On products of permutations with the most uncontaminated cycles by designated labels

There is a growing interest in studying the distribution of certain labels products permutations since work Stanley addressing conjecture Bóna. This paper concerned with problem that direction. Let D be permutation on set $$[n]=\{1,2,\ldots , n\}$$ and $$E\subset [n]$$ . Suppose maximum possible number cycles uncontaminated by E-labels product cyclic [n] $$\theta $$ (depending E). We prove for arbitrary E few exceptions, $$\gamma such $$D\circ \gamma has exactly -1$$ E-label free at least 1/2 to have cycles, where best possible. An even more general result also conjectured.