Update on Conjectures 4.47 and 4.48

In their combinatorial study of the third order mock theta function $\omega(q)$ due to Ramanujan [Ramanujan88] and Watson [Watson36], Andrews, Dixit and Yee [ADY15] defined the partition function $p_\omega(n)$, closely related to $\omega(q)$, as the number of partitions of $n$ in which each odd part is less than twice the smallest part. Recall that \begin{align*} \omega(q)=\sum_{n=0}^\infty \frac{q^{2n^2+2n}}{(q; q^2)_{n+1}^2}. \end{align*} They showed that \[\sum_{n=1}^\infty p_\omega(n)q^n=q\omega(q).\] Let $\mathrm{spt}_\omega(n)$ be the number of smallest parts in the partitions enumerated by $p_{\omega}(n)$. Andrews, Dixit and Yee obtained that \begin{align}\label{GF-spt-pw} \sum_{n=1}^\infty\textrm{spt}_\omega(n)q^n =\sum_{n=1}^\infty \frac{q^n}{(1-q^n)^2(q^{n+1};q)_n(q^{2n+2};q^2)_\infty}. \end{align}

Wang [Wang17, Conjecture 4.1] posed the following conjectures (Conjectures 4.47 and 4.48 in [Chen17]):

Conjecture 1. For $k\geq 1$ and $n\geq 0$, we have \begin{align}\label{eq-conj1} \mathrm{spt}_{\omega}\left(2 \cdot 5^{2k-1}n+ \frac{7\cdot 5^{2k-1}+1}{12} \right) \equiv 0 \pmod{5^{2k-1}}. \end{align}

Conjecture 2. For $k\geq 1$ and $n\geq 0$, we have \begin{align}\label{eq-conj2} \mathrm{spt}_{\omega} \left(2 \cdot 5^{2k}n+\frac{11\cdot 5^{2k}+1}{12}\right) \equiv 0 \pmod{5^{2k}}. \end{align}

The above conjectures were confirmed by Wang and Yang [WY18] by establishing congruence relations for an $\mathrm{spt}$-type function $\mathrm{spt}_{C5}(n)$, introduced by Garvan and Jennings-Shaffer. More precisely, \begin{align*} \sum_{n=1}^\infty \mathrm{spt}_{C5}(n)q^n =\sum_{n=1}^\infty \frac{q^{(n^2+n)/2}}{(1-q^n)^2(q^{n+1};q)_n (q^{2n+2};q^2)_\infty}. \end{align*}

To connect $\mathrm{spt}_\omega(n)$ to $\mathrm{spt}_{C5}(n)$, we need another spt-type function $\mathrm{spt}_{C1}(n)$ defined by Garvan and Jennings-Shaffer [GJ16]: \begin{align}\label{GF-spt-C1} \sum_{n=1}^\infty \mathrm{spt}_{C1}(n)q^n=\sum_{n=1}^\infty \frac{q^n}{(1-q^n)^2(q^{n+1};q)_n(q^{2n+2};q^2)_\infty}. \end{align} As observed by Wang and Yang, we have \begin{align}\label{omega-C1} \mathrm{spt}_\omega(n)=\mathrm{spt}_{C1}(n), \end{align} since the generating function in \eqref{GF-spt-pw} coincides with the generating function in \eqref{GF-spt-C1}.

Garvan and Jennings-Shaffer [GJ16, Corollary 2.10] showed that \begin{align} \mathrm{spt}(n)=\mathrm{spt}_{C1}(2n)- \mathrm{spt}_{C5}(2n) \end{align} and \begin{align} \mathrm{spt}_{C1}(2n+1)=\mathrm{spt}_{C5}(2n+1) \label{spt-C1-C5}, \end{align} where $\mathrm{spt}(n)$ is the spt-function introduced by Andrews [Andrews08], namely, the total number of appearances of the smallest parts in partitions of $n$.

Combining \eqref{omega-C1} and \eqref{spt-C1-C5}, one sees that \begin{align}\label{omega-C5} \mathrm{spt}_\omega(2n+1)=\mathrm{spt}_{C5}(2n+1). \end{align}

Wang and Yang [WY18] established the following congruences for $\mathrm{spt}_{C5}(n)$.

Theorem 3. For $k\geq 1$ and $n\geq 0$, \begin{align} \mathrm{spt}_{C5}\left(5^{2k-1}n+\frac{7\cdot 5^{2k-1}+1}{12} \right) &\equiv 0 \pmod{5^{2k-1}}, \label{C5-cong-1}\\ \mathrm{spt}_{C5}\left( 5^{2k}n+\frac{11\cdot 5^{2k}+1}{12}\right) &\equiv 0 \pmod{5^{2k}}.\label{C5-cong-2} \end{align}

In view of \eqref{omega-C5}, Conjecture 1 follows from \eqref{C5-cong-1}, and Conjecture 2 follows from \eqref{C5-cong-2}.

To prove Theorem 3, Wang and Yang [WY18] considered the Fourier coefficients $c(n)$ defined by \begin{align*} \sum_{n=0}^\infty c(n)q^n=q^{1/12}\frac{2E_2(2\tau)-E_2(\tau)}{\eta(2\tau)}, \end{align*} where $E_2(\tau)$ is the weight 2 Eisenstein series \begin{align*} E_2(\tau)=1-24\sum_{n=1}^\infty\dfrac{nq^n}{1-q^n}, \quad q=e^{2\pi i\tau},\quad\mathrm{Im}\:\tau>0, \end{align*} and $\eta(\tau)$ is the Dedekind eta function \begin{align*} \eta(\tau)=q^{1/24}(q;q)_\infty. \end{align*}

They obtained the following relations: \begin{align}\label{C5-p1} 24\:\mathrm{spt}_{C5}(2n)=c(2n) +(24 n-1)p(n) \end{align} and \begin{align}\label{C5-p2} 24\:\mathrm{spt}_{C5}(2n+1)=c(2n+1), \end{align} where $p(n)$ is the partition function.

Using the theory of modular forms, Wang and Yang derived the following congruences for $c(n)$.

Theorem 4. For $k\geq 1$ and $n\geq 0$, \begin{align} c\left(5^{2k-1}n+\frac{7\cdot 5^{2k-1}+1}{12} \right) &\equiv 0 \pmod{5^{2k-1}},\label{cong-1-odd} \\ c\left(5^{2k}n+\frac{11\cdot 5^{2k}+1}{12} \right) &\equiv 0 \pmod{5^{2k}}. \label{cong-1-even} \end{align}

Theorem 3 follows from \eqref{C5-p1}—\eqref{cong-1-even} and the Ramanujan congruences [Ramanujan19] \begin{align}\label{con-Ram} p(5^k n+\delta_k)\equiv 0\pmod{5^k}, \end{align} where $\delta_k$ is the least nonnegative integer such that $24\delta_k\equiv 1\pmod{5^k}$.


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