# 1st Place Solution for PSG competition with ECCV’22 SenseHuman Workshop

Qixun Wang, Xiaofeng Guo, Haofan Wang
Xiaohongshu Inc; Institute of Optics and Electronics, Chinese Academy of
Sciences
<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

Panoptic Scene Graph (PSG) generation aims to generate scene graph
representations based on panoptic segmentation instead of rigid bounding
boxes. Existing PSG methods utilize one-stage paradigm which simultaneously
generates scene graphs and predicts semantic segmentation masks or two-stage
paradigm that first adopt an off-the-shelf panoptic segmentor, then pairwise
relationship prediction between these predicted objects. One-stage approach
despite having a simplified training paradigm, its segmentation results are
usually under-satisfactory, while two-stage approach lacks global context and
leads to low performance on relation prediction. To bridge this gap, in this
paper, we propose GRNet, a Global Relation Network in two-stage paradigm,
where the pre-extracted local object features and their corresponding masks
are fed into a transformer with class embeddings. To handle relation ambiguity
and predicate classification bias caused by long-tailed distribution, we
formulate relation prediction in the second stage as a multi-class
classification task with soft label. We conduct comprehensive experiments on
OpenPSG dataset and achieve the state-of-art performance on the
leadboard111https://www.cvmart.net/race/10349/base. We also show the
effectiveness of our soft label strategy for long-tailed classes in ablation
studies. Our code has been released in https://github.com/wangqixun/mfpsg.

##  1 Introduction

Scene graph generation (SGG) task [3] typically aims to generate comprehensive
and structured representations for localized objects grounded by bounding
boxes together with their pairwise relationships, which is critical for scene
understanding in images. However, such a bounding box-based paradigm from a
detection perspective suffers from several issues. First, only a coarse
localization of objects are provided and there exists ambiguous pixels
belonging to different objects or categories. Second, context regions such as
background usually cannot be covered by bounding boxes. Third, there are
inconsistent and redundant labels in SGG datasets. To address these
limitations, [12] proposes the new task, panoptic scene graph generation (PSG)
along with a large dataset OpenPSG222https://psgdataset.org based on COCO [7]
and Visual Genome (VG) [4], to facilitate the progress of scene graph
representations based panoptic segmentations instead of rigid bounding boxes.
Existing PSG methods tackle the problem in one-stage paradigm or two-stage
paradigm. IMP [11], MOTIFS [13], VC-Tree [9], and GPSNet [8] are from the SGG
task and are adapted for the PSG task as two-stage baselines. Recently, [12]
designs a one-stage method named PSGTR based on the end-to-end DETR [1] to
predict triples and localizations simultaneously. PSGFormer [12] is further
proposed on the top of PSGTR with an explicit relation modeling with a
prompting-like matching mechanism and has shown competitive performance.

Figure 1: Panoptic Scene graph Generation. PSG presents a more comprehensive
and clean scene graph representation, with more accurate localization of
objects and including relationships with the background (known as stuff).

Despite of the great potential of one-stage paradigm, we claim that the choice
of one-stage and two-stage is still an open question for PSG task given the
fact that two-stage paradigm still dominates SSG task. We analyze these two
paradigms and find that one-stage methods benefit from global context and lead
to better relation prediction performance, but suffer from unsatisfactory
segmentation and instable training. While two-stage methods disentangle
segmentation and relation prediction, thus have better segmentation results,
nevertheless, they only extract local region features without global context
and are not suitable for predicting relation between two distance objects. The
trade-off between these paradigm inspires us to bridge such a gap.

In this work, we revive the two-stage paradigm by equipping it with ability to
obtain global context as one-stage paradigm. Specifically, we first adopt
Mask2Former [2] as our off-the-shelf panoptic segmentor, which generates masks
for each object. The intermediate feature map of a specific object from
segmentor and its corresponding mask are fused as object-level features.
Instead of handling pairwise objects individually, we propose to build the
global context utilizing a transformer that is fed with all found objects. A
class embedding is added to indicate object class. Therefore, the $N\times N$
relations are jointly optimized in the training process, where $N$ is the
number of objects.

Furthermore, we find that the performance of existing methods are also
severely limited by issues associated with dataset. In concise, the relation
ambiguity and inevitable bias caused by long-tailed distribution. To handle
these, we re-formulate the relation prediction in second stage from
traditional classification task to multi-class classification [10] supervised
with self-distilled soft label[5] instead of one-hot label. This also
alleviates the negative effect of missing label in the dataset. Meanwhile, a
focal loss [6] is also adopted to re-balance loss from rare categories.

We evaluate our proposed framework on OpenPSG dataset following the default
setting in [12] and conduct ablation studies to show the effectiveness of
self-distilled soft label and focal loss to handle dataset associated issues.
We claim that GRNet achieves the-state-of-art performance on the leaderboard
and exceeds previous methods by a large margin. We will release the code upon
acceptance.

In summary, our contributions are from three-fold:

  * •

We propose a new method called GRNet with a global relation module to bridge
the gap between existing two-stage and one-stage methods.

  * •

We formulate the relation prediction as a multi-class classification with soft
label, which effectively alleviates relation ambiguity and long-tailed issues.

  * •

We report the state-of-art performance on OpenPSG leaderboard, and exceed
previous both two-stage and one-stage methods by a large margin.

##  2 Methodology

Figure 2: The pipeline of our proposed GRNet. We follow two-stage paradigm
where an off-theshelf panoptic segmentor is used to generate object mask and
then pairwise relationship prediction between these predicted objects.

Existing approaches tackle the problem in one stage or two stages. For one-
stage methods, they are designed to predict relation triples and masks
simultaneously. While two-stage methods usually adopt an off-the-shelf mask
predictor first, then pairwise relationship prediction between these predicted
objects. However, there is still a clear trade-off between prediction
performance and global relation construction. In this section, we first
briefly recap the formulation of PSG task, and introduce our two-stage
approach with the ability to construct global relation. To handle relation
ambiguity and missing labels, we further propose self-distillation to produce
soft labels.

###  2.1 Problem Formulation

The panoptic scene graph generation (PSG) task aims to model the following
distribution:

$Pr(G|I)=Pr(M,O,R|I)$ (1)

where $I\in R^{H\times W\times 3}$ is the given input image, and $G$ is
desired scene graph which comprises the object masks
$M=\\{m_{1},m_{2},...,m_{n}\\}$ and class labels
$O=\\{o_{1},o_{2},...,o_{n}\\}$ that correspond to each of the $n$ objects in
the image, and their relations in the set $R=\\{r_{1},r_{2},...,r_{l}\\}$.
Each mask is associated with an object with class label $o_{i}$. The masks do
not overlap. More specifically, $m_{i}\in\\{0,1\\}^{H\times W}$ represents the
binary mask, $o_{i}$ and $r_{i}$ belong to the set of all object and relation
classes. For conciseness, both objects and background are referred as objects.

###  2.2 Global Context Modeling

Figure 3: Illustration of the necessity of global contextual information (non-
object entities) for relation prediction.

Context Matters. Previous two-stage methods only build pairwise relations
between subjects and objects in a triplet, while ignoring the influence of
rich global context from other neighborhoods. Let’s take Figure 3 as an
example, the subject and object are “person” and “football” respectively.
However, their relationship varies from context to context. When there are
adverbials of grassland, the predicate is likely to be the relationship of
“kicking”. When there are adverbials of sofa, beer, and TV, the predicate is
likely to be “holding”. Therefore, we claim that contextual information from
non-object entities are also significant for predicting complex scene graph.

Object Feature Tokens. In our work, instead of pairwise subject and object, we
construct global relation among uncertain entities via sequences modeling. The
structure of our global context module is shown in Figure. Specifically, given
feature map $f\in R^{H\times W}$ and a set of masks $M$ from our off-the-shelf
mask predictor, we first obtain object-level feature $f_{i}=f\cdot m_{i}$,
where $\cdot$ denotes dot product. As $f_{i}$ is already in hidden space, we
do not use extra linear project layer. To tokenize each object feature,
$f_{i}$ is flatten and divided into $L$ patches. Each patch is then applied
with an average pooling to obtain its token. In order to perform
classification, we use the standard approach of adding an extra learnable
“classification token” to the token sequence.

Global Context Module. After tokenization, we obtain $N\times(L+1)$ tokens
that represent all objects within the given image, where $N$ is the count of
objects. The token sequence is fed into a transformer to conduct $N\times N$
global modeling. Finally, for each object-level feature, we take global
average pooling to obtain $N$ object embeddings with contextual enhancement.
The message passing is achieved over all neighborhoods that makes use of the
edge direction information and global contextual information.

###  2.3 Multi-label Classification

Predicates Ambiguity. The rigorous annotation process is conducted in the PSG
task to ensure precision and salience, where the annotators strictly not to
annotate using general relations like “on” and “in” when a more precise
predicate like “parked on” is applicable, with the assumption that there
exists only a single relationship for a particular object pair. However, it is
usually invalid in common phenomenon that a pair of objects can be attached by
multiple relationships. In other words, although the different relationships
are hierarchical, they are not mutually exclusive. We term this problem as
predicate ambiguity. At the same time, we believe that this labeling ambiguity
will make it difficult for the model to converge. A natural solution is that
we convert the relationship prediction problem from the original one-label
classification problem to a multi-label classification problem.

Self-Distillation with Soft labels. Due to the task properties, there is only
a more accurate relation annotation for each paired object. To generate
multiple labels in practise, we lean on self-distillation to generate soft
labels in the training process. To be more specific, we make use of the
exponential moving average (EMA) to dynamic update our self-distilled
predictor. Given an initialized relation predictor $Pr$, we first regularly
train it with the hard labels at the beginning epochs until it converges to a
local minima. Then, we set $Pr_{ema}=Pr$ and update our $Pr_{ema}$ as
following, where $\alpha$ is the weight decay.

$Pr_{ema}=Pr_{ema}*\alpha+Pr*(1-\alpha)$ (2)

We use $Pr_{ema}$ as our pseudo-label generator to obtain multi-labels on the
training samples and finetune on soft labels for several epochs. We adopt
multi-head attention layers in $Pr$ to construct relations between different
entities, and the number of head is set to the number of relations.

|  |  |  | Scene Graph Generation
---|---|---|---|---
Backbone | Paradigm | Segmentor | Method | R/mR@20 | R/mR@50 | R/mR@100
|  |  | IMP | 16.5/6.52 | 18.2/7.05 | 18.6/7.23
|  |  | MOTIFS | 20.0/9.10 | 21.7/9.57 | 22.0/9.69
|  |  | VCTree | 20.6/9.70 | 22.1/10.2 | 22.5/10.2
|  |  | GPSNet | 17.8/7.03 | 19.6/7.49 | 20.1/7.67
|  | Panoptic-FPN | GRNet (Ours) | 31.0/25.9 | 37.2/34.0 | 40.3/37.5
| Two-Stage | Mask2former | GRNet (Ours) | 38.1/32.5 | 46.2/41.7 | 50.3/48.9
|  | - | PGSTR | 28.4/16.6 | 34.4/20.8 | 36.3/22.1
ResNet-50 | One-Stage | - | PSGFormer | 18.0/14.8 | 19.6/17.0 | 20.1/17.6
Swin-T | Two-Stage | Mask2former | GRNet (Ours) | 39.2/31.9 | 47.7/43.2 | 51.9/49.5
Swin-B | Two-Stage | Mask2former | GRNet (Ours) | 40.8/34.2 | 50.2/45.8 | 55.0/52.1
Table 1: Comparison between all baselines and our GCNet. Recall (R) and mean
recall (mR) are reported.

##  3 Experiments

In this section, we first introduce the dataset and the evaluation protocol
for the PSG task in Section 3.1. Implementation details are available in
Section 3.2. The comparisons with state-of-the-arts are provided in Section
3.3.

###  3.1 Dataset and Metric

Dataset. The OpenPSG dataset has 48,749 images with 133 object classes (80
objects and 53 stuff) and 56 predicate classes. It annotates inter-segment
relations based on COCO panoptic segmentation.

Metric. PSG task comprises two sub-tasks: predicate classification (when
applicable) and scene graph generation (main task) to evaluate the PSG models.
Predicate classification (PredCls) aims to generate a scene graph given the
ground-truth object labels and localization. Scene graph generation (SGDet)
aims to generate scene graphs from scratch, which is the main result for the
PSG task.

The classic metrics of R@K and mR@K are used to evaluate the previous two sub-
tasks, which calculates the triplet recall and mean recall for every predicate
category, given the top K triplets from the PSG model. Notice that PSG grounds
objects with segmentation, a successful recall requires both subject and
object to have mask-based IOU larger than 0.5 compared to their ground-truth
counterparts, with the correct classification on every position in the S-V-O
triplet. The panoptic segmentation evaluation protocol PQ can be also
considered as an auxiliary metric.

###  3.2 Implementation Details

We adopt Mask2former with Swin-B backbone from
MMDetection333https://github.com/open-mmlab/mmdetection to obtain object masks
and extract their RoI features. Since the performance of two-stage paradigm
highly depends on the underlying segmentor, we also use the same set of
parameters (ResNet backbone and Panoptic FPN) as previous works for fair
comparison. The parameters of the pre-trained Mask2former are not frozen
during training. We first train Mask2former individually for 50 epochs, and
then jointly train the full GRNet for extra 12 epochs. AdamW is adopted with
lr=0.0001 and weight_decay=0.05. The learning rate is decayed 10x at 6th and
10th epoch. The weight of self-attention layer is adopted from hfl/chinese-
roberta-wwm-ext-large444https://huggingface.co/hfl/chinese-roberta-wwm-ext-
large.

###  3.3 Comparisons with State-of-the-Arts

Comparing Methods. We select baselines in two different paradigms. For one-
stage approaches, we compare with PSGTR and PSGFormer. For two-stage
approaches, we compare with IMP, MOTIFS, VCTREE and GPSNet.

Quantitative Analysis. We compared the proposed GCNet with state-of-the-art
methods in Table 1. As shown, GRNet (ResNet-50, Panopitic-FPN) outputperforms
all two-stage methods by large margin under the same footprint, which
validates the effectiveness of our designed modules straightforwardly. We also
show the impressive performance gain brought by better backbone (Swin-T and
Swin-B) and segmentor (Mask2former).

##  4 Conclusion

In our submission to the PSG Challenge @ ECCV’22 SenseHuman Workshop, we
propose GR-Net with global context to boost the performance of two-stage
methods. We formulate relation prediction as a multi-class classification task
with soft label to further handle long-tailed issues. We claim that GR-Net
achieves the-state-of-art performance on the leaderboard and exceeds previous
methods (both two-stages and one-stage) by a large margin.

## References

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# IdeaReader: A Machine Reading System for Understanding the Idea Flow of
Scientific Publications

Qi Li1, Yuyang Ren1, Xingli Wang1, Luoyi Fu1, Jiaxin Ding1, Xinde Cao1,
Xinbing Wang1222Corresponding authors., Chenghu Zhou2

###### Abstract

Understanding the origin and influence of the publication’s idea is critical
to conducting scientific research. However, the proliferation of scientific
publications makes it difficult for researchers to sort out the evolution of
all relevant literature. To this end, we present IdeaReader, a machine reading
system that finds out which papers are most likely to inspire or be influenced
by the target publication and summarizes the ideas of these papers in natural
language. Specifically, IdeaReader first clusters the references and citations
(first-order or higher-order) of the target publication, and the obtained
clusters are regarded as the topics that inspire or are influenced by the
target publication. It then picks out the important papers from each cluster
to extract the skeleton of the idea flow. Finally, IdeaReader automatically
generates a literature review of the important papers in each topic. Our
system can help researchers gain insight into how scientific ideas flow from
the target publication’s references to citations by the automatically
generated survey and the visualization of idea flow. Code is available at
https://github.com/liqilcn/IdeaReader.

## Introduction

An excellent scientific paper usually serves as a connecting link in the
development of science. It is usually first inspired by previous work,
published after incorporating the authors’ idea and then influences subsequent
work. Understanding the origin and influence of the publication’s idea is an
indispensable step in conducting scientific research. For novice or cross-
disciplinary researchers, they need to sort out the evolution of the massive
literature’s idea to enter a new field (Zha et al. 2019). When writing a
paper, researchers also need to figure out the evolution of the related work’s
idea to show the advantages of their work in the article (Hoang and Kan 2010;
Hu and Wan 2014). However, the proliferation of scientific publications has
made it increasingly difficult for scholars to understand the contextual
research of all relevant papers (Chu and Evans 2021).

In this paper, we propose IdeaReader, a machine reading system that helps
researchers quickly understand the evolution of scientific publications’
ideas. Our system (available at https://agkg.acemap.cn) has the following
three characteristics: (1) IdeaReader can find out which articles are most
likely to inspire or be influenced by the target publication from the
references and citations of the target publication (first-order or higher-
order). (2) IdeaReader can utilize natural language to automatically summarize
the ideas of the target publication’s contextual research. (3) IdeaReader can
visualize the flow of the ideas in important literature associated with the
target publication.

## Related Work

Our work is mainly related to the construction of scientific evolution
roadmaps and automatic related work generation. Construction of scientific
evolution roadmaps: He et al. (2009) adapt the LDA model to the citation
network to construct topic evolution roadmaps of scientific literature. Zha et
al. (2019) use the tabular data in the paper as weakly supervised signals to
extract algorithmic roadmaps from massive publications. Yin et al. (2021)
utilize spectral clustering to build paper evolution roadmaps from the query
paper’s reference. Automatic related work generation: Hoang and Kan (2010) use
two separate strategies to generate general and detailed sentences of the
related work text. Hu and Wan (2014) propose an optimization method to select
important sentences from both the target paper and the reference papers to
generate a related work section for a target paper. Chen et al. (2021) design
a abstractive related work generation model by automatically capturing the
content dependency between multiple input documents during training.

However, the scientific evolution roadmaps can not reveal the specific
research content of the corresponding papers. Our system utilizes the
extracted skeleton of idea flow to guide survey generation, which can directly
present the ideas of the target paper’s contextual research.

## System Description

Figure 1: The algorithm pipeline of the IdeaReader.

We first introduce the algorithm pipeline (Fig. 1) of the IdeaReader, and then
we introduce the front-end interface (Fig. 2) of our system.

Paper query. First, our system queries the candidate reference and citation
papers of the target publication in the Acemap database (Tan et al. 2016).
Specifically, taking reference papers as an example, IdeaReader first selects
all first-order reference papers of the target publication. If the number of
these papers does not exceed 100, it then selects all second-order reference
papers (the papers cited by the first-order reference papers). Repeat this
process until more than 100 papers are obtained. PageRank (Page et al. 1999)
are used to obtain the top 100 papers as candidate papers among all the first-
order or higher-order reference papers obtained. The selection of candidate
citation papers follows a similar rule.

Paper clustering and relevance scoring. We then follow the approach in MRT
(Yin et al. 2021) to cluster and rank the candidate reference and citation
papers separately. In this step, IdeaReader first combines TF-IDF with
Sentence-BERT (Reimers and Gurevych 2019) to encode the abstracts of the
papers and uses ProNE (Zhang et al. 2019) to incorporate the resulting
embeddings according to the citation structure associated with the papers.
After obtaining representations of papers, Kernel k-means (Kulis et al. 2009)
is used to cluster candidate papers, and we treat each resulting cluster as a
topic to inspire or be influenced by the target publication. To find out which
papers are most likely to inspire or be influenced by the target publication
under a specific topic, it utilizes the vector inner product and the actual
citation relationship to calculate the relevance score between the candidate
papers and the target publication. Finally, our system ranks the papers within
each cluster according to their relevance scores.

Survey generation under specific topics. Finally, our system automatically
summarizes the ideas of the top five papers with relevance scores in each
cluster. The generated survey contains a heading, a general sentence, and
summary sentences of the selected papers. IdeaReader uses automatic text
annotation (Mei, Shen, and Zhai 2007) to extract the label from the title and
abstract of the selected papers as the heading of the corresponding topic’s
survey. After comparison and human evaluation, we use BertSumABS (Liu and
Lapata 2019) to generate the general sentence from the abstracts of the
selected papers. Our model is fine-tuned by the related work generation
dataset provided by Chen et al. (2021). Since we find that the results
generated by BertSumABS are difficult to effectively summarize the input
articles’ ideas, we only keep the first sentence as the final general
sentence. As for the summary sentences, we use SciBERT (Beltagy, Lo, and Cohan
2019) to train a binary classification model to extract these sentences from
the abstracts of the input articles. Our dataset is constructed from the
scientific abstract sentence classification dataset provided by Cohan et al.
(2019) and Gonçalves, Cortez, and Moro (2020). We utilize the sentences
labeled as the ‘OBJECTIVE’ class in the datasets as summary sentences, while
the other sentences are used as negative samples to train the binary
classification model. For readability, we finally align the subject and tense
of the generated text.

Figure 2: The front-end interface of the system.

Front-end interface. Our front-end interface is shown in Fig. 2 and consists
of four parts: (1) Target paper information panel. This part includes meta
information of the target paper, references and citations statistics, and the
topics that inspire or are influenced by the target publication. (2) Survey of
the papers most likely to inspire the target paper. (3) Survey of the papers
most likely to be influenced by the target paper. The panels in parts (2) and
(3) contain surveys on multiple topics, each of which is presented in a single
card. (4) Idea flow visualization. We call the result of visualization
‘Tracing and evolution tree’. In the tree, the root node is the target paper,
the left branch is the topics and papers that inspire the target paper, the
right branch is the topics and papers that are influenced by the target paper,
and the direction of idea flow is from the left to right.

Our system supports the download of machine reading results as pdf files. The
corresponding machine reading interface can be accessed via paper retrieval.

## Conclusion

We presented IdeaReader, a machine reading system that helps researchers
understand how scientific ideas flow between the target publication’s
contextual research. In the future, we will promote IdeaReader to scholars in
different fields to get more comprehensive feedback for improving our system.

## References

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# Type-$B$ analogue of Bell numbers
using Rota’s Umbral calculus approach

Eli Bagno
Jerusalem College of Technology,
21 HaVaad HaLeumi St.,
Jerusalem, Israel,
and Michlalah College Jerusalem,
36 Barukh Duvdevani
St., Jerusalem, Israel<EMAIL_ADDRESS>Holon Institute of Technology,
52 Golomb St., P.O.Box 305,
5810201 Holon, Israel David Garber
Holon Institute of Technology,
52 Golomb St., P.O.Box 305,
5810201 Holon, Israel<EMAIL_ADDRESS>
###### Abstract

Rota used the functional $L$ to recover old properties and obtain some new
formulas for the Bell numbers. Tanny used Rota’s functional $L$ and the
celebrated Worpitzky identity to obtain some expression for the ordered Bell
numbers, which can be seen as an evident to the fact that the ordered Bell
numbers are gamma-positive. In this paper, we extend some of Rota’s and
Tanny’s results to the framework of the set partitions of Coxeter type $B$.

##  1 Introduction

Rota [13] declares:

“It is the author’s conviction that formula (4), which we derive below, is the
natural description of the exponential numbers. The basic idea is a general
one, and can be applied to a variety of other combinatorial investigations. We
shall see that it easily leads to quick derivations of the properties of the
$B_{n}$.”

The ‘exponential numbers’ mentioned in the cited paragraph are the obsolete
name for what we call today Bell numbers, which count set partitions of the
set $[n]=\\{1,\dots,n\\}$. Rota’s Formula (4) reads: $B_{n}=L(u^{n})$, where
$L$ is a linear functional defined on the vector space of polynomials in the
indeterminate $u$ and $B_{n}$ is the $n$-th Bell number.

Rota [13] used the functional $L$ to recover old properties and obtain some
new formulas for the Bell numbers. Explicitly, let $V=\mathbb{R}[u]$ be the
vector space of all real polynomials in the single variable $u$. Then any
sequence of polynomials of degree $0,1,2,\dots$ is a basis for this vector
space, in particular the sequence of the falling factorials
$(u)_{0}=1,(u)_{1}=u,(u)_{2}=u(u-1),\cdots$ is a basis as well. Let
$L:\mathbb{R}[u]\to\mathbb{R}$ be the linear functional that is uniquely
defined by $L((u)_{j})=1$, for all $j\geq 0$. Rota [13] states the following
theorem:

######  Theorem 1.1.

Let $n\geq 0$. Then:
(1) $B_{n}=L(u^{n})$,
(2) $B_{n+1}=\sum\limits_{j=0}^{n}{n\choose j}B_{j}$,
(3) $B_{n}=\frac{1}{e}\sum\limits_{j\geq 0}\frac{j^{n}}{j!}$.

Tanny [15] used Rota’s functional $L$ and the celebrated Worpitzky identity
$m^{n}=\sum\limits_{k=0}^{n}{{n+m-k\choose{n}}}a_{n,k},$ to obtain the
following expression for the ordered Bell numbers :

$F_{n}(x)=\sum\limits_{k=1}^{n}a_{n,k}x^{n-k+1}(1+x)^{k-1},$

where $F_{n}(x)$ is the generating function of the number of ordered set
partitions of the set $[n]$ and $a_{n,k}$ are Eulerian numbers, a.k.a. the
number of permutations in the symmetric group $S_{n}$ having $k$ descents, see
[10]. In modern terms, this expression can be seen as an evident to the fact
that the ordered Bell numbers are gamma-positive.

In this paper, we extend some of Rota’s and Tanny’s results to the framework
of the set partitions of Coxeter type $B$.

In Section 2, we recall the definition of set partitions (and ordered set
partitions) of type $B$ and introduce the definition of Bell numbers and Bell
polynomials of type $B$. In Section 3, we extend the first two parts of Rota’s
Theorem 1.1 to Bell numbers of type $B$. Actually part $(2)$ of that result
will be proven both by Rota’s method and by a counting argument. In Section 4,
we use Brenti’s generalization of Worpitzky’s identity to obtain a gamma-
positivity result for the ordered Bell polynomials of type $B$.

##  2 Set partitions and Bell numbers of type B

###  2.1 Set partitions of type B

We now recall the definition of set partitions of type $B$ (see Dolgachev-
Lunts [6, p. 755] and Reiner [11, Section 2]; mentioned implicitly in Dowling
[7] and Zaslavsky [17] in the form of signed graphs):

######  Definition 2.1.

Denote: $[\pm n]:=\\{\pm 1,\dots,\pm n\\}$. A set partition of $[n]$ of type
$B$ or a signed set partition is a set partition of the set $[\pm n]$ such
that the following conditions are satisfied:

  * •

If $B$ appears as a block in the set partition, then $-B$ (which is obtained
from $B$ by negating all its elements) also appears in that partition.

  * •

There exists at most one block satisfying $-B=B$. This block is called the
zero block (if it exists, it is a subset of $[\pm n]$ of the form $\\{\pm
i\mid i\in C\\}$ for some $C\subseteq[n]$).

For example, the following is a set partition of $[6]$ of type $B$:

$\\{\\{1,-1,4,-4\\},\\{2,3,-5\\},\\{-2,-3,5\\},\\{6\\},\\{-6\\}\\}.$

Note that every non-zero block $B$ has a corresponding block $-B$ attached to
it. For the sake of convenience, we write for the pair of blocks $B,-B$, only
the representative block containing the minimal positive number appearing in
$B\cup-B$. For example, the pair of blocks $B=\\{-2,-3,5\\},-B=\\{2,3,-5\\}$
will be represented by the single block $\\{\mathbf{2},3,-5\\}$.

Our convention will be to write first the zero block and denote it by $B_{0}$
if exists and then the non-zero blocks of a set partition of type $B$ in such
a way that the sequence of absolute values of the minimal elements of the
blocks is increasing. We call this the standard presentation.

For example, the following is a set partition of $[6]$ of type $B$ in its
standard presentation:

$\left\\{B_{0}=\\{1,-1,4,-4\\},B_{1}=\\{\mathbf{2},3,-5\\},B_{2}=\\{\mathbf{6}\\}\right\\}.$

###  2.2 Stirling numbers and Bell numbers of type B

######  Definition 2.2.

Let $S^{B}(n,k)$ be the number of set partitions of type $B$ having $k$
representative non-zero blocks. This is known as the Stirling number of type
$B$ of the second kind (see sequence A085483 in OEIS [8]).

It is easy to see that $S^{B}(n,n)=S^{B}(n,0)=1$ for each $n\geq 0$. The
following recursion for $S^{B}(n,k)$ is well-known ([7, Theorem 7; see the
Erratum], [4, Corollary 3], for $m=2$, and [16, Equation (1)], for $m=2,c=1$):

######  Proposition 2.3.

For each $1\leq k<n$,

$S^{B}(n,k)=S^{B}(n-1,k-1)+(2k+1)S^{B}(n-1,k).$ (1)

The following result was proved combinatorially in [2] using a ‘balls into
urns’ approach:

######  Theorem 2.4.

Let $x\in\mathbb{R}$ and let $n\in\mathbb{N}$. Then we have:

$x^{n}=\sum\limits_{k=0}^{n}S^{B}(n,k)(x)^{B}_{k},$ (2)

where $(x)^{B}_{k}:=(x-1)(x-3)\cdots(x-2k+1)$ and $(x)^{B}_{0}:=1$, called the
falling factorial of type $B$.

######  Remark 2.5.

There is a simple connection between the falling factorials of types $A$ and
$B$:

$\left(\frac{x-1}{2}\right)_{n}=\left(\frac{x-1}{2}\right)\left(\frac{x-1}{2}-1\right)\cdots\left(\frac{x-1}{2}-n+1\right)=\frac{1}{2^{n}}(x-1)(x-3)\cdots(x-2n+1)=\frac{1}{2^{n}}(x)_{n}^{B},$

where the falling factorial of type $A$ is defined as follows:
$(x)_{k}:=x(x-1)(x-2)\cdots(x-k+1)$.

We define the Bell number of type $B$ as follows:
$B_{n}^{B}=\sum\limits_{k=0}^{n}S^{B}(n,k).$ Obviously, the Bell number counts
all the set partitions of the set $[n]$ of type $B$. A similar definition
appears in Mező and Ramírez [9].

We define also the Bell polynomial of type $B$ as follows:
$B_{n}^{B}(u)=\sum\limits_{k=0}^{n}S^{B}(n,k)u^{k}.$

Sagan and Swanson [14] discussed ordered set partitions of type $B$, which are
defined as follows:

######  Definition 2.6.

An ordered set partition of $[n]$ of type $B$ having $k$ non-zero blocks is a
sequence of sets $(S_{0},S_{1},\dots,S_{2k+1})$ which form a set partition of
$[n]$ of type $B$, such that the following two order conditions are satisfied:

  1. 1.

For each $1\leq i\leq n$, $i\in S_{0}$ if and only if $-i\in S_{0}$ (i.e. the
zero block $S_{0}$ is always the first block).

  2. 2.

For each $1\leq j\leq k$, we have $S_{2j}=-S_{2j-1}$.

Similar to the ordinary Bell number of type $B$, we define the ordered Bell
number of type $B$ and its associated ordered Bell polynomial as follows:

######  Definition 2.7.

$B_{n}^{B,o}=\sum\limits_{k=0}^{n}{2^{k}k!S^{B}(n,k)}$ and
$B_{n}^{B,o}(x)=\sum\limits_{k=0}^{n}{2^{k}k!S^{B}(n,k)x^{k}}$

##  3 Type-B analogue of Rota’s result

Let $V$ be the vector space of all polynomials in the variable $x$. It is easy
to see that the set $\left\\{(x)^{B}_{k}\right\\}_{k\in\mathbb{Z}}$ is a basis
of $V$. We use this basis to define a functional $L:V\rightarrow\mathbb{R}$ by
$L((x)_{k}^{B})=u^{k}$ for each $k\in\mathbb{Z}$, where $u$ is a fixed real
number.

We will use the following lemma in the proof of Theorem 3.2(2):

######  Lemma 3.1.

Let $p(x)$ be any polynomial. Then: $L((x-1)p(x-2))=u\cdot L(p(x)).$

###### Proof.

Since the set of falling factorials of type $B$

$\\{1,(x-1),(x-1)(x-3),\dots,(x-1)(x-3)\cdots(x-2n+1)\\}$

is a basis for $\mathbb{R}_{n}[x]$, we can write any polynomial $p(x)$ as a
linear combination of this set as follows:

$p(x)=a_{0}+a_{1}(x-1)+a_{2}(x-1)(x-3)+a_{n}(x-1)(x-3)\cdots(x-2n+1).$ (3)

Substituting $x-2$ for $x$, we have now:

$p(x-2)=a_{0}+a_{1}(x-3)+a_{2}(x-3)(x-5)+\cdots+a_{n}(x-3)(x-5)\cdots(x-2n+3).$

Multiplying the last equality by $(x-1)$ yields:

$(x-1)p(x-2)=a_{0}(x-1)+a_{1}(x-1)(x-3)+\cdots+a_{n}(x-1)\cdots(x-2n+3).$ (4)

Applying the operator $L$ on Equations (3) and (4), we get the desired result.
∎

We present here the analogue for type $B$ of the celebrated results by Rota
[13] (for part (2), see also Mező and Ramírez [9, p. 258]):

######  Theorem 3.2.

For each $n\in\mathbb{N}$, we have:
(1) $B_{n}^{B}(u)=L(x^{n})$.
(2) $B_{n+1}^{B}(u)=B_{n}^{B}(u)+u\cdot\sum\limits_{j=0}^{n}{2^{n-j}{n\choose
j}B_{j}^{B}(u)}.$

###### Proof.

(1) By Theorem 2.4, we have
$x^{n}=\sum\limits_{P\in\Pi_{n}^{B}}(x)_{N(P)}^{B}$, where $\Pi_{n}^{B}$ is
the set of set partitions of the set $[n]$ of type $B$ and $N(P)$ is the
number of non-zero representative blocks in the set partition $P$. Now apply
the functional $L$ on both sides of this equation and use the linearity of $L$
to get

$L(x^{n})=\sum\limits_{P\in\Pi_{n}^{B}}L\left((x)_{N(P)}^{B}\right)=\sum\limits_{P\in\Pi_{n}^{B}}u^{N(P)}=\sum\limits_{k=0}^{n}{S^{B}(n,k)u^{k}}=B_{n}^{B}(u).$

(2) For this result, we supply two different proofs: an algebraic one and a
combinatorial one for $u=1$. We start with the algebraic proof: Applying Lemma
3.1 to $p(x)=(x+2)^{n}$, we get:

$\displaystyle B_{n+1}^{B}(u)-B_{n}^{B}(u)$ $\displaystyle=$ $\displaystyle
L(x^{n+1})-L(x^{n})=L((x-1)x^{n})\stackrel{{\scriptstyle{\rm Lemma}\
\ref{property of L}}}{{=}}u\cdot L((x+2)^{n})=$ $\displaystyle=$
$\displaystyle u\cdot L\left(\sum\limits_{j=0}^{n}{n\choose
j}x^{j}2^{n-j}\right)=u\cdot\sum\limits_{j=0}^{n}{n\choose
j}2^{n-j}L(x^{j})=u\cdot\sum\limits_{j=0}^{n}{n\choose j}2^{n-j}B_{j}^{B}(u).$

The combinatorial proof for $u=1$ is as follows: The left hand side counts the
total number of set partitions of the set $[n]$ of type $B$ for any number of
non-zero representative blocks. we show that the right hand side counts the
same thing in a different way. We divide in two cases according to the
location of $n+1$:

  1. 1.

If $n+1$ is located in the zero-block, then we have $B_{n}^{B}$ possibilities
to locate the other elements.

  2. 2.

Otherwise, $n+1$ is located in a non-zero block. Then, for each $0\leq k\leq
n$, assume that $n-k$ is the number of elements that share a block with $n+1$.
The rest $k$ elements can be located in $B_{k}^{B}$ ways. Finally, we have
$2^{n-k}$ possibilities to sign the elements in the block containing $n+1$.

∎

##  4 Gamma-positivity of ordered Bell polynomial of type B

Let $V$ be the vector space of all polynomials in the variable $x$. It is easy
to see that the set $\left\\{(x)^{B}_{k}\right\\}_{k\in\mathbb{Z}}$ is a basis
of $V$. We use this basis to define a new functional
$L^{o}:V\rightarrow\mathbb{R}$ by

$L^{o}((x)_{k}^{B})=2^{k}k!u^{k}$ (5)

for each $k\in\mathbb{Z}$, where $u$ is a fixed real number. By slightly
modifying the proof of Theorem 3.2 we get that $L^{o}(x^{n})=B_{n}^{B,o}(u)$.

Brenti [5] obtained a Worpitzky-like identity for type $B$ as follows: For
each $m,n\in\mathbb{N}$, one has

$(1+2m)^{n}=\sum\limits_{k=0}^{n}{{n+m-k\choose{n}}}E^{B}_{n,k},$

where $E^{B}_{n,k}$ are the Eulerian numbers of type $B$, which counts the
number of signed permutations in the Coxeter group of type $B$ having $k$
descents; this set of numbers constitutes the sequence A060187 in OEIS [8].
For a combinatorial proof of this identity, see [3].

The following result shows the gamma-positivity of the ordered Bell
polynomials of type $B$:

######  Proposition 4.1.

$B_{n}^{B,o}(u)=\sum\limits_{k=0}^{n}E^{B}_{n,k}u^{k}(1+u)^{n-k}$.

###### Proof.

If we write $x=1+2m$, then we have:

$\displaystyle B_{n}^{B,o}(u)$ $\displaystyle=$ $\displaystyle
L^{o}(x^{n})\quad=\quad\sum\limits_{k=0}^{n}L^{o}\left[{n+m-k\choose{n}}\right]E^{B}_{n,k}=$
$\displaystyle=$
$\displaystyle\sum\limits_{k=0}^{n}L^{o}\left[{n+\frac{x-1}{2}-k\choose{n}}\right]E^{B}_{n,k}\quad=\quad\sum\limits_{k=0}^{n}\frac{E^{B}_{n,k}}{n!}L^{o}\left[\left(n+\frac{x-1}{2}-k\right)_{n}\right]=$
$\displaystyle\stackrel{{\scriptstyle(*)}}{{=}}$
$\displaystyle\sum\limits_{k=0}^{n}\frac{E^{B}_{n,k}}{n!}L^{o}\left[\sum\limits_{\ell=0}^{n}{n\choose\ell}(n-k)_{n-\ell}\left(\frac{x-1}{2}\right)_{\ell}\right]=$
$\displaystyle\stackrel{{\scriptstyle{\rm Rem.}\ \ref{connection}}}{{=}}$
$\displaystyle\sum\limits_{k=0}^{n}\frac{E^{B}_{n,k}}{n!}\sum\limits_{\ell=0}^{n}{n\choose\ell}(n-k)_{n-\ell}L^{o}\left[\frac{(x)^{B}_{\ell}}{2^{\ell}}\right]=$
$\displaystyle\stackrel{{\scriptstyle{\rm Eqn.}\ (\ref{def ordered L})}}{{=}}$
$\displaystyle\sum\limits_{k=0}^{n}E^{B}_{n,k}\sum\limits_{\ell=0}^{n}{n\choose\ell}\frac{(n-k)_{n-\ell}}{n!}\ell!u^{\ell}\quad=\quad\sum\limits_{k=0}^{n}E^{B}_{n,k}\sum\limits_{\ell=0}^{n}{n-k\choose
n-\ell}u^{\ell}=$ $\displaystyle=$
$\displaystyle\sum\limits_{k=0}^{n}E^{B}_{n,k}u^{k}\sum\limits_{\ell=0}^{n}{n-k\choose\ell-k}u^{\ell-k}\
\ =\ \ \sum\limits_{k=0}^{n}E^{B}_{n,k}u^{k}(1+u)^{n-k},$

where Equality $(*)$ is based on the binomial Umbral identity of the falling
factorials of type $A$:$(a+b)_{n}=\sum\limits_{k=0}^{n}{n\choose
k}(a)_{k}(b)_{n-k}$, see e.g. Roman [12, p. 29]. ∎

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