Gram matrices and Stirling numbers of a class of diagram algebras, I

Abstract

In this paper, we introduce Gram matrices for the signed partition algebras, the algebra of Z₂-relations and the partition algebras. The nondegeneracy and symmetic nature of these Gram matrices are establised. Also, (s₁, s₂, r₁, r₂, p₁, p₂)-Stirling numbers of the second kind for the signed partition algebras, the algebra of Z₂-relations are introduced and their identities are established. Stirling numbers of the second kind for the partition algebras are introduced and their identities are established.