Mex-Related Partition Functions of Andrews and Newman

Rupam Barman and Ajit Singh Department of Mathematics Indian Institute of Technology Guwahati–781039 India

                   rupam@iitg.ac.in
                   ajit18@iitg.ac.in

Abstract

In a recent paper, Andrews and Newman introduced certain families of partition functions using the minimal excludant or “mex” function. In this article, we study two of the families of functions Andrews and Newman introduced, namely p_{t,t}(n) and p_{2t,t}(n). We establish identities connecting the ordinary partition function p(n) to p_{t,t}(n) and p_{2t,t}(n) for all t ≥ 1. Using these identities, we prove that Ramanujan’s famous congruences for p(n) are also satisfied by p_{t,t}(n) and p_{2t,t}(n) for infinitely many values of t.

Very recently, da Silva and Sellers provided complete parity characterizations of p_{1,1}(n) and p_{3,3}(n). We prove that p_{t,t}(n) ≡ \overline{C}_{4t,t}(n) (mod 2) for all n ≥ 0 and t ≥ 1, where \overline{C}_{4t,t}(n) is Andrews’ singular overpartition function. Using this congruence, the parity characterization of p_{1,1}(n) given by da Silva and Sellers follows from that of \overline{C}_{4,1}(n).

We also give elementary proofs of certain congruences already proved by da Silva and Sellers.

1 Introduction

For each set S of positive integers the minimal excludant function (mex-function) is defined as follows:

mex(S) = min(ℤ_{>0} \ S).

Andrews and Newman [3] recently generalized this function to integer partitions. Given a partition λ of n, they defined the mex-function mex_{A,a}(λ) to be the smallest positive integer congruent to a modulo A that is not part of λ. They then defined p_{A,a}(n) to be the number of partitions λ of n satisfying

mex_{A,a}(λ) ≡ a (mod 2A).

For example, consider n = 5, A = 2, and a = 2. In the table below, we list the seven partitions λ of 5 and the corresponding values of mex_{2,2}(λ) for each λ:

Partition λ  mex_{2,2}(λ)
5  2
4 + 1  2
3 + 2  4
3 + 1 + 1  2
2 + 2 + 1  4
2 + 1 + 1 + 1  4
1 + 1 + 1 + 1 + 1  2

We see that four of the partitions of 5 satisfy mex_{2,2}(λ) ≡ 2 (mod 4). Therefore, p_{2,2}(5) = 4. Andrews and Newman [3, Lemma 9] proved that the generating function for p_{t,t}(n) is given by

∑_{n=0}^{∞} p_{t,t}(n)q^{n} = \frac{1}{(q; q)_{∞}} ∑_{n=0}^{∞}(−1)^{n}q^{tn(n+1)/2}  (1)

and the generating function for p_{2t,t}(n) is given by

∑_{n=0}^{∞} p_{2t,t}(n)q^{n} = \frac{1}{(q; q)_{∞}} ∑_{n=0}^{∞}(−1)^{n}q^{tn^{2}},  (2)

where (a; q)_{∞} := ∏_{j=0}^{∞}(1 − aq^{j}).

The rank of a partition is the largest part minus the number of parts. The crank of a partition is the largest part of the partition if there are no ones as parts, and otherwise is the number of parts larger than the number of ones minus the numbers of ones. Andrews and Newman [3] proved that p_{1,1}(n) equals the number of partitions of n with non-negative crank and p_{3,3}(n) equals the number of partitions of n with rank ≥ −1. They also proved that p_{2,1}(n) is equal to the number of partitions of n into even parts. They further proved that p_{4,2}(n) − p_{o}(n) equals the number of partitions of n into parts congruent to ±4, ±6, ±8, ±10 modulo 32 and p_{6,3}(n) − p_{o}(n) equals the number of partitions of n into parts congruent to ±2, ±4, ±5, ±6, ±7, ±8 modulo 24. Here p_{o}(n) denotes the number of partitions of n into odd parts. In a very recent paper [4], we have proved that p_{2^{α},2^{α}}(n) and p_{3·2^{α},3·2^{α}}(n) are almost always even for all α ≥ 1. Using a result of Ono and Taguchi on nilpotency of Hecke operators, we have also found infinite families of congruences modulo 2 satisfied by p_{2^{α},2^{α}}(n) and p_{3·2^{α},3·2^{α}}(n) for all α ≥ 1. In this article, we express p_{t,t}(n) and p_{2t,t}(n) in terms of the ordinary partition function p(n) for all t ≥ 1. Using our identities, we find that the partition functions p_{t,t}(n) and p_{2t,t}(n) satisfy Ramanujan’s famous congruences for p(n) for infinitely many values of t.

Beginning with the paper [6], Corteel and Lovejoy introduced and developed the theory of overpartitions. An overpartition of n is a non-increasing sequence of natural numbers whose sum is n in which the first occurrence of a number may be overlined. In order to give overpartition analogues of Rogers-Ramanujan type theorems for the ordinary partition function with restricted successive ranks, Andrews [2] defined the so-called singular overpartitions. Andrews’ singular overpartition function \overline{C}_{k,i}(n) counts the number of overpartitions of n in which no part is divisible by k and only parts ≡ ±i (mod k) may be overlined. For example, \overline{C}_{3,1}(4) = 10 with the relevant partitions being 4, \overline{4}, 2 + 2, \overline{2} + 2, 2 + 1 + 1, \overline{2} + 1 + 1, 2 + \overline{1} + 1, \overline{2} + \overline{1} + 1, 1 + 1 + 1 + 1, \overline{1} + 1 + 1 + 1. For k ≥ 3 and 1 ≤ i ≤ ⌊\frac{k}{2}⌋, the generating function for \overline{C}_{k,i}(n) is given by

∑_{n=0}^{∞} \overline{C}_{k,i}(n)q^{n} = \frac{(q^{k}; q^{k})_{∞}(−q^{i}; q^{k})_{∞}(−q^{k−i}; q^{k})_{∞}}{(q; q)_{∞}} .  (3)

In this article, we prove that p_{t,t}(n) and \overline{C}_{4t,t}(n) have the same parity for all t ≥ 1. Using this, we find new congruences satisfied by p_{t,t}(n). We also give elementary proofs of certain congruences proved by da Silva and Sellers [7].

2 Mex-related partitions and relations to ordinary partition

Let p(n) denote the ordinary partition function. We adopt the convention that p(n) = 0 when n is a negative integer. In the following theorem, we express p_{t,t}(n) and p_{2t,t}(n) in terms of p(n).

Theorem 1. Let t be a positive integer. Then, for all non-negative integers n, we have

p_{t,t}(n) = p(n) + ∑_{r=1}^{∞} p(n − tr(2r + 1)) − ∑_{s=1}^{∞} p(n − ts(2s − 1))  (4)

and

p_{2t,t}(n) = p(n) + ∑_{r=1}^{∞} p(n − 4tr^{2}) − ∑_{s=1}^{∞} p(n − t(2s − 1)^{2}).  (5)

Proof. We know that the generating function for the partition function p(n) is given by

∑_{n=0}^{∞} p(n)q^{n} = \frac{1}{(q; q)_{∞}} .

From (1), we find that

∑_{n=0}^{∞} p_{t,t}(n)q^{n} = \frac{1}{(q; q)_{∞}} ∑_{n=0}^{∞}(−1)^{n}q^{tn(n+1)/2}

= (∑_{n=0}^{∞} p(n)q^{n}) (1 + ∑_{r=1}^{∞} q^{tr(2r+1)} − ∑_{s=1}^{∞} q^{ts(2s−1)})

= ∑_{n=0}^{∞} (p(n) + ∑_{r=1}^{∞} p(n − tr(2r + 1)) − ∑_{s=1}^{∞} p(n − ts(2s − 1))) q^{n}.

Thus, for all non-negative integers n, we have

p_{t,t}(n) = p(n) + ∑_{r=1}^{∞} p(n − tr(2r + 1)) − ∑_{s=1}^{∞} p(n − ts(2s − 1)).  (6)

Again, from (2), we find that

∑_{n=0}^{∞} p_{2t,t}(n)q^{n} = \frac{1}{(q; q)_{∞}} ∑_{n=0}^{∞}(−1)^{n}q^{tn^{2}}

= (∑_{n=0}^{∞} p(n)q^{n}) (1 + ∑_{r=1}^{∞} q^{4tr^{2}} − ∑_{s=1}^{∞} q^{t(2s−1)^{2}})

= ∑_{n=0}^{∞} (p(n) + ∑_{r=1}^{∞} p(n − 4tr^{2}) − ∑_{s=1}^{∞} p(n − t(2s − 1)^{2})) q^{n}.

Thus, for all non-negative integers n, we have

p_{2t,t}(n) = p(n) + ∑_{r=1}^{∞} p(n − 4tr^{2}) − ∑_{s=1}^{∞} p(n − t(2s − 1)^{2}).

This completes the proof of the theorem.

In the following theorem, we prove that p_{t,t}(n) and p_{2t,t}(n) satisfy Ramanujan-type congruences, and these congruences follow from those satisfied by the ordinary partition function p(n).

Theorem 2. Let m, a ≥ 1 and b be integers. Suppose that p(an + b) ≡ 0 (mod m) for all non-negative integers n. Then, for all t ≥ 1, we have

p_{at,at}(an + b) ≡ 0 (mod m)

and

p_{2at,at}(an + b) ≡ 0 (mod m)

for all non-negative integers n.

Proof. Let n ≥ 0. From (4), we obtain

p_{at,at}(an + b)
= p(an + b) + ∑_{r=1}^{∞} p(a(n − tr(2r + 1)) + b) − ∑_{s=1}^{∞} p(a(n − ts(2s − 1)) + b).  (7)

We note that the terms remaining in the sums in (6) satisfy that n−tr(2r+1) and n−ts(2s−1) are non-negative. Hence, the same is true in (7). Now, if p(ℓa + b) ≡ 0 (mod m) for every non-negative integer ℓ, then (7) yields that p_{at,at}(an + b) ≡ 0 (mod m). This completes the proof of the first congruence of the theorem.

Using (5) and proceeding along similar lines, we prove the second congruence. This completes the proof of the theorem.

As an application of Theorem 2, we find that p_{t,t}(n) and p_{2t,t}(n) satisfy Ramanujan’s famous congruences for certain infinite families of t. Much to Ramanujan’s credit, the “Ramanujan congruences” for p(n) are given below. If k ≥ 1, then for every non-negative integer n, we have

p (5^{k}n + δ_{5,k}) ≡ 0 (mod 5^{k});
p (7^{k}n + δ_{7,k}) ≡ 0 (mod 7^{⌊k/2⌋+1});
p (11^{k}n + δ_{11,k}) ≡ 0 (mod 11^{k}),

where δ_{p,k} := 1/24 mod p^{k} for p = 5, 7, 11. In the following, we prove that p_{at,at}(n) and p_{2at,at}(n) satisfy the Ramanujan congruences when a = 5^{k}, 7^{k}, 11^{k}.

Corollary 3. For all k, t ≥ 1 and for every non-negative integer n, we have

p_{5^{k}t,5^{k}t} (5^{k}n + δ_{5,k}) ≡ 0 (mod 5^{k});
p_{7^{k}t,7^{k}t} (7^{k}n + δ_{7,k}) ≡ 0 (mod 7^{⌊k/2⌋+1});
p_{11^{k}t,11^{k}t} (11^{k}n + δ_{11,k}) ≡ 0 (mod 11^{k});
p_{2·5^{k}t,5^{k}t} (5^{k}n + δ_{5,k}) ≡ 0 (mod 5^{k});
p_{2·7^{k}t,7^{k}t} (7^{k}n + δ_{7,k}) ≡ 0 (mod 7^{⌊k/2⌋+1});
p_{2·11^{k}t,11^{k}t} (11^{k}n + δ_{11,k}) ≡ 0 (mod 11^{k}).

Proof. Combining Ramanujan congruences for p(n) and Theorem 2 we readily obtain that p_{at,at}(n) and p_{2at,at}(n) satisfy the Ramanujan congruences when a = 5^{k}, 7^{k}, 11^{k}. This completes the proof.

Remark 4. Da Silva and Sellers [7] proposed to undertake a more extensive investigation of the properties of p_{2t,t}(n). Based on extensive computations, they remarked that p_{2t,t}(n) need not satisfy Ramanujan-like congruences. But Corollary 3 disproves their observation. We have also verified Corollary 3 numerically for large values of the parameters.

3 Mex-related partitions and relations to singular overpartitions

In this section we relate the mex-related partition functions to Andrews’ singular overpartion functions. This helps us to find new congruences satisfied by the mex-related partition functions. In the following theorem, we prove that both p_{t,t}(n) and \overline{C}_{4t,t}(n) have the same parity.

Theorem 5. Let t be a positive integer. Then, for all n ≥ 0, we have

p_{t,t}(n) ≡ \overline{C}_{4t,t}(n) (mod 2).

Proof. From (1), we have

∑_{n=0}^{∞} p_{t,t}(n)q^{n} = \frac{1}{(q; q)_{∞}} ∑_{n=0}^{∞}(−1)^{n}q^{tn(n+1)/2}.  (8)

Employing Ramanujan’s theta function

ψ(q) := ∑_{n=0}^{∞} q^{n(n+1)/2} = \frac{(q^{2}; q^{2})_{∞}^{2}}{(q; q)_{∞}}

into (8), we find that

∑_{n=0}^{∞} p_{t,t}(n)q^{n} ≡ \frac{1}{(q; q)_{∞}} ∑_{n=0}^{∞} q^{tn(n+1)/2} (mod 2)

= \frac{1}{(q; q)_{∞}} \begin{matrix}(q^{2t}; q^{2t}) & {2 \atop ∞} \\ (q^{t}; q^{t}) & ∞ \end{matrix}

≡ \frac{1}{(q; q)_{∞}} \begin{matrix}(q^{t}; q^{t}) & {4 \atop ∞} \\ (q^{t}; q^{t}) & ∞ \end{matrix} (mod 2)

≡ \frac{(q^{t}; q^{t})_{∞}^{3}}{(q; q)_{∞}} (mod 2).  (9)

From (3), we find that

∑_{n=0}^{∞} \overline{C}_{4t,t}(n)q^{n} = \frac{(q^{4t}; q^{4t})_{∞} (−q^{t}; q^{4t})_{∞} (−q^{3t}; q^{4t})_{∞}}{(q; q)_{∞}}

= \frac{(q^{4t}; q^{4t})_{∞} (−q^{t}; q^{4t})_{∞} (−q^{3t}; q^{4t})_{∞} (−q^{2t}; q^{4t})_{∞} (−q^{4t}; q^{4t})_{∞}}{(q; q)_{∞} (−q^{2t}; q^{4t})_{∞} (−q^{4t}; q^{4t})_{∞}}

= \frac{(q^{4t}; q^{4t})_{∞} (−q^{t}; q^{t})_{∞}}{(q; q)_{∞} (−q^{2t}; q^{2t})_{∞}}

= \frac{(q^{4t}; q^{4t})_{∞} (q^{2t}; q^{2t})_{∞}^{2}}{(q; q)_{∞} (q^{t}; q^{t})_{∞} (q^{4t}; q^{4t})_{∞}}

≡ \frac{(q^{t}; q^{t})_{∞}^{3}}{(q; q)_{∞}} (mod 2).  (10)

Thus, combining (9) and (10), we have

∑_{n=0}^{∞} p_{t,t}(n) ≡ ∑_{n=0}^{∞} \overline{C}_{4t,t}(n) (mod 2).

This completes the proof of the theorem.

Chen, Hirschhorn, and Sellers [5] proved that, for all n ≥ 1,

\overline{C}_{4,1}(n) ≡ \begin{cases}1 (mod 2), & if n = k(3k − 1) for some k; \\ 0 (mod 2), & otherwise. \end{cases}

Due to Theorem 5, we have the same parity characterization for p_{1,1}(n). Recently, da Silva and Sellers [7, Theorem 4] also found the same parity characterization for p_{1,1}(n). To the best of our knowledge, the parity characterization for \overline{C}_{12,3}(n) is not known till date. da Silva and Sellers [7, Theorem 7] found the parity characterization for p_{3,3}(n). Combining [7, Theorem 7] and Theorem 5, we have the following parity characterization for \overline{C}_{12,3}(n):

Corollary 6. For all n ≥ 1, we have

\overline{C}_{12,3}(n) ≡ \begin{cases}1 (mod 2), & if 3n + 1 is a square; \\ 0 (mod 2), & otherwise. \end{cases}

We next use known congruences for Andrews’ singular overpartition functions \overline{C}_{4t,t}(n) and combine them with Theorem 5 to deduce new congruences for p_{t,t}(n) for different values of t. Our list of congruences obtained this way need not be exhaustive.

Theorem 7. Let p ≥ 5 be a prime and p ≢ 1 (mod 12). Then, for all k, n ≥ 0 with p ∤ n, we have

p_{1,1} (p^{2k+1}n + \frac{p^{2k+2} − 1}{12}) ≡ 0 (mod 2).

Proof. Taking t = 1 in Theorem 5, we have

p_{1,1}(n) ≡ \overline{C}_{4,1}(n) (mod 2).  (11)

Thanks to Chen, Hirschhorn and Sellers ([5, Corollary 3.6]) we know that, for all n ≥ 0, we have

\overline{C}_{4,1} (p^{2k+1}n + \frac{p^{2k+2} − 1}{12}) ≡ 0 (mod 4).  (12)

Combining (11) and (12), we deduce the required congruence.

Theorem 8. If p is a prime such that p ≡ 3 (mod 4) and 1 ≤ j ≤ p − 1, then for all non-negative integers α and n, we have

p_{2,2} (p^{2α+1}(pn + j) + \frac{5 (p^{2(α+1)} − 1)}{24}) ≡ 0 (mod 2).

Proof. Ahmed and Baruah [1, Theorem 1.7] proved that, if p is a prime such that p ≡ 3 (mod 4) and 1 ≤ j ≤ p − 1, then for all non-negative integers α and n,

\overline{C}_{8,2} (p^{2α+1}(pn + j) + \frac{5 (p^{2(α+1)} − 1)}{24}) ≡ 0 (mod 2).

Now, Theorem 5 yields that the same congruence is also satisfied by p_{2,2}(n).

In the following theorem, we find congruences satisfied by p_{3,3}(n).

Theorem 9. We have:

1. For all n ≥ 0,

p_{3,3}(16n + 11) ≡ p_{3,3}(16n + 15) ≡ 0 (mod 2).  (13)

2. If n cannot be represented as the sum of a pentagonal number and four times a pentagonal number, then

p_{3,3}(16n + 3) ≡ 0 (mod 2).  (14)

3. If n cannot be represented as the sum of two times a pentagonal number and three times a triangular number, then

p_{3,3}(16n + 7) ≡ 0 (mod 2).  (15)

Proof. Taking t = 3 in Theorem 5, for all n ≥ 0, we have

p_{3,3}(n) ≡ \overline{C}_{12,3}(n) (mod 2).  (16)

By Theorems 4.1, 4.2 and 4.3 of Li and Yao [8] we know that, for all n ≥ 0,

\overline{C}_{12,3}(16n + 11) ≡ \overline{C}_{12,3}(16n + 15) ≡ 0 (mod 8);  (17)
\overline{C}_{12,3}(16n + 3) ≡ 0 (mod 8);  (18)

and

\overline{C}_{12,3}(16n + 7) ≡ 0 (mod 8).  (19)

Combining (16), (17), (18) and (19) we complete the proof of the theorem. Corollary 10. We have:

1. Let p ≥ 5 be a prime with p ≡ 3 (mod 4). For α, n ≥ 0, if p ∤ n then

p_{3,3} (16p^{2α+1}n + \frac{10p^{2α+2} − 1}{3}) ≡ 0 (mod 2).

2. Let p ≥ 5 be a prime with (\frac{−2}{p}) = −1. For α, n ≥ 0, if p ∤ n then

p_{3,3} (16p^{2α+1}n + \frac{22p^{2α+2} − 1}{3}) ≡ 0 (mod 2).

Proof. By Corollary 4.2 and Corollary 4.3 of Li and Yao [8], for all n, we have

\overline{C}_{12,3} (16p^{2α+1}n + \frac{10p^{2α+2} − 1}{3}) ≡ 0 (mod 8)  (20)

and

\overline{C}_{12,3} (16p^{2α+1}n + \frac{22p^{2α+2} − 1}{3}) ≡ 0 (mod 8).  (21)

Now, combining (16), (20) and (21) we complete the proof.

Pore and Fathima [9] found congruences for \overline{C}_{20,5}(n). Combining their results and Theorem 5 for t = 5, we obtain the following two theorems.

Theorem 11. For all α, n ≥ 0, we have

p_{5,5} (2 · 5^{2α+1}n + \frac{31 · 5^{2α} − 7}{12}) ≡ 0 (mod 2);  (22)
p_{5,5} (2 · 5^{2α+1}n + \frac{79 · 5^{2α} − 7}{12}) ≡ 0 (mod 2);  (23)

p_{5,5} (2 · 5^{2α+2}n + \frac{83 · 5^{2α+1} − 7}{12}) ≡ 0 (mod 2);

p_{5,5} (2 · 5^{2α+2}n + \frac{107 · 5^{2α+1} − 7}{12}) ≡ 0 (mod 2).

Proof. Combining [9, Theorem 1.6] and Theorem 5 for t = 5, we obtain the desired congruences satisfied by p_{5,5}(n).

Theorem 12. Let p ≥ 5 be a prime such that (\frac{−10}{p}) = −1 and j = 1, 2, …, p − 1. Then, for all α, n ≥ 0,

p_{5,5} (2p^{2α+1}(pn + j) + 7 × \frac{p^{2α+2} − 1}{12}) ≡ 0 (mod 2).

Proof. Combining [9, Theorem 1.7] and Theorem 5 for t = 5, we obtain the desired congruences satisfied by p_{5,5}(n).

Theorem 13. For all α ≥ 0, we have

p_{7,7} (2 · 7^{2α+1}n + \frac{(11 + 12r) · 49^{α} − 5}{6}) ≡ 0 (mod 2), r ∈ \{3, 4, 6\}  (24)

and

p_{7,7} (2 · 49^{α+1}n + \frac{(12s + 5) · 7^{2α+1} − 5}{6}) ≡ 0 (mod 2), s ∈ \{2, 4, 5\}  (25)

Proof. In the proof of Theorem 5.1, Li and Yao [8] found two congruences satisfied by \overline{C}_{28,7}(n), for example see [8, (5.22) & (5.23)]. Combining these two congruences and Theorem 5 for t = 7, we complete the proof of the theorem.

Remark 14. Da Silva and Sellers [7, Theorem 11] found relations between p_{t,t}(n) and the t-core partition functions. They used certain congruences for t-core partition functions (obtained by Radu and Sellers [10] using modular forms) to find several congruences satisfied by p_{t,t}(n) when t = 5, 7, 11, 13, 17, 19, 23. They also proposed to find a fully elementary proof of their congruences listed in Theorem 11. Putting α = 0 in (22) and (23), we obtain the congruences for p_{5,5}(n) listed in [7, Theorem 11]. Again, putting α = 0 in (24), we obtain the congruences for p_{7,7}(n) listed in [7, Theorem 11].

Theorem 15. Let p ≥ 5 be a prime with (\frac{−21}{p}) = −1. We have:

1. For n, α, β ≥ 0,

p_{7,7} (2 · 7^{2α+1}p^{2β}n + \frac{(11 + 12r) · 49^{α}p^{2β} − 5}{6}) ≡ 0 (mod 2)

and

p_{7,7} (2 · 49^{α+1}p^{2β}n + \frac{(5 + 12s) · 7^{2α+1}p^{2β} − 5}{6}) ≡ 0 (mod 2),

where r ∈ {3, 4, 6} and s ∈ {2, 4, 5}.

2. For n, α, β ≥ 0, if p ∤ n then

p_{7,7} (2 · 49^{α}p^{2β+1}n + \frac{11 · 49^{α}p^{2β+2} − 5}{6}) ≡ 0 (mod 2).

Proof. Li and Yao [8, Theorem 5.1 and 5.2 ] proved that, for all n ≥ 0,

\overline{C}_{28,7} (2 · 7^{2α+1}p^{2β}n + \frac{(11 + 12r) · 49^{α}p^{2β} − 5}{6}) ≡ 0 (mod 4);  (26)
\overline{C}_{28,7} (2 · 49^{α+1}p^{2β}n + \frac{(5 + 12s) · 7^{2α+1}p^{2β} − 5}{6}) ≡ 0 (mod 4)  (27)

and

\overline{C}_{28,7} (2 · 49^{α}p^{2β+1}n + \frac{11 · 49^{α}p^{2β+2} − 5}{6}) ≡ 0 (mod 4).  (28)

Combining (26), (27), (28), and Theorem 5 with t = 7, we find the desired congruences satisfied by p_{7,7}(n).

References

[1] Z. Ahmed and N. D. Baruah, New congruences for Andrews’ singular overpartitions, Int. J. Number Theory 11 (2015), 2247–2264.

[2] G. E. Andrews, Singular overpartitions, Int. J. Number Theory 11 (2015), 1523–1533.

[3] G. E. Andrews and D. Newman, The minimal excludant in integer partitions, J. Integer Sequences 23 (2020), Article 20.2.3.

[4] R. Barman and A. Singh, On Mex-related partition functions of Andrews and Newman,

preprint, 2020. Available at http://arxiv.org/abs/2009.11602.

[5] S. C. Chen, M. D. Hirschhorn and J. A. Sellers, Arithmetic properties of Andrews’ singular overpartitions, Int. J. Number Theory 11 (2015), 1463–1476.

[6] S. Corteel and J. Lovejoy, Overpartitions, Trans. Amer. Math. Soc. 356 (2004), 1623– 1635.

[7] R. da Silva and J. A. Sellers, Parity considerations for the Mex-related partition functions of Andrews and Newman, J. Integer Sequences 23 (2020), Article 20.5.7.

[8] X. Li, O. X. M. Yao, New infinite families of congruences for Andrews’ (K, I)-singular overpartitions, Quaest. Math. 41 (2018), 1005–1019.

[9] U. Pore and S. N. Fathima, Some congruences modulo 2, 8 and 12 for Andrews’ singular overpartitions, Note Mat. 38 (2018), 101–114.

[10] S. Radu and J. A. Sellers, Parity results for broken k-diamond partitions and (2k + 1)- cores, Acta Arith. 146 (2011), 43–52.

2010 Mathematics Subject Classification: Primary 05A17; Secondary 11P83. Keywords: minimal excludant, mex function, partition, singular overpartition.

Received February 22 2021; revised versions received February 23 2021; May 17 2021; May 18 2021. Published in Journal of Integer Sequences, May 18 2021.

Return to Journal of Integer Sequences home page.