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

In this paper, we introduce Gram matrices for the signed partition algebras, the algebra of $\mathbb{Z}_2$-relations and the partition algebras. The nondegeneracy and symmetic nature of these Gram matrices are establised. Also, $(s_1, s_2, r_1, r_2, p_1, p_2)$-Stirling numbers of the second kind for the signed partition algebras, the algebra of $\mathbb{Z}_2$-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.