OEIS A000181: Coefficients of ménage hit polynomials

We explore the classic ménage problem—seating n couples around a circular table with men and women alternating and no adjacent partners. We spotlight OEIS A000181, the ménage hit polynomials, whose coefficients give a refined count: 2, 15, 60, 469, ... by how many couples sit together. We’ll trace the history to Lucas and Tate, explain the connection to constrained permutations, and peek at deeper math: recurrences (Mothar), divisibility conjectures (Van Hooge), and links to matrices and knot theory, all woven together in the OEIS tapestry.

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