Statistics on Linear Chord Diagrams

Linear chord diagrams are partitions of $\left[2n\right]$ into $n$ blocks of size two called chords. We refer to a block of the form $\{i,i+1\}$ as a short chord. In this paper, we study the distribution of the number of short chords on the set of linear chord diagrams, as a generalization of the Narayana distribution obtained when restricted to the set of noncrossing linear chord diagrams. We provide a combinatorial proof that this distribution is unimodal and has an expected value of one. We also study the number of pairs $(i,i+1)$ where $i$ is the minimal element of a chord and $i+1$ is the maximal element of a chord. We show that the distribution of this statistic on linear chord diagrams corresponds to the second-order Eulerian triangle and is log-concave.Comment: 10 pages, final revision

Inversion generating functions for signed pattern avoiding permutations

We consider the classical Mahonian statistics on the set B (Σ) of signed per- mutations in the hyperoctahedral group B which avoid all patterns in Σ, where Σ is a set of patterns of length two. In 2000, Simion gave the cardinality of B (Σ) in the cases where Σ contains either one or two patterns of length two and showed that |B (Σ]| is constant whenever |Σ| = 1, whereas in most but not all instances where |Σ| = 2, |B (Σ)| = (n + 1)!. We answer an open question of Simion by providing bijections from B (Σ) to S in these cases where |B (Σ)| = (n + 1)!. In addition, we extend Simion’s work by providing a combinatorial proof in the language of signed permutations for the major index on B (21,21) and by giving the major index on D (Σ) for Σ = {21,21} and Σ = {21,21}. The main result of this paper is to give the inversion generating functions for B (Σ) for almost all sets Σ with |Σ| ≤ 2. n n n n n n n+1 n n n

Statistics on Linear Chord Diagrams

Linear chord diagrams are partitions of $\left[2n\right]$ into $n$ blocks of size two called chords. We refer to a block of the form $\{i,i+1\}$ as a short chord. In this paper, we study the distribution of the number of short chords on the set of linear chord diagrams, as a generalization of the Narayana distribution obtained when restricted to the set of noncrossing linear chord diagrams. We provide a combinatorial proof that this distribution is unimodal and has an expected value of one. We also study the number of pairs $(i,i+1)$ where $i$ is the minimal element of a chord and $i+1$ is the maximal element of a chord. We show that the distribution of this statistic on linear chord diagrams corresponds to the second-order Eulerian triangle and is log-concave

Incidence and prevalence of NMOSD in Australia and New Zealand

Objectives We have undertaken a clinic-based survey of neuromyelitis optica spectrum disorders (NMOSDs) in Australia and New Zealand to establish incidence and prevalence across the region and in populations of differing ancestry. Background NMOSD is a recently defined demyelinating disease of the central nervous system (CNS). The incidence and prevalence of NMOSD in Australia and New Zealand has not been established. Methods Centres managing patients with demyelinating disease of the CNS across Australia and New Zealand reported patients with clinical and laboratory features that were suspicious for NMOSD. Testing for aquaporin 4 antibodies was undertaken in all suspected cases. From this group, cases were identified who fulfilled the 2015 Wingerchuk diagnostic criteria for NMOSD. A capture–recapture methodology was used to estimate incidence and prevalence, based on additional laboratory identified cases. Results NMOSD was confirmed in 81/170 (48%) cases referred. Capture–recapture analysis gave an adjusted incidence estimate of 0.37 (95% CI 0.35 to 0.39) per million per year and a prevalence estimate for NMOSD of 0.70 (95% CI 0.61 to 0.78) per 100 000. NMOSD was three times more common in the Asian population (1.57 (95% CI 1.15 to 1.98) per 100 000) compared with the remainder of the population (0.57 (95% CI 0.50 to 0.65) per 100 000). The latitudinal gradient evident in multiple sclerosis was not seen in NMOSD. Conclusions NMOSD incidence and prevalence in Australia and New Zealand are comparable with figures from other populations of largely European ancestry. We found NMOSD to be more common in the population with Asian ancestry.</p

Incidence and prevalence of NMOSD in Australia and New Zealand

OBJECTIVES: We have undertaken a clinic-based survey of neuromyelitis optica spectrum disorders (NMOSDs) in Australia and New Zealand to establish incidence and prevalence across the region and in populations of differing ancestry. BACKGROUND: NMOSD is a recently defined demyelinating disease of the central nervous system (CNS). The incidence and prevalence of NMOSD in Australia and New Zealand has not been established. METHODS: Centres managing patients with demyelinating disease of the CNS across Australia and New Zealand reported patients with clinical and laboratory features that were suspicious for NMOSD. Testing for aquaporin 4 antibodies was undertaken in all suspected cases. From this group, cases were identified who fulfilled the 2015 Wingerchuk diagnostic criteria for NMOSD. A capture-recapture methodology was used to estimate incidence and prevalence, based on additional laboratory identified cases. RESULTS: NMOSD was confirmed in 81/170 (48%) cases referred. Capture-recapture analysis gave an adjusted incidence estimate of 0.37 (95% CI 0.35 to 0.39) per million per year and a prevalence estimate for NMOSD of 0.70 (95% CI 0.61 to 0.78) per 100 000. NMOSD was three times more common in the Asian population (1.57 (95% CI 1.15 to 1.98) per 100 000) compared with the remainder of the population (0.57 (95% CI 0.50 to 0.65) per 100 000). The latitudinal gradient evident in multiple sclerosis was not seen in NMOSD. CONCLUSIONS: NMOSD incidence and prevalence in Australia and New Zealand are comparable with figures from other populations of largely European ancestry. We found NMOSD to be more common in the population with Asian ancestry