William Y. C. Chen
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William Y. C. Chen has a total of 118 co-authors. Here are the top co-authors by number of shared publications:
- Larry X. W. Wang
Nankai University
15 shared publications - Arthur Li Bo Yang
Center for Combinatorics, LPMC, Nankai University, Tianjin, China
12 shared publications - Kathy Qing Ji
Center for Applied Mathematics KL-AAGDM Tianjin University 300072 Tianjin P.R. China
12 shared publications - Lisa Hui Sun
Center for Combinatorics LPMC Nankai University 300071 Tianjin P.R. China
8 shared publications - Qing-Hu Hou
Center for Applied Mathematics, Tianjin University, Tianjin, P.R. China
8 shared publications - Ernest X. W. Xia
Center for Combinatorics, LPMC-TJKLC, Nankai University, Tianjin 300071, People’s Republic of China
6 shared publications
Latest Publications
- Open Access
- 26 December 2018
- Published by American Mathematical Society (AMS) in Transactions of the American Mathematical Society
- Vol. 372 (3) , 2143-2165
Higher order Turán inequalities for the partition function
The Turán inequalities and the higher order Turán inequalities arise in the study of the Maclaurin coefficients of real entire functions in the Laguerre–Pólya class. A sequence { a n } n ≥ 0 \{a_{n}\}_{n\geq 0} of real numbers is said to satisfy the Turán inequalities or to be log-concave if for n ≥ 1 n\geq 1 , a n 2 − a n − 1 a n + 1 ≥ 0 a_n^2-a_{n-1}a_{n+1}\geq 0 . It is said to satisfy the higher order Turán inequalities if for n ≥ 1 n\geq 1 , 4 ( a n 2 − a n − 1 a n + 1 ) ( a n + 1 2 − a n a n + 2 ) − ( a n a n + 1 − a n − 1 a n + 2 ) 2 ≥ 0 4(a_{n}^2-a_{n-1}a_{n+1})(a_{n+1}^2-a_{n}a_{n+2})-(a_{n}a_{n+1}-a_{n-1}a_{n+2})^2\geq 0 . For the partition function p ( n ) p(n) , DeSalvo and Pak showed that for n > 25 n>25 , the sequence { p ( n ) } n > 25 \{ p(n)\}_{n> 25} is log-concave, that is, p ( n ) 2 − p ( n − 1 ) p ( n + 1 ) > 0 p(n)^2-p(n-1)p(n+1)>0 for n > 25 n> 25 . It was conjectured by the first author that p ( n ) p(n) satisfies the higher order Turán inequalities for n ≥ 95 n\geq 95 . In this paper, we prove this conjecture by using the Hardy–Ramanujan–Rademacher formula to derive an upper bound and a lower bound for p ( n + 1 ) p ( n − 1 ) / p ( n ) 2 p(n+1)p(n-1)/p(n)^2 . Consequently, for n ≥ 95 n\geq 95 , the Jensen polynomials p ( n − 1 <mo...
- 26 December 2018
- Published by American Mathematical Society (AMS) in Transactions of the American Mathematical Society
- Vol. 372 (3) , 2143-2165
Open AccessHigher order Turán inequalities for the partition function
The Turán inequalities and the higher order Turán inequalities arise in the study of the Maclaurin coefficients of real entire functions in the Laguerre–Pólya class. A sequence { a n } n ≥ 0 \{a_{n}\}_{n\geq 0} of real numbers is said to satisfy the Turán inequalities or to be log-concave if for n ≥ 1 n\geq 1 , a n 2 − a n − 1 a n + 1 ≥ 0 a_n^2-a_{n-1}a_{n+1}\geq 0 . It is said to satisfy the higher order Turán inequalities if for n ≥ 1 n\geq 1 , 4 ( a n 2 − a n − 1 a n + 1 ) ( a n + 1 2 − a n a n + 2 ) − ( a n a n + 1 − a n − 1 a n + 2 ) 2 ≥ 0 4(a_{n}^2-a_{n-1}a_{n+1})(a_{n+1}^2-a_{n}a_{n+2})-(a_{n}a_{n+1}-a_{n-1}a_{n+2})^2\geq 0 . For the partition function p ( n ) p(n) , DeSalvo and Pak showed that for n > 25 n>25 , the sequence { p ( n ) } n > 25 \{ p(n)\}_{n> 25} is log-concave, that is, p ( n ) 2 − p ( n − 1 ) p ( n + 1 ) > 0 p(n)^2-p(n-1)p(n+1)>0 for n > 25 n> 25 . It was conjectured by the first author that p ( n ) p(n) satisfies the higher order Turán inequalities for n ≥ 95 n\geq 95 . In this paper, we prove this conjecture by using the Hardy–Ramanujan–Rademacher formula to derive an upper bound and a lower bound for p ( n + 1 ) p ( n − 1 ) / p ( n ) 2 p(n+1)p(n-1)/p(n)^2 . Consequently, for n ≥ 95 n\geq 95 , the Jensen polynomials p ( n − 1 <mo...
- Open Access
The spt-Function of Andrews
The spt-function spt($n$) was introduced by Andrews as the weighted counting of partitions of $n$ with respect to the number of occurrences of the smallest part. In this survey, we summarize recent developments in the study of spt($n$), including congruence properties established by Andrews, Bringmann, Folsom, Garvan, Lovejoy and Ono et al., a constructive proof of the Andrews-Dyson-Rhoades conjecture given by Chen, Ji and Zang, generalizations and variations of the spt-function. We also give an overview of asymptotic formulas of spt($n$) obtained by Ahlgren, Andersen and Rhoades et al. We conclude with some conjectures on inequalities on spt($n$), which are reminiscent of those on $p(n)$ due to DeSalvo and Pak, and Bessenrodt and Ono. Furthermore, we observe that, beyond the log-concavity, $p(n)$ and spt($n$) satisfy higher order inequalities based on polynomials arising in the invariant theory of binary forms. In particular, we conjecture that the higher order Tur\'{a}n inequality $4(a_n^2-a_{n-1}a_{n+1})(a_{n+1}^2-a_{n}a_{n+2})-(a_na_{n+1}-a_{n-1}a_{n+2})^2>0$ holds for $p(n)$ when $n\geq 95$ and for spt($n$) when $n\geq 108$.Open AccessThe spt-Function of Andrews
The spt-function spt($n$) was introduced by Andrews as the weighted counting of partitions of $n$ with respect to the number of occurrences of the smallest part. In this survey, we summarize recent developments in the study of spt($n$), including congruence properties established by Andrews, Bringmann, Folsom, Garvan, Lovejoy and Ono et al., a constructive proof of the Andrews-Dyson-Rhoades conjecture given by Chen, Ji and Zang, generalizations and variations of the spt-function. We also give an overview of asymptotic formulas of spt($n$) obtained by Ahlgren, Andersen and Rhoades et al. We conclude with some conjectures on inequalities on spt($n$), which are reminiscent of those on $p(n)$ due to DeSalvo and Pak, and Bessenrodt and Ono. Furthermore, we observe that, beyond the log-concavity, $p(n)$ and spt($n$) satisfy higher order inequalities based on polynomials arising in the invariant theory of binary forms. In particular, we conjecture that the higher order Tur\'{a}n inequality $4(a_n^2-a_{n-1}a_{n+1})(a_{n+1}^2-a_{n}a_{n+2})-(a_na_{n+1}-a_{n-1}a_{n+2})^2>0$ holds for $p(n)$ when $n\geq 95$ and for spt($n$) when $n\geq 108$. - Open Access
Stanley's Lemma and Multiple Theta Functions
We present an algorithmic approach to the verification of identities on multiple theta functions in the form of products of theta functions $[(-1)^{\delta}a_1^{\alpha_1}a_2^{\alpha_2}\cdots a_r^{\alpha_r}q^{s}; q^{t}]_\infty$, where $\alpha_i$ are integers, $\delta=0$ or $1$, $s\in \mathbb{Q}$, $t\in \mathbb{Q}^{+}$, and the exponent vectors $(\alpha_1,\alpha_2,\ldots,\alpha_r)$ are linearly independent over $\mathbb{Q}$. For an identity on such multiple theta functions, we provide an algorithmic approach for computing a system of contiguous relations satisfied by all the involved multiple theta functions. Using Stanley's Lemma on the fundamental parallelepiped, we show that a multiple theta function can be determined by a finite number of its coefficients. Thus such an identity can be reduced to a finite number of simpler relations. Many classical multiple theta function identities fall into this framework, including Riemann's addition formula and the extended Riemann identity.Open AccessStanley's Lemma and Multiple Theta Functions
We present an algorithmic approach to the verification of identities on multiple theta functions in the form of products of theta functions $[(-1)^{\delta}a_1^{\alpha_1}a_2^{\alpha_2}\cdots a_r^{\alpha_r}q^{s}; q^{t}]_\infty$, where $\alpha_i$ are integers, $\delta=0$ or $1$, $s\in \mathbb{Q}$, $t\in \mathbb{Q}^{+}$, and the exponent vectors $(\alpha_1,\alpha_2,\ldots,\alpha_r)$ are linearly independent over $\mathbb{Q}$. For an identity on such multiple theta functions, we provide an algorithmic approach for computing a system of contiguous relations satisfied by all the involved multiple theta functions. Using Stanley's Lemma on the fundamental parallelepiped, we show that a multiple theta function can be determined by a finite number of its coefficients. Thus such an identity can be reduced to a finite number of simpler relations. Many classical multiple theta function identities fall into this framework, including Riemann's addition formula and the extended Riemann identity. - Open Access
Higher Order Turán Inequalities for the Partition Function
The Tur\'{a}n inequalities and the higher order Tur\'{a}n inequalities arise in the study of Maclaurin coefficients of an entire function in the Laguerre-P\'{o}lya class. A real sequence $\{a_{n}\}$ is said to satisfy the Tur\'{a}n inequalities if for $n\geq 1$, $a_n^2-a_{n-1}a_{n+1}\geq 0$. It is said to satisfy the higher order Tur\'{a}n inequalities if for $n\geq 1$, $4(a_{n}^2-a_{n-1}a_{n+1})(a_{n+1}^2-a_{n}a_{n+2})-(a_{n}a_{n+1}-a_{n-1}a_{n+2})^2\geq 0$. A sequence satisfying the Tur\'an inequalities is also called log-concave. For the partition function $p(n)$, DeSalvo and Pak showed that for $n>25$, the sequence $\{ p(n)\}_{n> 25}$ is log-concave, that is, $p(n)^2-p(n-1)p(n+1)>0$ for $n> 25$. It was conjectured by Chen that $p(n)$ satisfies the higher order Tur\'{a}n inequalities for $n\geq 95$. In this paper, we prove this conjecture by using the Hardy-Ramanujan-Rademacher formula to derive an upper bound and a lower bound for $p(n+1)p(n-1)/p(n)^2$. Consequently, for $n\geq 95$, the Jensen polynomials $g_{3,n-1}(x)=p(n-1)+3p(n)x+3p(n+1)x^2+p(n+2)x^3$ have only real zeros. We conjecture that for any positive integer $m\geq 4$ there exists an integer $N(m)$ such that for $n\geq N(m) $, the polynomials $\sum_{k=0}^m {m\choose k}p(n+k)x^k$ have only real zeros. This conjecture was independently posed by Ono.Open AccessHigher Order Turán Inequalities for the Partition Function
The Tur\'{a}n inequalities and the higher order Tur\'{a}n inequalities arise in the study of Maclaurin coefficients of an entire function in the Laguerre-P\'{o}lya class. A real sequence $\{a_{n}\}$ is said to satisfy the Tur\'{a}n inequalities if for $n\geq 1$, $a_n^2-a_{n-1}a_{n+1}\geq 0$. It is said to satisfy the higher order Tur\'{a}n inequalities if for $n\geq 1$, $4(a_{n}^2-a_{n-1}a_{n+1})(a_{n+1}^2-a_{n}a_{n+2})-(a_{n}a_{n+1}-a_{n-1}a_{n+2})^2\geq 0$. A sequence satisfying the Tur\'an inequalities is also called log-concave. For the partition function $p(n)$, DeSalvo and Pak showed that for $n>25$, the sequence $\{ p(n)\}_{n> 25}$ is log-concave, that is, $p(n)^2-p(n-1)p(n+1)>0$ for $n> 25$. It was conjectured by Chen that $p(n)$ satisfies the higher order Tur\'{a}n inequalities for $n\geq 95$. In this paper, we prove this conjecture by using the Hardy-Ramanujan-Rademacher formula to derive an upper bound and a lower bound for $p(n+1)p(n-1)/p(n)^2$. Consequently, for $n\geq 95$, the Jensen polynomials $g_{3,n-1}(x)=p(n-1)+3p(n)x+3p(n+1)x^2+p(n+2)x^3$ have only real zeros. We conjecture that for any positive integer $m\geq 4$ there exists an integer $N(m)$ such that for $n\geq N(m) $, the polynomials $\sum_{k=0}^m {m\choose k}p(n+k)x^k$ have only real zeros. This conjecture was independently posed by Ono. - 17 February 2016
- Published by Taylor & Francis in Journal of Difference Equations and Applications
- Vol. 22 (6) , 780-788
Automated discovery and proof of congruence theorems for partial sums of combinatorial sequences
Many combinatorial sequences (e.g. the Catalan and the Motzkin numbers) may be expressed as the constant term of , for some Laurent polynomials P(x) and Q(x) in the variable x with integer coefficients. Denoting such a sequence by , we obtain a general formula that determines the congruence class, modulo p, of the indefinite sum , for any prime p, and any positive integer r, as a linear combination of sequences that satisfy linear recurrence (alias difference) equations with constant coefficients. This enables us (or rather, our computers) to automatically discover and prove congruence theorems for such partial sums. Moreover, we show that in many cases, the set of the residues is finite, regardless of the prime p.- 17 February 2016
- Published by Taylor & Francis in Journal of Difference Equations and Applications
- Vol. 22 (6) , 780-788
Automated discovery and proof of congruence theorems for partial sums of combinatorial sequences
Many combinatorial sequences (e.g. the Catalan and the Motzkin numbers) may be expressed as the constant term of , for some Laurent polynomials P(x) and Q(x) in the variable x with integer coefficients. Denoting such a sequence by , we obtain a general formula that determines the congruence class, modulo p, of the indefinite sum , for any prime p, and any positive integer r, as a linear combination of sequences that satisfy linear recurrence (alias difference) equations with constant coefficients. This enables us (or rather, our computers) to automatically discover and prove congruence theorems for such partial sums. Moreover, we show that in many cases, the set of the residues is finite, regardless of the prime p.- 1 May 2015
- Published by Springer Nature in The Ramanujan Journal
- Vol. 40 (2) , 311-322
Ramanujan-type congruences for overpartitions modulo 16
- 1 May 2015
- Published by Springer Nature in The Ramanujan Journal
- Vol. 40 (2) , 311-322
Ramanujan-type congruences for overpartitions modulo 16
- 19 September 2014
- Published by Springer Nature in Discrete & Computational Geometry
- Vol. 52 (4) , 630-662
Equivalence Classes of Full-Dimensional $0/1$ 0 / 1 -Polytopes with Many Vertices- 19 September 2014
- Published by Springer Nature in Discrete & Computational Geometry
- Vol. 52 (4) , 630-662
Equivalence Classes of Full-Dimensional $0/1$ 0 / 1 -Polytopes with Many Vertices- Open Access
Ramanujan-type Congruences for Overpartitions Modulo 16
Let $\overline{p}(n)$ denote the number of overpartitions of $n$. Recently, Fortin-Jacob-Mathieu and Hirschhorn-Sellers independently obtained 2-, 3- and 4-dissections of the generating function for $\overline{p}(n)$ and derived a number of congruences for $\overline{p}(n)$ modulo $4$, $8$ and $64$ including $\overline{p}(5n+2)\equiv 0 \pmod{4}$, $\overline{p}(4n+3)\equiv 0 \pmod{8}$ and $\overline{p}(8n+7)\equiv 0 \pmod{64}$. By employing dissection techniques, Yao and Xia obtained congruences for $\overline{p}(n)$ modulo $8, 16$ and $32$, such as $\overline{p}(48n+26) \equiv 0 \pmod{8}$, $\overline{p}(24n+17)\equiv 0 \pmod{16}$ and $\overline{p}(72n+69)\equiv 0 \pmod{32}$. In this paper, we give a 16-dissection of the generating function for $\overline{p}(n)$ modulo 16 and we show that $\overline{p}(16n+14)\equiv0\pmod{16}$ for $n\ge 0$. Moreover, by using the $2$-adic expansion of the generating function of $\overline{p}(n)$ due to Mahlburg, we obtain that $\overline{p}(\ell^2n+r\ell)\equiv0\pmod{16}$, where $n\ge 0$, $\ell \equiv -1\pmod{8}$ is an odd prime and $r$ is a positive integer with $\ell \nmid r$. In particular, for $\ell=7$, we get $\overline{p}(49n+7)\equiv0\pmod{16}$ and $\overline{p}(49n+14)\equiv0\pmod{16}$ for $n\geq 0$. We also find four congruence relations: $\overline{p}(4n)\equiv(-1)^n\overline{p}(n) \pmod{16}$ for $n\ge 0$, $\overline{p}(4n)\equiv(-1)^n\overline{p}(n)\pmod{32}$ for $n$ being not a square of an odd positive integer, $\overline{p}(4n)\equiv(-1)^n\overline{p}(n)\pmod{64}$ for $n\not\equiv 1,2,5\pmod{8}$ and $\overline{p}(4n)\equiv(-1)^n\overline{p}(n)\pmod{128}$ for $n\equiv 0\pmod{4}$.Open AccessRamanujan-type Congruences for Overpartitions Modulo 16
Let $\overline{p}(n)$ denote the number of overpartitions of $n$. Recently, Fortin-Jacob-Mathieu and Hirschhorn-Sellers independently obtained 2-, 3- and 4-dissections of the generating function for $\overline{p}(n)$ and derived a number of congruences for $\overline{p}(n)$ modulo $4$, $8$ and $64$ including $\overline{p}(5n+2)\equiv 0 \pmod{4}$, $\overline{p}(4n+3)\equiv 0 \pmod{8}$ and $\overline{p}(8n+7)\equiv 0 \pmod{64}$. By employing dissection techniques, Yao and Xia obtained congruences for $\overline{p}(n)$ modulo $8, 16$ and $32$, such as $\overline{p}(48n+26) \equiv 0 \pmod{8}$, $\overline{p}(24n+17)\equiv 0 \pmod{16}$ and $\overline{p}(72n+69)\equiv 0 \pmod{32}$. In this paper, we give a 16-dissection of the generating function for $\overline{p}(n)$ modulo 16 and we show that $\overline{p}(16n+14)\equiv0\pmod{16}$ for $n\ge 0$. Moreover, by using the $2$-adic expansion of the generating function of $\overline{p}(n)$ due to Mahlburg, we obtain that $\overline{p}(\ell^2n+r\ell)\equiv0\pmod{16}$, where $n\ge 0$, $\ell \equiv -1\pmod{8}$ is an odd prime and $r$ is a positive integer with $\ell \nmid r$. In particular, for $\ell=7$, we get $\overline{p}(49n+7)\equiv0\pmod{16}$ and $\overline{p}(49n+14)\equiv0\pmod{16}$ for $n\geq 0$. We also find four congruence relations: $\overline{p}(4n)\equiv(-1)^n\overline{p}(n) \pmod{16}$ for $n\ge 0$, $\overline{p}(4n)\equiv(-1)^n\overline{p}(n)\pmod{32}$ for $n$ being not a square of an odd positive integer, $\overline{p}(4n)\equiv(-1)^n\overline{p}(n)\pmod{64}$ for $n\not\equiv 1,2,5\pmod{8}$ and $\overline{p}(4n)\equiv(-1)^n\overline{p}(n)\pmod{128}$ for $n\equiv 0\pmod{4}$. - Open Access
Finite Differences of the Logarithm of the Partition Function
Let $p(n)$ denote the partition function. DeSalvo and Pak proved that $\frac{p(n-1)}{p(n)}\left(1+\frac{1}{n}\right)> \frac{p(n)}{p(n+1)}$ for $n\geq 2$, as conjectured by Chen. Moreover, they conjectured that a sharper inequality $\frac{p(n-1)}{p(n)}\left( 1+\frac{\pi}{\sqrt{24}n^{3/2}}\right) > \frac{p(n)}{p(n+1)}$ holds for $n\geq 45$. In this paper, we prove the conjecture of Desalvo and Pak by giving an upper bound for $-\Delta^{2} \log p(n-1)$, where $\Delta$ is the difference operator with respect to $n$. We also show that for given $r\geq 1$ and sufficiently large $n$, $(-1)^{r-1}\Delta^{r} \log p(n)>0$. This is analogous to the positivity of finite differences of the partition function. It was conjectured by Good and proved by Gupta that for given $r\geq 1$, $\Delta^{r} p(n)>0$ for sufficiently large $n$.Open AccessFinite Differences of the Logarithm of the Partition Function
Let $p(n)$ denote the partition function. DeSalvo and Pak proved that $\frac{p(n-1)}{p(n)}\left(1+\frac{1}{n}\right)> \frac{p(n)}{p(n+1)}$ for $n\geq 2$, as conjectured by Chen. Moreover, they conjectured that a sharper inequality $\frac{p(n-1)}{p(n)}\left( 1+\frac{\pi}{\sqrt{24}n^{3/2}}\right) > \frac{p(n)}{p(n+1)}$ holds for $n\geq 45$. In this paper, we prove the conjecture of Desalvo and Pak by giving an upper bound for $-\Delta^{2} \log p(n-1)$, where $\Delta$ is the difference operator with respect to $n$. We also show that for given $r\geq 1$ and sufficiently large $n$, $(-1)^{r-1}\Delta^{r} \log p(n)>0$. This is analogous to the positivity of finite differences of the partition function. It was conjectured by Good and proved by Gupta that for given $r\geq 1$, $\Delta^{r} p(n)>0$ for sufficiently large $n$. - Open Access
Ramanujan-type Congruences for Overpartitions Modulo 5
Let $\overline{p}(n)$ denote the number of overpartitions of $n$. Hirschhorn and Sellers showed that $\overline{p}(4n+3)\equiv 0 \pmod{8}$ for $n\geq 0$. They also conjectured that $\overline{p}(40n+35)\equiv 0 \pmod{40}$ for $n\geq 0$. Chen and Xia proved this conjecture by using the $(p,k)$-parametrization of theta functions given by Alaca, Alaca and Williams. In this paper, we show that $\overline{p}(5n)\equiv (-1)^{n}\overline{p}(4\cdot 5n) \pmod{5}$ for $n \geq 0$ and $\overline{p}(n)\equiv (-1)^{n}\overline{p}(4n)\pmod{8}$ for $n \geq 0$ by using the relation of the generating function of $\overline{p}(5n)$ modulo $5$ found by Treneer and the $2$-adic expansion of the generating function of $\overline{p}(n)$ due to Mahlburg. As a consequence, we deduce that $\overline{p}(4^k(40n+35))\equiv 0 \pmod{40}$ for $n,k\geq 0$. Furthermore, applying the Hecke operator on $\phi(q)^3$ and the fact that $\phi(q)^3$ is a Hecke eigenform, we obtain an infinite family of congrences $\overline{p}(4^k \cdot5\ell^2n)\equiv 0 \pmod{5}$, where $k\ge 0$ and $\ell$ is a prime such that $\ell\equiv3 \pmod{5}$ and $\left(\frac{-n}{\ell}\right)=-1$. Moreover, we show that $\overline{p}(5^{2}n)\equiv \overline{p}(5^{4}n) \pmod{5}$ for $n \ge 0$. So we are led to the congruences $\overline{p}\big(4^k5^{2i+3}(5n\pm1)\big)\equiv 0 \pmod{5}$ for $n, k, i\ge 0$. In this way, we obtain various Ramanujan-type congruences for $\overline{p}(n)$ modulo $5$ such as $\overline{p}(45(3n+1))\equiv 0 \pmod{5}$ and $\overline{p}(125(5n\pm 1))\equiv 0 \pmod{5}$ for $n\geq 0$.Open AccessRamanujan-type Congruences for Overpartitions Modulo 5
Let $\overline{p}(n)$ denote the number of overpartitions of $n$. Hirschhorn and Sellers showed that $\overline{p}(4n+3)\equiv 0 \pmod{8}$ for $n\geq 0$. They also conjectured that $\overline{p}(40n+35)\equiv 0 \pmod{40}$ for $n\geq 0$. Chen and Xia proved this conjecture by using the $(p,k)$-parametrization of theta functions given by Alaca, Alaca and Williams. In this paper, we show that $\overline{p}(5n)\equiv (-1)^{n}\overline{p}(4\cdot 5n) \pmod{5}$ for $n \geq 0$ and $\overline{p}(n)\equiv (-1)^{n}\overline{p}(4n)\pmod{8}$ for $n \geq 0$ by using the relation of the generating function of $\overline{p}(5n)$ modulo $5$ found by Treneer and the $2$-adic expansion of the generating function of $\overline{p}(n)$ due to Mahlburg. As a consequence, we deduce that $\overline{p}(4^k(40n+35))\equiv 0 \pmod{40}$ for $n,k\geq 0$. Furthermore, applying the Hecke operator on $\phi(q)^3$ and the fact that $\phi(q)^3$ is a Hecke eigenform, we obtain an infinite family of congrences $\overline{p}(4^k \cdot5\ell^2n)\equiv 0 \pmod{5}$, where $k\ge 0$ and $\ell$ is a prime such that $\ell\equiv3 \pmod{5}$ and $\left(\frac{-n}{\ell}\right)=-1$. Moreover, we show that $\overline{p}(5^{2}n)\equiv \overline{p}(5^{4}n) \pmod{5}$ for $n \ge 0$. So we are led to the congruences $\overline{p}\big(4^k5^{2i+3}(5n\pm1)\big)\equiv 0 \pmod{5}$ for $n, k, i\ge 0$. In this way, we obtain various Ramanujan-type congruences for $\overline{p}(n)$ modulo $5$ such as $\overline{p}(45(3n+1))\equiv 0 \pmod{5}$ and $\overline{p}(125(5n\pm 1))\equiv 0 \pmod{5}$ for $n\geq 0$.