Antichains and chains of finite posets

Theorem 1.1. Let P be a finite poset with ]P = n. Take i ∈ P. For all positive integers i, let di be the maximal number of elements of P contained in a disjoint union of i chains in P , and define d0 = 0. Let also gi be the maximal number of elements of P contained in a disjoint union of i antichains in P for i > 0 and g0 = 0. Define δi = di − di−1 and γi = gi − gi−1 for all positive integers i. Then, the sequences δ = {δi} and γ = {γi} are non-increasing. Furthermore, the partitions of n defined by δ and γ are conjugate to each other.