Arithmetical properties of the number of \(t\)-core partitions
Let \(\Lambda=\{\lambda_1\geq\ldots\geq \lambda_s\geq 1 \}\) be a partition of an integer \(n\). The Ferrers-Young diagram of \(\Lambda\) is an array of nodes with \(\lambda_i\) nodes in the \(i\)th row. Let \(\lambda'_j\) denote the number of nodes in column \(j\) in the Ferrers--Young diagram of \(\Lambda\). The hook number of the \((i,j)\) node in the Ferrers-Young diagram of \(\Lambda\) is denoted by \(H(i,j):=\lambda_i+\lambda'_j-i-j+1\). A partition of \(n\) is called a \(t\)-core partition if none of the hook numbers is a multiple of \(t\). The number of \(t\)-core partitions of \(n\) is denoted by \(a(t;n)\). The author proves: Theorem 1. Let \(t\geq 3\) and \(l\geq 3\) be integers. Suppose that for each \(1\leq r\leq 2^l-1\) there exists an integer \(n_r\equiv 4^t-1\pmod{24}\) such that \[ a\left(2^t;\frac{n_r-4^t+1}{24}\equiv r\pmod{2^l}\right). \] Then \[ \#\{1\leq n\leq x\mid a(2^t;n)\equiv r\pmod{2^l}\}\gg \begin{cases} x &\text{if \(r=0\)},\\ \frac{\sqrt{x}}{\log x} &\text{if \(1\leq r\leq 2^l-1\).} \end{cases} \] Theorem 2. Let \(l\geq 1\) and \(t\geq 1\) be two integers. Suppose that \(m\) is a square-free odd integer with at least \(\frac{l(4^t-1)}{3}\) prime divisors. Then \[ a\left(2^t;\frac{nm-\frac{4^t-1}{3}}{8}\equiv 0\pmod{2^l}\right) \] for all \((n,m)=1\). Theorem 3. For any positive integers \(l,t\) we have \[ \sum_{m=0}^{t^2l-1}a(t;m)\sigma(t^2l-m)\equiv 0\pmod{t^2}, \] where \(\sigma(n)=\sum_{d|n}d\).
scientific article; zbMATH DE number 5636937
scientific article