GCD matrices, posets, and nonintersecting paths

We show that with any finite partially ordered set P (which need not be a lattice) one can associate a matrix whose determinant factors nicely. This was also noted by D.A. Smith, although his proof uses manipulations in the incidence algebra of P while ours is combinatorial, using nonintersecting pa...

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Bibliographic Details
Published in:Linear & multilinear algebra 2005-03, Vol.53 (2), p.75-84
Main Authors: Altinisik, Ercan, Sagan, Bruce E., Tuglu, Naim
Format: Article
Language:English
Subjects:
1991 Mathematics Subject Classifications: Primary 11C20
Determinant
GCD matrix
Nonintersecting paths
Poset
Secondary 05E99
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Summary:We show that with any finite partially ordered set P (which need not be a lattice) one can associate a matrix whose determinant factors nicely. This was also noted by D.A. Smith, although his proof uses manipulations in the incidence algebra of P while ours is combinatorial, using nonintersecting paths in a digraph. As corollaries, we obtain new proofs for and generalizations of a number of results in the literature about GCD matrices and their relatives.
ISSN:0308-1087
1563-5139
DOI:10.1080/03081080500054612