GCD matrices, posets, and nonintersecting paths
  We show that with any finite partially ordered set one can associate a matrix whose determinant factors nicely. As corollaries, we obtain a number of results in the literature about GCD matrices and their relatives. Our main theorem is proved combinatorially using nonintersecting paths in a directed graph. 
Ercan Altinisik (Gazi U.), Bruce E. Sagan (Michigan State U.), Naim   Tuglu (Gazi U.)