Symmetric Chain Decompositions of Products of Posets with Long Chains

Abstract

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 $(a_1,\dots ,a_k,0)\prec \dots \prec (a_1,\dots ,a_k,n-1)$. 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 \cite{orth}, making progress on a conjecture of Shearer and Kleitman. 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 $m,n\ge \text{rk}(P)+1$, that preserves the existence of taut chains. If $P$ has a unique maximal and minimal element, then we also produce a canonical $(\text{rk}(P)+1)$ to $1$ surjection from symmetric chain decompositions of $P\times (\text{rk}(P)+1)$ to symmetric chain decompositions of $P\times \text{rk}(P)$ which sends decompositions with taut chains to decompositions with taut chains.

arXiv comment11 pages, 1 figure, 3 tables, fixed arXiv reference