Ménage numbers: seat n couples around a circular table, men and women alternating, no husband next to his wife. M_n = 2·n!·Σ_{k=0}^{n}(-1)^k·(2n/(2n-k))·C(2n-k,k)·(n-k)!. M_3=12, M_4=96, M_5=3120.
How to Use
Enter n couples:
- M_n: Valid arrangements
- Formula: Inclusion-exclusion
- Sequence: First values
History
Posed by Édouard Lucas (1891). Touchard (1934) gave the closed formula. The problem combines circular permutations with a 'no-fixed-point' constraint, making it much harder than ordinary derangements. It's a masterpiece of inclusion-exclusion.
Connections
For linear tables: discordant permutations. The ménage numbers satisfy: M_n = n!·D_n - correction terms, where D_n = derangements. The sequence grows like 2·n!·e^{-2} for large n.
Step-by-Step Instructions
- 1Enter n.
- 2Compute M_n.
- 3View formula.
- 4Check sequence.
- 5Compare to n!.