An integer partition $\lambda$ of $n$ is called a binary partition provided that its parts are powers of $2$ (dyadic). Example: Let $n=3$. The binary partitions are $\lambda=(2,1)$ and $\lambda=(1,1)$ but not $\lambda=(3)$. If $b(n)$ counts the number of binary partitions of $n$, then $$\sum_{n\geq0}b(n)x^n=\prod_{k=0}^{\infty}\frac1{1-x^{2^k}}.$$
Given such a partition $\lambda$, let $Y_{\lambda}$ be its corresponding Young diagram. If $\square$ is a cell in $Y_{\lambda}$, construct its hook-lengths $h_{\square}$ in the usual manner. Example: the multi-set of hooks of $\lambda=(2,1)$ is $\{h_{\square}: \square\in Y_{\lambda}\}=\{3,1,1\}$.
A partition $\lambda$ is called an $(s,t)$-core if both $s$ and $t$ are absent from the hook-lengths in $Y_{\lambda}$. There has been a flurry of activity regarding this notion.
Recall 1. The number of $(s,s+1)$-core (unrestricted) partitions is the Catalan number $\frac1{s+1}\binom{2s}s$.
Recall 2. The number of $(s,s+1)$-core partitions with distinct parts is the Fibonacci number $F_{s+1}$.
I like to ask:
QUESTION. What is the total number $f_{s,s+1}$ of $(s,s+1)$-core binary partitions?
Example. The first few values I can compute: $f_{1,2}=1, f_{2,3}=2, f_{3,4}=4, f_{4,5}=9, f_{5,6}=19$.
