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I am looking for a proof that:

$$\prod_\limits{m=0}^\infty \dfrac{1}{(1-x^{5m+1})}=\sum_\limits{i=0}^\infty \dfrac{x^i}{\prod_\limits{j=1}^i (1-x^{5j})}$$

The left hand side expands into:

$$\dfrac{1}{(1-x)(1-x^6)(1-x^{11})\dots}$$

which gives the number of partitions into parts $1\mod5$.

The right hand side expands into:

$$1+\dfrac{x}{(1-x^5)}+\dfrac{x^2}{(1-x^5)(1-x^{10})}+\dfrac{x^3}{(1-x^5)(1-x^{10})(1-x^{15})}+\dots$$.

The only proof I have is a fairly brute force method.

EDIT: I think we have:

$$\dfrac{1}{(1-x)(1-ax)(1-a^2x)\dots}\\=1+\dfrac{x}{(1-a)}+\dfrac{x^2}{(1-a)(1-a^2)}+\dfrac{x^3}{(1-a)(1-a^2)(1-a^3)}+\dots$$

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Isn't each term of the right side the generating function of the number of partitions into $n$ terms of size 1 modulo 5? – Anthony Quas Feb 5 at 16:33
    
@AnthonyQuas; I think so, so there is a 1-1 relation, my proof consists of fairly bluntly matching each partition to its LHS counterpart – Jon Mark Perry Feb 5 at 16:39
    
I have a strong feeling this is in Stanleys book... – Per Alexandersson Feb 5 at 20:53
up vote 11 down vote accepted

More generally, there is an equality of power series in two variables $$ \prod_\limits{m=0}^\infty \dfrac{1}{(1-t x^{5m+1})}=\sum_\limits{i=0}^\infty \dfrac{t^i x^i}{\prod_\limits{j=1}^i (1-x^{5j})}. $$ The coefficient of $t^i x^n$ on the left hand side is the number of partitions of $\frac{n-i}{5}$ into at most $i$ pieces (by convention the number of partitions of a non-integer in $0$). The coefficient of $t^i x^n$ on the right hand side is the number of partitions of $\frac{n-i}{5}$ into pieces of size at most $i$. Conjugation gives a bijection between the two kinds of partitions.

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It is not about five at all, as may be seen from Julian Rosen's answer, but let me say it in more explicit form.

Simply denote $x^5=y$ and get more general identity $$\prod_{m=0}^\infty \frac{1}{1-xy^m}=\sum_{i=0}^\infty \frac{x^i}{(1-y)(1-y^2)\dots (1-y^i)}.\,\,\,(*)$$ Coefficient of $x^a y^b$:

on the left - is the number of partitions of $b$ into at most $a$ parts;

on the right - is the number of partitions of $b$ into parts not exceeding $a$.

Yes, it is the same by conjugation.

If you ask for purely algebraic solution (conjugation is rather combinatorial bijection), you may note that each part of $(*)$ satisfies initial condition $F(0,0)=1$ and the functional equation $F(x,y)(1-x)=F(xy,y)$, which determine unique formal power series.

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1  
разбиение на n частей = partition into n parts. – KConrad Feb 5 at 18:50
    
Спасибо, Keith! – Fedor Petrov Feb 5 at 19:07

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