The number of elements in a generalized partition semilattice

4Citations
Citations of this article
5Readers
Mendeley users who have this article in their library.

Abstract

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.

Cite

CITATION STYLE

APA

Gill, R. (1998). The number of elements in a generalized partition semilattice. Discrete Mathematics, 186(1–3), 125–134. https://doi.org/10.1016/S0012-365X(97)00187-8

Register to see more suggestions

Mendeley helps you to discover research relevant for your work.

Already have an account?

Save time finding and organizing research with Mendeley

Sign up for free