• Journal of the Korean Mathematical Society
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• v.54 no.4
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• pp.1149-1161
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• 2017

An ($n_1,\;n_2,\;{\ldots},\;n_k$)-colored permutation is a permutation of $n_1+n_2+{\cdots}+n_k$ in which $1,\;2,\;{\ldots},\;n_1$ have color 1, and $n_1+1,\;n_1+2,\;{\ldots},\;n_1+n_2$ have color 2, and so on. We give a bijective proof of Steinhardt's result: the number of colored permutations with no monochromatic cycles is equal to the number of permutations with no fixed points after reordering the first $n_1$ elements, the next $n_2$ element, and so on, in ascending order. We then find the generating function for colored permutations with no monochromatic cycles. As an application we give a new proof of the well known generating function for colored permutations with no fixed colors, also known as multi-derangements.

• Journal of Oral Medicine and Pain
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• v.45 no.3
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• pp.56-64
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• 2020

Purpose: Temporomandibular disorders (TMD) is a mosaic of clinical signs and symptoms that can be regarded as a set of phenotypes that are affected by various factors including pain sensitivity, pain disability, sleep and psychological functioning. The aims of this study were to evaluate association of pain experience, sleep quality and psychological distress with different phenotypes of TMD patients. Methods: This retrospective study included a cohort (n=1,858; 63.8% for female, mean age=34.9±15.9 years) of patients with TMD. A set of self-administered questionnaires concerning pain interference (Brief Pain Inventory), pain disability (Graded Chronic Pain Scale), sleep quality (Pittsburg Sleep Questionnaire Index), psychological distress (Symptom Checklist-90 revised), and pain catastrophizing (Pain Catastrophizing Scale) were administered to all participants at the first consultation. All TMD patients were classified into four groups including TMD with internal derangement without pain (TMD_ID, n=370), TMD with joint pain (TMD_J, n=571), TMD with muscle pain (TMD_M, n=541) and TMD with muscle-joint combined pain (TMD_MJ, n=376). Results: The female ratio was particularly high in the group with TMD_MJ (p=0.001). The patients with muscle pain and both muscle and joint pain had longer symptom duration (p=0.004) and presented significantly higher scores in pain experience (p<0.001), subjective sleep quality (p<0.001), pain catastrophizing (p<0.001) and psychological distress (p<0.05) except for paranoid-ideation than the groups with only joint problems. Conclusions: The results of this study highlight the importance of multi-dimensional approach that consider pain disability, sleep quality, and psychological functioning in the management of TMD with muscle component. This study would contribute to a better understanding of interaction between heterogeneous TMD and multiple risk factors in order to build tailored treatment based on different phenotypes.