Integer strings such as: 4,1,1,3,4,2,3,2

What you are describing are known as Langford sequences. An Internet search will give you http://legacy.lclark.edu/~miller/langford.html and other links.

(According to the Internet Archive that page has moved to http://dialectrix.com/langford.html).

Skolem or near Skolem sequences may also be of interest to you. I have a specialization of this I am studying: see Has anyone seen this version of ring toss (combinatorial object) before? .