Intersection matrices revisited

Details Intersection matrices revisited Intersection matrices revisited

2009-02-25

arxiv.org/abs/0902.4367v4

Mathematics

Combinatorics

17 pages, revised version

Scientific paper

Several intersection matrices of $s$-subsets vs. $k$-subsets of a $v$-set are introduced in the literature. We study these matrices systematically through counting arguments and generating function techniques. A number of new or known identities appear as natural consequences of this viewpoint; especially, appearance of the derivative operator $d/dz$ and some related operators reveals some connections between intersection matrices and the "combinatorics of creation-annihilation". As application, the eigenvalues of several intersection matrices including some generalizations of the adjacency matrices of the Johnson scheme are derived; two new bases for the Bose--Mesner algebra of the Johnson scheme are introduced and the associated intersection numbers are obtained as well. Finally, we determine the rank of some intersection matrices.

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