capbtabboxtable[][]

11institutetext: Radboud University, Nijmegen, The Netherlands
11email:
<EMAIL_ADDRESS>CISPA Helmholtz Centre for Information Security, Saarbrücken, Germany
22email<EMAIL_ADDRESS>Equal contribution

# Generative Poisoning Using Random Discriminators

Dirren van Vlijmen 11 Alex Kolmus∗ 11 Zhuoran Liu∗ 11 Zhengyu Zhao 22 Martha
Larson 11

###### Abstract

We introduce ShortcutGen, a new data poisoning attack that generates sample-
dependent, error-minimizing perturbations by learning a generator. The key
novelty of ShortcutGen is the use of a randomly-initialized discriminator,
which provides spurious shortcuts needed for generating poisons. Different
from recent, iterative methods, our ShortcutGen can generate perturbations
with only one forward pass in a label-free manner, and compared to the only
existing generative method, DeepConfuse, our ShortcutGen is faster and simpler
to train while remaining competitive. We also demonstrate that integrating a
simple augmentation strategy can further boost the robustness of ShortcutGen
against early stopping, and combining augmentation and non-augmentation leads
to new state-of-the-art results in terms of final validation accuracy,
especially in the challenging, transfer scenario. Lastly, we speculate,
through uncovering its working mechanism, that learning a more general
representation space could allow ShortcutGen to work for unseen data.

##  1 Introduction

The field of protective poisoning attacks has been gathering attention [7, 10,
4, 6, 13] with the rise of the non-consensual use of social images to train
deep learning models [1, 23, 15, 9]. Poisoning is conventionally achieved by
injecting additional poisoning images [4, 16] into the training set, and more
recently, by slightly perturbing images already in the training set [6, 7, 10,
13], making it less noticeable [21, 22]. Most existing studies on
perturbation-based poisoning [6, 7, 10] use an iterative creation process,
which is relatively inefficient. Moreover, these studies use loss functions
that require knowledge of the labels for specific images, which might not be
desirable in privacy-sensitive scenarios.

In this work, to address these limitations, we propose ShortcutGen, a new
generative poisoning method capable of generating sample-dependent poisoning
shortcuts. Once trained, the generator can generate perturbations for any
given input image with only one forward pass, which is naturally more
efficient than iterative methods. In addition, the use of a generator
eliminates the need for label knowledge during the creation process. The main
difference between our ShortcutGen and the only existing generative method,
DeepConfuse, is that our ShortcutGen only trains the generator with a static
discriminator, while DeepConfuse requires time-consuming alternating updates
between the discriminator and generator. This difference makes our ShortcutGen
about $\times 100$ faster than DeepConfuse.

In particular, we demonstrate that using a well-trained discriminator,
following recent advances on generative adversarial examples [19, 18, 17],
cannot work but a randomly-initialized discriminator is needed. This insight
is based on our new conjecture that a randomly-initialized discriminator
provides a noisy mapping between images and labels, encouraging the generator
to learn spurious shortcuts instead of generalizable features.

We also explore possible ways to further improve ShortcutGen. Specifically, we
show that using a simple augmentation strategy boosts the robustness of
ShortcutGen against early stopping [10, 20], and that using composite
perturbations from ShortcutGen with and without this augmentation yields
better performance than existing methods in terms of the final validation
accuracy, especially in the challenging, transfer scenario. Finally, we
provide additional insights into the working mechanism of ShortcutGen and its
potential to generate poisons for unseen data based on learning a stronger
generator with a more general latent representation space.

Figure 1: Training pipeline of our ShortcutGen for generating poisoning
shortcuts.

##  2 ShortcutGen

First, we briefly introduce the general formulation of the data poisoning
problem studied in this work. The general goal of data poisoning is to
compromise the availability of a victim model by reducing its accuracy on
clean test data111A thorough review of availability poisoning attacks can be
found in [8].. Following recent work [6, 7, 10, 13], we achieve this goal by
adding perturbations $\mathcal{P}=\\{\boldsymbol{\delta}_{i}\\}$ into the
original (clean) training dataset
$\mathcal{D}_{o}=\\{(\boldsymbol{x}_{i},y_{i})\\}$ [10, 7] to obtain a
poisoned training dataset
$\mathcal{D}_{p}=\\{(\boldsymbol{x}^{\prime}_{i},y_{i})\\}$. Specifically, the
generated perturbations are sample-dependent, i.e.,
$\boldsymbol{x}^{\prime}_{i}=\boldsymbol{x}_{i}+\boldsymbol{\delta}_{i}$, and
bounded by $||\boldsymbol{\delta}_{i}||\leq~{}\epsilon=8/255$.

Next, we present technical details of ShortcutGen, which is inspired by
existing work on generative adversarial perturbations [19, 17, 18]. Fig. 1
presents the training pipeline of our ShortcutGen. ShortcutGen learns a
ResNet-based generator $\mathcal{G}_{\Psi}$, which takes as input a specific
image sample $\boldsymbol{x}_{i}$ and outputs its corresponding bounded
perturbations $\boldsymbol{\delta}_{i}$, against a discriminator $F_{\theta}$.
During training, only the generator is updated but the discriminator is
static. Specifically, ShortcutGen uses an error-minimizing loss [10] that aims
to associate shortcut features with the true class of each image. The
generator is trained on the clean dataset $\mathcal{D}_{o}$ using the loss
function Eq. 1.

$\min_{\Psi}\mathcal{L}(F_{\theta}(\mathcal{G}_{\Psi}(\boldsymbol{x})+\boldsymbol{x}),y)$
(1)
$\min_{\Psi}\mathcal{L}(F_{\textrm{init}}(\textrm{Aug}(\mathcal{G}_{\Psi}(\boldsymbol{x})+\boldsymbol{x})),y)$
(2)

We adopt the cross-entropy loss for $\mathcal{L}$ and use the Change of
Variables [3] to softly clip the perturbations for more accurate gradients.

Random Discriminator. We find that directly adopting the framework in [18],
thus using a well-trained discriminator, can only reduce the model accuracy by
2%. We assume a well-trained discriminator provides the possibility of using
highly-generalizable (yet not robust) features as the shortcuts. To address
this problem, we propose to instead use a randomly-initialized discriminator
($F_{\textrm{init}}$), which helps the generator learn spurious shortcuts,
which are hardly generalizable. This is supported by the fact that random
networks can detect random, foreground objects but have low discriminative
capability [2]. Using a random discriminator can also simulate the early
training stage of the victim model, thus it can generate faster learned
poisons that should be more effective [20].

Input Augmentation Against Discriminator. Another important change we made
beyond the framework in [18] is to augment the input of the discriminator for
improving the robustness of ShortcutGen against a potential early stopping
defense [10, 20]. This is motivated by our finding that a victim model trained
with data augmentations is capable of maintaining the model accuracy in the
early training stage, in contrast to a victim model without data
augmentations. Our input augmentation is expected to reduce the impact of the
victim model’s data augmentations, relative to the literature, because it
forces the generator to learn augmentation-invariant perturbations.

Overall, the loss function in Eq. 1 is modified to obtain Eq. 2 by integrating
a random discriminator $F_{\textrm{init}}$ and input augmentation
$\textrm{Aug}(\cdot)$. We also combine the perturbations from the augmented
and non-augmented variations (using maximal pixel values) to further boost the
performance in terms of final validation accuracy, while remaining competitive
in the peak validation accuracy.

##  3 Results

We first present the evidence of ShortcutGen’s training efficiency. As can be
seen from Table 1, our ShortcutGen is roughly 100 times more efficient than
DeepConfuse [6], the only existing generative poisoning method.

Method | Training Time
---|---
DeepConfuse [6] | 10 days
ShortcutGen no Aug |  2 hours
ShortcutGen |  2.5 hours
Table 1: Training time of compared generative poisoning frameworks for the
training set of CIFAR10 [11] using the original code and hyperparameters. All
methods were trained on the same hardware, which included 1 GeForce RTX 2080
Ti GPU.

Then, we compare the performance of our ShortcutGen (SG) to the only existing
generative poisoning attack method, DeepConfuse (DC) [6], and also other
state-of-the-art methods: Error-Minimizing (E-Min) [10] and Error-Maximizing
(E-Max, targeted) [7].

Method | Final | Peak (# epoch)
---|---|---
Clean | 94.09 | 94.17 (58)
E-Min [10] | 22.16 $\pm$ 0.33 |  24.3 $\pm$ 0.93 (28)
E-Max [7] | 9.14 $\pm$ 0.44 | 50.52 $\pm$ 3.93 (2)
DC [6] | 13.34 $\pm$ 1.05 | 32.55 $\pm$ 3.34 (1)
SG no Aug | 10.45 $\pm$ 2.49 | 60.96 $\pm$ 2.97 (4)
SG | 36.04 $\pm$ 4.23 | 39.19 $\pm$ 3.67 (36)
Combined | 6.42 $\pm$ 0.42 | 41.02 $\pm$ 1.45 (2)

Figure 2: Validation accuracy (%) in terms of final (60 epochs) and peak (#
epoch) accuracy for a RN18 victim model on CIFAR10. Results are averaged over
5 runs.

.  Method VGG11 CNN MN WRN Clean 91.42/91.42 67.56/67.67 77.37/77.61
95.64/95.64 E-Min [10] 21.00/23.06 21.61/23.03 19.87/28.02 20.56/23.48 E-Max
[7] 21.93/50.37 28.28/46.34 14.54/39.32 8.78/49.34 DC [6] 15.94/24.25
16.66/24.19 15.40/38.05 13.20/18.94 SG no Aug 20.26/52.64 20.31/46.10
17.90/55.91 13.61/53.01 SG 55.11/56.56 34.64/34.86 38.86/38.83 38.63/40.44
Combined 9.91/30.83 13.58/46.54 10.95/43.18 5.87/38.26

Figure 3: Final/peak transferability (%) of a VGG11, MobileNetV2 (MN), or
WideResNet (WRN) victim model when RN18 is adopted as the discriminator.

As can be seen from Table 3, SG (no Aug) reduces the final accuracy to random
guessing (10 % for CIFAR10) and further the best results are achieved by the
combined method. The superior performance in terms of final validation
accuracy of the combined method also generalizes to the challenging, transfer
scenario, as shown by Table 3. This performance gain may be explained by the
fact that the combination inherits the strong shortcuts from no augmentation
but with a lower start-point accuracy provided by augmentation. In addition,
as expected, incorporating the augmentation improves ShortcutGen on the peak
accuracy.

| MNIST | ImageNet
---|---|---
Clean | 100.00/100.00 | 89.60/89.60
DC [6] | 97.39/98.66 | 84.40/86.20
SG no Aug | 95.91/98.15 |  83.60/84.40
SG | 99.03/99.06 | 84.80/85.80
Combined |  86.59/97.55 | 85.40/86.00

Figure 4: Validation final/peak accuracy (%) of a RN18 victim model trained on
CIFAR10 poisons but tested on unseen (ImageNet or MNIST) data.

Figure 5: The t-SNE visualizations for the latent space of our generator (no
Aug) on CIFAR10 [11] (left) , MNIST [12] (middle) and subset of ImageNet [5]
(right).

In order to be truly useful in privacy-sensitive scenarios, generative
poisoning should be successful when generating perturbations for data not seen
during training. As can be seen from Table 5, our combined method has the
potential for this purpose since it reduces the final model accuracy on
ImageNet and MNIST. Specifically, the results on MNIST are better than those
on ImageNet. This might be explained by the t-SNE [14] visualizations of the
last non-upsampling layer of the generator in Fig. 5, which shows that the
feature space of the generator on MNIST is more separable than that on
ImageNet, and fully-separable on CIFAR10. These visualizations also highlight
the working mechanism of our ShortcutGen and imply that learning a more
general representation space allows it to work on unseen data.

##  4 Conclusion

In this paper, we propose a novel generative poisoning method, ShortcutGen,
for learning spurious shortcuts based on a randomly-initialized discriminator.
We demonstrate that integrating input augmentation makes ShortcutGen more
robust against the early stopping defense, and combining an augmented and non-
augmentation variant yields new state-of-the-art results in terms of the final
validation accuracy, especially in the challenging, transfer scenario.
Finally, we uncover the working mechanism of ShortcutGen and demonstrate its
potential to work on unseen data.

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# Heavy quark potential and LQCD based quark condensate at finite magnetic
field

Indrani Nilima  Indian Institute of Technology Bhilai, GEC Campus, Sejbahar,
Raipur - 492015, Chhattisgarh, India  Aritra Bandyopadhyay  Guangdong
Provincial Key Laboratory of Nuclear Science, Institute of Quantum Matter,
South China Normal University, Guangzhou 510006, China  Guangdong-Hong Kong
Joint Laboratory of Quantum Matter, Southern Nuclear Science Computing Center,
South China Normal University, Guangzhou 510006, China  Ritesh Ghosh
<EMAIL_ADDRESS>Theory Division, Saha Institute of Nuclear Physics,
HBNI,
1/AF, Bidhannagar, Kolkata 700064, India  Homi Bhabha National Institute,
Anushaktinagar,
Mumbai, Maharashtra 400094, India  Sabyasachi Ghosh  Indian Institute of
Technology Bhilai, GEC Campus, Sejbahar, Raipur - 492015, Chhattisgarh, India

###### Abstract

We have studied various properties of heavy quarkonia in hot and magnetized
quark gluon plasma. Inverse magnetic catalysis (IMC) effect is incorporated by
modifying the effective quark masses. Then we obtain the real and imaginary
part of the heavy quark potential. After evaluating binding energy and decay
width we obtain the dissociation temperature of heavy quarkonia in presence of
magnetic field.

##  I Introduction

A lot of information is being provided by the ongoing relativistic heavy-ion
collisions (HIC) in respect of deconfined state of matter, i.e. Quark Gluon
Plasma (QGP). QGP at sufficiently high temperature behaves like a weakly
interacting gas of quarks and gluons, which can also be studied perturbatively
by using hard thermal loop (HTL) resummation Weldon:1982aq ; Braaten:1989mz ;
Frenkel:1989br ; Braaten:1991gm  techniques, apart from the first principle
lattice QCD estimations. Depending on the non-centrality, HIC can also produce
a very strong magnetic field in the direction perpendicular to the reaction
plane, as has been realized in recent studies Kharzeev:2007jp . At the RHIC
energies, the estimated strength of the magnetic field is around $B\sim
m_{\pi}^{2}\equiv 10^{18}$ Gauss whereas at the LHC the estimated strength is
around $B\sim 15m_{\pi}^{2}\equiv 1.5\times 10^{19}$ Gauss Kharzeev:2007jp ;
Skokov:2009qp , where $m_{\pi}$ is the pion mass. The issue - whether this
large initial magnetic field will decay very fast McLerran:2013hla  or slow
Tuchin:2010gx ; Das:2017qfi  is still probably an open problem. Latter
possibility provide the opportunity to revisit entire QGP phenomenology in
presence of magnetic field like heavy quark phenomenology, which is attempted
in present work. Such an extensive magnetic field might even have survived
from the very early stages of universe Vachaspati:1991nm ; Grasso:2000wj .

In the quarks / antiquarks, the magnetic field effects enter in the degrees of
freedom through various Landau levels, which signifies the quantization of the
transverse momenta in the dispersion relation. Inclusion of the external
magnetic field in the theoretical studies of quark matter have produced a
large body of work and various novel and interesting features in recent years.
One of the most interesting aspects of a magnetized medium is the discovery of
a pair of contrasting effects, i.e. magnetic catalysis (MC) and inverse
magnetic catalysis (IMC) Bali1 ; Bali2 . MC shows the low temperature
enhancements in the values of quark condensates with increasing magnetic field
and extensively studied through lattice QCD and effective models. On the other
hand, IMC shows a decreasing behaviour of the condensates with increasing
magnetic field close to the transition temperature. IMC was first discovered
by lattice QCD simulations using physical values of pion mass and for the
light quarks. Since then there have been many efforts to implement IMC in the
effective models. In our present work, we have implemented this important IMC
effect in our calculation through the constituent quark mass generated by the
lattice QCD simulations, which incorporates the complex and non-monotonic
temperature and magnetic field dependence within the quark condensates.

One of the useful probe of the QGP formation is heavy quarkonium which is a
bound state of $Q\bar{Q}$ pair McLerran:1986zb ; Back:2004je . After the
discovery of $J/\psi$ ( a bound state of $c{\bar{c}}$), Aubert:1974js ;
Augustin:1974xw , in 1974, a large number of excellent articles have been
published that proposed several essential refinements in the study of heavy
quark potential. The first work to study the quarkonia at finite temperatures
using potential models have been done by Karsch, Mehr, and Satz mehr . Whereas
a pioneering work to study the dissociation of quarkonia was firstly carried
out by Matsui and Satz Matsui:1986dk  due to color screening in the deconfined
medium with finite temperature. Recently different studies have been executed
to see the impact of magnetic field on heavy quark phenomenology
Alford:2013jva ; Marasinghe:2011bt ; Cho:2014exa ; Guo:2015nsa ;
Bonati:2015dka ; Rougemont:2014efa ; Dudal:2014jfa ; Sadofyev:2015hxa ;
Machado:2013rta ; Machado:2013yaa ; Gubler:2015qok ; Fukushima:2015wck ;
Das:2016cwd ; Singh:2017nfa ; Kurain:2019 ; Hasan:2017fmf . On quarkonia
production, the effects of the external magnetic field have been discussed in
Refs. Machado:2013rta ; Guo:2015nsa . The physics about the fate of quarkonia
in the QGP medium, at zero temperature can be understood with the help of non-
relativistic potential models. Since masses of heavy quarks ($m_{Q}$) are much
larger than QCD scale ($\Lambda_{QCD}$) and velocity of the quarks in the
bound state is small, $v\ll 1$ Lucha:1991vn , to understand the binding
effects in quarkonia generally one uses the Cornell potential which is a type
of non-relativistic potential modelsEichten:1979ms  and can be derived
directly from QCD using the effective field theories (EFTs) Eichten:1979ms ;
Lucha:1991vn ; Brambilla:2004jw . In Ref. Das:2016cwd  studied the directed
flow of charm quarks which is considered as an efficient probe to characterize
the evolving magnetic field produced in ultra-relativistic HIC. In Ref.
Machado:2013yaa  they had investigated QCD sum rules in calculation of the
mass of heavy mesons to estimate the modification of the charged B meson mass,
($m_{B}$), in the presence of an external Abelian magnetic field, eB. Ref.
Fukushima:2015wck  had computed the momentum diffusion coefficients of heavy
quarks, in a strong magnetic field B along the directions parallel and
perpendicular to B, respectively, at the leading order in QCD coupling
constant. Refs. Singh:2017nfa ; Hasan:2017fmf  has studied the effect of a
strong magnetic field on the heavy quark complex potential with the lowest
Landau level (LLL) approximation. In this regards, present work has moved
beyond LLL estimation and considered all Landau level summation, which valid
for entire range of magnetic field from weak to strong. This is one of the new
ingredient of present work. Apart from it, another major goal of the present
work is to incorporate IMC effect in heavy quark potential problems. These two
components are mainly introduced in the standard formalism of heavy quark
potential Agotiya:2008ie ; Thakur:2012eb ; Thakur:2013nia ; Kakade:2015xua ;
Agotiya:2016bqr , which is the sum of both Coulomb and string terms
Eichten:1974af .

The paper is organized as follows. In section II, we will discuss the basic
formalism of our present work which includes discussions about the real and
imaginary parts of the heavy quark potential, decay width, binding energy and
the Debye screening mass. In section III we will show our results for the same
as well as find out the dissociation temperatures and discuss their magnetic
field dependence. Finally in section IV, we shall conclude the present work.

##  II Formalism

In this section, we will described entire formalism, required for our current
study. In the first subsection (II.1), we have addressed the standard
framework of heavy quark potential in presence of an external magnetic field,
obtained from the gluon propagator. A temperature and magnetic field dependent
Debye mass is entered into heavy quark potential through gluon propagator.
This Debye mass is calculated from semi-classical transport theory, whose
magneto-thermodynamical phase space can be obtained by projecting the
temperature and magnetic field dependent condensate from lattice quantum
chromodynamics (LQCD) calculation. This part is discussed in subsection
(II.2). Next in subsection (II.3), we have discussed about the framework of
thermal width and binding energy of heavy quarkonia (e.g. $J/\psi$ or
$\Upsilon$) from imaginary and real part of heavy quark potential
respectively.

###  II.1 In-medium heavy quark potential in presence of magnetic field

Let us start our discussion with the full Cornell potential Eichten:1978tg ;
Eichten:1979ms , that contains the Coulombic as well as the string part given
as,

$\displaystyle{\text{V}(r)}=-\frac{\alpha}{r}+\sigma r~{}.$ (1)

Here, $r$ is the effective radius of the corresponding quarkonia state,
$\alpha$ is the strong coupling constant given by
$({\alpha}={\alpha}_{s}C_{F}=\frac{g_{s}^{2}C_{F}}{4\pi};C_{F}=4/3)$ and
$\sigma$ is the string tension.

The Fourier transform of ${\text{V}(r)}$ is

$\displaystyle{\bar{\text{V}}}(k)=-\sqrt{\frac{2}{\pi}}\bigg{(}\frac{\alpha}{k^{2}}+2\frac{\sigma}{k^{4}}\bigg{)}~{}.$
(2)

The assumption given in Ref. Agotiya:2008ie  has been followed which says
that, the in-medium modification can be obtained in the Fourier space by
dividing the heavy-quark potential from the medium dielectric permittivity,
$\epsilon({\bf k},T,eB)$ as,

$\displaystyle{\tilde{V}}(k,T,eB)=\frac{{\bar{\text{V}}}(k)}{\epsilon(k,T,eB)}~{},$
(3)

where $\epsilon(k,T,eB)$ can be obtained from the static limit of the
longitudinal part of gluon self-energy schneider . So, information of
temperature $T$ and magnetic field $eB$ (in M2 dimension) are entered through
this medium dielectric permittivity $\epsilon({\bf k},T,eB)$. By making the
inverse Fourier transform, we can obtain the modified potential at finite $T$
and $eB$ as,

$\displaystyle\bar{{\text{V}}}(r,T,eB)=\int\frac{d^{3}\mathbf{k}}{(2\pi)^{3/2}}(e^{i\mathbf{k}\cdot\mathbf{r}}-1)\tilde{V}(k,T,eB)~{}.$
(4)

Next, the dielectric permittivity can be obtained in the static limit, in the
Fourier space, from the temporal component of the propagator ($\Delta^{00}$)
as Weldon:1982aq ; schneider ,

$\displaystyle\epsilon^{-1}({\bf k},T,eB)=-\lim_{\omega\to
0}k^{2}\Delta^{00}(\omega,{\bf k},T,eB).$ (5)

Now, to obtain the real part of the inter-quark potential in the static limit,
the temporal component of real part of the retarded propagator in the Fourier
space is demanded, which is given as

$\displaystyle Re[\Delta^{00}(\omega=0,{\bf k},T,eB)]$ $\displaystyle=$
$\displaystyle\frac{-1}{k^{2}+m_{D}^{2}(T,eB)}$ (6)

The imaginary part of the same can be derived from the imaginary part of the
temporal component of symmetric propagator in the static limit which is given
as Sym_Prop

$\displaystyle Im[\Delta^{00}(\omega=0,{\bf k},T,eB)]$ $\displaystyle=$
$\displaystyle\pi~{}T~{}m_{D}^{2}\bigg{\\{}\frac{-1}{k\\{k^{2}+m_{D}^{2}(T,eB)\\}^{2}}\Big{\\}}$

Thus in the short-distance limit ($\hat{r}\ll 1$), the sum of Coulomb and
string term gives the real and imaginary part of the potential in terms of
modified coordinate space $\hat{r}=rm_{D}$ given as Agotiya:2016bqr ; V_mD1 ;
V_mD2

$\displaystyle ReV(\hat{r};T,B)=\left(\frac{2\sigma}{m_{D}}-\alpha
m_{D}\right)\frac{e^{-\hat{r}}}{\hat{r}}-\frac{2\sigma}{m_{D}\hat{r}}+\frac{2\sigma}{m_{D}}-\alpha
m_{D}~{},$ (7)

and

$\displaystyle
ImV(\hat{r};T,B)=T\left(\frac{\alpha{\hat{r}^{2}}}{3}-\frac{\sigma{\hat{r}}^{4}}{30m_{D}^{2}}\right)\log(\frac{1}{\hat{r}})$
(8)

respectively. Reader can notice that the information of $T$ and $B$ are
entering through Debye mass $m_{D}=m_{D}(T,B)$, whose mathematical derivation
is addressed in next subsection.

###  II.2 Debye mass in presence of magnetic field

Debye screening mass is an important observable in the context of heavy ion
collisions which also acts as a QGP signature through heavy quarkonia (i.e.
$J/\Psi$ and $\Upsilon$) suppression. Debye screening mass can be directly
evaluated from the temporal component of the gluon self energy tensor
($\Pi_{00}(p_{0},\vec{p})$) by employing the static limit
($|\vec{p}|=0,p_{0}\rightarrow 0$) through a perturbative order by order
evaluation. On the other hand one can also determine the Debye screening mass
through the semi-classical transport theory by using the relation Carrington ;
Chandra_Ravi ; Agotiya:2016bqr

$m_{D}^{2}=g_{s}^{2}C_{q/g}\int\frac{d^{3}p}{(2\pi)^{3}}~{}\frac{d}{dE}f_{q/g}(E),$
(9)

where $g_{s}^{2}\equiv 4\pi\alpha_{s}$ is the strong coupling constant,
$C_{q/g}$ are the Casimir constants for quarks and gluons and
$f_{q/g}=\frac{1}{e^{\beta E}\pm 1}$ are their respective distribution
functions - Fermi-Dirac (FD) for quark and Bose-Einstein (BE) for gluons. For
an ideal non interacting QGP medium, one can readily trace back the leading
order hard thermal loop (HTL) expression of the Debye mass, i.e.
$m_{D}^{2}=\frac{g_{s}^{2}T^{2}}{3}(N_{c}+N_{f}/2)$. There are several
investigations on the Debye screening masses of the QGP as a function of the
magnetic field from the temporal component of the gluon polarization in
perturbative QCD (pQCD) calculation Bandyopadhyay:2016fyd ; Bonati:2017uvz ;
Singh:2017nfa ; Kurian:2017yxj ; Mitra:2017sjo ; Kurain:2019 ;
Karmakar:2019tdp , whose equivalence anatomy in semi-classical transport
theory has been shown by Ref. SG_VC . The gluonic distribution function
remains unchanged in presence of an external anisotropic magnetic field along
the $z$ direction ($\vec{B}=B\hat{z}$), whereas the quark distribution
function gets modified to:

$f^{l}_{q}=\dfrac{1}{\exp{(\beta E_{f}^{l})}+1},$ (10)

where the Landau quantized dispersion relation reads as
$E^{l}_{f}=\sqrt{p_{z}^{2}+m^{2}_{f}+2l|q_{f}eB|}$, with $l=0,1,2,..$ being
the number of Landau levels and $q_{f}=+\frac{2}{3},-\frac{1}{3}$ being the
fractional charge of the $u$ and $d$ quarks respectively. Again, in a
magnetized medium the phase space quantization Bruckmann:2017pft ;
Tawfik:2015apa ; Gusynin:1995nb  can be represented as

$\int{\dfrac{d^{3}p}{(2\pi)^{3}}}\rightarrow\dfrac{\mid
q_{f}eB\mid}{2\pi}\sum_{l=0}^{\infty}\int{\dfrac{dp_{z}}{2\pi}}(2-\delta_{0l}).$
(11)

Incorporating these, the expression for the Debye screening mass $m_{D}$ for a
magnetized QGP medium becomes,

$m_{D}^{2}=g_{s}^{2}T^{2}\frac{N_{c}}{3}+\dfrac{g_{s}^{2}|q_{f}eB|}{\pi^{2}T}\int_{0}^{\infty}dp_{z}\sum_{l=0}^{\infty}(2-\delta_{l0})f^{l}_{q}(1-f^{l}_{q}).$
(12)

Figure 1: Variation of the constituent quark masses ($M_{u}$ and $M_{d}$) with
temperature for different values of magnetic field $eB=0.2$ (solid line),
$0.4$ (dashed line), $0.6$ (dotted line) GeV2 in left panel and with magnetic
field for different values of $T=100$ (solid line), $150$ (dash line) MeV in
right panel. Black and Red curves denotes respectively $u$ and $d$ quarks.

Now, in the earlier studies of the Debye screening mass in a magnetized medium
Bandyopadhyay:2016fyd ; Bonati:2017uvz ; Singh:2017nfa ; Kurian:2017yxj ;
Mitra:2017sjo ; Kurain:2019 ; Karmakar:2019tdp ; SG_VC , one of the important
aspect of QCD at finite magnetic field have not been considered. It is Inverse
Magnetic Catalysis (IMC) phenomenon near quark-hadron phase transition
temperature, recently revealed from lattice QCD (LQCD) calculations Bali1 ;
Bali2 . Their observed phenomenon and connected physics can be described as
follows. We know that QCD vacuum or quark condensate in vacuum is non-zero,
for which current quark mass $m_{f}\sim 5-10$ MeV get an effective constituent
quark mass $M_{f}$ (more than $300$ MeV) and fusion of three constituent
quarks can able justify the origin of nucleon mass ($\sim 939$ MeV). Now, in
presence of magnetic field, condensate value in vacuum or $T=0$ enhances,
which is known as magnetic catalysis (MC) phenomenon. LQCD calculations Bali1
; Bali2  found this MC effect within low $T$ domain but near transition $T$
range, its inverse nature is observed, which is called as IMC. To capture this
detailed MC and IMC aspects of QCD through $T$ and $B$ dependent quark
condensate and constituent quark mass $M_{f}(T,B)$ in the present work, we
have redefined the dispersion relation in a magnetized medium as

$\bar{E}_{f}^{l}=\sqrt{p_{z}^{2}+2l|q_{f}eB|+M_{f}(T,eB)^{2}},$ (13)

where $M_{f}(T,eB)$ represents the effective constituent mass in terms of LQCD
Bali1 ; Bali2  predicted normalized quark condensate $\langle
q{\bar{q}}\rangle_{f}(T,eB)$, which varies from 1 to 0 during the transition
from hadronic to quark temperature regions for $eB=0$ case. Corresponding
constituent quark mass will vary from $M_{f}(T=0,eB=0)$ to $m_{f}(T,eB=0)$ and
owing to effective QCD model, we can build a connecting relation between
$M_{f}(T,eB)$ and LQCD data $\langle q{\bar{q}}\rangle_{f}(T,eB)$ as

$\displaystyle M_{f}(T,eB)$ $\displaystyle=$ $\displaystyle
M_{f}(T=0,eB=0)\times\langle q{\bar{q}}\rangle_{f}(T,eB)+m_{f}(T,eB=0)$ (14)
$\displaystyle\approx$ $\displaystyle M_{f}(T=0,eB=0)\times\langle
q{\bar{q}}\rangle_{f}(T,eB)~{}.$

Using the LQCD data of $\langle q{\bar{q}}\rangle_{f}(T,eB)$ from Ref. Bali1
and using those in Eq. (14), we have plotted constituent quark mass
$M_{f=u,d}$ against $T$-axis (left panel) and $eB$-axis (right panel) in Fig.
(1). With this modified Lattice QCD inspired dispersion relation we have
evaluated the Debye screening mass as

$m_{D}^{2}=g_{s}^{2}(T)T^{2}\frac{N_{c}}{3}+\sum_{f}\dfrac{g_{s}^{2}(T)|q_{f}eB|}{\pi^{2}T}\int_{0}^{\infty}dp_{z}\sum_{l=0}^{\infty}(2-\delta_{l0})~{}{f}^{l}_{q}(\bar{E}_{f}^{l})~{}\left(1-{f}^{l}_{q}(\bar{E}_{f}^{l})\right),$
(15)

where we have considered $g_{s}(T)$ \- temperature dependent one loop running
coupling Haque:2018eph :

$\displaystyle
g_{s}^{2}(T)=4\pi\alpha_{s}(T)=\frac{24\pi^{2}}{\left(11N_{c}-2N_{f}\right)\ln\left(\frac{2\pi
T}{\Lambda_{\overline{\rm MS}}}\right)}.$ (16)

###  II.3 Thermal Width and Binding Energy

Next, we focus on other relevant quantities like thermal width or dissociation
rate, binding energy in the context of heavy quark potential. Their working
formulas are described below.

Let us first discuss about thermal width or dissociation rate $\Gamma$, which
can be formulated from the imaginary part of potential Im$~{}V(r,T,B)$,
discussed in Sec. (II.1). In fact, the quantity Im $V(r,T,B)$ provide more
detailed structure of dissociation rate along $r$, $T$ and $B$ axes. Now, if
we know the $r$-profile of quarkonia wave function $\Psi({{r}})$, then the
thermal width $\Gamma$ can be computed as Agotiya:2016bqr

$\displaystyle\Gamma(T,B)=-\int
d^{3}{\bf{r}}\,\left|\Psi({{r}})\right|^{2}{\rm{Im}}~{}V({\hat{r};T,B})~{},$
(17)

where imaginary part of the potential is folded by the unperturbed (1S)
Coulomb wave function, owing to the first-order perturbation theory. Here we
take $\Psi(r)$ as the Coulombic wave function for ground state
($1s$,corresponding to $n=1$ ($J/\psi$ and $\Upsilon$)) given as

$\displaystyle\Psi_{1s}(r)$ $\displaystyle=$ $\displaystyle\frac{1}{\sqrt{\pi
r_{B}^{3}}}e^{\frac{-r}{r_{B}}}$

where, $r_{B}$=$2/(\alpha_{s}~{}m_{Q})$ is the Bohr radius of the quarkonia
system. By solving Eq.17, we have Agotiya:2016bqr

$\displaystyle\Gamma(T,B)=\left(1+\frac{3\sigma}{\alpha_{s}m_{Q}^{2}}\right)~{}\frac{4T}{\alpha_{s}}{\frac{m_{D}^{2}}{m_{Q}^{2}}}~{}\log\frac{\alpha_{s}m_{Q}}{2m_{D}}~{}.$
(18)

Here also, one can see that the $T$ and $B$ dependent profile of $\Gamma$ is
coming through $m_{D}(T,B)$.

Following Ref. Agotiya:2008ie , we can consider simple Culombic type real part
of heavy quark potential and then by solving its corresponding Schrödinger
equation, we can get the eigenvalues for charmonium and bottomonium system as

$\displaystyle E_{n}=-\frac{1}{n^{2}}\frac{m_{Q}\sigma^{2}}{m^{4}_{D}}~{},$
(19)

where $m_{Q}$ is the mass of the heavy quark $c$ and $b$ respectively. Now its
ground state eigen value for $n=1$ can be considered as binding energies of
lowest possible charmonium and bottomonium states i.e. $J/\psi$ and $\Upsilon$
respectively. Similar to thermal width $\Gamma(T,B)$, binding energy
$E_{n=1}(T,B)$ of $J/\psi$ and $\Upsilon$ will carry magnetic field dependent
information through Debye mass $m_{D}(T,B)$.

##  III Results

In this section we will show and discuss about our results corresponding to
heavy quark potential in a magnetized medium incorporating IMC based quark
condensate. In the present work we have used $N_{c}=3,N_{f}=2$ and
$\Lambda_{\overline{\rm MS}}=0.176$ GeV Haque:2014rua , the string tension,
$\sigma=0.184~{}GeV^{2}$ nilima_aniso  and the value of $\alpha_{s}$ from
Equation 16.

In the left panel of Fig. (2), we have plotted the variation of Debye mass
with $T$ for different values of magnetic field $eB=0.2$ GeV2 (black solid
line), $0.4$ (dash line), $0.6$ (dotted line) GeV2. Whearas, the right panel
of the Fig. (2) shows the variation of Debye mass with $eB$ for different
values of temperature $T=100$ (black solid line), and $200$ (dash line) MeV.
If we follow the Debye screening mass expression, given in Eq. (15), then we
can identify the $T$ and $eB$ dependent components. Sources of $T$ are FD
distribution function $f^{l}_{q}(T)$ and coupling constant $g_{s}(T)$, which
provide increasing and decreasing $T$ profile. Hence, due to their collective
role, we notice that $m_{D}(T)$ decreases first and then increase. On the
other hand, we can notice a straight forward $m_{D}\propto eB$ relation in Eq.
(15). However, right panel of Fig. (2) shows a deviation from the proportional
relation, which is because a non-trivial $eB$ dependency is entering through
the $eB$ dependent constituent quark mass $M(eB)$, located in FD distribution
function. Also we notice that the effect of magnetic field is stronger at a
lower temperature and becomes weaker at a higher temperature.

Figure 2: Variation of the Debye mass ($m_{D}$) with temperature for different
values of magnetic field $eB=0.2$ (black solid line), $0.4$ (dash line), $0.6$
(dotted line) GeV2 in left panel and with magnetic field for different values
of $T=100$ (black solid line), $200$ (dash line) MeV in right panel. Red solid
line in left panel denotes for current quark mass or approximately massless
quark limit.

Now, if we compare with earlier $m_{D}(T,B)$ calculations, done by Refs.
Bandyopadhyay:2016fyd ; Bonati:2017uvz ; Singh:2017nfa ; Kurian:2017yxj ;
Mitra:2017sjo ; Kurain:2019 ; Karmakar:2019tdp ; SG_VC , then one can find
that the new aspect of present work is the adoption of IMC based constituent
quark mass. Though quark condensate and constituent quark mass are the good
quantities, where we can see the IMC effect near transition temperature, but
for the quantity like Debye mass $m_{D}(T,B)$, it is not zoomed in as it
becomes fade due to integration of thermal distribution. For thermodynamical
quantities like pressure, energy density or transport coefficients like
electrical conductivity Aritra_condB , we get similar fact. So we will
continue to present our final results as a IMC based outcome instead of
zooming the actual modification, which is quite mild in graphical
presentation. For seeing this modification in the left panel of Fig. (2), we
have plotted red solid line by considering current quark mass $m_{f}=10$ MeV
(close to massless quark limit) and compared it with black solid line, which
has considered IMC based constituent quark mass $M_{f}(T,B)$. Within the
$T=0.080$-$0.160$ GeV, a mild suppression of Debye mass is noticed due to
considering the IMC based constituent quark mass $M_{f}(T,B)$ in place of
current quark mass. This suppression of Debye mass is connected with the
magnetized quark condensate in non-perturbative QCD (non-pQCD) domain,
obtained from LQCD calculation Bali1 ; Bali2 , which reveals IMC effect near
transition temperature. So, this effect will propagate to other quantities of
heavy quark phenomenology, as discussed next.

Figure 3: Variation of the real part of potential with separation distance $r$
between $Q\bar{Q}$ for three different values of the magnetic field and with
fixed temperature $T=100$ MeV (left) and $T=200$ MeV (right).

We had plotted the variation of the real part of the potential with the
separation distance ($r$) between the Q$\bar{Q}$ pair for different values of
magnetic field (eB= $0.2~{}GeV^{2}$ and $0.4~{}GeV^{2}$) at $T=100$ MeV (left)
and $T=200$ MeV (right) in Figure 3 and compared it with the absence of
magnetic field i.e $eB=0$. We also observed that the real part of potential
without the magnetic field decreases slightly as compared to the one in
presence of magnetic field.

Figure 4: Variation of the imaginary part of potential with separation
distance $r$ between $Q\bar{Q}$ for various values of magnetic field $T=100$
MeV (left) and $T=200$ MeV (right).

Similar to the real-part of the potential we have plotted the imaginary part
of the potential with the separation distance ($r$) for different values of
magnetic field (eB= $0.2~{}GeV^{2}$ and $0.4~{}GeV^{2}$) at $T=100$ MeV (left)
and $T=200$ MeV (right) in Fig. 4 and compared it with the absence of magnetic
field i.e $eB=0$. As we can see from the figure that magnitude of the
imaginary part of the potential increases with the increase in magnetic field
and hence it provides more contribution to the thermal width obtained from the
imaginary part of the potential. We also find that the magnitude of imaginary
part of potential is more at a higher temperature ($T=200$ MeV) as compared to
a lower temperature ($T=100$ MeV) for a given $eB$ and $r$. This observation
is coming from the fact that Debye mass which is a function of temperature and
magnetic field is found to be increased with temperature and magnetic field.

Next, let us proceed to compute thermal width or dissociation probability
$\Gamma(T,B)$ by using Eq. (18) and binding energy using the Eq. (19).
Imaginary and real part of heavy quark potential are their respective sources,
whose $T$ and $eB$ profiles are mainly coming from $m_{D}(T,eB)$.

Figure 5: Variation of binding energy(in $GeV$) with temperature with
different values of magnetic field (eB) for $J/\psi$ (left panel) and
$\Upsilon$ (right panel).

Let us first discuss about binding energy (BE), plotted in Fig.5. Its left
panel shows the binding energies of $J/\psi$ as a function of $T$ for various
values of magnetic field (eB=$0.2~{}GeV^{2}$ and $0.4~{}GeV^{2}$) and compared
it with the results, considering current quark mass (red line). We observe
that binding energy is decreasing as the temperature and magnetic field both
are increasing.

In the right panel of Fig.5 we have plotted the binding energies of $\Upsilon$
as a function of $T$ for various values of magnetic field (eB=$0.2~{}GeV^{2}$
and $0.4~{}GeV^{2}$) and compared it with without magnetic field ($eB=0$).The
similar behaviour has been observed for $\Upsilon$ also, except that the value
of binding energy for $J/\Psi$ is lower as compared to the value for
$\Upsilon$.

Figure 6:  Variation of $\Gamma$, 2BE($E_{b}$) with Temperature with different
values of magnetic field (eB) for $J/\psi$

Figure 7: Variation of $\Gamma$, 2BE($E_{b}$) with Temperature with different
values of magnetic field (eB) for $\Upsilon$

Next, let us come to the estimation of thermal width or dissociation
probability, given in Eq. (18), which is obtained by substituting Eq. ((8)) in
Eq. ((17)). So Eq. ((8)) represents a detailed tomography of dissociations of
quarkonia state, while Eq. ((17)) present its integrated values. Now this
heavy quark bound states, called quarkonia, can be dissociate completely, when
thermal width will be greater than twice of the the binding energy i.e.
$\Gamma\geq 2BE$ utt . Now, if one plots twice the binding energy ($2BE$)
along with the thermal width ($\Gamma$) against $T$-axis, then their
intersection point can be considered as dissociation temperature ($T_{d}$). We
have done this in fig. 6 and in fig. 7 for ground state of charmonium and
bottomonium respectively with different values of magnetic field. Here we take
charmonium and bottomonium masses as $m_{c}=1.5$ GeV and $m_{b}=4.5$ GeV
respectively.

If we analyse the fig. 6 and fig. 7 we can find that the thermal width is
increasing with the increase in magnetic field. Also we can observe that the
width for the $J/\psi$ is much larger than the $\Upsilon$, because charmonium
states are larger in size and smaller in masses as compared to bottomonium
states which are smaller in size and larger in masses and hence will get
dissociated at higher temperatures. So we can conclude that as $\Gamma$ is a
increasing function of magnetic field which results in the early dissociation
of quarkonium states. Our findings of the dissociation temperatures from
intersection points in graphs are enlisted in table 1.

State |  $eB=0$ |  $eB=0.2$ |  $eB=0.4$
---|---|---|---
$J/\psi$ |  297 |  186 |  158
$\Upsilon$ |  580 |  403 |  388
Table 1: The dissociation temperature($T_{D}$) for the quarkonia states (in
units of MeV), when $\Gamma$= 2BE

We notice that dissociation temperature of quarkonia states decreases as
magnetic field increases. This fact is exactly similar with the reduction of
quark-hadron phase transition temperature due to magnetic field, which is
connected with IMC aspect of QCD. If QCD follow MC near transition
temperature, then magnetic field will push the location of transition
temperature towards the higher values. However, reduction of dissociation
temperature with magnetic field can not be linked with IMC as constituent
quark mass has very mild impact on Debye mass and heavy quark phenomenology.
In other word, the fact of reduction of dissociation temperature with magnetic
field remain same for both MC and IMC. This conclusion can be established from
Figs. (6) and (7), where we notice that $T_{d}$ remain almost same for IMC-
based constituent quark mass (black line) and current quark mass (red line).

##  IV Conclusions

We have revisited the medium modified heavy quark potential which is the sum
of both Coulombic and string terms and obtain its the real and imaginary part.
This is done by obtaining the real and imaginary parts of the resummed gluon
propagator, which in turn gives the real and imaginary parts of the dielectric
permittivity. A temperature and magnetic field dependent Debye mass, capturing
the information of inverse magnetic catalysis, is enerting into the gluon
propagator. Now the real and imaginary parts of the dielectric permittivity
will be used to evaluate the real and imaginary parts of the complex heavy
quark potential. With respect to earlier works, present work has incorporated
two new ingredients - inverse magnetic catalysis information and all Landau
level summations.

The real part of the potential is used in the time-independent Schrödinger
equation for the radial wave function to obtain the binding energy of heavy
quarkonia, whereas the imaginary part is used to calculate the thermal width.
We observe that the binding energies of $J/\psi$ and $\Upsilon$ is decreasing
as the temperature and magnetic field both are increasing. We also found that
binding energy in absence of magnetic field is greater than the binding energy
in presence of magnetic field. This is because in the presence of magnetic
field, real-part of the potential becomes more screened as compared to one in
absence of magnetic field.

Finally we have plotted the twice of binding energy along with the thermal
width and obtain the dissociation temperature as a point of their
intersection. We find that the thermal width is increasing with the increase
in magnetic field. Also we can observe that the width for the $J/\psi$ is much
larger than the $\Upsilon$. We can observed that the dissociation temperatures
become slightly higher in the absence of magnetic field as compared to the
presence of magnetic field

## Acknowledgement

I. Nilima acknowledges IIT Bhilai for the academic visit and hospitality
during the course of this work. A.B. acknowledges the support from Guangdong
Major Project of Basic and Applied Basic Research No. 2020B0301030008 and
Science and Technology Program of Guangzhou Project No. 2019050001. Authors
are thankful to Mujeeb Hassan for useful discussion.

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# Fuss-Catalan numbers and planar partitions

F. Aicardi  Sistiana 56, Trieste IT<EMAIL_ADDRESS>
###### Abstract.

We show how the Fuss-Catalan numbers $\frac{1}{pn+1}\binom{pn+1}{n}$ enter
different problems of counting simple and multiple planar partitions.

###### 1991 Mathematics Subject Classification:

05A10,05A19,57M50

##  1\. Results

The Catalan numbers occur in numerous counting problems as well as their
generalizations, the Fuss-Catalan numbers with one parameter, $p$

$A_{n}^{p}:=\frac{1}{pn+1}\binom{pn+1}{n}.$

The Catalan numbers correspond to the case $p=2$.

A planar partition (or non crossing partition) is a set partition that can be
represented by a diagram whose arcs, defining the blocks, do not cross each
other, see Section 2.1.

In the context of planar partitions, the numbers $A^{p}_{n}$ turn out in
different situations, some of which we highlight here. The set of integers
$\\{1,\dots,n\\}$ will be abbreviated as $[n]$.

R1. The Catalan number counts the number of planar partitions of $[n]$.

R2.[6] The Catalan number counts the number of planar partitions of $[2n]$
whose blocks contain exactly two elements.

R3.[4] The Fuss-Catalan number $A^{3}_{n}$ counts the number of planar
partitions of $[2n]$ in which each block contains an even number of elements.

We recall that a double planar partition of $[n]$ is a pair of planar
partitions in which the first one is a refinement of the second one, i.e.,
each block of the first partition is contained ia a block of the second
partition.

In [1], I have proved that

R4. the Fuss-Catalan number $A^{4}_{n}$ counts the number of double planar
partitions of [2n], in which each block of the first partition has two
elements, and the second partition is simply planar.

This was done by introducing by recurrence a triangle of integers whose
elements of the $n$-th row add to $A_{n}^{4}$.

In [2] I have generalized this recurrence to every integer $p>0$ obtaining all
Fuss-Catalan numbers $A_{n}^{p}$. In the same note I proved, using the
recurrence corresponding to the case $p=3$, that

R5. $A^{3}_{n}$ counts the number of double planar partitions of $[n]$.

So, the question arises whether the mentioned results R1–R5 are special cases
of more general statements.

This note answers this question, proving in particular four statements, in
which the following notations are used.

  1. (1)

A p-partition is a partition of $[pn]$ whose blocks have cardinality $p$.

  2. (2)

A m-tuple planar partition of $[n]$ is an ordered set of $m$ planar partitions
of $[n]$ such that each one of the first $m-1$ partitions is a refinement of
the successive.

######  Theorem 1.

For every integer $p>0$, the Fuss-Catalan number $A_{n}^{p}$ counts the number
of planar $p$-partitions of $[pn]$.

This generalizes result R2, see Section 2.

######  Theorem 2.

For $p=2q$, $A_{n}^{p}$ counts the double planar partitions of $[qn]$, in
which the first one is a q-partition and the second one is simply planar.

So, this theorem generalizes R4 ($q=2$) and likewise it has a generalization:

######  Theorem 3.

For every pair $(m,p)$ of positive integers, $A_{n}^{mp}$ counts the $m$-tuple
planar partitions of $[pn]$, in which the first one is a $p$-partition and the
other ones are simply planar.

Theorem 3 has the following

######  Corollary 1.

For every integer $m>1$ $A_{n}^{m}$ counts the $(m-1)$-tuple planar partitions
of $[n]$.

This generalize result R1 ($m=2$) and R5 ($m=3$), see Section 4.

Let $\mathcal{M}^{p}_{n}$ be the set of planar partitions of $[pn]$, in which
each block contains a number of elements multiple of $p$.

######  Theorem 4.

For every integer $p>0$, the cardinality of $\mathcal{M}^{p}_{n}$ is
$A_{n}^{p+1}$.

This generalizes result R3 ($p=2$).

In Figure 1 we illustrate the results above by giving different examples of
the occurrence of $A_{2}^{6}=6$: (a) by Theorem 1, it is the number of
$6$-planar partitions of $[12]$; (b) by Theorem 2, the number of double
partitions of $[6]$ the first of which is a planar $3$-partition; (c) by
Theorem 3, the number of triple partitions of $[4]$ the first of which is a
planar $2$-partition; (d) by Corollary 1, the number of 5-tuple planar
partitions of $[2]$; (e) by Theorem 4, the number of planar partitions of [10]
whose blocks have a number of elements multiple of $5$.

Figure 1. Five examples of the occurrence of $A^{6}_{2}=6$.

In Appendix we show also a bijection between the set of the planar
$p$-partitions of $[pn]$ and the set of the $p$-ary trees with $n$ internal
nodes.

##  2\. Counting planar $p$-partitions: proof of Theorem 1

###  2.1. Planar partitions and arc diagrams

A partition of $[n]$ is represented here by a diagram, consisting of $n$
points on a line, labeled 1 to $n$, and arcs connecting pairs of points. If
two points are connected by an arc, then they belong to the same block of the
partition. The diagram representing a partition is evidently non unique. A
diagram is said standard if for every pair of points $(i<j)$ belonging to the
same block, there is an arc $(i,j)$ if and only if in the same block there is
no some $k$ such that $i<k<j$. The standard diagram is unique. In what follows
we deal only with standard diagrams, unless it be specified, see Figure 2b.

######  Notation 1.

A partition is said planar if its diagram is planar, i.e., its arcs do no
cross each other. The set of planar partitions of $[n]$ is denoted by
$\mathcal{P}_{n}$, and the set of planar $p$-partitions of $[pn]$ is denoted
by $\mathcal{P}^{p}_{n}$.

Figure 2. a) Diagram of a non planar partition; b) the standard diagram of a
planar 4-partition; c) a partition of $[11]$ with 3 boxes

Here we prove Theorem 1, generalizing R1.

######  Remark 1.

For $p=1$ a planar 1-partition of $[n]$ consists of $n$ blocks, each one with
1 element. Therefore it is unique and indeed $A_{n}^{1}=1$ for every $n$.

###  2.2. Proof of Theorem 1

In [2] I have introduced, for every integer $p>0$, a triangle $T^{p}(n,k)$,
whose rows add to $A^{p}_{n}$.

Here we need to consider, for every $p$, the triangle $F^{p}(n,k)$, obtained
by $T^{p}(n,k)$ by:

(1) $F^{p}(n,k):=T^{p}(n,n-k),\quad k=0,\dots,n.$

The integers $F^{p}(n,k)$ for $n\geq 0$, $0\leq k\leq n$, result to be given
by the initial conditions

(2) $F^{p}(0,0)=1;\quad F^{p}(n,0)=0\quad\text{ for}\quad n>0$

and the recurrence, for $n>0$ and $0<k\leq n$:

(3) $F^{p}(n,k)=\sum_{j=k-1}^{n-1}\binom{j-k+p-1}{p-2}F^{p}(n-1,j).$

We recall also the notion of box in a diagram of a planar partition. Observe
that two blocks $A,B$, satisfying $min(A)<min(B)$ may satisfy either
$max(A)<min(B)$ or $max(A)>min(B)$. In the last case we have also, by
planarity, that $max(A)>max(B)$, and we say that $B$ is nested in $A$. A box
is a block which is not nested. A planar partition of $[n]$ with $m$ blocks
has $1\leq k\leq m$ boxes, see Figure 2.

We denote by $N^{p}(n,k)$ the number of $p$-partitions of $[pn]$ having $k$
boxes. We see that $N^{p}(n,k)$ fulfills (2): indeed, the void partition has
no boxes, so $N^{p}(0,0)=1$; also, there are no $p$-partitions of $[pn]$ with
no boxes, so $N^{p}(n,0)=0$.

We see now that $N^{p}(n,k)$ satisfies recurrence (3). We consider the set of
partitions of $\mathcal{P}^{p}_{n}$ having $k$ boxes. We observe that each one
of them can be obtained from a unique partition $Q\in\mathcal{P}^{p}_{n-1}$
having $j$ boxes if $j\geq k-1$ and $j\leq n-1$, by adding a new block $B$ to
$Q$ that has the last element at right of $Q$. On the other hand, given such a
partition $Q$ with $j$ boxes, we may get different partitions with $k$ boxes.
Indeed, the first $k-1$ boxes of $Q$ remain unaltered in the partition of
$\mathcal{P}^{p}_{n}$, say $P$, while $j-k+1$ boxes must be nested in the new
block $B$ of $P$. So, the first point of $B$ is necessarily between the first
$k-1$ boxes and the nested $j-k+1$ boxes of $Q$. Now, between the first and
the last point of the new block, there are $j-k+1$ boxes and $p-2$ points of
$B$, see Figure 3. The partition $P$ is uniquely defined by the ordered set
containing the $p-2$ points and the $j-k+1$ boxes: for a fixed $Q$, the number
of possibilities is given by $\binom{j-k+1+p-2}{p-2}$. This gives the
recurrence (3). Therefore $N^{p}(n,k)=F^{p}(n,k)$. Since
$|\mathcal{P}^{p}_{n}|$ is the sum of the numbers $N^{p}(n,k)$ for
$k=1,\dots,n$, this sum is $A^{p}_{n}$ by (1) and (2).

Figure 3. From a partition with $j=5$ boxes and $n$ blocks to three
3-partitions with $k=4$ boxes and $n+1$ blocks.

##  3\. Counting double planar partitions

###  3.1. Double planar partitions and tied arc diagrams

In this section we consider special double planar partitions $(P_{1},P_{2})$
of $[pn]$ in which $P_{1}\in\mathcal{P}^{p}_{n}$ and
$P_{2}\in\mathcal{P}_{pn}$. The set of such double partitions is denoted by
$\mathcal{P}^{II,p}_{n}$.

A double partition is represented by a sole diagram of arcs and ties in the
following way, see also [3]. The partition $P_{1}$ is represented by $n$
blocks. Remember that each block of $P_{1}$ consists of $p-1$ consecutive
arcs. A block is labeled by its point at left, so that the blocks of $P_{1}$
are ordered by this labeling. A tie is an arc, drawn as a dotted line, with
endpoints on two blocks of $P_{1}$. Suppose that
$B_{k_{1}},B_{k_{2}},\dots,B_{k_{m}}$ are the ordered blocks of $P_{1}$
contained in a same block of $P_{2}$. Then we put a tie from $B_{k_{i}}$ to
$B_{k_{i+1}}$, for $i=1,\dots,m-1$. The planarity of $P_{2}$ guarantees that
the ties can be drawn without crossing neither the arcs nor each other.
Moreover, a double partition is uniquely represented by arcs and ties in this
way, see Figure 4 for an example.

Now we will prove that, for $p>0$, the cardinality of the set
$\mathcal{P}^{II,p}_{n}$ is the Fuss Catalan number $A^{2p}_{n}$.

###  3.2. Proof of Theorem 2

The proof given in [1] for the case $p=4$ uses the corresponding Fuss-Catalan
triangle and here we could proceed the same way for any value of $p$. However,
we prefer to proceed by using Theorem 1 an by proving that there is a
bijection between $\mathcal{P}^{2q}_{n}$ and $\mathcal{P}^{II,q}_{n}$.

We need to introduce some basic notations. The notion of refinement provides
the set of partitions of $[n]$ with a partial order, that can be restrict to
$\mathcal{P}_{n}$: $P_{1}\leq P_{2}$ if $P_{1}$ is a refinement of $P_{2}$.
Also, we call product of two planar partitions $P_{1}$ and $P_{2}$ the planar
partition $P_{3}$ which is the minimal partition in $\mathcal{P}_{n}$
satisfying $P_{1}\leq P_{3}$ and $P_{2}\leq P_{3}$. The product, indicated by
the symbol $*$, is evidently commutative.

######  Remark 2.

Observe that the partition $I\in\mathcal{P}_{n}$ consisting of $n$ blocks with
a unique element, satisfies $I*P=P$ and $I\leq P$ for every
$P\in\mathcal{P}_{n}$. Moreover, for any set of partition $\mathcal{Q}$, the
set of double partitions $(I,Q)$ with $Q\in\mathcal{Q}$ is in bijection with
$\mathcal{Q}$.

Finally, given $P\in\mathcal{P}_{2n}$, we denote by $P^{/2}$ a partition of
$[n]$ obtained by $P$ eliminating from all blocks of $P$ the even numbers and
sending the odd numbers $2j-1$ to $j$.

Consider a partition $P\in\mathcal{P}_{n}^{2q}$. The elements of every block
are ordered by the natural order in $[2n]$. Let us define the partition
$P^{\prime}$ of $[qn]$ as $P^{\prime}=P^{/2}$.

######  Proposition 1.

The partition $P^{\prime}$ is a $q$-partition.

###### Proof.

The planarity of $P^{\prime}$ follows from the planarity of $P$. The fact that
$P^{\prime}$ has exactly $n$ blocks with $q$ elements follows from the fact
that the parity of the elements $e_{1},\dots,e_{2q}$ of every block of $P$
alternates, since, by planarity, $e_{i+1}=e_{i}+1+2kq$, for some $k\geq 0$.
Therefore each block of the partition $P^{\prime}$ has exactly $q$ elements. ∎

We have to define the second partition, $P^{\prime\prime}\in\mathcal{P}_{qn}$,
of the double partition.

Denote $I_{2}$ the partition of $[2nq]$ with $qn$ blocks of 2 elements
$\\{2j-1,2j\\}_{j\in[n]}$. Then consider the partition $\overline{P}=I_{2}*P$.
Clearly $P\leq\overline{P}$. The partition $P^{\prime\prime}$ of $[qn]$ is
thus defined as $P^{\prime\prime}:=(\overline{P})^{/2}$ Diagrammatically, we
will put a tie between two blocks $B^{\prime}_{i}$ and $B^{\prime}_{j}$ of
$P^{\prime}$ if $B_{i}$ and $B_{j}$ result in the same block of
$\overline{P}$.

This procedure defines uniquely the partition $P^{\prime\prime}$ and hence the
double partition. Example. In Figure 4 we show how from the diagram of the
partition $P\in{\mathcal{P}^{4}_{3}}$,
$P=\\{\\{1,2,7,12\\},\\{3,4,5,6\\},\\{8,9,10,11\\}\\}$, we get the diagram of
the double partition $(P^{\prime},P^{\prime\prime})\in\mathcal{P}^{II,2}_{3}$,
where $P^{\prime}=\\{\\{1,4\\},\\{2,3\\},\\{5,6\\}\\}$, and
$P^{\prime\prime}=\\{\\{1,4,5,6\\},\\{2,3\\}\\}$.

Figure 4. From the diagram of $P$ to the diagram of
$(P^{\prime},P^{\prime\prime})$. The arcs of $I_{2}$ are drawn below the
horizontal line. The diagram of $\overline{P}$ is non standard

The next observations follow from the procedure above introduced, as well as
from the remark given proving Proposition 1.

######  Remark 3.

  1. (1)

A block $B_{i}^{\prime}$ of $P^{\prime}$ is non tied to other preceding blocks
if and only if the first point of $B_{i}$ is odd;

  2. (2)

If a block $B^{\prime}_{j}$ is tied to a block $B^{\prime}_{i}$, with $j>i$,
then a) the first point of $B_{j}$ is even and is the next point of an odd
point $y$ of $B_{i}$; the last point of $B_{j}$ is odd and its next point is
either the even point of $B_{i}$ successive to $y$, or the first point of
another block tied to $B_{j}$.

  3. (3)

The parity of the last element of a block is opposite to that of the first
one.

To prove that the procedure above defines a bijection, we need to verify that
the inverse procedure defines a unique planar $2q$-partition $P$ starting from
a double planar partition in which the first partition $P^{\prime}$ is a
$q$-partition and the second one, $P^{\prime\prime}$, is planar. Let us start
from the partition $P^{\prime}$ of a double partition in
$\mathcal{P}^{II,q}_{n}$. We enumerate $B^{\prime}_{1},\dots,B^{\prime}_{n}$
its blocks according to their first points $x_{1},\dots,x_{n}$, and
$B_{1},\dots,B_{n}$ the corresponding blocks of $P$. Remember that the block
$B_{i}$ surely contains the odd points $2j-1$ for all $j\in B^{\prime}_{i}$.

To define $P$, we have therefore to put the even points in each block $B_{i}$.
We say that Remark 3 is sufficient to do this. Indeed, if two consecutive
points $j,j+1$ belong to a block $B^{\prime}$, the even point $2j$ is
evidently in $B$. The problem arises when we have to put in a block of $P$ a
point $2j$ such that $2j-1$ and $2j+1$ belong to different blocks
$B^{\prime}_{r}$ and $B^{\prime}_{s}$. In this case, we use Remark 3. For
instance, let us start from the double partition
$(P^{\prime},P^{\prime\prime})$ in Figure 4. From
$P^{\prime}=\\{\\{1,4\\},\\{2,3\\},\\{5,6\\}\\}$, we get the odd points of the
three blocks of $P$: $B_{1}:\\{1,7\\},B_{2}:\\{3,5\\},B_{3}:\\{9,11\\}\\}$.
Now, points 4 and 10 are put in $B_{2}$ and $B_{3}$. Since $B_{1}$ and $B_{3}$
are tied, points 8 and 12 are put in $B_{2}$ and $B_{1}$ by using item (2),
the remaining point 2 belongs necessarily to $B_{1}$.

##  4\. Counting multiple planar partitions

###  4.1. Proof of Theorem 3

The proof of Theorem 3 uses the same argument as the proof of Theorem 2: we
define a bijection between $\mathcal{P}_{n}^{mp}$ and
$\mathcal{P}^{(m),p}_{n}$, the set of $m$-tuple partitions of $[pn]$, of which
the first one is a $p$-partition, and the others are planar.

For any $k>2$, given $P\in\mathcal{P}_{kn}$ we denote $P^{/k}$ the partition
of $[n]$ obtained by $P$ sending, for every block, each element of the form
$kj+1$ to the element $j+1$ of a block of $P^{/k}$. Observe that $0\leq j\leq
n-1$.

Let $P\in\mathcal{P}_{n}^{mp}$, i.e. $P$ is a planar partition of $[mpn]$,
with $n$ blocks of $mp$ elements.

Now, we define the partitions $I_{r}$, $r=1,\dots,m$ as the partitions of
$[mpn]$ containing the blocks $\\{mk+1,mk+2,\dots,mk+r\\}$ for
$k=0,\dots,pn-1$, and coinciding with $I$ for the remaining elements. Observe
that $I_{1}=I$.

Then we define, for every $r=1,\dots,m$, the partition of $[pn]$:

$P^{(r)}=(P*I_{r})^{/m}$

Observe that $I_{1}=I$, and that $P^{(1)}$ is a $p$-partition. This last fact
follows again from the planarity of $P$: the number of elements of each block
of $P^{(1)}$ is $p$ since there are $p$ integers in a sequence of $mp$
integers $x_{i}$ satisfying $x_{i}=1\mod m$ if these integer satisfy
$x_{i+1}=x_{i}+1+kmp$ for $k\geq 0$.

Since $(P*I_{r})\leq(P*I_{r+1})$, the set $\\{P^{(r)}\\}_{r=1,\dots,m}$ is a
set of planar partitions ordered by refinement, in which $P^{(1)}$ is a
$p$-partition.

The proof that this procedure defines a bijection between
$\mathcal{P}^{mp}_{n}$ and $\mathcal{P}^{(m),p}_{n}$ is based essentially on
the fact that the number of partitions
$\mathcal{P}_{i}\in\mathcal{P}^{mp}_{n}$ that coincide except on two blocks,
which are sent by $P^{/m}$ to the same two blocks of a partition
$P^{(1)}\in\mathcal{P}^{p}_{n}$ is $m$ if and only if these blocks of $Q$ can
be tied. So, the $m$ partitions $P_{i}$ give rise to the different partition
$P_{i}^{(r)}$ in which they become tied. The proof verifies that the products
of $P_{i}$ with $I_{r}$ cover all these possibilities.

In Figure 5 we show an example with $m=3$, $p=2$ and $n=3$. Starting from a
$6$-partition of $[18]$, we get a triple partition of [6], of which the first
one is a 2-partition.

######  Remark 4.

An $m$-tuple partition, instead of a set of $m$ partitions ordered by
refinement, is represented by a unique diagram with arcs and ties, like the
double partition: the first partition is represented as usually by an arc
diagram, and every tie between two arcs is labeled by the ordinal of the
partition where this arc appears.

Figure 5. From the diagram of a 6-partition to the diagram of a triple
partition. The arcs of $I_{2}$ and $I_{3}$ are drawn below the horizontal
line. The diagrams of $P*I_{j}$ are non standard

###### Proof of Corollary 1.

When $p=1$, Theorem 3 says that $A^{m}_{n}$ counts the $m$-tuple planar
partitions of $[n]$ in which the first partition is the 1-partition, therefore
they are equivalent to the $(m-1)$-th planar partitions of $[n]$, see Remarks
1 and 2. ∎

###  4.2. Proof of Theorem 4

We define a bijection between $\mathcal{M}^{p}_{n}$ and the set of $p$-tuple
planar partitions of $[n]$. Consider a planar partition
$M\in\mathcal{M}^{p}_{n}$. As in the proof of Theorem 3, we define the
partitions

$M^{(r)}:=(M*I_{r})^{/p},\quad r=1,\dots,p.$

They are evidently $p$ planar partitions of $[n]$ ordered by refinement. The
fact that $M^{(1)}$ is a planar partition of $[n]$ follows again from the
planarity of $M$ and from the fact that in each block of $M$ with $kp$
elements, $k$ of them are congruent to 1 $\mod p$, see proof of Theorem 3.

Therefore the cardinality of $\mathcal{M}^{p}_{n}$ is $A^{p+1}_{n}$ by
Corollary 1.

##  5\. Appendix. Planar p-partitions and full p-ary trees

######  Definition 1.

We call for short $p$-tree an ordered tree where every node has 0 or $p$
children (such trees are often called full $p$-ary trees). The set of
$p$-trees with $n$ internal nodes (i.e.,nodes with $p$ children) is denoted
$\mathcal{T}^{p}_{n}$.

In [5], it is shown that the cardinality of $\mathcal{T}^{p}_{n}$ is
$A^{p}_{n}$. So, another proof of Theorem 1 is the following

######  Proposition 2.

There is a bijection between the set $\mathcal{T}^{p}_{n}$ of $p$-trees with
$n$ internal nodes and $\mathcal{P}^{p}_{n}$

###### Proof.

Consider the map $\tau$ from $\mathcal{P}^{p}_{n}$ to $\mathcal{T}^{p}_{n}$
defined in this way. Let $D$ be the arc diagram representing a $p$-partition
of [pn]. It consists of $pn$ points and $n$ chains of $p-1$ consecutive arcs,
that connect the $p$ points of a same block. For any two adjacent points
$x,x+1$ we say that $x+1$ is the successive of $x$. We enumerate 1 to $n$ the
blocks according to the order of their first points. Then, for the $k$-th
block, we denote by $(k,m)$, $m=1,\dots,p$, its points in the order from left
to right. Now, we associate to $D$ a $p$-tree. To the first block we associate
the root of the $p$-tree with $p$ children. Observe now that the point (2,1)
is necessarily at right of (1,1), but is the successive of (1,m), for some
$1\leq m\leq p$. Then we label by 2 the $m$-th child of the root, and we give
to it $p$ children. We say that 2 is the child of 1 of ordinal $m$. Similarly,
point (3,1) is the successive either of $(1,j)$ for $j>m$, and in this case we
label by 3 the $j$-th child of the root, or of $(2,h)$ for some $1\leq h\leq
p$, in which case we label by 3 the $h$-th child of node 2. When we give a
label to a node, we give to it $p$ children. We proceed this way with all
points $(k,1)$, till $k=p$, assigning the label $k$ to the $i$-th child of the
node $k^{\prime}<k$ such that $(k,1)$ is the successive of $(k^{\prime},i)$.
Evidently the label $k$ is assigned to a node which is internal by
construction. So, at the end we have a $p$ tree with $n$ internal nodes. The
map $\tau$ is bijective since a $p$-tree is uniquely defined by the $n-1$
pairs $(a,b)_{k}$, $k=2,\dots,n$, where $a\in[n]$, $a<k$, is the father of the
node $k$ and $b\in[p]$ is the ordinal of the child $k$; on the other hand, the
diagram of a planar $p$-partition is uniquely determined by the same pairs,
meaning that the point $(k,1)$ is the successive of point $(a,b)$. ∎

Example. See Figure 6

Figure 6. The diagram of a partition of $\mathcal{P}^{4}_{5}$ and the
corresponding 4-tree with 5 internal nodes

######  Remark 5.

A box of a $p$-partition $P$ correspond evidently to a node of $\tau(P)$ that
is either the root, or a node having all ancestors with maximal ordinal, e.g.
nodes 1 and 5 in the figure above.

## References

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