On the exact number of possibilities for cutting and reconnecting the   tour of a traveling salesman with Lin-$k$-Opts
  When trying to find approximate solutions for the Traveling Salesman Problem with heuristic optimization algorithms, small moves called Lin-$k$-Opts are often used. In our paper, we provide exact formulas for the numbers of possible tours into which a randomly chosen tour can be changed with a Lin-$k$-Opt. 
Guenther Stattenberger, Markus Dankesreiter, Florian Baumgartner,   Johannes J. Schneider