On the number of partitions into an even and odd number of parts (Q2785028)

Let \(q_i^e(n)\) and \(q_i^o(n)\) denote the number of partitions of \(n\) into an even and odd number of parts, respectively, where each part occurs at most \(i\) times, and put \(\Delta_i(n)=q_i^e(n)-q_i^o(n)\). Formulae for \(\Delta_i(n)\) with \(i\geq 2\) were obtained by Hickerson, Alder and Muwafi in 1970's. In this paper the author presents new expressions for \(\Delta_i(n)\) with \(i\) odd. Put \(\omega(j)=j(3j-1)/2\), and let \(b_r(n)\) be the number of \(r\)-regular partitions of \(n\), that is, the number of partitions of \(n\) such that each part occurs less than \(r\) times. Define \(b_r(z)=0\) when \(z\) is not a nonnegative integer. Then, by handling several product formulae concerning partitions, it is shown for \(r\geq 2\) that NEWLINE\[NEWLINE \Delta_{2r-1}(n)=b_r(n/2)+ \sum_{k\geq 1}(-1)^k\{b_r((n- \omega(k))/2)+ b_r((n-\omega(-k))/2)\}. NEWLINE\]NEWLINE From this formula with \(r=2\), moreover, the author derives some recurrence formulae for the familiar partition function \(q(n)\), that is the same thing as \(b_2(n)\).