3a+1)/2+b(3b+1)/2+c(3c+1)/2=n} (-1)^{a+b+c} \left( a(a-2)+b(b-2)+c(c-2)-3abc+2ab+2bc+2ac\right) $$ is divisible by 9.

The summand is $(a+b+c) (a+b+c-2)-3abc$. If $a=b=c$ this is $9a^2 - 3(a^3+2a)$, divisible by 9. Other triples partition by permuting the variables onto 3-tuples and 6-tuples, so the sum of $3abc$ is of course divisible by 9. As for $a+b+c$, it is congruent to $2n$ modulo 3, thus unless $n=3t+2$ the expression $(a+b+c) (a+b+c-2)$ is divisible by 3, as we need.

It remains to show that for $n=3m+2$ the sum of $(-1)^{a+b+c}$ over our triples is divisible by 9. In other words, the coefficient of $x^{3m+2} $ in $f^3$ must be divisible by 9. We have $$ f^3=\prod (1-x^k)^3 =\prod (1-x^{3k}+3(x^k-x^{2k})). $$ Expanding the brackets and reducing modulo 9 we should prove that the expression $$ [x^{3m+2} ] 3 f(x^3) \sum_k \frac{x^k-x^{2k} } {1 - x^{3k} } $$ is divisible by 9. But in the latter sum $\sum_k (x^k-x^{2k}+x^{4k}-x^{5k}+\dots) $ all coefficients of powers $x^{3s+2} $ do cancel, since if $3s+2=kr$, the guys $x^{kr} $ and $x^{rk} $ go with different signs.

Answer 2

The function $\,h(x) := \prod_{n=1}^\infty (1 - x^n)\,$ is known as a Ramanujan theta function. It is essentially the Dedekind $\eta$ function. The connection is $\,f(q) := q h(q^{24}) = \eta(24\tau)\,$ where $\,q=\exp(2\pi i \tau).\,$ Differentiating we get $\, dq = (2\pi i)\, q\, d\tau,\,$ and $$ \frac{d}{d\tau} f(q) = (2\pi i)\, q\, f'(q).$$ Since all powers of $\,q\,$ in $\,f(q)\,$ are of the form $\,q^{24n+1}\,$ then the congruence $\,f'(q) - f(q) \equiv 0 \pmod 3\,$ holds.

If we use the original $\,h(x)\,$ then there is similar but more complicated result using modulo $3$ exponents. Let$\, A = A(x) := h(x)\,$ and let $\,A_0\,$ be all the terms of $\,A\,$ where the exponent of $\,x\,$ is $0$ modulo $3$ and similarly for $\,A_1\,$ and $\,A_2\,$ so that $\,A = A_0+A_1+A_2.\,$ This is the trisection of the power series $\,A.\,$ Refer to my essay A Multisection of q-Series for the nice identity $$ 0 = A_2 A_0^2 + A_0 A_1^2 + A_1 A_2^2. \tag{1}$$ Now let $\, B = B(x) := A'(x)\,$ and let $\,B_0,B_1,B_2\,$ be the trisection of $\,B.\,$ It is easy to show that $$ q\,B_0 \equiv A_1,\quad q\,B_1 \equiv -A_2, \quad B_2 \equiv 0 \pmod 3.$$ Now let $\, C = C(q) := A''(x)\,$ and let $\,C_0,C_1,C_2\,$ be the trisection of $\,C.\,$ It is easy to show that $$ q^2\,C_0 \equiv -A_2, \quad C_1 \equiv C_2 \equiv 0 \pmod 3.$$ When we make these substitutions in the expression $$ h''(x) h(x)^2 + x\, h'(x)^3 + 2\,h(x)h'(x)^2 \tag{2}$$ and reduce modulo $3$ we get the nice identity $(1)$.