Further combinatorial results for the symmetric inverse monoid

Let In be the set of partial one-to-one transformations on the chain Xn={1,2, . . . , n} and, for each α in In, let h(α)=|Imα|, f(α)=|{x∈Xn:xα=x}| and w(α)=max(Imα). In this note, we obtain formulae involving binomial coefficients of F(n; p, m, k)=|{α ∈ In:h(α)=p∧f(α)=m∧w(α)=k}| and F(n;·, m, k)=|{α ∈ In:f(α)=m∧w(α)=k}| and analogous results on the set of partial derangements of In.