On a Conjecture of Andrews

Following G.E. Andrews, let $q^*_d(n)$ (resp. $Q^*_d(n)$) be the number of partitions of $n$ into $d$-distinct parts with difference at least $2d$ between multiples of $d$ (resp. into parts which are $\pm 1, \pm (d+2) \pmod{4d}$). Andrews conjectured that $q^*_d(n)-Q^*_d(n) \geq 0$ for all $n$. We prove that this conjecture is true for sufficiently large $n$ by establishing that $\lim_{n\to\infty}\left(q^*_d(n)-Q^*_d(n)\right)=+\infty.$

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Published 1 January 2010