# Identification of a curious function

The following question was asked on MSE but there were no replies.

During computation of some Shapley values (details below), I encountered the following function: $$f\left(\sum_{k \geq 0} 2^{-p_k}\right) = \sum_{k \geq 0} \frac{1}{(p_k+1)\binom{p_k}{k}},$$ where $$p_0 > 0$$ and $$p_{k+1} > p_k$$ for all $$k$$. In other words, the input to $$f$$ is the binary expansion of a real number in the range $$[0,1]$$, and the $$p_k$$ correspond to the positions of $$1$$s in the binary expansion.

For example, $$f(2^{-t}) = 1/(t+1)$$, so $$f(1/2) = 1/2$$, $$f(1/4) = 1/3$$ and so on. More complicated examples are $$f(5/8) = f(2^{-1} + 2^{-3}) = 1/2 + 1/(4\cdot 3) = 7/12$$ and $$f(2/3) = f\left(\sum_{k \geq 0}2^{-(2k+1)}\right) = \sum_{k \geq 0} \frac{1}{(2k+2)\binom{2k+1}{k}} = \frac{\pi}{\sqrt{27}}.$$

The function $$f$$ is a continuous increasing function satisfying $$f(0) = 0$$, $$f(1) = 1$$, and $$f(1-t) = 1-f(t)$$ for $$t \in [0,1]$$. It has vertical asymptotes at dyadic points.

Here is a plot of $$f$$:

Is the function $$f$$ known?

Here is where $$f$$ came from. Let $$n \geq 1$$ be an integer and let $$t \in [0,1]$$. For a permutation $$\pi$$ of the numbers $$\{ 2^{-m} : 0 \leq m \leq n-1 \}$$ satisfying $$\pi^{-1}(1) = i$$, we say that $$\pi$$ is pivotal if $$\sum_{j. Let $$f_n(t)$$ be the probability that a random $$\pi$$ is pivotal. Then $$f(t) = \lim_{n \rightarrow \infty} f_n(t)$$.

For example, take $$n = 4$$. The permutation $$1/8,1/2,1,1/4$$ is pivotal for $$t \in (5/8,1]$$. For all $$n \geq 2$$ we have $$f_n(1/2) = 1/2$$, since $$\pi$$ is pivotal iff $$1$$ appears before $$1/2$$ in $$\pi$$. The general formula for $$f$$ is derived in a similar way.

We leave it to the reader to figure out how $$f_n$$ measures some Shapley value. The functions $$f_n$$ are step functions with steps of length $$1/2^{n-1}$$. They are left-continuous, and are equal to $$f$$ at the breakpoints.

• I'll take the responsibility for suggesting that the long silence can be interpreted as "Not to MO participants" at this point. Is there anything else you want to ask about it? Aug 29, 2013 at 1:00
• It is not known to me; I feel however that fedja is rushing things a bit. Perhaps there are other functions using binary expansions for which more is known? Aug 29, 2013 at 2:07
• There is some resemblance (in the definition and in the look) to en.wikipedia.org/wiki/Minkowski%27s_question_mark_function Aug 29, 2013 at 19:33
• in fact, even more to the inverse function of it Aug 30, 2013 at 7:50
• "Perhaps there are other functions using binary expansions" I'm even guilty of creating one myself with Vasyunin (the Bellman function for the weak type bound for a function whose dyadic square function is bounded by $1$; we have never published it though and, probably, never will). I can confirm that its graph also "has some resemblance" to the graph posted (especially on low resolution). However this is not what the question was about... Aug 30, 2013 at 19:11