  The initial purpose of this paper is to provide a combinatorial proof of the
minor summation formula of Pfaffians based on the lattice path method. There we
related Pl\"ucker relations with the minor summation formula of Pfaffians to
simplify its proof by lattice paths. The second aim is to find a various
applications of the minor summation formula. First we studied a variant of the
Sundquist formula established in J. Alg. Combin. {\bf 5} (1996). Next we gave a
simple proof of Kawanaka's formula concerning a $q$-series identity involving
Schur functions in Osaka J. Math. {\bf 36} (1999). We also establish a certain
identity similar to the Kawanaka formula, and give a combinatorial proof of the
determinant version of his formula in Osaka J. Math. {\bf 38} (2001).

Combinatorics;Representation Theory