On inequalities and linear relations for 7-core partitions
A partition of the positive integer \(n\) is said to be a \(t\)-core one if there are no hook numbers that are multiples of \(t\). Let \(a_t(n)\) be the number of \(t\)-core partitions of \(n\). Using Ramanujan's modular equations \textit{N. D. Baruah} and \textit{B. C. Berndt} [J. Comb. Theory, Ser. A 114, No. 6, 1024--1045 (2007; Zbl 1206.11132)] have obtained the recurrence \(a_5(4n+3)=a_5(2n+1)+2a_5(n)\) and \textit{A. Berkovich} and \textit{H. Yesilyurt} [Discrete Math. 308, No. 22, 5246--5259 (2008; Zbl 1223.05010)] have proved the inequalities \(a_7(2n+2)\geq 2a_7(n)\) and \(a_7(4n+6)\geq 10a_7(n)\). In this present paper the author generalizes these results and proves conjectures of Berkovich and Yesilyurt. The proofs are based on the theory of modular forms.
scientific article; zbMATH DE number 5701943
scientific article