I am going through the solution of relaxed Ménage problem which is given https://math.dartmouth.edu/~doyle/docs/menage/menage/menage.html where he solved using Principle of Inclusion Exclusion.
While calculating the value of $w_k$ which is the number of ways of seating the couples such that k specified couples sit together, He expanded the expression as $d_k \times k! \times 2^k \times (2n-2k)!$ He interpreted $d_k$ using some dominos which I know nothing and I also don't understand how the terms after $d_k$ also came into the expression. Can some one explain the value of $w_k$ elaborately in simple terms.