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Graphs and Combinatorics

, Volume 9, Issue 2–4, pp 117–134 | Cite as

A problem related to Foulkes’s conjecture

  • C. Coker
Graphs and Combinatoric

Abstract

In this paper we study a class of symmetric matricesT indexed by positive integers m≥ n≥2 and defined as follows: for any positive integersp andq let ℬp,q be the set of partitions ofU = {1,2,3, ...,pq} into p blocks each of sizeq. Letmn ≥ 2 be positive integers. By atransversal of α = A1/A2/.../An ∈ ℬn,m we mean a partitionß = B1/B2/.../Bm m,n such that ‖A i B j = 1 for every i= 1,2, ...,n and everyj = 1,2, ...,m. LetM be the zero-one matrix with rows indexed by the elements of ℬn,m and columns indexed by the elements of ℬm,n such that Mαß = 1 iffß is a transversal of α. We are interested in finding the eigenvalues and eigenspaces of the symmetric matrixT = MMt. The nonsingularity ofT implies Foulkes’s Conjecture (for these values of m andn). In the casen = 2 we completely determine the eigenvalues and eigenspaces of T and in so doing demonstrate the non-singularity ofT. Forn = 3 we develop a fast algorithm for computing the eigenvalues ofT, and give numerical results in the cases m = 3,4, 5, 6.

Keywords

Symmetric Group Irreducible Character Wreath Product Association Scheme Real Vector Space 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Copyright information

© Springer-Verlag 1993

Authors and Affiliations

  • C. Coker
    • 1
  1. 1.SUNY College at OneontaOneontaUSA

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