Vol. 96, No. 2, 1981

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The fixed-point partition lattices

Philip Hanlon

Vol. 96 (1981), No. 2, 319–341
Abstract

Let σ be a permutation of the set {1,2,,n} and let Π(N) denote the lattice of partitions of {1,2,,n}. There is an obvious induced action of σ on Π(N); let Π(N)σ = L denote the lattice of partitions fixed by σ.

The structure of L is analyzed with particular attention paid to , the meet sublattice of L consisting of 1 together with all elements of L which are meets of coatoms of L. It is shown that is supersolvable, and that there exists a pregeometry on the set of atoms of whose lattice of flats G is a meet sublattice of . It is shown that G is supersolvable and results of Stanley are used to show that the Birkhoff polynomials B(λ) and BG(λ) are

BG (λ) = (λ− 1)(λ− j)⋅⋅⋅(λ − (m − 1)j)

and

B ℳ(λ) = (λ − 1)r−1BG (λ).

Here m is the number of cycles of σ, j is square-free part of the greatest common divisor of the lengths of σ and r is the number of prime divisors of j. coincides with G exactly when j is prime.

Mathematical Subject Classification 2000
Primary: 06C10
Secondary: 05B35
Milestones
Received: 16 July 1980
Published: 1 October 1981
Authors
Philip Hanlon
University of Michigan, Ann Arbor
Ann Arbor MI
United States