Intersection matrices revisited

TL;DR

This paper systematically studies intersection matrices of subsets using counting and generating functions, revealing new identities, eigenvalues, bases, and ranks, and connecting to creation-annihilation combinatorics.

Contribution

It introduces new identities, eigenvalues, bases, and rank results for intersection matrices, linking combinatorics with algebraic and spectral properties of association schemes.

Findings

Derived eigenvalues for intersection matrices including Johnson scheme generalizations

Introduced two new bases for the Bose--Mesner algebra of the Johnson scheme

Determined the rank of several intersection matrices

Abstract

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.