Application of Marks to Computation of Ranks and Subdegrees of the Symmetric Group Acting on Ordered Pairs and on Ordered Triples

Ranks and subdegrees can be computed using combinatorial arguments, the Cauchy-Frobenius lemma and use of the concept of marks. However the concept of Marks has been given very little attention. In this paper we will apply the concept of marks to compute the ranks and subdegrees of the symmetric..........

Cycle Index Formulas for Dn Acting on Unordered Pairs

The cycle index of dihedral group.......... More information can be found in the full paper

Application of Marks to Computation of Ranks and Subdegrees of the Symmetric Group Acting on Ordered 4-Element and 5- Element Subsets

Abstract Ranks and subdegrees can be computed using combinatorial arguments, the Cauchy-Frobenius lemma and use of the concept of marks. However the concept of Marks has been given very little attention. In this paper we will apply the concept of marks to compute the ranks and subdegrees of the symmetric group ( = 7,8,9) acting on ordered 4-element subsets and ( = 8,9,10) acting on ordered 5-element subsets

Combined Nivolumab and Ipilimumab in Melanoma Metastatic to the Brain

BackgroundBrain metastases are a common cause of disabling neurologic complications and death in patients with metastatic melanoma. Previous studies of nivolumab combined with ipilimumab in metastatic melanoma have excluded patients with untreated brain metastases. We evaluated the efficacy and safety of nivolumab plus ipilimumab in patients with melanoma who had untreated brain metastases.MethodsIn this open-label, multicenter, phase 2 study, patients with metastatic melanoma and at least one measurable, nonirradiated brain metastasis (tumor diameter, 0.5 to 3 cm) and no neurologic symptoms received nivolumab (1 mg per kilogram of body weight) plus ipilimumab (3 mg per kilogram) every 3 weeks for up to four doses, followed by nivolumab (3 mg per kilogram) every 2 weeks until progression or unacceptable toxic effects. The primary end point was the rate of intracranial clinical benefit, defined as the percentage of patients who had stable disease for at least 6 months, complete response, or partial response.ResultsAmong 94 patients with a median follow-up of 14.0 months, the rate of intracranial clinical benefit was 57% (95% confidence interval [CI], 47 to 68); the rate of complete response was 26%, the rate of partial response was 30%, and the rate of stable disease for at least 6 months was 2%. The rate of extracranial clinical benefit was 56% (95% CI, 46 to 67). Treatment-related grade 3 or 4 adverse events were reported in 55% of patients, including events involving the central nervous system in 7%. One patient died from immune-related myocarditis. The safety profile of the regimen was similar to that reported in patients with melanoma who do not have brain metastases.ConclusionsNivolumab combined with ipilimumab had clinically meaningful intracranial efficacy, concordant with extracranial activity, in patients with melanoma who had untreated brain metastases. (Funded by Bristol-Myers Squibb and the National Cancer Institute; CheckMate 204 ClinicalTrials.gov number, NCT02320058 .)

Relatlimab and Nivolumab versus Nivolumab in Untreated Advanced Melanoma.

BACKGROUND: Lymphocyte-activation gene 3 (LAG-3) and programmed death 1 (PD-1) are distinct inhibitory immune checkpoints that contribute to T-cell exhaustion. The combination of relatlimab, a LAG-3-blocking antibody, and nivolumab, a PD-1-blocking antibody, has been shown to be safe and to have antitumor activity in patients with previously treated melanoma, but the safety and activity in patients with previously untreated melanoma need investigation. METHODS: In this phase 2-3, global, double-blind, randomized trial, we evaluated relatlimab and nivolumab as a fixed-dose combination as compared with nivolumab alone when administered intravenously every 4 weeks to patients with previously untreated metastatic or unresectable melanoma. The primary end point was progression-free survival as assessed by blinded independent central review. RESULTS: The median progression-free survival was 10.1 months (95% confidence interval [CI], 6.4 to 15.7) with relatlimab-nivolumab as compared with 4.6 months (95% CI, 3.4 to 5.6) with nivolumab (hazard ratio for progression or death, 0.75 [95% CI, 0.62 to 0.92]; P = 0.006 by the log-rank test). Progression-free survival at 12 months was 47.7% (95% CI, 41.8 to 53.2) with relatlimab-nivolumab as compared with 36.0% (95% CI, 30.5 to 41.6) with nivolumab. Progression-free survival across key subgroups favored relatlimab-nivolumab over nivolumab. Grade 3 or 4 treatment-related adverse events occurred in 18.9% of patients in the relatlimab-nivolumab group and in 9.7% of patients in the nivolumab group. CONCLUSIONS: The inhibition of two immune checkpoints, LAG-3 and PD-1, provided a greater benefit with regard to progression-free survival than inhibition of PD-1 alone in patients with previously untreated metastatic or unresectable melanoma. Relatlimab and nivolumab in combination showed no new safety signals. (Funded by Bristol Myers Squibb; RELATIVITY-047 ClinicalTrials.gov number, NCT03470922.)