Rearrangálás

Rearrangálás is a Hungarian term that translates to “rearrangement” in English. In mathematics it refers to the process of permuting the elements of a set or sequence. The concept is used in combinatorics, where the number of distinct rearrangements of a set of \(n\) distinct objects is \(n!\). When some elements are identical, the count is reduced by dividing by the factorials of the multiplicities, yielding the multinomial coefficient.

In the context of inequality theory, the rearrangement inequality is a classical result. It states that for two real sequences sorted in opposite order, the sum of the products of corresponding terms is minimal, whereas the sum is maximal when the sequences are sorted in the same order. This principle underlies many proofs in analysis and optimization.

Rearrangálás is also central to computer science, especially in sorting algorithms. The operation of rearranging an array into a specific order is the core of algorithms such as quicksort, mergesort, and heapsort. In data structures, rebalancing often involves a rearrangement of nodes to maintain optimal access times.

Outside of pure mathematics and computer science, rearrangement appears in linguistics, where word order changes can alter meaning or emphasis. In engineering, assembly lines sometimes rearrange components to improve workflow efficiency.

The term also arises in everyday usage, meaning the act of moving objects to new positions, e.g., rearranging furniture in a room. This broad sense captures both the abstract combinatorial notion and concrete physical actions.