Let $Z$ be a finite set and $\mathcal S$ be a finite set of vectors in $Z^n$.


For each $z\in Z$ let $x\_z$ be a formal variable. A **distribution** (please excuse abuse of language) $d \in \mathbf N^n$ is a vector with nonnegative integer entries, not all zero, such that the set $\{\sum\_{i=1}^n d\_i x\_{s\_i} | s\in \mathcal S\}$ consists of a single element.


For example, if
$$\mathcal S = \{(1, 2, 2, 3), (1, 2, 3, 2), (2, 1, 3, 1)\},$$ then $d=(1,1,0,0)$ is a distribution, and so is $(1,0,1,1)$. If $$\mathcal S = \{(1,1,2,3),(1,2,1,3),(2,3,3,1)\},$$ then $(1,1,1,2)$ is a distribution.


A distribution $d$ is **minimal** if there is no distribution $d'$ such that $d'\_i \leq d\_i$ for all $i\in\{1,\dots,n\}$ and $d' \neq d$.


The distributions in the examples above are minimal.


I am looking for a set $\mathcal S$ with the following properties:


* there are no two positions $1\leq i < j\leq n$ with $s\_i = s\_j$ for all $s\in\mathcal S$.
* there are two minimal distributions $d\neq d'$ on $\mathcal S$ such that $d\_i > 0$ if and only if $d'\_i > 0$ for all $i\in\{1,\dots,n\}$.


Note that, without the first requirement, there are trivial examples of such sets.
For example, let $$\mathcal S = \{(1,1,2,2,2,1), (1,1,2,2,1,2), (1,1,2,1,2,2), (1,1,1,2,2,2), (2,2,1,1,1,1)\}.$$ Then $d = (2,1,1,1,1,1)$ and $d'=(1,2,1,1,1,1)$ are minimal distributions. The first condition is violated, because $s\_1 = s\_2$ for all $s\in\mathcal S$.



are minimal subdistributions determined by their support?
Extracted from the comments:


there are examples of the following type: $Z=\{0,1\}$, the set $\mathcal{S} $ consists of sequences $(t\_1,\ldots,t\_n)$ satisfying two different relations of the form $\sum c\_i t\_i=C$, where $c\_i$ are positive integers and $C$ is a positive integer (it is natural to choose it close to $\frac12 \sum c\_i$). Then $(c\_1,\ldots,c\_n)$ is a distribution, and it is easy to come up with example when two such distributions are both minimal and other conditions hold.


