User:Dvberkel/sandbox

Examples
We can define a equivalence relation $$L$$ on the Dyck language $$\mathcal{D}$$. For $$u,v\in\mathcal{D}$$ we have $$(u,v)\in L$$ if and only if $$|u| = |v|$$, i.e. $$u$$ and $$v$$ have the same length. This relation partitions the Dyck language $$\mathcal{D} / L = \mathcal{D}_{0} \cup \mathcal{D}_{2} \cup \mathcal{D}_{4} \cup \ldots = \bigcup_{n=0}^{\infty} \mathcal{D}_{n}$$ where $$\mathcal{D}_{n} = \{ u\in\mathcal{D} \mid |u| = n\}$$. Note that $$\mathcal{D}_{n}$$ is empty for odd $$n$$. E.g there are 14 words in $$\mathcal{D}_{8}$$, i.e., [][], , ][[], [][], ][][, [][], ][[], , []][, [][], [][[]][], [][][[]], [][][][].

Having introduced the Dyck words of length $$n$$, we can introduce a relationship on them. For every $$n \in \mathbb{N}$$ we define a relation $$S_{n}$$ on $$\mathcal{D}_{n}$$; for $$u,v\in\mathcal{D}_{n}$$ we have $$(u,v)\in S_{n}$$ if and only if $$v$$ can be reached from $$u$$ by a series of proper swaps. A proper swap in a word $$u\in\mathcal{D}_{n}$$ swaps an occurrence of '][' with '[]'. For each $$n\in\mathbb{N}$$ the relation $$S_{n}$$ makes $$\mathcal{D}_{n}$$ into a partially ordered set. The relation $$S_{n}$$ is reflexive because an empty sequence of proper swaps takes $$u$$ to $$u$$. Transitivity follows because we can extend a sequence of proper swaps that takes $$u$$ to $$v$$ by concatenating it with a sequence of proper swaps that takes $$v$$ to $$w$$ forming a sequence that takes $$u$$ into $$w$$. To see that $$S_{n}$$ is also antisymmetric we introduce an auxilliary function $$\sigma_{n}:\mathcal{D}_{n}\rightarrow\mathbb{N}:u\mapsto\sum_{vw=u} imbalance(v)$$ where $$v$$ ranges over all prefixes of $$u$$. The following table illustrates that $$\sigma_{n}$$ is strictly monotonic with respect to proper swaps.

Hence $$\sigma_{n}(u') - \sigma_{n}(u) = 2 > 0$$ so $$\sigma_{n}(u) < \sigma_{n}(u')$$ when there is a poper swap that takes $$u$$ into $$u'$$. Now if we assume that both $$(u,v), (v,u)\in S_{n}$$ and $$u\ne v$$, then there are non-empty sequences of proper swaps such $$u$$ is taken into $$v$$ and vice versa. But then $$\sigma_{n}(u) < \sigma_{n}(v) < \sigma_{n}(u)$$ which is nonsensical. Therefore, whenever both $$(u,v)$$ and $$(v,u)$$ are in $$S_{n}$$, we have $$u = v$$, hence $$S_{n}$$ is antisymmetric.

The partial ordered set $$D_{8}$$ is shown in the illustration accompanying the introduction.