sequencesplaced

Sequencesplaced is a term used in combinatorics and puzzle design to denote the problem of placing multiple sequences into a fixed container, such as a line or grid, while preserving the internal order of each sequence and avoiding overlap between the sequences. The concept is often explored in abstract settings as well as in practical puzzle and packing contexts.

Formally, consider sequences s1, s2, ..., sk, where each si has length li and consists of elements drawn from a finite alphabet. A placement of the sequences into a line of length L is a collection of strictly increasing maps fi: {1, ..., li} -> {1, ..., L} for each i, such that the sets {fi(1), ..., fi(li)} are pairwise disjoint. Variants may impose additional constraints, such as minimum gaps between consecutive elements within a sequence or allowable overlap when symbols match.

The problem has several common forms. In the 1D version, all placements occur along a single line; in the 2D version, sequences are placed on a grid, potentially in multiple rows or columns. Computational questions include counting the number of valid placements, finding an arrangement that minimizes the overall span, or optimizing some objective under the placement constraints. These problems are typically combinatorial and can be NP-hard in general, though efficient solutions exist for restricted cases or fixed numbers of sequences.

Example: with a line of length 5, s1 = ABC (length 3) and s2 = DE (length 2). A valid placement is s1 at positions 1,2,3 and s2 at 4,5; another valid placement is s1 at 3,4,5 and s2 at 1,2.

See also: permutations, combinatorics, packing problems, sequence alignment, scheduling.