Some parity results for 16-cores
A partition is said to be a \(t\)-core if its Ferrers graph does not contain a hook whose length is a multiple of \(t\). Let \(a_t(n)\) denote the number of \(t\)-cores of \(n\) and \((q)_\infty = \prod_{n\geq 1}(1-q^n).\) Using an identity discovered by F. G. Garvan, D. Kim and D. Stanton, namely, \[ \sum_{n\geq 0} a_t(n) q^n = \dfrac{(q^t)_\infty^t}{(q)_\infty}, \] the authors show that for \(\lambda\geq 1\) and \(n\geq 0\), \[ a_{16}\left(6561^\lambda n +\dfrac{7\cdot 6561^\lambda -255}{24}\right) \quad\text{and}\quad a_{16}\left(6561^\lambda n +\dfrac{23\cdot 6561^\lambda -255}{24}\right) \] are even.
scientific article; zbMATH DE number 1371465