A Permutation Cycle Arising from Alternating Position Sorting

I define and analyze a permutation transformation on the first positive integers, governed by alternating index sorting. The transformation constructs a new sequence by retaining the first element, then appending selected elements based on position: those at even-numbered indices (starting from index 2) are preserved in order, followed by elements at odd-numbered indices (starting from index 1) in reverse order. Iterating this process produces a deterministic cycle whose length depends on n. I explore the behavior of this cycle length, provide computational data for values up to , and raise questions about its underlying structure and potential periodicities.