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Covariance within Random Integer Compositions

Fix a positive integer N. Select an additive composition ξ of N uniformly out of 2^N-1 possibilities. The interplay between the number of parts in ξ and the maximum part in ξ is our focus. It is not surprising that correlations ρ(N) between these quantities are negative; we earlier gave inconclusive evidence that lim_N →∞ρ(N) is strictly less than zero. A proof of this result would imply asymptotic dependence. We now retract our presumption in such an unforeseen outcome. Similar experimental findings apply when ξ is a 1-free composition, i.e., possessing only parts ≥ 2.

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1 Unconstrained and Pinned Solus Bitstrings

Given a random unconstrained bitstring of length , we have

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

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

i.e., parts of correspond to “waiting times” for each in .  The number of parts in is equal to the number of s in and the maximum part in is equal to the duration of the longest run of s in , 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 and its last bit is .

  • A bitstring is solus if all of its s are isolated.

The latter was discussed in [2, 3]; additionally imposing the former is new.  Given a random pinned solus bitstring of length , formulas for number of s and number of s are best expressed using generating functions.

If we append the string with to construct , then the associated is a composition of with all parts .  For example, if ,

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

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

Nej & Satyanarayana Reddy [8] gave an impressive recursion for the number of unconstrained bitstrings of length containing exactly s and a longest run of exactly s:

where (of course) and ,

and

Consequently, the numerator of for -bitstrings is

equivalently, the numerator of for -compositions is

The denominator is .  Returning to the unrestricted example, the covariance for is .  Correlations for selected small turn out to be

and Table 1 exhibits values for larger .

By a similar argument, we deduce the number of pinned solus bitstrings of length containing exactly s and a longest run of exactly s.  The recursion is identical to before (with replaced by ) but possesses different initial conditions

where , with a different :

and a different :

Consequently, the numerator of under constraints is

equivalently, the numerator of under restrictions is

The denominator is .  Returning to the -free example, the covariance for is .  Correlations for selected small turn out to be

i.e., dependency is more significant than earlier.  Table 1 exhibits values for larger .

2 Sketches of Proofs

Let be a set of finite bitstrings and be the subset of consisting of strings of length containing exactly s and a longest run of exactly s.  Let and be the subset of of strings starting with and respectively.

Assume that consists of all unconstrained strings.  If , then is of the form where .  If , then is of the form

or

We have

(1)

hence

upon addition.  This proof of the recurrence for appeared in [8].

Assume instead that consists of all solus strings.  If , then is of the form where .  We have

that is,

From formula (1) in the preceding,

which gives a recurrence underlying what we called in [3].

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

if is odd, then

if is even, then

These imply the expression for . For pinned strings, the latter two results hold, but the former becomes .  The expression for comes from [8]

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

Table 2: Correlation between number of s and longest run of s within random bitstrings as a function of .

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