The number of elements in a generalized partition semilattice

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Let ∏n,k be the partially ordered set whose elements are all nonempty intersections of the affine hyperplanes Hi,j,r = {x ∈ ℝn: xi = xj + r} for integers i,j,k,r such that 1 ≤i, j ≤n and |r| ≤k, ordered by reverse inclusion. First we show that for a fixed k, the exponential generating function Mk(x) of the number of maximal elements in this poset is Mk(x)=ex - 1/(1 + k) - kex, and then from this, it follows immediately, using species, that the number of elements in this poset which have a given dimension d is the coefficient of tdxn/n! in Nk(x, t) = etMk(x). After we do this, we use the fact that Mk+1(x) can be expressed in terms of Mk(x) for each k to show that this implies that there is a bijection between the set of maximal elements of ∏n,k+1 and a certain other set. © 1998 Published by Elsevier Science B.V. All rights reserved.




Gill, R. (1998). The number of elements in a generalized partition semilattice. Discrete Mathematics, 186(1–3), 125–134.

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