Counting gradings on block-triangular matrix algebras

Let G be a finite group, let \(\mathbb {F}\) $\mathbb{F}$ be an arbitrary field and let \(\textbf{m}=(m_1,\dots , m_s)\) $\mathbf{m}=(m_1,\cdots ,m_s)$ be an s-tuple of positive integers. The number \(E(G,\textbf{m})\) $E(G,\mathbf{m})$ of isomorphism classes of the algebra \(UT(\textbf{m})\) $UT(\mathbf{m})$ over \(\mathbb {F}\) $\mathbb{F}$ is finite and \(E(G,m_1,\dots , m_s)\sim \frac{1}{|G|\cdot [(|G|-1)!]^{s}}(m_{1}\cdots m_{s})^{|G|-1}.\) $E(G,m_1,\cdots ,m_s)\sim \frac{1}{|G|·[(|G|-{1)!]}^s}{(m_1\cdots m_s)}^{|G|-1}.$ As a consequence of this we prove that an abelian group G is determined up to isomorphism by the s-sequence \(E(G,\cdot )\) $E(G,·)$ . If the field \(\mathbb {F}\) $\mathbb{F}$ is algebraically closed then the number \(N(G,\textbf{m})\) $N(G,\mathbf{m})$ of isomorphism classes of G-gradings \(UT(\textbf{m})\) $UT(\mathbf{m})$ is finite and \(N(G,\textbf{m})\sim E(G,\textbf{m})\) $N(G,\mathbf{m})\sim E(G,\mathbf{m})$ . The same result holds if the field \(\mathbb {F}\) $\mathbb{F}$ is finite or if G is abelian and \(\mathbb {F}=\mathbb {R}\) $\mathbb{F}=\mathbb{R}$ .