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Let $$\{d_{n}^{(v)}\}_{n\ge 0}$$ be the sequence of generalized derangement numbers, where v is a nonnegative integer. In this paper, we mainly study the log-balancedness of $$\{d_{n}^{(v)}\}_{n\ge 0}$$ , where $$v\ge 1$$ . We prove that $$\{d_{n}^{(1)}\}_{n\ge 1}$$ and $$\{d_{n}^{(v)}\}_{n\ge 0}$$ ( $$v\ge 2$$ ) are log-balanced. In addition, we discuss the log-balancedness of some sequences involving $$d_{n}^{(v)}$$ . For example, we show that $$\{d_{n+1}^{(1)}-d_{n}^{(1)}\}_{n\ge 1}$$ is log-balanced.

The aim of this clinical study was to evaluate the level of Interleukine-6 (IL-6), pre and post arthrocentesis to validate it as a biomarker in the Internal Derangement (ID) of TMJ. This study included 30 patients (20 females and 10 males) of Temporo-Mandibular Dysfunction (TMD) with Disc displacement without reduction (DDwoR) Wilkes stage III, who were refractory to conservative management. Arthrocentesis was performed as a therapeutic modality. Synovial fluid aspirates were obtained prior to arthrocentesis and post arthrocentesis with 300 ml of Ringer Lactate solution into the superior joint compartment for the assessment of level of IL-6. The clinical parameters used for correlating the level of IL-6 were degree of pain (VAS I), chewing ability (VAS II), Maximal Mouth Opening (MMO) in both pre and post op phase with the follow-up period of 01 day, 01 week, 01 month, 03 month and 06 month and the results were compared. ELISA was performed to analyze the levels of IL-6 in the aspirates. The clinical parameters and the level of IL-6 were recorded and analyzed statistically. The study showed ID of TMJ (Wilkes stage III) s are more prevalent in females especially in the fourth decades of life with the mean age of 38.4 years. The post operative assessment in terms of pain, maximum mouth opening, lateral movements of the mandible and the levels of IL-6 were found to be statistically significant with a P value

The purpose of this study was to (1) report on the incidence of concurrent surgical pathology at the time of adolescent ACL reconstruction, (2) evaluate patient risk factors for concurrent pathology, and (3) measure the effect of BMI on operating room (OR) time. A retrospective analysis of the NSQIP database for the years 2005–2017 was conducted. Nine-hundred and seventeen patients 18 years of age and younger who underwent ACL reconstruction (ACLR) were identified using CPT code 29888 and patients undergoing surgery for multi-ligamentous knee injuries were excluded. The mean patient age was 17.6 years (range 14–18, standard deviation 0.52) and consisted of 546 males (59.5%) and 371 females (40.5%). Logistic regression was used to assess the relationship between BMI and additional CPT codes for internal derangement at the time of surgery. Internal derangement was defined as any procedure for the treatment of a meniscal tear, chondral lesion, or loose body removal. Linear regression analysis was then performed to evaluate the effect of BMI on operative time. 43.7% of patients undergoing ACLR required an associated procedure for internal derangement. Additionally, the risk of requiring additional procedures for internal derangement increased by 3.1% per BMI point. BMI was also predictive of operative time, independent of the number of additional procedures. Specifically, the operative time increased by nearly one minute for every point increase in BMI (58.0 s). Adolescent patients with an elevated BMI were much more likely to require additional surgical procedures for internal derangement at the time of ACL reconstruction. Additionally, BMI was a significant predictor for longer operative times. Level III.

Arthrocentesis is a simple minimally invasive procedure with less risk of complications and significant benefits in patients with internal derangement of temporomandibular joint. Here we report a study on four patients to evaluate the efficacy of arthrocentesis in the treatment of internal derangement of temporomandibular joint.

The Eulerian polynomial $$ \mathrm {AExc}_n(t)$$ enumerating excedances in the symmetric group $$\mathfrak {S}_n$$ is known to be gamma positive for all n. When enumeration is done over the type B and type D Coxeter groups, the type B and type D Eulerian polynomials are also gamma positive for all n. We consider $$ \mathrm {AExc}_n^+(t)$$ and $$ \mathrm {AExc}_n^-(t)$$ , the polynomials which enumerate excedance in the alternating group $$\mathcal {A}_n$$ and in $$\mathfrak {S}_n - \mathcal {A}_n$$ , respectively. We show that $$ \mathrm {AExc}_n^+(t)$$ is gamma positive iff $$n \ge 5$$ is odd. When $$n \ge 4$$ is even, $$ \mathrm {AExc}_n^+(t)$$ is not even palindromic, but we show that it is the sum of two gamma positive summands. An identical statement is true about $$ \mathrm {AExc}_n^-(t)$$ . We extend similar results to the excedance based Eulerian polynomial when enumeration is done over the positive elements in both type B and type D Coxeter groups. Gamma positivity results are known when excedance is enumerated over derangements in $$\mathfrak {S}_n$$ . We extend some of these to the case when enumeration is done over even and odd derangements in $$\mathfrak {S}_n$$ .

It is necessary to clarify that temporomandibular joint disorders (TMDs) are one of the most misdiagnosed and mistreated maladies in the medical practice. It is an umbrella term, embracing conditions which involve the temporomandibular joint (TMJ) and related muscles. One of these common irritating disorders is the internal derangement which is used specifically to describe the displacement of the TMJ disc. The treatment can vary according to the severity and chronicity into non-invasive, minimally invasive, and invasive procedures. However, permanent recovery is rarely obtained. This study was established to compare the effectiveness of two minimally invasive procedures: arthrocentesis and glucocorticosteroid (GCS) local single joint injection in the management of internal derangement of the TMJ. Thirty patients aged from 18 to 42 years were included in this study with internal derangement which was confirmed clinically and with a cone beam CT scan. The patients were divided into two groups of 15 patients. Arthrocentesis was performed to one group (group A) by using Shepard’s cannula and lactated Ringer’s solution. Glucocorticosteroid injection was done to the other group (group B) using a 1 ml/40 mg methylprednisolone acetate vial. The study was performed in the Department of Oral and Maxillofacial Surgery, Ghazi Alhareery Hospital—Medical City, from October 2017 to September 2018. After 4 months of clinical follow-up, the results revealed that the GCS injection has minimal outcomes in the treatment of TMJ internal derangement compared to arthrocentesis. On the other hand, arthrocentesis and lavage had dedicated promising outcomes.

This prospective clinical study aimed to establish a new classification system for TMJ internal derangement based on MRI in correlation with clinical findings contributing to a nonsurgical treatment protocol. A consecutive sample of 435 internal derangement patients was enrolled in the study. Clinical and MRI studies were used to establish the new classification system. A total of 747 joints were classified according to our staging system and received treatment according to the associated nonsurgical treatment protocol. The primary outcome variables were maximum voluntary mouth opening and visual analogue scale pain scores. The secondary outcome variable was joint sound. Statistical analysis of the differences between pretreatment and posttreatment measurements showed an increase in mouth opening throughout the study period (P VAS scores showed a statistically significant decrease in all study groups during all study periods, with P joint sounds showed significant improvement during all study periods. The new classification system is a simple, & reasonable including a detailed description of all the pathologic changes of the joint. The nonsurgical treatment protocol was Simple, effective and specific depending on the pathological changes in joint.

Background Rapid response systems aim to achieve a timely response to the deteriorating patient; however, the existing literature varies on whether timing of escalation directly affects patient outcomes. Prior studies have been limited to using ‘decision to admit’ to critical care, or arrival in the emergency department as ‘time zero’, rather than the onset of physiological deterioration. The aim of this study is to establish if duration of abnormal physiology prior to critical care admission [‘Score to Door’ (STD) time] impacts on patient outcomes. Methods A retrospective cross-sectional analysis of data from pooled electronic medical records from a multi-site academic hospital was performed. All unplanned adult admissions to critical care from the ward with persistent physiological derangement [defined as sustained high National Early Warning Score (NEWS) > / = 7 that did not decrease below 5] were eligible for inclusion. The primary outcome was critical care mortality. Secondary outcomes were length of critical care admission and hospital mortality. The impact of STD time was adjusted for patient factors (demographics, sickness severity, frailty, and co-morbidity) and logistic factors (timing of high NEWS, and out of hours status) utilising logistic and linear regression models. Results Six hundred and thirty-two patients were included over the 4-year study period, 16.3% died in critical care. STD time demonstrated a small but significant association with critical care mortality [adjusted odds ratio of 1.02 (95% CI 1.0–1.04, p = 0.01)]. It was also associated with hospital mortality (adjusted OR 1.02, 95% CI 1.0–1.04, p = 0.026), and critical care length of stay. Each hour from onset of physiological derangement increased critical care length of stay by 1.2%. STD time was influenced by the initial NEWS, but not by logistic factors such as out-of-hours status, or pre-existing patient factors such as co-morbidity or frailty. Conclusion In a strictly defined population of high NEWS patients, the time from onset of sustained physiological derangement to critical care admission was associated with increased critical care and hospital mortality. If corroborated in further studies, this cohort definition could be utilised alongside the ‘Score to Door’ concept as a clinical indicator within rapid response systems.

In the original article there were capitalization errors in two of the references. The corrected references follow

We derive new congruences bounding r-Bell and derangement polynomials, which generalize the existing ones, while the presented approach is significantly simpler and, at the same time, more informative. Namely, we provide precise identities that imply the congruences and explain somehow their nature.

We provide a very short proof of a Qi-Zhao-Guo closed form for derangements numbers based on the determinants of certain Hessenberg matrices. The proof is grounded on a basic result on finite sequences and an inductive argument.

In this paper we consider the derangement graph for the group $$\mathop {\text {PSU}}(3,q)$$ , where q is a prime power. We calculate all eigenvalues for this derangement graph and use the eigenvalues to prove that $$\mathop {\text {PSU}}(3,q)$$ , under its two-transitive action on a set of size $$q^3+1$$ , has the Erdős-Ko-Rado property and, provided that $$q\ne 2, 5$$ , another property that we call the Erdős-Ko-Rado module property.

Let $$D_h$$ , $$E_k$$ and $$F_{\alpha }$$ be sets of size $$h,k,\alpha $$ respectively, with $$k \le h$$ . We define a strongly widened derangement to be a permutation of $$D_h\cup E_k\cup F_{\alpha }$$ such that the elements of $$D_h$$ are not fixed and the elements of $$E_k$$ cannot occupy a site originally occupied by an object of the same type or by an object of $$F_{\alpha }$$ . We show a connection between strongly widened derangements and generalized Laguerre polynomials that provides a generalization, for integer values of $$\alpha $$ , of Even and Gillis (Math Proc Camb Philos Soc, 79:135–143, 1976) different from the one presented in Foata and Zeilberger (SIAM J Discrete Math, 1:425–433, 1988).

In this paper, we answer two questions on local $h$-vectors, which were asked by Athanasiadis. First, we characterize all possible local $h$-vectors of quasi-geometric subdivisions of a simplex. Second, we prove that the local $\gamma$-vector of the barycentric subdivision of any CW-regular subdivision of a simplex is nonnegative. Along the way, we derive a new recurrence formula for the derangement polynomials., Comment: 15 pages, 4 figures

In this paper, we derive a formula for the eigenvalues of the matching derangement graph. The formula gives an insight regarding the alternating sign conjecture for the eigenvalues of the matching derangement graph. In particular, we show that the alternating sign property holds for certain partitions.

The aim of this study was to investigate possible correlation of specific skeletal or dental class in children and adolescents with clinical signs of temporomandibular dysfunction (TMD) with the severity of internal derangement (ID) of the temporomandibular joint.Based on MRI images, the ID of 232 juvenile temporomandibular joints in 116 patients were retrospectively recorded. The distribution of the ID stages within the skeletal and dental classes was compared by means of the χExcluding the comparison between skeletal Class I (S I) and skeletal Class II (S II; p 0.05), no statistically significant differences in the distribution of the ID stages were found between the skeletal classes (p 0.05). No statistically significant differences were found when comparing the distribution of the ID stages between the dental classes (p 0.05).According to these findings, there is no skeletal or dental class that is related to higher degrees of internal derangement in the TMJs of children and adolescents presenting clinical signs of TMD. Therefore, it is not possible to draw conclusions about the severity of the ID in relation to the dental and skeletal class in symptomatic juvenile TMJs.

A derangement is a permutation that has no fixed points. In this paper, we are interested in the proportion of derangements of the finite affine general linear groups. We prove a remarkably simple and explicit formula for this proportion. We also give a formula for the proportion of derangements of prime power order. Both formulae rely on a result of independent interest on partitions: we determine the generating function for the partitions with m parts and with the kth largest part not k, for every $$k\in \mathbb {N}$$kźN.

The derangement graph is the Cayley graph on the symmetric group $$\mathcal {S}_{n}$$Sn whose generating set $$D_{n}$$Dn is the set of permutations on $$[n]=\{1, \ldots , n\}$$[n]={1,?,n} without any 1-cycle. For any fixed positive integer $$k \le n$$k≤n, the Cayley graph generated by the subset of $$D_{n}$$Dn consisting of permutations without any i-cycles for all $$1 \le i \le k$$1≤i≤k is a regular subgraph of the derangement graph. In this paper, we determine the smallest eigenvalue of these subgraphs and show that the set of all the largest independent sets in these subgraphs is equal to the set of all the largest independent sets in the derangement graph, provided n is sufficiently large in terms of k.

The problem of counting derangements was initiated by Pierre Rémond de Montmort in 1708 (see Carlitz in Fibonacci Q. 16(3):255–258, 1978, Clarke and Sved in Math. Mag. 66(5):299–303, 1993, Kim, Kim and Kwon in Adv. Stud. Contemp. Math. (Kyungshang) 28(1):1–11 2018. A derangement is a permutation that has no fixed points, and the derangement number \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$d_{n}$\end{document}dn is the number of fixed-point-free permutations on an n element set. In this paper, we study the derangement polynomials and investigate some interesting properties which are related to derangement numbers. Also, we study two generalizations of derangement polynomials, namely higher-order and r-derangement polynomials, and show some relations between them. In addition, we express several special polynomials in terms of the higher-order derangement polynomials by using umbral calculus.

Let $G$ be a transitive permutation group on a finite set of size at least $2$. By a well known theorem of Fein, Kantor and Schacher, $G$ contains a derangement of prime power order. In this paper, we study the finite primitive permutation groups with the extremal property that the order of every derangement is an $r$-power, for some fixed prime $r$. First we show that these groups are either almost simple or affine, and we determine all the almost simple groups with this property. We also prove that an affine group $G$ has this property if and only if every two-point stabilizer is an $r$-group. Here the structure of $G$ has been extensively studied in work of Guralnick and Wiegand on the multiplicative structure of Galois field extensions, and in later work of Fleischmann, Lempken and Tiep on $r'$-semiregular pairs., 30 pages; to appear in Manuscripta Math

We derive the asymptotics of certain combinatorial numbers defined on multi-sets when the number of sets tends to infinity but the sizes of the sets remain fixed. This includes the asymptotics of generalized derangements, numbers related to k-partite graphs, and exponentially weighted derangements. The asymptotics use integral and sum representations of the numbers involved. We also explore the combinatorial implications of the asymptotic results. In fact we first derive general asymptotic formulas for integrals and sums of certain types and then we specialize them to study the asymptotics of the combinatorial numbers.

The aim of the paper is a new transformation of certain infinite series based on an identity from Sylvester also known as the non-local derangement identity. Surprisingly, the result found can be applied to evaluate certain types of the generalized hypergeometric function p F q (z) or infinite series involving the combination of hyperbolic functions, irrational functions and polynomials like some of which seem to be new. On the other hand, for those which are known, our approach provides a new method to derive the discussed identities using a simple algorithm.

The paper proposes a general approach to generating permutations that contain cycles, based on constructive tools introduced to describe combinatorial sets. Different generation problems for permutations of definite class are formulated and solved. A combinatorial set is introduced to define permutations represented as the multiplication of a definite number of cycles. For this set, combinatorial species and associated generating series are constructed.

We count derangements, involutions and unimodal elements in the wreath product Cr ≀ Sn by the numbers of excedances, fixed points and 2-cycles. Properties of the generating functions, including combinatorial formulas, recurrence relations and real-rootedness are studied. The results obtained specialize to those on the symmetric group Sn and on the hyperoctahedral group Bn when r = 1, 2, respectively.

The term internal derangement was first adopted to describe any pathologic entity that interfered with the smooth function of the temporomandibular joint (TMJ). The term is currently used exclusively to describe alterations in disc–fossa relations. Internal derangement of the TMJ does not always cause pain, although when the disc becomes displaced, noises and locking can occur and ligamentous, capsular or retrodiscal pain may dominate the clinical picture. Meniscopexy of TMJ was done in ten patients where clinical features suggest true internal derangement of the TMJ refractory to conservative treatment with restricted movement and pain identified as arising primarily from the joint. Magnetic resonance imaging was done to identify the cause of problem.

Given a permutation τ of length j, we say that a permutation σ has a τ-match starting at position i, if the elements in positions i, i+1, . . . , i+j−1 in σ have the same relative order as the elements of τ. We have been able to take advantage of the results of Mendes and Remmel [1] to obtain a generating function for the number of permutations with no τ-matches for several new classes of permutations. These new classes include a large class of permutations which are shuffles of an increasing sequence 1 2 · · · n with an arbitrary permutation σ of the integers {n + 1, . . . , n + m}. Finally we give a formula for the generating function for the number of permutations which have no 1 3 2 4-matches.

Many measures have been developed to determine the extent of disc displacement in internal derangements of the temporomandibular joint (TMJ) using magnetic resonance imaging. The purpose of this study was to develop a quantitative method of analyzing disc position and to evaluate the positions of the disc in internal derangements of the TMJ (group 1, with reduction; group 2, without reduction). Magnetic resonance images of 150 TMJs in 20 healthy volunteers and 55 patients with internal derangements were evaluated. The anatomical points of interest of the TMJ, including the anterior (DA) and posterior (DP) points of the disc, were marked on parasagittal magnetic resonance images of the TMJ disc taken in both the closed- and the open-mouth positions. All points were recorded using an x-y coordinate system, with reference to a referral line. In the closed-mouth position, the DP in patients in group 1 was situated in a more-anterior direction than the DP in volunteers. The DP in group 2 was located further anterior and inferior than the DP in group 1. However, the position of the DA did not differ between group 1 and group 2. In the open-mouth position, the DP was displaced anteroinferiorly to a greater extent in group 2 than in group 1 (one-way ANOVA, followed by Scheffe's test; P < 0.0001). The distance between the disc points in the closed- and open-mouth positions was also evaluated. Comparison of the disc point position in the closed- and open-mouth positions in symptomatic and asymptomatic displaced TMJ discs revealed no significant difference. In conclusion, most of our results quantitatively support previously reported findings in imaging, surgical, and histopathological studies of TMJ internal derangement. We suggest that our measure of disc position of the TMJ would be useful to assess the status and response to treatment of internal derangements of the TMJ.

In this paper, we study the numbers D n,k which are defined as the number of permutations σ of the symmetric group S n such that σ has no cycles of length j for j ≤ k. In the case k = 1, D n,1 is simply the number of derangements of an n-element set. As such, we shall call the numbers D n,k generalized derangement numbers. Garsia and Remmel [4] defined some natural q-analogues of D n,1, denoted by D n,1(q), which give rise to natural q-analogues of the two classical recursions of the number of derangements. The method of Garsia and Remmel can be easily extended to give natural p, q-analogues D n,1(p, q) which satisfy natural p, q-analogues of the two classical recursions for the number of derangements. In [4], Garsia and Remmel also suggested an approach to define q-analogues of the numbers D n,k . In this paper, we show that their ideas can be extended to give a p, q-analogue of the generalized derangements numbers. Again there are two classical recursions for eneralized derangement numbers. However, the p, q-analogues of the two classical recursions are not as straightforward when k ≥ 2.

PURPOSE: The purposes of this study were to examine the occurrence of different types of mandibular lateral translation (ΔYMLT) using condylography and magnetic resonance imaging (MRI), to investigate the relationship between ΔYMLT and the disunion pattern in intracapsular derangement patients with ΔYMLT, and to consider the three-dimensional therapeutic position of the condyle in the mandibular fossa. MATERIAL AND METHODS: Sixty-two patients with ΔYMLT, or side shift, during symmetrical mandibular movements were analyzed with computerized condylography and MRI to determine the occurrence of different types of MLT and the relationship between the ΔYMLT and the intracapsular derangement pattern. RESULTS: Of a total of 62 subjects, we classified 43 (69.4%), 12 (9.4%), and 7 (11.2%) of them as ΔYMLT type, closed lock type, and excessive condylar rotation type, respectively. There was a high prevalence of ΔYMLT for subjects who had one side antero-medial (including medial sideway) and the other side antero-lateral disk displacement. The directions of ΔYMLT and disk displacement coincided in 95.3% of cases; this means recapturing of the displaced disk by the condyle in lower joint translation. CONCLUSION: Results indicate that mandibular lateral displacement causes the condyle to be displaced in the same direction, and the disk displacement to the opposite side; hence it creates ΔYMLT in the disk displacement direction. On the basis of these findings, it is concluded that the ΔYMLT is a useful indicator for early detection of intracapsular derangement and that the ΔYMLT allows estimation of the therapeutic condylar position.

The paper deals with a new method of solving a combinatorial problem with account for the properties of the set of permutations and its structure. Using this method, the values of the linear objective function are sequenced and the set of permutations is decomposed over hyperplanes, with account of element recurrences. This makes it possible to develop an algorithm of finding the point (an element of the set of permutations) at which the objective function attains a given value.

We show that some properties of permutation groups can be applied to the construction of combinatorial objects with given properties.

The flag-major index "fmaj" and the classical length function "$\ell$" are used to construct two $q$-analogs of the generating polynomial for the hyperoctahedral group~$B_n$ by number of positive and negative fixed points (resp. pixed points). Specializations of those $q$-analogs are also derived dealing with signed derangements and desarrangements, as well as several classical results that were previously proved for the symmetric group., Comment: 21 pages

A pattern class is a set of permutations closed under the formation of subpermutations. Such classes can be characterized as those permutations not involving a particular set of forbidden permutations. A simple collection of necessary and sufficient conditions on sets of forbidden permutations which ensure that the associated pattern class is of polynomial growth is determined. A catalogue of all such sets of forbidden permutations having three or fewer elements is provided together with bounds on the degrees of the associated enumerating polynomials.

The purpose of this study was to compare the accuracy of two diagnostic modalities: clinical examination and condylography with magnetic resonance imaging (MRI), in diagnosing intracapsular derangement of temporomandibular joint (TMJ). A total of 38 symptomatic (mean age of 26 years) and 20 asymptomatic (mean age of 21 years) subjects underwent thorough evaluation by the three diagnostic modalities. Findings were interpreted by three experienced investigators using diagnostic categories, then the agreement between the diagnostic modalities was studied. Clinical examination and condylography both alone and in combination, was not reliable for structural diagnostic. However with regard of the condyle-disk disunion without reposition, the two diagnostic modalities demonstrated a better agreement with MRI and the combination of the two diagnostic modalities increased significantly the percentage of agreement with MRI (89% to 96% of agreement). With regard of disunion without reposition, the combination of clinical examination with condylography is a valid diagnostic modality. The low reliability of clinical examination and condylography with regard of both normality and disunion with reposition could probably be explained by the young age of the symptomatic population.

In this note we obtain and discuss formulae for the number of partial one-one transformations (of an n-element set) of height (equivalently, width) r and having exactly k fixed points. Moreover, we consider a generalization of the 'Hat Problem' or 'Probleme des Rencontres'.

Billey, Jockusch, and Stanley characterized 321-avoiding permutations by a property of their reduced decompositions. This paper generalizes that result with a detailed study of permutations via their reduced decompositions and the notion of pattern containment. These techniques are used to prove a new characterization of vexillary permutations in terms of their principal dual order ideals in a particular poset. Additionally, the combined frameworks yield several new results about the commutation classes of a permutation. In particular, these describe structural aspects of the corresponding graph of the classes and the zonotopal tilings of a polygon defined by Elnitsky that is associated with the permutation., Comment: 19 pages, 6 figures; to appear in J. Alg. Combin

C. Radoux (J. Comput. Appl. Math.115 (2000) 471–477) obtained a computational formula of Hankel determinants on some classical combinatorial sequences such as Catalan numbers and polynomials, Bell polynomials, Hermite polynomials, Derangement polynomials etc. From a pair of matrices this paper introduces two kinds of numbers. Using the first kind of numbers we give a unified treatment of Hankel determinants on those sequences, i.e., to consider a general representation of Hankel matrices on the first kind of numbers. It is interesting that the Hankel determinant of the first kind of numbers has a close relation that of the second kind of numbers.

We focus on derangement characters of GL(n,q) which depend solely on the dimension of the space of fixed vectors. This family includes Thoma characters which become asymptotically irreducible as n→∞. We find explicit decomposition of Thoma characters into irreducibles, construct further derangement characters and seek for extremes in the family derangement characters.

The statement of a problem of Euclidean combinatorial optimization with a fractional-linear objective function on a common set of permutations and with additional linear constraints is formulated. A problem with a fractional-linear objective function is transformed into that with a linear objective function. An approach is proposed to the solution of such problems, and a method of combinatorial truncation of solutions of problems of combinatorial type with fractional-linear objective functions on permutations is developed.

We obtain the decomposition of the tensor space \frak s\frak li>n⊗i>k as a module for \frak s\frak li>n, find an explicit formula for the multiplicities of its irreducible summands, and (when i>n ≥ 2i>k) describe the centralizer algebra \cal C e End_{\frak s\frak l_n}(\frak s\frak li>n⊗i>k) and its representations. The multiplicities of the irreducible summands are derangement numbers in several important instances, and the dimension of \cal C is given by the number of derangements of a set of 2i>k elements.

Das Internal Derangement ist eine der haufigsten Erkrankungen des Temporomandibulargelenks. Es ist definiert als eine abnorme anatomische Beziehung des Discus articularis zu den ubrigen gelenkbildenden Strukturen. Bei mehr als 2/3 der Patienten mit Kiefergelenkbeschwerden kann ein Internal Derangement nachgewiesen werden. Die haufigste Form ist die Ventralverlagerung des Diskus bei erhaltener Reduktion wahrend der Mundoffnung. Bei schwereren Formen kommt es zu totalen Ventralverlagerungen ohne Reduktion. Das Internal Derangement begunstigt die Entwicklung einer Arthrose mit entsprechenden Veranderungen am Caput mandibulae sowie an der Fossa mandibularis. Die Palette der bildgebenden Diagnostik reicht von der Arthrographie uber die CT zur MRT. Durch die modernen Schnittbildverfahren wurde die Abklarung der kraniofazialen Dysfunktion revolutioniert und sind heute essentieller Bestandteil der Initialdiagnostik sowie der Verlaufsbeurteilung nach kieferorthopadischer oder chirurgischer Therapie. Die MRT hat sich als Goldstandard etabliert.

The computation time for counting “good” permutations rapidly grows as the length of permutations increases. The paper presents algorithms for enumeration of “good” permutations. Algorithms reducing twice the number of “good” permutations that should be counted are considered along with the algorithm employing the concept of weight of a “good” permutation.

We analyze random walks on a class of semigroups called “left-regular bands.” These walks include the hyperplane chamber walks of Bidigare, Hanlon, and Rockmore. Using methods of ring theory, we show that the transition matrices are diagonalizable and we calculate the eigenvalues and multiplicities. The methods lead to explicit formulas for the projections onto the eigenspaces. As examples of these semigroup walks, we construct a random walk on the maximal chains of any distributive lattice, as well as two random walks associated with any matroid. The examples include a q-analogue of the Tsetlin library. The multiplicities of the eigenvalues in the matroid walks are “generalized derangement numbers,” which may be of independent interest.

In [8] Dumont and Randrianarivony have given several combinatorial interpretations for the coefficients of the Euler-Seidel matrix associated withn!. In this paper we consider aq-analogue of their results, which leads to the discovery of a new Mahonian statistic “maf” on the symmetric group. We then give new proofs and generalizations of some results of Gessel and Reutenauer [12] and Wachs [17].

The paper gives a description the permutations from the alternating groupA n that, for a given positive integerk≥4, cannot be presented as a product of two permutations each of which contains only cycles of lengths 1 and 4 in the expansion into independent cycles. We construct a setQ k such that, for eachn fromQ k , the groupA n contains a permutation not representable in the above form. We give answers to two questions of Brenner and Evans on the representability of even permutations in the form of a product of two permutations of a given orderk.

Perpendicular Arrays are ordered combinatorial structures, which recently have found applications in cryptography. A fundamental construction uses as ingredients combinatorial designs and uniformly t-homogeneous sets of permutations. We study the latter type of objects. These may also be viewed as generalizations of t-homogeneous groups of permutations. Several construction techniques are given. Here we concentrate on the optimal case, where the number of permutations attains the lower bound. We obtain several new optimal such sets of permutations. Each example allows the construction of infinite families of perpendicular arrays.

The clinical and arthrographic findings were evaluated on the basis of surgical procedures, performed on 120 symptomatic TMJs of 100 patients. The pre-operative arthrographic findings had significant correlation with surgical findings with respect to the presence of disc displacement, adhesion, and perforation. The MR findings of 10 TMJs of 5 patients were observed and studied. The diagnostic usefulness of MR imaging was considered and evaluated by observing the MR imagings of symptomatic TMJs.

A set S of permutations of k objects is μ-uniform, t-homogeneous if for every pair A, B of t-subsets of the ground set, there are exactly μ permutations in S mapping A onto B. Arithmetical conditions and symmetries are discussed. We describe the character-theoretic method which is useful if S is contained in a permutation group. A main result is the construction of a 2-uniform, 2-homogeneous set of permutations on 6 objects and of a 3-uniform, 3-homogeneous set of permutations on 9 objects. These are contained in the simple permutation groups PSL2(5) and PSL2(8), respectively. The result is useful in the framework of theoretical secrecy and authentication (see Stinson 1990, Bierbrauer and Tran 1991).

In this paper we investigate the combinatorial structure of then/m/P/Cmax permutation flow shop problem. To each solution (a permutation ofn elements) we introduce a network which represents the job and machine orders. Based on the introduced generalized distance between a permutation and a path of a network we derive combinatorial properties with respect to special neighbourhoods which are important for developing iterative heuristics for the considered problem.