### Gram matrices and Stirling numbers of a class of diagram algebras, II

#### Abstract

In the paper [6], we introduced Gram matrices for the signed partition algebras, the algebra of \(\mathbb{Z}_2\)-relations and the partition algebras. \((s_1, s_2, r_1, r_2, p_1, p_2)\)-Stirling numbers of the second kind are also introduced and their identities are established. In this paper, we prove that the Gram matrix is similar to a matrix which is a direct sum of block submatrices. As a consequence, the semisimplicity of a signed partition algebra is established.

#### Keywords

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

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