scholarly journals Symmetric Chain Decomposition of Necklace Posets

10.37236/1178 ◽  
2012 ◽  
Vol 19 (1) ◽  
Author(s):  
Vivek Dhand
Keyword(s):  
Disjoint Union ◽  
Cyclic Permutation ◽  
Chain Decomposition ◽  
Chain Order ◽  
Symmetric Chain Decomposition ◽  
Symmetric Chain ◽  
Symmetric Chain Order

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.

scholarly journals Symmetric Chain Decompositions of Products of Posets with Long Chains

10.37236/7124 ◽  
2018 ◽  
Vol 25 (1) ◽  
Author(s):  
Stefan David ◽  
Hunter Spink ◽  
Marius Tiba
Keyword(s):  
Minimal Element ◽  
Difficult Case ◽  
Chain Decomposition ◽  
Graded Poset ◽  
Canonical Bijection ◽  
Symmetric Chain Decomposition ◽  
Symmetric Chain ◽  
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.

scholarly journals Venn Diagrams and Symmetric Chain Decompositions in the Boolean Lattice

10.37236/1755 ◽  
2004 ◽  
Vol 11 (1) ◽  
Author(s):  
Jerrold Griggs ◽  
Charles E. Killian ◽  
Carla D. Savage
Keyword(s):  
Boolean Lattice ◽  
Constructive Proof ◽  
Existence Proof ◽  
Chain Decomposition ◽  
Venn Diagrams ◽  
Minimum Number ◽  
Symmetric Chain Decomposition ◽  
Open Question ◽  
Symmetric Chain

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.

Applications of the symmetric chain decomposition of the lattice of divisors

Order ◽  
1994 ◽  
Vol 11 (1) ◽  
pp. 41-46
Author(s):  
Jerrold R. Griggs ◽  
Chuanzhong Zhu
Keyword(s):  
Chain Decomposition ◽  
Symmetric Chain Decomposition ◽  
Symmetric Chain

scholarly journals Some Quotients of Chain Products are Symmetric Chain Orders

10.37236/2430 ◽  
2012 ◽  
Vol 19 (2) ◽  
Author(s):  
Dwight Duffus ◽  
Jeremy McKibben-Sanders ◽  
Kyle Thayer
Keyword(s):  
Automorphism Group ◽  
Symmetric Group ◽  
Boolean Lattice ◽  
Disjoint Cycles ◽  
Chain Order ◽  
Symmetric Chain ◽  
Straightforward Proof ◽  
Symmetric Chain Order

Canfield and Mason have conjectured that for all subgroups $G$ of the automorphism group of the Boolean lattice $B_n$ (which can be regarded as the symmetric group $S_n$), the quotient order $B_n/G$ is a symmetric chain order.  We provide a straightforward proof of a generalization of a result of K. K. Jordan: namely, $B_n/G$ is an SCO whenever $G$ is generated by powers of disjoint cycles. In addition, the Boolean lattice $B_n$ can be replaced by any product of finite chains. The symmetric chain decompositions of Greene and Kleitman provide the basis for partitions of these quotients.

scholarly journals Symmetric Chain Decompositions of Quotients by Wreath Products

10.37236/5073 ◽  
2015 ◽  
Vol 22 (2) ◽  
Author(s):  
Dwight Duffus ◽  
Kyle Thayer
Keyword(s):  
Symmetric Group ◽  
Elementary Proof ◽  
Boolean Lattice ◽  
Wreath Products ◽  
Cyclic Groups ◽  
Chain Order ◽  
A Chain ◽  
Symmetric Chain ◽  
Symmetric Chain Order

Subgroups of the symmetric group $S_n$ act on $C^n$ (the $n$-fold product $C \times \cdots \times C$ of a chain $C$) by permuting coordinates, and induce automorphisms of the power $C^n$. For certain families of subgroups of $S_n$, the quotients defined by these groups can be shown to have symmetric chain decompositions (SCDs). These SCDs allow us to enlarge the collection of subgroups $G$ of $S_n$ for which the quotient $\mathbf{2}^n/G$ on the Boolean lattice $\mathbf{2}^n$ is a symmetric chain order (SCO). The methods are also used to provide an elementary proof that quotients of powers of SCOs by cyclic groups are SCOs.

scholarly journals A Decomposition of Parking Functions by Undesired Spaces

10.37236/5940 ◽  
2016 ◽  
Vol 23 (3) ◽  
Author(s):  
Melody Bruce ◽  
Michael Dougherty ◽  
Max Hlavacek ◽  
Ryo Kudo ◽  
Ian Nicolas
Keyword(s):  
Parking Functions ◽  
Partition Lattice ◽  
Self Duality ◽  
Noncrossing Partition ◽  
Symmetric Chain Decomposition ◽  
Order Complexes ◽  
Maximal Chains ◽  
Symmetric Chain ◽  
Noncrossing Partition Lattice

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.

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