1 Constrained Cases
The first two examples of constrained bitstrings were introduced in [8].

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

A bitstring is multus if each of its s possess at least one neighboring .
Counts of solus bitstrings have a quadratic character:
whereas counts of multus bitstrings have a cubic character:
The remaining two examples are new, as far as is known.

A bitstring is bimultus if each of its s possess at least one neighboring and each of its s possess at least one neighboring . Both isolated bits and isolated bits are avoided in such bitstrings; a certain symmetry holds here.

A bitstring is persolus if all of its s are isolated and each of its s possess at least one neighboring . That is, while s in solus bitstrings are alone, s in persolus bitstrings are very alone.
Counts of bimultus bitstrings have a quadratic character [9]
whereas counts of persolus bitstrings have a cubic character
2 Bitsums
Given a set of finite bitstrings, what can be said about the bitsum of a random of length ? If is unconstrained, i.e., if all strings are included in the sample, then
because a sum of independent Bernoulli() variables is Binomial(,). Expressed differently, the average density of s in a random unconstrained string is , with a corresponding variance .
We previously covered solus and multus bitstrings in [8]. If consists of bimultus bitstrings, then the total bitsum of all of length has generating function [9]
and the total bitsum squared has generating function
hence has generating function
Standard techniques [1] give asymptotics
for the average density of s in a random bimultus string and corresponding variance.
If instead consists of persolus bitstrings, then the total bitsum of all of length has generating function [9]
and the total bitsum squared has generating function
hence has generating function
We obtain asymptotics
for the average density of s in a random persolus string and corresponding variance. Unsurprisingly and , where
are estimates associated with solus strings and
are estimates associated with multus strings [8]. While insisting on symmetry forces equiprobability, it also increases the variance, but only slightly.3 Longest Bitruns
Given a set of finite bitstrings, what can be said about the duration of the longest run of s in a random of length ? We have already discussed the case when is unconstrained. Preliminary coverage for constrained (for means, but not mean squares) occurred in [8].
If consists of solus bitstrings, then it makes little sense to talk about runs. For runs, over all , we have
the Taylor expansion of the numerator series for is
and the Taylor expansion of the numerator series for is
Let us abbreviate such series as and for simplicity – likewise and – and let denote the Golden mean. It is conjectured that, up to small periodic fluctuations,
as .
If consists of multus bitstrings, then we can talk both about runs:
and runs:
Letting
it is conjectured that
as .
If consists of bimultus strings, there is symmetry (just as for the unconstrained case). We have
It is conjectured that, up to small periodic fluctuations,
as . The same asymptotic variance occurred for solus bitstrings.
If consists of persolus strings, then it makes little sense to talk about runs. For runs, we have
4 CrossCovariances I
Let us return to the unconstrained case. Exhibiting in a manner parallel to old formulas in our introduction seems impossible: no analogous summation identity for apparently exists. Thus new formulas are somewhat less tidy, but nevertheless workable. The number of bitstrings with no runs of s and no runs of s has generating function
hence
where . The Taylor expansion of the numerator series for is
and the correlation coefficient
is prescribed numerically in Table 1 for . These results complement those in [11].
For multus bitstrings, since s are clumped (but s are not necessarily so), the associated generating function
is unsurprisingly asymmetric in and . The associated Taylor expansion is
and, again, the corresponding is prescribed in Table 1. Correlations are all negative but approach zero as increases. We observe a slightly stronger dependency between and for multus strings than for unconstrained strings. Calculating for bimultus strings remains open. Simulation suggests that dependence is greater still for the bimultus case.
Table 1: Correlation as a function of .
5 CrossCovariances II
Let us return to the solus case. Being isolated, each acts as barrier to gathering s; we wonder to what extent the (random) number of such walls affects the largest crowd size. To calculate seems to be difficult. The number of bitstrings with less than two s and no runs of s has generating function
(a polynomial!) with
assuming . For example,
when .
The number of bitstrings with less than three s and no runs of s has generating function with
assuming . For example,
when .
The number of bitstrings with less than four s and no runs of s has generating function with
assuming , where
is when and is otherwise. For example,
when .
An expression for , the generating function corresponding to bitstrings with less than five s and no runs of s, exists but awaits simplication. For example,
when . For arbitrary , clearly is equal to
for and
for . Also,
where
The interval deserves more attention. From the approximation
we obtain the Taylor expansion of the numerator series for :
(every exhibited coefficient is correct). We would need , , … to achieve the precision necessary to adequately estimate for large . Simulation suggests that correlations are all negative but, unlike the previous section, tend to a nonzero quantity (possibly ?) as approaches infinity. We have not yet attempted to study the persolus case.
6 CrossCovariances III
This section is an addendum to the preceding. A recent paper [12] gave an impressive recursion for the number of unconstrained bitstrings of length containing exactly s and a longest run of exactly s:
where ,
By a similar argument, we deduce the number of solus bitstrings of length containing exactly s and a longest run of exactly s:
where is defined recursively as before, with the same but with and a different :
Consequently, the number of solus bitstrings of length with less than s and no runs of s is
and our prior results for and are easily verified. As more examples, we have