# Human-AI Collaboration in Thematic Analysis using ChatGPT: A User Study and
Design Recommendations

Lixiang Yan  Monash UniversityAustralia ,  Vanessa Echeverria  Monash
UniversityAustralia ,  Gloria Fernandez Nieto  Monash UniversityAustralia ,
Yueqiao Jin  Monash UniversityAustralia ,  Zachari Swiecki  Monash
UniversityAustralia ,  Linxuan Zhao  Monash UniversityAustralia ,  Dragan
Gašević  Monash UniversityAustralia and  Roberto Martinez-Maldonado  Monash
UniversityAustralia

(2024)

###### Abstract.

Generative artificial intelligence (GenAI) offers promising potential for
advancing human-AI collaboration in qualitative research. However, existing
works focused on conventional machine-learning and pattern-based AI systems,
and little is known about how researchers interact with GenAI in qualitative
research. This work delves into researchers’ perceptions of their
collaboration with GenAI, specifically ChatGPT. Through a user study involving
ten qualitative researchers, we found ChatGPT to be a valuable collaborator
for thematic analysis, enhancing coding efficiency, aiding initial data
exploration, offering granular quantitative insights, and assisting
comprehension for non-native speakers and non-experts. Yet, concerns about its
trustworthiness and accuracy, reliability and consistency, limited contextual
understanding, and broader acceptance within the research community persist.
We contribute five actionable design recommendations to foster effective
human-AI collaboration. These include incorporating transparent explanatory
mechanisms, enhancing interface and integration capabilities, prioritising
contextual understanding and customisation, embedding human-AI feedback loops
and iterative functionality, and strengthening trust through validation
mechanisms.

Human-AI Collaboration, Thematic Analysis, Qualitative Research, Generative
Artificial Intelligence, ChatGPT

††copyright: acmcopyright††journalyear: 2024††doi: XXXXXXX.XXXXXXX††ccs:
Human-centered computing User studies††ccs: Human-centered computing
Collaborative interaction

##  1\. Introduction

Human-AI collaboration in qualitative research is gaining traction within the
human-computer interaction (HCI) community (Amershi et al., 2019; Wang et al.,
2020; Yang et al., 2020). This collaboration reflects the paradigm shift from
complete automation by AI to a more collaborative role that AI plays in human
workflows, where humans and AI work together to achieve a shared goal (Harper,
2019; Wang et al., 2020; Lai et al., 2022). Such collaborations present
opportunities for both enhancing research efficiency and deepening analytical
insights (Lennon et al., 2021). Specifically, in the domain of thematic
analysis, various prototype systems have been introduced, demonstrating
different automation levels through the integration of pattern-based and
machine-learning methods (Rietz and Maedche, 2021; Gebreegziabher et al.,
2023; Gao et al., 2023a). While these systems have demonstrated efficacy in
enhancing coding quality and improving inter-coder reliability (Rietz and
Maedche, 2021; Gebreegziabher et al., 2023), they were difficult to scale with
diverse linguistic features due to limited pattern rules or model training
requirements, which is one of the unique features of qualitative data (Braun
and Clarke, 2021).

Recently, the emergence of generative artificial intelligence (GenAI),
exemplified by ChatGPT and other state-of-the-art large language models,
underscores the capabilities of advanced AI to process and analyze
unstructured text-based qualitative data without additional model training or
human intervention (Van Dis et al., 2023; Li et al., 2023; Yan et al., [n.
d.]). Such technologies, equipped with extensive pre-trained knowledge, have
the capacity to identify data patterns that may not be immediately apparent to
researchers (Zambrano et al., 2023; Gao et al., 2023b). Yet, the implications
of integrating GenAI into the qualitative analysis workflow, inherently
iterative and human-centred, remain underexplored. Additionally, little is
known about how to optimise researchers’ collaborative experience with GenAI
in thematic analysis as such systems offer a high degree of user autonomy
through natural language interaction (Pavlik, 2023; Sallam, 2023), which is
substantially different from prior machine-learning and pattern-based systems.
In light of this, our user study delves into the dynamics of how qualitative
researchers engage with ChatGPT in the context of thematic analysis. The
primary contribution of our research includes five actionable design
recommendations synthesised from the benefits and challenges that qualitative
researchers experienced when collaborating with ChatGPT for thematic analysis.
These recommendations can offer guidance to HCI scholars in designing and
developing applications that not only optimise the process of human-AI
collaboration in thematic analysis but also fortify user trust and confidence
in the technology.

##  2\. Related Work

###  2.1. Thematic Analysis in Qualitative Research

Thematic analysis is a qualitative research method that involves uncovering
and analyzing themes or patterns within qualitative data, such as interview
transcripts and open-ended responses (Braun and Clarke, 2006). As
conceptualized by Braun and Clarke (Braun and Clarke, 2021), depending on the
epistemological assumptions, this multifaceted method can be undertaken
deductively, where researchers analyze and interpret data through the lens of
existing research and theory, or inductively, where the analysis is grounded
in the data. Regardless of the approach, it is an iterative method, where
researchers need to go over the data multiple times to gain an in-depth
understanding of the data, articulate, refine, and confirm the themes or
patterns that they have identified (Boyatzis, 1998). Consequently, thematic
analysis is a time-demanding and skill-intensive task that may prove
challenging to scale with a large corpus of qualitative data (Marathe and
Toyama, 2018; Chen et al., 2018).

###  2.2. Human-AI Collaboration in Thematic Analysis

Research on human-AI collaboration in thematic analysis has produced several
prototype systems to support automated or semi-automated coding, including
text classification, topic modelling, and annotations of qualitative data
(Rietz and Maedche, 2021; Gebreegziabher et al., 2023; Lennon et al., 2021;
Goudjil et al., 2018). These systems have leveraged a mix of techniques,
ranging from machine-learning techniques for text classification (Blackwell et
al., 2018; Rietz and Maedche, 2021; Yan et al., 2014; Goudjil et al., 2018) to
interactive topic modelling algorithms (Bakharia et al., 2016; Baumer et al.,
2017; Leeson et al., 2019; Lennon et al., 2021; Hong et al., 2022) and
adaptive pattern-based rules (Crowston et al., 2010; Gebreegziabher et al.,
2023; Kaufmann et al., 2020; Marathe and Toyama, 2018; Rietz and Maedche,
2021). Systems based on machine-learning techniques like logistic regression
(Rietz and Maedche, 2021) and support vector machine models (Yan et al., 2014)
have demonstrated potential in automating large-scale text data annotation.
Topic modelling algorithms, especially variations of Latent Dirichlet
Allocation, have shown performance comparable to open coding in identifying
underlying topics in qualitative data (Bakharia et al., 2016; Hong et al.,
2022; Lennon et al., 2021). These techniques, however, raise concerns about
transparency in their decision-making, which may challenge the primary
objective of thematic analysis: gaining an in-depth understanding of
qualitative data (Chen et al., 2018; Marathe and Toyama, 2018). In contrast,
adaptive pattern-based systems such as Cody (Rietz and Maedche, 2021) and
PaTAT (Gebreegziabher et al., 2023) adopted user-interpretable patterns to
enhance user understanding and trust. Yet, their reliance on pattern-based
rules might constrain their analysis of data with varied linguistic features
(Gebreegziabher et al., 2023). Additionally, these pattern-based systems may
also be unsustainable considering the nuances in the linguistic and semantic
features of different languages (Eisenschlos et al., 2019).

###  2.3. Generative Artificial Intelligence

GenAI has witnessed significant progress in recent years, providing a
foundation for developing solutions capable of generating human-like contents
(Sallam, 2023). The core idea behind GenAI is to train models on vast amounts
of data, enabling them to produce original content that closely mirrors human
language patterns (Pavlik, 2023; Van Dis et al., 2023). Among the notable
advancements in this field are large language models like ChatGPT, which have
been trained on diverse textual data sources, allowing them to comprehend and
generate a wide range of language structures (Sallam, 2023; Yan et al., [n.
d.]). Beyond text generation, these models are also capable of understanding
context, offering explanations, and assisting in tasks such as thematic
analysis (Zambrano et al., 2023; Amarasinghe et al., 2023). The vast knowledge
encoded in these models, coupled with their ability to generalize from
training data, positions them as potential collaborative partners in
qualitative research (Gao et al., 2023a). Their ability to capture nuanced
language structures and provide contextually relevant content has the
potential to support qualitative researchers in the labour-intensive process
of thematic analysis (Zambrano et al., 2023). However, while GenAI exhibits
promise, it also raises questions about its suitability and reliability in
qualitative research contexts (Bang et al., 2023; Reiss, 2023). For instance,
the reliability of the decisions made by these models and their
trustworthiness remain topics of discussion (Van Dis et al., 2023), especially
when used in thematic analysis, where understanding and transparency are
paramount (Braun and Clarke, 2021; Gebreegziabher et al., 2023). Thus, the
interplay between GenAI technologies and qualitative researchers requires
further investigation to uncover collaborative potential and areas of concern
following a human-centred AI approach (Shneiderman, 2020b, a).

###  2.4. Research Questions

The current user study aims to address the aforementioned gaps by delving into
qualitative researchers’ perceptions of their collaboration with GenAI,
specifically ChatGPT, and highlighting actionable design recommendations to
foster effective human-AI collaboration with this novel technology.
Specifically, the following research questions were addressed.

  * •

RQ1: In the context of thematic analysis, what opportunities do qualitative
researchers identify when collaborating with ChatGPT?

  * •

RQ2: While collaborating with ChatGPT for thematic analysis, what challenges
do qualitative researchers encounter?

##  3\. Methods

###  3.1. Participants

We conducted a user study to investigate the experience of qualitative
researchers when collaborating with ChatGPT to conduct thematic analysis, with
a specific focus on uncovering the opportunities and challenges that they
perceived during this human-AI collaboration process. We recruited
participants following a criterion-based sampling via a faculty and a personal
mailing list. Participants were doctoral students or full-time academics with
at least one year of experience in qualitative research and had conducted
thematic analysis, either inductively or deductively, themselves. Three full-
time academics (P1–P3) and seven doctoral students (P4–P10) with different
years of experience (M = 4.0, SD = 3.3; four females) in thematic analysis
participated in this study. Ethics approval was obtained from Anonymous
University (Project ID: Anonymized).

###  3.2. Study Design

We adopted a contextual inquiry approach to guide the study design (Karen and
Sandra, 2017). This user-centred approach has been used in prior HCI research
to uncover enriched insights regarding users’ authentic experiences and needs
of specific tasks in their natural environment or context (Rietz and Maedche,
2021). Each session was around one hour and contained two main sections:

  1. (1)

Thematic Analysis Exercise (20 mins): Participants were first asked to
manually perform inductive thematic analysis on a short transcript containing
15 utterances. Following Braun and Clarke’s (Braun and Clarke, 2021)
guidelines, the coding process was open and organic without any pre-existing
coding framework and the final outcome was the themes that participants
identified. This practical exercise aimed to provide participants with
contextual knowledge of the transcript, which was required for them to
evaluate the outputs from ChatGPT in the second task.

  2. (2)

Collaboration with ChatGPT (40 mins): After analyzing the short transcript,
participants were given an introduction to ChatGPT (5 mins). We then provided
participants with an extended version of the short transcript, containing 15
additional utterances (30 utterances in total). Participants were given access
to ChatGPT (GPT-4 Default mode) and were asked to collaborate with ChatGPT to
perform thematic analysis on the extended transcript while conducting an in-
situ evaluation of the outputs with the think-aloud method (15 mins). Finally,
we evaluated participants’ experience of collaborating with ChatGPT for
thematic analysis through a semi-structured interview (20 mins). Specifically,
participants were asked to reflect on:

     * •

Accuracy of the AI-generated results.

     * •

Trustworthiness of the AI-generated results.

     * •

Helpfulness of ChatGPT for thematic analysis.

Participants were also asked about how the current interface and functionality
of ChatGPT can be improved to better accommodate their research needs for
thematic analysis. All sessions were conducted remotely via Zoom, during which
we captured audio, shared screen visuals, and archived the chat history
between participants and ChatGPT.

###  3.3. Analysis

Using Otter AI (otter.ai), we transcribed each session’s audio recording.
Addressing our research questions, the primary author conducted a reflexive
thematic analysis (Braun and Clarke, 2021) of these transcripts in conjunction
with each participant’s ChatGPT chat history. This initial analysis was
augmented by discussions with a co-author, facilitating the iterative
refinement of emergent themes related to participants’ perceived opportunities
(RQ1) and challenges (RQ2) when collaborating with ChatGPT to conduct thematic
analysis. We commenced our analysis on an individual participant basis,
extracting preliminary codes pertinent to each RQ. Subsequently, we aggregated
these preliminary codes into overarching themes for every RQ. The goal of this
analysis was to comprehend the nature of participants’ interactions with
ChatGPT and inform design recommendations for future HCI studies.

##  4\. Findings and Discussion

In this section, we first reported the opportunities and challenges
participants experienced when collaborating with ChatGPT to conduct thematic
analysis. Based on these findings, we synthesised five actionable design
recommendations to foster effective human-AI collaboration.

###  4.1. Human-AI Collaboration Opportunities (RQ1)

Participants reported several multifaceted opportunities for human-AI
collaboration, particularly leveraging tools like ChatGPT, in thematic
analysis.

####  4.1.1. Efficiency in Processing and Analysis

The efficiency of ChatGPT in processing large datasets and distilling themes
was a salient observation. Such proficiency offers a promising avenue for the
swift analysis of unstructured qualitative data. P1 articulated: ”ChatGPT
provides an efficient method to identify key themes from qualitative data.”
This sentiment was echoed by P7, who observed: ”For this task, I think, yeah,
it definitely can save my time. Yeah, to identify emerging themes.” P10 went a
step further, emphasizing the technology’s superiority in certain aspects of
analysis: ”I think it’s good at doing this kind of summarization task. Even
better than human.” The advantages of ChatGPT become even more pronounced when
handling voluminous datasets. P8 remarked: ”Especially if I have a lot of data
and I really don’t have time commitment or to analyze it by myself.” Beyond
its efficiency, ChatGPT’s capabilities in transforming unstructured datasets
into structured formats were also recognized. P2 commented: ”I was impressed
at its ability to, at least for some of the data, format it in a way that I
asked it to right with, like the table and stuff.”

####  4.1.2. Facilitation of Initial Exploration

Participants recognized the instrumental value of ChatGPT during the
preliminary stages of both inductive and deductive analytical processes. They
posited that ChatGPT could offer a foundational coding framework that would be
particularly beneficial during the initial stages of code generation,
exploration, and ideation. P2, for instance, highlighted the tool’s capacity
for summation and ideation by stating: ”I have identified that ChatGPT is very
useful to summarize general topics, so I will use it for inspiration, and to
identify codes in the process.” In the context of the inductive coding
approach, P8 underscored ChatGPT’s potential for exploration: ”So, for
instance, we were talking about the inductive way of coding, I will ask
ChatGPT, can you give me the list of the most prominent themes which are
presented in the piece of text.” Extending on this, P8 also suggested that the
themes generated by ChatGPT could set the stage for developing an initial
coding framework for deductive analysis: ”When we’re doing deductive analysis,
it [ChatGPT] will give the list of themes, and if we ask ChatGPT, okay, can
you try to find instances [utterance] of those themes. It [ChatGPT] also will
work fine.” This potential for initiating the coding process in deductive
analysis was also evidenced by P9, who highlighted the potential for coding
interview transcripts: ”I will have multiple interview transcripts, and I will
code maybe one or two transcripts, and then I will probably pass it to
ChatGPT, and maybe it will learn the patterns and help me to code the
transcripts.”

####  4.1.3. Quantitative Insights and Detailed Metrics

Beyond assisting with the initial exploration stages, ChatGPT’s ability to
offer quantitative insights and detailed metrics like coverage is appreciated.
For example, P1 highlighted its strength in providing a detailed breakdown of
theme coverage through prompting: ”I think that’s really useful getting all
these metrics of coverage.” Such metrics can help researchers understand the
extent to which a particular theme or topic is represented within the dataset.
P2 echoed this sentiment, emphasizing ChatGPT’s capacity to shed light on ”the
frequency or popularity of certain themes or topics.” This capability can be
instrumental in discerning recurrent themes, thereby paving the way for deeper
qualitative investigations, such as identifying sub-themes. Meanwhile, P7 also
saw the potential of ChatGPT in analyzing coded data: ”But after coding, we
still need to do some analysis, right? We are not only going to do some
frequency tables. If it can help me with analyzing coded data to identify some
patterns or some preliminary analysis, it will be good.”

####  4.1.4. Support Language Comprehension

Participants also appreciated ChatGPT’s assistance in language comprehension,
especially for non-native English speakers. For example, P4 reflected that:
”Not only about time-wise, but also sometimes because, for example, for me,
English is not my first language. Sometimes the interviewees may say something
too casual [or] too verbal that I couldn’t understand, [but] I think ChatGPT
could.” P10 pointed out ChatGPT’s value in understanding language usage in
complex or unfamiliar domains: ”I think a human has some limitation of
knowledge. For example, maybe I don’t understand when talking about politics,
but ChatGPT can understand it better.”

####  4.1.5. Discussion of RQ1

The findings on the efficiency and efficacy of ChatGPT in supporting the
initial exploration of qualitative data resonated with prior works (Zambrano
et al., 2023; Goudjil et al., 2018; Rietz and Maedche, 2021; Gebreegziabher et
al., 2023), illustrating the collaborative synergy between researchers and AI,
where they both contribute to the process, instead of completely outsourcing
the task to AI. The appreciation for quantitative insights and detailed
metrics further strengthened such collaborations where researchers can
leverage these additional insights offered by AI to conduct further in-depth
analysis. This finding also reinforced the need for human-centred AI where the
goal of well-designed AI systems should be supporting humans instead of
replacing them (Shneiderman, 2020b, a). Apart from these findings, we also
identified a unique benefit of GenAI, that is, supporting the language
comprehension of non-native speakers and non-experts. Such a novel finding is
unseen in prior works on machine-learning and pattern-based AI systems (Rietz
and Maedche, 2021; Gebreegziabher et al., 2023; Bakharia et al., 2016;
Kaufmann et al., 2020). This finding further highlights the augmentation of
human capabilities that human-AI collaboration enables, especially in skill-
intensive tasks like thematic analysis, where experience and expertise are
essential for ensuring quality results (Boyatzis, 1998; Braun and Clarke,
2021).

###  4.2. Human-AI Collaboration Challenges (RQ2)

In exploring the challenges of human-AI collaboration for thematic analysis,
several consistent themes emerged across participants’ experiences.

####  4.2.1. Trustworthiness and Accuracy Concerns

Participants consistently expressed concerns about the trustworthiness and
accuracy of ChatGPT’s outputs. They were sceptical of the results produced and
often felt the need to manually verify them. P2 articulated this scepticism:
”If I notice something is not very well performed, I will stop trusting the
technology, and I will maybe have to go on my own and check manually.” This
sentiment was resonated by P4, who expressed a balanced stance: ”I have mutual
experience, a neutral perception on the extent I trust ChatGPT.” Moreover, P6
underscored the indispensable role of human oversight: ”So it needs to be
checked by a human before it can be used for any research.” Interestingly,
while participants generally acknowledged ChatGPT’s proficiency in certain
tasks like summarization, doubts emerged when it came to numerical data. As P5
observed: ”It has a good side on the summarizing and like integrating codes
like that. I trust that part. But about the coverage and like any number,
basically the numbers, I do not trust it.” A point of contention arose around
the potential for ChatGPT to generate hallucinated results. P8 cautioned:
”First of all is that, for instance, it might be inaccurate in the sense that
the codes might be made up.” Yet, P10 offered a more nuanced view, suggesting
that while ChatGPT might occasionally deviate from accuracy in some contexts,
thematic analysis might not be one of them: ”When it hallucinates, it’s
usually under the context when I ask for some specific literature, but in this
task [thematic analysis] we give it [ChatGPT] the material and let it
summarise, seems like an easier task.” These varied perspectives accentuated
P9’s call for validation works: ”So one thing, we need some literature to show
the ChatGPT can actually be used for thematic analysis.”

####  4.2.2. Reliability and Consistency Issues

Apart from participants’ inherent trust in the technology, the challenge of
obtaining consistent results from ChatGPT was another common concern. For
example, P3 expressed concerns about the temporal consistency of the system:
”I’m wondering how consistent ChatGPT is over time, you know, with the same
stuff.” This sentiment hints at potential intra-reliability issues, as P3
further explained: ”It [ChatGPT] might need some refinement with itself. There
are intra-reliability problems that might be happening.” This concern was
echoed by P6, who observed variances in the system’s outputs: ”I know
sometimes the answers can be quite random and inconsistent, no matter how you
prompt the request.” Intriguingly, P6 also noted potential differences in
results even when using identical prompts: ”Yeah, sometimes with the same
prompt. If you use it for the first time, you get great results. But when you
use it for even the same transcript, it can produce different results.”
Highlighting the broader implications of this inconsistency, P8 underscored
the user’s inclination towards more deterministic systems: ”You’re more
inclined to serve the trust to the software, which is more deterministic.”

####  4.2.3. Limited Data Capacity and Contextual Understanding

Several participants pointed out ChatGPT’s limitations in terms of data
processing capacity and its understanding of context. P3 mentioned: ”It
[ChatGPT] has some limitations to the amount of data you can give it.” P9
resonated with this concern: ”From my previous experience with ChatGPT, it has
a problem with the length of the token [input text].” This capacity issue may
limit the length of the transcript that ChatGPT can analyze, as P9 elucidated:
”The transcripts for thematic analysis [are] usually very long. And in that
case, we can only input one session to make it [ChatGPT] learn.” This
segmented input could compromise the quality of outcomes, with P9 noted: ”If
it [transcripts] cannot be inputted as one message, from my previous
experience, the learning efficiency [of ChatGPT] will be harmed.” Another
concern is ChatGPT’s ability to infer deeper contexts. P4’s critique explained
this: ”Sometimes ChatGPT only provides information based on the text, and
sometimes the text may not be able to 100% reflect on the key things that the
[participant] wants to express.” Drawing from personal experiences in domain-
specific research, P4 underscored the necessity of a contextualized
understanding, especially in thematic analysis. For example, P4 explained the
following coming from the context of law education: ”I found that if you had
experience learning or teaching experience in law, you would have a better
understanding about what the [study participants] want to express.” Meanwhile,
P7 highlighted the lack of academic focus of ChatGPT: ”I think it only can
identify some emerging themes based on the transcripts, but it’s not really
academic oriented.” The imperative for deep comprehension of existing academic
literature was also highlighted by P7: ”I think it [ChatGPT] needs to
understand the literature. So I probably need to upload my literature review
or at least what articles that I used to create the coding scheme?” Without
such an enriched contextual foundation, there could be a misalignment between
humans and AI. As P10 summarized, ”I feel like it’s about the weight of
importance that ChatGPT has maybe misaligned with researchers.”

####  4.2.4. Interface and Integration Challenges

Other than the aforementioned functionality challenges, participants also
discussed the difficulties and inefficiencies in interacting with the current
interface of ChatGPT for thematic analysis. P3 highlighted the suboptimal
experience: ”And obviously, it’s not ideal to be interacting with it in the
ChatGPT window.” Expanding on this, P3 underscored the system’s inefficiencies
when adopting an iterative methodology: ”I will probably do [the analysis
using ChatGPT] like 2 or 3 more times, but at least in the way that the
interface is set up, it’s not conducive to me doing it quickly and easily.” On
the same note, P9 emphasized the need for better integration and formatting
tools: ”Maybe we can also format the input as a table; maybe we can also
import the transcription table directly as an input to the AI.”

####  4.2.5. Acceptance within the Research Community

Participants expressed concerns about the adoption and disclosure of human-AI
collaboration in qualitative research. A predominant challenge arises from
potential scrutiny by reviewers regarding the tool’s reliability and the
validation process. This is coupled with the broader acceptance of such
technologies within the scientific community. P6 highlighted the dilemma: ”We
use ChatGPT. We want to disclose it. However, we can imagine there could be
challenges from reviewers about reliability and accuracy. And how do you
validate the process?” This participant further commented on the societal
perceptions around the use of GenAI technologies like ChatGPT, especially in
academic contexts. As P6 explained: ”I think when I publish something, I’ll
tell everyone. Okay. I used ChatGPT to polish my articles. That just sounds
sort of weird.” P1 echoed the sentiment of uncertainty but also expressed
optimism about the future adoption of these technologies in research: ”But if
it is about the use of AI in research, I think that’s a big question, and we
don’t know what needs to be done at this point.” P1 further articulated hope
for a future where the use of GenAI is normalized: ”I guess it will be more
accepted as well, and it will be just another method being used like any other
methods. So it’s not about disclosing if you’re using AI or not, but just
simply describing the method and the tool that was used.” However, not all
views were optimistic. P9 highlighted potential biases within the reviewer
community: ”I know some reviewers actually, very hate the AI in doing stuff.”

####  4.2.6. Discussion of RQ2

The concerns regarding the trustworthiness, accuracy, reliability, and
consistent of AI-generated resonated with previous findings (Rietz and
Maedche, 2021; Gebreegziabher et al., 2023; Lennon et al., 2021; Hong et al.,
2022), indicating the need to include trust-assuring mechanisms when designing
human-AI collaboration. Whereas, the limited data capacity and contextual
understanding seem to be a novel challenge for GenAI. While data capacity
issues might be resolvable through model advancements and algorithm
innovations (Chen et al., 2023), contextual understanding can only be achieved
through effective human-AI collaboration, where humans actively provide and
refine the contextual knowledge of AI during an iterative circle (Boyatzis,
1998). Such a synergy would require designing an interface that prioritises
the continued and mutual communication between humans and AI (Shneiderman,
2020b; Schmidt, 2020). Lastly, an ethical dilemma exists between adopting a
human-AI collaborative approach and adequately reporting such usages in
academic publications. While this issue remains an ongoing debate, there is an
increasing consensus on the need for transparency, disclosure, and clear
guidelines to ensure that AI contributions are acknowledged without
undermining the authenticity and integrity of the research (Dwivedi et al.,
2023; Van Dis et al., 2023). Apart from guidelines, in the context of thematic
analysis, empirical evidence is urgently needed to evaluate the validity of
the AI-generated results.

###  4.3. Design Recommendations

Building on our findings, we have synthesised five actionable design
recommendations to support future HCI studies in designing and developing AI
solutions that streamline the process of human-AI collaboration in thematic
analysis while also ensuring user trust and confidence in the technology.

####  4.3.1. Incorporate Transparent Explanatory Mechanisms

To increase user confidence and understanding of the technology, it is crucial
to design human-AI collaboration solutions with transparent explanatory
mechanisms. The AI system should offer detailed rationales behind the
generation of specific themes. For instance, by providing relevant data
metrics (P2: ”it would be good to see the actual coverage and see it
everywhere”) or highlighting illustrative examples for each theme (P5: ”having
quotes… yeah, like for each line, this is the code. And this is the quote”),
users can discern the AI’s decision-making process. Incorporating features
that allow the AI to show its ”thought process” or its basis for conclusions
can demystify its operations, ensure that generated themes align with data
nuances, and potentially improve its sense of trust.

####  4.3.2. Enhance Interface and Integration Capabilities

A human-centred interface redesign is critical. The user interface should be
intuitive, allowing for easy navigation and operations such as data uploads
and theme suggestions (P3: ”Imagine a step like, I upload my data, and then I
click a button and to say like, what do you think the themes are in this?”).
Integration features, like spreadsheet compatibility (P1: ”Maybe a way to
submit this spreadsheet… and then the output to be the same spreadsheet”), can
streamline the thematic analysis process, making it more efficient for users.
Moreover, considering compatibility with other research tools and platforms
can elevate the user experience (P6: ”I wish it could be an all-in-one system
that, for example, after this interview… can automatically summarize what we
said on Zoom.”), ensuring that human-AI collaboration seamlessly fits into the
researcher’s workflow.

####  4.3.3. Prioritize Contextual Understanding and Customization

The design of human-AI collaboration solutions should prioritize a deep
understanding of the data’s unique context. Features that allow users to
provide contextual cues or specific research backgrounds can enhance the AI’s
sensitivity to nuances (P4: ”I prefer to have some kind of tools that I can
provide more contextual information to when generating the automatic analysis.
Now it’s just a transcript.”). Furthermore, customization options, where users
can adjust parameters or guide the AI’s theme generation based on their
specific needs (P5: ”I have the next step, which is to introduce some of my
understanding or my knowledge.”), can ensure that the generated themes are
rooted in the data’s specific context.

####  4.3.4. Embed Feedback Loops and Iterative Functionality

A dynamic and collaborative relationship between the user and AI can be
fostered by embedding feedback loops within the system. By allowing users to
provide feedback on generated themes, and then having AI adjust its outputs
accordingly in real-time, the tool can evolve and refine its understanding
(P2: ”If I have identified something which is not correct from the tool. If I
have some chance to provide additional feedback to the tool to recalculate and
to improve.”). Such iterative functionality underscores the idea that thematic
analysis is a continuous process (P3: ”Obviously, this could be an iterative
process.”), which can benefit immensely from ongoing human-AI collaboration.

####  4.3.5. Strengthen Trust through Validation Mechanisms

Building user trust is foundational for effective human-AI collaboration.
Designing AI systems with robust validation mechanisms can address concerns
regarding their reliability and accuracy. While, the aforementioned
transparent explanatory mechanisms, such as coverage metrics, can be an
enabler of such validations, more rigorous and empirically-orientated
mechanisms are required. For example, features that allow side-by-side
comparisons of AI-generated themes with human analysis or that offer manual
coding checks can enhance user confidence (P10: ”I need to compare the results
generated by ChatGPT and human.”). Such mechanisms could also contribute
empirical evidence that is required to convey the validity and reliability of
this human-AI collaborative approach to the academic community and external
reviewers.

##  5\. Limitations and Future Works

This study has several limitations. First, ChatGPT’s interface is not
optimized for thematic analysis, yet interactions with this GenAI provide
insights for future interface design. Second, the interview transcript used
has 30 utterances (1,685 tokens) to avoid context length issues (e.g., GPT-4’s
8192-token limit), potentially not reflecting challenges with larger datasets.
Only two participants (P3, P9) with ChatGPT experience highlighted this issue.
Third, the lack of research questions may have limited insights into deductive
thematic analysis challenges. Lastly, we identified opportunities and
challenges but did not explore the specific processes researchers used during
thematic analysis. Future work should design a system informed by this study’s
findings to support thematic analysis and human-AI collaboration. Evaluating
such a system against Cody (Rietz and Maedche, 2021) and PaTAT (Gebreegziabher
et al., 2023) could reveal GenAI’s advantages over prior systems. System
architectures addressing context length limitations, like assessing chunking
strategies’ efficacy (Chen and Yu, 2023), also require more investigation.

##  6\. Conclusion

This study investigated the collaborative experience of qualitative
researchers using the leading GenAI application, ChatGPT, for thematic
analysis. Our findings indicate that researchers value ChatGPT’s potential to
enhance efficiency and deepen qualitative data comprehension. However,
concerns exist regarding trust assurance mechanisms and the practicality of
the current interface for interactive analysis. Drawing from these insights,
we synthesized five design recommendations that aim to inform the development
of future GenAI systems that not only facilitate seamless human-AI
collaboration in qualitative research but also enhance user trust and
confidence in the technology.

###### Acknowledgements.

This research was at least in part funded by the Australian Research Council
(DP210100060) and Jacobs Foundation (Research Fellowship).

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# Schur Polynomials through Lindström Gessel Viennot Lemma

XIONG Rui

###### Abstract

In this article, we use Lindström Gessel Viennot Lemma to give a short,
combinatorial, visualizable proof of the identity of Schur polynomials — the
sum of monomials of Young tableaux equals to the quotient of determinants. As
a by-product, we have a proof of Vandermonde determinant without words. We
also prove the cauchy identity. In the remarks, we discuss factorial Schur
polynomials, dual Cauchy identity and the relation beteen Newton interpolation
formula.

##  1 Introduction

The main purpose of this article is to give a combinatorial proof of the
identity of Schur polynomials

$\sum_{T\textrm{ Young
tableaux}}x^{T}=\frac{\det(x_{i}^{\lambda_{j}+n-j})}{\det(x_{i}^{n-j})}$

using this Lindström Gessel Viennot lemma. The classic algebraic proof, see
Fulton and Harris [1] page 462, proving
$\det(h_{\lambda_{i}+i-j})=\frac{\det(x_{i}^{\lambda_{j}+n-j})}{\det(x_{i}^{n-j})}$.
The classic combinatorial proof (analysing a bijection carefully), see Mendes,
and Remmel [3], page 40.

### Lindström Gessel Viennot Lemma.

Let $\Gamma=(V,E)$ be a locally finite directed acyclic graph. Let
$\\{X_{e}:e\in E\\}$ be a set of pairwise commutative indeterminants. For each
path $P:v_{0}\stackrel{{\scriptstyle
e_{1}}}{{\to}}v_{1}\to\cdots\stackrel{{\scriptstyle e_{k}}}{{\to}}v_{k}$, we
define $X_{P}=X_{e_{1}}\cdots X_{e_{k}}$. For two vertices $u,v\in V$, we
define $e(u,v)$ to be the sum of $X_{P}$ with $P$ going through all path from
$u$ to $v$.

Now, fix some integer $n>0$, and two subsets of order $n$ of $V$, say
$A=\\{a_{1},\ldots,a_{n}\\}$ and $B=\\{b_{1},\ldots,b_{n}\\}$. We have the
following lemma.

######  Lemma 1 (Lindström Gessel Viennot Lemma, [2])

As notations above,

$\det(e(a_{i},b_{j}))=\sum_{\sigma\in\mathfrak{S}_{n}}\operatorname{sgn}\sigma\sum_{\\{P_{1},\ldots,P_{n}\\}}X_{P_{1}}\cdots
X_{P_{n}}$

where $\\{P_{1},\ldots,P_{n}\\}$ goes through all pairwise non-intersecting
paths with each $P_{i}$ from $a_{i}$ to $b_{\sigma(i)}$.

### Schur polynomials.

For $n>0$, the $n$-variable _Schur polynomial_ of a partition
$\lambda:\lambda_{1}\geq\cdots\geq\lambda_{n}$ is defined to be

$S_{\lambda}=\sum_{T}x^{T},$

where $T$ goes through all Young tableaux (weakly increasing in row and
strictly increasing in column) of alphabet $\\{1,\cdots,n\\}$ with shape
$\lambda$. If we denote $N_{i}=\\#T^{-1}\\{i\\}$, then $x^{T}$ is defined to
be $x_{1}^{N_{1}}\cdots x_{n}^{N_{n}}$.

We know the following identity.

######  Theorem 2 (Jacobi-Trudy identity)

As notation above,

$S_{\lambda}=\det(h_{\lambda_{i}+i-j}),$

where $h_{k}$ is the sum of all monomials of degree $k$ in $n$ variables.

One of the combinatorial proof is to use Lindström Gessel Viennot lemma 1
above. Since we will use the proof of it, let me state the proof briefly.

Consider the _lattice graph_ $\Gamma=(V,E)$, with
$V=\mathbb{Z}\times\mathbb{Z}$ the plane lattice points, and $E$ all the edge
from $(i,j)$ to $(i,j+1)$ and $(i+1,j)$. Assign the edge $(i,j)\to(i+1,j)$ by
weight $x_{j}$, the rest by $1$, and consider

$\begin{array}[]{l}A=\\{a_{1}=(1,1),a_{2}=(2,1),\ldots,a_{n}=(n,1)\\};\\\
B=\\{b_{1}=(1+\lambda_{n},n),b_{2}=(2+\lambda_{n-1},n),\ldots,b_{n}=(n+\lambda_{1},n)\\}.\end{array}$

Then the left hand side of Lindström Gessel Viennot Lemma is exactly
$\det(h_{\lambda_{i}+i-j})$, and right hand side is exactly Schur polynomial.
The proof is contained in wikipedia, [6]. For a complete proof, see for
example Prasad [4].

##  2 Main Lemma

Here we consider the same graph $\Gamma=(V,E)$, the lattice graph, as above
but of different weight. More precisely, assign the edge $(i,j)\to(i+1,j)$ by
$x_{j}-x_{i+j}$, and the rest by $1$.

###### Lemma 3

Let $a=(1,1)$ and $b=(m,n)$, with $m,n>0$, then

$e(a,b)=(x_{1}-x_{m+n-1})\cdots(x_{1}-x_{n+2})(x_{1}-x_{n+1}).$

We use the convention that when $m+n-1<n+1$ the above expression equals to
$1$.

This follows from induction. Firstly, the expression holds when $m=1$ or
$n=1$. Generally, if $m,n>1$, let $b^{\prime}=(m-1,n)$ and
$b^{\prime\prime}=(m,n-1)$, as below

$\textstyle{e(a,b^{\prime})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{(x_{m}-x_{m+n-1})}$$\textstyle{e(a,b)}$$\textstyle{\cdots}$$\textstyle{e(a,b^{\prime\prime})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$

Then

$\begin{array}[]{rll}e(a,b)&=e(a,b^{\prime})(x_{n}-x_{m+n-1})+e(a,b^{\prime\prime})\\\
&=(x_{1}-x_{m+n-2})\cdots(x_{1}-x_{n+2})(x_{1}-x_{n+1})(x_{n}-x_{m+n-1})\\\
&+(x_{1}-x_{m+n-2})\cdots(x_{1}-x_{n+2})(x_{1}-x_{n+1})(x_{1}-x_{n})&\textrm{by
induction}\\\
&=(x_{1}-x_{m+n-1})\cdots(x_{1}-x_{n+2})(x_{1}-x_{n+1}).\end{array}$

The proof is complete.

See figure 1 for first several terms.

Figure 1: The table

###### Corollary 4

Let $a=(1,t)$ and $b=(m,n)$, with $m>s>0,n>t>0$. Then

$e(a,b)=(x_{t}-x_{m+n-1})\cdots(x_{t}-x_{n+2})(x_{t}-x_{n+1}).$

In particular, when $0=x_{n+1}=x_{n+2}=\cdots$, $e(a,b)=x_{t}^{m-1}$.

By replacing $(x_{1},x_{2},\ldots)$ by $(x_{t},x_{t+1},\ldots)$.

As an application, we can get the classic Vandermonde determinant from above
computation and Lindström Gessel Viennot Lemma 1.

######  Theorem 5 (Vandermonde)

For $n$ variables, the determinant
$\det(x_{i}^{n-j})=\prod_{i<j}(x_{i}-x_{j})$.

Consider the graph above, with $0=x_{n+1}=x_{n+2}=\cdots$ and consider

$\begin{array}[]{l}A=\\{a_{1}=(1,1),a_{2}=(1,2),\cdots,a_{n}=(1,n)\\}\\\
B=\\{b_{1}=(n,n),b_{2}=(n-1,n),\cdots,b_{n}=(1,n)\\}\end{array}$

Then by 4 above $e(a_{i},b_{j})=x_{i}^{n-j}$. But the only non-intersecting
paths are as figure 2. Apply Lindström Gessel Viennot Lemma 1, we see
$\det(x_{i}^{n-j})=\prod_{i<j}(x_{i}-x_{j})$.

Figure 2: Prove Vandermonde determinant without words

##  3 Schur polynomial by quotient of determinants

###### Theorem 6

The $n$-variable Schur polynomial of a partition
$\lambda:\lambda_{1}\geq\cdots\geq\lambda_{n}$

$S_{\lambda}=\frac{\det(x_{i}^{\lambda_{j}+n-j})}{\det(x_{i}^{n-j})}.$

Step 1.  Remind the proof of Jacobi Trudy identity 2. Since the path from
$(i,1)$ in any set of non-intersecting paths only went vertically before
height $n-i+1$, so it is equivalent to change $(i,1)$ by
$a_{i}^{\prime}=(i,n-i+1)$. That is,

$\det(e(a^{\prime}_{i},b_{j}))=S_{\lambda}.$ $None$

See figure 3.

Figure 3: First step of reduction

Now, if we apply $0=x_{n+1}=x_{n+2}=\cdots$ in the weight of last section,
then in the region $\\{(i,j):i+j\geq n\\}$, it coincides with that in the
proof of Jacobi Trudy identity 2.

Step 2.  Let us consider $a_{i}^{\prime\prime}=(1,n-i+1)$. It is easy to see

$e(a_{i}^{\prime\prime},b_{j})=\sum_{k}e(a_{i}^{\prime\prime},a_{k}^{\prime})\cdot
e(a_{k}^{\prime},b_{j}),$

so

$\det(e(a_{i}^{\prime\prime},b_{j}))=\det(e(a_{i}^{\prime\prime},a_{j}^{\prime}))\cdot\det(e(a_{i}^{\prime},b_{j})).$
$None$

And trivially,

$\det(e(a_{i}^{\prime\prime},a_{j}^{\prime}))=\prod_{i<j}(x_{i}-x_{j})=\det(x_{i}^{n-j}),$
$None$

the Vandermonde determinant 5. See figure 4.

Figure 4: Second step of reduction

Step 3.  By corollary 4,
$e(a_{n+1-i}^{\prime\prime},b_{n+1-j})=x_{i}^{\lambda_{j}+n-j}$, so

$\det(e(a_{i}^{\prime\prime},b_{j}))=\det(x_{i}^{\lambda_{j}+n-j})$ $None$

See figure 5.

Hence the equation $(1,2,3,4)$ proves the assertion.

Figure 5: Last step of reduction

##  4 Cauchy identity

We can use the identical way to prove Cauchy identity.

######  Theorem 7 (Cauchy)

For fixed $n>0$, and $x_{1},\ldots,x_{n},y_{1},\ldots,y_{n}$ two sets of
variables,

$\frac{1}{\prod_{i<j}(x_{i}-x_{j})\prod_{i<j}(y_{i}-y_{j})}\det\left(\frac{1}{1-x_{i}y_{j}}\right)=\sum_{\lambda}S_{\lambda}(x)S_{\lambda}(y),$

with $\lambda$ going through all young diagrams with no more than $n$ rows.

Here we consider a variant of our graph. Consider the graph $\Gamma=(V,E)$,

$V=\\{1,2,3,\ldots\\}\times\\{1,2,\cdots,2n-1,2n\\}$

with edge $(i,j)\to(i,j+1)$ with weight $1$, and $(i,j)\to(i+1,j)$ of weight
$x_{j}-x_{i+j}$ if $j\leq n$, $(i+1,j)\to(i,j)$ of weight
$y_{2n-j}-y_{i+2n-j}$ if $j\geq n+1$. Then apply $0=x_{n+1}=x_{n+2}=\cdots$
and $0=y_{n+1}=y_{n+2}=\cdots$. Consider the points

$a_{1}=(1,1),a_{2}=(1,2),\cdots,a_{n}=(1,n)$

and

$b_{n}=(1,n+1),b_{n-1}=(1,n+2),\cdots,b_{1}=(1,2n).$

Then cutting the line $y=n+1/2$, by corollary 4 we see (see figure 6)

$e(a_{i},b_{j})=\sum_{k=0}^{\infty}x_{i}^{k}y_{j}^{k}=\frac{1}{1-x_{i}y_{j}}.$

Apply Lindström Gessel Viennot Lemma 1. By the same argument of cut, we see
the right hand side is actually $\sum_{\lambda}S_{\lambda}(x)S_{\lambda}(y)$,
see figure 7. The proof is complete.

Figure 6: Cutting the line $y=n+1/2$
Figure 7: Cutting the line $y=n+1/2$

Actually, the left hand side of above identity turns out to be
$\prod_{i,j=1}^{n}\frac{1}{1-x_{i}y_{j}}$, known as Cauchy determinant.

For the classic proof, see Richard and Sergey [5] theorem 7.12.1, using
Robinson Schensted Knuth Algorithm.

##  5 Remarks

### Factorial Schur polynomials.

Firstly, there are also _factorial Schur polynomials_

$S_{\lambda}(x\mid
a)=\sum_{T}\prod_{(i,j)\in\lambda}(x_{T(i,j)}-a_{T(i,j)+i-j})$

with $A$ going through all Young tableaux of $\lambda$. By a completely same
argument, but $x_{n+i}=a_{i}$,

$S_{\lambda}(x\mid a)=\frac{\det[(x_{j}\mid
a)^{\lambda_{i}+n-i}]}{\det(x_{i}^{n-j})}$

with $(y\mid a)^{k}=(y-a_{1})...(y-a_{k})$.

### Dual Cauchy identity.

Secondly, there is also _dual Cauchy identity_ ,

$\prod_{i=1}^{n}\prod_{j=1}^{m}(1+x_{i}y_{j})=\sum_{\lambda}S_{\lambda}(x)S_{\lambda^{\prime}}(y)$

where $\lambda$ goes through all young diagrams of at lost $n$ rows and $m$
columns, and $\lambda^{\prime}$ is the conjugation of $\lambda$. The classic
proof, see Richard and Sergey [5] theorem 7.14.3, using dual Robinson
Schensted Knuth Algorithm. This can be proven like what we did in proof of
Cauchy identity 7 but a little careful about the sign. More precisely,
consider the graph $\Gamma=(V,E)$,

$V=\\{1,2,3,\cdots,m+n\\}\times\\{1,2,3\cdots,m+n-1\\}$

with edge $(i,j)\to(i,j+1)$ with weight $1$, and $(i,j)\to(i+1,j)$ of weight
$x_{j}-x_{i+j}$ if $j\leq n$, $(i,j)\to(i+1,j)$ of weight
$-(y_{m+n-j}-y_{(m+n-i)+(m+n-j)})$ if $j\geq n+1$. Similar, we apply
$0=x_{n+1}=x_{n+2}=\cdots$ and $0=y_{m+1}=y_{m+2}=\cdots$ Then consider

$(1,1),(1,2),\ldots,(1,n),\quad(m+n,n),(m+n,n+1),\ldots,(m+n,m+n-1)$

and

$(1,n),(2,n),\ldots,(m+n,n).$

See figure 8. The sign of permutation cancels the sign of $y_{\bullet}$’s.
Apply Lindström Gessel Viennot Lemma, 1, the left hand side is

$\det\left(\begin{matrix}x_{1}^{m+n-1}&\cdots&x_{n}^{m+n-1}&(-y_{1})^{0}&\cdots&(-y_{m})^{0}\\\
\vdots&\ddots&\vdots&\vdots&\ddots&\vdots\\\
x_{1}^{0}&\cdots&x_{n}^{0}&(-y_{n})^{m+n-1}&\cdots&(-y_{m})^{m+n-1}\end{matrix}\right)$
$None$

and the right hand side is

$\prod_{1\leq i<j\leq n}(x_{i}-x_{j})\prod_{1\leq i<j\leq
m}(y_{i}-y_{j})\sum_{\lambda}S_{\lambda}(x)S_{\lambda^{\prime}}(y).$

By a direct computation of $(*)$ (a good exercise of linear algebra), we get
dual Cauchy identity.

Figure 8: Dual Cauchy identity

### Newton interpolation formula.

Last but no mean least, our weight chosen is relative to _Newton interpolation
formula_. If we view $x_{0}=x$ as variable, and $x_{1},x_{2},\ldots,x_{n}$ as
data points, as usual $0=x_{n+1}=x_{n+2}=\cdots$. Cutting the path from
$(1,0)$ to $(n,n)$ by $y=\frac{1}{2}$, see figure 9, by corollary 4, we have
the following identity

$\begin{array}[]{rl}x^{n}&=e(a_{0},b)\\\ &\quad+e(a_{1},b)\cdot(x-x_{1})\\\
&\qquad+e(a_{2},b)\cdot(x-x_{1})(x-x_{2})+\cdots\\\
&\qquad\quad+e(a_{n},b)\cdot(x-x_{1})\cdots(x-x_{n}),\end{array}$

with $a_{i}=(i,1)$, $N=(n,n)$.

Figure 9: Newton interpolation formula

One can show by induction that for each point $(i,j)$ with $0\leq i,j\leq n$,
$e((i,j),b)$ corresponds to a divided differences of $x^{n}$. More exactly,
consider the inverse diagram, and stare at the following diagram.

$\textstyle{\vdots\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\vdots\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{x_{2}^{n}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\diamondsuit\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{(x_{2}-x_{3})}$$\textstyle{\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\cdots}$$\textstyle{x_{1}^{n}}$$\textstyle{\heartsuit\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{(x_{1}-x_{2})}$$\textstyle{\ignorespaces\ignorespaces\ignorespaces\ignorespaces\square}$$\scriptstyle{(x_{1}-x_{3})}$$\textstyle{\ignorespaces\ignorespaces\ignorespaces\ignorespaces\cdots}$

Since $\heartsuit\cdot(x_{1}-x_{2})+x_{2}^{n}=x_{1}^{n}$, so
$\heartsuit=\frac{x_{1}^{n}-x_{2}^{n}}{x_{1}-x_{2}}$. The same reason,
$\diamondsuit=\frac{x_{2}^{2}-x_{3}^{n}}{x_{2}-x_{3}}$. Therefore
$\square=\frac{\heartsuit-\diamondsuit}{x_{1}-x_{3}}$, and so on.

## References

  * [1] William Fulton and Joe Harris.  Representation theory. A first course., volume 129.  New York etc.: Springer-Verlag, 1991.
  * [2] Ira M Gessel and Xavier Viennot.  Determinants, paths, and plane partitions.  preprint, 132(197.15), 1989.
  * [3] Anthony Mendes and Jeffrey Remmel.  Counting with symmetric functions.  Springer, 2015.
  * [4] Amritanshu Prasad.  An introduction to schur polynomials.  2018\.
  * [5] Richard P. Stanley and Sergey Fomin.  Enumerative Combinatorics, volume 2 of Cambridge Studies in Advanced Mathematics.  Cambridge University Press, 1999.
  * [6] Wikipedia contributors.  Lindström–gessel–viennot lemma — Wikipedia, the free encyclopedia, 2019.  [Online; accessed 19-March-2020].