Counting complements in the partition lattice, and hypertrees

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Abstract

Given two partitions π, σ of the set [n] = {1, ..., n} we call π and σ complements if their only common refinement is the partition {{1}, ..., {n}} and the only partition refined by both π and σ is {[n]}. If π = {A1, ..., Am} then we write ∣π∣ = m. We prove that the number of complements σ of π satisfying ∣σ∣ = nm + 1 is i=1mAi(nm+1)m2.

For the proof we assign to each σ a hypertree describing the pattern of intersections of blocks of π and σ and then count the number of hypertrees and the number of σ corresponding to each hypertree.

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Current address: Department of Mathematics, University of California at Los Angeles, Los Angeles, California 90024.