On a Ramanujan-type Congruence for Bipartitions with 5-Cores

Ranganatha Dasappa Department of Studies in Mathematics University of Mysore, Manasagangotri Mysuru-570006 Karnataka India

                    ddranganatha@gmail.com

Abstract In this short note, we prove a Ramanujan-type congruence modulo 5^{α} (α ≥ 1) for

A_{5}(n), which counts the number of 5-core bipartitions of n.

1 Introduction

A partition of a positive integer n is a finite non-increasing sequence of positive integers λ_{1}, λ_{2}, …, λ_{r} such that ∑_{i=1}^{r} λ_{i} = n. The λ_{i} are called the parts of the partition. For example, the partitions of 4 are 4, 3 + 1, 2 + 2, 2 + 1 + 1, 1 + 1 + 1 + 1.

Given a partition [λ] = λ_{1} + λ_{2} + ⋯ + λ_{r} of n, where λ_{1} ≥ λ_{2} ≥ ⋯ ≥ λ_{r}, the Ferrers-Young diagram of [λ] is an array of nodes with λ_{i} nodes in the i^{th} row. The (i, j) hook is the set of nodes directly below, together with the set of nodes directly to the right of the (i, j) nodes, as well as the (i, j) node itself. The hook number of (i, j), denoted by H(i, j), is the total number of nodes in the (i, j) hook. For a positive integer t ≥ 2, a partition of n is said to be t-core if it has no hook numbers that are multiples of t.

Example 1. The Ferrers-Young diagram of the partition λ = 5 + 3 + 2 of 10 is

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The nodes (1, 1), (1, 2), (1, 3), (1, 4), (1, 5), (2, 1), (2, 2), (2, 3), (3, 1) and (3, 2) have hook numbers 7, 6, 4, 2, 1, 4, 3, 1, 2 and 1, respectively. Therefore λ is 5-core. Note that λ is t-core for t ≥ 8.

A bipartition of n is a pair of partitions (λ, μ) such that the sum of all the parts of λ and μ equals n. A bipartition with t-core of n is a bipartition (λ, μ) of n such that λ and μ are both t-cores. Let A_{t}(n) denote the number of bipartitions with t-cores of n. The generating function for A_{t}(n) is given by

∑_{n=0}^{∞} A_{t}(n)q^{n} = \frac{(q^{t}; q^{t})_{∞}^{2t}}{(q; q)_{∞}^{2}} .  (1)

Here and throughout this note, we assume that |q| < 1 and we follow standard q-series notation:

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

Motivated by the work of Ramanujan on congruences for unrestricted partition function p(n), many mathematicians considered the function A_{t}(n) and studied its congruence properties. For example, Lin [3] established several congruences for A_{3}(n). Soon after, Xia [8] and Yao [9] extended the list of congruences for A_{3}(n).

The main aim of this note is to prove the following Ramanujan-type congruence modulo 5^{α} (α ≥ 1) for A_{5}(n):

A_{5}(5^{α}n + 5^{α} − 2) ≡ 0 (mod 5^{α}), α ≥ 1.

The following 5-dissection formula for (q; q)_{∞} was first stated by Ramanujan [4, p. 212] without proof.

Lemma 2. [4, p. 212] We have

(q; q)_{∞} = (q^{25}; q^{25})_{∞}(\frac{1}{R(q^{5})} − q − q^{2}R(q^{5})),  (2)

where R(q) = \frac{(q;q^{5})_{∞}(q^{4};q^{5})_{∞}}{(q^{2};q^{5})_{∞}(q^{3};q^{5})_{∞}}.

Watson [5] presented a proof of (2) using the quintuple product identity.

Lemma 3. [2, eq. (7.4.14), p. 165 ] We have

\frac{1}{(q; q)_{∞}} =\frac{(q^{25}; q^{25})_{∞}^{5}}{(q^{5}; q^{5})_{∞}^{6}} (\frac{1}{R(q^{5})^{4}} + \frac{q}{R(q^{5})^{3}} + \frac{2q^{2}}{R(q^{5})^{2}} + \frac{3q^{3}}{R(q^{5})} + 5q^{4} − 3q^{5}R(q^{5})
+ 2q^{6}R(q^{5})^{2} − q^{7}R(q^{5})^{3} + q^{8}R(q^{5})^{4}).  (3)

The following lemmas are useful to prove our main congruence for A_{5}(n): Lemma 4. Let ∑_{n=0}^{∞} a(n)q^{n} = \frac{q}{(q;q)_{∞}^{2}} . Then

∑_{n=0}^{∞} a(5n + 4)q^{n} = 125q \frac{(q^{5}; q^{5})_{∞}^{10}}{(q; q)_{∞}^{12}} + 10\frac{(q^{5}; q^{5})_{∞}^{4}}{(q; q)_{∞}^{6}} .  (4)

Proof. In view of (3), we have

∑_{n=0}^{∞} a(n)q^{n} =\frac{(q^{25}; q^{25})_{∞}^{10}}{(q^{5}; q^{5})_{∞}^{12}} (\frac{q}{R(q^{5})^{8}} + \frac{2q^{2}}{R(q^{5})^{7}} + \frac{5q^{3}}{R(q^{5})^{6}} + \frac{10q^{4}}{R(q^{5})^{5}} + \frac{20q^{5}}{R(q^{5})^{4}} + \frac{16q^{6}}{R(q^{5})^{3}}

+ \frac{27q^{7}}{R(q^{5})^{2}} + \frac{20q^{8}}{R(q^{5})} + 15q^{9} − 20q^{10}R(q^{5}) + 27q^{11}R(q^{5})^{2} − 16q^{12}R(q^{5})^{3}

+ 20q^{13}R(q^{5})^{4} − 10q^{14}R(q^{5})^{5} + 5q^{15}R(q^{5})^{6} − 2q^{16}R(q^{5})^{7} + q^{17}R(q^{5})^{8}). (5)

Extracting the terms involving q^{5n+4} in (5), dividing by q^{4} and replacing q^{5} by q, we obtain

∑_{n=0}^{∞} a(5n + 4)q^{n} =\frac{(q^{5}; q^{5})_{∞}^{10}}{(q; q)_{∞}^{12}} (\frac{10}{R(q)^{5}} + 15q − 10q^{2}R(q)^{5})
=\frac{(q^{5}; q^{5})_{∞}^{10}}{(q; q)_{∞}^{12}} (10(\frac{1}{R(q)^{5}} − 11q − q^{2}R(q)^{5}) + 125q).  (6)

Berndt [2, Thm. 7.4.4] proved the following identity:

\frac{1}{R(q)^{5}} − 11q − q^{2}R(q)^{5} = \frac{(q; q)_{∞}^{6}}{(q^{5}; q^{5})_{∞}^{6}} .  (7)

Employing (7) in (6), we obtain (4).

In a similar way, we have the following: Lemma 5. Let ∑_{n=0}^{∞} b(n)q^{n} = \frac{1}{(q;q)_{∞}^{2}} . Then

∑_{n=0}^{∞} b(5n + 3)q^{n} = 125q \frac{(q^{5}; q^{5})_{∞}^{10}}{(q; q)_{∞}^{12}} + 10\frac{(q^{5}; q^{5})_{∞}^{4}}{(q; q)_{∞}^{6}} .

Lemma 6. Let ∑_{n=0}^{∞} c(n)q^{n} = (q; q)_{∞}^{4}. Then

∑_{n=0}^{∞} c(5n + 4)q^{n} = −5(q^{5}; q^{5})_{∞}^{4}.

Theorem 7. Let α be a integer ≥ 1. Then

∑_{n=0}^{∞} A_{5}(5^{α}n + 5^{α} − 2)q^{n} = 5^{3α}q \frac{(q^{5}; q^{5})_{∞}^{10}}{(q; q)_{∞}^{2}} + 5^{α}(\frac{25^{α} − (−1)^{α}}{13} )(q; q)_{∞}^{4}(q^{5}; q^{5})_{∞}^{4}.  (8)

Proof. From (1) and Lemma 5, we have

∑_{n=0}^{∞} A_{5}(5n + 3)q^{n} =(q; q)_{∞}^{10}(125q \frac{(q^{5}; q^{5})_{∞}^{10}}{(q; q)_{∞}^{12}} + 10\frac{(q^{5}; q^{5})_{∞}^{4}}{(q; q)_{∞}^{6}} )

=10(q; q)_{∞}^{4}(q^{5}; q^{5})_{∞}^{4} + 125q \frac{(q^{5}; q^{5})_{∞}^{10}}{(q; q)_{∞}^{2}} ,

which is same as (8) with α = 1. Suppose that (8) holds for some α ≥ 1. From (8), Lemmas 4 and 6, we deduce

∑_{n=0}^{∞} A_{5}(5^{α+1}n + 5^{α+1} − 2)q^{n}

= 5^{3α}(q; q)_{∞}^{10}(125q \frac{(q^{5}; q^{5})_{∞}^{10}}{(q; q)_{∞}^{12}} + 10\frac{(q^{5}; q^{5})_{∞}^{4}}{(q; q)_{∞}^{6}} ) − 5^{α+1}(\frac{25^{α} − (−1)^{α}}{13} )(q; q)_{∞}^{4}(q^{5}; q^{5})_{∞}^{4}

= 5^{3α+3}q \frac{(q^{5}; q^{5})_{∞}^{10}}{(q; q)_{∞}^{2}} + (10 · 5^{3α} − 5^{α+1}(\frac{25^{α} − (−1)^{α}}{13} ))(q; q)_{∞}^{4}(q^{5}; q^{5})_{∞}^{4}

= 5^{3α+3}q \frac{(q^{5}; q^{5})_{∞}^{10}}{(q; q)_{∞}^{2}} + 5^{α+1}(\frac{25^{α+1} − (−1)^{α+1}}{13} )(q; q)_{∞}^{4}(q^{5}; q^{5})_{∞}^{4}.

That is, (8) holds for α + 1. This completes the proof by induction of (8).

From (8), we have the following congruence relation:

Theorem 8. For all integers n ≥ 0 and α ≥ 1,

A_{5}(5^{α}n + 5^{α} − 2) ≡ 0 (mod 5^{α}).  (9)

2 Acknowledgments

The author would like to thank the referee for his/her valuable comments. The author would also like to thank Prof. Chadrashekar Adiga for his advice and guidance.

References

[1] C. Adiga, B. C. Berndt, S. Bhargava, and G. N. Watson, Chapter 16 of Ramanujan’s Second Notebook: Theta Functions and q-Series, Mem. Amer. Math. Soc., 1985.

[2] B. C. Berndt, Number Theory in the Spirit of Ramanujan, Amer. Math. Soc., 2006.

[3] B. L. S. Lin, Some results on bipartitions with 3-core, J. Number Theory 139 (2014), 44–52.

[4] S. Ramanujan, Collected Papers, Cambridge University Press, Cambridge, 1927.

[5] G. N. Watson, Ramanujan’s Vermutung über Zerfällungsanzahlem, J. reine angew. Math. 179 (1938), 97–128.

[6] G. N. Watson, Theorems stated by Ramanujan (VII): theorems on contuined fractions, J. London Math. Soc. 4 (1929), 39–48.

[7] G. N. Watson, Theorems stated by Ramanujan (IX): two continued fractions, J. London Math. Soc. 4 (1929), 231–237.

[8] E. X. W. Xia, Arithmetic properties of bipartitions with 3-cores, Ramanujan J. 38 (2015), 529–548.

[9] O. Y. M. Yao, Infinite families of congruences modulo 3 and 9 for bipartitions with 3-cores, Bull. Aust. Math. Soc. 91 (2015), 47–52.

2010 Mathematics Subject Classification: Primary 05A17; Secondary 11P83. Keywords: congruence, partition, bipartition, core partition.

Received September 5 2016; revised versions received September 15 2016; September 16 2016; September 19 2016. Published in Journal of Integer Sequences, October 1 2016.

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