Descent c-Wilf Equivalence

Let $S_n$ denote the symmetric group. For any $\sigma \in S_n$, we let $\mathrm{des}(\sigma)$ denote the number of descents of $\sigma$, $\mathrm{inv}(\sigma)$ denote the number of inversions of $\sigma$, and $\mathrm{LRmin}(\sigma)$ denote the number of left-to-right minima of $\sigma$. For any sequence of statistics $\mathrm{stat}_1, \ldots \mathrm{stat}_k$ on permutations, we say two permutations $\alpha$ and $\beta$ in $S_j$ are $(\mathrm{stat}_1, \ldots \mathrm{stat}_k)$-c-Wilf equivalent if the generating function of $\prod_{i=1}^k x_i^{\mathrm{stat}_i}$ over all permutations which have no consecutive occurrences of $\alpha$ equals the generating function of $\prod_{i=1}^k x_i^{\mathrm{stat}_i}$ over all permutations which have no consecutive occurrences of $\beta$. We give many examples of pairs of permutations $\alpha$ and $\beta$ in $S_j$ which are $\mathrm{des}$-c-Wilf equivalent, $(\mathrm{des},\mathrm{inv})$-c-Wilf equivalent, and $(\mathrm{des},\mathrm{inv},\mathrm{LRmin})$-c-Wilf equivalent. For example, we will show that if $\alpha$ and $\beta$ are minimally overlapping permutations in $S_j$ which start with 1 and end with the same element and $\mathrm{des}(\alpha) = \mathrm{des}(\beta)$ and $\mathrm{inv}(\alpha) = \mathrm{inv}(\beta)$, then $\alpha$ and $\beta$ are $(\mathrm{des},\mathrm{inv})$-c-Wilf equivalent.Comment: arXiv admin note: text overlap with arXiv:1510.0431

Split-disk micro-lasers: Tunable whispering gallery mode cavities

Optical micro-cavities of various types have emerged as promising photonic structures, for both the investigation of fundamental science in cavity quantum electrodynamics and simultaneously for various applications, e.g., lasers, filters, or modulators. In either branch a demand for adjustable and tunable photonic devices becomes apparent, which has been mainly based on the modification of the refractive index of the micro-resonators so far. In this paper, we report on a novel type of whispering gallery mode resonator where resonance tuning is achieved by modification of the configuration. This is realized by polymeric split-disks consisting of opposing half-disks with an intermediate air gap. Functionality of the split-disk concept and its figures of merit like low-threshold lasing are demonstrated for laser dye-doped split-disks fabricated by electron beam lithography on Si substrates. Reversible resonance tuning is achieved for split-disks structured onto elastomeric substrates by direct laser writing. The gap width and hence the resonance wavelength can be well-controlled by mechanically stretching the elastomer and exploiting the lateral shrinkage of the substrate. We demonstrate a broad spectral tunability of laser modes by more than three times the free spectral range. These cavities have the potential to form a key element of flexible and tunable photonic circuits based on polymers

Some identities on derangement and degenerate derangement polynomials

In combinatorics, a derangement is a permutation that has no fixed points. The number of derangements of an n-element set is called the n-th derangement number. In this paper, as natural companions to derangement numbers and degenerate versions of the companions we introduce derangement polynomials and degenerate derangement polynomials. We give some of their properties, recurrence relations and identities for those polynomials which are related to some special numbers and polynomials.Comment: 12 page

An interleukin-1 polymorphism additionally intensified by atopy as prognostic factor for aseptic non-mechanical complications in metal knee and hip arthroplasty

Background: In contrast to infection or mechanical issues joint replacement failure following inflammatory adverse reactions is poorly understood. Objective: To assess the association of IL-1β polymorphisms and history of allergy with aseptic non-mechanical complications following arthroplasty. Methods: In 102 patients with aseptic non-mechanically caused symptomatic knee or hip arthroplasty (SA) and 93 patients with asymptomatic arthroplasty (AA) questionnaire-based history, patch test with at least standard series, lymphocyte transformation test (LTT) with nickel, cobalt and chromium and interleukin-1 polymorphism analysis were done. Three polymorphisms of the IL1B gene [IL-1b -3954 (rs1143634), IL-1b -511 (rs16944) and IL-1b -31 (rs1143627)] and one polymorphism of the IL1RN gene [IL1RN intron 2, variable number of tandem repeats, VNTR (rs2234663)] were assessed by PCR and gel electrophoresis. Results: We found no significant difference in smoking history and atopy but 25% versus 10% of self-reported metal allergy in SA versus AA; the patch test (respective, LTT) for metal sensitivity was more often positive in SA patients. The allele 498 bp of the IL1RN polymorphism occurred significantly more often in the SA group (37% versus 11%; p < 0.0001). Upon additional presence of atopy, the difference was even greater (60% vs 10%) (p < 0.000001). There was no association of IL-1 polymorphisms with metal allergy. Conclusion: The IL1RN VNTR allele 498 bp was strongly associated with SA. In patients with a history of atopy, presence of the IL1RN VNTR allele 498 bp led to a four-fold higher SA prevalence compared to patients without this allele