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 $24\delta_k\equiv 1\pmod{5^k}$ .

References

[Andrews08]	G.E. Andrews, The number of smallest parts in the partitions of $n$ , J. Reine Angew. Math. 624 (2008), 133-142.
[ADY15]	G.E. Andrews, A. Dixit and A.J. Yee, Partitions associated with Ramanujan/Watson mock theta functions $\omega(q)$ , $\nu(q)$ and $\varphi(q)$ , Res. Number Theory (2015), 1-19.
[Chen17]	W.Y.C. Chen, The $\mathrm{spt}$ -Function of Andrews, In: A. Claesson, M. Dukes, S. Kitaev, D. Manlove and K. Meeks (eds.), Surveys in Combinatorics 2017, 141-203, Cambridge Univ. Press, Cambridge, 2017.
[GJ16]	F. Garvan and C. Jennings-Shaffer, Exotic Bailey-Slater SPT-functions II: Hecke-Rogers-type double sums and Bailey pairs from groups A, C, E, Adv. Math. 299 (2016), 605-639.
[Ramanujan19]	S. Ramanujan, Some properties of $p(n)$ , the number of partitions of $n$ , Proc. Cambridge Phil. Soc. 19 (1919), 207-210.
[Ramanujan88]	S. Ramanujan, The Lost Notebook and Other Unpublished Papers, Narosa Publishing House, New Delhi, 1988.
[Wang17]	L. Wang, New congruences for partitions related to mock theta functions, J. Number Theory 175 (2017), 51-65.
[WY18]	L. Wang and Y. Yang, The smallest parts function associated with $\omega(q)$ , arXiv:1812.00379.
[Watson36]	G.N. Watson, The final problem: an account of the mock theta functions, J. London Math. Soc. 11(1) (1936), 55-80.