Are there any binomial poset which has non-isomorphic interval of the same length? - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-20T22:40:19Z http://mathoverflow.net/feeds/question/116714 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/116714/are-there-any-binomial-poset-which-has-non-isomorphic-interval-of-the-same-length Are there any binomial poset which has non-isomorphic interval of the same length? Michael Zhong 2012-12-18T15:28:47Z 2012-12-20T02:44:34Z <p>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. </p> http://mathoverflow.net/questions/116714/are-there-any-binomial-poset-which-has-non-isomorphic-interval-of-the-same-length/116836#116836 Answer by Richard Stanley for Are there any binomial poset which has non-isomorphic interval of the same length? Richard Stanley 2012-12-20T02:22:46Z 2012-12-20T02:44:34Z <p>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 $\hat{0}&lt; x _0&lt; x _1&lt; \cdots$ 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 <a href="http://front.math.ucdavis.edu/0508.5397" rel="nofollow">http://front.math.ucdavis.edu/0508.5397</a>.</p>