# Symmetric Chain Decompositions of Products of Posets with Long Chains

## TL;DR

This paper establishes conditions for the existence of symmetric chain decompositions without taut chains in product posets, resolving a key case in orthogonal chain decompositions and advancing conjectures in the field.

## Contribution

It proves the existence of taut-chain-free symmetric decompositions for certain product posets and introduces canonical bijections and surjections that preserve taut chains.

## Key findings

- Existence of taut-chain-free decompositions for k ≥ 5 and n ≥ 3
- Canonical bijections between decompositions of P×m and P×n for m,n ≥ rank(P)+1
- A surjection from decompositions of P×(rank(P)+1) to P×rank(P) preserving taut 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 $(a_1,\ldots, a_k,0)\prec \ldots\prec (a_1,\ldots,a_k,n-1)$. 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 \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.