User:RandomUndergrad/Sandbox/Number of Partitions with no Multiple of n equals Number of Partitions where Parts appear less than n times

A generalisation of Number of Partitions with no Multiple of 3 equals Number of Partitions where Parts appear No More than Twice,

which results from a generalisation of a proof by Gauss which shows that the number of partitions with odd parts equals the number of partitions with distinct parts.

Given any $n\ge 2$, let $m$ be a positive integer.

$(\Rightarrow)$For each partition of $m$ with no multiple of $n$, we can group the parts to form the equation


 * $m=a_1p_1+\dots+a_kp_k$, where $p_i$ are the distinct sizes of parts, and $a_i$ is the number of parts of size $p_i$. Then $n\not\vert p_i$.

By Basis Representation Theorem, each $a_i$ can be uniquely represented in base $n$.

In other words, there exists unique $t_i\in\N$, $r_{1_i},\dots,r_{t_i}\in\N_n$, $r_{t_i}\neq 0$ such that


 * $a_i=\displaystyle\sum_{j=0}^{t_i}r_{j_i}n^j$

Then

Since $0\le r_{j_i}<n$, this is a partition of $m$ with parts (of size $n^jp_i$) appearing less than $n$ times, provided that each $n^jp_i$ is distinct.

Suppose there exists some $i_1,i_2,j_1,j_2\in\N$ where $n^{j_1}p_{i_1}=n^{j_2}p_{j_2}$.

Without loss of generality, assume that $j_1\ge j_2$.

Then $n^{j_1-j_2}p_{i_1}=p_{i_2}$.

Since $p_{i_1}$ and $p_{i_2}$ are not multiples of $n$, $n^{j_1-j_2}$ cannot be a multiple of $n$.

Therefore $n^{j_1-j_2}=1$, which gives $j_1=j_2$, $i_1=i_2$. And hence each $n^jp_i$ is distinct.

By uniqueness of $t_i$, $r_{j_i}$, this is an injection.

$(\Leftarrow)$ Now suppose that we have a partition of $m$ with parts appearing less than n times.

Write $m=b_1q_1+\dots+b_lq_l$, where each $b_iq_i$,

So $\dfrac{q_i}{n^x}$ cannot be an integer.

If $n\biggr\vert\dfrac{q_i}{n^{d_i}}$ then $\dfrac{q_i}{n^{d_i+1}}$ is an integer, which contradicts our maximal assumption.

By splitting each part of size $q_i$ into $n^{d_i}$ parts of size $\dfrac{q_i}{n^{d_i}}$,

we obtain a partition of $m$ with no multiples of $n$ as its parts.

By the uniqueness of $d_i$, this is also an injection.

Cantor-Bernstein-Schröder Theorem guarentees there exists a bijection between the set of partitions with no multiples of $n$ and the set of partitions where parts appear less than n times.

Therefore the two sets must have equal cardinality.