# Some Elementary Partition Inequalities and Their Implications

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## Abstract

We prove various inequalities between the number of partitions with the bound on the largest part and some restrictions on occurrences of parts. We explore many interesting consequences of these partition inequalities. In particular, we show that for \(L\ge 1\), the number of partitions with \(l-s \le L\) and \(s=1\) is greater than the number of partitions with \(l-s\le L\) and \(s>1\). Here *l* and *s* are the largest part and the smallest part of the partition, respectively.

## Keywords

Partition inequalities Partitions with bounded differences between largest and smallest parts Non-negative*q*-series expansions Injective maps

*q*-Binomial theorem Heine transformations Jackson transformation

## Mathematics Subject Classification

05A15 05A17 05A19 05A20 11B65 11P81 11P84 33D15## 1 Introduction

*partition*(shown in

*frequency representation*[2]), where the exponents \(f_i\) are the number of occurrences of

*i*. The numbers

*i*with non-zero frequencies in \(\pi \) are called

*parts*of \(\pi \). Since there are only finitely many non-zero frequencies in a partition \(\pi \), the sum

*norm*of the partition \(\pi \). To shorten the notation, one can ignore the zero frequencies; we keep the option of writing any zero frequencies that need emphasizing. As an example \(\pi = (1^4, 3^2,4^0,10^1)\) is a partition of 20 (meaning \(|\pi | = 20\)), where 1 appears as a part with frequency 4, 3 appears twice, 4 is not a part, and part 10 only appears once in \(\pi \). The partition where all the frequencies are equal to zero is a conventional and unique partition of 0.

- (i)
Let \({\mathcal {A}}_{L,1}\) be the set of partitions with the smallest part being 1, where all the parts \(\le L+1\) and \(f_L = \delta _{L,1}\),

- (ii)
and let \({\mathcal {A}}_{L,2}\) be the set of non-empty partitions where the parts are in the domain \(\{2,3,\dots , L+1 \}\).

### Theorem 1.1

Elementary combinatorial inequalities, such as (1.1), have interesting implications for *q*-series and the theory of partitions. This simple observation about the magnitude of sets, in this case, implies non-negativity results for a refinement of an earlier discussed-weighted partition identity result [10]. We introduce that result and its refinement here.

- (i)
\(s(\pi )\) denote the smallest part of the partition \(\pi \),

- (ii)
\(l(\pi )\) denote the largest part of \(\pi \),

- (iii)
\(\nu (\pi ):= \sum _{i\ge 1} f_i\) denote the total number of parts in \(\pi \),

- (iv)
\(r(\pi ):=l(\pi )-\nu (\pi )\), rank of \(\pi \).

- (i)
\(f_i\equiv 1\mod 2\), for \(1\le i\le t(\pi )\),

- (ii)
\(f_{t(\pi )+1}\equiv 0 \mod 2\).

*length of the initial odd-frequency chain*. With this new statistic, the authors have proven a new combinatorial identity of partitions.

### Theorem 1.2

One example of Theorem 1.2 is given in Table 1.

The total count of partitions of a positive integerN,counted with the weight1if the smallest part is odd, and\(-1\)if the smallest part is even, is the same as the total of all odd-frequency chain lengths of partitions ofN.

Example of Theorem 1.2 with \(|\pi |=6\)

\(\pi \in \mathcal {U}\) | \((-1)^{s(\pi )+1}\) | \(t(\pi )\) |
---|---|---|

\((1^6)\) | 1 | 0 |

\((1^4,2^1)\) | 1 | 0 |

\((1^3,3^1)\) | 1 | 1 |

\((1^2,2^2)\) | 1 | 0 |

\((1^2,4^1)\) | 1 | 0 |

\((1^1,2^1,3^1)\) | 1 | 3 |

\((1^1,5^1)\) | 1 | 1 |

\((2^3)\) | \(-1\) | 0 |

\((2^1,4^1)\) | \(-1\) | 0 |

\((3^2)\) | 1 | 0 |

\((6^1)\) | \(-1\) | 0 |

Total | 5 | 5 |

*non-negativity*of a series

*n*, \(a_n\ge 0\), where

*q*is a formal summation variable. We denote the non-negativity by the notationOne important observation about Theorem 1.2 is that the statistic \(t(\pi )\) is non-negative for any partition \(\pi \). It is clear thatwhere \(p_t(n)\) is the total weighted count of partitions with the

*t*statistic. This implies that the series in (1.2) is non-negative. Written in analytic form, the identity (1.2) is equivalent to

In this work, we introduce a refinement of Theorem 1.2 where we put a bound on the difference between the largest and the smallest parts of partitions. We prove that

### Theorem 1.3

Section 2 has a short repertoire of basic hypergeometric identities that will be referred to later. In Sect. 3, we are going to prove two inequalities between sets of partitions (Theorem 1.1 and an analogue) using only injections between sets, and later state some related open questions. We will state the analytic versions of some of the theorems of Sect. 3 and their implications in Sect. 4. We later will use the complements of the range of the injective maps of Sect. 3 to get new *q*-series summation formulas. Theorem 1.3 will be proven in Sect. 5. Section 6 has an excursion in different representations and an observably non-negative expression for the analytic expression of (1.4) of Theorem 1.3. An outlook section finishes the paper with a summary of open questions that arise from this study.

## 2 Some *q*-Hypergeometric Identities

*q*-hypergeometric functions and some of their related formulas that will be used later are stated here. Let

*r*and

*s*be non-negative integers and \(a_1,a_2,\dots ,a_r,b_1,b_2,\dots ,b_s,q,\) and

*z*be variables. Then

*a*,

*b*,

*c*,

*q*, and

*z*be variables. The

*q*-binomial theorem [13, II.3, P. 236] is

## 3 Two New Partition Inequalities

We start our discussion with a proof of Theorem 1.1. Recall

**Theorem**1.1 For any \(L\ge 2\) and \(N\ge 1 \),

For \(L\ge 2\), we prove the inequality (1.1) in an injective manner.

### Proof of Theorem 1.1

- (i)
if \(f_2>0\), then \(\gamma ^*(\pi ) = \big (1^{2f_2},2^0,3^{f_3}\big )\),

- (ii)
if \(f_2=0\) and \(f_3>0\), then \(\gamma ^*(\pi ) = \big (1^{3},2^0,3^{f_3-1}\big )\).

- (i)if \(2< s(\pi ) <L+1\), then$$\begin{aligned}&\gamma (\pi ) = \big (1^{[(f_L-\delta _{L,s(\pi )})\cdot L + 1]},(s(\pi )-1)^{1},s(\pi )^{f_{s(\pi )}-1},\dots , L^{0},(L+1)^{f_{L+1}}\big ), \end{aligned}$$
- (ii)
if \(s(\pi ) = L+1\), then \(\gamma (\pi ) = \big (1^{L+1},(L+1)^{f_{L+1}-1}\big )\),

- (iii)
if \(s(\pi ) = 2\), then \(\gamma (\pi ) = \big (1^{(f_L\cdot L + 2)},2^{f_2-1},\dots , L^{0},(L+1)^{f_{L+1}}\big )\).

*L*in the image is either 1 or 2. The remainder 2 comes from a unique case. In the remainder being 1 case, one can uniquely identify the pre-image by looking at the smallest part size that is greater than 1. This proves that \(\gamma \) is an injection and it is enough to show (1.1). \(\square \)

The interested reader is invited to examine [3, 6, 7, 8, 9] and [14] for other examples of injective combinatorial arguments and inequalities between the sizes of sets of partitions.

Example of Theorem 1.1 with \(L=3\) and \(N=12\), where the images of the map \(\gamma \) are also indicated

\(\pi \in {\mathcal {A}}_{3,2}\) | \(\gamma \) | \(\pi \in {\mathcal {A}}_{3,1}\) |
---|---|---|

\((1^{12})\) | ||

\((3^4)\) | \(\rightarrow \) | \((1^{10},2)\) |

\((2,3^2,4)\) | \(\rightarrow \) | \((1^8,4)\) |

\((2^3,3^2)\) | \(\rightarrow \) | \((1^8,2^2)\) |

\((1^6,2^3)\) | ||

\((1^6,2,4)\) | ||

\((4^3)\) | \(\rightarrow \) | \((1^4,4^2)\) |

\((1^4,2^2, 4)\) | ||

\((1^4, 2^4)\) | ||

\((2^2,4^2)\) | \(\rightarrow \) | \((1^2,2,4^2)\) |

\((2^4,4)\) | \(\rightarrow \) | \((1^2,2^3,4)\) |

\((2^6)\) | \(\rightarrow \) | \((1^2,2^5)\) |

- (i)
\({\mathcal {B}}_{L,1}\) to be the set of partitions such that the smallest part is 2, all the parts are \(\le L+2\) and \(f_{L+1} = \delta _{L,1}\),

- (ii)
\({\mathcal {B}}_{L,2}\) to be the set of non-empty partitions where the parts are in the domain \(\{3,4,\dots ,L+2 \}\).

### Theorem 3.1

Before the proof of Theorem 3.1, we examine the excluded initial cases of *L*. In the case \(L=1\), \({\mathcal {B}}_{1,1}\) is the set of partitions of type \(\big (2^1,3^{f_3}\big )\). Hence, all partitions of \({\mathcal {B}}_{1,1}\) have norm 2 modulo 3. The set \({\mathcal {B}}_{1,2}\) contains partitions only of the type \(\big (3^{1+f_3}\big )\), which 0 modulo 3 norm. Therefore, the inequality (3.1) cannot hold for all *N*. The sets \({\mathcal {B}}_{2,1}\) and \({\mathcal {B}}_{2,2}\) contain partitions exclusively of the type \(\big (2^{1+f_2},4^{f_4}\big )\) and \(\big (3^{f_3},4^{f_4}\big )\) with \(f_3+f_4>0\), respectively. It is easy to see that all the partitions in \({\mathcal {B}}_{2,1}\) have even norms, but for any \(k\ge 0\), there are partitions of odd norm \(4k+3\) in \({\mathcal {B}}_{2,2}\). Therefore, for \(L=2\), the inequality (3.1) does not hold for all *N* either.

### Proof of Theorem 3.1

- (i)
if \(f_4 >0\), then \(\pi \mapsto \big (2^{2f_4}, 3^{f_3},5^{f_5}\big )\),

- (ii)
if \(f_3=f_4=0\), then \(f_5>0\) and define \(\pi \mapsto \big (2^1,3^1,5^{f_5-1}\big )\),

- (iii)
if \(f_4=0\) and \(f_3>1\), then \(\pi \mapsto \big (2^3,3^{f_3 -2},5^{f_5}\big )\),

- (iv)
if \(f_4=0\) and \(f_3=1\), since \(|\pi |>3\), \(f_5>0\), then \(\pi \mapsto \big (2^1,3^{2},5^{f_5-1}\big )\).

- (i)
If \(f_{L+1}=f_{2m}>0\), then \(\pi \mapsto \big (2^{f_{2m}\cdot m},3^{f_3},\dots , (L+1)^0, (L+2)^{f_{L+2}}\big )\),

- (ii)if \(\exists i\in \{2,\dots , m-1\}\) such that \(f_{2i}>0\) and \(f_{2j}=0\), \(\forall j> i\), then$$\begin{aligned}&\pi \mapsto \big (2^{i},3^{f_3},\dots ,(2i)^{f_{2i}-1},\\&\quad (2i+1)^{f_{2i+1}},(2i+2)^0,\dots ,(L+1)^0, (L+2)^{f_{L+2}}\big ), \end{aligned}$$
- (iii)if \(\forall i \in \{2,\dots , m\}\), \(f_{2i}=0\) and \(s(\pi )\) is odd \(> 3\), then$$\begin{aligned} \pi \mapsto \big (2^{1},(s(\pi )-2)^1,(s(\pi )-1)^0,s(\pi )^{f_{s(\pi )}-1},\dots \big ), \end{aligned}$$
- (iv)
if \(\forall i \in \{2,\dots , m\}\), \(f_{2i}=0\) and \(f_3\ge 2\), then \(\pi \mapsto \big (2^{1},3^{f_3-2},4^1,5^{f_5},\dots \big )\),

- (v)
if \(\forall i \in \{2,\dots , m\}\), \(f_{2i}=0\) and \(f_3=1\), then since \(|\pi |>3\) there is a smallest positive \(j>1\) such that \(f_{2j+1}>0\), then \(\pi \mapsto \big (2^{1},(j+1)^2,(2j+1)^{f_{2j+1}-1},\dots \big ).\)

One can also view \(\Gamma _1^*\) for \(L=3\) as a derivation of \(\Gamma _1\). We use the Cases (i), (iii) and (v) of \(\Gamma _1\) as they are and modify the Case (iv) as \(\pi \mapsto \big (2^3,3^{f_3-2},5^{f_5}\big )\). The Case (ii) of \(\Gamma _1\) does not apply for \(L=3\).

- (i)
if \(f_5\) is positive even, then \(\pi \mapsto \big (2^{(f_5/2)5},3^{f_3},4^{f_4},5^{0},6^{f_6}\big )\),

- (ii)
if \(f_5\) is positive odd and if \(f_3>0\), then \(\pi \mapsto \big (2^{((f_5-1)/2)5+4},3^{f_3-1},4^{f_4},5^{0},6^{f_6}\big )\),

- (iii)
if \(f_5\) is positive odd and if \(f_3{=}0\), then \(\pi \mapsto \big (2^{((f_5-1)/2)5+1},3^{1},4^{f_4},5^{0},6^{f_6}\big ),\)

- (iv)if \(f_5=0\),
- (1)
and \(f_6>0\), then \(\pi \mapsto \big (2^{3},3^{1},4^{f_4},5^{0},6^{f_6-1}\big )\),

- (2)
or \(f_6=0\) and \(f_4>0\), then \(\pi \mapsto \big (2^{2},3^{1},4^{f_4-1},5^{0},6^{0}\big )\),

- (1)
- (v)if \(f_4=f_5=f_6=0\), since \(|\pi |\not = 3\),
- (1)
either \(f_3=2\), then \(\pi \mapsto \big (2^{1},3^{0},4^{1},5^{0},6^{0}\big )\),

- (2)
or \(f_3 \ge 3\), then \(\pi \mapsto \big (2^{1},3^{f_3-2},4^{1},5^{0},6^{0}\big )\).

- (1)

*L*. We define \(\Gamma _2\) for all even \(L= 2m \ge 6\). Let \(\pi = \big (3^{f_3},\dots ,(L+2)^{f_{L+2}}\big )\) be a partition in \({\mathcal {B}}_{L,2}\), with norm \(> 3\):

- (i)if \(f_{L+1}=f_{2m+1}\) is positive even, then$$\begin{aligned} \pi \mapsto \big (2^{(f_{2m+1}/2)(2m+1)},3^{f_3},\dots , (L+1)^0, (L+2)^{f_{L+2}}\big ), \end{aligned}$$
- (ii)if \(f_{2m+1}\) is positive odd and if \(\exists k \in \{2,\dots ,m\}\) with \(f_{2k-1}>0\), where \(\forall k<j<m+1\), \(f_{2j-1}=0\), then$$\begin{aligned}&\pi \mapsto \big (2^{[(f_{2m+1}-1)/2](2m+1) + (m+k)},\dots ,\\&\quad (2k-1)^{f_{2k-1}-1},\dots ,(L+1)^0, (L+2)^{f_{L+2}}\big ), \end{aligned}$$
- (iii)if \(f_{2m+1}\) is positive odd and \(\forall k \in \{2,\dots ,m\}\), \(f_{2k-1}=0\), then$$\begin{aligned}&\pi \mapsto \big (2^{[(f_{2m+1}-1)/2](2m+1) + 1}, \dots ,\\&\quad (2m-1)^{1},(2m)^{f_{2m}},(L+1)^0, (L+2)^{f_{L+2}}\big ), \end{aligned}$$
- (iv)
if \(f_{2m+1}=0\), and there exists the largest integer \(k \in \{2,\dots ,m +1\}\) such that \(f_{2k} >0\), then \(\pi \mapsto \left( 2^{k},3^{f_3}, \dots ,(2k)^{f_{2k}-1},\dots \right) ,\)

- (v)
if \(f_{2m+1}=0\), and \(\forall k \in \{2,\dots ,m +1\}\), \(f_{2k} = 0\) and if \(f_3=1\), then since \(|\pi |>3\) there exists the smallest positive integer \(i>1\) such that \(f_{2i+1}>0\), then \(\pi \mapsto (2^{1},(i+1)^2,(2i+1)^{f_{2i+1}-1},\dots )\),

- (vi)
if \(f_{2m+1}=0\), and \(\forall k \in \{2,\dots ,m +1\}\) such that \(f_{2k} = 0\) and if \(f_3>1\), then \(\pi \mapsto \big (2^{1},3^{f_3-2},4^1,\dots \big )\),

- (vii)
if \(f_{2m+1}=0\), and \(\forall k \in \{2,\dots ,m +1\}\) such that \(f_{2k} = 0\) and if \(f_{3}=0\) then there exists the smallest integer \(m>i>1\) such that \(f_{2i+1}>0\), then \(\pi \mapsto \big (2^{1},(2i-1)^1,(2i+1)^{f_{2i+1}-1},\dots \big )\).

For the \(L=4\) case, our injection \(\Gamma _2^*\) for partitions \(|\pi |\) with norm not equal to 3 or 9 can be related with \(\Gamma _2\), where we use Cases (i)–(iv) and (vi) with \(m=2\), where Case (vi) comes with the extra assertion that \(f_3-2 \not = 1\).

\(\square \)

Example of Theorem 3.1 with \(L=5\) and \(N=12\), where the images of the map \(\Gamma _1\) are also indicated

\(\pi \in {\mathcal {B}}_{5,2}\) | \(\Gamma _1\) | \(\pi \in {\mathcal {B}}_{5,1}\) |
---|---|---|

\((6^2)\) | \(\rightarrow \) | \((2^6)\) |

\((3^2,6)\) | \(\rightarrow \) | \((2^3,3^2)\) |

\((3^4)\) | \(\rightarrow \) | \((2,3^2,4)\) |

(3, 4, 5) | \(\rightarrow \) | \((2^2,3,5)\) |

(5, 7) | \(\rightarrow \) | (2, 3, 7) |

\((2^4,4)\) | ||

\((4^3)\) | \(\rightarrow \) | \((2^2,4^2)\) |

\((2,5^2)\) |

- (i)
\({\mathcal {C}}_{L,s,1}\) denotes the set of partitions where the smallest part is

*s*, all the parts are \(\le L+s\) and \(L+s-1\) does not appear as a part, - (ii)
\({\mathcal {C}}_{L,s,2}\) denotes the set of non-empty partitions where the parts are in the domain \(\{s+1,\dots ,L+s \}\).

### Conjecture 3.2

*s*, there exists

*M*, which only depends on

*s*, such that

The first two initial families of cases for \(s=1\) and 2 are Theorems 1.1 and 3.1 with \(M=1\) and \(M=10\). It should be noted that in a case when *L* tends to \(\infty \), this conjecture is nothing but a tautology.

In the definition of \({\mathcal {B}}_{L,i}\), we shifted the permissible part sizes of \({\mathcal {A}}_{L,i}\) up by one. Another route to take would be shifting the sets, but keeping the impermissible part *L* of \({\mathcal {A}}_{L,i}\) the same. For \(L\ge s+1\), let \({\mathcal {C}}^*_{L,s,1}\) be the set of partitions where the smallest part is *s*, all the parts are \(\le L+s\) and *L* does not appear as a part.

Similar to Conjecture 3.2, we also claim that

### Conjecture 3.3

*s*, there exists

*M*, which only depends on

*s*, such that

In Sect. 7, we will be reiterating these conjectures and state their analytic versions.

## 4 Some Analytic Non-negativity Results

Theorems 1.1 and 3.1 lead to new non-negativity results and some new summation formulas. The analytic analogue of Theorem 1.1 is the following:

### Theorem 4.1

We can, and will, extend the definition of \(H_{L,1}(q)\) for \(L=1\) case, but in this case, the expression simplifies to \(q/(1+q)\) and is not non-negative.

### Proof of Theorem 4.1

\(\square \)

*“reciprocal product take away one”*expressions of the form (4.2) will appear in our future calculations, and they are always going to be non-negative by this observation.

*L*,

- (i)
\(\big (1^1,2^0,3^{f_3}\big )\), where \(f_3\) is a non-negative integer,

- (ii)
\(\big (1^{2j+5},2^0,3^{f_3}\big )\) for

*j*and \(f_3\) non-negative integers

- (i)
for \(2< s < L+1\), \(\big (1^{tL+1},(s-1)^{k+2},\dots ,L^0,(L+1)^{f_{L+1}}\big )\), where

*t*and*k*are non-negative integers, - (ii)
\(\big (1^{kL+1},L^0,(L+1)^{f_{L+1}}\big )\), where \(k \ge 2\) or 0,

- (iii)
\(\big (1^{kL+r},\dots ,L^0,(L+1)^{f_{L+1}}\big )\), where \(r\in \{3,\dots ,L\}\) and

*k*is a non-negative integer.

Equating (4.1) and (4.5) yields the formula using combinatorial means only.

### Theorem 4.2

*L*,

A direct proof can also be given.

### Proof of Theorem 4.2

Taking the limit \(L\rightarrow \infty \) in (4.6), it is easy to get:

### Corollary 4.3

One can also give a direct *q*-hypergeometric proof of Corollary 4.3. This proof amounts to using the second Heine transformation (2.3) followed by the *q*-binomial theorem (2.1).

Similar to Theorems 1.1, 3.1 also has a *q*-theoretic equivalent.

### Theorem 4.4

### Proof

Similar to the treatment of the injection \(\gamma \) after Theorem 4.1, one can look for the partitions that are outside of the image of \(\Gamma ^*_1\), \(\Gamma _1\), etc. and write the (4.8) expression with manifestly non-negative terms. Yet, the increase in the number of cases are making this study not necessarily harder, but messier.

- (i)
\(\big (2^{2j+5},3^{f_3},5^{f_5}\big )\), where

*j*is a non-negative integer, - (ii)
or \(\big (2^1,3^{f_3},5^{f_5}\big )\), where \(f_3 \ge 3\),

- (iii)
or \(\big (2^1,5^{f_5}\big )\),

*i*. For larger odd

*L*values that fall under the injective map \(\Gamma _1\), we can repeat this process and write \(H_{L,2}\) as a sum of manifestly non-negative terms. Needless to say, \(f_{L+1}=0\) in these cases. The partitions in \({\mathcal {B}}_{L,1} {\backslash } \Gamma _1({\mathcal {B}}_{L,2})\), for odd \(L>3\), are ones of the form:

- (i)
\(\left( 2^{i+k(L+1)/2},3^{f_3},\dots ,(L+1)^0, (L+2)^{f_{L+2}}\right) \), where \((L-1)/2\ge i \ge 2\) and \(k~+~\sum _{j=i+1}^{(L-1)/2}~f_{2j}~>~0\),

- (ii)
\(\left( 2^{1+k(L+1)/2},(L+2)^{f_{L+2}}\right) \), where

*k*is a non-negative integer, - (iii)
\(\left( 2^{1+k(L+1)/2}, s^{3+f_s},\dots ,(L+1)^0 ,(L+2)^{f_{L+2}}\right) \), where \(L\ge s\ge 4\),

- (iv)
\(\left( 2^{1+k(L+1)/2}, s^2,\dots ,(L+1)^0 ,(L+2)^{f_{L+2}}\right) \), where \(L \ge s\ge (L+5)/2\),

- (v)\(\left( 2^{1+k(L+1)/2}, s^2,\dots ,(L+1)^0 ,(L+2)^{f_{L+2}}\right) \), where \((L+3)/2 \ge s\ge 4\) and$$\begin{aligned} k + \sum _{j=s+1}^{2s-2}f_j+\sum _{j=s}^{(L-1)/2}f_{2j}>0, \end{aligned}$$
- (vi)\(\left( 2^{1+k(L+1)/2}, s^1,\dots , (L+1)^0,(L+2)^{f_{L+2}}\right) \), where
*s*is odd and \(L\ge s\ge ~5\),$$\begin{aligned} k+\sum _{j=(s+1)/2}^{(L-1)/2} f_j>0, \end{aligned}$$ - (vii)
\(\left( 2^{1+k(L+1)/2}, s^1,\dots ,(L+1)^0,(L+2)^{f_{L+2}}\right) \), where \((L-1)/2\ge s>4\) even,

- (viii)
\(\left( 2^{1+k(L+1)/2},4^1,\dots ,(L+1)^0,(L+2)^{f_{L+2}}\right) \), where \(k+\sum _{j=3}^{(L-1)/2}f_{2j}>0\),

- (ix)
\(\left( 2^{1+k(L+1)/2},3^{1+f_3},4^{2+f_4},\dots ,(L+1)^0,(L+2)^{f_{L+2}}\right) \),

- (x)
\(\left( 2^{1+k(L+1)/2},3^{1+f_3},4^{\alpha },\dots ,(L{+}1)^0,(L{+}2)^{f_{L+2}}\right) \), where \(k+\sum _{j=3}^{(L-1)/2}f_{2j}{>}0\) and \(\alpha =0\) or 1,

- (xi)
\(\left( 2^{1},3^{3+f_3},\dots \right) \), where \(\sum _{i=2}^{(L-1)/2} f_{2i} =0\).

*k*th term is the generating function for the number of partitions from the

*k*th \(\Gamma _1\) unmapped case described above, where \(k\in \{1,2,\dots ,11\}\). Also note that as

*L*tends to infinity (4.10) simplifies significantly, and can be reduced to (4.7) after some labor.

The interested reader can also write \(H_{L,2}(q)\) with only non-negative terms for even choices of *L* with the same type of argument for \(\Gamma _2^*\) and \(\Gamma _2\) injections.

## 5 An Alternative Proof of Theorems 1.2 and 1.3

We start this section by recalling Theorem 1.2:

**Theorem**1.2 We have

*q*-factorial \((q;q)_\infty \), the right-hand side series of (5.1) becomes the right-hand side of (1.3). Also recall that

We would also like to remind the reader of the non-negativity question:

**Theorem**1.3 For \(L\ge 1\),We define the following difference of generating functions:

*n*parts where the smallest part \(\ge 2\). We can clearly understand that \(|\pi |\ge 2n\) since there are

*n*parts and all the parts are \(\ge 2\). The whole column over the smallest part of the partition \(\pi \) is generated by the

*q*-factor

*q*-binomial coefficient

### Theorem 5.1

### Proof

*q*-binomial theorem (2.1), we can verify the formula of \(G_{L,1}(q)\):The positivity claim on \(G_{L,1}(q)\), for \(L\ge 2\) follows from Theorem 4.1 as \(1/(1-q^L)\) and \(H_{L,1}(q)\) both have non-negative series, their multiplication has non-negative series. The \(L=1\) case can be directly/algebraically checked from (5.6). For \(L=1\), the expression (5.6) reduces to \(q/(1-q^2)\) and hence, is represented by a power series with non-negative coefficients. \(\square \)

Theorem 5.1 implies Theorem 1.3.

### Proof of Theorem 1.3

### Theorem 5.2

With this definition, we can make a similar claim to Conjecture 3.3 about \(G_{L,2}(q)\).

### Conjecture 5.3

A more general analytic conjecture, which contains Conjecture 5.3, is discussed in Sect. 7.

## 6 Transformations of the Analytic Refined Weighted Identity

### Theorem 6.1

*L*, We have

*q*-hypergeometrically. Note that

*q*-series of (1.4).

### Theorem 6.2

## 7 Outlook

One project to pursue is to identify the statistics \(t_L(\pi )\) for partitions, which would be the refined statistics of \(t(\pi )\) of Theorem 1.2 for partitions with the difference between the largest and the smallest parts bounded by *L*. As it stands, going from Theorems 1.2 to 1.3 we lose grasp of the non-negative statistic \(t(\pi )\).

*eventually positive*if there is some

*k*such that \(a_n >0\) for all \(n > k\). Theorems 4.1 and 4.4 seem to be the initial steps of a eventually positive family of

*q*-products. Let

### Conjecture 7.1

For positive integers \(L \ge 3\), *s* and \(k\ge s+1\), \(H_{L,s,k}(q)\) is eventually positive.

We have already proven the conjecture for the \((L,s,k)=(L,1,L)\) and \((L,2,L+1)\) families in Theorems 4.1 and 4.4. The particular branch \(H_{L,s,L+s-1}(q)\) is a natural generalization of the functions \(H_{L,1}\) and \(H_{L,2}\) mentioned in Sect. 4, and the non-negativity claim related to Conjecture 3.2. All other triplets with \(L=k\ge s+1\) are related to the Conjectures 3.3 and 5.3. Therefore, one can view Conjecture 7.1 with the above relations as natural extension of these observations. For all other triplets (*L*, *s*, *k*) are experimental.

The number of exceptional cases increases with *s*, making it less feasible to combinatorially study these functions for larger starting values *s*. More interestingly, the presence of a one-time exception at \(q^9\) for the \((L,s,k)=(4,2,5)\) case (\(H_{L,2}(q)\)), which was handled in Theorem 3.1, also hints a higher degree of underlying complexity.

## Notes

### Acknowledgements

Open access funding provided by Johannes Kepler University Linz. The authors would like to thank George Andrews for interest and helpful insights. We are grateful to William Severa and the anonymous referee for the careful reading of the manuscript. Research of the first author is partly supported by the Simons Foundation, Award ID: 308929. Research of the second author is supported by the Austrian Science Fund FWF, SFB50-07 Project.

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