# Inequalities for Plane Partitions

Bernhard Heim  Lehrstuhl A für Mathematik, RWTH Aachen University, 52056
Aachen, Germany<EMAIL_ADDRESS>,  Markus Neuhauser  Kutaisi
International University, 5/7, Youth Avenue, Kutaisi, 4600 Georgia
<EMAIL_ADDRESS>and  Robert Tröger<EMAIL_ADDRESS>
###### Abstract.

Inequalities are important features in the context of sequences of numbers and
polynomials. The Bessenrodt–Ono inequality for partition numbers and
Nekrasov–Okounkov polynomials has only recently been discovered. In this paper
we study the log-concavity (Turán inequality) and Bessenrodt–Ono inequality
for plane partitions and their polynomization.

###### Key words and phrases:

Inequalities, Plane Partitions, Polynomials

###### 2010 Mathematics Subject Classification:

Primary 05A17, 11P82; Secondary 05A20

##  1\. Introduction and Main Results

In this paper we address inequalities for plane partitions and their
polynomization. Plane partitions are, according to Stanley, fascinating
generalizations of partitions of integers ([St99], Section 7.20). Andrews
[An98] gave an excellent introduction of plane partitions in the context of
higher-dimensional partitions. We also refer to Krattenthaler’s survey on
plane partitions in the work of Stanley and his school [Kr16].

A plane partition $\pi$ of $n$ is an array $\pi=\left(\pi_{ij}\right)_{i,j\geq
1}$ of non-negative integers $\pi_{ij}$ with finite sum
$|\pi|:=\sum_{i,j=1}\pi_{ij}=n$, which is weakly decreasing in rows and
columns. It can be considered as the filling of a Ferrers diagram with weakly
decreasing rows and columns, where the sum of all these numbers is equal to
$n$. Let the numbers in the filling represent the heights for stacks of blocks
placed on each cell of the diagram (Figure 1). This is a natural
generalization of the concept of classical partitions [An98, On03].

$\phantom{xxx}\young(5443321,432,21)$

$\longrightarrow$ xx xx

Figure 1. Representation of plane partitions.

Let ${p}(n)$ denote the number of partitions of $n$ and $\mathop{\rm pp}(n)$
the number of plane partitions of $n$. As usual, ${p}\left(0\right):=1$ and
$\mathop{\rm pp}\left(0\right):=1$. In Table 1 we have listed the first values
of the partition and plane partition function.

$\begin{array}[]{cccccccccccc}\hline\cr n&0&1&2&3&4&5&6&7&8&9&10\\\
\hline\cr\hline\cr p\left(n\right)&1&1&2&3&5&7&11&15&22&30&42\\\ \mathop{\rm
pp}\left(n\right)&1&1&3&6&13&24&48&86&160&282&500\\\ \hline\cr\end{array}$
Table 1.  Values for $0\leq n\leq 10$.

We investigate the log-concavity [Br89, St89] and the Bessenrodt–Ono
inequality [BO16, HNT20] of plane partitions. A sequence of real, non-negative
numbers $\\{\alpha(n)\\}_{n=0}^{\infty}$ is log-concave if for all
$n\in\mathbb{N}$:

(1.1) $\alpha(n)^{2}>\alpha(n-1)\,\alpha(n+1).$

We say log-concave at $n$ if (1.1) is satisfied for a specific $n$. A sequence
of real non-negative numbers $\\{\alpha(n)\\}_{n=0}^{\infty}$ satisfies the
Bessenrodt–Ono inequality for a set $S\subset\mathbb{N}\times\mathbb{N}$, if
for all $(a,b)\in S$:

(1.2) $\alpha(a)\,\alpha(b)>\alpha(a+b).$

###  1.1. Related Work and Recent Results

We recall that Nicolas [Ni78] proved that the partition function is log-
concave for $n>25$:

(1.3) ${p}(n)^{2}>{p}(n-1)\,{p}(n+1).$

It is valid for all $n$ even and fails for $1\leq n\leq 25$ odd. In the course
of proving several conjectures of Chen and Sun, DeSalvo and Pak [DP15]
reproved this result. They remarked that due to the Hardy–Ramanujan asymptotic
formula

$p(n)\sim\frac{1}{4\sqrt{3n}}\,e^{\pi\sqrt{\frac{2}{3}n}}\quad\text{ as
}n\rightarrow\infty,$

there is no way of knowing precisely when the asymptotic formula dominates the
calculation. Their proof is based on Rademacher type estimates [Ra37] by
Lehmer (e. g. [Le38, Le39]), which provide the demanded, explicit, guaranteed
error estimate. This proves the log-concavity for $n\geq 2,600$ ([DP15]). More
generally, let $p_{k}(n)$ be the $k$-colored partition function, obtained for
every $k\in\mathbb{N}$ by the generating function

$\prod_{n=1}^{\infty}\left(1-q^{n}\right)^{-k}=\sum_{n=0}^{\infty}p_{k}(n)\,q^{n},$

which is essentially the $k$th power of the reciprocal of the Dedekind eta
function [On03]. Then (1.3) was extended by Chern–Fu–Tang [CFT18] to an
interesting conjecture: Suppose $k,\ell,n\in\mathbb{N}$ with $k\geq 2$ and
$n>\ell$. Let $\left(k,n,\ell\right)\neq(2,6,4)$, then

$p_{k}(n-1)\,p_{k}\left(\ell+1\right)\geq p_{k}(n)\,p_{k}\left(\ell\right).$

The conjecture was extended to $k\in\mathbb{R}_{\geq 2}$ [HN21A], which
involves the so-called D’Arcais polynomials or Nekrasov–Okounkov polynomials
[NO06, Ha10]. The Conjecture by Chern–Fu–Tang and some portion of the
Conjecture by Heim–Neuhauser was recently proven by Bringmann, Kane, Rolen,
and Tripp [BKRT21]. The proof is based on Rademacher type formulas for the
coefficients of powers of the Dedekind eta function, utilizing the weak
modularity property.

Bessenrodt and Ono [BO16] discovered a beautiful and simple inequality for
partition numbers. Let $a,b\in\mathbb{N}$. Suppose $a,b\geq 2$ and $a+b\geq
10$, then

(1.4) $p(a)\,p(b)>p(a+b).$

The inequality is symmetric and always fails for $a=1$ or $b=1$. Let $2\leq
b\leq a$. There is equality for the pairs $(4,3),(6,2),(7,2)$ and the opposite
inequality of (1.4) is exactly true for the pairs
$(2,2),(3,2),(4,2),(5,2),(3,3),(5,3)$. Bessenrodt and Ono’s proof is based on
an analytic result of Lehmer [Le39] of Rademacher type, similar to the proof
of the log-concavity of $p(n)$ [Ni78, DP15]. Shortly after the result was
published, Alanazi, Gangola III, and Munagi [AGM17] came up with a subtle
combinatorial proof. Chern, Fu, and Tang [CFT18] generalized and proved the
Bessenrodt–Ono inequality to $k$-colored partitions. In [HNT20] this was
extended to $k$ real, again involving polynomials. The proof was given by
induction and involving derivatives. Further, the work of Bessenrodt and Ono
triggered the results of Beckwith and Bessenrodt [BB16] on $k$-regular
partitions, Hou and Jagadeesan [HJ18] on the numbers of partitions with ranks
in a given residue class modulo $3$ and Males [Ma21] for general $t$, and Heim
and Neuhauser [HN19], and Dawsey and Masri [DM19] for the Andrews spt-
function.

We performed several numerical experiments and are convinced that some of the
recorded results can be transferred to plane partitions $\mathop{\rm pp}(n)$
and its generalization. MacMahon [Ma97, Ma99, Ma60] proved the following non-
trivial result, which took him several years. The generating function of the
plane partition is given by

$\prod_{n=1}^{\infty}\left(1-q^{n}\right)^{-n}=\sum_{n=0}^{\infty}\mathop{\rm
pp}(n)\,q^{n}.$

Since this generating function is not related to a weakly modular form, in
contrast to the partition numbers, we have only an asymptotic formula provided
by Wright [Wr31] based on the circle and saddle point method for plane
partition numbers. Wright proved the following asymptotic behavior as $n$ goes
to infinity:

(1.5) $\mathop{\rm
pp}(n)\sim\frac{\zeta(3)^{\frac{7}{36}}}{\sqrt{12\pi}}\left(\frac{2}{n}\right)^{\frac{25}{36}}\exp\left(3\,\zeta(3)^{\frac{1}{3}}\left(\frac{n}{2}\right)^{\frac{2}{3}}+\zeta^{\prime}(-1)\right).$

Here $\exp\left(z\right)=\mathrm{e}^{z}$ and $\zeta(s)$ denotes the Riemann
zeta function. Thus, we are in a similar situtation as described before by
DeSalvo and Pak [DP15] for partition numbers.

Recently, we invented a new proof method [HN21B] and reproved some known
results related to the Bessenrodt–Ono inequality for the partition function,
the $k$-colored partitions and extension to the D’Arcais polynomials.

###  1.2. Main Results: Plane Partitions

We first start with the Bessenrodt–Ono inequality (1.2).

######  Theorem 1.1 (Bessenrodt–Ono inequality).

Let $a$ and $b$ be positive integers. Let $a,b\geq 2$ and $a+b\geq 12$. Then

$\mathop{\rm pp}(a)\,\mathop{\rm pp}(b)>\mathop{\rm pp}(a+b).$

Equality is never satisfied.

Due to symmetry, let us assume that $2\leq b\leq a$. Then $\mathop{\rm
pp}(a)\,\mathop{\rm pp}(b)<\mathop{\rm pp}(a+b)$ for
$(a,b)\in\left\\{\left(a,2\right)\,:\,2\leq a\leq
9\right\\}\,\cup\left\\{\left(a,3\right)\,:\,2\leq a\leq 5\right\\}$. Note
that $\mathop{\rm pp}(n)<\mathop{\rm pp}(n+1)$, similar to $p(n)<p(n+1)$. We
have

$\left(\mathop{\rm pp}(3)\right)^{2}<\mathop{\rm pp}(3+3)\text{ and
}\left(\mathop{\rm pp}(4)\right)^{2}>\mathop{\rm pp}(4+4).$

Based on our investigations we state the following

######  Conjecture 1.

Let $n\geq 12$. Then the sequence $\\{\mathop{\rm pp}(n)\\}_{n}$ of plane
partitions is log-concave.

(1.6) $\mathop{\rm pp}(n)^{2}>\mathop{\rm pp}(n-1)\,\mathop{\rm pp}(n+1).$

We can show with Wright’s formula (1.5), that (1.6) is true for large $n$, and
it seems that this is already true for all $n$ even and for all $n\geq 12$.

######  Theorem 1.2 (Log-Concavity).

Let $12\leq n\leq 10^{5}$. Then Conjecture 1 is true. It is further true for
all $n$ even and false for all odd $n$ below $12$. Furthermore there is an $N$
such that it is true for all $n>N$.

It would be very interesting to determine such a $N$ of reasonable size and to
finally prove the conjecture.

###  1.3. Main Results: Polynomization

It is possible to consider $\\{\mathop{\rm pp}(n)\\}_{n}$ as special values of
a family of polynomials $\\{P_{n}(x)\\}_{n}$. We will have $\mathop{\rm
pp}(n)=P_{n}(1)$. This makes it possible to generalize the Bessenrodt–Ono
inequality and the log-concavity. We view the inequalities as a property of
the largest positive real zeros of new polynomials associated with
$\\{P_{n}(x)\\}_{n}$.

######  Definition.

Let $\sigma_{2}(n):=\sum_{d\mid n}d^{2}$. Let $P_{0}(x):=1$ and

(1.7) $P_{n}(x):=\frac{x}{n}\sum_{k=1}^{n}\sigma_{2}(k)\,P_{n-k}(x).$

We have listed the first polynomials in Table 2.

$\begin{array}[]{rl}\hline\cr n&P_{n}\left(x\right)\\\ \hline\cr\hline\cr
1&x\\\ 2&\frac{1}{2}\\*x^{2}+\frac{5}{2}\\*x\\\
3&\frac{1}{6}\\*x^{3}+\frac{5}{2}\\*x^{2}+\frac{10}{3}\\*x\\\
4&\frac{1}{24}\\*x^{4}+\frac{5}{4}\\*x^{3}+\frac{155}{24}\\*x^{2}+\frac{21}{4}\\*x\\\
5&\frac{1}{120}\\*x^{5}+\frac{5}{12}\\*x^{4}+\frac{115}{24}\\*x^{3}+\frac{163}{12}\\*x^{2}+\frac{26}{5}\\*x\\\
\hline\cr\end{array}$ Table 2. Polynomials $P_{n}\left(x\right)$ for
$n\in\left\\{1,2,3,4,5\right\\}$.

Let $q,z\in\mathbb{C}$ and $|q|<1$. It is a standard procedure to show that

$\sum_{n=0}^{\infty}P_{n}(z)\,q^{n}=\prod_{n=1}^{\infty}\left(1-q^{n}\right)^{-n\,z}=\mathop{\rm
exp}\left(z\,\sum_{n=1}^{\infty}\sigma_{2}(n)\,\frac{q^{n}}{n}\right).$

Thus, $n\,\mathop{\rm pp}(n)=\sum_{k=1}^{n}\sigma_{2}(k)\,\mathop{\rm
pp}(n-k)$, applying MacMahon’s discovery. Let $k$ be a positive integer. We
would like to call $P_{n}(k)$ the $k$-_colored_ plane partitions (compare
[BBPT19]), but at the moment there is no combinatorial interpretation
available, as in the case of partitions [HNT20, BKRT21]. The topic is quite
complicated, since MacMahon’s result is already non-trivial and $\mathop{\rm
pp}(n)$ can also be identified with the number of all partitions of $n$, where
each part $n_{j}$ is allowed to have $n_{j}$ colors.

####  1.3.1. Bessenrodt–Ono Inequalities

######  Theorem 1.3.

Let $x\in\mathbb{R}$ and $x>5$. Then

$P_{a}(x)\,P_{b}(x)>P_{a+b}(x)$

for all positive integers $a$ and $b$.

It is also possible to get results for $x=1,2,3,4,5$. This leads to
restrictions on $a$ and $b$, reflected in Table 3, where we have recorded the
largest real zero of

$P_{a,b}(x):=P_{a}(x)\,P_{b}(x)-P_{a+b}(x).$

Thus, studying the polynomials $P_{n}(x)$ and $P_{a,b}(x)$ and their leading
coefficients and zeros, provides the big picture and reveals information on
the original task, studying properties of plane partitions $\mathop{\rm
pp}(n)=P_{n}(1)$. Note that $P_{a,b}(x)$ goes to infinity for $a,b\geq 1$, as
$x$ goes to infinity.

######  Theorem 1.4.

Let $x\in\mathbb{R}$ and $x\geq 2$. Then

$P_{a}(x)\,P_{b}(x)>P_{a+b}(x)$

for all positive integers $a$ and $b$ satisfying $a+b\geq 12$.

It would be interesting to search for a combinatorial proof for the plane
partition numbers (Theorem 1.1) and their generalization to $k$-colored plane
partitions (Theorems 1.3 and 1.4).

$\begin{array}[]{rcccccccccccc}\hline\cr a\backslash
b&1&2&3&4&5&6&7&8&9&10&11&12\\\ \hline\cr\hline\cr
1&5.0&3.2&3.0&2.6&2.5&2.3&2.2&2.1&2.1&2.0&2.0&1.9\\\
2&3.2&1.9&1.6&1.4&1.3&1.2&1.1&1.1&1.0&1.0&1.0&0.9\\\
3&3.0&1.6&1.5&1.2&1.1&1.0&1.0&0.9&0.9&0.8&0.8&0.8\\\
4&2.6&1.4&1.2&0.9&0.9&0.8&0.7&0.7&0.6&0.6&0.6&0.6\\\
5&2.5&1.3&1.1&0.9&0.9&0.7&0.7&0.6&0.6&0.6&0.6&0.5\\\
6&2.3&1.2&1.0&0.8&0.7&0.6&0.6&0.5&0.5&0.5&0.5&0.4\\\
7&2.2&1.1&1.0&0.7&0.7&0.6&0.6&0.5&0.5&0.5&0.5&0.4\\\
8&2.1&1.1&0.9&0.7&0.6&0.5&0.5&0.4&0.4&0.4&0.4&0.4\\\
9&2.1&1.0&0.9&0.6&0.6&0.5&0.5&0.4&0.4&0.4&0.4&0.3\\\
10&2.0&1.0&0.8&0.6&0.6&0.5&0.5&0.4&0.4&0.4&0.3&0.3\\\
11&2.0&1.0&0.8&0.6&0.6&0.5&0.5&0.4&0.4&0.3&0.3&0.3\\\
12&1.9&0.9&0.8&0.6&0.5&0.4&0.4&0.4&0.3&0.3&0.3&0.3\\\ \hline\cr\end{array}$
Table 3.  Approximative largest real zeros of $P_{a,b}\left(x\right)$ for
$1\leq a,b\leq 12$.

####  1.3.2. Log-Concavity

In the spirit of the Chern–Fu–Tang Conjecture [CFT18] on $k$-colored
partitions and their polynomization [HN21A] (see also [BKRT21]), we consider
the polynomials

$\Delta_{a,b}(x):=P_{a-1}(x)P_{b+1}(x)-P_{a}(x)\,P_{b}(x).$

Note that $\Delta_{a+1,a-1}(x)=P_{a}(x)^{2}-P_{a-1}(x)\,P_{a+1}(x)$. Then
$\Delta_{a+1,a-1}(1)>0$ is the log-concavity condition for $\mathop{\rm
pp}(a)$.

Figure 2.  Zeros of $\Delta_{a+1,a-1}(x)$ with positive real part for $1\leq
a\leq 100$. Blue labels the real zeros and red the imaginary zeros.

This leads to the following conjecture on the log-concavity of the polynomials
$P_{n}(x)$. For orthogonal polynomials, such kinds of inequalities are called
Turán inequality.

######  Conjecture 2 (Turán inequality).

Let $a$ be an integer with $a\geq 12$ and let $x$ be a real number with $x\geq
1$. Then

$P_{a}(x)^{2}>P_{a-1}(x)\,P_{a+1}(x).$

This is a natural extension of Conjecture 1 on the log-concavity of plane
partitions.

It would also be interesting to study the hyperbolicity of the associated
Jensen polynomials [GORZ19] and higher Turán inequalities [CJW19]. Let
$\alpha(m):=\mathop{\rm pp}(m)$ or more general, $\alpha(m):=P_{m}(x)$,
$x\in\mathbb{R}_{>0}$. The Jensen polynomial of degree $d$ and shift $n$
attached to the sequence $\\{\alpha(0),\alpha(1),\alpha(2),\ldots\\}$ of non-
negative real numbers is the polynomial

$J_{\alpha}^{d,n}(X):=\sum_{k=0}^{d}\binom{d}{k}\,\alpha(n+k)\,X^{k}.$

It follows from the zero distribution of the polynomial
$\Delta_{a+1,a-1}(x):=P_{a}(x)^{2}-P_{a-1}(x)\,P_{a+1}(x)$ (see Figure 2) that
Conjecture 2 is true for $12\leq a\leq 100$, since the polynomial goes to
infinity, as $x$ goes to infinity. Let $a$ be even and $2\leq a\leq 1000$,
then the Turán inequality in Conjecture 2 is already valid for $x>0$.
Actually, in this case all coefficients of $\Delta_{a+1,a-1}(x)$ are non-
negative.

The generalization of the former Chern–Fu–Tang conjecture of $k$-colored
partitions and its polynomization [HN21A] leads to:

######  Conjecture 3.

Let $a$ and $b$ be integers. Suppose $a-1>b\geq 0$ and
$(a,b)\not\in\\{(4,2),(6,4)\\}$. Then for all real numbers $x\geq 2$:

$\Delta_{a,b}(x):=P_{a-1}(x)\,P_{b+1}(x)-P_{a}(x)\,P_{b}(x)>0.$

We refer to Table 8, where we recorded the largest real zero for
$\Delta_{a,b}$ for all pairs $(a,b)$ with $a-1>b\geq 0$ and $2\leq a\leq 19$,
which shows that the Conjecture 3 is valid for all admissible pairs $(a,b)$ in
this range.

##  2\. Basic Properties of $\sigma_{2}(n)$ and $P_{n}(x)$

For proof of Theorem 1.1 and Theorem 1.3 we need some elementary properties of
$\sigma_{2}(n)$ and $P_{n}(x)$. As a further evidence that Conjecture 2 most
likely to be true for even arguments $n$, we prove that $\sigma_{2}(n)/n$ is
log-concave for $n$ even. Note that this is false for
$\frac{1}{n}\sum_{d|n}d$, where $n$ is even (which is related to the
Chern–Fu–Tang conjecture for $k$-colored partitions).

######  Proposition 2.1.

Let $n$ be an even, positive integer. Then

$\left(\frac{\sigma_{2}(n)}{n}\right)^{2}>\frac{\sigma_{2}(n-1)}{n-1}\,\,\frac{\sigma_{2}(n+1)}{n+1}.$

###### Proof.

For $n$ odd we estimate

$\sigma_{2}\left(n\right)=\sum_{t\mid n}t^{2}=\sum_{t\mid
n}\left(\frac{n}{t}\right)^{2}=n^{2}\sum_{t\mid n}t^{-2}\leq
n^{2}\left(1-\frac{1}{4}\right)\sum_{t}t^{-2}\leq\frac{\pi^{2}}{8}n^{2}<\frac{5}{4}n^{2}.$

Now, let $n\geq 6$ be even. Then $\sigma_{2}\left(n\right)\geq
n^{2}+\left(\frac{n}{2}\right)^{2}+4+1=\frac{5}{4}\left(n^{2}+4\right)$.
Therefore,

$\frac{\sigma_{2}\left(n-1\right)\sigma_{2}\left(n+1\right)}{n^{2}-1}<\frac{25}{16}\left(n^{2}-1\right)<\left(\frac{5}{4}\frac{n^{2}+4}{n}\right)^{2}\leq\left(\frac{\sigma_{2}\left(n\right)}{n}\right)^{2}.$

The inequality holds also for even $n\leq 4$ (Table 4). ∎

$\begin{array}[]{cccccc}\hline\cr n&1&2&3&4&5\\\
\hline\cr\hline\cr\sigma_{2}\left(n\right)&1&5&10&21&26\\\
\hline\cr\end{array}$ Table 4. Values of $\sigma_{2}\left(n\right)$ for
$n\in\left\\{1,2,3,4,5\right\\}$.

Note, that $\sigma_{2}(n)<\tilde{\sigma}_{2}(n):=2\,n^{2}$ for all
$n\in\mathbb{N}$. This follows from

$\sum_{t\mid n}t^{2}=n^{2}\sum_{t\mid n}\frac{1}{t^{2}}\leq
n^{2}\left(1+\int_{1}^{n}t^{-2}\,\mathrm{d}t\right)<2n^{2}.$

The functions $P_{n}(x)$ are polynomials of degree $n$. This can be deduced
directly from the recurrence formula (1.7). We have
$P_{n}(x)=\frac{1}{n!}\,\sum_{m=1}^{n}A_{n,m}\,x^{m}$, where
$A_{n,m}\in\mathbb{N}$ for $n\geq 1$. We later use the fact that
$A_{n,1}=n!\,\frac{\sigma_{2}(n)}{n}$. Further, we have the following
properties.

######  Proposition 2.2.

Let $n,m$ be natural numbers and $x\geq 1$ real. Then

(2.1) $\displaystyle P_{n}^{\prime}(x)$ $\displaystyle=$
$\displaystyle\sum_{k=1}^{n}\frac{\sigma_{2}\left(k\right)}{k}\,P_{n-k}\left(x\right),$
(2.2) $\displaystyle P_{n+1}(x)$ $\displaystyle>$ $\displaystyle
P_{n}\left(x\right),$ (2.3) $\displaystyle P_{n}(x)$ $\displaystyle>$
$\displaystyle\sum_{\ell=1}^{m}\frac{1}{\ell!}\binom{n+\ell-1}{2\ell-1}\,x^{\ell}\text{
for }n>1.$

###### Proof.

The formula for the derivative $P_{n}^{\prime}(x)$ is given similarly as that
obtained for the polynomials attached to $\sigma(n)=\sum_{d\mid n}d$ [HN18].
We prove (2.2) by induction. Let
$\Delta_{n}\left(x\right):=P_{n}\left(x\right)-P_{n-1}\left(x\right)$.

Let $n=1$. Then we have $\Delta_{1}(x)>0$ for all $x>1$. Suppose
$\Delta_{m}(x)>0$ for all $1\leq m\leq n-1$. We obtain for
$P_{n+1}^{\prime}(x)$ the strict lower bound

$\sum_{k=1}^{n}\frac{\sigma_{2}\left(k\right)}{k}\,P_{n+1-k}(x)\geq\sum_{k=1}^{n}\frac{\sigma_{2}\left(k\right)}{k}\,P_{n-k}(x).$

Thus, $P_{n+1}^{\prime}(x)>P_{n}^{\prime}(x)$. The plane partition function is
strictly increasing:

$\mathop{\rm pp}\left(n\right)<\mathop{\rm pp}\left(n+1\right)$

for all $n\in\mathbb{N}$, since every plane partition of $n$ can be lifted to
a plane partition of $n+1$. This provides

$P_{n+1}(1)=\mathop{\rm pp}\left(n+1\right)>\mathop{\rm
pp}\left(n\right)=P_{n}(1).$

Thus, the claim is proven. The lower bound given in (2.3) is obtained in the
following way. We have $n^{2}<\sigma_{2}(n)$ for $n>1$ and $\sigma_{2}(1)=1$.
Thus, the coefficients of the polynomial $S_{n}(x)$, defined by $S_{0}(x):=1$
and $S_{n}(x)=\frac{x}{n}\sum_{k=1}^{n}k^{2}\,S_{n-k}(x)$, are smaller than
the coefficients of $P_{n}(x)$ for $n>1$. The $m$th coefficient is given by
$\frac{1}{m!}\binom{n+m-1}{2m-1}$. This can be deduced from [HLN19], Section
$4$. ∎

######  Corollary 2.3.

Let $n\in\mathbb{N}$ and $x\geq 1$. Then $\Delta_{n}^{\prime}(x)>0$.

##  3\. Proof Strategy for the Bessenrodt–Ono Inequality for Plane Partitions

In this section we lay out a general strategy for proving Bessenrodt–Ono type
inequalities. This also makes the appearance of exceptions transparent.

###  3.1. Input Data and Proof by Induction

Let $A,B,N_{0}\in\mathbb{N}$ be given with $2\leq B<N_{0}$ and
$x_{0}\in\mathbb{R}_{\geq 1}$. Let $n\geq B$. Let $\mathrm{S}(n)$ be the
following mathematical statement for one of the three cases: $x\in\mathbb{R}$
with $x=x_{0}$, $x>x_{0}$, or $x\geq x_{0}$. For all $A\leq b\leq a$ with
$a+b=n$ and all $x$ with fixed case, we have

$P_{a,b}(x):=P_{a}(x)\,P_{b}(x)-P_{a+b}(x)>0.$

Note that $P_{a,b}(x_{0})=P_{b,a}(x_{0})$ and that
$\lim_{x\rightarrow\infty}P_{a,b}(x)=+\infty$.

Given input data $A,B,x_{0}$ by induction we prove $\mathrm{S}(n)$ for one of
the given cases. We choose $N_{0}$ and prove first manually or by numerical
calculation (utilizing PARI/GP) that $\mathrm{S}\left(\ell\right)$ is true for
all $B\leq\ell\leq N_{0}$. In the case of $x\geq x_{0}$ or $x>x_{0}$ we study
the real zeros of the polynomial $P_{a,b}(x)$ for all $A\leq b\leq a$ with
$a+b=\ell$. Let $n\geq N_{0}$. Then we prove $\mathrm{S}(n)$ by assuming that
$\mathrm{S}(m)$ is true for all $B\leq m\leq n-1$, the induction hypothesis.

###  3.2. Basic Decomposition

Let $A,B\in\mathbb{N}$ be given with $B\geq 2$. Further, let
$a,b\in\mathbb{N}$ satisfy $A\leq b\leq a$ and $a+b\geq B$. We define

$\displaystyle k_{0}$ $\displaystyle:=$ $\displaystyle a-\mathop{\rm
max}\\{B-b,A\\}+1,$ $\displaystyle f_{k}(a,b,x)$ $\displaystyle:=$
$\displaystyle\sigma_{2}(k)\left(\frac{P_{a-k}(x)\,P_{b}(x)}{a}-\frac{P_{a+b-k}(x)}{a+b}\right).$

We consider the decomposition $P_{a,b}(x)=L_{a,b}(x)+R_{a,b}(x)$, where

$\displaystyle L=L_{a,b}(x)$ $\displaystyle:=$
$\displaystyle-\sum_{k=1}^{b}\frac{\sigma_{2}\left(k+a\right)}{a+b}P_{b-k}\left(x\right),$
$\displaystyle R=R_{a,b}(x)$ $\displaystyle:=$
$\displaystyle\sum_{k=1}^{a}f_{k}\left(a,b,x\right).$

Utilizing (2.2) leads immediately to

$L>-4\,b\,a\,P_{b}(x).$

Further, let $R=R_{1}+R_{2}+R_{3}$, where

$R_{1}:=f_{1}\left(a,b,x\right),\,\,R_{2}:=\sum_{k=2}^{k_{0}-1}\,f_{k}(a,b,x),\,\,R_{3}:=\sum_{k=k_{0}}^{a}\,f_{k}(a,b,x).$

Suppose $P_{a+b-k}(x)<P_{a-k}(x)\,P_{b}(x)$ for $1\leq k\leq k_{0}-1$, then

$R_{1}>\frac{b}{2\,a^{2}}P_{a-1}(x)\,P_{b}(x)\text{ and }R_{2}>0.$

Further, we put $R_{3}=R_{31}+R_{32}+R_{33}$, where

$R_{31}:=\sum_{k=k_{0}}^{a-A}\,f_{k}(a,b,x),\,\,R_{32}:=\sum_{k=a-A+1}^{a-1}\,f_{k}(a,b,x),\,\,R_{33}:=f_{a}\left(a,b,x\right).$

Then $R_{33}>0$ and $R_{32}=0$ if $A=1$. Moreover, let $B-b\leq A$, then
$R_{31}=0$. Thus, $R_{3}$ can only have a negative contribution to $R$ if
$A\geq 2$ (by $R_{32}$) and if $B-b>A$ (by $R_{31}$). Note that
$n^{2}\leq\sigma_{2}(n)<\tilde{\sigma}_{2}(n):=2\,n^{2}$. We obtain by
straight-forward estimations the following.

######  Lemma 3.1.

Let $B-b>A$. Then

$R_{31}>\frac{a-A-
k_{0}+1}{a}\,\big{(}k_{0}^{2}\,P_{A}(x)\,P_{b}(x)-\tilde{\sigma}_{2}(a-A)P_{a-b-
k_{0}}(x)\big{)}.$

If we know that some
$P_{k}\left(x\right)P_{b}\left(x\right)-P_{b+k}\left(x\right)$ is negative we
can use the estimate

(3.1)
$R_{31}>\frac{\tilde{\sigma}_{2}\left(a\right)}{a}\sum_{k=A}^{B-b}\left(P_{k}\left(x\right)P_{b}\left(x\right)-P_{k+b}\left(x\right)\right))_{-}$

where $v_{-}=v$ if $v<0$, otherwise $v_{-}=0$.

##  4\. Proof of Theorem 1.3 and Theorem 1.4

###### Proof of Theorem 1.3.

Let $x>5$. We prove that $P_{a,b}(x)>0$ for all $a,b\geq 1$. Due to symmetry,
we can assume $b\leq a$. We follow the strategy presented in Section 3. Let
$A=1$ and $B=2$. Let $x_{0}=5$ and $x>x_{0}$. Note that $x=x_{0}$ does not
work, since $P_{1,1}(5)=0$. Let $N_{0}=12$. Then the mathematical statement
$\mathrm{S}(n)$ is true for all $A\leq a,b\leq N_{0}$ (see Table 3). Let
$n>N_{0}$, we assume that $\mathrm{S}(m)$ is true for all $B\leq m\leq n-1$.
Let $a+b=n$ with $A\leq b\leq a$. Then $P_{a,b}(x)>L+R_{1}$. Note that
$R_{2},R_{3}\geq 0$. With (2.2) we have

$\displaystyle L$ $\displaystyle>$
$\displaystyle-4\,a\,b\,P_{b}\left(x\right),$ $\displaystyle R$
$\displaystyle>$ $\displaystyle\frac{b}{2\,a^{2}}P_{a-1}(x)\,P_{b}(x).$

Final step. Putting things together and estimating $P_{a-1}(x)$ from below by
(2.3), we obtain

(4.1)
$P_{a,b}(x)>\frac{b\,P_{b}(x)}{2\,a^{2}}\left(-8\,a^{3}+\sum_{\ell=1}^{5}\binom{a+\ell-2}{2\,\ell-1}\,\frac{5^{\ell}}{\ell!}\right).$

The right hand side of (4.1) is polynomial in $a$ of degree $9$ with a
positive leading coefficient. Calculating the largest real zero shows that the
right hand side is positive for $a\geq 7$. ∎

###### Proof of Theorem 1.4.

Let $x\geq 2$. We put $A=1$ and $B=12$. Let $N_{0}=58$. Then by induction, as
before, we obtain

$P_{a,b}(x)>L+R_{1}+R_{31},$

where for $b\geq 12$ we can apply the induction hypothesis and obtain
$R_{31}>0$. In case $1\leq b\leq 11$ we find from Table 3 that
$P_{a-k,b}\left(2\right)>0$ for $b=1$ and
$a-k\notin\left\\{1,2,\ldots,11\right\\}$ or $k=a-1$ and
$b\notin\left\\{1,2,\ldots,11\right\\}$. Therefore,

$R_{31}>\sum_{k=a-11}^{a-1}\frac{\sigma\left(k\right)P_{a-k,1}\left(x\right)}{a}.$

It can be checked that the polynomials $P_{k,1}\left(x\right)$, $1\leq k\leq
10$, are monotonically increasing for $x\geq 2$. Thus,

$R>R_{31}>-641\cdot 2a\geq-641\,a\,b\,P_{b}\left(x\right)$

from (3.1) and Table 5.

Final step: Putting everything together leads to

(4.2)
$P_{a,b}\left(x\right)>\frac{b\,P_{b}\left(x\right)}{2\,a^{2}}\left(-1290a^{3}+\sum_{\ell=1}^{8}\binom{a+\ell-2}{2\ell-1}\frac{2^{\ell}}{\ell!}\right).$

In the last step we used the property (1.7) and that $x\geq 2$. We obtain that
the expression (4.2) is positive for all $a\geq 27$. Since the leading
coefficient of $P_{a,b}\left(x\right)$ is positive we only have to determine
the largest real zero of all remaining $P_{a,b}\left(x\right)$. We checked
this for $12\leq b+a\leq 52$ with PARI/GP (compare Table 3). ∎

$\begin{array}[]{crrrrrrrrrr}\hline\cr b&1&2&3&4&5&6&7&8&9&10\\\
\hline\cr\hline\cr&-641&-4&-11&-16&-38&-52&-101&-126&-180&-110\\\
\hline\cr\end{array}$ Table 5. Values of
$\sum_{k=1}^{11-b}\left(P_{k}\left(2\right)P_{b}\left(2\right)-P_{k+b}\left(2\right)\right)_{-}$
for $b\leq 10$.

##  5\.  Proof of the Bessenrodt–Ono Inequality: Theorem 1.1

We start with the following auxiliary result.

######  Lemma 5.1.

We have $\mathop{\rm pp}(2)=3\,\mathop{\rm pp}(1)$ and for $n\neq 1$:

(5.1) $\mathop{\rm pp}\left(n+1\right)<3\mathop{\rm pp}\left(n\right).$

###### Proof.

The proof is by mathematical induction. We checked with PARI/GP that (5.1)
holds for $n\leq N_{0}=24$.

Now let $n>N_{0}$. Then

$\displaystyle 3\mathop{\rm pp}\left(n\right)-\mathop{\rm pp}\left(n+1\right)$
$\displaystyle=$
$\displaystyle\frac{3}{n}\sum_{k=1}^{n}\sigma_{2}\left(k\right)\mathop{\rm
pp}\left(n-k\right)-\frac{1}{n+1}\sum_{k=1}^{n+1}\sigma_{2}\left(k\right)\mathop{\rm
pp}\left(n+1-k\right)$ $\displaystyle=$
$\displaystyle-\frac{\sigma_{2}\left(n+1\right)}{n+1}+\sum_{k=1}^{n}\sigma_{2}\left(k\right)\left(\frac{3}{n}\mathop{\rm
pp}\left(n-k\right)-\frac{1}{n+1}\mathop{\rm pp}\left(n-k+1\right)\right)$
$\displaystyle>$
$\displaystyle-2\frac{\left(n+1\right)^{2}}{n+1}+\left(\frac{3}{n}-\frac{3}{n+1}\right)\mathop{\rm
pp}\left(n-1\right)$ $\displaystyle\geq$
$\displaystyle-2\left(n+1\right)+\frac{3}{\left(n+1\right)n}\sum_{\ell=1}^{3}\binom{n+\ell-2}{2\ell-1}\frac{1}{\ell!}>0.$

We simplified (5) by the induction hypothesis. ∎

###### Proof of Theorem 1.1.

The proof is again by induction. We follow the proof strategy stated in
Section 3. See also Section 4.

Let $A=2$, $B=12$ and $x=1$. Then
$k_{0}=a-\max\left\\{B-b,A\right\\}+1=a-\max\left\\{11-b,1\right\\}$.
Furthermore, $L>-4ab\mathop{\rm pp}\left(b\right)$,
$R_{1}>\frac{b}{2a^{2}}\mathop{\rm pp}\left(a-1\right)\mathop{\rm
pp}\left(b\right)$, and $R_{2}>0$. For $R_{33}$ we obtain the lower bound $0$
and for $R_{32}$ we obtain

$\sigma_{2}\left(a-1\right)\left(\frac{\mathop{\rm
pp}\left(b\right)}{a}-\frac{\mathop{\rm
pp}\left(b+1\right)}{a+b}\right)\geq-\sigma_{2}\left(a-1\right)\frac{2\mathop{\rm
pp}\left(b\right)}{a}>-4a\mathop{\rm pp}\left(b\right).$

Finally,

$\displaystyle R_{31}$ $\displaystyle>$
$\displaystyle\sum_{k=k_{0}}^{a-A}\frac{\sigma\left(k\right)}{a}\left(\mathop{\rm
pp}\left(a-k\right)\mathop{\rm pp}\left(b\right)-\mathop{\rm
pp}\left(a+b-k\right)\right)$ $\displaystyle\geq$ $\displaystyle-106\cdot
2a=-212a$

(Table 6). Therefore, note $R_{3}>-4a\mathop{\rm
pp}\left(b\right)-212a>-38ab\mathop{\rm pp}\left(b\right)$. Putting everything
together leads to

$\displaystyle\mathop{\rm pp}\left(a\right)\mathop{\rm
pp}\left(b\right)-\mathop{\rm pp}\left(a+b\right)$ $\displaystyle>$
$\displaystyle\frac{b\mathop{\rm
pp}\left(b\right)}{2a^{2}}\left(-76a^{3}+\mathop{\rm
pp}\left(a-1\right)\right)$ $\displaystyle>$ $\displaystyle\frac{b\mathop{\rm
pp}\left(b\right)}{2a^{2}}\left(-76a^{3}+\sum_{\ell=1}^{3}\frac{1}{\ell!}\binom{a-\ell-2}{2\ell-1}\right)$

using (1.7). This is positive for $a\geq 237$. We have checked with PARI/GP
that $\mathop{\rm pp}\left(a\right)\mathop{\rm pp}\left(b\right)-\mathop{\rm
pp}\left(a+b\right)>0$ for $B\leq a+b\leq N_{0}=472$. ∎

$\begin{array}[]{crrrrrrrr}\hline\cr b&2&3&4&5&6&7&8&9\\\
\hline\cr\hline\cr&-106&-42&-17&-30&-16&-24&-20&-13\\\ \hline\cr\end{array}$
Table 6. Values of $\sum_{k=2}^{11-b}\left(\mathop{\rm
pp}\left(k\right)\mathop{\rm pp}\left(b\right)-\mathop{\rm
pp}\left(k+b\right)\right)$ for $2\leq b\leq 9$.

##  6\. Proof of Theorem 1.2

######  Lemma 6.1.

Let $s\in\mathbb{R}$. There are $C_{0},N>0$ such that

(6.1)
$2n^{s}-\left(n+1\right)^{s}-\left(n-1\right)^{s}=\left(1-s\right)sn^{s-2}+E_{n}n^{s-3}$

for $n>N$ with $\left|E_{n}\right|<C_{0}$.

###### Proof.

We have

$\displaystyle 2n^{s}-\left(n-1\right)^{s}-\left(n+1\right)^{s}$
$\displaystyle=$
$\displaystyle\left(2-\left(1-\frac{1}{n}\right)^{s}-\left(1+\frac{1}{n}\right)^{s}\right)n^{s}$
$\displaystyle=$
$\displaystyle\left(2-\left(1-\frac{s}{n}-\frac{\left(1-s\right)s}{2n^{2}}+\frac{D_{1,n}}{n^{3}}\right)-\left(1+\frac{s}{n}-\frac{\left(1-s\right)s}{2n^{2}}+\frac{D_{2,n}}{n^{3}}\right)\right)n^{s}$
$\displaystyle=$ $\displaystyle\left(1-s\right)sn^{s-2}-E_{n}n^{s-3}$

with $\left|D_{1,n}\right|,\left|D_{2,n}\right|,\left|E_{n}\right|<C_{0}$ for
some $C_{0},N>0$ and all $n>N$. ∎

######  Corollary 6.2.

Let $C_{1}>0$. There is $N>0$ such that

(6.2)
$1+\frac{C_{1}}{9}n^{-4/3}<\exp\left(2C_{1}n^{2/3}-C_{1}\left(n+1\right)^{2/3}-C_{1}\left(n-1\right)^{2/3}\right)<1+\frac{4C_{1}}{9}n^{-4/3}$

for all $n>N$.

###### Proof.

For example for the lower bound we obtain from (6.1)

$2n^{2/3}-\left(n-1\right)^{2/3}-\left(n+1\right)^{2/3}\geq\frac{2}{9}n^{-4/3}-2C_{0}n^{-7/3}>\frac{1}{9}n^{-4/3}$

for some $C_{0}>0$ and all $n>N$ for some $N$. Therefore

$\displaystyle\exp\left(2C_{1}n^{2/3}-C_{1}\left(n-1\right)^{2/3}-C_{1}\left(n+1\right)^{2/3}\right)$
$\displaystyle>$ $\displaystyle\exp\left(\frac{C_{1}}{9}n^{-4/3}\right)$
$\displaystyle>$ $\displaystyle 1+\frac{C_{1}}{9}n^{-4/3}.$

∎

######  Theorem 6.3.

Let $C_{1},C_{2},C_{3}>0$, $r,\gamma_{1},\gamma_{2}\in\mathbb{R}$,
$\beta\left(n\right)=C_{2}n^{r}\mathrm{e}^{C_{1}n^{2/3}}$, and

$\left|\frac{\alpha\left(n\right)}{\beta\left(n\right)}-1-\gamma_{1}n^{-2/3}-\gamma_{2}n^{-4/3}\right|\leq
C_{3}n^{-2}$

for $n>N_{0}$ for some $N_{0}$. There is an $N\geq N_{0}$ such that
$\left(\alpha\left(n\right)\right)_{n\geq N_{0}}$ is log-concave for $n>N$.

###### Proof.

Let $f_{\pm}\left(n\right)=\gamma_{1}n^{-2/3}+\gamma_{2}n^{-4/3}\pm
C_{3}n^{-2}$. Then there is an $N_{1}\geq N_{0}$ such that

$\left(1+f_{-}\left(n\right)\right)\beta\left(n\right)<\alpha\left(n\right)<\left(1+f_{+}\left(n\right)\right)\beta\left(n\right)$

for all $n>N_{1}$. Therefore

$\displaystyle\left(\alpha\left(n\right)\right)^{2}-\alpha\left(n-1\right)\alpha\left(n+1\right)$
$\displaystyle>$
$\displaystyle\left(\left(1+f_{-}\left(n\right)\right)\beta\left(n\right)\right)^{2}-\left(1+f_{+}\left(n-1\right)\right)\left(1+f_{+}\left(n+1\right)\right)\beta\left(n-1\right)\beta\left(n+1\right)$
$\displaystyle=$
$\displaystyle\left(1+f_{-}\left(n\right)\right)^{2}\beta\left(n+1\right)\beta\left(n-1\right)\cdot{}$
$\displaystyle{}\cdot(\exp\left(2Bn^{2/3}-B\left(n-1\right)^{2/3}-B\left(n+1\right)^{2/3}\right)-{}$
$\displaystyle{}-\frac{\left(1+f_{+}\left(n-1\right)\right)\left(1+f_{+}\left(n+1\right)\right)}{\left(1+f_{-}\left(n\right)\right)^{2}}\left(\frac{n^{2}-1}{n^{2}}\right)^{r}).$

Now

$\displaystyle\frac{\left(1+f_{+}\left(n-1\right)\right)\left(1+f_{+}\left(n+1\right)\right)}{\left(1+f_{-}\left(n\right)\right)^{2}}$
$\displaystyle=$ $\displaystyle
1+\frac{f_{+}\left(n-1\right)+f_{+}\left(n+1\right)-2f_{-}\left(n\right)}{1+f_{-}\left(n\right)}+{}$
$\displaystyle{}+\frac{\left(f_{+}\left(n-1\right)-f_{-}\left(n\right)\right)\left(f_{+}\left(n+1\right)-f_{-}\left(n\right)\right)}{\left(1+f_{-}\left(n\right)\right)^{2}}.$

With (6.1) we obtain

$\left|f_{+}\left(n-1\right)+f_{+}\left(n+1\right)-2f_{-}\left(n\right)\right|=\left|-\frac{10}{9}\gamma_{1}n^{-8/3}+E_{n}n^{-10/3}\right|<C_{5}n^{-8/3}$

with $\left|E_{n}\right|<C_{4}$, $C_{4},C_{5}>0$, and

$\left|\left(n+v\right)^{u}-n^{u}\right|=\left|1+\frac{uv}{n}+\frac{U_{n}}{n^{2}}-1\right|n^{u}=\left|uvn^{u-1}+U_{n}n^{u-2}\right|<C_{6}n^{-5/3}$

with $\left|U_{n}\right|<C_{7}$, $C_{6},C_{7}>0$, and $u\leq-2/3$. Therefore

$\left|\frac{\left(1+f_{+}\left(n-1\right)\right)\left(1+f_{+}\left(n+1\right)\right)}{\left(1+f_{-}\left(n\right)\right)^{2}}\right|<1+C_{8}n^{-8/3}$

for some $C_{8}>C_{5}$ which implies with (6.2)

$\displaystyle\exp\left(2C_{1}n^{2/3}-C_{1}\left(n-1\right)^{2/3}-C_{1}\left(n+1\right)^{2/3}\right)-{}$
$\displaystyle{}-\frac{\left(1+f_{+}\left(n-1\right)\right)\left(1+f_{+}\left(n+1\right)\right)}{\left(1+f_{-}\left(n\right)\right)^{2}}\left(1+\frac{1}{n^{2}-1}\right)^{-r}$
$\displaystyle>$ $\displaystyle
1+\frac{C_{1}}{9}n^{-4/3}-(1+C_{8}n^{-8/3})\left(1+\frac{C_{9}}{n^{2}}\right)>0$

for some $C_{9}>0$. This is positive for $n>N$ and $N$ sufficiently large. ∎

###### Proof of Theorem 1.2.

Wright’s formula ([Wr31], Formula (2.21)) tells us that there are
$C_{1},C_{2},C_{3}>0$, $r=-25/36$, and $\gamma_{1},\gamma_{2}\in\mathbb{R}$
such that

$\mathop{\rm
pp}\left(n\right)=C_{2}n^{-\frac{25}{36}}\mathrm{e}^{C_{1}{n}^{{2}/{3}}}\left(1+\frac{\gamma_{1}}{n^{\frac{2}{3}}}+\frac{\gamma_{2}}{n^{\frac{4}{3}}}+\frac{G_{n}}{n^{2}}\right)$

with $\left|G_{n}\right|<C_{3}$ for all $n>N_{0}$, $N_{0}$ sufficiently large.
∎

##  7\. Conjecture 2 and Conjecture 3

In this section we provide evidence for the Conjectures by information on the
zeros of the underlying polynomials. Here we also use the crucial property
that the leading coefficient of these polynomials always has positive sign.

###  7.1. Conjecture 2

Figure 2 indicates that it is most likely that if one considers the sequence
of real parts of the zero of $\\{\Delta_{a+1,a-1}(x)\\}$ with the largest real
part, that these numbers tend in the limit to zero. Another aspect is given by
coefficients of these polynomials (Table 7).

$\begin{array}[]{rl}\hline\cr a&\Delta_{a+1,a-1}\left(x\right)\\\
\hline\cr\hline\cr 2&\frac{1}{12}\\*x^{4}+\frac{35}{12}\\*x^{2}\\\
3&\frac{1}{144}\\*x^{6}+\frac{5}{48}\\*x^{5}+\frac{145}{144}\\*x^{4}-\frac{101}{48}\\*x^{3}-\frac{145}{72}\\*x^{2}\\\
4&\frac{1}{2880}\\*x^{8}+\frac{1}{72}\\*x^{7}+\frac{67}{288}\\*x^{6}+\frac{137}{144}\\*x^{5}+\frac{11629}{2880}\\*x^{4}+\frac{1373}{144}\\*x^{3}+\frac{491}{48}\\*x^{2}\\\
5&\frac{1}{86400}\\*x^{10}+\frac{1}{1152}\\*x^{9}+\frac{5}{192}\\*x^{8}+\frac{941}{2880}\\*x^{7}+\frac{63131}{28800}\\*x^{6}+\frac{6449}{1152}\\*x^{5}-\frac{1303}{864}\\*x^{4}-\frac{54613}{1440}\\*x^{3}-\frac{1671}{100}\\*x^{2}\\\
\hline\cr\end{array}$ Table 7. Polynomials $\Delta_{a+1,a-1}\left(x\right)$
for $a\in\left\\{2,3,4,5\right\\}$.

We observe that for $a$ odd there are coefficients, which are negative. In the
case $a$ even, as already mentioned in the introduction, we have calculated
the polynomials for $2\leq a\leq 1000$ and observed that the coefficients are
all non-negative. Let

$\Delta_{a+1,a-1}\left(x\right)=\sum_{k=2}^{2a}B_{2a,k}\,x^{k},$

then we deduce from Proposition 2.1 that $B_{2a,2}>0$.

###  7.2. Conjecture 3

Theorem 1.4 implies that $\Delta_{a,0}(x)=P_{a-1}(x)\,P_{1}(x)-P_{a}(x)>0$ for
$a\geq 12$ and $x\geq 2$. Since $\Delta_{a,0}(x)>0$ also for $3\leq a\leq 11$
(Table 8), we obtain:

######  Corollary 7.1.

Let $x\geq 2$. Then Conjecture 3 is valid for all $a\geq 3$ and $b=0$.

Next, we prove Conjecture 3 for $x\geq 1$ and all pairs $(a,1)$ with $a\geq
3$. We refer also to Figure 3 for the pairs $(a,b)$ with $2\leq b\leq 4$.
Note, for $(a,1)$ and $3\leq a\leq 100$, there are no zeros with a positive
real part.

Figure 3. Zeros with the largest real part of $\Delta_{a,b}(x)$ for $2\leq
b\leq 4$,
blue = real zero, red = imaginary zero.

######  Proposition 7.2.

Let $x\geq 1$ and $a\geq 3$. Then $\Delta_{a,1}(x)>0$.

###### Proof.

We deduce from Table 8 that $\Delta_{a,1}(x)>0$ for $x>0$ and $a\leq 20$.
Moreover, $\Delta_{a,1}(1)=3\,\mathop{\rm pp}(a-1)-\mathop{\rm pp}(a)>0$ for
$a\geq 3$. This follows from Lemma 5.1. Let $\Delta_{a,1}(x)=x\,F_{a}(x)$. We
prove that $F_{a}(x)>0$ for $x\geq 1$ and $a\geq 3$ by induction on $a$. Let
$a\geq 6$ and $F_{m}(x)>0$ for $3\leq m<a$ and $x\geq 1$. We show that
$F_{a}^{\prime}(x)>0$, which completes the proof, since $F_{a}(1)>1$, for
$a\geq 3$. Recall Formula (2.1). Then the derivative $F_{a}^{\prime}(x)$ is
equal to

$\frac{x+5}{2}\,\sum_{k=1}^{a-1}\frac{\sigma_{2}(k)}{k}P_{a-1-k}(x)\,-\sum_{k=1}^{a}\frac{\sigma_{2}(k)}{k}P_{a-k}(x)+\frac{P_{a-1}(x)}{2}.$

By the induction hypothesis we obtain

$\frac{x+5}{2}\sum_{k=1}^{a-1}\frac{\sigma_{2}(k)}{k}P_{a-1-k}(x)\,>\,\frac{x+5}{2}\frac{\sigma_{2}(a-1)}{a-1}+\sum_{k=1}^{a-2}\frac{\sigma_{2}(k)}{k}P_{a-k}(x).$

This leads to

$\displaystyle F_{a}^{\prime}\left(x\right)$ $\displaystyle>$
$\displaystyle\frac{x+5}{2}\frac{\sigma_{2}(a-1)}{a-1}-\sum_{k=a-1}^{a}\frac{\sigma_{2}(k)}{k}P_{a-k}(x)+\frac{P_{a-1}(x)}{2}$
$\displaystyle=$
$\displaystyle\frac{P_{a-1}(x)}{2}-\frac{x\,\sigma_{2}(a-1)}{2(a-1)}+\frac{5\,\sigma_{2}(a-1)}{2\,(a-1)}-\frac{\sigma_{2}(a)}{a}.$

Moreover, we have

$2\,F_{a}^{\prime}(x)>\left(P_{a-1}(x)-\frac{\sigma_{2}(a-1)}{a-1}\,x\right)+a-5,$

since $a^{2}<\sigma_{2}(a)<2\,a^{2}$. We observe that
$P_{a}(x)-\frac{\sigma_{2}(a)}{a}\,x$ has non-negative coefficients. Finally,
this implies that $F_{a}^{\prime}(x)>0$ for all $x>1$. Thus, $F_{a}(x)>0$
since $F_{a}(1)>0$. ∎

######  Corollary 7.3.

Let $x\geq 1$ and $a\geq 3$. Then $\Delta_{a,1}^{\prime}(x)>0$.

######  Remarks.

a) The real part of the zeros of $\Delta_{a,1}(x)/\,x$ is negative for $3\leq
a\leq 100$.
b) The method of the proof is similar to the one outlined in [HN21A].

2 | 5.00 |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---
3 | 3.16 |  -–- |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |
4 | 3.00 |  -–- | 2.31 |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |
5 | 2.57 |  -–- |  -–- |  -–- |  |  |  |  |  |  |  |  |  |  |  |  |  |  |
6 | 2.50 |  -–- | 1.00 | 0.09 | 2.05 |  |  |  |  |  |  |  |  |  |  |  |  |  |
7 | 2.30 |  -–- |  -–- |  -–- |  -–- |  -–- |  |  |  |  |  |  |  |  |  |  |  |  |
8 | 2.25 |  -–- | 0.47 |  -–- | 1.06 |  -–- | 1.64 |  |  |  |  |  |  |  |  |  |  |  |
9 | 2.14 |  -–- |  -–- |  -–- | 0.55 |  -–- | 0.73 |  -–- |  |  |  |  |  |  |  |  |  |  |
10 | 2.09 |  -–- |  -–- |  -–- | 0.73 |  -–- | 0.89 |  -–- | 1.17 |  |  |  |  |  |  |  |  |  |
11 | 2.03 |  -–- |  -–- |  -–- | 0.45 |  -–- | 0.55 |  -–- |  -–- |  -–- |  |  |  |  |  |  |  |  |
12 | 1.99 |  -–- | 0.16 | 0.00 | 0.62 |  -–- | 0.71 | 0.04 | 0.69 | 0.28 | 1.07 |  |  |  |  |  |  |  |
13 | 1.94 |  -–- |  -–- |  -–- | 0.36 |  -–- | 0.46 |  -–- |  -–- |  -–- |  -–- |  -–- |  |  |  |  |  |  |
14 | 1.91 |  -–- | 0.02 |  -–- | 0.51 |  -–- | 0.57 |  -–- | 0.48 | 0.06 | 0.59 |  -–- | 0.92 |  |  |  |  |  |
15 | 1.88 |  -–- |  -–- |  -–- | 0.39 |  -–- | 0.46 |  -–- | 0.27 |  -–- | 0.33 |  -–- | 0.46 |  -–- |  |  |  |  |
16 | 1.85 |  -–- |  -–- |  -–- | 0.44 |  -–- | 0.49 |  -–- | 0.37 |  -–- | 0.43 |  -–- | 0.55 |  -–- | 0.70 |  |  |  |
17 | 1.82 |  -–- |  -–- |  -–- | 0.35 |  -–- | 0.41 |  -–- | 0.19 |  -–- | 0.25 |  -–- | 0.33 |  -–- |  -–- |  -–- |  |  |
18 | 1.80 |  -–- | 0.03 |  -–- | 0.40 |  -–- | 0.45 |  -–- | 0.33 | 0.05 | 0.37 |  -–- | 0.44 | 0.04 | 0.43 | 0.18 | 0.68 |  |
19 | 1.78 |  -–- |  -–- |  -–- | 0.32 |  -–- | 0.38 |  -–- | 0.17 |  -–- | 0.22 |  -–- | 0.29 |  -–- |  -–- |  -–- |  -–- |  -–- |
20 | 1.76 |  -–- | 0.01 |  -–- | 0.36 |  -–- | 0.41 |  -–- | 0.29 | 0.03 | 0.32 |  -–- | 0.37 | 0.01 | 0.31 | 0.08 | 0.41 |  -–- | 0.61
$x_{a,b}$ | 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | 13 | 14 | 15 | 16 | 17 | 18

Table 8.  Approximative largest positive real zeros of
$\Delta_{a,b}\left(x\right)$ for $0\leq b<a-1\leq 19$.

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# Dependency Learning for Legal Judgment Prediction with a Unified Text-to-
Text Transformer

Yunyun Huang*, Xiaoyu Shen*, Chuanyi Li, Jidong Ge, Bin Luo  Y. Huang, C. Li,
J. Ge, and B. Luo are with State Key Laboratory for Novel Software Technology,
Software Institute, Nanjing University, Nanjing 210093, China. E-mails:
<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>X. Shen is
with Amazon Alexa AI, Berlin, Germany (Work done before joining). E-mail:
<EMAIL_ADDRESS>* Authors contributed equally.

###### Abstract

Given the fact of a case, Legal Judgment Prediction (LJP) involves a series of
sub-tasks such as predicting violated law articles, charges and term of
penalty. We propose leveraging a _unified text-to-text Transformer_ for LJP,
where the dependencies among sub-tasks can be naturally established within the
auto-regressive decoder. Compared with previous works, it has three
advantages: (1) it fits in the pretraining pattern of masked language models,
and thereby can benefit from the semantic prompts of each sub-task rather than
treating them as atomic labels, (2) it utilizes a single unified architecture,
enabling full parameter sharing across all sub-tasks, and (3) it can
incorporate both classification and generative sub-tasks. We show that this
unified transformer, albeit pretrained on general-domain text, outperforms
pretrained models tailored specifically for the legal domain. Through an
extensive set of experiments, we find that the best order to capture
dependencies is _different_ from human intuitions, and the most reasonable
logical order for humans can be sub-optimal for the model. We further include
two more auxiliary tasks: court view generation and article content
prediction, showing they can not only improve the prediction accuracy, but
also provide interpretable explanations for model outputs even when an error
is made. With the best configuration, our model outperforms both previous SOTA
and a single-tasked version of the unified transformer by a large margin. Code
and dataset are available at https://github.com/oli-yun/Dependency-LJP.

###### Index Terms:

NLP in Law, Legal Judgment Prediction, Dependency Learning, Neural Networks.

##  1 Introduction

Legal Judgment Prediction (LJP) aims to predict the judgment results of legal
cases according to the fact descriptions, which involves multiple sub-tasks
depending on country-specific standards. Under the Civil Law system, these
sub-tasks usually include (1) finding the violated law articles, (2) defining
the charge, and (3) deciding the term of penalty. Automating these sub-tasks
is of great interest in that it can not only improve the working efficiency of
judges and lawyers, but also provide basic legal assistance to non-
professionals.

Earlier works solved these sub-tasks as independent text classification
problems, while ignoring the close dependencies among different sub-tasks [1,
2, 3, 4, 5]. Recently, many works have demonstrated benefits of modelling such
dependencies. The simplest way is a pipeline method that conditions each sub-
task on the prediction of its dependent sub-tasks [6, 7, 8, 9]. However, it
requires an independent sub-module for each sub-task, which complicates the
overall system and prevents information sharing across sub-tasks. Another line
of work is multi-tasking by sharing the same model parameters among all sub-
tasks [10, 11]. Nonetheless, the prediction of each sub-task is still based
solely on the fact description. The dependency learning is only implicitly
reflected through parameter sharing while ignoring the logical orders of sub-
tasks. A few works enable both parameter sharing and dependency learning under
specified logical orders [12, 13, 14]. They design different architectures for
each purpose, e.g., CNN for fact encoding and LSTM for dependency learning,
such that all sub-tasks can share the same fact encoder while maintaining
their own task-specific representations.

We argue that this way of combining different architectures is by no means the
optimal solution, and instead propose leveraging a unified text-to-text
Transformer for dependency learning in LJP. Specifically, we implement our
model based on the pretrained T5 [15] architecture, where all sub-tasks are
considered as “masked spans” from the original legal judgment document. In the
training stage, the model learns all sub-tasks by reconstructing “masked
spans” sequentially. The generated sequence can be easily mapped to
classification labels for the evaluation purpose. Compared with previous
works, it has the following advantages: (1) It fits in the same pattern as the
pretraining objective of the T5 model. The structure of the legal judgment
documents can naturally serve as semantic prompts [16, 17] for different sub-
tasks, which provides the model with a much better initialization especially
under the fewshot scenario. (2) There will be no separate architectures or
task-specific representations. It utilizes a single unified Transformer,
enabling full parameter sharing across all sub-tasks. (3) It applies to both
classification and generative tasks, so that there will be no limitation on
the sub-task types.

Experiments on real-world criminal cases show that our proposed unified
architecture, albeit pretrained on general-domain text, outperforms pretrained
models tailored specifically for the legal domain. It performs especially well
on the Macro-F1 metric, implying a strong generalization on low-frequency
labels. In contrast, all previous works overfit to high-frequency labels
without exception. It also demonstrates strong fewshot learning capability
with only hundreds of training examples. We conducted an extensive set of
experiments on the effects of sub-task orders under dependency learning. We
find that the best order is _different_ from human intuitions, and the most
reasonable logical order for humans can be sub-optimal for the model. The
model is seeking a sweet spot to balance well the dependency on other sub-
tasks and error propagation. When the risk of error propagation outweighs the
benefit from dependency learning, it prefers putting ahead the sub-task, even
if it should be solved in a later logical order for humans.

Apart from the three commonly used sub-tasks, we introduce two more auxiliary
tasks: court view generation and law article content prediction. For the
former, we show it is able to not only improve the prediction accuracy, but
also provide interpretable explanations to justify the logic behind, even when
the model makes a wrong prediction. For the latter, we find that it boosts the
prediction accuracy when combining with every single task. In the law article
prediction sub-task, it even outperforms the text-matching model which learns
a full interaction between facts and law article contents, while being
hundreds of times faster.

Figure 1: Process of LJP. Lines indicate dependency relations.

In short, this work makes the following contributions:

  * •

We propose leveraging a unified text-to-text Transformer for dependency
learning in LJP. We show it significantly outperforms previous SOTA with a
much stronger generalization on low-frequency labels, and performs decently
even under the fewshot scenario.

  * •

We perform an extensive set of experiments to find the optimal order for
dependency modelling. We find the best order for the model depends on the
trade-off between sub-task dependency and error propagation, and can be
different from human intuitions.

  * •

We introduce two new auxiliary tasks: court view generation and law article
content prediction. They can further enhance the prediction accuracy, while
providing interpretable explanations to the model outputs.

Figure 2: Example of processing the legal judgment document to get the input-
output formats. Spans in red are main tasks in LJP and spans in blue are
auxiliary tasks. We mask colored words from the original document as the
input. The output is a sequence of masked spans. The order of spans in the
output can be adjusted to model different dependency relations.

##  2 Problem Formulation

We focus on LJP tasks under the civil law system, but similar techniques can
be easily extended to other systems as well. Given factual descriptions of one
case, LJP under the civil law system involves a set of sub-tasks, including
predicting the violated law articles, charges and term of penalty. The
violated law article is legal basis for judgment results and charge is the
category of the committed crime, e.g., theft, intentional injury, etc. We
illustrate the process of LJP in Figure 1. For human judges, there exist
dependencies with a strict order among these sub-tasks. A judge will always
first decide the violated law articles, then determine the charges according
to the instructions of relevant law articles. Finally, the term of penalty
will be confirmed based on these.

The violated law article and charge have a pre-defined fixed set of possible
labels, but the term of penalty can be rather diverse across different crimes.
Following common practices, we simplify it into three main types: fixed-term
imprisonment, life imprisonment and death penalty. For fixed-term
imprisonment, the model needs to predict the exact length in the unit of
month. Since it is stipulated that the maximum period of fixed-term
imprisonment is 15 years and that of the criminal who commits several crimes
is 25 years, we set the value of life imprisonment as $350$ and death penalty
as $400$ in accordance with [12].

Formally speaking, we denote our dataset with $n$ cases as
$D=(F^{i},\\{A_{1}^{i},\dots,A_{k}^{i}\\},\\{C_{1}^{i},\dots,C_{l}^{i}\\},P^{i})_{i=1}^{n}$
where each case $D^{i}$ consists of a fact description $F^{i}$, a set of $k$
violated law articles $\\{A_{1}^{i},\dots,A_{k}^{i}\\}$, a set of $l$ accused
charges $\\{C_{1}^{i},\dots,C_{l}^{i}\\}$ and a penalty term $P^{i}$. All of
these items are composed of a sequence of words. Law articles and charges have
their corresponding sets of categories and each law article has its own
textual definition. Our goal is to learn a classifier $\zeta$ which is able to
predict the judgment results for a given fact, i.e.,
$(\\{A_{1}^{i},\dots,A_{k}^{i}\\},\\{C_{1}^{i},\dots,C_{l}^{i}\\},P^{i})=\zeta(F^{i})$.

##  3 Methods

In this section, we first introduce the T5 model and semantic prompts, then
describe how we can leverage the existing legal judgment documents to
construct semantic prompts for LJP based on T5. Finally, we explain how it
enables easy dependency learning for arbitrary sub-tasks.

###  3.1 Introduction of T5

We construct our model based on T5 [15], a large-scaled pretrained language
model (PLM) based on the Transformer architecture [18]. At each time step $t$,
it uses a left-to-right language model $\mathcal{D}$ to predict
$\mathbf{y}_{t}$ conditioned on previous generated tokens and hidden states
$\mathbf{h}$ from a separate self-attention-based encoder $\mathcal{E}$ for
input $\mathbf{x}$:

$\mathbf{h}=\mathcal{E}(\mathbf{x})\quad\mathbf{y}_{t}=\mathcal{D}(\mathbf{h},\mathbf{y}_{1},\dots,\mathbf{y}_{t-1})$

T5 unifies the training framework of natural language understanding and
natural language generation into the text-to-text form. It is pretrained with
the span-corruption objective, where consecutive spans of input tokens are
replaced with a mask token and the model is trained to reconstruct the masked-
out tokens. During pretraining, the input is the masked sentence and the
target is a sequence of masked spans where each span starts with its indicator
(e.g., $<$extra_id_0$>$). The mean length of a span is 3 and 15% of the
original text sequence are masked out.

###  3.2 Semantic prompts

Applying semantic prompts to PLMs has become increasingly popular. Under this
paradigm, downstream tasks are reformulated to look more like those solved
during the original PLM pretraining objective with the help of a textual
prompt. For example, when recognizing the emotion of a social media post, “I
missed the bus today.”, we may continue with a prompt “I felt so  ”, and ask
the LM to fill the blank with an emotion-bearing word [19]. The filled word
can be mapped to emotion labels by pre-defined rules. It has shown consistent
improvement over the traditional supervised finetuning especially in the
fewshot scenario [20, 16] because it can map tasks into existing semantic
knowledge inside the PLM instead of learning from scratch.

###  3.3 Prompts For LJP

It is quite straightforward to construct prompts for LJP. Since all sub-task
labels are themselves extracted from legal judgment documents by regex rules
[21], we can directly leverage the structure of the document as prompts for
different sub-tasks. A legal judgment document starts with a fact, following
by a set of judgment results and explanations. We can mask out the words
describing the judgment results from the document the same regex rule as used
in [21], input this corrupted document into the T5 model, then ask the model
to predict original spans in order. An illustration is shown in Figure 2. By
this means, the training shares exactly the same format as the original
pretraining objective of T5, such that we can make the most use of the
knowledge inside the pretrained model. After the model predicts the sequence,
we can convert it into sub-task labels with a one-to-one mapping.

###  3.4 Dependency Learning

The dependency learning across sub-tasks is naturally established with the
autoregressive decoder from the T5 model, where the prediction of one span
conditions on all the previously decoded spans. Taking predicting law articles
and term of penalty as example, we suppose that the latter depends on the
former here. We first use a prompting function to modify the input fact
$F^{i}$ into a prompt:

$\tilde{F}^{i}=f_{prompt}(F^{i})$

and the output is the concatenation of different sub-tasks labels which are
connected by their indicators:

$Y^{i}=\\{I_{1};A_{1}^{i};\dots;A_{k}^{i};I_{2};P_{i};I_{3}\\}$

where $I_{k}$ represents the indicator of the $k$-th sub-task and the last one
is the end signal. Then we pass $\tilde{F}^{i}$ into T5 and get output tokens
step by step. When the model decode $I_{2}$ at time step $t$, we can use
previous generated tokens $\\{\mathbf{y}_{1},\dots,\mathbf{y}_{t-1}\\}$ to map
corresponding law articles $\\{A_{1}^{i},\dots,A_{k}^{i}\\}$, and these tokens
are also used to predict term of penalty which is equivalent to:

$P^{i}=\mathcal{D}(\mathcal{E}(\tilde{F}^{i}),A_{1}^{i},\dots,A_{k}^{i})$

In this way, we build dependencies between different sub-tasks. Same as in
text generation tasks, the decoder conditions on the ground-truth spans during
training, and on the self-predicted spans during inference. The order of spans
in the decoding side reflects the dependency relations. We can simply adjust
the order of decoded spans (together with its indicator) to control the
logical order of sub-tasks.

(a) Single-task Learning

(b) Multi-task Learning

(c) Dependency Learning

Figure 3: Three types of learning methods used in LJP. Single-task learning
applies independent models for different tasks. Multi-task learning applies a
single model for all tasks, but there is no dependency across tasks.
Dependency-learning applies a single model to capture all dependencies within
tasks.

###  3.5 Auxiliary Tasks

The above framework provides us a unified way to incorporate arbitrary sub-
tasks existing in the legal judgment document. We can easily add on more tasks
by masking out the corresponding spans from the original document. In this
work, we experiment with two more auxiliary tasks: court view generation and
law article content explanation. Court view can be regarded as the
interpretation for the sentence of a case. It summarizes the fact description
and specifies related charges, then explains the rationale to derive the
judgment. Law article content includes a complete description of the premises
for violation and the scope of sentence. Both of them can be useful for LJP
since they provide the background basis and explanations on how the judgment
results are derived. By co-training with these two auxiliary tasks, the model
is able to “explain” its own predictions by generating the corresponding court
view and article content, which is crucial for a trustworthy LJP system.

##  4 Experiments

We conduct our experiments on a dataset collected from the China Judgments
Online 111https://wenshu.court.gov.cn/. In this section, we first explain how
we collect such dataset, then go over a set of baseline systems we compare
with, and finally present the findings.

###  4.1 Dataset

In the field of LJP, existing works often experimented on the CAIL dataset
[21] or extracted task-specific texts from the published judgment documents to
construct their own dataset [6, 22, 10]. However, none of them contains all
the 5 tasks we need. Furthermore, these datasets have been through a series of
pre-processing which makes it difficult to map each case to the original legal
judgment document. Therefore, we construct a dataset ourselves following the
same mechanism as in [21]. First of all, we filtered those cases involving
multiple defendants because different defendants correspond to different trial
results. Besides, the top 102 law articles in Chinese Criminal Law are not
relevant to specific charges, we also filtered these labels. In this way, we
got a dataset involving 200 charges and 183 law articles from 225,843 criminal
judgment documents. We randomly selected 12,810 cases to form the test set and
ensured it covered all types of charges and law articles. We also selected
12,634 cases to construct our validation set and the rest is training set.
Each case refers to 1.14 law articles and involves 1.06 charges on average,
but has strictly one term of penalty. We provide more detail statistics in
Table I.

TABLE I: Statistics of our dataset. We show the number of unique facts,
articles, etc. and the average number of words in each of them.

| Fact | Article | Charge | Penalty | View | Content
---|---|---|---|---|---|---
# Unique | 225,843 | 183 | 200 | 207 | 225,843 | 183
# Words | 199.46 | 3 | 7.26 | 6.41 | 148.28 | 137.9

###  4.2 Experimental Settings

####  4.2.1 Model configuration

We fine-tune our model based on mT5 base [23], a multilingual variant of T5
that can be used to fine-tune on Chinese tasks. For training, we set batch
size as 128 and adopt the gradient accumulation strategy. All models are
trained for a maximum of 30 epochs with early-stopping, and the model which
performs best on the validation set will be selected. Max length of input and
output are both set to 512. We use the AdaFactor optimizer [24] which can
relieve memory pressure to a certain extent.

####  4.2.2 Metrics

For law article and charge, we employ micro-F1 and macro-F1 as evaluation
metrics. Since penalty is regarded as regression task, we calculate the log-
distance between prediction and ground-truth following [25], a smaller log-
distance indicates better prediction.

####  4.2.3 Baselines

We compare with the following baselines:

  * •

LSTM [26]: We employ two-layer bidirectional LSTM to encode factual
description and then make prediction for each task with full-connected neural
network.

  * •

CNN [27]: We employ CNN with multiple filter widths as fact encoder and then
make prediction.

  * •

FactLaw [6]: It predicts top-k law articles with SVM, then encodes fact and
law article contents with HAN [28] and predicts the result.

  * •

TopJudge [12]: It encodes facts with CNN, and then models task dependencies
with LSTM.

  * •

LBERT [29]: It pretrained BERT on a large scale number of Chinese criminal
judgment documents which shows better performance than original BERT [30] in
LJP. We use this model to encode fact following by three different fully
connected networks for prediction of main tasks respectively.

TABLE II: Main results of different models trained on 10,000 samples.

|  | Dev | Test
---|---|---|---
|  | Article | Charge | Penalty | Article | Charge | Penalty
Method |  Model | MiF | MaF | MiF | MaF | Dis $\downarrow$ | MiF | MaF | MiF | MaF | Dis $\downarrow$
Single-task Learning | STL-LSTM | 0.697 | 0.304 | 0.709 | 0.293 | 2.170 | 0.688 | 0.250 | 0.706 | 0.243 | 2.178
STL-CNN | 0.730 | 0.330 | 0.730 | 0.315 | 2.177 | 0.718 | 0.275 | 0.717 | 0.261 | 2.188
STL-TopJudge | 0.760 | 0.353 | 0.775 | 0.338 | 2.139 | 0.753 | 0.315 | 0.777 | 0.330 | 2.097
STL-LBERT | 0.817 | 0.438 | 0.803 | 0.428 | 2.053 | 0.809 | 0.394 | 0.800 | 0.386 | 2.063
STL-T5 | 0.763 | 0.488 | 0.818 | 0.550 | 1.980 | 0.757 | 0.429 | 0.807 | 0.499 | 1.969
Multi-task Learning | MTL-LSTM | 0.719 | 0.320 | 0.720 | 0.296 | 2.185 | 0.708 | 0.265 | 0.708 | 0.247 | 2.185
MTL-CNN | 0.736 | 0.345 | 0.740 | 0.323 | 2.183 | 0.719 | 0.286 | 0.725 | 0.274 | 2.201
MTL-TopJudge | 0.774 | 0.365 | 0.777 | 0.339 | 2.093 | 0.771 | 0.313 | 0.767 | 0.289 | 2.110
MTL-LBERT | 0.805 | 0.418 | 0.810 | 0.418 | 2.115 | 0.801 | 0.380 | 0.801 | 0.372 | 2.123
MTL-T5 | 0.813 | 0.483 | 0.826 | 0.506 | 1.936 | 0.808 | 0.431 | 0.814 | 0.455 | 1.927
Dependency Learning | FactLaw | 0.733 | 0.416 | 0.535 | 0.220 | 2.087 | 0.717 | 0.403 | 0.536 | 0.217 | 2.096
Dependent-TopJudge | 0.777 | 0.381 | 0.787 | 0.393 | 2.047 | 0.772 | 0.327 | 0.781 | 0.335 | 2.054
Dependent-T5★ | 0.825 | 0.526 | 0.836 | 0.558 | 1.877 | 0.816 | 0.469 | 0.824 | 0.512 | 1.878
\+ Auxiliary Tasks |  $\bigstar$ \+ View | 0.827 | 0.538 | 0.843 | 0.581 | 1.854 | 0.823 | 0.498 | 0.839 | 0.530 | 1.846
$\bigstar$ \+ Content | 0.828 | 0.532 | 0.838 | 0.543 | 1.842 | 0.821 | 0.486 | 0.826 | 0.515 | 1.844
$\bigstar$ \+ View & Content | 0.827 | 0.543 | 0.841 | 0.567 | 1.867 | 0.819 | 0.481 | 0.830 | 0.521 | 1.840

FactLaw did not release the code, we set $k=5$ and reproduce it ourselves.
Other baselines are implemented with the open-sourced code in [25]. For all
above models, we compare them under the following settings:

  * •

Single-task Learning (STL): As shown in Figure 3a, it applies independent
models for different tasks. There is not any correlation among tasks.

  * •

Multi-task Learning (MTL): As shown in Figure 3b, it trains all tasks with a
single model with parameter sharing, but there is no dependency across tasks.

  * •

Dependency Learning: As shown in Figure 3c, it trains all tasks with parameter
sharing plus explicit dependency modelling. Only FactLaw and TopJudge can
apply to this setting.

###  4.3 Main Results

We show the main comparison results in Table II. All models are trained on the
same 10,000 subset sampled from the whole training data as a pilot study. For
the dependency-learning models, we use the order of article-charge-penalty for
model prediction, which, as explained, is the order that humans follow to make
the judgment results. For the experiments with auxiliary tasks, we append the
auxiliary tasks in the end of the three main tasks.

As for the model architecture, pretrained _Lbert and T5 have a clear
advantage_ compared with other models that are trained from scratch. T5 also
significantly outperforms LBert, even though it has never been tailored for
legal-specific text. T5 performs especially well on the Macro-F1 metric, with
a consistent lead of $8\sim 30\%$, suggesting it is able to generalize to low-
frequency labels effectively, while all other models overfit only to high-
frequency labels. For all architectures, _MTL does not lead to significant
improvement over STL_ on most tasks, suggesting parameter sharing only is not
an effective way for dependency learning.

When applying dependency learning, both the performance of TopJudge and T5 get
improved. However, the improvement of Dependent-T5 over STL/MTL-T5 is much
larger than that of Dependent-TopJudge over STL/MTL-TopJudge, indicating _our
proposed structure is a more efficient way to enable dependency learning than
TopJudge_.

By adding the two auxiliary tasks individually, the performance of
Dependency-T5 is again improved. However, combining both of them did not help
further. The benefit that could get from auxiliary tasks could have been
saturated.

Figure 4: Effects of decoding orders. For visualization, we divided the term
of penalty into 11 categories to calculate Macro-F1 too. Bars in different
colors indicate different orders of decoding article ($\mathcal{A}$), charge
($\mathcal{C}$), penalty ($\mathcal{P}$), court view ($\mathcal{V}$) and
article content ($\mathcal{A^{\prime}}$). Upper bars in lighter colors are
results of making prediction based on the ground-truths of predecessor tasks.
Dashed lines indicate single-task results.

###  4.4 Analysis

####  4.4.1 Orders and Dependencies

We show the effects of decoding orders in Figure 4, where we modify the orders
of three main tasks and insert two auxiliary tasks into different positions.
As can be observed, _dependency learning benefits not only successor tasks,
but also predecessor tasks_. Law article prediction under all the 6 order
combinations outperform the single-task performance, suggesting _the unified
dependency learning framework provides a more effective way of parameter
sharing than multi-task learning_. Even for the first decoding task that has
no dependency on other tasks, it can still outperform multi-task learning
under the same setting.

The prediction of article and charge correlate more with each other, and
knowing one of them helps significantly the prediction of the other. Penalty
can benefit more from article/charge than the other way around. This coincides
with humans that the decision of penalty should be left at the end.

_The best decoding order for each task can be different from human
intuitions_. The reason is that there exists error propagation for machines.
When the downside of error propagation outweighs the dependency of other
tasks, machines will prefer putting ahead the task to maximize the
performance. For example, humans tend to decide the penalty after confirming
the charge, yet the model prefers predicting the penalty before the charge
because the error propagation from charges will negatively impact the
prediction.

This is more obvious for auxiliary tasks. For example, humans will first come
with the court view because it includes the derivation that we need to decide
the penalty. For the model, however, putting the court view before the penalty
significantly deteriorates the results because the court view is long and
prone to error propagation. The same holds for law article contents. _For both
auxiliary tasks, the best order is to put them at the end_ so that they will
not incur any error propagation for the main tasks.

TABLE III: Comparison of leveraging law article contents for article
prediction (all models are based on mT5). Dependent-T5 is more effective than
text matching while being significantly faster.

Methods | MiF | MaF | Speed/s
---|---|---|---
STL (article) | 0.757 | 0.429 | 0.37
Text Matching | 0.770 | 0.431 | 64.74
Dependent (article + content) | 0.817 | 0.488 | 0.37

####  4.4.2 Law Article Content

The law article item and content have a strictly one-to-one relation. A more
common way to leverage the context for article prediction is by text matching,
where we turn it from multi-label classification to multiple binary
classification problems. Following the common practice for text matching, we
concatenate the fact with every single law article content as the input of T5,
and decode a binary relevant/irrelevant output. We take the positive-negative
ratio as 1:31 to construct training set and train it with cross-entropy loss
following [31]. When testing, the classification threshold is set as 0.7
(tuned to maximized the micro-F1). We compare it with the STL-T5 (predict
article only) and Dependent-T5 (decode article then content). Results are
shown in Table III. We can see that Dependent-T5 outperforms the text-matching
method without affecting the inference speed because the content prediction is
only used in training. Text matching, on the contrary, slows down the
inference significantly as it needs to match over all articles one by one.
Even with parallel batching, it still doubles the time with hundreds more
memory cost.

Figure 5: Performance by training on varying data sizes.

Figure 6: Visualization of cross attention between the encoder and decoder.
The decoder can learn to attend to proper key words in the fact to predict
corresponding judgement results. Figure 7: Point-wise mutual information
between different classes of LJP tasks. From left to right are counted on (1)
ground-truth dataset, (2) results predicted by Dependent-T5 and (3)
mispredicted results by Dependent-T5 with error of penalty $\geq$ 1 year.

####  4.4.3 Data Size

To observe the effect of data size on the dependency learning advantage, we
trained TopJudge, STL-T5, and dependent-T5 (with the identified best decoding
order $\mathcal{A},\mathcal{C},\mathcal{P},\mathcal{V}$) on training sets of
varying sizes. The results on test set are shown in Figure 5. With the
increase of training data, performances of three models continue to improve.
When training data grows, the gap between STL-T5 and Dependent-T5 becomes
smaller. We conjecture that the effects of parameter sharing and dependency
learning will saturate with sufficient data. Nonetheless, both outperform
TopJudge by a large margin. Dependent-T5 is especially data efficient and
achieves decent accuracy with only hundreds of training samples.

####  4.4.4 Interpretability

Figure 6 depicts the cross attention weight distribution of each token in the
input (x-axis) when generating output tokens (y-axis). Each score is obtained
by summing over the attention score of each head. It shows which parts of the
fact description play an important role in the generation of judgment results
and also provides interpretability for the prediction.

In Figure 7, we visualize the point-wise mutual information (PMI) between any
two categories in LJP tasks under three scenarios: (1) ground-truth data, (2)
predicted results and (3) wrongly predicted results where the predicted
penalty has an error of more than 1 year. Due to the large number of
categories involved in our dataset, we selected top-10 law articles and top-11
(two have equal probabilities) charges with the highest co-occurrence
probabilities visualization. We also divided term of penalty into 11
categories as mentioned earlier. We can see that _the PMI heatmap shares the
similar shape under all three scenarios_. The model prediction follows the
same correlation as the ground-truth data. Even when the model prediction
makes significant errors, the correlation is still maintained. The strong
correlation suggests the model has indeed learnt the proper dependency
relationships, and _we can use the dependent tasks to interpret the model
predictions_ , e.g., the wrong prediction of the following task is because of
the wrong prediction of the predecessor tasks.

####  4.4.5 Case Study

We provide an example predicted with our method in Figure 8. Since the
defendant causes the burn of the face, neck and hands of the victim, the model
makes the wrong judgment prediction which relates the case to the crime of
arson. The generated court view and relevant law article content clearly
explain why the model makes such a prediction. When we replace the first
decoded law article with ground-truth, results of following tasks are also
corrected accordingly, and the error of predicted penalty is greatly reduced
from 28 months to only 2 months. The generated court view is also changed
which can provide supports for the new judgment results. Similarly, the law
article content also explains why the model prediction is changed from 36
months to 6 months because the penalty of arson is stronger than the penalty
of intentional injury by law. The example illustrates how we can let the model
“explain” itself by decoding the whole line of tasks, which is crucial for a
transparent and trustworthy LJP system.

Figure 8: A wrongly predicted example. When we replace the predicted law
article with ground-truth, subsequent judgment results are corrected. The
model-generated court view and law article content clearly explain why the
model makes such a prediction.

##  5 Related Work

Research on LJP has been studied for decades. Early researches relied
primarily on hand-craft features and applied statistical [1, 2] or machine
learning methods [3, 4, 32, 5] for it. Due to limitations of data, they
focused on a tiny subset of case categories.

Recently, with the accessibility of large-scale judicial datasets [21, 33, 34]
and the development of deep learning [35, 30, 36, chang2020dart,
chang2021neural, chang2021training], rapid progress has been made by treating
each sub-task in LJP as a text classification problem [37, 38]. Some works
experimented conditioning a certain sub-task on other sub-tasks and showed
improvement. For example, [7] applied reading comprehension framework to
incorporate law articles for the task of predicting judgments of civil cases.
[8] utilized given law articles as supplementary information to predict
related charges. [9] proposed a deep gating network for charge-based prison
term prediction. All these works rely on the ground-truth dependent sub-tasks,
which are usually not available in real scenarios. When replacing the ground-
truth labels with predicted ones, there is a significant drop of performance
[12].

There have been works applying multi-task learning to share model parameters
among sub-tasks [10, 11]. As all sub-tasks are classification problems, all
parameters can be shared except for the task-specific prediction heads.
Nevertheless, there exists a strict logical order when humans make LJP
decisions, and the multi-task framework lacks the mechanism to reflect this
ordered dependency. Therefore, some researchers try to establish explicit
ordered dependencies among sub-tasks. For example, after encoding facts with
neural network, [6] extracted top k law articles with SVM and utilized them to
assist the prediction of charges. [12, 13] built dependencies among all three
sub-tasks. They apply CNN for fact encoding and use LSTM to model the ordered
dependencies. Each sub-task gets a task-specific representation to predict the
result. In comparison, our model leverages a single unified architecture that
can naturally build dependencies for arbitrary types of sub-tasks.

There have also been works focusing on court view generation [22, 39, 40], but
did not leverage it to improve the LJP results. Some works utilized law
article contents to assist charge prediction [41, 42]. They design complex
architectures to incorporate the contents. In contrast, our framework is
simple and _does not affect the inference speed at all_ since the content is
only used for training.

Leveraging a single unified text-to-text Transformer has also been applied in
other NLP tasks like dialogue generation [43, 44] and question answering [45].
We adopt a similar approach in our work and further show its flexibility of
enabling effective dependency learning.

##  6 Ethical Statement

Since LJP is an emerging but sensitive technology, we would like to discuss
ethical concerns of our work. Firstly, the corpus is created from publicly
available data and the personal private information (e.g., name, plate number,
etc.) has been anonymized. In addition, the proposed method aims to assist
legal professionals in their research and decision-making instead of replacing
them. Therefore, ethical considerations such as allowing legal rights and
obligations of human beings to be decided by non-human intelligence are not
breached by the system.

##  7 Conclusion

In this paper, we propose learning dependencies among sub-tasks of LJP with a
unified text-to-text Transformer. It directly leverages the original legal
judgment document as the prompt without handcrafting, enables full parameter
sharing and supports arbitrary types and amounts of sub-tasks. We show it
significantly outperforms SOTA, and the model can learn the task dependencies
more effectively than previous methods. We analyze the effects of decoding
order, data size and illustrate it can provide explanations to its own
prediction results, even when an error is made. The proposed framework is
simple and flexible. It can be easily extended to cover more legal tasks,
which we leave for future work.

## Acknowledgments

This work was supported by the National Key R&D Program of China
(2016YFC0800803 ). Chuanyi Li is the corresponding author.

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