### 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 \(\mathbb{Z}_2\)-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.

#### Keywords

Gram matrices, partition algebras, signed partition algebras and the algebra of \(\mathbb{Z}_2\)-relations

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