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A non-crossing partition of the set $\mathbb{Z}_{mn}=\{ 1, 2,\dots,m n\}=\bigcup X_i$ is a disjoint union of sets such that

  • if $ a < b \in X_1$ and $ c < d \in X_2$ then we can't have $a < c < b < d$.

If additionally I require that $x \mapsto mx$ takes $X_i \to \{0, m, 2m, \dots, (n-1)m\}$ is there a name for this kind of partition?

I'd like to know more about the limit as $n \mapsto \infty$ which I've been calling "congruent non-crossing partitions of $S^1$"

  • n-tuples $(X_1, \dots, X_n)$ of disjoint unions of $ S^1$

  • $X_1 \cup \dots \cup X_n = S^1$ and $X_i \cap X_j = \varnothing$

  • the map $x \mapsto nx (\mod 1)$ takes $X_i \to S^1$

Is there a name for this space? Does it now have nice cell decomposition?

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  • $\begingroup$ What topology are you putting on it? $\endgroup$
    – HJRW
    Commented Sep 20, 2013 at 4:16
  • $\begingroup$ Unless I misunderstood your construction, then for each $n\geq2$, your 'space' of partitions is precisely two copies of $S^1$. Each $X_i$ looks like a half-open interval $[a_i,a_{i+1})$ or $(a_i,a_{i+1}]$ that covers precisely $1/n$ of $S^1$, so the whole partition is determined by $a_1$ and the choice of whether to use the first or second sort of half-open interval. As you haven't explained what topology you want to use, I don't know how you want to piece this together as $n\to\infty$. But, thinking of this as a collection of coverings of the circle, perhaps the limiting object is a solenoid? $\endgroup$
    – HJRW
    Commented Sep 20, 2013 at 9:48

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Am I right that you consider $x \mapsto mx$ modulo $mn$? This property is then equivalent to simply consider noncrossing partitions of $\{1,\ldots,mn\}$ where each block has size $n$.

Given any multiset of block sizes, Kreweras counted the number of noncrossing set partitions with these block sizes, see [Arm, Theorem 4.4.2].

Drew Armstrong studied then a variant of this in Section 3.3 thereof where he studied $n$-divisible noncrossing partitions where each block size is divisible by $n$.

In an upcoming paper, Christian Krattenthaler (see an abstract here) promised to study as well noncrossing partitions with completely fixed block sizes which are moreover rotationally invariant under $r^k$ for some $k$ where $r$ is the primitive rotation given by $x \mapsto x+1 (\text{mod }mn)$.

I have never heart of any limit construction for $n \rightarrow \infty$, but others might...

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