permutationdepends

Permutationdepends is a formal framework in combinatorics and theoretical computer science that studies how the ordering of a finite set, as specified by a permutation, determines or constrains dependencies among its elements. In this framework, a permutation pi of a set S induces a dependency structure D_pi on S, where the admissibility or outcome for each element may depend on the elements that precede it in pi.

Formally, let S be a finite set and Pi(S) the set of all permutations of S. A permutationdepends specification consists of a family of local rules R_pi for each pi in Pi(S). Each rule associates to a current partial assignment of values to the subset of elements seen so far in pi a Boolean decision about whether the extension to the next element is allowed. The defining feature is that the dependency pattern can vary with pi rather than being fixed across all orders.

Example: consider S = {A, B, C}. Under pi = (A, B, C), C may be activated only after A has been activated; under pi = (B, A, C), C may be activated only after B has been activated. This shows how the same elements can exhibit different constraints under different permutations.

Properties include that the set of admissible assignments can change with pi, and certain symmetries of pi may yield equivalent dependency patterns. The framework relates to dependency graphs and sequential circuits with order-dependent constraints.

Applications span scheduling with order-sensitive prerequisites, reversible circuits where input order affects dependencies, and combinatorial puzzles where the legality of a move depends on the permutation of previous moves. See also: permutation, dependency graph, permutation pattern, sequential constraints.