Symmetric Chain Decompositions of Products of Posets with Long Chains

We ask if there exists a symmetric chain decomposition of the cuboid $Q_k \times n$ such that no chain is taut, i.e. no chain has a subchain of the form $(a_1,\ldots, a_k,0)\prec \cdots\prec (a_1,\ldots,a_k,n-1)$. In this paper, we show this is true precisely when $k \ge 5$ and $n\ge 3$. This question arises naturally when considering products of symmetric chain decompositions which induce orthogonal chain decompositions — the existence of the decompositions provided in this paper unexpectedly resolves the most difficult case of previous work by the second author on almost orthogonal symmetric chain decompositions (2017), making progress on a conjecture of Shearer and Kleitman (1979). In general, we show that for a finite graded poset $P$, there exists a canonical bijection between symmetric chain decompositions of $P \times m$ and $P \times n$ for $m, n\ge rk(P) + 1$, that preserves the existence of taut chains. If $P$ has a unique maximal and minimal element, then we also produce a canonical $(rk(P) +1)$ to $1$ surjection from symmetric chain decompositions of $P \times (rk(P) + 1)$ to symmetric chain decompositions of $P \times rk(P)$ which sends decompositions with taut chains to decompositions with taut chains.

A Decomposition of Parking Functions by Undesired Spaces

There is a well-known bijection between parking functions of a fixed length and maximal chains of the noncrossing partition lattice which we can use to associate to each set of parking functions a poset whose Hasse diagram is the union of the corresponding maximal chains. We introduce a decomposition of parking functions based on the largest number omitted and prove several theorems about the corresponding posets. In particular, they share properties with the noncrossing partition lattice such as local self-duality, a nice characterization of intervals, a readily computable Möbius function, and a symmetric chain decomposition. We also explore connections with order complexes, labeled Dyck paths, and rooted forests.

Symmetric Chain Decomposition of Necklace Posets

A finite ranked poset is called a symmetric chain order if it can be written as a disjoint union of rank-symmetric, saturated chains. If $\mathcal{P}$ is any symmetric chain order, we prove that $\mathcal{P}^n/\mathbb{Z}_n$ is also a symmetric chain order, where $\mathbb{Z}_n$ acts on $\mathcal{P}^n$ by cyclic permutation of the factors.

Venn Diagrams and Symmetric Chain Decompositions in the Boolean Lattice

We show that symmetric Venn diagrams for $n$ sets exist for every prime $n$, settling an open question. Until this time, $n=11$ was the largest prime for which the existence of such diagrams had been proven, a result of Peter Hamburger. We show that the problem can be reduced to finding a symmetric chain decomposition, satisfying a certain cover property, in a subposet of the Boolean lattice ${\cal B}_n$, and prove that such decompositions exist for all prime $n$. A consequence of the approach is a constructive proof that the quotient poset of ${\cal B}_n$, under the relation "equivalence under rotation", has a symmetric chain decomposition whenever $n$ is prime. We also show how symmetric chain decompositions can be used to construct, for all $n$, monotone Venn diagrams with the minimum number of vertices, giving a simpler existence proof.

Symmetric Chain Decompositions of Linear Lattices

It has been known for several years that the lattice of subspaces of a finite vector space has a decomposition into symmetric chains, i.e. a decomposition into disjoint chains that are symmetric with respect to the rank function of the lattice. This paper gives a positive answer to the long-standing open problem of providing an explicit construction of such a symmetric chain decomposition for a given lattice of subspaces of a finite (dimensional) vector space. The construction is done inductively using Schubert normal forms and results in a bracketing algorithm similar to the well-known algorithm for Boolean lattices.