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.
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$\begingroup$ Is this a question? $\endgroup$– Anthony QuasCommented Dec 18, 2012 at 18:51
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$\begingroup$ The question is the title. $\endgroup$– Ira GesselCommented Dec 19, 2012 at 3:23
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$\begingroup$ Let $c_n$ be the number of maximal chains in each n-interval. Does the list of chain-counts $c_1,c_2,\cdots$ determine a binomial poset $P$ up to isomorphism? If not then it seems = we can take non-isomorphic $P$ and $P'$ with the same chain-counts and identify the $\hat{0}$ to get a new poset with the same counts and non-isomorphic intervals. $\endgroup$– Aaron MeyerowitzCommented Dec 19, 2012 at 7:09
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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.
answered Dec 20, 2012 at 2:22