Further combinatorial results for the symmetric inverse monoid^∗

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