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of n is said to be nearly self-conjugate if the Ferrers graph of
and its transpose have exactly
cells in common. The generating function of the number of such partitions was first conjectured by Campbell and recently confirmed by Campbell and Chern (‘Nearly self-conjugate integer partitions’, submitted for publication). We present a simple and direct analytic proof and a combinatorial proof of an equivalent statement.
Sleep disturbance is an important factor in the pathophysiology and progression of psychiatric disorders, but whether it is a cause, or a downstream effect is still not clear.
To investigate causal relationships between three sleep-associated traits and seven psychiatric diseases, we used genetic variants related to insomnia, chronotype and sleep duration to perform a two-sample bidirectional Mendelian randomisation analysis. Summary-level data on psychiatric disorders were extracted from the Psychiatric Genomics Consortium. Effect estimates were obtained by using the inverse-variance-weighted (IVW), weights modified IVW, weighted-median methods, MR-Egger regression, MR pleiotropy residual sum and outlier (MR-PRESSO) test and Robust Adjusted Profile Score (RAPS).
The causal odds ratio (OR) estimate of genetically determined insomnia was 1.33 (95% confidence interval (CI) 1.22–1.45; p = 5.03 × 10−11) for attention-deficit/hyperactivity disorder (ADHD), 1.31 (95% CI 1.25–1.37; p = 6.88 × 10−31) for major depressive disorder (MDD) and 1.32 (95% CI 1.23–1.40; p = 1.42 × 10−16) for post-traumatic stress disorder (PTSD). There were suggestive inverse associations of morningness chronotype with risk of MDD and schizophrenia (SCZ). Genetically predicted sleep duration was also nominally associated with the risk of bipolar disorder (BD). Conversely, PTSD and MDD were associated with an increased risk of insomnia (OR = 1.06, 95% CI 1.03–1.10, p = 7.85 × 10−4 for PTSD; OR = 1.37, 95% CI 1.14–1.64; p = 0.001 for MDD). A suggestive inverse association of ADHD and MDD with sleep duration was also observed.
Our findings provide evidence of potential causal relationships between sleep disturbance and psychiatric disorders. This suggests that abnormal sleep patterns may serve as markers for psychiatric disorders and offer opportunities for prevention and management in psychiatric disorders.
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