On the Average Size of an $(\overline{s},\overline{t})$-Core Partition

Wang, Joseph L. P.; Yang, Jane Y. X.

doi:10.11650/tjm/181006

Cited by 6 publications

(5 citation statements)

References 8 publications

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“…According to the Ministry of Finance statistics, Taiwan's apparel retail industry reached NTD 108.3 billion in 2018. In 2019, Kantar Insights Taiwan conduct a survey questionnaire [71]. The results found that Taiwanese consumers buy an average of 11 pieces of clothing per year.…”

Section: Research Design and Datamentioning

confidence: 99%

Exploring the Key Elements of Sustainable Design from a Social Responsibility Perspective: A Case Study of Fast Fashion Consumers’ Evaluation of Green Projects

Fan

Chang

2023

Sustainability

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With the growing global awareness of sustainable development, the impact of “fast fashion” on the environment has drawn much attention. This study investigated 365 fast fashion consumers with higher-than-average spending on fast fashion garments in Taiwan and analyzed the data with the analytic network process. From the perspective of social responsibility theory, the triple bottom line framework, and the concept of cradle-to-cradle, this study conducted in-depth interviews with 10 experts in the field of clothing. Based on the results, this study constructed a model with 4 core dimensions and 12 key criteria of sustainable clothing design, with the enterprise, design, and consumer sides as the decision-making options. Moreover, it was found that consumers believe that the first decision-making key that affects sustainable design is the enterprise side, and the most important green item that affects sustainable design is “cherishing”. The findings of this study can provide an important reference standard for sustainable design in the apparel industry.

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Section: Research Design and Datamentioning

confidence: 99%

Exploring the Key Elements of Sustainable Design from a Social Responsibility Perspective: A Case Study of Fast Fashion Consumers’ Evaluation of Green Projects

Fan

Chang

2023

Sustainability

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“…In Figure 2, one can see that λ = (7, 6, 3, 2) is a 4-core partition, but h * 2,3 (λ) = 8. Later, Wang and Yang [11] gave a characterization of s-core partitions. Proposition 2.2.…”

Section: Properties and Generating Functionsmentioning

confidence: 99%

“…For (s, t)-cores with coprime even s and odd t, Wang and Yang [11] defined the Yin-Yang diagram to be an array B(s, t), where…”

Section: Enumeration On (S T)-coresmentioning

confidence: 99%

“…There are a couple of results on counting the number of simultaneous core partitions of the three objects, bar-cores, CSYDs, and doubled distinct cores. Bessenrodt and Olsson [3] adopted the Yin-Yang diagram to count the number of (s, t)-core partitions for odd numbers s and t, Wang and Yang [11] counted the same object when s and t are in different parity, and Ding [5] counted the number of (s, s + 1)-CSYDs (as far as the authors know these are the only counting results on the three objects known until now). Our main goal is to fill out all the possible results we could get on (s, t)-cores and (s, s + d, s + 2d)-cores for the three objects by constructing some bijections.…”

Section: Introductionmentioning

confidence: 99%

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Results on bar-core partitions, core shifted Young diagrams, and doubled distinct cores

Cho¹,

Nam²,

Sohn³

2022

Preprint

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Simultaneous bar-cores, core shifted Young diagrams (or CSYDs), and doubled distinct cores have been studied since Morris and Yaseen introduced the concept of bar-cores. In this paper, our goal is to give a formula for the number of these core partitions on (s, t)-cores and (s, s + d, s + 2d)-cores for the remaining cases that are not covered yet. In order to achieve this goal, we observe a characterization of s-core partitions to obtain characterizations of doubled distinct s-core partitions and s-CSYDs. By using them, we construct N E lattice path interpretations of these core partitions on (s, t)-cores. Also, we give free Motzkin path interpretations of these core partitions on (s, s + d, s + 2d)-cores.

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“…[We note that J. Wang and J. Yang [30] have recently extended the Yin-Yang diagram of Bessenrodt and Olsson to the case when s is even and t is odd. We do not consider this case here.]…”

Section: Bijectionsmentioning

confidence: 99%

Simultaneous core partitions with nontrivial common divisor

Gramain¹,

Nath²,

Sellers³

2019

Preprint

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A tremendous amount of research has been done in the last two decades on (s, t)-core partitions when s and t are positive integers with no common divisor. Here we change perspective slightly and explore properties of (s, t)-core and (s, t)-core partitions for s and t with nontrivial common divisor g.We begin by revisiting work by D. Aukerman, D. Kane and L. Sze on (s, t)core partitions for nontrivial g before obtaining a generating function for the number of (s, t)-core partitions of n under the same conditions. Our approach, using the g-core, g-quotient and bar-analogues, allows for new results on tcores and self-conjugate t-cores that are not g-cores and t-cores that are not ḡ-cores, thus strengthening positivity results of K. Ono and A. Granville, J. Baldwin et. al., and I. Kiming.We then detail a new bijection between self-conjugate (s, t)-core and (s, t)core partitions for s and t odd with odd, nontrivial common divisor g. Here the core-quotient construction fits remarkably well with certain lattice-path labelings due to B. Ford, H. Mai, and L. Sze and C. Bessenrodt and J. Olsson. Along the way we give a new proof of a correspondence of J. Yang between self-conjugate t-core and t-core partitions when t is odd and positive.We end by noting (s, t)-core and (s, t)-core partitions inherit Ramanujantype congruences from those of g-core and ḡ-core partitions.

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On the Average Size of an $(\overline{s},\overline{t})$-Core Partition

Cited by 6 publications

References 8 publications

Exploring the Key Elements of Sustainable Design from a Social Responsibility Perspective: A Case Study of Fast Fashion Consumers’ Evaluation of Green Projects

Exploring the Key Elements of Sustainable Design from a Social Responsibility Perspective: A Case Study of Fast Fashion Consumers’ Evaluation of Green Projects

Results on bar-core partitions, core shifted Young diagrams, and doubled distinct cores

Simultaneous core partitions with nontrivial common divisor

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