#  Socialite-llama: An Instruction-Tuned Model
for Social Scientific Tasks

Gourab Dey1 , Adithya V Ganesan1∗, Yash Kumar Lal1∗,
Manal Shah2, Shreyashee Sinha3, Matthew Matero4
Salvatore Giorgi5,6, Vivek Kulkarni7, H. Andrew Schwartz1
1Stony Brook University, 2JP Morgan Chase, 3Bloomberg
4Preferabli, 5University of Pennsylvania, 6National Institute of Health,
7Grammarly
<EMAIL_ADDRESS>

Equal contribution

###### Abstract

Social science NLP tasks, such as emotion or humor detection, are required to
capture the semantics along with the implicit pragmatics from text, often with
limited amounts of training data. Instruction tuning has been shown to improve
the many capabilities of large language models (LLMs) such as commonsense
reasoning, reading comprehension, and computer programming. However, little is
known about the effectiveness of instruction tuning on the social domain where
implicit pragmatic cues are often needed to be captured. We explore the use of
instruction tuning for social science NLP tasks and introduce Socialite-llama—
an open-source, instruction-tuned Llama2. On a suite of 20 social science
tasks, Socialite-llama improves upon the performance of Llama2 as well as
matches or improves upon the performance of a state-of-the-art, multi-task
finetuned model on a majority of them. Further, Socialite-llama also leads to
improvement on 5 out of 6 related social tasks as compared to Llama2,
suggesting instruction tuning can lead to generalized social understanding.
All resources including our code, model and dataset can be found through
bit.ly/socialitellama.

##  1 Introduction

Instruction finetuned large language models (LLMs) have demonstrated
impressive performance on many standard NLP tasks Wei et al. (2022a); Chung et
al. (2022), but these models tuned on non-social tasks seem to have poor
social pragmatics Ziems et al. (2023); Choi et al. (2023); Havaldar et al.
(2023); V Ganesan et al. (2023). The fact that instruction tuning has been
successful with limited amounts of data Gupta et al. (2023) and that
instruction tuned models have the ability to generalize to new tasks in both
few- and zero-shot settings Wei et al. (2022a), suggests that LLMs could
become more socially capable by instruction tuning them on a wide variety of
social NLP tasks.

Here, we introduce Socialite-llama, a Llama2 7B-based LLM Touvron et al.
(2023) instruction tuned on a suite of social scientific classification tasks
spanning 5 broad categories for which we hand-craft instructions. We evaluate
its performance on both zero- and few-shot settings on seen and related social
tasks, demonstrating that our model significantly outperforms prior open
models. Our results support prior research highlighting the effectiveness of
instruction tuning when applying LLMs to a new domain.

Social and psychological factors have been shown important and beneficial to
model in past interdisciplinary NLP studies Lynn et al. (2017); Flek (2020);
Hovy and Yang (2021). Modeling human factors and social context can not only
improve performance on primarily non-social NLP tasks Lynn et al. (2017); Flek
(2020); Hovy and Yang (2021), but can also prove to be beneficial for a number
of psychological and social scientific tasks Garten et al. (2019); Matero et
al. (2021). However, a major limitation of prior models has been that they
have been task-specific and do not generalize well to new tasks. Many such
models also are trained from scratch and typically need an extensive amount of
human-annotated training data. With the development of LLMs that exhibit the
capability to learn from instructions, we posit that this new capability can
be leveraged to address these two major limitations.

Our contributions include: (1) we develop and systematically evaluate
Socialite-llama– an instruction-tuned language model for social science tasks
– across 20 seen and 6 related social scientific tasks, (2) we show Socialite-
llama consistently improved over Llama2 in all seen tasks and the improvement
generalized to 5 of 6 related tasks; In fact, it matched the performance of a
state-of-the-art multi-task tuned DeBERTa on a majority of seen tasks, (3) we
suggest that the benefits of few-shot examples (over zero-shot) become
negligible on tasks seen during instruction tuning as opposed to related tasks
where few-shot still provided a benefit, and (4) we release Socialite-llama
111Model available on huggingface models as well as its instructions corpora,
SocialiteInstructions 222Dataset available on huggingface datasets, as open-
source resources for the community.

##  2 Related Work

Language is inherently social — the underlying meaning is constructed through
social interactions Wittgenstein (1953); Clark and Schober (1992); Hovy and
Spruit (2016). Understanding communication requires reasoning about the social
implications drawn from that message Halliday (2004). Prior work has sought to
build language models for social scientific tasks, which we discuss next.

#### LMs for Social Scientific NLP

Social science NLP models are usually built by fine-tuning Sap et al. (2020);
Matero et al. (2019); V Ganesan et al. (2021) for specific tasks or
pretraining language models on the the domain of language that captures the
social factors. BERTweet Nguyen et al. (2020), the first public large-scale
pre-trained language model on English tweets, was trained using the BERT
architecture with the RoBERTa pre-training procedure Liu et al. (2019) for
this purpose. DeLucia et al. (2022) built Bernice, a multilingual model for
social science NLP tasks using the RoBERTa architecture and pretrained from
scratch on 2.5 billion tweets. It outperforms a variety of models adapted to
social media data as well as strong multilingual baselines. Despite the
strength of such existing task-specific models, there exists no base
instruction-following LLM tailored to the domain of social science tasks.

#### Instruction-Tuning

Instruction tuning, in general, refers to the practice of finetuning pre-
trained language models to better understand and respond to a wide variety of
human requests that are expressed in natural language Wei et al. (2022a);
Mishra et al. (2022). The success of instruction tuning requires two key
components: 1) a powerful pre-trained language model like Llama2 Touvron et
al. (2023), and 2) an instruction dataset that is diverse and representative
enough to adapt the LLM to potential downstream usage. However, existing
instruction-tuning datasets Longpre et al. (2023); Sanh et al. (2022); Wang et
al. (2023, 2022) are general-purpose and do not contain a significant amount
of social science tasks. Consequently, models built using this data have
limited usability for social scientific tasks Choi et al. (2023); Ziems et al.
(2023). To address this gap, we curate an instruction-tuning dataset aimed at
modeling social scientific knowledge and use it to train our model.

##  3 Datasets and Experiments

We use a diverse collection of English social science NLP tasks to create
SocialiteInstructions by (re-)framing each task into a binary or multi-class
classification problem. For each task, we hand-craft instructions for the
model to follow. These instructions were composed with task descriptive which
included examples in some cases (e.g. BiasedImplication from Table 9),
followed by the classification label choices it has to pick from. The
instructions and the dataset sizes for all the tasks have been tabulated in
Appendix A.

###  3.1 Training Tasks

We draw on datasets mentioned in Choi et al. (2023) for compiling our
instruction following dataset. Our train set spans 20 different datasets
across 5 broad categories of tasks: Humor, Offensiveness, Sentiment and
Emotion, Trustworthiness, and other social factors. To ensure broader
applicability, we frame the non-classification datasets as classification
tasks. We refer to these as seen tasks going forward.

Humor We use SemEval 2021 Task 7 data Meaney et al. (2021) to capture humor.
We use the binary humor detection task as is (Humor). Additionally, we convert
the humor rating task into a binary classification problem collapsing ground
truth labels higher than 3 into high humor, and the others into low humor
(HumourRating).

Offensiveness We use Social Bias Frames Sap et al. (2020) as a benchmark to
detect offense and bias directed towards people and groups. Social Bias Frames
comprises of 4 binary classification tasks - Offensive to discern whether a
given text exhibits rudeness, disrespect, or toxicity, Sexist to determine if
a text contains lewd or sexual references which can be considered as
offensive, IntentToOffend to capture whether the perceived motivation of the
author was to indeed offend, and BiasedImplication to identify any forms of
prejudice or group-based discrimination within the text.

Sentiment and Emotion For emotion classification, we use SemEval 2018 Task 1
data Mohammad et al. (2018). Emotion involves classifying a tweet into anger,
joy, optimism or sadness displayed by the author. We also include Emobank
Buechel and Hahn (2017) to infer the valence, arousal and dominance levels
that a text would invoke in a reader. We convert these regression tasks to
binary classification - by transforming a score greater than 4 to ’high’ and
less than 4 to ’low’, for each of the three datasets respectively (ValenceCLS,
ArousalCLS, DominanceCLS). We use SemEval 2017 Task 4 data for sentiment
classification Rosenthal et al. (2017) to identify the overall sentiment of a
text as positive, negative or neutral. Further, we also include SameSideStance
Körner et al. (2021), the task of classifying whether two different texts are
on the same side of an argument.

Trustworthiness We use Hyperbole Zhang and Wan (2022) to detect hyperbolic
language present in a text, and SubjectiveBias Pryzant et al. (2020), to
detect which sentence exhibits subjective bias among two pieces of text.

Other Social Factors We also include other discursive and rhetorical type
tasks — empathy scored on multi-item scales (EmpathyExplorations) Sharma et
al. (2020), self-rated empathy (EmpathySelfRated) Buechel et al. (2018),
distress (DistressSelfRated) Buechel et al. (2018), figurative speech
detection (FLUTE) Chakrabarty et al. (2022), politeness (PolitenessHayati)
Hayati et al. (2021) and intimacy (Intimacy) Pei and Jurgens (2020).

###  3.2 Evaluation Tasks

For evaluation, we use the same twenty seen datasets as well as six datasets
for related social tasks. The process we used to choose the ‘seen tasks’ for
instruction tuning was towards the goal of training on a very broad category
of tasks and consequently, any task would be related to these fundamental
categories seen during training. Hence for evaluation we choose six tasks,
half of which are more directly related to seen tasks than the other. The
datasets used for evaluation contained no data overlap with the training
tasks’ data. Among the related tasks’ datasets, we first select three datasets
directly related to our training tasks — HateSpeech Basile et al. (2019) to
detect hate speech in a tweet, PolitenessStanford Fu et al. (2020) to judge
whether a text is polite or impolite, and AgreeDisagree Varadarajan et al.
(2022) to classify whether two texts agree, disagree or N/A on a particular
topic. To strengthen our evaluation, we also evaluate on three other tasks
which are less related to the seen set — Irony Van Hee et al. (2018) to
classify if a tweet is ironic, Optimism Ruan et al. (2016) to categorize a
text as optimistic, pessimistic or neutral, and Complaints Preoţiuc-Pietro et
al. (2019) to judge if a text is a complaint or not. We refer to these six as
related social tasks.

###  3.3 Task Selection Criterion

The above tasks were chosen from SocKET (a) to be representative of each
social category and (b) if the document lengths weren’t too long, i.e.,
utterance level tasks Ziems et al. (2023).

Our initial experiments suggested that skew in sample sizes from different
tasks affected the overall performance. Particularly, using all 35k examples
from Social Bias Frames for all the four tasks limited the performance on
PolitenessHayati and SameSideStance. Hence, we picked datasets from SocKET
based on 2 additional factors: recency and minimum number of 8k examples. This
is why we chose Sap et al. (2020) over other toxicity datasets (further down
sampled 8k examples for each task), and Rosenthal et al. (2017) over Socher et
al. (2013) for sentiment task.

Task |  | Socialite-
---
Llama
Llama2
| Few-shot | Zero-shot | Few-shot | Zero-shot
Seen Tasks
Sentiment | 72.69 | 73.43 | 46.40 | 18.28
Emotion | 79.68 | 81.08 | 66.84 | 23.92
ValenceCLS | 82.34 | 83.84 | 44.52 | 31.59
ArousalCLS | 60.69 | 64.22 | 52.69 | 34.42
DominanceCLS | 60.88 | 66.63 | 43.05 | 19.33
EmpathyExplorations | 30.10 | 56.99 | 29.43 | 5.89
EmpathySelfRated | 63.59 | 63.97 | 58.99 | 8.75
DistressSelfRated | 68.45 | 70.96 | 42.36 | 9.80
FLUTE | 80.77 | 96.27 | 55.13 | 4.81
Hyperbole | 65.62 | 69.23 | 59.20 | 25.43
SameSideStance | 88.19 | 88.19 | 82.10 | 25.00
Humor | 90.48 | 95.29 | 44.44 | 38.8
HumourRating | 52.81 | 48.66 | 43.51 | 34.84
PolitenessHayati | 89.36 | 84.54 | 83.58 | 24.34
Intimacy | 26.25 | 33.21 | 13.24 | 3.17
SubjectiveBias | 52.04 | 87.13 | 51.18 | 40.98
Offensive | 80.47 | 83.11 | 71.78 | 35.76
Sexist | 72.61 | 73.85 | 52.81 | 6.25
IntentToOffend | 74.17 | 75.65 | 32.09 | 21.34
BiasedImplication | 79.66 | 85.72 | 56.41 | 26.16
Related Tasks
HateSpeech | 42.13 | 39.63 | 68.66 | 31.34
Irony | 42.99 | 59.04 | 47.11 | 32.24
PolitenessStanford | 63.22 | 64.89 | 58.64 | 1.2
Optimism | 62.24 | 36.96 | 56.61 | 44.49
Complaints | 79.35 | 85.37 | 58.23 | 22.8
AgreeDisagree | 39.54 | 19.3 | 28.1 | 10.01
Table 1: Evaluation of Socialite-llama and Llama2 on 20 seen tasks and 6
unseen tasks using macro-F1 scores. We report few-shot performance with $k=5$
for Socialite-llama on seen tasks and $k=15$ on unseen tasks. For Llama2, we
use $k=15$ for all tasks since it is not trained for social scientific tasks.
$k$ refers to the number of examples used in the few-shot setting. All
differences between Socialite-llama and Llama2 are statistically significant
$(p<0.001)$. Bold indicates the performance with the best setting for that
task.

###  3.4 Training Procedure

We transform each of the 20 training datasets using hand-crafted instructions.
To reduce the skew caused by the disproportionate number of examples present
in a few tasks, we limit the number of training examples from Offensive,
Sexist, IntentToOffend, BiasedImplication, Sentiment and SubjectiveBias to 8k
and use the original number of training data points for the rest. This
resulted in ~108k data points. We employ Low-Rank Adaptation Hu et al. (2022),
with rank set to 8 applied on the key and query projection matrices reducing
the number of trainable parameters to 4.1 million, and use AdamW Loshchilov
and Hutter (2019) with a learning rate of $\eta=1e-4$ and a batch size of $64$
for a maximum of $7$ epochs with early stopping. We expand the context length
to 3k using mixed precision training, gradient checkpointing and DeepSpeed
Zero Rajbhandari et al. (2020) Stage 2 which enables finetuning on 2 48GB
A6000 GPUs in ~21 hours and train using Huggingface Wolf et al. (2020).
Training Socialite-llama on Lambda333https://lambdalabs.com would cost $33.60,
making it very cost-effective.

##  4 Results

We analyze the effect of instruction tuning Llama2 on 20 seen and 6 related
social scientific tasks and report macro F1 scores in Table 1.

Instruction tuning improves social understanding. We show that Socialite-llama
outperforms Llama2 on all seen tasks in both zero- and few-shot setting. In
fact, Socialite-llama used in a zero-shot setting is even better than Llama2
in few-shot setting. Surprisingly, we find that the benefit of few-shot
learning over zero-shot is much less on seen tasks as opposed to the related
tasks with the instruction tuned model.

Particularly, performance of our model is better for all tasks that are
proximate to personal factors or how one feels, such as the affective
dimensions, sentiment, emotion and empathy/distress. We also see consistent
improvements over Llama2 on offensiveness related tasks. Despite the
tremendous general-purpose capabilities of Llama2, it appears to lack social
understanding from text, and can be improved using instruction tuning on
social scientific tasks.

Task |  | Socialite-
---
LLama
| DeBERTa
---
(multi-task FT)
Sentiment | 73.43 | 69.00
Emotion | 81.08 | 80.00
DistressSelfRated | 70.96 | 65.00
EmpathySelfRated | 63.97 | 59.00
Hyperbole | 69.23 | 69.00
SameSideStance | 88.19 | 76.00
Humor | 95.29 | 91.00
PolitenessHayati | 84.54 | 89.00
Intimacy | 33.21 | 46.00
SubjectiveBias | 87.13 | 96.00
Offensive | 83.11 | 83.00
IntentToOffend | 75.65 | 74.00
Sexist | 73.85 | 79.00
BiasedImplication | 85.72 | 87.00
Table 2:  We compare Socialite-llama with a multi-task finetuned (FT) DeBERTa
reported in Choi et al. (2023). Socialite-llama matches or outperforms it on
10 out of 14 tasks that both models have been trained on, even though our
training data size is much lower. Bold macro F1 represents improvement or
equivalent performance of Socialite-llama with DeBERTa (statistically
significant results performed using bootstrapped resampling; $p<.05$ ).
Because DeBERTa is not generative, it is not possible to apply to tasks for
which it was not finetuned.

Instruction tuning helps generalization. In related social science tasks,
which are composed of new instructions and language samples, we find that
Socialite-llama is better on 4 out of 6 under few-shot setting and 5 out of 6
overall. These 6 related tasks were on a varying degree of relatedness to the
ones in the seen tasks. Socialite-llama shows significant improvement in
performance over Llama2 on the related social science tasks. Through
instruction tuning, our model generalizes social aspects of language to
perform well on new related social tasks and task categories.

To assess the generalization emerging from the commonalities present between
the task, we applied the instructions from the seen tasks’ set on a related
task and compared its performance with the related tasks’ instruction (which
was unseen to Socialite-llama). A non-author NLP expert suggested 4 seen tasks
(Offensive, Sexist, IntentToOffend and BiasedImplication) that they thought
were very similar to the HateSpeech task. Of those, instructions from
Offensive and IntentToOffend gave improvement over zero-shot HateSpeech
performance, while the other two led to worse performance. This suggests it’s
possible to do better on zero-shot by using a related task prompt from
instruction tuning but it can also hurt performance.

Socialite-llama is state-of-the-art overall on SocKET. We compare Socialite-
llama with a state-of-the-art multi-task finetuned DeBERTa model presented in
Choi et al. (2023) for a subset of the seen tasks since their DeBERTa model
can’t be applied on the related social tasks. Despite being trained on
significantly less data than DeBERTa model (2.1 million data points and 58
tasks as compared to 108k data points and 20 tasks), Table 2 shows that our
model is better or equivalent in performance on 10 out 14 tasks. This result
highlights a large reduction in the performance separation between LLMs and
smaller task finetuned language models in Social Science tasks established by
previous works Ziems et al. (2023); Choi et al. (2023).

##  5 Conclusion

We introduce Socialite-llama, an instruction tuned Llama2 7B for social
science applications. Socialite-llama is trained using SocialiteInstructions,
a diverse collection of 20 social scientific tasks and tested on those plus an
additional 6 related social tasks. It consistently performs better than Llama2
on all the seen tasks and all but 1 related task, demonstrating its strong
generalization abilities. Instruction tuning on diverse social scientific
tasks helps our model achieve state-of-the-art overall. Our results provide
valuable insights to improve the social understanding of LLMs.

## Limitations

The improvements brought through instruction tuning of Llama2 on social
scientific tasks highlights the prevailing neglect of these tasks and the
importance of doing this. However, there are some important limitations. For
one, the tests for generalization to unseen tasks were limited to those that
are semantically related to seen tasks (many tasks in SocKET are related to
each other). Future work would need to evaluate this model and process for
generalizing to more distant tasks as well as across other points in time.

This work was informed by a number of works in the NLP and deep learning
literature, was largely facilitated by various open sourced datasets Choi et
al. (2023) and code Wolf et al. (2020). However, the number of tasks in our
work is not close to the number of tasks found in the prior instruction tuning
literature Wang et al. (2022); Sanh et al. (2022). This work would likely
benefit more tasks within the realm of social science NLP.

Owing to limited access to computational resources, we could only focus on
classification tasks on the smallest available Llama2 model and Low-Rank
Adaptation as training strategy. However, literature on scaling models Brown
et al. (2020); Wei et al. (2022b) suggests that these performances likely go
up with increasing the model size, datasets, instructions, which would make it
more capable for regression and generation tasks with increased input lengths.
Inclusion of few-shot examples in the training set could also improve the few-
shot generalization capability of the model, which could not be performed with
the resources available.

## Ethics Statement

Large language models (LLMs) like GPT-4 OpenAI (2023) have been deployed to
tens of millions of consumers in different forms Heaven (2023); Hu (2023).
While a number of works have already shown its limited social understanding
Ziems et al. (2023); Choi et al. (2023); Havaldar et al. (2023); V Ganesan et
al. (2023), their rapid adoption and proliferation necessitates improving
their social sensibility to make it reliable and useful. This work is aimed
towards building a socially sensible language model for the practitioners of
social science NLP research.

We hope our work improves NLP’s benefit for social scientific pursuits and
caution against the use of such models for non-scientific pursuits like
targeting of individuals without their consent or awareness. This work has
been built for research and can be re-distributed as per the policy laid out
by Meta for Llama2 444https://ai.meta.com/llama/license/. We strongly believe
that these models have to be tested for failure modes and harmful biases, and
should be further be adjusted through appropriate methods Ouyang et al. (2022)
before deploying it into the hands of users.

## Acknowledgements

We would like to thank the reviewers for their valuable feedback that helped
us improving this paper. This work was supported in part by a grant from the
CDC/NIOSH (U01 OH012476), a grant from IARPA (585968), a grant from the NIH-
NIAAA (R01 AA028032). YKL was supported in part by the Air Force Research
Laboratory (AFRL), DARPA, for the KAIROS program under agreement number
FA8750-19-2-1003 and in part by the NSF under the award IIS #2007290. We
approve this work for public release and redistribution with attribution. The
conclusions and views in this paper are those of the authors, and should not
be seen as representing the official policies expressed or implied by CDC,
NIOSH, NIH, NIAAA, IARPA, DARPA, ARFL, NSF or the U.S. government.

## References

  * Basile et al. (2019) Valerio Basile, Cristina Bosco, Elisabetta Fersini, Debora Nozza, Viviana Patti, Francisco Manuel Rangel Pardo, Paolo Rosso, and Manuela Sanguinetti. 2019.  SemEval-2019 task 5: Multilingual detection of hate speech against immigrants and women in Twitter.  In _Proceedings of the 13th International Workshop on Semantic Evaluation_ , pages 54–63, Minneapolis, Minnesota, USA. Association for Computational Linguistics.
  * Brown et al. (2020) Tom Brown, Benjamin Mann, Nick Ryder, Melanie Subbiah, Jared D Kaplan, Prafulla Dhariwal, Arvind Neelakantan, Pranav Shyam, Girish Sastry, Amanda Askell, Sandhini Agarwal, Ariel Herbert-Voss, Gretchen Krueger, Tom Henighan, Rewon Child, Aditya Ramesh, Daniel Ziegler, Jeffrey Wu, Clemens Winter, Chris Hesse, Mark Chen, Eric Sigler, Mateusz Litwin, Scott Gray, Benjamin Chess, Jack Clark, Christopher Berner, Sam McCandlish, Alec Radford, Ilya Sutskever, and Dario Amodei. 2020.  Language models are few-shot learners.  In _Advances in Neural Information Processing Systems_ , volume 33, pages 1877–1901. Curran Associates, Inc.
  * Buechel et al. (2018) Sven Buechel, Anneke Buffone, Barry Slaff, Lyle Ungar, and João Sedoc. 2018.  Modeling empathy and distress in reaction to news stories.  In _Proceedings of the 2018 Conference on Empirical Methods in Natural Language Processing_ , pages 4758–4765, Brussels, Belgium. Association for Computational Linguistics.
  * Buechel and Hahn (2017) Sven Buechel and Udo Hahn. 2017.  EmoBank: Studying the impact of annotation perspective and representation format on dimensional emotion analysis.  In _Proceedings of the 15th Conference of the European Chapter of the Association for Computational Linguistics: Volume 2, Short Papers_ , pages 578–585, Valencia, Spain. Association for Computational Linguistics.
  * Chakrabarty et al. (2022) Tuhin Chakrabarty, Arkadiy Saakyan, Debanjan Ghosh, and Smaranda Muresan. 2022.  FLUTE: Figurative language understanding through textual explanations.  In _Proceedings of the 2022 Conference on Empirical Methods in Natural Language Processing_ , pages 7139–7159, Abu Dhabi, United Arab Emirates. Association for Computational Linguistics.
  * Choi et al. (2023) Minje Choi, Jiaxin Pei, Sagar Kumar, Chang Shu, and David Jurgens. 2023.  Do LLMs understand social knowledge? evaluating the sociability of large language models with SocKET benchmark.  In _Proceedings of the 2023 Conference on Empirical Methods in Natural Language Processing_ , pages 11370–11403, Singapore. Association for Computational Linguistics.
  * Chung et al. (2022) Hyung Won Chung, Le Hou, Shayne Longpre, Barret Zoph, Yi Tay, William Fedus, Yunxuan Li, Xuezhi Wang, Mostafa Dehghani, Siddhartha Brahma, Albert Webson, Shixiang Shane Gu, Zhuyun Dai, Mirac Suzgun, Xinyun Chen, Aakanksha Chowdhery, Alex Castro-Ros, Marie Pellat, Kevin Robinson, Dasha Valter, Sharan Narang, Gaurav Mishra, Adams Yu, Vincent Zhao, Yanping Huang, Andrew Dai, Hongkun Yu, Slav Petrov, Ed H. Chi, Jeff Dean, Jacob Devlin, Adam Roberts, Denny Zhou, Quoc V. Le, and Jason Wei. 2022.  Scaling instruction-finetuned language models.
  * Clark and Schober (1992) Herbert H. Clark and Michael F. Schober. 1992.  Asking questions and influencing answers.
  * DeLucia et al. (2022) Alexandra DeLucia, Shijie Wu, Aaron Mueller, Carlos Aguirre, Philip Resnik, and Mark Dredze. 2022.  Bernice: A multilingual pre-trained encoder for Twitter.  In _Proceedings of the 2022 Conference on Empirical Methods in Natural Language Processing_ , pages 6191–6205, Abu Dhabi, United Arab Emirates. Association for Computational Linguistics.
  * Flek (2020) Lucie Flek. 2020.  Returning the N to NLP: Towards contextually personalized classification models.  In _Proceedings of the 58th Annual Meeting of the Association for Computational Linguistics_ , pages 7828–7838, Online. Association for Computational Linguistics.
  * Fu et al. (2020) Liye Fu, Susan Fussell, and Cristian Danescu-Niculescu-Mizil. 2020.  Facilitating the communication of politeness through fine-grained paraphrasing.  In _Proceedings of the 2020 Conference on Empirical Methods in Natural Language Processing (EMNLP)_ , pages 5127–5140, Online. Association for Computational Linguistics.
  * Garten et al. (2019) Justin Garten, Brendan Kennedy, Joe Hoover, Kenji Sagae, and Morteza Dehghani. 2019.  Incorporating demographic embeddings into language understanding.  _Cognitive Science_ , 43(1):e12701.
  * Gupta et al. (2023) Himanshu Gupta, Saurabh Arjun Sawant, Swaroop Mishra, Mutsumi Nakamura, Arindam Mitra, Santosh Mashetty, and Chitta Baral. 2023.  Instruction tuned models are quick learners.
  * Halliday (2004) Michael AK Halliday. 2004.  Introduction: How big is a language? on the power of language.  _The language of science_ , 5:19–32.
  * Havaldar et al. (2023) Shreya Havaldar, Bhumika Singhal, Sunny Rai, Langchen Liu, Sharath Chandra Guntuku, and Lyle Ungar. 2023.  Multilingual language models are not multicultural: A case study in emotion.  In _Proceedings of the 13th Workshop on Computational Approaches to Subjectivity, Sentiment, & Social Media Analysis_, pages 202–214, Toronto, Canada. Association for Computational Linguistics.
  * Hayati et al. (2021) Shirley Anugrah Hayati, Dongyeop Kang, and Lyle Ungar. 2021.  Does BERT learn as humans perceive? understanding linguistic styles through lexica.  In _Proceedings of the 2021 Conference on Empirical Methods in Natural Language Processing_ , pages 6323–6331, Online and Punta Cana, Dominican Republic. Association for Computational Linguistics.
  * Heaven (2023) William Heaven. 2023.  Chatgpt is everywhere. here’s where it came from.  _MIT Technology Review_.
  * Hovy and Spruit (2016) Dirk Hovy and Shannon L. Spruit. 2016.  The social impact of natural language processing.  In _Proceedings of the 54th Annual Meeting of the Association for Computational Linguistics (Volume 2: Short Papers)_ , pages 591–598, Berlin, Germany. Association for Computational Linguistics.
  * Hovy and Yang (2021) Dirk Hovy and Diyi Yang. 2021.  The importance of modeling social factors of language: Theory and practice.  In _Proceedings of the 2021 Conference of the North American Chapter of the Association for Computational Linguistics: Human Language Technologies_ , pages 588–602, Online. Association for Computational Linguistics.
  * Hu et al. (2022) Edward J Hu, Yelong Shen, Phillip Wallis, Zeyuan Allen-Zhu, Yuanzhi Li, Shean Wang, Lu Wang, and Weizhu Chen. 2022.  LoRA: Low-rank adaptation of large language models.  In _International Conference on Learning Representations_.
  * Hu (2023) Krystal Hu. 2023.  Chatgpt sets record for fastest-growing user base - analyst note.  _Reuters_.
  * Körner et al. (2021) Erik Körner, Gregor Wiedemann, Ahmad Dawar Hakimi, Gerhard Heyer, and Martin Potthast. 2021.  On classifying whether two texts are on the same side of an argument.  In _Proceedings of the 2021 Conference on Empirical Methods in Natural Language Processing_ , pages 10130–10138, Online and Punta Cana, Dominican Republic. Association for Computational Linguistics.
  * Liu et al. (2019) Yinhan Liu, Myle Ott, Naman Goyal, Jingfei Du, Mandar Joshi, Danqi Chen, Omer Levy, Mike Lewis, Luke Zettlemoyer, and Veselin Stoyanov. 2019.  Roberta: A robustly optimized bert pretraining approach.
  * Longpre et al. (2023) Shayne Longpre, Le Hou, Tu Vu, Albert Webson, Hyung Won Chung, Yi Tay, Denny Zhou, Quoc V Le, Barret Zoph, Jason Wei, et al. 2023.  The flan collection: Designing data and methods for effective instruction tuning.  _arXiv preprint arXiv:2301.13688_.
  * Loshchilov and Hutter (2019) Ilya Loshchilov and Frank Hutter. 2019.  Decoupled weight decay regularization.
  * Lynn et al. (2017) Veronica Lynn, Youngseo Son, Vivek Kulkarni, Niranjan Balasubramanian, and H. Andrew Schwartz. 2017.  Human centered NLP with user-factor adaptation.  In _Proceedings of the 2017 Conference on Empirical Methods in Natural Language Processing_ , pages 1146–1155, Copenhagen, Denmark. Association for Computational Linguistics.
  * Matero et al. (2019) Matthew Matero, Akash Idnani, Youngseo Son, Salvatore Giorgi, Huy Vu, Mohammad Zamani, Parth Limbachiya, Sharath Chandra Guntuku, and H Andrew Schwartz. 2019.  Suicide risk assessment with multi-level dual-context language and bert.  In _Proceedings of the sixth workshop on computational linguistics and clinical psychology_ , pages 39–44.
  * Matero et al. (2021) Matthew Matero, Nikita Soni, Niranjan Balasubramanian, and H Andrew Schwartz. 2021.  Melt: Message-level transformer with masked document representations as pre-training for stance detection.  _arXiv preprint arXiv:2109.08113_.
  * Meaney et al. (2021) J. A. Meaney, Steven Wilson, Luis Chiruzzo, Adam Lopez, and Walid Magdy. 2021.  SemEval 2021 task 7: HaHackathon, detecting and rating humor and offense.  In _Proceedings of the 15th International Workshop on Semantic Evaluation (SemEval-2021)_ , pages 105–119, Online. Association for Computational Linguistics.
  * Mishra et al. (2022) Swaroop Mishra, Daniel Khashabi, Chitta Baral, and Hannaneh Hajishirzi. 2022.  Cross-task generalization via natural language crowdsourcing instructions.  In _Proceedings of the 60th Annual Meeting of the Association for Computational Linguistics (Volume 1: Long Papers)_ , pages 3470–3487, Dublin, Ireland. Association for Computational Linguistics.
  * Mohammad et al. (2018) Saif Mohammad, Felipe Bravo-Marquez, Mohammad Salameh, and Svetlana Kiritchenko. 2018.  SemEval-2018 task 1: Affect in tweets.  In _Proceedings of the 12th International Workshop on Semantic Evaluation_ , pages 1–17, New Orleans, Louisiana. Association for Computational Linguistics.
  * Nguyen et al. (2020) Dat Quoc Nguyen, Thanh Vu, and Anh Tuan Nguyen. 2020.  BERTweet: A pre-trained language model for English tweets.  In _Proceedings of the 2020 Conference on Empirical Methods in Natural Language Processing: System Demonstrations_ , pages 9–14, Online. Association for Computational Linguistics.
  * OpenAI (2023) OpenAI. 2023.  Gpt-4 technical report.
  * Ouyang et al. (2022) Long Ouyang, Jeffrey Wu, Xu Jiang, Diogo Almeida, Carroll Wainwright, Pamela Mishkin, Chong Zhang, Sandhini Agarwal, Katarina Slama, Alex Ray, et al. 2022.  Training language models to follow instructions with human feedback.  _Advances in Neural Information Processing Systems_ , 35:27730–27744.
  * Pei and Jurgens (2020) Jiaxin Pei and David Jurgens. 2020.  Quantifying intimacy in language.  In _Proceedings of the 2020 Conference on Empirical Methods in Natural Language Processing (EMNLP)_ , pages 5307–5326, Online. Association for Computational Linguistics.
  * Preoţiuc-Pietro et al. (2019) Daniel Preoţiuc-Pietro, Mihaela Gaman, and Nikolaos Aletras. 2019.  Automatically identifying complaints in social media.  In _Proceedings of the 57th Annual Meeting of the Association for Computational Linguistics_ , pages 5008–5019, Florence, Italy. Association for Computational Linguistics.
  * Pryzant et al. (2020) Reid Pryzant, Richard Diehl Martinez, Nathan Dass, Sadao Kurohashi, Dan Jurafsky, and Diyi Yang. 2020.  Automatically neutralizing subjective bias in text.  _arXiv preprint arXiv:1911.09709_.
  * Rajbhandari et al. (2020) Samyam Rajbhandari, Jeff Rasley, Olatunji Ruwase, and Yuxiong He. 2020.  Zero: Memory optimizations toward training trillion parameter models.
  * Rosenthal et al. (2017) Sara Rosenthal, Noura Farra, and Preslav Nakov. 2017.  SemEval-2017 task 4: Sentiment analysis in Twitter.  In _Proceedings of the 11th International Workshop on Semantic Evaluation (SemEval-2017)_ , pages 502–518, Vancouver, Canada. Association for Computational Linguistics.
  * Ruan et al. (2016) Xianzhi Ruan, Steven Wilson, and Rada Mihalcea. 2016.  Finding optimists and pessimists on Twitter.  In _Proceedings of the 54th Annual Meeting of the Association for Computational Linguistics (Volume 2: Short Papers)_ , pages 320–325, Berlin, Germany. Association for Computational Linguistics.
  * Sanh et al. (2022) Victor Sanh, Albert Webson, Colin Raffel, Stephen Bach, Lintang Sutawika, Zaid Alyafeai, Antoine Chaffin, Arnaud Stiegler, Arun Raja, Manan Dey, M Saiful Bari, Canwen Xu, Urmish Thakker, Shanya Sharma Sharma, Eliza Szczechla, Taewoon Kim, Gunjan Chhablani, Nihal Nayak, Debajyoti Datta, Jonathan Chang, Mike Tian-Jian Jiang, Han Wang, Matteo Manica, Sheng Shen, Zheng Xin Yong, Harshit Pandey, Rachel Bawden, Thomas Wang, Trishala Neeraj, Jos Rozen, Abheesht Sharma, Andrea Santilli, Thibault Fevry, Jason Alan Fries, Ryan Teehan, Teven Le Scao, Stella Biderman, Leo Gao, Thomas Wolf, and Alexander M Rush. 2022.  Multitask prompted training enables zero-shot task generalization.  In _International Conference on Learning Representations_.
  * Sap et al. (2020) Maarten Sap, Saadia Gabriel, Lianhui Qin, Dan Jurafsky, Noah A. Smith, and Yejin Choi. 2020.  Social bias frames: Reasoning about social and power implications of language.  In _Proceedings of the 58th Annual Meeting of the Association for Computational Linguistics_ , pages 5477–5490, Online. Association for Computational Linguistics.
  * Sharma et al. (2020) Ashish Sharma, Adam Miner, David Atkins, and Tim Althoff. 2020.  A computational approach to understanding empathy expressed in text-based mental health support.  In _Proceedings of the 2020 Conference on Empirical Methods in Natural Language Processing (EMNLP)_ , pages 5263–5276, Online. Association for Computational Linguistics.
  * Socher et al. (2013) Richard Socher, Alex Perelygin, Jean Wu, Jason Chuang, Christopher D. Manning, Andrew Ng, and Christopher Potts. 2013.  Recursive deep models for semantic compositionality over a sentiment treebank.  In _Proceedings of the 2013 Conference on Empirical Methods in Natural Language Processing_ , pages 1631–1642, Seattle, Washington, USA. Association for Computational Linguistics.
  * Touvron et al. (2023) Hugo Touvron, Louis Martin, Kevin Stone, Peter Albert, Amjad Almahairi, Yasmine Babaei, Nikolay Bashlykov, Soumya Batra, Prajjwal Bhargava, Shruti Bhosale, et al. 2023.  Llama 2: Open foundation and fine-tuned chat models.  _arXiv preprint arXiv:2307.09288_.
  * V Ganesan et al. (2023) Adithya V Ganesan, Yash Kumar Lal, August Nilsson, and H. Schwartz. 2023.  Systematic evaluation of GPT-3 for zero-shot personality estimation.  In _Proceedings of the 13th Workshop on Computational Approaches to Subjectivity, Sentiment, & Social Media Analysis_, pages 390–400, Toronto, Canada. Association for Computational Linguistics.
  * V Ganesan et al. (2021) Adithya V Ganesan, Matthew Matero, Aravind Reddy Ravula, Huy Vu, and H. Andrew Schwartz. 2021.  Empirical evaluation of pre-trained transformers for human-level NLP: The role of sample size and dimensionality.  In _Proceedings of the 2021 Conference of the North American Chapter of the Association for Computational Linguistics: Human Language Technologies_ , pages 4515–4532, Online. Association for Computational Linguistics.
  * Van Hee et al. (2018) Cynthia Van Hee, Els Lefever, and Véronique Hoste. 2018.  SemEval-2018 task 3: Irony detection in English tweets.  In _Proceedings of the 12th International Workshop on Semantic Evaluation_ , pages 39–50, New Orleans, Louisiana. Association for Computational Linguistics.
  * Varadarajan et al. (2022) Vasudha Varadarajan, Nikita Soni, Weixi Wang, Christian Luhmann, H. Andrew Schwartz, and Naoya Inoue. 2022.  Detecting dissonant stance in social media: The role of topic exposure.  In _Proceedings of the Fifth Workshop on Natural Language Processing and Computational Social Science (NLP+CSS)_ , pages 151–156, Abu Dhabi, UAE. Association for Computational Linguistics.
  * Wang et al. (2023) Yizhong Wang, Yeganeh Kordi, Swaroop Mishra, Alisa Liu, Noah A. Smith, Daniel Khashabi, and Hannaneh Hajishirzi. 2023.  Self-instruct: Aligning language models with self-generated instructions.  In _Proceedings of the 61st Annual Meeting of the Association for Computational Linguistics (Volume 1: Long Papers)_ , pages 13484–13508, Toronto, Canada. Association for Computational Linguistics.
  * Wang et al. (2022) Yizhong Wang, Swaroop Mishra, Pegah Alipoormolabashi, Yeganeh Kordi, Amirreza Mirzaei, Atharva Naik, Arjun Ashok, Arut Selvan Dhanasekaran, Anjana Arunkumar, David Stap, Eshaan Pathak, Giannis Karamanolakis, Haizhi Lai, Ishan Purohit, Ishani Mondal, Jacob Anderson, Kirby Kuznia, Krima Doshi, Kuntal Kumar Pal, Maitreya Patel, Mehrad Moradshahi, Mihir Parmar, Mirali Purohit, Neeraj Varshney, Phani Rohitha Kaza, Pulkit Verma, Ravsehaj Singh Puri, Rushang Karia, Savan Doshi, Shailaja Keyur Sampat, Siddhartha Mishra, Sujan Reddy A, Sumanta Patro, Tanay Dixit, and Xudong Shen. 2022.  Super-NaturalInstructions: Generalization via declarative instructions on 1600+ NLP tasks.  In _Proceedings of the 2022 Conference on Empirical Methods in Natural Language Processing_ , pages 5085–5109, Abu Dhabi, United Arab Emirates. Association for Computational Linguistics.
  * Wei et al. (2022a) Jason Wei, Maarten Bosma, Vincent Y. Zhao, Kelvin Guu, Adams Wei Yu, Brian Lester, Nan Du, Andrew M. Dai, and Quoc V. Le. 2022a.  Finetuned language models are zero-shot learners.
  * Wei et al. (2022b) Jason Wei, Yi Tay, Rishi Bommasani, Colin Raffel, Barret Zoph, Sebastian Borgeaud, Dani Yogatama, Maarten Bosma, Denny Zhou, Donald Metzler, Ed H. Chi, Tatsunori Hashimoto, Oriol Vinyals, Percy Liang, Jeff Dean, and William Fedus. 2022b.  Emergent abilities of large language models.  _Transactions on Machine Learning Research_.  Survey Certification.
  * Wittgenstein (1953) Ludwig Wittgenstein. 1953.  _Philosophical investigations_.
  * Wolf et al. (2020) Thomas Wolf, Lysandre Debut, Victor Sanh, Julien Chaumond, Clement Delangue, Anthony Moi, Pierric Cistac, Tim Rault, Remi Louf, Morgan Funtowicz, Joe Davison, Sam Shleifer, Patrick von Platen, Clara Ma, Yacine Jernite, Julien Plu, Canwen Xu, Teven Le Scao, Sylvain Gugger, Mariama Drame, Quentin Lhoest, and Alexander Rush. 2020.  Transformers: State-of-the-art natural language processing.  In _Proceedings of the 2020 Conference on Empirical Methods in Natural Language Processing: System Demonstrations_ , pages 38–45, Online. Association for Computational Linguistics.
  * Zhang and Wan (2022) Yunxiang Zhang and Xiaojun Wan. 2022.  MOVER: Mask, over-generate and rank for hyperbole generation.  In _Proceedings of the 2022 Conference of the North American Chapter of the Association for Computational Linguistics: Human Language Technologies_ , pages 6018–6030, Seattle, United States. Association for Computational Linguistics.
  * Ziems et al. (2023) Caleb Ziems, William Held, Omar Shaikh, Jiaao Chen, Zhehao Zhang, and Diyi Yang. 2023.  Can Large Language Models Transform Computational Social Science? _Computational Linguistics_ , pages 1–53.

##  Appendix A Appendix

We first provide statistics about SocialiteInstructions which has been used to
instruction tune Socialite-llama. Table 3 includes information about amount of
data used per task as well as the number of labels used.

Next, we provide descriptions of the instructions and prompt templates used
for each task in SocialiteInstructions. Please note that this may contain
examples of potentially dangerous and harmful text.

Dataset | Train Set | Validation Set | Test Set | Num_classes
---|---|---|---|---
Emotion | 3257 | 374 | 1421 | 4
FLUTE | 6780 | 754 | 1498 | 4
EmpathyExplorations | 2220 | 247 | 617 | 3
Humor | 8000 | 1000 | 1000 | 2
Offensive | 8000 | 4666 | 4691 | 2
Sexist | 7999 | 4666 | 4691 | 2
IntentToOffend | 7999 | 4666 | 4691 | 2
BiasedImplication | 7999 | 4666 | 4691 | 2
PolitenessHayati | 256 | 32 | 32 | 2
Hyperbole | 2580 | 323 | 323 | 2
SameSideStance | 140 | 18 | 17 | 2
Sentiment | 8000 | 2000 | 12284 | 3
Intimacy | 1797 | 225 | 225 | 6
SubjectiveBias | 8000 | 9379 | 9379 | 2
ValenceCLS | 9002 | 510 | 550 | 2
ArousalCLS | 9002 | 510 | 550 | 2
DominanceCLS | 9002 | 510 | 550 | 2
EmpathySelfRated | 1487 | 186 | 186 | 2
DistressSelfRated | 1487 | 186 | 186 | 2
HumourRating | 4932 | 632 | 615 | 2
HateSpeech | - | - | 2970 | 2
Irony | - | - | 784 | 2
PolitenessStanford | - | - | 567 | 2
Optimism | - | - | 1495 | 3
Complaints | - | - | 345 | 2
AgreeDisagree | - | - | 4760 | 3
Table 3: Training, validation and test set statistics SocialiteInstructions. ‘-’ denotes that the dataset was not used in training Socialite-llama but we create instructions for its test set. Overall, SocialiteInstructions contains ~202k data points. Dataset |  Instruction Format Example
---|---
Humor |  Instruction: Upon receiving a piece of text, your task is to analyze and determine whether the intention of the text was to be humorous. You are instructed to look at the text and identify the structure of the joke, e.g. setup and punchline, or the content of the joke, e.g. absurdity, in order to determine if the intention of the text was to be humorous. If you think the intention of the text was to be humorous, classify it as ‘humorous’, else classfy it as ‘non-humorous’. Input: TENNESSEE: We’re the best state. Nobody even comes close. *Elevennessee walks into the room* TENNESSEE: Oh shit… Output: humorous
HumourRating |  Instruction: Upon receiving a piece of text, your task is to assess its comedic quality and categorize it as either ‘low humor’ or ‘high humor’. Input: How many men does it take to open a can of beer? None. It should be open by the time she brings it to the couch. Output: low humor
Table 4: Instruction prompts and output examples for humour task  Dataset |  Instruction Format Example
---|---
Sentiment |  Instruction: Evaluate the sentiment conveyed in the input text and determine whether it is positive, negative, or neutral. This sentiment assessment should encompass the overall sentiment of the event described within the context of the topic mentioned in the text. Your options for classification are confined to positive, negative or neutral. Input: Few more hours to iPhone 6s launch and im still using the 4th generation Output: positive
Emotion |  Instruction: Analyze the following sentence and determine the predominant emotion it displays. Your options for classification are confined to anger, joy, optimism, or sadness. Please select one emotion from the given alternatives that you believe best epitomizes the emotional context of the sentence. Input: Worry is a down payment on a problem you may never have’. Joyce Meyer. #motivation #leadership #worry Output: Optimism
ValenceCLS |  Instruction: Analyze the provided text using the Valence-Arousal-Dominance model for emotional assessment. Your task is to classify the valence level it would likely elicit in an average reader, where ‘Low Valence’ indicates a low level of pleasant feelings and ‘High Valence’ indicates a high level of pleasant feelings. Remember, the valence scale is used to measure the degree of pleasure or displeasure a person may feel towards something. Your options for classification are confined to ‘Low Valence’ or ‘High Valence’. Input: Remember what she said in my last letter? Output: High Valence
ArousalCLS |  Instruction: Analyze the provided text using the Valence-Arousal-Dominance (VAD) emotional model. Your task is to classify the arousal level it might trigger in an average reader. Arousal, in this context, refers to the degree of energy or lethargy the text might induce. ’Low Arousal’ indicates a low arousal level, suggesting the text is likely to make the reader feel calm or lethargic. Conversely, ’High Arousal’ indicates a high arousal level, suggesting the tweet is likely to energize or excite the reader. Your options for classification are confined to ‘Low Arousal’ or ‘High Arousal’. Input: Remember what she said in my last letter? Output: High Arousal
DominanceCLS |  Instruction: Please analyze the provided text using the Valence-Arousal-Dominance (VAD) model for emotional response. Specifically, we’re interested in the Dominance aspect of this model. This involves assessing the level of control or dominance the text might make an average reader feel, versus feelings of being controlled or submissive.\nPlease classify this dominance level as ‘Low Dominance’ or ‘High dominance’. ‘Low Dominance’ indicates that the text is likely to evoke a low level of dominance or control in the reader, making them feel more submissive or controlled. Conversely, ‘High Dominance’ suggests that the text would make the reader feel highly dominant or in control.\nYour options for classification are confined to ‘Low Dominance’ or ‘High Dominance’. Input: Remember what she said in my last letter? Output: High Dominance
SameSideStance |  Instruction: You are provided with two pieces of text sourced from an online debate forum. Your task is to analyze and categorize these texts based on their argumentative stance. Determine whether both texts are arguing in favor of the same viewpoint or if they are opposing each other. Your options for classification are confined to ‘same side’ or ‘not same side’. Input: Legalizing gay marriage will not destroy man/woman relationships. [SEP] Why should gay marriage destroy man/woman relationships? It would just give gays the option to marry. Output: same side
Table 5: Instruction prompts and output examples for sentiment and emotion task  Dataset |  Instruction Format Example
---|---
SubjectiveBias |  Instruction: Given two pieces of text, your objective is to detect subjective bias, which manifests when language that should remain neutral and impartial is influenced by feelings, opinions, or personal preferences, whether intentionally or unintentionally. If you find bias in the first sentence, indicate ‘first sentence’ as the output; otherwise, specify ‘second sentence’. Input: the term finds widespread usage among members of the educational establishment who see students as tools of social change. [SEP] the term finds widespread usage among members of the educational establishment who see students as agents of social change. Output: first sentence
Hyperbole |  Instruction: Upon receiving a piece of text, your task is to analyze and determine whether it contains hyperbolic language, which is an exaggerated statement or claim not meant to be taken literally, or if it does not. Your options for classification are confined to ‘hyperbole’ or ‘not hyperbole’. Input: He looks great and yet he must be pushing sixty by now. Output: not hyperbole
Table 6: Instruction prompts and output examples for trustworthiness task  Dataset |  Instruction Format Example
---|---
EmpathyExplorations |  Instruction: Evaluate the degree of inquiry exhibited in the counselor’s response provided below, categorizing it as either “Strong Exploration", “Weak Exploration" or “No Exploration". We define ‘exploration’ as instances where a mental health counselor displays keen interest in a patient by asking about experiences that haven’t been explicitly mentioned. Input: Patient: I’m finally on medication and found a therapist who makes me feel hopeful for the first time in years. It has been a while since I’ve felt sincere hope. Counselor’s response: im envious of the fact that you want hope. Output: No Exploration
EmpathySelfRated |  Instruction: Please carefully peruse the subsequent text, which is a personal account penned by an individual expressing their emotions and reflections after reading a news article. This account is directed towards their friends. After reading, your task is to accurately classify the level of empathetic concern demonstrated by the author. Your options for classification are ‘low empathy’ which indicates low empathetic concern or ‘high empathy’ which signifies a high degree of empathetic concern. Input: This sounds like a horrible accident. I can’t even imagine what that family is going through. The ones that were involved in the accident but survived probably saw some horrible things, and those kids who witnessed their mother die. As a parent myself that’s a terrifying thought I would never want my son to go through something like that. Output: high empathy
DistressSelfRated |  Instruction: Please carefully peruse the subsequent text, which is a personal account written by an individual to their friends. This account details their emotional reactions and cognitive responses upon reading a specific news article. Your task is to accurately classify the level of personal distress experienced by the author. Your options for classification are ‘low distress’ or ‘high distress’. Input: This sounds like a horrible accident. I can’t even imagine what that family is going through. The ones that were involved in the accident but survived probably saw some horrible things, and those kids who witnessed their mother die. As a parent myself that’s a terrifying thought I would never want my son to go through something like that. Output: low distress
FLUTE |  Instruction: Please follow these steps: 1\. First, you’ll be presented with a premise and a hypothesis in the input section. 2\. Your task is to determine and categorize the type of figurative language utilized in the hypothesis. 3\. Finally, based on your assessment, respond with a single answer that most accurately represents the figurative language detected in the hypothesis. Choose from these four classifications: Idiom, Metaphor, Sarcasm, or Simile. Input: Premise: he was flying high happy and joy. Hypothesis: he was feeling down hurt and sad. Output: Metaphor
Table 7: Instruction prompts and output examples for other social factors  Dataset |  Instruction Format Example
---|---
PolitenessHayati |  Instruction: Upon receiving a piece of text, your task is to analyze and determine whether the language used within it is courteous and respectful, indicating politeness, or if it contains disrespectful or rude elements, indicating impoliteness. Your options for classification are confined to ‘impolite’ or ‘polite’. Input: @selfiequeenbri: cause I’m tired of you big bitches coming for us skinny girls!! Output: impolite
Intimacy |  Instruction: Assess the degree of intimacy expressed in the input text, taking into account the social context within the text. Your options for classification are confined to ‘very intimate‘, ‘intimate’, ‘somewhat intimate’, ‘not very intimate’, ‘not intimate’ or ‘not intimate at all’. Input: What are the most mediocre animals in the animal kingdom? Output: not intimate
Table 8: Instruction prompts and output examples for other social factors  Dataset |  Instruction Format Example
---|---
Offensive |  Instruction: Evaluate the given text for any offensive content, which includes rudeness, disrespect, or toxicity. This assessment should consider if the text could potentially offend anyone, based on previous studies indicating a higher recall rate. Identify any disrespectful, inappropriate, or harmful language, phrases, or sentiments. If these elements exist, label the text as ‘offensive’. If these elements are absent, mark the text as ‘not offensive’. Input: jack and jill went up a hill . <repeat> jill came down , i went up , and blew jack . ooohh ! <allcaps> it do n’t rhyme . it just happened that way. Output: not offensive
Sexist |  Instruction: Analyze the provided sentence and evaluate if it contains any elements that could be considered as gender-based discrimination. You are required to categorize the sentence into one of two classifications: ‘sexism’ if it exhibits gender discrimination, or ‘not sexism’ if it does not. Input: "Alternative Theory: They had to change the timeline to explain why the building engineer was up there–had he heard shots, he would not have gone to 32nd floor– in order to have #StephenSchuck corroborate #JesusCampos story, it had to have been before shooting started. Output: not sexism
IntentToOffend |  Instruction: Your task is to analyze the given text and identify if it intentionally promotes social biases and stereotypes. If you perceive that the author’s motive was to deliberately offend, categorize the text as ‘intentional’. If there’s no apparent intent to offend, classify it as ‘not intentional’. Remember, the focus is on the author’s intent to offend, not just the offensiveness of the content. Input: should i start discourse with eileen and hope it makes them stop telling my friends im a pedophile cus i’m fucking not? Output: not intentional
BiasedImplication |  Instruction: Your task is to scrutinize a given text and identify any signs of prejudice. You need to differentiate between statements that merely insult an individual and those that imply group discrimination, such as comparing ‘F*ck you’ (an individual insult) with ‘F*ck you, f*ggot’ (a group implication). Based on your analysis, categorize the text as either ‘biased’ or ‘not biased’. Input: Obviously everyone you disagree with on twitter is a manipulation bot & everyone you agree with is real, right? :p Output: biased
Table 9: Instruction prompts and output examples for offensive task  Dataset |  Instruction Format Example
---|---
HateSpeech |  Instruction: Analyze the given text for hateful contents against women or immigrants. Your options for classification are confined to ‘hate speech’ or ‘not hate speech’. Input: Send home migrants not in need of protection, Peter Dutton tells UN, HEY DUTTON HOW ABOUT THE ONES THAT HAVE STAYED AND NOT LEFT THE COUNTRY WHEN THEY SHOULD OVERSTAYERS ? WHY DONT YOU GO AND ROUND ALL THEM UP ? Output: hate speech
Irony |  Instruction: Classify the input sentence as ironic or not ironic. Here ironic can be defined as the use of words or situations to convey a meaning that is opposite to what is expected. Your options for classification are confined to ‘ironic’ or ‘not ironic’. Input: Now I remember why I buy books online @user #servicewithasmile Output: ironic
PolitenessStanford |  Instruction: Analyze the provided text, considering its tone and language, and categorize it as either polite or impolite. Your options for classification are confined to ’impolite’ or ’polite’. Input: Hi, I made on comment on the image page about this being different than Sbarro3 image. Please… comment? Output: polite
Optimism |  Instruction: Analyze the sentiment of the provided text and classify it as ‘optimistic’, ‘pessimistic’ or ‘neutral’. Input: fuck you fuck you fuck you Output: pessimistic
Complaints |  Instruction: Given an input text, identify if it contains a complaint or not. Complaining is a basic speech act used to express a negative mismatch between reality and expectations towards a state of affairs, product, organization or event. Your options for classification are confined to ‘complaint’ or ‘not complaint’. Input: @SamsungSupport Can someone please help me? I’ve already sent a DM. Output: complaint
AgreeDisagree |  Instruction: You are provided with two pieces of text and your task is to analyze and categorize these texts based on their argumentative stance. Determine whether both texts are arguing in favor of the same viewpoint, if they are opposing each other or if they are talking about two different topics. Your options for classification are confined to ‘agree’, ‘disagree’ or ‘N/A’. Input: President Trump also intentionally lied to hide the truth from the American people and make himself look better. [SEP] President Trump told public lies every single day for his first 40 days. Output: agree
Table 10: Instruction prompts and output examples for related social tasks
tasks


Binomial Coefficients in a Row of Pascal’s Triangle from Extension of Power of
Eleven: Newton’s Unfinished Work
Md. Shariful Islam1 Md. Robiul Islam3 Md. Shorif Hossan2 and Md. Hasan Kibria1

1Department of Mathematics, University of Dhaka, Bangladesh

2Department of Applied Mathematics, University of Dhaka, Bangladesh

3Department of Computer Science and Engineering, Green University of
Bangladesh, Dhaka, Bangladesh

E-mail<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>and
<EMAIL_ADDRESS>
$\ast$Author for correspondence<EMAIL_ADDRESS>
###### Abstract

The aim of this paper is to find a general formula to generate any row of
Pascal’s triangle as an extension of the concept of $\left(11\right)^{n}$. In
this study, the visualization of each row of Pascal’s triangle has been
presented by extending the concept of the power of 11 to the power of 101,
1001, 10001, and so on. We briefly discuss how our proposed concept works for
any $n$ by inserting an appropriate number of zeros between $1$ and $1$
(eleven), that is the concept of $\left(11\right)^{n}$ has been extended to
$\left(1\Theta 1\right)^{n}$, where $\Theta$ represents the number of zeros.
We have proposed a formula for obtaining the value of $\Theta$. The proposed
concept has been verified with Pascal’s triangle and matched successfully.
Finally, Pascal’s triangle for a large n has been presented considering the
$51^{\text{st}}$ row as an example.

Keywords: Binomial Coefficients, Pascal’s triangle, Logarithm, Modular
Arithematic.

## I. Introduction

Algebra is a spacious part of the science of mathematics which provides the
opportunity to express mathematical ideas precisely. In algebra, the Binomial
expansion and Pascal’s triangle are considered important. Pascal’s triangle is
an arrangement of the binomial coefficients and one of the most known integer
models. Though it was named after French scientist Blaise Pascal, it was
studied in ancient India, Persia, China, Germany, and Italy [1].
In reality, the definition of the triangle was made centuries ago. In 450 BC,
an Indian mathematician named Pingala is said to have introduced the
definition of this triangle in a Sanskrit poetry book. At the same time, the
commentators of this book acquainted with the diagonal surface of the
triangle, which is the sum of the Fibonacci numbers. Chinese mathematicians
had the same idea and named the triangle as “Yang Hui’s triangle”. Later,
Persian mathematician Al-Karaji and Persian astronomer-poet Omar Khayyam named
the triangle as the “Khayyam triangle”. It also has multi-dimensional shapes,
the three-dimensional shape is referred to as Pascal’s pyramid or Pascal’s
tetrahedron, while the other general-shaped ones are called Pascal’s
simplifications.
Various studies have been conducted in many different disciplines about
Pascal’s triangle. For the construction of Pascal’s triangle, Sgroi [2] stated
that each line starts with 1 and ends with 1, and this series can be expanded
with simple cross-joints. Jansson [3] developed three geometric forms related
to Pascal’s triangle and included examples on each form. Toschi [4] used
various permutations to generate new forms of Pascal’s triangles and
generalized them. Duncan and Litwiller [5] addressed the reconstruction of
Pascal’s triangle with the individuals. Here they collected data on the
opinions of individuals using qualitative methods, and determined the methods
of constructing the Pascal’s triangle in different ways with the attained
findings.
Researches worked on Pascal’s fascinating characteristics. Using the principle
of permutation, Putz [6] designed the Pascal Polytope and linked it to the
Fibonacci concept. Houghton [7] gave the concept of the relationship between
successive differential operation of a function and Pascal’s triangle. With an
application, he attempted to incorporate the idea of a differentiable function
into Pascal’s triangle. The relationship between Pascal’s triangle and the
Tower of Hanoi has been elucidated by Andreas M Hinz [8]. Finding diagonal sum
[9], k-Fibonacci sequence [10], recurrence relations [11], finding exponential
$(e)$ [12] were a part of those to describe the work that generates from the
Pascal’s triangle. Some fascinating properties of Pascal’s triangle are
available in [13, 14]. In 1956, Freund [15] elicited that the generalized
Pascal’s triangles of $s^{\text{th}}$ order can be constructed from the
generalized binomial coefficients of order $s$. Bankier [16] gave the Freud’s
alternative proof. Kallós generalized Pascal’s triangle from algebraic point
of view by different bases. He tried to generalize Pascal’s triangle using the
power of integers [17], powers of base numbers [18] and their connections with
prime number [19]. kuhlmann tried to generate Pascal’s triangle using the
T-triangle concept [20].
The concept of the power of $11$ was first introduced by Sir Isaac Newton. He
noticed that first five rows of Pascal’s triangle are formed by the power of
$11$ and claimed (without proof) that subsequent rows can also be generated by
the power of eleven as well [21]. Arnold et al. [22] showed if one assigns a
place value to each of the individual terms in a certain row of the triangle,
the pattern can be seen again. Morton [23] noted the Pascal’s triangle
property by the power of 11 for 10 base numerals system. Mueller [24] noted
that one can get the $n^{\text{th}}$ power of $11$ from the $n^{\text{th}}$
row of the Pascal’s triangle with positional addition.
It is clearly concluded that above mentioned works did not express the full
row of Pascal’s triangle from the power of $11$, or some variant of it, as Sir
Isaac Newton hinted . This paper has worked on the generalization of Pascal’s
triangle by extending the power of eleven idea. Here, we have extend the
concept of power of $11$ to the power of $101,1001,10001,\ldots$ and proved a
general formula to achieve any row of Pascal’s triangle. Using our formula one
can generate any row of Pascal’s triangle, regardless of the number of rows
one can imagine.

## II. Methods

The very basic definition to get any element of a row of the Pascal’s triangle
is the summation of two adjacent elements of the previous row. Each number in
Pascal’s triangle is the sum of two numbers above that number. Usually, the
lines of Pascal’s triangle are numbered starting from $n=0$ on the top and the
numbers in each line are starting from $k=0$ on the left. For $k=0$, their is
only one value 1. As the next lines are created, The remaining right most and
left most element for new row is 1.

$\begin{array}[]{ccccccccccccc}{}\hfil&{}\hfil&{}\hfil&{}\hfil&{}\hfil&{}\hfil&{1}&{}\hfil&{}\hfil&{}\hfil&{}\hfil&{}\hfil&{}\hfil\\\
{}\hfil&{}\hfil&{}\hfil&{}\hfil&{}\hfil&{1}&{}\hfil&{1}&{}\hfil&{}\hfil&{}\hfil&{}\hfil&{}\hfil\\\
{}\hfil&{}\hfil&{}\hfil&{}\hfil&{1}&{}\hfil&{2}&{}\hfil&{1}&{}\hfil&{}\hfil&{}\hfil&{}\hfil\\\
{}\hfil&{}\hfil&{}\hfil&{1}&{}\hfil&{3}&{}\hfil&{3}&{}\hfil&{1}&{}\hfil&{}\hfil&{}\hfil\\\
{}\hfil&{}\hfil&{1}&{}\hfil&{4}&{}\hfil&{6}&{}\hfil&{4}&{}\hfil&{1}&{}\hfil&{}\hfil\\\
{}\hfil&{1}&{}\hfil&{5}&{}\hfil&{10}&{}\hfil&{10}&{}\hfil&{5}&{}\hfil&{1}&{}\hfil\\\
{1}&{}\hfil&{6}&{}\hfil&{15}&{}\hfil&{20}&{}\hfil&{15}&{}\hfil&{6}&{}\hfil&{1}\\\
{}\hfil&{}\hfil&{\ldots}&{}\hfil&{}\hfil&{}\hfil&{\ldots}&{}\hfil&{}\hfil&{}\hfil&{\ldots}&{}\hfil&{}\hfil\end{array}$

Figure 1: Pascal’s triangle

The concept of the power of $11$ leads to us $11^{1}=11$, $1^{\text{st}}$ row
of Pascal’s triangle and so $11^{2}=121$, $11^{3}=1331$ and $11^{4}=14641$
reveal $2^{\text{nd}}$, $3^{\text{rd}}$ and $4^{\text{th}}$ row respectively.
Before finding the general rule for subsequent rows, we first elaborate the
previous concept of power 11. The reason behind getting Pascal’s triangle by
the power of 11 lies on the general rule of multiplication. What do we get
from multiplication of a number by $11?$

$\begin{array}[]{@{}r@{}}2^{nd}\text{ row of Pascal's triangle}\rightarrow
121\\\ {}\times 11\\\ \hline\cr
1${\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}2}$1\\\
\text{left shift of all digits by 1 place}\rightarrow
12${\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}1}$0\\\
\hline\cr 3^{rd}\text{ row of Pascal's triangle}\rightarrow 13$\textul{3}$1\\\
\end{array}$

Figure 2: Results after multiplication by $11$

Figure 2 shows that multiplication of a number by $11$ gives an output which
is the sum of the two adjacent numbers of previous row of Pascal’s triangle.

Patently $11^{5}=161051$ and $11^{6}=1771561$, but the $5^{\text{th}}$ and
$6^{\text{th}}$ row of Pascal’s triangle are

1 5 10 10 5 1

and

1 6 15 20 15 6 1

respectively. The above scheme fails for $11^{5}$ or $11^{6}$. Why are we not
getting the $5^{th}$ row or why does the power of $11$ fail here? The answer
is the middle values of the $5^{\text{th}}$ row of Pascal’s triangle are two
digit numbers, whereas the power of $11$ represents Pascal’s row as a
representation of one decimal place. So for finding $5^{\text{th}}$ or any row
onward, we need a formula that can represent the number generated from the
power of 11 have two or more digits. Now, we will endeavor to formulate a
specific rule that generates the required number of digits for the
representation of a row of the Pascal’s triangle.
At first, we attempt to generate the number of two digits using the very basic
rules of multiplication. Figure 3, displays the impact of multiplication by
$101.$

$\begin{array}[]{@{}r@{}}101\\\ {}\times 101\\\
\hline\cr${\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}1}$01\\\
\text{zeros cause the left shift of all digits by 1
place}\rightarrow$\leavevmode\nobreak\
{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}00}$00\\\
\text{left shift of all digits by 2 places}\rightarrow
1${\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}01}$00\\\
\hline\cr 1$\textul{02}$01\\\ {}\times 101\\\
\hline\cr${\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}1}{\color[rgb]{0,1,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,1,0}02}$01\\\
${\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}00}{\color[rgb]{0,1,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,1,0}00}$00\\\
\text{left shift of all digits by 2 places}\rightarrow
1${\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}02}{\color[rgb]{0,1,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,1,0}01}$00\\\
\hline\cr 1$\textul{03}\textul{03}$01\\\ \end{array}$

Figure 3: Results after multiplication by $101$
The underlined numbers are same as the summation of two adjacent numbers of
the previous row, but multiplication by 101 displays the rows as a
representation of two digits number.
Now, $101^{5}=10510100501$, from which we can construct $5^{\text{th}}$ row of
Pascal’s triangle by omitting extra zeros and separating the digits.

1 5 10 10 5 1

Similarly from $101^{6}=1061520150601$ and $101^{7}=107213535210701$, we can
easily construct $6^{\text{th}}$ and $7^{\text{th}}$ row respectively.

1 06 15 20 15 06 01

and

1 07 21 35 35 21 07 01

$101^{5}$, $101^{6}$ and $101^{7}$ all are representing $5^{th}$, $6^{th}$ and
$7^{th}$ row of Pascal’s triangle respectively as a representation of two
digit numbers due to the insertion of one zero between $1$ and
$1\leavevmode\nobreak\ $ in 11 such that $101$. $11^{5}=161051$,
$11^{6}=1771561$ and $11^{7}=19487171$ could also represent the respective
rows according to the Newton’s claim but $101^{n}$ makes it precise.
Can a conclusion be drawn for generating any row of Pascal’s triangle with the
help of extended concept of the power of 11 such as $101^{n}$? The
$9^{\text{th}}$ row of Pascal’s triangle is

$1\leavevmode\nobreak\ \leavevmode\nobreak\ 9\leavevmode\nobreak\
\leavevmode\nobreak\ 36\leavevmode\nobreak\ 84\leavevmode\nobreak\
\leavevmode\nobreak\ 126\leavevmode\nobreak\ \leavevmode\nobreak\
126\leavevmode\nobreak\ \leavevmode\nobreak\ 84\leavevmode\nobreak\
\leavevmode\nobreak\ 36\leavevmode\nobreak\ 9\leavevmode\nobreak\
\leavevmode\nobreak\ 1$

Plainly, $101^{9}=1093685272684360901$ does not give the $9^{\text{th}}$ row
because of the central element of this row contains three digits.

So the representation of three decimal places for each entry of Pascal’s
triangle requires a new formula to be generated. The previous context directed
that multiplication of a number by $11$ and $101$ makes the left shift of all
digits by one and two places respectively. Therefore three digits
representation requires multiplication by $1001$.
Figure 4, proofs the left shift of all digits by $3$ times when a number is
multiplied by $1001.$

$\begin{array}[]{@{}r@{}}1001\\\ {}\times 1001\\\
\hline\cr${\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}1}$001\\\
${\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}00}$000\\\
${\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}000}$000\\\
\text{left shift of all digits by 3 time}\rightarrow
1${\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}001}$000\\\
\hline\cr 1$\textul{002}$001\\\ {}\times 1001\\\
\hline\cr${\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}1}{\color[rgb]{0,1,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,1,0}002}$001\\\
${\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}00}{\color[rgb]{0,1,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,1,0}000}$000\\\
${\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}000}{\color[rgb]{0,1,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,1,0}000}$000\\\
\text{left shift of all digits by 3 time}\rightarrow
1${\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}002}{\color[rgb]{0,1,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,1,0}001}$000\\\
\hline\cr 1$\textul{003}\textul{003}$001\end{array}$

Figure 4: Results after multiplication by $1001$

By continuing the multiplication by 1001 in Figure 4, we get

$1001^{9}=1009036084126126084036009001$

from which one may form the $9^{th}$ row of Pascal’s triangle by partitioning
the digits of the number from the right three digits in each partition.

$1\leavevmode\nobreak\ \leavevmode\nobreak\ 009\leavevmode\nobreak\
\leavevmode\nobreak\ 036\leavevmode\nobreak\ \leavevmode\nobreak\
084\leavevmode\nobreak\ \leavevmode\nobreak\ 126\leavevmode\nobreak\
\leavevmode\nobreak\ 126\leavevmode\nobreak\ \leavevmode\nobreak\
084\leavevmode\nobreak\ \leavevmode\nobreak\ 036\leavevmode\nobreak\
\leavevmode\nobreak\ 009\leavevmode\nobreak\ \leavevmode\nobreak\ 001$

Similarly, $(1001)^{10}=1010045120210252210120045010001$,

$1010045120210252210120045010001\longmapsto 1\leavevmode\nobreak\
\leavevmode\nobreak\ 010\leavevmode\nobreak\ \leavevmode\nobreak\
045\leavevmode\nobreak\ \leavevmode\nobreak\ 120\leavevmode\nobreak\
\leavevmode\nobreak\ 210\leavevmode\nobreak\ \leavevmode\nobreak\
252\leavevmode\nobreak\ \leavevmode\nobreak\ 210\leavevmode\nobreak\
\leavevmode\nobreak\ 120\leavevmode\nobreak\ \leavevmode\nobreak\
045\leavevmode\nobreak\ \leavevmode\nobreak\ 010\leavevmode\nobreak\
\leavevmode\nobreak\ 001$

the $10^{\text{th}}$ row of the Pascal’s triangle.

## III. Results and discussion

From the above study, it may be concluded that the representation of three
decimal places requires the left shift of all digits by three places, and the
left shift of all digits requires two zeros between $1$ and
$1\leavevmode\nobreak\ $ in 11, that is $1001$. Why do we require three
decimal places representation for $9^{th}$ and $10^{th}$ rows of Pascal’s
triangle? Because the central elements of $9^{\text{th}}$ and $10^{\text{th}}$
rows are of three digits. Similarly, we need two digits representation for
$5^{th}$ to $8^{th}$ rows since the central element of these rows are numbers
of two digits. And, the first four rows satisfy $11^{n}$ since the central
element of the first four rows contains one digit only. So for any row, the
number of decimal places representation should be equal to the number of
digits in the central value of that row.
The above discussion compels to generate a formula to find the central value
of any row of the Pascal’s triangle. For an odd number, say $n=9,$ we get
$n+1=10$ elements in $9^{th}$ row and so the central value should be
$\left(\frac{10}{2}\right)^{th}=5^{th}$ observation of that row, which is
$\binom{9}{5-1}=\binom{9}{4}=126$. For an even number, say $n=10$ we get
$n+1=11$ elements and the central value should be
$\lceil\frac{11}{2}\rceil^{th}=6^{th}$ observation, which is
$\binom{10}{6-1}=\binom{10}{5}=252$.
By taking the floor value of $\frac{n}{2}$, a formula for central value of
$n^{th}$ row is $\binom{n}{\lfloor\frac{n}{2}\rfloor}$. But we never need a
central element rather it is necessary to know how many digits the central
element has. Applying the property of Logarithmic function, one can identify
how many digits (or decimal places) of the central element has without knowing
it. Therefore the number of digits in the central value is given by

$\lceil\log_{10}\binom{n}{\lfloor\frac{n}{2}\rfloor}\rceil$

Since $\lceil\log_{10}(X)\rceil$ represents the number of digits of $X$ when
$X\neq 10^{n}$, for $n\in\mathbb{N}$. For a central value of $d$ decimal
places we require $d-1$ zeros between $1\leavevmode\nobreak\
\text{and}\leavevmode\nobreak\ 1\leavevmode\nobreak\ $ in 11, such that
$\left(1\leavevmode\nobreak\ \left(d-1\right)\leavevmode\nobreak\
zeros\leavevmode\nobreak\ 1\right)^{n}$. So, the required number of zeros
between $1\leavevmode\nobreak\ \text{and}\leavevmode\nobreak\
1\leavevmode\nobreak\ $ in 11 can be obtained by taking the floor value of
$\log_{10}\binom{n}{\lfloor\frac{n}{2}\rfloor}$.
If $\Theta$ represents the number of zeros between $1$ and
$1\leavevmode\nobreak\ $ in 11. Then

$\Theta=\lfloor\log_{10}\binom{n}{\lfloor\frac{n}{2}\rfloor}\rfloor$

. We now verify it for an odd number $n=9$ and an even number $n=10$.
If, $n=9$ then $\Theta=2$, and if $n=10$ then $\Theta=2$.
For both of the numbers we need $2$ zeros between $1$ and
$1\leavevmode\nobreak\ $ in 11. So, to get $9^{th}$ and $10^{th}$ rows we have
to calculate $1001^{9}$ and $1001^{10}$ respectively. Both of these cases have
been discussed above.
It’s time to generate the formula to find any row of Pascal’s triangle. The
general formula for generating $n^{th}$ row of Pascal’s triangle is $(1\Theta
1)^{n}$. For a random number such as $n=15$ we get $\Theta=3$.
So, we have to insert 3 zeros and the $15^{th}$ row can be constructed from
the following

$10001^{15}=1001501050455136530035005643564355005300313650455010500150001$

Partitioning the digits from the right four digits at a time as shown below

$1\leavevmode\nobreak\ 0015\leavevmode\nobreak\ 0105\leavevmode\nobreak\
0455\leavevmode\nobreak\ 1365\leavevmode\nobreak\ 3003\leavevmode\nobreak\
5005\leavevmode\nobreak\ 6435\leavevmode\nobreak\ 6435\leavevmode\nobreak\
5005\leavevmode\nobreak\ 3003\leavevmode\nobreak\ 1365\leavevmode\nobreak\
0455\leavevmode\nobreak\ 0105\leavevmode\nobreak\ 0015\leavevmode\nobreak\
0001$

Notice the partitioning yields the $15^{\text{th}}$ row of the Pascal’s
triangle

$1\leavevmode\nobreak\ 15\leavevmode\nobreak\ 105\leavevmode\nobreak\
455\leavevmode\nobreak\ 1365\leavevmode\nobreak\ 3003\leavevmode\nobreak\
5005\leavevmode\nobreak\ 6435\leavevmode\nobreak\ 6435\leavevmode\nobreak\
5005\leavevmode\nobreak\ 3003\leavevmode\nobreak\ 1365\leavevmode\nobreak\
455\leavevmode\nobreak\ 105\leavevmode\nobreak\ 15\leavevmode\nobreak\ 1$

Similarly, we may verify for $n=16$, $\Theta=4$ and
$(100001)^{16}=\seqsplit{100016001200056001820043680800811440128701144008008043680182000560001200001600001}$
This $16^{\text{th}}$ row can also be verified from the existing Pascal’s
triangle. The above formula can be used for a large $n$. We now exemplify
$51^{st}$ row of Pascal’s triangle. Hence $n=51$ gives $\Theta=14$
We have to put $14$ zeros between $1\leavevmode\nobreak\
\text{and}\leavevmode\nobreak\ 1\leavevmode\nobreak\ $ in 11, that is
$(1000000000000001)^{51}$.
$(1000000000000001)^{51}=1{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}000000000000051}{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}000000000001275}{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}000000000020825}\\\
{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}000000000249900}{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}000000002349060}{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}000000018009460}{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}000000115775100}{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}0000006367630}\\\
{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}50}{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}000003042312350}{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}000012777711870}{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}000047626016970}{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}000158753389900}{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}00047626016}\\\
{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}9700}{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}001292706174900}{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}003188675231420}{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}007174519270695}{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}014771069086725}{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}027900908}\\\
{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}274925}{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}048459472266975}{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}077535155627160}{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}114456658306760}{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}156077261327400}{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}1967930}\\\
{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}68630200}{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}229591913401900}{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}247959266474052}{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}247959266474052}{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}229591913401900}{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}19679}\\\
{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}3068630200}{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}156077261327400}{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}114456658306760}{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}077535155627160}{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}048459472266975}{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}027}\\\
{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}900908274925}{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}014771069086725}{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}007174519270695}{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}003188675231420}{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}001292706174900}{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}0}\\\
{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}00476260169700}{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}00158753389900}{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}000047626016970}{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}000012777711870}{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}000003042312350}\\\
{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}000000636763050}{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}000000115775100}{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}000000018009460}{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}000000002349060}{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}0000000002499}\\\
{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}00}{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}000000000020825}{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}000000000001275}{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}000000000000051}{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}000000000000001}$

The desired $51^{\text{st}}$ row can be obtained by partitioning each $15$
digits from the right. For readers convenience, we marked each segment with
different colors and showing that the above formula generates the
$51^{\text{st}}$ row of the Pascal’s triangle. Now we give a proof of the fact
observed above.

We have the following inequalities

$\displaystyle\Theta=\lfloor\log_{10}\binom{n}{\lfloor\frac{n}{2}\rfloor}\rfloor$
(1) $\displaystyle\Rightarrow
10^{\Theta+1}>\binom{n}{\lfloor\frac{n}{2}\rfloor}$ $\displaystyle\Rightarrow
10^{\Theta+1}-1\geq\binom{n}{\lfloor\frac{n}{2}\rfloor}$ (2)
$\displaystyle\binom{n}{\lfloor\frac{n}{2}\rfloor}\geq\binom{n}{r}$ (3)
$\displaystyle 10^{\Theta}\leq\binom{n}{\lfloor\frac{n}{2}\rfloor}$ (4)

And notice that $\left(1\Theta 1\right)^{n}=\left(10^{\Theta+1}+1\right)^{n}$.

######  Lemma 1.

If $n,r\in\mathbb{N}$ and $0\leq r\leq n$, then

$\displaystyle\binom{n}{n-(r-1)}10^{(r-1)(\Theta+1)}+\binom{n}{n-(r-2)}10^{(r-2)(\Theta+1)}+\cdots+1<10^{r(\Theta+1)}.$

Proof. By inequality 3, we have

$\displaystyle\binom{n}{n-(r-1)}10^{(r-1)(\Theta+1)}+\binom{n}{n-(r-2)}10^{(r-2)(\Theta+1)}+\cdots+1$
$\displaystyle<\binom{n}{\lfloor\frac{n}{2}\rfloor}\left(10^{(r-1)(\Theta+1)}+10^{(r-2)(\Theta+1)}+\cdots+1\right)$
$\displaystyle=\binom{n}{\lfloor\frac{n}{2}\rfloor}\left(\frac{10^{r(\Theta+1)}-1}{10^{(\Theta+1)}-1}\right)<\binom{n}{\lfloor\frac{n}{2}\rfloor}\left(\frac{10^{r(\Theta+1)}}{10^{(\Theta+1)}-1}\right).$

Dividing the last quantity by $10^{r(\Theta+1)}$ and using the inequality 2,
we have

$\displaystyle\binom{n}{\lfloor\frac{n}{2}\rfloor}\left(\frac{10^{r(\Theta+1)}}{10^{(\Theta+1)}-1}\right)\left(\frac{1}{10^{r(\Theta+1)}}\right)=\binom{n}{\lfloor\frac{n}{2}\rfloor}\left(\frac{1}{10^{(\theta+1)}-1}\right)$
$\displaystyle\leq\binom{n}{\lfloor\frac{n}{2}\rfloor}\frac{1}{\binom{n}{\lfloor\frac{n}{2}\rfloor}}=1.$

Therefore

$\displaystyle\binom{n}{n-(r-1)}10^{(r-1)(\Theta+1)}+\binom{n}{n-(r-2)}10^{(r-2)(\Theta+1)}+\cdots+1<10^{r(\Theta+1)}.$

######  Proposition 1.

If $n,r\in\mathbb{N}$ and $0\leq r\leq n$, then

$\displaystyle\left(10^{\Theta+1}+1\right)^{n}\mod 10^{r(\Theta+1)}$
$\displaystyle=\binom{n}{n-(r-1)}10^{(r-1)(\Theta+1)}+\binom{n}{n-(r-2)}10^{(r-2)(\Theta+1)}+\cdots+1$

Proof. Expanding $\left(10^{\Theta+1}+1\right)^{n}$ by binomial theorem, we
have

$\displaystyle\left(10^{\Theta+1}+1\right)^{n}\mod 10^{r(\Theta+1)}$
$\displaystyle=[10^{n(\Theta+1)}+\binom{n}{1}10^{(n-1)(\Theta+1)}+\cdots+\binom{n}{n-r}10^{(r)(\Theta+1)}+$
$\displaystyle\binom{n}{n-(r-1)}10^{(r-1)(\Theta+1)}+\binom{n}{n-(r-2)}10^{(r-2)(\Theta+1)}+\cdots+1]\mod
10^{r(\Theta+1)}$
$\displaystyle=\binom{n}{n-(r-1)}10^{(r-1)(\Theta+1)}+\binom{n}{n-(r-2)}10^{(r-2)(\Theta+1)}+\cdots+1,$

since by the Lemma 1, we have

$\displaystyle\binom{n}{n-(r-1)}10^{(r-1)(\Theta+1)}+\binom{n}{n-(r-2)}10^{(r-2)(\Theta+1)}+\cdots+1<10^{r(\Theta+1)}.$

######  Corollary 1.

The integer
$\binom{n}{n-(r-1)}10^{(r-1)(\Theta+1)}+\binom{n}{n-(r-2)}10^{(r-2)(\Theta+1)}+\cdots+1$
has at best $r(\Theta+1)$ significant digits.

Proof. Since
$\binom{n}{n-(r-1)}10^{(r-1)(\Theta+1)}+\binom{n}{n-(r-2)}10^{(r-2)(\Theta+1)}+\cdots+1$
is the remainder when $\left(10^{(\Theta+1)}+1\right)^{n}$ is moded out by
$10^{r(\Theta+1)}$.

######  Corollary 2.

The left most $\left(\Theta+1\right)$ digits of the integer

$\displaystyle\binom{n}{n-(r-1)}10^{(r-1)(\Theta+1)}+\binom{n}{n-(r-2)}10^{(r-2)(\Theta+1)}+\cdots+1$

is

$\displaystyle\binom{n}{n-(r-1)}=\binom{n}{r-1}.$

Proof. By Corollary 1, the integer

$\displaystyle\binom{n}{n-(r-1)}10^{(r-1)(\Theta+1)}+\binom{n}{n-(r-2)}10^{(r-2)(\Theta+1)}+\cdots+1$

has at most $r(\Theta+1)$ significant digits, and similarly

$\displaystyle\binom{n}{n-(r-2)}10^{(r-2)(\Theta+1)}+\binom{n}{n-(r-3)}10^{(r-3)(\Theta+1)}+\cdots+1$

has at most $(r-1)(\Theta+1)$ significant digits. But
$\binom{n}{n-(r-1)}10^{(r-1)(\Theta+1)}$ has $(r-1)(\Theta+1)$ zeros to the
right, the left most $(r)(\Theta+1)-(r-1)(\Theta+1)=\Theta+1$ digits gives

$\displaystyle\binom{n}{n-(r-1)}=\binom{n}{r-1}.$

######  Theorem 1.

The $r$-th block of $(\Theta+1)$ digits from the right of the integer
$(1\Theta 1)^{n}$ is the binomial coefficient $\binom{n}{r-1}$, where
$\Theta=\lfloor\log_{10}\binom{n}{\lfloor\frac{n}{2}\rfloor}\rfloor$.

Proof. The proof follows from the Corollary 1 and the Corollary 2. Hence
partitioning the digits of the integer $(1\Theta 1)^{n}$ generates all the
binomial coefficients or the $(n+1)^{\text{th}}$ row of the Pascal’s triangle.

## IV. Conclusion

Sir Isaac Newton hinted that binomial coefficients in the $(n+1)^{\text{th}}$
row of the Pascal’s triangle may be achieved from partitioning the digits in
the $n^{\text{th}}$ power of some variation of $11$ [21]. It has been shown
earlier the weighted sum of the values in the $(n+1)^{\text{th}}$ row of the
Pascal’s triangle is $(11)^{n}$ [22]. We have shown that
$\left(\Theta+1\right)$ digit partition of $(1\Theta 1)^{n}$ from the right
gives the values of the $(n+1)^{\text{th}}$ row of the Pascal’s triangle.

## References

  * [1] A. W. F. Edwards, Pascal’s arithmetical triangle, Oxford University Press, 1987.
  * [2] James T Sgroi, Pascal’s Triangle: A Different Approach to Subsets. The Mathematics Teacher, 60(4): 352–353, 1967.
  * [3] Lars C Jansson, Spaces, Functions, Polygons, and Pascal’s Triangle. Mathematics Teacher, 66(1): 71–77, 1973.
  * [4] Larry M toschi, The permutations triangle. The Mathematics Teacher, 67(3):210–214, 1974.
  * [5] David R Duncan and Bonnie H Litwiller,  A Pattern: Pascal’s Triangle and Prime Numbers. Mathematics Teacher, 68(1):23–26, 1975..
  * [6] John F Putz, The Pascal polytope: An extension of Pascal’s triangle to N dimensions. The College Mathematics Journal, 17(2):144–155, 1986.
  * [7] Chris Houghton, Differences and Pascal’s Triangle. Mathematics in School, 20(4):37–38, 1991.
  * [8] Andreas M Hinz, Pascal’s Triangle and the Tower of Hanoi. The American mathematical monthly, 99(6):538–544, 1992.
  * [9] Vo E Hoggatt Jr and Marjorie Bicknell, Diagonal sums of generalized Pascal triangles. The Fibonacci Quarterly, 7(4):341–358, 1969.
  * [10] Sergio Falcon and Ángel Plaza, The k-Fibonacci sequence and the Pascal 2-triangle. Chaos, Solitons & Fractals, 33(1):38–49, 2007.
  * [11] Thomas M Green, Recurrent sequences and Pascal’s triangle. Mathematics Magazine, 41(1):13–21, 1968.
  * [12] Harlan J Brothers, Math Bite: Finding e in Pascal’s Triangle. Mathematics Magazine, 85(1):51–51,2012.
  * [13] Boris A Bondarenko, Generalized Pascal triangles and pyramids: their fractals, graphs, and applications. Fibonacci Association Santa Clara, CA, 1993.
  * [14] Ivan Korec, Theories of generalized Pascal triangles. Annals of Pure and Applied Logic, 89(1):45–52,1997.
  * [15] John E Freund, Restricted occupancy theory—a generalization of Pascal’s triangle. The American Mathematical Monthly, 63(1):20–27,1956.
  * [16] JD Bankier, Generalizations of Pascal’s triangle. The American Mathematical Monthly, 64(6):416–419,1957.
  * [17] Gábor Kallós, The generalization of Pascal’s triangle from algebraic point of view. Acta Academiae Paedagogicae Agriensis, Sectio Mathematicae, 24:11–18,1997.
  * [18] Gábor Kallós, A generalization of Pascal’s triangle using powers of base numbers. Annales mathématiques Blaise Pascal, 13(1):1–15, 2006.
  * [19] Gabor Farkas and Gabor Kallos, Prime numbers in generalized Pascal triangles. Acta Technica Jaurinensis, 1(1):109–117, 2008.
  * [20] Michael Anton Kuhlmann, Generalizations of Pascal’s Triangle: A Construction Based Approach. University of Nevada, Las Vegas (PhD dissertation), 2013.
  * [21] Isaac Newton, A Treatise of the Method of Fluxions and infinite Series. The Mathematical Works of Isaac Newton, 1:31–33 ,1736.
  * [22] Robert Arnold, Tom Attenweiler, Christopher Brockman, Bethany Lesko, Christine Martinek, Colleen McCormick, Jessica McQuiston, Jessica Parker and Amy Rohmiller, Newton’s Unfinished Business: Uncovering the Hidden Powers of Eleven in Pascal’s Triangle. 2004.
  * [23] Robert L Morton, Pascal’s Triangle and Powers of 11. The Mathematics Teacher, 57(6):392–394, 1964.
  * [24] Francis J Mueller, More on Pascal’s Triangle and Powers of 11. The Mathematics Teacher, 58(5):425–428, 1965.