We initiate the study of a class of polytopes, which we coin polypositroids, defined to be those polytopes that are simultaneously generalized permutohedra (or polymatroids) and alcoved polytopes. Whereas positroids are the matroids arising from the totally nonnegative Grassmannian, polypositroids are “positive” polymatroids. We parametrize polypositroids using Coxeter necklaces and balanced graphs, and describe the cone of polypositroids by extremal rays and facet inequalities. We introduce a notion of $(W,c)$-polypositroid for a finite Weyl group W and a choice of Coxeter element c. We connect the theory of $(W,c)$-polypositroids to cluster algebras of finite type and to generalized associahedra. We discuss membranes, which are certain triangulated 2-dimensional surfaces inside polypositroids. Membranes extend the notion of plabic graphs from positroids to polypositroids.

We study the ring of regular functions on the space of planar electrical networks, which we coin the grove algebra. This algebra is an electrical analog of the Plücker ring studied classically in invariant theory. We develop the combinatorics of double groves to study the grove algebra, and find a quadratic Gröbner basis for the grove ideal.

Mounting evidence suggests that the first few months of life are critical for the development of obesity. The relationships between the timing of solid food introduction and the risk of childhood obesity have been examined previously; however, evidence for the association of timing of infant formula introduction remains scarce. This study aimed to examine whether the timing of infant formula introduction is associated with growth z-scores and overweight at ages 1 and 3 years. This study included 5733 full-term (≥ 37 gestational weeks) and normal birth weight (≥ 2500 and < 4000 g) children in the Born in Guangzhou Cohort Study, a prospective cohort study with data collected at 6 weeks, 6, 12 and 36 months. Compared with infant formula introduction at 0–3 months, introduction at 4–6 months was associated with the lower BMI, weight-for-age and weight-for-length z-scores at 1 and 3 years old. Also, introduction at 4–6 months was associated with the lower odds of at-risk of overweight at age 1 (adjusted OR 0·72, 95 % CI 0·55, 0·94) and 3 years (adjusted OR 0·50, 95 % CI 0·30, 0·85). Introduction at 4–6 months also decreased the odds of overweight at age 1 year (adjusted OR 0·42, 95 % CI 0·21, 0·84) but not at age 3 years. Based on our findings, compared with introduction within the first 3 months, introduction at 4–6 months has a reduction on later high BMI risk and at-risk of overweight. However, these results need to be replicated in other well-designed studies before more firm recommendations can be made.

We study the back stable Schubert calculus of the infinite flag variety. Our main results are:

• – a formula for back stable (double) Schubert classes expressing them in terms of a symmetric function part and a finite part;

• – a novel definition of double and triple Stanley symmetric functions;

• – a proof of the positivity of double Edelman–Greene coefficients generalizing the results of Edelman–Greene and Lascoux–Schützenberger;

• – the definition of a new class of bumpless pipedreams, giving new formulae for double Schubert polynomials, back stable double Schubert polynomials, and a new form of the Edelman–Greene insertion algorithm;

• – the construction of the Peterson subalgebra of the infinite nilHecke algebra, extending work of Peterson in the affine case;

• – equivariant Pieri rules for the homology of the infinite Grassmannian;

• – homology divided difference operators that create the equivariant homology Schubert classes of the infinite Grassmannian.

The (tree) amplituhedron $\mathcal {A}_{n,k,m}(Z)$ is a certain subset of the Grassmannian introduced by Arkani-Hamed and Trnka in 2013 in order to study scattering amplitudes in $N=4$ supersymmetric Yang–Mills theory. Confirming a conjecture of the first author, we show that when $m$ is even, a collection of affine permutations yields a triangulation of $\mathcal {A}_{n,k,m}(Z)$ for any $Z\in \operatorname {Gr}_{>0}(k+m,n)$ if and only if the collection of their inverses yields a triangulation of $\mathcal {A}_{n,n-m-k,m}(Z)$ for any $Z\in \operatorname {Gr}_{>0}(n-k,n)$. We prove this duality using the twist map of Marsh and Scott. We also show that this map preserves the canonical differential forms associated with the corresponding positroid cells, and hence obtain a parity duality for amplituhedron differential forms.

Hill (Twin Research and Human Genetics, Vol. 21, 2018, 84–88) presented a critique of our recently published paper in Cell Reports entitled ‘Large-Scale Cognitive GWAS Meta-Analysis Reveals Tissue-Specific Neural Expression and Potential Nootropic Drug Targets’ (Lam et al., Cell Reports, Vol. 21, 2017, 2597–2613). Specifically, Hill offered several interrelated comments suggesting potential problems with our use of a new analytic method called Multi-Trait Analysis of GWAS (MTAG) (Turley et al., Nature Genetics, Vol. 50, 2018, 229–237). In this brief article, we respond to each of these concerns. Using empirical data, we conclude that our MTAG results do not suffer from ‘inflation in the FDR [false discovery rate]’, as suggested by Hill (Twin Research and Human Genetics, Vol. 21, 2018, 84–88), and are not ‘more relevant to the genetic contributions to education than they are to the genetic contributions to intelligence’.

In Hong Kong, universal varicella vaccination started in July 2014. Before this, children could receive varicella vaccine via the private market. We analysed the epidemiology of varicella and zoster before universal vaccination. We estimated varicella vaccination coverage through surveys in preschool children. We estimated the burden of varicella and zoster with varicella notifications from 1999/00 to 2013/14, Accident and Emergency Department (A&E) attendance and inpatient admissions to public hospitals from 2004/05 to 2013/14. We fitted a catalytic model to serological data on antibodies against varicella-zoster virus to estimate the force of infection. We found that varicella vaccination coverage gradually increased to about 50% before programme inception. In children younger than 5 years, the annual rate of varicella notifications, varicella admission and zoster A&E attendance generally declined. The annual notification, A&E attendance and hospitalisation rate of varicella and zoster generally increased for individuals between 10 and 59 years old. Varicella serology indicated an age shift during the study period towards a higher proportion of infections in slightly older individuals, but the change was most notable before vaccine licensure. In conclusion, we observed a shift in the burden of varicella to slightly older age groups with a corresponding increase in incidence but it cannot necessarily be attributed to private market vaccine coverage alone. Increasing varicella vaccination uptake in the private market might affect varicella transmission and epidemiology, but not to the level of interrupting transmission.

While the intersection of the Grassmannian Bruhat decompositions for all coordinate flags is an intractable mess, it turns out that the intersection of only the cyclic shifts of one Bruhat decomposition has many of the good properties of the Bruhat and Richardson decompositions. This decomposition coincides with the projection of the Richardson stratification of the flag manifold, studied by Lusztig, Rietsch, Brown–Goodearl–Yakimov and the present authors. However, its cyclic-invariance is hidden in this description. Postnikov gave many cyclic-invariant ways to index the strata, and we give a new one, by a subset of the affine Weyl group we call bounded juggling patterns. We call the strata positroid varieties. Applying results from [A. Knutson, T. Lam and D. Speyer, Projections of Richardson varieties, J. Reine Angew. Math., to appear, arXiv:1008.3939 [math.AG]], we show that positroid varieties are normal, Cohen–Macaulay, have rational singularities, and are defined as schemes by the vanishing of Plücker coordinates. We prove that their associated cohomology classes are represented by affine Stanley functions. This latter fact lets us connect Postnikov’s and Buch–Kresch–Tamvakis’ approaches to quantum Schubert calculus.

We construct the Schubert basis of the torus-equivariant K-homology of the affine Grassmannian of a simple algebraic group G, using the K-theoretic NilHecke ring of Kostant and Kumar. This is the K-theoretic analogue of a construction of Peterson in equivariant homology. For the case where G=SLn, the K-homology of the affine Grassmannian is identified with a sub-Hopf algebra of the ring of symmetric functions. The Schubert basis is represented by inhomogeneous symmetric functions, calledK-k-Schur functions, whose highest-degree term is a k-Schur function. The dual basis in K-cohomology is given by the affine stable Grothendieck polynomials, verifying a conjecture of Lam. In addition, we give a Pieri rule in K-homology. Many of our constructions have geometric interpretations by means of Kashiwara’s thick affine flag manifold.

Dedicated to George Szekeres on his ninetieth birthday

In this paper we prove that a bipartite graph with parts of sizes M and N, having no cycles of even length less that or equal to 2(2k + 1), where k is a positive integer, has at most (NM)(k+1)/(2k+1) + Dk(N + M) edges, where Dk only depends on k.

In particular, we show that when k = 1, D1 = 1 is possible.