Definition: A poset $P$ is called a binomial poset if it satisfy 
a. $P$ is locally finite with a $\hat{0}$, and contains a infinite chain.
b. Every interval $[x, y]$ of $P$ is graded. If $l(x,y)$ = n, then we call $[x,y]$ an 
n-interval. 
c. For all $n \in \mathbb{N}$, any two $n$-intervals contain the same number of maximal chains. 



Are there any binomial poset which has non-isomorphic interval of the same length?
To avoid creating new binomial posets by taking two of them with the
same factorial function (i.e., number of maximal chains in an $n$-interval)
and identifying their least elements, we
should add the extra condition that there exists a maximal chain
$\hat{0}< x \_0< x \_1< \cdots$ such that every element $x$ satisfies $x\leq
x\_n$ for some $n$. Backelin has constructed an uncountable number of nonisomorphic
such binomial posets, all with the same factorial function. Most of
these have the property that they contain two nonisomorphic intervals
of the same length. See


* Jörgen Backelin, *Binomial posets with non-isomorphic intervals*
<https://arxiv.org/abs/math/0508397>.


