The semigroup algebra of the Catalan monoid is isomorphic to the incidence algebra of $P_n$, where $P_n$ is the poset consisting of subsets of { 1,...,n } where for two subsets $X \leq Y$ if and only if $X$ and $Y$ have the same cardinality and if X= {x_1 < ... < x_k } and Y= {y_1 < ... < y_k } we have $x_i \leq y_i$ for $i=1,...,k$. This is for example proved in https://arxiv.org/pdf/1806.06531.pdf. Recall that the width of a poset is the maximum size of an antichain.

I noted that the width of the poset $P_n$ starts with 2, 3, 4, 6, 8, 13, 20, 32, 52, 90, 152 for $n=1,...,11$ and this leads to the sequence https://oeis.org/A084239 of the rank of K-groups of the Furstenberg transformation group of the $n$-torus. See table 1 in https://arxiv.org/pdf/1109.4473.pdf .

Question 1: Is this true for all $n$? Is there a deeper explanation? Does the width have a homoogical interpretation for the Catalan monoid?

The poset $P_n$ has $n+1$ connected components, for each of the $k$-subsets and one can restrict to find antichains in those subsets and put them together. But I am more curious whether there is a deeper connection to the $K$-group sequence or is this just random? One might also ask about other nice properties of $P_n$. I noted that appending a minimum and maximum to $P_n$, one obtains a lattice.

The width of $P_n$ is equal to the maximal number of covers an element can have in the distributive lattice of order ideals $L(P_n)$ of $P_n$.

Question 2: Does the incidence algebra of $L(P_n)$ have an algebraic meaning in relation to the Catalan monoid?

Question 3: Is the Coxeter matrix of $L(P_n)$ periodic?

Question 3 has a positive answer for $n=1,2,3,4$ and the periods are given by 6,12,30,42 in that case.

(Small values suggest that also the Coxeter matrix of $P_n$ might be periodic, but that might be not so good evidence since the connected compoenent have not many points for small $n$.)

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