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A sequence $(P_0,P_1,\ldots)$ of finite posets is called uniform if: each $P_n$ is graded of rank $n$ with a minimum $\hat{0}_n$ and a maximum $\hat{1}_n$; for any $p \in P_n$ with $\mathrm{rank}(p)=n-i$ we have $[p,\hat{1}_n]\simeq P_i$. This is a slight adaptation of the definition that appears in Stanley, "Enumerative Combinatorics," Vol. 1, 2nd edition, Exercise 3.130: instead of defining "uniform" for a single poset, I am defining it for an infinite sequence.

Exercise 3.130(a) says that for $(P_0,P_1,\ldots)$ uniform, the matrices of Whitney numbers of the 1st and 2nd kind for the $P_i$ are inverses to one another (this generalizes the fact that the Stirling numbers of the 1st and 2nd kind are inverses, which is the case with $P_i=\Pi_{i+1}$, the lattice of set partitions of $\{1,2,\ldots,i+1\}$). Apparently this is a result of Dowling.

Exercise 3.130(b) asks for interesting examples of uniform posets.

Question: Are there any updates on this problem of finding interesting examples of (sequences of) uniform posets? Maybe even a classification under certain additional hypotheses?

Some examples I know:

  • $P_i=B_i$, the Boolean lattice of subsets of $\{1,2,\ldots,i\}$;
  • $P_i=B_i(q)$, the lattice of subspaces of $\mathbb{F}_q^{i}$, for $q=p^d$ a prime power and $\mathbb{F}_q$ the finite field with $q$ elements;
  • $P_i=\Pi_{i+1}$, the lattice of set partitions of $\{1,2,\ldots,i+1\}$, as mentioned before.
  • $P_i =$ the lattice of faces of the $i$-dimensional cross polytope (this is a kind of Type B version of the Boolean lattice).
  • $P_i =$ the bond lattice of the $i$-cycle graph (or more generally the lattice of flats of the uniform matroid).

I am interested in lattices, and especially non-Eulerian lattices (e.g., not face lattices) so that $|\mu(\hat{0}_n,\hat{1}_n)|$ grows unboundedly.

EDIT: I realized that I'm really interested in a certain combinatorial phenomenon connected to many of these examples, which I inquired about in a separate question.

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    $\begingroup$ The lattice of affine subspaces of $\mathbb{F}_{q}^{i}$ is another example. $\endgroup$
    – Joseph Van Name
    Commented Mar 1, 2022 at 20:06
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    $\begingroup$ By the affine subspaces, I mean the collection of all cosets of the vector subspaces of $\mathbb{F}_{q}^{i}$. The minimal object is $\emptyset$ while the maximal object is $\mathbb{F}_{q}^{i}$ itself. In the case where $q=p$ (so $q$ is prime), the affine subspaces are simply the subsets which are closed under the operation $(x,y,z)\mapsto x-y+z$. This operation is the heap operation on $F_{p}^{i}$ en.wikipedia.org/wiki/Heap_%28mathematics%29. $\endgroup$
    – Joseph Van Name
    Commented Mar 1, 2022 at 20:47
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    $\begingroup$ Let $G$ be a finite directly indecomposable group (directly indecomposable means that the group cannot be written as a direct product of subgroups). Let $P_{n}$ be the collection of all subgroups of $G^{n}$ that are internal direct factors of $G$. Then as a consequence of the Krull-Schmidt theorem, all groups in $P_{n}$ are isomorphic to the group $G^{m}$ for some $0\leq m\leq n$, so the posets $(P_{n})_{n}$ are uniform. $\endgroup$
    – Joseph Van Name
    Commented Mar 1, 2022 at 23:24
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    $\begingroup$ The Krull-Schmidt theorem also applies to modules. If $M$ is a finite indecomposable module, and $P_{n}$ is the collection of all submodules of $M^{n}$ that are direct factors of $M^{n}$, then $P_{n}$ is uniform since each element of $P^{n}$ is of the form $M^{m}$ for some $m\leq n$. One can produce more examples of uniform posets by letting $P_{n}$ denote the collection of all cosets of direct factors of $M^{n}$ (or of $G^{ n}$ in the group case). $\endgroup$
    – Joseph Van Name
    Commented Mar 1, 2022 at 23:31
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    $\begingroup$ Example 3.18.3 in EC1 leads to lots of uniform sequences of posets. In fact, they have the stronger property that all intervals of $P_n$ of length $m$ are isomorphic to $P_m$. Not all binomial posets have the property that all intervals of length $n$ are isomorphic, but those of Example 3.18.3 have this property if in (d) we take $P_1,\dots,P_k$ to have this property, and similarly for $P$ in (f). $\endgroup$
    – Richard Stanley
    Commented Mar 2, 2022 at 2:20

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