The following combinatorial problem popped up in a totally uncombinatorial context:
Let $\mathcal{P}$ denote the power set of a set and let $k \in \mathbb{N}$. Is there a map $c: \mathcal{P}(\{1,2,\dots,k\}) \to \mathbb{Q}$ with the following properties: \begin{align*} &c(\emptyset)=1 \ , \\ &c(\{i,i+1,\dots,j\}) = \sum_{\nu=i}^{j} (c(\{i,i+1,\dots,\nu-1\}) + c(\{\nu+1,\dots,j\} ) \quad \forall i \leq j \in \{1,2,\dots,k\} \ . \end{align*}
I am searching for an explicit description of a map with these properties. I have tried to define such a map using factorials and binomial coefficients.
(This is a more special version of a map I was searching for in this earlier question: Find a map on a power set with certain combinatorial properties )
Let's make the additional assumption that $c(S)$ depends only on the size of $S$. Let $c_\ell=c(S)$ for any set $S$ with $\ell$ elements.
Then your conditions reduce to: \begin{align} c_0&=1\\ c_\ell&=\sum_{n = 0}^{\ell - 1} c_n + c_{\ell - 1 - n}=2\sum_{n=0}^{\ell-1} c_n \quad \text{for } \ell \geq 1 \end{align} after some rearrangement.
This is already enough to allow us to compute $c_\ell$ for any $\ell$. To find an explicit formula, notice that, for $\ell \geq 2$, we have $$ c_\ell=2c_{\ell-1} + 2 \sum_{n=0}^{\ell-2}c_n = 3c_{\ell-1} $$ and so $$ c_\ell=3^{\ell -1} c_1 = 2 \cdot 3^{\ell-1} $$ for all $\ell \geq 1$.