The level polynomials of the free distributive lattices

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Abstract

We show that there exist a set of polynomials {Lkk = 0, 1⋯} such that Lk(n) is the number of elements of rank k in the free distributive lattice on n generators. L0(n) = L1(n) = 1 for all n and the degree of Lk is k−1 for k⩾1. We show that the coefficients of the Lk can be calculated using another family of polynomials, Pj. We show how to calculate Lk for k = 1,…,16 and Pj for j = 0,…,10. These calculations are enough to determine the number of elements of each rank in the free distributive lattice on 5 generators a result first obtained by Church [2]. We also calculate the asymptotic behavior of the Lk's and Pj's.

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Supported in part by ONR Contract N00014-67-A-0298-0015.