155 通りのルーク配置を持つ順列格子

A permutation determines a crossword grid by placing one black square in each row and column. A complete rook placement chooses white squares so that every maximal horizontal or vertical white interval contains exactly one rook. Lewis and Won conjectured that the positive integers which occur as complete placement counts are precisely those other than 4, 12, and the integers congruent to 3 modulo 4. We disprove the proposed congruence restriction: the permutation 27481635 has exactly 155 complete rook placements. The count is obtained directly from the grid. Five rooks are forced; a further reversible reduction leaves a weighted bipartite graph with eight vertices in each part. Its matching count is a squarefree coefficient, which we evaluate as 116+39. We also give a bijection showing that prepending an initial fixed point preserves the number of complete placements. Repeatedly applying this operation produces counterexamples in every order at least eight. The argument uses only explicit word intervals, matching reductions, and a finite polynomial calculation.