Reorderings

Reorderings refer to the operation or result of changing the arrangement of elements within a collection. In mathematics, a reordering of a finite sequence x1, x2, ..., xn is produced by a permutation π of {1,...,n}, yielding a reordered sequence x_{π(1)}, x_{π(2)}, ..., x_{π(n)}. Reorderings preserve the multiset of elements but not their positions. In combinatorics, counting reorderings corresponds to counting permutations; some problems impose constraints such as monotone or adjacent restrictions.

In order theory, a related concept is the linear extension or total ordering that reorders the elements to be compatible with a partial order; the set of all linear extensions is sometimes described as reorderings of the poset.

In sorting and data processing, reordering refers to arranging items into a specified order, such as nondecreasing. Sorting algorithms perform reordering and are classified by stability: a stable sort preserves the relative order of equal elements. Reordering is also important when combining sequences or aligning data, as the order of elements affects comparisons and computed statistics.

In probability and statistics, the rearrangement (or reordering) inequality describes how pairing of two sequences can maximize or minimize sums when one sequence is sorted in the opposite order of the other; permutation tests rely on reordering data to assess significance without strong distributional assumptions.

In computing and communications, reordering describes the correction of out-of-sequence data units, such as packets that arrive out of order, typically handled by buffers that reorder data to restore original sequencing.

See also: permutation, sorting, linear extension, rearrangement inequality, packet reordering.