---
tag: AEDP5ANYTW
kind: statement
role: proposition
name: Touchard's formula
sources:
  - wikipedia: en:21068755@r1337397118
    locator: Ménage problem, section Touchard's formula
    note: adopted
    mention: wikipedia://en/21068755#PHMN
    sections: [Statement]
inventory: [N77Y0G6022]
---

## Statement

Let *M*<sub>*n*</sub> denote the number of seating arrangements for *n* couples. (Touchard 1934) derived the formula

$$M_n = 2 \cdot n! \sum_{k=0}^n (-1)^k \frac{2n}{2n-k} {2n-k\choose k} (n-k)!.$$

Much subsequent work has gone into alternative [[proofs|0B94TKFQJX]] for this formula and into various generalized versions of the problem.

A different umbral formula for *M*<sub>*n*</sub> involving Chebyshev polynomials of first kind was given by (Wyman Moser 1958).