Given a partition $\lambda$ and its Young diagram $\pmb{Y}\_{\lambda}$, we say $\lambda$ is a $(t,s)$-core partition provided that neither $t$ nor $s$ is a hook length in $\pmb{Y}\_{\lambda}$. We now recall a conjecture (now a theorem) of D. Armstrong which states "the total number $(s,t)$-core partitions is $\frac1{s+t}\binom{s+t}s$".


Focusing on the special case of $(s,s+1)$-core partitions, [Stanley and Zanello](http://math.mit.edu/%7Erstan/papers/core.pdf) proved the enumeration $C\_s=\frac1{2s+1}\binom{2s+1}s$. Their argument utilizes a result of J. Anderson: $(s,s+1)$-cores correspond bijectively to the order ideals of
the poset $P(s,s+1)$ (that is, positive integers that are not contained in the numerical
semigroup generated by $s$ and $s+1$) by associating the lower ideal $\{a\_1,\dots,a\_j\}$, where $a\_1>\cdots>a\_j$, to the $(s,s+1)$-core partition $(a\_1-(j-1), a\_2-(j-2),\dots,a\_{j-1}-1,a\_j)$.


A few years ago, I considered core-partitions [**with distinct parts**](http://dauns01.math.tulane.edu/%7Etamdeberhan/conjectures.pdf) (see page 13) to the effect that the number of such $(s,s+1)$-core partitions equals the Fibonacci number $F\_{s+1}$. This conjecture is now a theorem due to different people ([see this paper](https://www.combinatorics.org/ojs/index.php/eljc/article/view/v25i1p57) and references therein).



> 
> **QUESTION.** Viewing $(s,s+1)$-cores of distinct partitions as embedded in the ordered ideals of the poset $P(s,s+1)$ (the above Catalan case), do they have any "interesting" structure?
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> 



Fibonacci embedded in Catalan?
The order ideal corresponding to a core partition with distinct parts cannot contain an element $x \geq s+2$, otherwise it must also contain $x-s, x-s-1$, resulting in two equal parts. For the same reason, it also can't contain two consecutive elements in $\{1,2,\dots, s-1\}$. Therefore the poset in questions is simply the poset of subsets of $\{1,2,\dots, s-1\}$ that do not contain consecutive elements, ordered by inclusion.


The underlying Hasse diagram of this poset is called the [Fibonacci cube](https://en.wikipedia.org/wiki/Fibonacci_cube).


