Rearrangementek

Rearrangementek is a theoretical construct used in combinatorial optimization and discrete mathematics to analyze processes that reorder elements of a fixed multiset under a defined energy function. The term emphasizes the role of rearrangement operations as the fundamental actions driving a system toward lower energy configurations. It is typically explored as a framework rather than a single algorithm.

Formally, a rearrangementek consists of a finite sequence of elements x = (x1,...,xn), a set M of allowed moves (such as adjacent swaps or transpositions), and an energy function E that assigns a nonnegative real number to each arrangement. A state is a particular ordering of the multiset; a move maps one state to another. A sequence of moves is pursuing a descent if E decreases with every step. An arrangement is stable if no move in M reduces E; if E is convex or well-behaved, descent sequences terminate at a global minimum.

Examples include sorting processes where E is the number of inversions, and moves are adjacent swaps; every descent reduces the inversion count by one, yielding a sorted arrangement with minimum energy. More generally, rearrangementek can model local-search algorithms, Monte Carlo methods on permutations, or network-flow inspired rearrangements, depending on how E and M are defined.

Origins of the term are postulated in the early 2020s as a unifying label for studies of rearrangement-based optimization. The concept connects to related ideas such as the rearrangement inequality, permutation groups, and energy-based combinatorial optimization. See also rearrangement inequality, permutation, sorting network.