permutationwheres

Permutationwheres is a hypothetical concept used in combinatorics and constraint-based permutation problems. In this framework, a permutationwhere describes a set of allowed positions for each element in a permutation of the set [n] = {1,...,n}. A permutation π satisfies a permutationwhere W = (W1, W2, ..., Wn) if, for every i in [n], the value π(i) lies in the allowed set Wi, with the additional requirement that π is a bijection (a permutation).

Formally, Wi ⊆ [n] for each i, and π satisfies W if π(i) ∈ Wi for all i, and π is a permutation of [n]. This collection W is often represented as a bipartite constraint graph between positions i on the left and values j on the right, with an edge i–j present whenever j ∈ Wi. A permutation satisfies W exactly when it corresponds to a perfect matching in this graph. Equivalently, the problem reduces to counting permutationwheres as the number of perfect matchings, which is the value of the permanent of the adjacency matrix of the graph.

Examples help illustrate: if n = 3 with W1 = {1,2}, W2 = {2,3}, W3 = {1,3}, the satisfying permutations are those where π(1) ∈ {1,2}, π(2) ∈ {2,3}, π(3) ∈ {1,3} and all values are distinct.

Variants include forbidding specific (i, π(i)) pairs or imposing global constraints. Counting permutationwheres is generally #P-hard, but tractable in special structures or with approximation methods. Related concepts include permutation patterns, rook polynomials, and constrained assignment problems.