R.P. Stanley [Adv. Appl. Math. 34, No. 4, 880–902 (2005; Zbl 1097.06004)] introduced a partition statistic \(\operatorname{srank}(\pi)=\mathcal{O}(\pi)-\mathcal{O}(\pi^\prime)\), where \(\mathcal{O}(\pi)\) denotes the number of odd parts of the partition \(\pi\), and \(\pi^\prime\) is the conjugate of \(\pi\). Let \(p_i(n)\) denote the number of partitions of \(n\) with srank congruent to \(i\) modulo \(4\). G. E. Andrews [Electron. J. Comb. 11, No. 2, Research paper R1, 10 p. (2004; Zbl 1067.05005)] proved the following refinement of Ramanujan’s classical partition congruence modulo 5: \[ p_0(5n+4)\equiv p_2(5n+4)\equiv0\pmod{5}. \]
The authors introduce a new partition statistic lrank, given by \[ \operatorname{lrank}(\pi)=\mathcal{O}(\pi)+\mathcal{O}(\pi^\prime). \] Let \(p_i^{+}(n)\) denote the number of partitions of \(n\) with lrank congruent to \(i\) modulo \(4\). The authors establish the generating function of \(p_0^{+}(n)\) and \(p_2^{+}(n)\) and prove that they satisfy similar properties to \(p_i(n)\). As consequence, they prove that for any \(n\geq0\), \[ p_0^{+}(5n+4)\equiv p_2^{+}(5n+4)\equiv0\pmod{5}.\tag{1} \] Moreover, the authors also present a direct proof of (1) by utilizing the following two \(q\)-series identities: \begin{align*} \sum_{n=0}^\infty a(5n+4)q^n &=-5\sum_{n=0}^\infty b(n)q^{5n+3},\\ \sum_{n=0}^\infty b(5n+4)q^n &=-5\sum_{n=0}^\infty a(n)q^{5n+3}, \end{align*} where \begin{align*} \sum_{n=0}^\infty a(n)q^n &=\prod_{n=1}^\infty(1+q^{8n-5})^2(1+q^{8n-3})^2(1-q^{8n})^4,\\ \sum_{n=0}^\infty b(n)q^n &=q\prod_{n=1}^\infty(1+q^{8n-7})^2(1+q^{8n-1})^2(1-q^{8n})^4. \end{align*}