Covariance within Random Integer Compositions

1 Unconstrained and Pinned Solus Bitstrings

Given a random unconstrained bitstring of length $n = N - 1$ , we have

\begin{matrix} E (number of 1 s) = n / 2, & V (number of 1 s) = n / 4 \end{matrix}

because a sum of $n$ independent Bernoulli( $1 / 2$ ) variables is Binomial( $n$ , $1 / 2$ ). Expressed differently, the average density of $1$ s in a string is $1 / 2$ , with a corresponding variance $1 / 4$ . The word “unconstrained” offers that, in the sampling process, all $2^{n}$ strings are included and equally weighted.

If we append the string with a $1$ , calling this $η$ , then there is a natural way [7] to associate $η$ with an additive composition $ξ$ of $N$ . For example, if $N = 10$ ,

η = 0110100111 ⟷ ξ = {2, 1, 2, 3, 1, 1}

i.e., parts of $ξ$ correspond to “waiting times” for each $1$ in $η$ . The number of parts in $ξ$ is equal to the number of $1$ s in $η$ and the maximum part in $ξ$ is equal to the duration of the longest run of $0$ s in $η$ , plus one.

In this paper, the word “constrained” refers to the logical conjunction of two requirements:

A bitstring is pinned if its first bit is $0$ and its last bit is $0$ .
A bitstring is solus if all of its $1$ s are isolated.

The latter was discussed in [2, 3]; additionally imposing the former is new. Given a random pinned solus bitstring of length $n = N - 1$ , formulas for $E ($ number of $1$ s $)$ and $V ($ number of $1$ s $)$ are best expressed using generating functions.

If we append the string with $1$ to construct $η$ , then the associated $ξ$ is a composition of $N$ with all parts $\geq 2$ . For example, if $N = 15$ ,

η = 010001010010101 ⟷ ξ = {2, 4, 2, 3, 2, 2} .

It should now be clear why, starting with the original $n$ -bitstring,

1 + E (number of 1 s) = E (number of parts),

1 + E (longest run of 0 s) = E (maximum part)

for both scenarios, but the corresponding variances are always equal.

Nej & Satyanarayana Reddy [8] gave an impressive recursion for the number $F_{n} (x, y)$ of unconstrained bitstrings of length $n$ containing exactly $x$ $0$ s and a longest run of exactly $y$ $0$ s:

F_{n} (x, y) = ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ \begin{matrix} y - 1 \sum i = κ F_{n - i - 1} (x - i, y) + y \sum j = 0 F_{n - y - 1} (x - y, j) & if 1 \leq x \leq n - 2 and ε_{n} (x, y) = 1, λ_{n} (y) & if x = n - 1 and ε_{n} (x, y) = 1, 0 & otherwise, \end{matrix}

\begin{matrix} F_{n} (0, 0) = 1 - κ, & F_{n} (n, n) = 1 \end{matrix}

where $n \geq x \geq y$ (of course) and $κ = 0$ ,

ε_{n} (x, y) = ⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩ \begin{matrix} 1 & if n \geq x % and ⌊ \frac{n}{n - x + 1} ⌋ \leq y \leq x, 0 & otherwise \end{matrix}

and

λ_{n} (y) = ⎧ ⎪ ⎨ ⎪ ⎩ \begin{matrix} 1 & if n is odd and y = \frac{n - 1}{2}, 2 & otherwise. \end{matrix}

Consequently, the numerator of $E [(number of 1 s) (longest run of 0 s)]$ for $n$ -bitstrings is

{n \sum x = 0 x \sum y = 0 (n - x) y F_{n} (x, y)}_{n}^{\infty} = {0, 2, 11, 40, 122, 338, 881, 2202, 5337, 12634, \dots};

equivalently, the numerator of $E [(number of parts) (maximum part)]$ for $N$ -compositions is

{N \sum x = 0 x \sum y = 0 (N - x) (y + 1) F_{n} (x, y)}_{n}^{\infty} = {1, 4, 14, 42, 115, 296, 732, 1757, 4125, 9516, \dots} .

The denominator is $2^{n}$ . Returning to the unrestricted example, the covariance for $N = 5$ is $\frac{40}{16} - (2) (\frac{27}{16}) = \frac{115}{16} - (1 + 2) (1 + \frac{27}{16})$ . Correlations for selected small $N$ turn out to be

{ρ (N) : N = 5, 11, 21, 51} = {- 0.890799, - 0.752444, - 0.654958, - 0.530128}

and Table 1 exhibits values for larger $N = 101, 201, \dots$ .

By a similar argument, we deduce the number $G_{n} (x, y)$ of pinned solus bitstrings of length $n$ containing exactly $x$ $0$ s and a longest run of exactly $y$ $0$ s. The recursion is identical to before (with $F$ replaced by $G$ ) but possesses different initial conditions

\begin{matrix} G_{n} (x, 0) = 1 - κ for all 0 \leq x \leq n, & G_{n} (n, n) = 1 \end{matrix}

where $κ = 1$ , with a different $ε_{n} (x, y)$ :

ε_{n} (x, y) = ⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩ \begin{matrix} 1 & if n \geq x % and (⌊ \frac{n}{n - x + 1} ⌋ \leq y < x or x = y = n), 0 & otherwise \end{matrix}

and a different $λ_{n} (y)$ :

λ_{n} (y) = ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ \begin{matrix} 1 & if n is odd and y = \frac{n - 1}{2}, 2 & if ⌊ \frac{n - 1}{2} ⌋ < y < n - 1, 0 & otherwise. \end{matrix}

Consequently, the numerator of $E [(number of 1 s) (longest run of 0 s)]$ under constraints is

{n \sum x = 0 x \sum y = 0 (n - x) y G_{n} (x, y)}_{n}^{\infty} = {0, 0, 1, 4, 10, 26, 54, 118, 230, 458, 864, 1632, \dots};

equivalently, the numerator of $E [(number of parts) (maximum part)]$ under restrictions is

{N \sum x = 0 x \sum y = 0 (N - x) (y + 1) G_{n} (x, y)}_{n}^{\infty} = {0, 2, 3, 8, 17, 34, 70, 131, 255, 466, 868, 1565, \dots} .

The denominator is $d_{n}$ . Returning to the $1$ -free example, the covariance for $N = 7$ is $\frac{26}{8} - (\frac{5}{4}) (\frac{13}{4}) = \frac{70}{8} - (1 + \frac{5}{4}) (1 + \frac{13}{4})$ . Correlations for selected small $N$ turn out to be

{ρ (N) : N = 7, 11, 21, 51} = {- 0.945611, - 0.860467, - 0.763395, - 0.629068},

i.e., dependency is more significant than earlier. Table 1 exhibits values for larger $N = 101, 201, \dots$ .

2 Sketches of Proofs

Let $Ω$ be a set of finite bitstrings and $Ω_{n}^{x, y}$ be the subset of $Ω$ consisting of strings of length $n$ containing exactly $x$ $0$ s and a longest run of exactly $y$ $0$ s. Let $Ω_{n, 0}^{x, y}$ and $Ω_{n, 1}^{x, y}$ be the subset of $Ω_{n}^{x, y}$ of strings starting with $0$ and $1$ respectively.

Assume that $Ω$ consists of all unconstrained strings. If $ω \in Ω_{n, 1}^{x, y}$ , then $ω$ is of the form $1 ω_{1}$ where $ω_{1} \in Ω_{n - 1, 0}^{x, y} \cup Ω_{n - 1, 1}^{x, y}$ . If $ω \in Ω_{n, 0}^{x, y}$ , then $ω$ is of the form

\begin{matrix} 00 \dots 0      i ω_{2} & where & ω_{2} \in Ω_{n - i, 1}^{x - i, y} & and & 1 \leq i \leq y - 1 \end{matrix}

\begin{matrix} 00 \dots 0      y ω_{3} & where & ω_{3} \in Ω_{n - y, 1}^{x - y, j} & and & 0 \leq j \leq y . \end{matrix}

We have

∣ ∣ Ω_{n, 1}^{x, y} ∣ ∣ = ∣ ∣ Ω_{n - 1, 0}^{x, y} ∣ ∣ + ∣ ∣ Ω_{n - 1, 1}^{x, y} ∣ ∣ = ∣ ∣ Ω_{n - 1}^{x, y} ∣ ∣,

∣ ∣ Ω_{n, 0}^{x, y} ∣ ∣ = y - 1 \sum i = 1 ∣ ∣ Ω_{n - i, 1}^{x - i, y} ∣ ∣ + y \sum j = 0 ∣ ∣ Ω_{n - y, 1}^{x - y, j} ∣ ∣ = y - 1 \sum i = 1 ∣ ∣ Ω_{n - i - 1}^{x - i, y} ∣ ∣ + y \sum j = 0 ∣ ∣ Ω_{n - y - 1}^{x - y, j} ∣ ∣

(1)

hence

∣ ∣ Ω_{n}^{x, y} ∣ ∣ = y - 1 \sum i = 0 ∣ ∣ Ω_{n - i - 1}^{x - i, y} ∣ ∣ + y \sum j = 0 ∣ ∣ Ω_{n - y - 1}^{x - y, j} ∣ ∣

upon addition. This proof of the recurrence for $F_{n} (x, y)$ appeared in [8].

Assume instead that $Ω$ consists of all solus strings. If $ω \in Ω_{n, 1}^{x, y}$ , then $ω$ is of the form $1 ω_{1}$ where $ω_{1} \in Ω_{n - 1, 0}^{x, y}$ . We have

∣ ∣ Ω_{n, 1}^{x, y} ∣ ∣ = ∣ ∣ Ω_{n - 1, 0}^{x, y} ∣ ∣ = ∣ ∣ Ω_{n - 1}^{x, y} ∣ ∣ - ∣ ∣ Ω_{n - 1, 1}^{x, y} ∣ ∣,

that is,

∣ ∣ Ω_{n}^{x, y} ∣ ∣ = ∣ ∣ Ω_{n, 1}^{x, y} ∣ ∣ + ∣ ∣ Ω_{n + 1, 1}^{x, y} ∣ ∣ = ∣ ∣ Ω_{n - 1, 0}^{x, y} ∣ ∣ + ∣ ∣ Ω_{n, 0}^{x, y} ∣ ∣ .

From formula (1) in the preceding,

∣ ∣ Ω_{n, 0}^{x, y} ∣ ∣ = y - 1 \sum i = 1 ∣ ∣ Ω_{n - i, 1}^{x - i, y} ∣ ∣ + y \sum j = 0 ∣ ∣ Ω_{n - y, 1}^{x - y, j} ∣ ∣ = y - 1 \sum i = 1 ∣ ∣ Ω_{n - i - 1, 0}^{x - i, y} ∣ ∣ + y \sum j = 0 ∣ ∣ Ω_{n - y - 1, 0}^{x - y, j} ∣ ∣

which gives a recurrence underlying what we called ${~ F}_{n} (x, y)$ in [3].

Let us turn attention to various boundary conditions. For either unconstrained or solus strings,

Ω_{n}^{n - 1, n - 1} = ⎧ ⎨ ⎩ 1 00 \dots 0      n - 1, 00 \dots 0      n - 1 1 ⎫ ⎬ ⎭;

if $n$ is odd, then

Ω_{n}^{n - 1, (n - 1) / 2} = ⎧ ⎪ ⎨ ⎪ ⎩ 00 \dots 0      (n - 1) / 2 1 00 \dots 0      (n - 1) / 2 ⎫ ⎪ ⎬ ⎪ ⎭;

if $n$ is even, then

Ω_{n}^{n - 1, n / 2} = ⎧ ⎪ ⎨ ⎪ ⎩ 00 \dots 0      (n - 2) / 2 1 00 \dots 0      n / 2, 00 \dots 0      n / 2 1 00 \dots 0      (n - 2) / 2 ⎫ ⎪ ⎬ ⎪ ⎭ .

These imply the expression for $λ_{n} (y)$ . For pinned strings, the latter two results hold, but the former becomes $Ω_{n}^{n - 1, n - 1} = \emptyset$ . The expression for $ε_{n} (x, y)$ comes from [8]

F_{n} (x, y) > 0 ⟺ ⎧ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎩ \begin{matrix} x + ⌊ \frac{x}{y} ⌋ \leq n & if y > 0 and y ∤ x, x + \frac{x}{y} - 1 \leq n & if y > 0 and y ∣ x; \end{matrix}

G_{n} (x, y) > 0 ⟺ ⎧ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎩ \begin{matrix} x + ⌊ \frac{x}{y} ⌋ \leq n & if y > 0 and y ∤ x, x + \frac{x}{y} - 1 \leq n & if (x > y > 0 or x = y = n) and y ∣ x . \end{matrix}

For completeness’ sake, we give the analog of Table 1 for pinned and solus strings.

Table 2: Correlation between $($ number of $1$ s $)$ and $($ longest run of $0$ s $)$ within random bitstrings as a function of $n$ .

3 Acknowledgements

R, Mathematica and Maple have been useful throughout. I am grateful to Ernst Joachim Weniger, Claude Brezinski and Jan Mangaldan for very helpful discussions about convergence acceleration. Dr. Weniger’s software code and numerical computations were especially appreciated.

References

[1] R. Sedgewick and P. Flajolet, Introduction to the Analysis of Algorithms, Addison-Wesley, 1996, pp. 120–121, 159–161, 366–373, 379.
[2] S. R. Finch, Cantor-solus and Cantor-multus distributions, arXiv:2003.09458.
[3] S. R. Finch, Variance of longest run duration in a random bitstring, arXiv:2005.12185.
[4] D. W. Boyd, Losing runs in Bernoulli trials, unpublished note (1975), https://www.math.ubc.ca/~boyd/bern.runs/bernoulli.html.
[5] M. F. Schilling, The longest run of heads, College Math. J. 21 (1990) 196–207; MR1070635.
[6] A. Alexandru, The longest run of heads, unpublished note (2011), https://alexamarioarei.github.io/Research/Research-en.html.
[7] P. Hitczenko and C. D. Savage, On the multiplicity of parts in a random composition of a large integer, SIAM J. Discrete Math. 18 (2004) 418–435; MR2112515.
[8] M. Nej and A. Satyanarayana Reddy, Binary strings of length $n$ with $x$ zeros and longest $k$ -runs of zeros, Indian J. Math. 61 (2019) 111–139; arXiv:1707.02187; MR3931610.

Steven Finch

MIT Sloan School of Management

Cambridge, MA, USA

steven_finch@harvard.edu

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1 Unconstrained and Pinned Solus Bitstrings

2 Sketches of Proofs

3 Acknowledgements

References

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References