On permutations involving pairs of twins (Q1186825)

We are concerned with permutations of a set of \(2n\) elements coupled into \(n\) pairs; the members of each pair are referred to as twins (two decks of cards serve as a model). Let \(p(n,k)\) be the probability that, in a randomly chosen permutation of such a set, exactly \(k\) pairs of twins are nonseparated (occupy neighbouring positions). To be more precise: assume \(S=\{a_ 1,b_ 1,a_ 2,b_ 2,\dots,a_ n,b_ n\}\) is the set, element \(a_ i\) matching \(b_ i\); define \(T(n,k)\) as the set of all maps \(\varphi\) of \(\{1,\dots,2n\}\) onto \(S\) such that \(|\varphi^{- 1}(a_ i)-\varphi^{-1}(b_ i)|=1\) holds for exactly \(k\) values of \(i\) (permutations regarded as enumerations). Then \(p(n,k)={1\over (2n)!}| T(n,k)|\) \((0\leq k\leq n)\) (here and in the sequel \(|\cdot|\) denotes the cardinality of a set). Evidently, \(\sum^ n_{k=0}p(n,k)=1\) for \(n=1,2,3,\dots\). It is not hard to show that \(p(n,0)<1/2\), for any \(n\). We examine the asymptotic behaviour of quantities \(p(n,k)\) for growing \(n\). Namely, we prove \(\lim_{n\to\infty}p(n,k)={1\over ek!}\) for \(k=0,1,2,\dots\).