Proving two partition identities

Proving two partition identities

R. da Silva^{I, 1}; J.C. Filho^{II, 2}; J.P.O. Santos^{III, 3}

^ICentro de Matemática, Computação e Cognição, UFABC, 09210-170 Santo André, SP, Brasil

^IIDepartamento de Matemática e Computação, ICE - UNIFEI, 37500-903 Itajubá, MG, Brasil

^IIIDepartamento de Matemática Aplicada, IMECC - UNICAMP, 13083-859 Campinas, SP, Brasil

ABSTRACT

In this paper we give combinatorial proofs for two partition identities. The first one solves a recent open question formulated by G. E. Andrews.

Keywords: Partition Identities, Combinatorics, Ferrers diagram.

RESUMO

Neste artigo fornecemos provas combinatórias para duas identidades de partições. A primeira resolve uma questão recentemente formulada por G. E. Andrews.

Palavras-chave: Identidades de partições, Combinatória, Diagrama de Ferrers.

1. Introduction

In [2] Andrews considers a variety of parity questions related to partition identities. We mention two of them just to illustrate results of this nature. The first one is a well known identity given by Euler.

Euler's Partition Identity. The number of partitions of any positive integer n into distinct parts equals the number of partitions of n into odd parts.

In terms of generating functions: for |q| < 1,

The second one is due to B. Gordon ([6], [7]) and H. Göllnitz ([4], [5]) that independently introduced parity considerations as follows:

First Gollnitz-Gordon Identity. The number of partitions of n into distinct non-consecutive parts with no even parts differing by exactly 2 equals the number of partitions of n into parts ≡ 1, 4, or 7 (mod 8).

At the end of [2], he presents a list of problems. In Section 2 we have a few definitions and the description of the Problem 5. In Section 3 we present our solution to the problem. In the last section we have a bijective proof for an identity related to the Durfee square of partitions.

2. Solving an Andrews's Problem

It is important to mention that a solution for Problem 5 of Andrews's paper [2] was given in [8]. However, Yee's proof is based on some results related to generating functions for partitions and q-series. Our proof, although equivalent to Yee's one, deals with combinatorial aspects of the Ferrers diagram of a partition.

Before presenting the proof we recall a few definitions from [1] and [2].

Definition 2.1. A partition of a positive integer n is a colection of positive integers λ₁, λ₂, ..., λ_ssuch that λ₁> λ₂> · · · > λ_s. Each λ_iis called a part of the partition. We denote a partition by λ₁ + λ₂ + · · · + λ_sor (λ₁, λ₂, ..., λ_s).

Example 2.1.The five partitions of 4 are: 4, 3 + 1, 2 + 2, 2 + 1 + 1, 1 + 1 + 1 + 1.

Definition 2.2. A Ferrers diagram of a partition λ₁ + λ₂ +· · ·+ λ_s is an array of dots left justified having λ₁dots in the first row, λ₂dots in the second row and so on.

Definition 2.3. Given a partition, the largest possible square of dots, starting in the upper left-hand corner, contained in its Ferrers diagram is called the Durfee square of this partition.

Example 2.2.Below we have the Ferrers diagram of the partition 8 + 7 + 4 + 3 with the Durfee square indicated.

Definition 2.4.Let λ = λ₁ + λ₂ + · · · + λ_jbe a partition, where λ₁> λ₂> · · · > λ_j . The lower odd parity index of λ, I_lo(λ), is defined as the maximum length of nondecreasing subsequences of {λ₁, λ₂, ..., λ_j} whose terms alternate in parity starting with an odd λ_i.

Example 2.3.

I_lo(11 + 11 + 7 + 6 + 6 + 3 + 2 + 1 + 1) = 5

I_lo(14 + 11 + 8 + 6 + 6 + 3 + 3 + 2) = 4

In [2] Andrews asks for a combinatorial proof for

where p_o(r, m, N) is the number of partitions of N into m distinct parts with lower odd parity index equal to r.

Consider, for example, the tables below. In the first columns we have partitions into distinct parts. The lower odd parity indices are shown in the second columns.

From these tables it is easy to see that and

In order to present a combinatorial proof of (2.2), we identify each partition with its Ferrers diagram, the standard graphical representation of a partition.

The combinatorial proof we are going to show is based on the one-to-one correspondence found by F. Franklin in 1881 to prove Legendre's combinatorial version of Euler's pentagonal number theorem (see [1]).

3. The Combinatorial Proof

Let D_e(N) (resp. D_o(N)) be the number of partitions of N into distinct parts having even (resp. odd) lower odd parity index. Then, we can rephrase (2.2) in the following theorem.

Theorem 3.1.

Proof. We shall establish a one-to-one correspondence between the partitions enumerated by D_e(N) and those enumerated by D_o(N). This correspondence will not work for some partitions of the numbers of the form N = n(3n+1)/2 and N = n(3n+5)/2 + 1.

Each partition λ = λ₁ + λ₂ + · · ·+ λ_s of N into distinct parts has a smallest part λ_s and the largest part λ₁ is the first element of a decreasing sequence of l consecutive integers that are parts of λ. For example, below we have the Ferrers diagrams of the partitions 8+7+4+3 and 8+7+6+3+2 with their parameter λ_s and l.

In order to establish the one-to-one correspondence we consider the following two cases.

Case 1: λ_s > l. In this case, we remove 1 from the parts λ₁, λ₂, ..., λ_l and form a new smallest part of size l. For example,

8+7+4+3 7+6+4+3+2,

or, in terms of the graphical representation,

Case 2: λ_s< l. In this case, we remove the smallest part and add 1 to the first λ_s of the l largest parts. For example,

8+7+6+3+2 9+8+6+3,

that is,

Note that exactly one case is applicable to a given partition into distinct parts. Then, it seems that the mapping establishes a one-to-one correspondence. This is true except for certain partitions like 5+4+3.

The procedure described in Case 1 is not applicable to those partitions having l parts and λ_s = l+1, in which case the number being partitioned is

While the procedure described in Case 2 is not applicable to those partitions having l parts and λ_s = l, in which case the number being partitioned is

where m = l - 1. For example, Case 1 and Case 2 are not applicable to

Note that partitions of the form λ = (2l - 1) + · · ·+ (l + 1) + l have an odd lower odd parity index: if l is odd, then I_lo(λ) = l; if l is even, then I_lo(λ) = l - 1. We also note that partitions of the form λ = 2l + · · ·+ (l + 2) + (l + 1) have an even lower odd parity index: if l is odd, then I_lo(λ) = l - 1; if l is even, then I_lo(λ) = l.

In order to finish the proof we shall show that the correspondence described above changes the parity of the index I_lo(λ), when λ = λ₁ + λ₂ + · · ·+ λ_s is neither a partition into l parts with λ_s = l nor a partition into l parts with λ_s = l + 1. Again, we consider the two cases above.

We will call m the image partition obtained from λ = λ₁ + λ₂ + · · ·+ λ_l + λ_l+1 + · · ·+ λ_s by the correspondence above, i.e.,

We shall show that I_lo(λ) and I_lo(µ) have different parities.

Case 1: λ_s > l. We split this case into two sub-cases.

Sub-case 1.1: s = l. In this sub-case, λ = λ₁ + λ₂ + · · ·+ λ_l and µ = (λ₁ - 1) + (λ₂ -1) + · · ·+ (λ_l - 1) + l. Hence, if I_lo(λ) = l, then λ_l is odd and, consequently,

• if l is odd, then I_lo(µ) = l + 1.

• if l is even, then I_lo(µ) = l - 1.

If I_lo(λ) = l - 1, then λ_l is even and, consequently,

• if l is odd, then I_lo(µ) = l.

• if l is even, then I_lo(µ) = l.

For example

Sub-case 1.2: s > l. The parts λ₁, λ₂, ..., λ_l of λ alternate in parity. The same is true for the parts (λ₁ - 1), (λ₂ - 1), ..., (λ_l - 1) of µ. Then in order to determine the index I_lo(µ) in terms of I_lo(λ) we have to look at the parity of λ_l and λ_l+1 as well as the parity of l and λ_s. While the parity of λ_l is changed by the correspondence, the same does not happen with λ_l+1. For example,

As there are two choices for the parities of l, λ_l, λ_l+1, and λ_s, then we have to verify, in this sub-case, all the 2⁴ possibilities. In the table below we summarise the results. In this table e means even and o means odd.

Case 2: λ_s< l. Again we consider two sub-cases.

Sub-case 2.1: λ_s < l. In this case the operation of removing the smallest part of

and adding 1 to each of its first λ_s largest parts produces

Since and have opposite parities, we have to consider the parities of λ_s, λ_s-1, and . For example, by this operation we can obtain

The table below shows the 2³ possibilities and the values of I_lo(µ) in terms of I_lo(λ).

Sub-case 2.2: λ_s = l. In this sub-case we have to look at the parity of λ_s, λ_s-1, λ_l, and λ_l+1. For instance, if λ_s and λ_s-1 are odd and both λ_l and λ_l+1 are even, then I_lo(µ) = I_lo(λ) + 1. We summarize all possibilities in the next table.

As one can see from the tables above the parity of the index

I_lo is always changed by the one-to-one correspondence described above. Therefore the theorem is proved.

4. The Second Bijection

In this section we present a bijective proof for the partition identity below, which is related to the Durfee square (the largest possible square contained within the Ferrers diagram of a partition starting in the upper left-hand corner). This identity comes from two different interpretations for the Fibonacci Numbers (see [3]).

Theorem 4.2. The number of partitions into at most n parts without gaps ⁴ and having each part appearing at least twice is equal to the number of partitions where the largest part is the side of its Durfee square and the largest part plus the number of parts is less than or equal to n.

Proof. Let n be a non-negative integer. Define as the set of partitions into at most n parts without gaps and having each part appearing at least twice. We also define as being the set of partitions where the largest part is the side of its Durfee square and the largest part plus the number of parts is less than or equal to n.

Note that the empty partition is in both sets and , consequently we associate these two partitions. Now we describe the bijection between and for non-empty partitions. In order to do this we associate each partition with its Ferrers diagrams.

Given a non-empty partition λ ∈ , with largest part p, we obtain a partition µ ∈ , with Durfee square of side p, by the following procedure:

• delete p dots from the first column of λ

• move dots from the columns to the columns , respectively.

For example, consider the partition λ = 7 + 7 + 6 + 6 + 6 + 6 + 5 + 5 + 4 + 4 + 4 + 3 + 3 + 3 + 3 + 2 + 2 + 2 + 1 + 1 + 1. The partition m obtained by the above procedure is µ = 7 + 7 + 7 + 7 + 7 + 7 + 7 + 6 + 6 + 4 + 3 + 3 + 2 + 1 or, in terms of Ferrers diagrams,

It is important to note that the resulting diagram still represents a partition. This is ensured by the condition that each part appears at least twice.

Given a partition σ ∈ we obtain a partition δ ∈ just by inverting the above procedure. As an example, consider σ = 6 + 6 + 6 + 6 + 6 + 6 + + 3 + 1; the partition associated to σ is δ = 6 + 6 + 5 + 5 + 4 + 4 + 3 + 3 + 3 + 2 + 2 + 1 + 1 + 1:

Recebido em 13 Setembro 2011; Aceito em 20 Maio 2012.

• [1] G.E. Andrews, "The Theory of Partitions", Cambridge University Press, 1984.
• [2] G.E. Andrews, Parity in partition identities, The Ramanujan Journal, 23 (2010), 45-90.
• [3] J.C. Filho, "Variações do Diagrama de Ferrers, Partições Planas e Funções Geradoras", Tese de Doutorado, IMECC, UNICAMP, Campinas, SP, 2006.
• [4] H. Göllnitz, Einfache Partionen, Diplomarbeit W.S. 1960, Göttingen, 65 pp.
• [5] H. Göllnitz, Partitionen mit Differenzenbedingungen, J. Reine Angew. Math, 225 (1967), 154-190.
• [6] B. Gordon, Some Ramanujan-like continued fractions, Abstracts of Short Com- munications, Int. Congr. of Math., 29-30, Stockholm, 1962.
• [7] B. Gordon, Some continued fractions of the Rogers-Ramanujan type, Duke Math. J., 31 (1965), 741-748.
• [8] A.J. Yee, Ramanujan's partial theta series and parity in partitions, The Ramanujan Journal, 23 (2010), 215-225.

4

all positive integer less than or equal to the largest part are parts of the partition

Publication Dates

• Publication in this collection
  15 Oct 2012
• Date of issue
  2012

History

• Received
  13 Sept 2011
• Accepted
  20 May 2012