The level polynomials of the free distributive lattices |
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Authors: | George Markowsky |
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Institution: | Computer Science Department, IBM Thomas J. Watson Research Center, Yorktown Heights, NY 10598, USA |
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Abstract: | We show that there exist a set of polynomials {Lk?k = 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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