Further Combinatorial Properties of the Symmetric Inverse Semigroup
نویسندگان
چکیده
Let Xn = {1, 2, · · · , n} and let α : Domα ⊆ Xn → Imα ⊆ Xn be a (partial) transformation on Xn. The height of a transformation α is h(α) = | Imα|, the right [left] waist of α is w+(α) = max( Imα) [w−(α) = min( Imα)], and fix of α is denoted by f(α), and defined by f(α) = |F (α)| = |{x ∈ Xn : xα = x}|. 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}|. 1 2
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