# Counting binary strings that have atmost k consecutive 0's

I know how to count how many binary strings with length n and having exactly k 0's are there but i am not able to find a way to count the number of binary strings that have exactly x 0's and y 1's and uses at most k1 consecutive 0's and k2 consecutive 1's.

Here is how I approach I can enumurate through all possible binary string that uses exactly x 0's and y 1's and subtract from those that have more than k1 consecutive 0's and k2 consecutive 1's but how do I find binary strings that have more than k1 consecutive 0's and k2 consecutive 1's.

Any other approach to solve the same problem.

## 2 Answers

Note: You could model the situation with generating functions. We start with some preliminary considerations, introducing a symbolic method for combinatorial structures from which we can easily derive corresponding generating functions.

Problem: Let's consider binary strings containing $0s$ and $1s$. We are looking for the number of strings with length $n$, having at most $k_1$ consecutive $0$s and $k_2$ consecutive $1$s. We want to count the strings with $x$ $0$s and $y$ $1$s.

We build them starting with simpler patterns. Let $$\text{SEQ}(1)=\{\varepsilon,1,11,111,\ldots\}$$ denote all strings containing only $1$s with length $\geq0$. The empty string is denoted with $\varepsilon$. The corresponding generating function is $$w^0+w^1+w^2+\ldots=\frac{1}{1-w},$$ with the exponent of $w^n$ marking the length $n$ of a string of $1s$ and the coefficient of $w^n$ marking the number of strings of length $n$. Similarly, we define $\text{SEQ}(0)=\{\varepsilon,0,00,000,\ldots\}$.

Let $\text{SEQ}^{\leq k}(1)$ be the set of all strings of $1$s with length $\leq k$. The generating function for $\text{SEQ}^{\leq k}(1)$ is $$1+w+w^2+\ldots+w^{k}=\frac{1-w^{k+1}}{1-w}$$

Simplified problem: First we consider the simplified variant: $$k_1=k_2=1,$$ namely strings having runs of $0$s and $1$s of maximum length $1$.

How could we describe all these strings? We may have a leading $1$. So, we start with zero or one $1$s. Formally: $$\text{SEQ}^{\leq 1}(1)=\{\varepsilon,1\}$$ with generating function $$1+w$$

Then we may have zero or more groups, each group containing a $0$ followed by a $1$. Formally:

$$\text{SEQ}(01)=\{\varepsilon,01,0101,010101,\ldots\}$$

with generating function $$1+uw+(uw)^2\ldots=\frac{1}{1-uw}$$ Observe, that since we want to count $x$ $0$s and $y$ $1$s separately, we encode them using a bivariate generating function with $u$ indicating the amount of $0$s and $w$ indicating the amount of $1$s. The strings may finish with zero or one $0$, formally $\text{SEQ}^{\leq 1}(0)$ with generating function $1+u$.

We conclude: The binary strings having runs of $0$s and $1$s of maximum length $1$ are modelled by

\begin{align*} \text{SEQ}^{\leq 1}(1)\text{SEQ}(01)\text{SEQ}^{\leq 1}(0)\tag{1} \end{align*}

with generating function

\begin{align*} \frac{(1+u)(1+w)}{1-uw}\tag{2} \end{align*}

General Problem: Now we consider the symbolic expression for the general problem. Since we have to consider runs of at most $k_1$ $0$s and runs of at most $k_2$ $1s$ we have to substitute the outer occurrences $0$ and $1$ by up to $k_1$ $0$s resp. $k_2$ $1$s and the inner $0$s and $1$s enlarge by up to $k-1$ $0$s resp. $1$s. So, we get

\begin{align*} \text{SEQ}^{\leq k_2}(1)\text{SEQ}\left(0\text{SEQ}^{\leq k_1-1}(0)1\text{SEQ}^{\leq k_2-1}(1)\right)\text{SEQ}^{\leq k_1}(0)\tag{3} \end{align*}

with the corresponding bivariate generating function $G(u,w)$

\begin{align*} G(u,w)=\frac{\left(\frac{1-u^{k_1+1}}{1-u}\right)\left(\frac{1-w^{k_2+1}}{1-w}\right)} {1-\left(u\frac{1-u^{k_1}}{1-u}\right)\left(w\frac{1-w^{k_2}}{1-w}\right)}\tag{4} \end{align*}

In order to simplify calculations we introduce:

\begin{align*} H_k(z)=\frac{1-z^k}{1-z} \end{align*}

We use the coefficient of operator $[z^n]$ to denote the coefficient of $z^n$ of the corresponding generating function and so we can conclude:

Result: The number of $x$ $0$s and $y$ $1$s of strings of length $n=x+y$ containing runs of at most $k_1$ $0$s and at most $k_2$ $1$s is the coefficient of $[u^xw^y]$ in $G(u,w)$ with

\begin{align*} [u^xw^y]G(u,w)&=[u^xw^y]\frac{H_{k_1+1}(u)H_{k_2+1}(w)}{1-uwH_{k_1}(u)H_{k_2}(w)}\tag{5}\\ &=[u^xw^y]{H_{k_1+1}(u)H_{k_2+1}(w)}\\ &\qquad\qquad\cdot\sum_{n\geq 0}(uw)^n\left(H_{k_1}(u)\right)^n\left(H_{k_2}(w)\right)^n \end{align*}

The coefficients $[u^x]$ and $[w^y]$ can be calculated separately due to the (relatively) nice structure of $G(u,w)$. We find

\begin{align*} [u^x]u^n\left(H_{k_1}(u)\right)^n&=[u^x]u^n\left(\frac{1-u^{k_1}}{1-u}\right)^n\\ &=[u^{x-n}]\sum_{j=0}^{n}\binom{n}{j}(-u)^{k_1j}\sum_{l\geq 0}\binom{-n}{l}(-u)^l\\ &=[u^{x-n}]\sum_{j=0}^{n}\binom{n}{j}(-u)^{k_1j}\sum_{l\geq 0}\binom{n+l-1}{l}u^l\\ &=\sum_{l\geq 0}\binom{n+l-1}{l}[u^{x-n-l}]\sum_{j=0}^{n}\binom{n}{j}(-u)^{k_1j}\\ &=\sum_{j=0}^{n}(-1)^{k_1j}\binom{n}{j}\binom{n+(x-n-k_1j)-1}{x-n-k_1j}\\ &=\sum_{j=0}^{n}(-1)^{k_1j}\binom{n}{j}\binom{x-k_1j-1}{n-1} \end{align*}

Similar calculations can be used to find the other parts of (5).

Simplification: If we consider the simplified problem of counting the number of runs of length at most $k$ in a binary alphabet, we can use a univariate generating function based upon (3) and (4).

• The symbolic equation (3) simplifies to

\begin{align*} \text{SEQ}^{\leq k}(1)\text{SEQ}\left(0\text{SEQ}^{\leq k-1}(0)1\text{SEQ}^{\leq k-1}(1)\right)\text{SEQ}^{\leq k}(0) \end{align*}

• and instead of $G(u,w)$ in (4) we can write:

\begin{align*} \widetilde{G}(u)&=\frac{\left(\frac{1-u^{k+1}}{1-u}\right)^2}{1-\left(u\frac{1-u^{k}}{1-u}\right)^2}\\ &=\frac{1-u^{k+1}}{1-2u+u^{k+1}}\\ \end{align*}

Reference: The generating function $\widetilde{G}(u)$ is Example 1.10 in Analytic Combinatorics the classic from P. Flajolet and R. Sedgewick. The symbolic method is described there in section I.1 and I.2.

• How am I suppose to program this? – akashchandrakar Dec 30 '14 at 16:28
• @Aksam: You could try to implement it by a recursive approach. First you start to implement a (recursive) function which generates strings according to $SEQ^{\leq k}(a)$. Then you could use this as building block and create successively more complex sets of strings accordingly to the symbolic equation above. Since design, implementation and tests are presumably somewhat effortful, you could consider the functions as part of a library for string generation which you could also use for similar problems. Regards, – Markus Scheuer Dec 30 '14 at 17:09
• @Blex: Thanx! I suppose you might also like this answer of mine! :-) – Markus Scheuer Jul 17 '15 at 19:07

A generating function is straightforward to compute using the Goulden-Jackson cluster method (which is designed to solve exactly this kind of problem). A canonical reference is [Noonan-Zeilberger 1998]; additionally, [Kupin-Yuster 1998] works out an example problem in exactly the generality you're looking for. The calculation itself is easy; the post is long as I include a brief exposition of the method.

We're given a set $\mathcal{B}$ of "bad words" over an alphabet $\mathcal{A}$; we'll take $\mathcal{A}=\{0,1\}$. We assume no bad word contains another. We want to enumerate strings containing no bad words according to their weight; the weight of a string $s$ is $w(s)=x^i y^j$, where $i$ is the number of $0$'s in the string and $j$ is the number of $1$'s. The weight enumerator of a set $S$ of strings is the sum of $w(s)$ over all $s\in S$.

A marked string is a string $s$ together with a collection $S$ of some instances of bad words contained in $s$; the weight of a marked string is $(-1)^{|S|} w(s)$.

A cluster is a string composed of an overlapping set of bad words; see the references for the precise definition. (In your application, the bad words are $0^{k_1+1}$ and $1^{k_2+1}$, so the only clusters have the form $0^{> k_1}$ or $1^{> k_2}$.)

Define $\mathcal{C}(x,y)$ to be the weight enumerator of all marked clusters; we have $\displaystyle\mathcal{C}(x,y) = \sum_{b\in\mathcal{B}}\mathcal{C}_b(x,y),$ where $\mathcal{C}_b$ is the weight enumerator of marked clusters in which the first bad word is $b$.

The Goulden-Jackson method rests on two observations. First, it's easy to write down a system of linear equations which determine the marked cluster enumerators, based on the overlaps among the bad words. Second, the weight enumerator for strings containing no bad words is given by $$\mathcal{G}(x,y) = \frac{1}{1-(x+y)-\mathcal{C}(x,y)}.$$

To compute the marked cluster enumerators, we have the following equations for each $b\in\mathcal{B}$: $$\mathcal{C}_b = -w(b) - \sum_{b'\in\mathcal{B}}\sum_r w(b_r)\mathcal{C}_{b'}$$ where $b_r$ denotes the $r$-long prefix of $b$, and $r$ ranges over values such that $b$ is the concatenation of $b_r$ with a prefix of $b'$. In the present example, the bad words overlap with themselves but not with each other, so the equations separate. In particular, for $b=0^{k_1+1}$, we get $$\mathcal{C}_b(x,y)=-x^{k_1+1} - \sum_{r=0}^{k_1} x^r \mathcal{C}_b(x,y),$$ which implies $$\mathcal{C}_b(x,y)= -\frac{x^{k_1+1}(1-x)}{1-x^{k_1+1}}.$$ You get an analogous formula for $b=1^{k_2+1}$. Thus, according to Goulden-Jackson, the generating function for strings containing neither $0^{k_1+1}$ nor $1^{k_2+1}$ is given by $$\mathcal{G}(x,y)=\frac{1}{1-(x+y)+\frac{x^{k_1+1}(1-x)}{1-x^{k_1+1}}+\frac{y^{k_2+1}(1-y)}{1-y^{k_2+1}}}.$$

This answer agrees with the one provided by @MarkusScheuer.

• Very nice answer! I also do appreciate the cluster method. +1 – Markus Scheuer Nov 10 '15 at 7:50