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I've simplified this a bit so that I can just get help with the basic steps.

Say we have a language of all words over $\{a,b,c,d\}$ where the only letters allowed to commute are $ab$. I need help understanding how to define equivalence classes of words in the language. Then, take a representative of each equivalence class and find a generating function for the sequence of these equivalence class representatives.

To make sure I understand the language, these would be the possible words, yes? Or am I missing some? I think maybe the restriction that only $ab$ commute is confusing me. $$a, b,c,d$$ $$ab, ac,ad,bc, bd, cd, ba$$ $$abc,abd,acd,bcd,bac, bad$$ $$abcd, bacd$$

What would each equivalence class be? Words of the same length? Then how do I select a representative from each? Can I just pick any one? So maybe these would be them? $$a,ab,abc,abcd$$ But then these would be the same representatives even if we said only $ab$ can commute so that doesn't seem right.

And then finally, once I find these, how do I find a generating function?

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  • $\begingroup$ Could you give the context of your question? As it stands, it is not clear at all. $\endgroup$ – J.-E. Pin Mar 4 '16 at 10:00
  • $\begingroup$ What is your definition of a generating function? $\endgroup$ – mrp Mar 4 '16 at 13:57
  • $\begingroup$ @User: I've added a somewhat more detailed treatment regarding generating functions. Regards, $\endgroup$ – Markus Scheuer Mar 5 '16 at 16:14
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Note: Main theme of this answer is to find a generating function for the language. We develop it by foot. Then we indicate a Rolls Royce variant.

If an alphabet $\mathcal{A}=\{a,b,c,d\}$ is given there are $4^n$ different words of length $n$, since for each of the $n$ positions within a word one out of four letters from $\mathcal{A}$ can be chosen.

The language $\mathcal{L}$

Let's assume that $ab$ commute, so that $ab$ and $ba$ are not distinguishable. This way we can choose either $ab$ or $ba$ as representative for the corresponding equivalence class. We select in the following lexicographically ordered $ab$ as representative for the equivalence class \begin{align*} [ab]=\{ab,ba\} \end{align*} The language $\mathcal{L}$ so defined consists of all words built from the alphabet $\mathcal{A}$ which does not contain the substring $ba$, since we always choose the representative $ab$ instead.

Let's look for all words in $\mathcal{L}$ of length $\leq 4$. We identify all words which contain a substring $ba$ since these are the words which are in a corresponding equivalence class with $ab$ and which are to exclude. Words of length $1$ and $2$ are simple.

Words in $\mathcal{L}$ with length $\leq 4$

  • Words of length $=1$: There are $4^1=4$ words of length $1$ which can be built from $\mathcal{A}$, namely \begin{align*} a,b,c,d \end{align*} and of course none of them contains a substring $ba$. So all of them are elements of the language $\mathcal{L}$

From now on we group words according to the number of occurrences of the substring $ba$. Words of length $2$ and $3$ have either zero or one occurrence of $ba$, words of length $4$ and $5$ have zero, one or two occurrences of $ba$.

  • Words of Length $=2$: Words of length $2$ have the shape \begin{align*} uv\qquad \color{blue}{ba} \qquad\qquad u,v \in\mathcal{A}\\ \end{align*} We have two groups. The number of different words in the first group $uv$, which are all words of length $2$ is $4\cdot4=16$. Since we want to count valid words only, we have to subtract the words from the second group, which is only $1$ in this case. We conclude

There are \begin{align*} 4^2-1=15 \end{align*} words in $\mathcal{L}$ with length $2$.

  • Length $=3$: Words of length $3$ have the shape \begin{array}{ccc} uvw\qquad &\color{blue}{ba}u\qquad&\qquad\qquad u,v \in\mathcal{A}\\ \qquad &u\color{blue}{ba}\qquad &\\ \end{array} We have two groups. The number of different words in the first group $uvw$, which are all words of length $3$ is $4\cdot3=64$. Since we want to count valid words only, we have to subtract all words from the second group, which is $\binom{2}{1}4=8$ in this case. There are $\binom{2}{1}$ possibilities to position $ba$ and $4$ possibilities for $u$. We conclude

There are \begin{align*} 4^3-\binom{2}{1}4=56 \end{align*} words in $\mathcal{L}$ with length $3$.

  • Length $=4$: Words of length $4$ have the shape \begin{array}{cccc} uvwx\qquad &\color{blue}{ba}uv\qquad &\color{blue}{ba}\color{blue}{ba}\\ \qquad &u\color{blue}{ba}v\qquad&&\qquad u,v \in\mathcal{A}\\ \qquad &uv\color{blue}{ba}\qquad&&\\ \end{array} We have three groups. The number of different words in the first group $uvwx$, which are all words of length $4$ is $4\cdot4=256$. Since we want to count valid words only, we have to subtract all words from the second group, which is $\binom{3}{1}4^2=8$ in this case. There are $\binom{3}{1}$ possibilities to position $ba$ and $4^2$ possibilities for $uv$. Since we count occurrences of $ba$ twice in a word more than once, we have to add all words with $ba$ twice as substring. We conclude

There are \begin{align*} 4^4-\binom{3}{1}4^2+\binom{2}{2}4^0=209 \end{align*} words in $\mathcal{L}$ with length $4$.

We obsere a nice scheme. According to the Inclusion-exclusion principle we subtract and add words containing an increasing number of substrings $ba$. In case words have length $n$ and $k$ occurrences of $ba$, there are $\binom{n}{k}$ possibilities to choose the positions of $ba$ and $n-2k$ positions left for the other letters which give a total of \begin{align*} \binom{n}{k}4^{n-2k} \end{align*} different words for $n\geq 0$ and $0\leq k\leq \left\lfloor\frac{n}{2}\right\rfloor$.

We conclude the following is valid. There are \begin{align*} \sum_{k=0}^{ \left\lfloor\frac{n}{2}\right\rfloor}\binom{n-k}{k}(-1)^k4^{n-2k}\qquad\qquad n\geq 0 \end{align*} different words of length $n$ in $\mathcal{L}$.

We allow $n=0$ which means that the empty word $\varepsilon$ of length zero is also element in $\mathcal{L}$.

Generating function $L(x)$

A generating function is a formal power series

\begin{align*} L(x)=\sum_{n=0}^{\infty}a_nx^n \end{align*} with the coefficient $a_n$ of $x^n$ denoting the number of words of length $n$ in $\mathcal{L}$.

Since we know the number of words of $\mathcal{L}$ we can write the generating function as \begin{align*} L(x)&=\sum_{n=0}^{\infty}\sum_{k=0}^{ \left\lfloor\frac{n}{2}\right\rfloor}\binom{n-k}{k}(-1)^k4^{n-2k}x^n\\ &=1+4x+15x^2+56x^3+209x^4+780x^5+\cdots \end{align*}

We can do even better. We derive an expression of $L(x)$ as rational function in $x$. In order to do so we use the coefficient of operator $[x^n]$ to denote the coefficient of $x^n$ in a generating series and transform the coefficients of $L(x)$ appropriately.

We obtain \begin{align*} \sum_{k=0}^n&\binom{n-k}{k}(-1)^k4^{n-2k}\tag{1}\\ &=\sum_{k=0}^n\binom{k}{n-k}(-1)^{n-k}4^{2k-n}\tag{2}\\ &=\sum_{k=0}^\infty\binom{k}{n-k}\left(-\frac{1}{4}\right)^{n-k}4^k\tag{3}\\ &=\sum_{k=0}^\infty4^k[x^{n-k}]\sum_{l=0}^k\binom{k}{l}\left(-\frac{1}{4}\right)^{l}x^l\tag{4}\\ &=\sum_{k=0}^\infty[x^{n}](4x)^k\left(1-\frac{1}{4}x\right)^k\tag{5}\\ &=[x^n]\sum_{k=0}^\infty\left(4x-x^2\right)^k\tag{6}\\ &=[x^n]\frac{1}{1-4x-x^2}\\ \end{align*}

Comment:

  • In (1) we put for convenience only the upper limit of the index $k$ to $n$ without changing anything, since we use the convention $\binom{n}{k}=0$ if $0<n<k$.

  • In (2) we exchange $k$ with $n-k$.

  • In (3) we do a small rearrangement and put the upper limit of the index $k$ to $\infty$ for convenience only. This changes nothing with the same reasoning as in (1).

  • In (4) we write $\binom{k}{n-k}\left(-\frac{1}{4}\right)^{n-k}$ as a Cauchy product $[x^{n-k}]\sum_{l=0}^k\binom{k}{l}\left(-\frac{1}{4}\right)^{l}x^l$. This is the core of the derivation since it allows a drastic simplification in the next step.

  • In (5) we can write the polynomial in $x$ as $k$-th power of a binom.

  • In (6) we obtain a geometric series representation which gives us in the next line a representation as rational function in $x$.

We conclude

\begin{align*} L(x)&=\frac{1}{1-4x+x^2}\tag{7}\\ &=1+4x+15x^2+56x^3+209x^4+780x^5+\cdots \end{align*}

Note: In fact, there is a powerful method to solve problems of this kind, called the Goulden-Jackson Cluster Method. In this context we regard $ba$ as bad word and derive immediately (see formula of $f(s)$ on page $9$) the rational expression (7) of $L(x)$.

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    $\begingroup$ It would also seem meaningful to ask about a language over the given alphabet where $a$ and $b$ cannot be distinguished. This can be computed, but the question has three close votes already. $\endgroup$ – Marko Riedel Mar 4 '16 at 22:02
  • $\begingroup$ @MarkoRiedel: The three close votes are not clear to me. Presumably there is enough time before this answer might change its status, so feel free to add some information. I'm curious about it! :-) $\endgroup$ – Markus Scheuer Mar 4 '16 at 22:18
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    $\begingroup$ Using Power Group Enumeration produces the sequence $$ 3, 10, 36, 136, 528, 2080, 8256, 32896,\ldots$$ which points us to OEIS A007582. On consulting the simple formula that appears there we realize that it has an elementary proof and we can do without groups and cycle indices. $\endgroup$ – Marko Riedel Mar 4 '16 at 22:46
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It looks like you're working with the set of ALL strings whose characters are $a,b,c,d$. You are defining an equivalence class by saying that $uabv \equiv ubav$ (where $u$ and $v$ are arbitrary strings), and then taking the transitive closure to make it an equivalence relation.

Hence, $ bacab \equiv bacba \equiv abcba \equiv abcab \equiv bacab$, so the equivalence class for $bacab$ would be $\{bacab ,bacba ,abcba ,abcab\}$. As for the representative, I would choose $abcab$, where there are no substrings of the form $ba$.

I'm not sure what you mean by a "generating function" in this context; generating functions usually are for sequences of real numbers.

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