Center symmetry of incidence matrices (Q2706482)

Let \(A\) be an algebra over a field \(F\) of characteristic \(0\) and let \(n>0\) be an integer. The Nagata-Higman theorem states that, if \(a^{n}=0\) for all \(a\in A,\) then \(A\) is nilpotent [see \textit{M. Nagata}, J. Math. Soc. Japan 4, 296-301 (1952; Zbl 0049.02402)]. It has been proved that the best possible value \(\mathcal{N} (n)\) of the index of nilpotence of \(A\) lies between \(n(n+1)/2\) and \(n^{2}\) but the precise value is not known in general. Let \(p_{n}(m)\) be the number of partitions of \(m\) into \(n\) parts. The author of the present paper observes that \(\mathcal{N}(n)\) is equal to the least integer \(m\) such that the rank of a certain \(p_{n}(m)\cdot m!\times m!\) incidence matrix has rank \(m!\), and finds some elementary properties of this matrix.