1 Introduction
Throughout this paper, we define
$q:=e^{2\pi i\tau }$
for an element
$\tau $
in the complex upper-half plane
$\mathbb {H}$
and
$f_m := \prod _{n=1}^\infty (1-q^{mn})$
for an integer
$m\geq 1$
. A partition
$\lambda $
of a positive integer n is a nonincreasing sequence of positive integers, known as the parts of
$\lambda $
, whose sum is n. The generating function for the number
$p(n)$
of partitions of n with
$p(0) := 1$
is
Ramanujan [Reference Ramanujan19, page 210, (18)] found the q-series identity
which yields the famous congruence
for all
$n\geq 0$
. Subsequently, Darling [Reference Darling10] obtained another proof of this congruence for
$p(n)$
.
Schmidt [Reference Schmidt20] posed a surprising description of
$p(n)$
as a problem in the American Mathematical Monthly:
$p(n)$
is also the number of partitions
$a_1+a_2+a_3+\cdots $
with
$a_1> a_2 > a_3 > \cdots $
such that
$n = a_1+a_3+a_5+\cdots $
. Andrews and Paule [Reference Andrews and Paule3] utilised MacMahon’s partition analysis to prove that the number of partitions
${a_1+a_2+a_3+\cdots }$
with
$a_1 \geq a_2 \geq a_3 \geq \cdots $
such that
$n = a_1+a_3+a_5+\cdots $
is the number
$p_2(n)$
of two-color partitions of n, whose generating function is
Consequently, in [Reference Andrews and Paule3, Reference Andrews and Paule4], they established analogous Schmidt-type results for plane partition diamonds and partitions with n copies of n.
Recently, Andrews and Paule [Reference Andrews and Paule5] studied the Schmidt phenomenon for partitions where the parity of the parts satisfies certain conditions. In particular, they proved that the number of partitions
$a_1+a_2+a_3+\cdots $
with
$a_1 \geq a_2 \geq a_3 \geq \cdots $
such that
${n = a_1+a_3+a_5+\cdots }$
and
$a_1, a_3, a_5, \ldots $
are all even is
$S(n)$
, where
For example,
$S(4) = 8$
as presented by the following Schmidt-type partitions of
$4$
:
Define
so that
$S(2n) = s(n)$
. Combinatorially,
$s(n)$
counts the number of partition triples
$(\pi _1,\pi _2,\pi _3)$
of n such that
$\pi _1$
and
$\pi _2$
have unrestricted parts,
$\pi _3$
has only odd parts, and the sum of the parts of
$\pi _1, \pi _2$
and
$\pi _3$
is n. Andrews and Paule [Reference Andrews and Paule5, page 36] remarked that
$s(n)$
and
$S(n)$
do not seem to satisfy Ramanujan-type congruences modulo small primes.
We establish in this paper that
$S(n)$
actually possesses Ramanujan-type congruences modulo
$7$
, as shown in our main results stated below.
Theorem 1.1. For
$r\in \{19,33,40,47\}$
and all integers
$n\geq 0$
,
Theorem 1.2. Let
$p\equiv 7, 11\pmod {12}$
be a prime. For all integers
$n\geq 0$
and
${r\in \{1,\ldots ,p-1\}}$
,
We organise the rest of the paper as follows. In Section 2, we give a short review of necessary results on modular functions and
$\eta $
-quotients for the congruence subgroup
$\Gamma _0(N)$
, as well as some useful properties of Atkin’s
$U_p$
operator for a prime p dividing N. We then deduce Theorems 1.1 and 1.2 in Section 3 by finding the action of
$U_7$
on an
$\eta $
-quotient for
$\Gamma _0(98)$
, which leads us to the generating function for
$s(7n+5)$
modulo
$7$
, and then applying a result of Ahlgren [Reference Ahlgren2]. We have performed our computations using Mathematica.
2 Modular functions on
$\Gamma _0(N)$
We recall in this section necessary results on modular functions and
$\eta $
-quotients for the congruence subgroup
Given a matrix
$\gamma :=[\begin {smallmatrix} a & b\\ c& d \end {smallmatrix}]\in \Gamma _0(N)$
and an element
$\tau \in \mathbb {H}^\ast := \mathbb {H}\cup \mathbb {Q}\cup \{\infty \}$
, we define the action of
$\gamma $
on
$\tau $
by the linear fractional transformation
We define the cusps of
$\Gamma _0(N)$
as the equivalence classes of
$\mathbb {Q}\cup \{\infty \}$
under this action. We call a meromorphic function
$f:\mathbb {H}\rightarrow \mathbb {C}$
modular for
$\Gamma _0(N)$
if
$f(\gamma \tau )=f(\tau )$
for all
$\gamma \in \Gamma _0(N)$
and, for every cusp
$a/c$
of
$\Gamma _0(N)$
and
$\gamma \in \text {SL}_2(\mathbb {Z})$
with
$\gamma (\infty )=a/c$
, the q-expansion of
$f(\gamma \tau )$
is given by
for some integer
$n_0$
with
$a_{n_0}\neq 0$
. Here,
$n_0$
is the order of vanishing of
$f(\tau )$
at
$a/c$
, denoted by
$\mathrm {ord}_{a/c}\,f(\tau )$
, and we call
$a/c$
a zero (respectively, a pole) of
$f(\tau )$
if
$\mathrm {ord}_{a/c}\,f(\tau )$
is positive (respectively, negative).
An
$\eta $
-quotient is a product of the form
for some index set
$\{r_\delta \in \mathbb {Z} : \delta \mid N\}$
, where
$\eta (\tau ) := q^{1/24}f_1$
is the Dedekind eta function. The following results give conditions for an
$\eta $
-quotient to be modular for
$\Gamma _0(N)$
.
Lemma 2.1. Let
$f(\tau ) = \prod _{\delta \mid N} \eta ^{r_{\delta }}(\delta \tau )$
be an
$\eta $
-quotient with
$\sum _{\delta \mid N} r_{\delta }=0$
. Then f is modular for
$\Gamma _0(N)$
if and only if:
-
(1) $\sum _{\delta \mid N} \delta r_{\delta } \equiv 0\pmod {24}$
; -
(2) $\sum _{\delta \mid N} {N}r_{\delta }/{\delta } \equiv 0\pmod {24}$
; and -
(3) $\prod _{\delta \mid N} \delta ^{r_\delta }$
is a rational perfect square.
Proof. This follows from [Reference Cohen and Strömberg9, Proposition 5.9.2].
Lemma 2.2. Let
$c, d$
and N be positive integers with
$d\mid N$
and
$\gcd (c,d)=1$
, and let
$f(\tau ) = \prod _{\delta \mid N} \eta ^{r_{\delta }}(\delta \tau )$
be an
$\eta $
-quotient satisfying the conditions of Lemma 2.1. Then the order of vanishing of
$f(\tau )$
at the cusp
$c/d$
of
$\Gamma _0(N)$
is
Proof. See [Reference Aygin7, Lemma 4.1.3].
Denote the algebra of all modular functions for
$\Gamma _0(N)$
having a pole only at
$\infty $
by
$\mathcal {M}^\infty (N)$
and the subalgebra of
$\mathcal {M}^\infty (N)$
consisting of all
$\eta $
-quotients for
$\Gamma _0(N)$
by
$\mathcal {E}^\infty (N)$
. Radu [Reference Radu18] devised Algorithm AB that provides an algebra basis for
$\langle \mathcal {E}^\infty (N)\rangle _{\mathbb {Q}}$
for
$N\geq 2$
, yielding the following result.
Lemma 2.3. For
$N\geq 2$
,
$\mathcal {E}^\infty (N)$
is a finitely generated monoid. Moreover, there exist modular functions
$t, u_1, u_2,\ldots , u_v\in \mathcal {M}^\infty (N)$
such that:
-
(1) $|\mathrm {ord}_\infty \, t| = v$
; -
(2) $|\mathrm {ord}_\infty \, u_i| < |\mathrm {ord}_\infty \, u_j|$
for
$1\leq i < j\leq v$
; -
(3) $\mathrm {ord}_\infty \, u_i \not \equiv \mathrm {ord}_\infty \, u_j\pmod {v}$
for
$1\leq i < j \leq v$
; -
(4) $\mathrm {ord}_\infty \, u_i \not \equiv 0\pmod {v}$
for
$1 \leq i \leq v$
; and -
(5) $\langle \mathcal {E}^\infty (N) \rangle _{\mathbb {Q}} = \langle 1,u_1,u_2,\ldots ,u_v\rangle _{\mathbb {Q}[t]}$
.
Proof. See [Reference Radu18, Sections 2.1 and 2.2]. We remark that the generators of
$\mathcal {E}^\infty (N)$
can be found using a terminating algorithm [Reference Radu18, Lemma 25] and the corresponding basis elements of
$\langle \mathcal {E}^\infty (N) \rangle _{\mathbb {Q}}$
can be obtained using the Algorithm AB [Reference Radu18, Theorem 16].
For a prime p and a function
$f(\tau ) = \sum _{n= n_0}^\infty a(n)q^n$
, we define Atkin’s
$U_p$
operator by
We note that
$U_p$
is a linear operator with the following properties (see [Reference Knopp15, Ch. 8] and [Reference Atkin and Lehner6, Lemma 7]):
-
(1) $U_p(f(\tau )) = {p}^{-1}\sum _{j=0}^{p-1}f({(\tau +j)}/{p})$
; -
(2) $U_p(f(p\tau )g(\tau )) = f(\tau )U_p(g(\tau ))$
for any function
$g(\tau )$
; and -
(3) if $p\mid N$
and
$f(\tau )$
is modular for
$\Gamma _0(pN)$
, then
$U_p(f(\tau ))$
is modular for
$\Gamma _0(N)$
.
The following lemma provides lower bounds for the orders of vanishing of
$U_p(f(\tau ))$
at the cusps of
$\Gamma _0(N)$
, where
$f(\tau )$
is modular for
$\Gamma _0(pN)$
for some prime p dividing N.
Lemma 2.4. Let p be a prime dividing N and
$f(\tau )$
be modular for
$\Gamma _0(pN)$
. Let
$\nu _p(m)$
be the largest exponent k with
$p^k\mid m$
. Suppose
$r=a/c$
is a cusp of
$\Gamma _0(N)$
where
$c\mid N$
and
$\gcd (a,c)=1$
. Then
Proof. See [Reference Gordon, Hughes and Knopp11, Theorem 4].
3 Proofs of Theorems 1.1 and 1.2
We establish Theorems 1.1 and 1.2 by deriving the generating function for
$s(7n+5)$
modulo
$7$
. Since the underlying modular curve
$X_0(14)$
has genus
$1$
, the Weierstrass gap theorem [Reference Paule and Radu16] dictates that any modular function for
$\Gamma _0(14)$
with pole only at
$\infty $
must have order of vanishing at least
$2$
. In view of the modified version of Newman’s conjecture by Paule and Radu [Reference Paule and Radu17], any function in
$\langle \mathcal {E}^\infty (14)\rangle _{\mathbb {Q}}$
can be expressed as a suitable combination of
$\eta $
-quotients with orders of vanishing
$2$
and
$3$
, which will suffice as an algebra basis for
$\langle \mathcal {E}^\infty (14)\rangle _{\mathbb {Q}}$
. Thus, applying Lemma 2.3 (or [Reference Ramanujan19, Theorem 16]) and [Reference Ramanujan19, Lemma 25], we find that
where
are generators of
$\mathcal {E}^\infty (14)$
. Define
By Lemmas 2.1 and 2.2,
$X^2Z, X^2Z^2\in \mathcal {E}^\infty (14)$
. From the relations involving
$X^2, X^3, X^2Z$
and
$X^2Z^2$
found in [Reference Guadalupe12, Section 3], we conclude that
We now use (3.1) to deduce the following generating function.
Proposition 3.1. We have
Proof. We follow [Reference Guadalupe12, Section 3] to search for an
$\eta $
-quotient of the form
for some
$a,b,c,d\in \mathbb {Z}$
such that
$g(\tau )\in \mathcal {E}^\infty (98)$
. The existence of
$a,b,c,d$
is guaranteed by [Reference Radu18, Lemma 20]. From [Reference Cho, Koo and Park8, Corollary 4(1)],
is the set of inequivalent cusps of
$\Gamma _0(98)$
. Employing Lemmas 2.1 and 2.2 and the set
$S_{\Gamma _0(98)}$
, we infer that
$(a,b,c,d)$
is a solution to the following system of equalities and inequalities
for some
$x_i\in \mathbb {Z}$
for
$1\leq i\leq 4$
. To solve this system, we use the zsolve command of the Mathematica interface math4ti2.m implemented by Hemmecke and Radu [Reference Hemmecke and Radu13], which requires the Diophantine software package 4ti2 [1]. We then obtain the solution
$(a,b,c,d)=(19,1,14,-32)$
, so that
with a pole at
$\infty \in S_{\Gamma _0(98)}$
with
$\mathrm {ord}_\infty g(\tau)=-96$
. Since
$\{\infty , 0, 1/2, 1/7\}$
is the set of inequivalent cusps of
$\Gamma _0(14)$
by [Reference Cho, Koo and Park8, Corollary 4(1)], we infer from Lemma 2.4 that
Using (3.1), we deduce that
We then combine (3.3) and (3.4), and use the definitions of
$X, Y$
and Z. Since
$f_{7k}\equiv f_k^7 \pmod {7}$
for all
$k\geq 1$
by the binomial theorem, we arrive at (3.2) as desired.
Proof of Theorem 1.1
We require the identity of Jacobi [Reference Hirschhorn14, page 11, (1.7.1)],
so that (3.2) becomes
Now observe that
$j(j+1)/2\equiv 0,1,3,6\pmod {7}$
and
$j(j+1)/2\equiv 6\pmod {7}$
if and only if
$2j+1\equiv 0$
. Thus, looking at the terms of (3.5) involving
$q^{7n+t}$
, where
$t\in \{2,4,5,6\}$
, we get
for all integers
$n\geq 0$
. The desired congruence now follows from (3.6).
Proof of Theorem 1.2
By the binomial theorem, we recast (3.2) as
Let
$p\equiv 7,11\pmod {12}$
be a prime. Substituting
$(r,s) = (3,7)$
in [Reference Ahlgren2, Theorem 1] yields
We replace n by
$pn+r$
in (3.8), where
$r\in \{1,\ldots ,p-1\}$
, so that
The desired congruence now follows from (3.10).
