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Let $a_{j,k}$ be the number of strings of length $k$, formed from an alphabet of size $n$, such that the $i$th character in the alphabet appears $i+jn$ times in the string.

For example, if $n=3,k=15,j=1$ and the three characters of the alphabet are $a,b,c$ then one such string is $aaaabbbbbcccccc$ and $a_{1,15}$ is the number of rearrangements of this string.

I am trying to find a generating function for the $a_{j,k}$.

Clearly, if $k\ne \sum_{1\le i\le n}(i+jn)$ then $a_{j,k}=0$

I am aware that the general function to generate $n$ elements is $G(x) = (1+x)^{n}$ derived from the binomial theorem.

I have already looked at this post and thought that I could simply plug $j$ into the function:

$G(x) = (x^{1+jn} + x^{2+jn}+...+x^{k})(x^{2+jn} + x^{3+jn}+...+x^{k})...(x^{n+jn} + x^{n+jn+1}+...+x^{k})$

I am unsure as to whether the above function is mathematically correct since I do not fully understand the steps to create a generating function which takes into account this condition.

How would you create a generating function that would satisfy the condition that the $i^{th}$ element must appear $i + jn$ times, assuming $1 ≤ i ≤ n$ and $j \geq 0$? What are the steps in creating this function?

Angela Pretorius
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Thicc
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  • Are there $n$ objects, or are there $n$ types of object and $\sum_{1\le i \le n}i+jn$ objects? Does it matter what order the $k$ chosen objects are chosen in? – Angela Pretorius Apr 08 '20 at 17:08
  • @AngelaRichardson There are only $n$ objects, without regard to type. I do not believe the order matters, since these generating functions deal with combinations rather than permutations. – Thicc Apr 08 '20 at 17:37
  • $i+jn>n$ so how can a subset of these $n$ objects have the same object appearing $i+jn$ times? – Angela Pretorius Apr 08 '20 at 17:52
  • @AngelaRichardson Sorry, I meant to say that the number of total objects to choose from is $n$, but any given subset (combination) that is chosen is of size $k$. $k$ may be less than, equal to, or greater than $n$, but must be greater than or equal to 0. The subset is of size $k$, not $i+jn$. I think $i$ is just to refer to a particular element in the generating function. – Thicc Apr 08 '20 at 18:05
  • So are you asking for the number of strings of length $k$ formed from an alphabet of $n$ characters such that the $i$th character in the alphabet appears $i+jn$ times in the string? – Angela Pretorius Apr 08 '20 at 19:40
  • @AngelaRichardson That is correct. More specifically, the generating function which represents this quantity. – Thicc Apr 08 '20 at 19:47
  • Let us continue this discussion in chat. – Angela Pretorius Apr 08 '20 at 19:57

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