A bit-string is any finite sequence of $1$s and $0$s. For example, $1011011$, $1011010$, and $000110$ are bit-strings.
In this post, I will refer to bit-strings as strings, to be concise.
I now introduce and define the following categories of strings: left-eager, right-eager, and doubly-eager.
A string is left-eager iff the number of $1$s always stays ahead of the number of $0$s when read from left to right. For example, $1101$ is left-eager, but $101$ is not left-eager.
A string is right-eager iff the number of $1$s always stays ahead of the number of $0$s when read from right to left. For example, $101011$ is right-eager, but $1110$ is not right-eager. These strings correspond to the OEIS sequence A036994.
Finally, a string is doubly-eager iff it is both left-eager and right-eager. For example, $11010111$ is doubly-eager.
Now, for any string, we may construct a corresponding set of nonnegative integers. We construct this set by indexing the digits of the string ($0,1,2,\ldots$, from left to right) and taking the set of indices at which the digit in the string is $1$. For example, $1101011$ corresponds to the set $\{0,1,3,5,6\}$ because $1101011$ has a $1$ at indices $0, 1, 3, 5,$ and $6$.
Consider the doubly-eager strings and their corresponding sets of nonnegative integers. The first few are:
$1$, which corresponds to $\{0\}$,
$11$, which corresponds to $\{0,1\}$,
$111$, which corresponds to $\{0,1,2\}$,
$1111$, which corresponds to $\{0,1,2,3\}$,
$11011$, which corresponds to $\{0,1,3,4\}$,
etc.
Interestingly, it appears that all of these sets are restricted finite additive 2-bases.
[A restricted finite additive $2$-basis is a set of nonnegative integers $S = \{a_1, a_2, \ldots a_n\}$ with $0 \leq a_1 < a_2 < \cdots < a_n$ such that $S+S = \{0,1,2,\ldots,2 a_n\}$, where $S+S$ is defined to be $\{a_i+a_j : i,j \in \{1,2,\ldots,n\} \}$. For more information, see this post.]
I conjecture that the set of nonnegative integers associated with any doubly-eager string is a restricted finite additive $2$-basis.
The converse of my conjecture is false; that is, not all restricted finite additive $2$-bases are associated with doubly-eager strings. The first counterexample is $\{0, 1, 2, 5, 8, 9, 10\}$, which is a restricted finite additive $2$-basis, yet is associated with the string $11100100111$, which is not doubly-eager (in fact, it is neither left-eager nor right-eager).
However, it appears that while being associated with a doubly-eager string is not a necessary condition for being a restricted finite additive $2$-basis, it does seem to be a sufficient condition for being a restricted finite additive $2$-basis.
To test my conjecture, I wrote a program. If it works properly, it verifies my conjecture for all strings of length $1$ to $18$.
My primary questions are:
Is my conjecture true?
Are there any heuristic reasons for why my conjecture may be true (or false)?
Has this been studied before?
Do you have any ideas about how to prove (or disprove) this conjecture?
[Update (10/26/2023): In the literature, "doubly-eager bit-strings" are referred to as "bidirectional ballot sequences".]
