Report
Symmetric Chain Decompositions of Products of Posets with Long Chains
| العنوان: | Symmetric Chain Decompositions of Products of Posets with Long Chains |
|---|---|
| المؤلفون: | David, Stefan, Spink, Hunter, Tiba, Marius |
| المصدر: | Electronic Journal of Combinatorics (2018) Vol. 25 P1.69 |
| سنة النشر: | 2017 |
| المجموعة: | Mathematics |
| مصطلحات موضوعية: | Mathematics - Combinatorics, 06A07 |
| الوصف: | 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. Comment: 11 pages, 1 figure, 3 tables, fixed arXiv reference |
| نوع الوثيقة: | Working Paper |
| URL الوصول: | http://arxiv.org/abs/1706.08546 |
| رقم الانضمام: | edsarx.1706.08546 |
| قاعدة البيانات: | arXiv |
| الوصف غير متاح. |