The Enumeration of Prudent Polygons by Area and its Unusual Asymptotics

Prudent walks are special self-avoiding walks that never take a step towards an already occupied site, and \emph{$k$-sided prudent walks} (with $k=1,2,3,4$) are, in essence, only allowed to grow along $k$ directions. Prudent polygons are prudent walks that return to a point adjacent to their starting point. Prudent walks and polygons have been previously enumerated by length and perimeter (Bousquet-M\'elou, Schwerdtfeger; 2010). We consider the enumeration of \emph{prudent polygons} by \emph{area}. For the 3-sided variety, we find that the generating function is expressed in terms of a $q$-hypergeometric function, with an accumulation of poles towards the dominant singularity. This expression reveals an unusual asymptotic structure of the number of polygons of area $n$, where the critical exponent is the transcendental number $\log_23$ and and the amplitude involves tiny oscillations. Based on numerical data, we also expect similar phenomena to occur for 4-sided polygons. The asymptotic methodology involves an original combination of Mellin transform techniques and singularity analysis, which is of potential interest in a number of other asymptotic enumeration problems.Comment: 27 pages, 6 figure

On consecutive pattern-avoiding permutations of length 4, 5 and beyond

We review and extend what is known about the generating functions for consecutive pattern-avoiding permutations of length 4, 5 and beyond, and their asymptotic behaviour. There are respectively, seven length-4 and twenty-five length-5 consecutive-Wilf classes. D-finite differential equations are known for the reciprocal of the exponential generating functions for four of the length-4 and eight of the length-5 classes. We give the solutions of some of these ODEs. An unsolved functional equation is known for one more class of length-4, length-5 and beyond. We give the solution of this functional equation, and use it to show that the solution is not D-finite. For three further length-5 c-Wilf classes we give recurrences for two and a differential-functional equation for a third. For a fourth class we find a new algebraic solution. We give a polynomial-time algorithm to generate the coefficients of the generating functions which is faster than existing algorithms, and use this to (a) calculate the asymptotics for all classes of length 4 and length 5 to significantly greater precision than previously, and (b) use these extended series to search, unsuccessfully, for D-finite solutions for the unsolved classes, leading us to conjecture that the solutions are not D-finite. We have also searched, unsuccessfully, for differentially algebraic solutions.Comment: 23 pages, 2 figures (update of references, plus web link to enumeration data). Minor update. Typos corrected. One additional referenc

Compressed self-avoiding walks, bridges and polygons

We study various self-avoiding walks (SAWs) which are constrained to lie in the upper half-plane and are subjected to a compressive force. This force is applied to the vertex or vertices of the walk located at the maximum distance above the boundary of the half-space. In the case of bridges, this is the unique end-point. In the case of SAWs or self-avoiding polygons, this corresponds to all vertices of maximal height. We first use the conjectured relation with the Schramm-Loewner evolution to predict the form of the partition function including the values of the exponents, and then we use series analysis to test these predictions.Comment: 29 pages, 6 figure

A numerical adaptation of SAW identities from the honeycomb to other 2D lattices

Recently, Duminil-Copin and Smirnov proved a long-standing conjecture by Nienhuis that the connective constant of self-avoiding walks on the honeycomb lattice is $\sqrt{2+\sqrt{2}}.$ A key identity used in that proof depends on the existence of a parafermionic observable for self-avoiding walks on the honeycomb lattice. Despite the absence of a corresponding observable for SAW on the square and triangular lattices, we show that in the limit of large lattices, some of the consequences observed on the honeycomb lattice persist on other lattices. This permits the accurate estimation, though not an exact evaluation, of certain critical amplitudes, as well as critical points, for these lattices. For the honeycomb lattice an exact amplitude for loops is proved.Comment: 21 pages, 7 figures. Changes in v2: Improved numerical analysis, giving greater precision. Explanation of why we observe what we do. Extra reference

On the enumeration of column-convex permutominoes

We study the enumeration of \emphcolumn-convex permutominoes, i.e. column-convex polyominoes defined by a pair of permutations. We provide a direct recursive construction for the column-convex permutominoes of a given size, based on the application of the ECO method and generating trees, which leads to a functional equation. Then we obtain some upper and lower bounds for the number of column-convex permutominoes, and conjecture its asymptotic behavior using numerical analysis

The critical fugacity for surface adsorption of self-avoiding walks on the honeycomb lattice is 1+21+\sqrt{2}

In 2010, Duminil-Copin and Smirnov proved a long-standing conjecture of Nienhuis, made in 1982, that the growth constant of self-avoiding walks on the hexagonal (a.k.a. honeycomb) lattice is $\mu=\sqrt{2+\sqrt{2}}.$ A key identity used in that proof was later generalised by Smirnov so as to apply to a general O(n) loop model with $n\in [-2,2]$ (the case $n=0$ corresponding to SAWs). We modify this model by restricting to a half-plane and introducing a surface fugacity $y$ associated with boundary sites (also called surface sites), and obtain a generalisation of Smirnov's identity. The critical value of the surface fugacity was conjectured by Batchelor and Yung in 1995 to be $y_{\rm c}=1+2/\sqrt{2-n}.$ This value plays a crucial role in our generalized identity, just as the value of growth constant did in Smirnov's identity. For the case $n=0$, corresponding to \saws\ interacting with a surface, we prove the conjectured value of the critical surface fugacity. A crucial part of the proof involves demonstrating that the generating function of self-avoiding bridges of height $T$, taken at its critical point $1/\mu$, tends to 0 as $T$ increases, as predicted from SLE theory.Comment: Major revision, references updated, 25 pages, 13 figure

Two-dimensional self-avoiding walks and polymer adsorption: Critical fugacity estimates

Recently Beaton, de Gier and Guttmann proved a conjecture of Batchelor and Yung that the critical fugacity of self-avoiding walks interacting with (alternate) sites on the surface of the honeycomb lattice is $1+\sqrt{2}$. A key identity used in that proof depends on the existence of a parafermionic observable for self-avoiding walks interacting with a surface on the honeycomb lattice. Despite the absence of a corresponding observable for SAW on the square and triangular lattices, we show that in the limit of large lattices, some of the consequences observed for the honeycomb lattice persist irrespective of lattice. This permits the accurate estimation of the critical fugacity for the corresponding problem for the square and triangular lattices. We consider both edge and site weighting, and results of unprecedented precision are achieved. We also \emph{prove} the corresponding result fo the edge-weighted case for the honeycomb lattice.Comment: 12 pages, 3 figures, 7 table

Addition of four doses of rituximab to standard induction chemotherapy in adult patients with precursor B-cell acute lymphoblastic leukaemia (UKALL14): a phase 3, multicentre, randomised controlled trial

BACKGROUND: Treatment for adults with acute lymphoblastic leukaemia requires improvement. UKALL14 was a UK National Cancer Research Institute Adult ALL group study that aimed to determine the benefit of adding the anti-CD20 monoclonal antibody, rituximab, to the therapy of adults with de novo B-precursor acute lymphoblastic leukaemia. METHODS: This was an investigator-initiated, phase 3, randomised controlled trial done in all UK National Health Service Centres treating patients with acute lymphoblastic leukaemia (65 centres). Patients were aged 25-65 years with de-novo BCR-ABL1-negative acute lymphoblastic leukaemia. Patients with de-novo BCR-ABL1-positive acute lymphoblastic leukaemia were eligible if they were aged 19-65 years. Participants were randomly assigned (1:1) to standard-of-care induction therapy or standard-of-care induction therapy plus four doses of intravenous rituximab (375 mg/m2 on days 3, 10, 17, and 24). Randomisation used minimisation and was stratified by sex, age, and white blood cell count. No masking was used for patients, clinicians, or staff (including the trial statistician), although the central laboratory analysing minimal residual disease and CD20 was masked to treatment allocation. The primary endpoint was event-free survival in the intention-to-treat population. Safety was assessed in all participants who started trial treatment. This study is registered with ClincialTrials.gov, NCT01085617. FINDINGS: Between April 19, 2012, and July 10, 2017, 586 patients were randomly assigned to standard of care (n=292) or standard of care plus rituximab (n=294). Nine patients were excluded from the final analysis due to misdiagnosis (standard of care n=4, standard of care plus rituximab n=5). In the standard-of-care group, median age was 45 years (IQR 22-65), 159 (55%) of 292 participants were male, 128 (44%) were female, one (<1%) was intersex, and 143 (59%) of 244 participants had high-risk cytogenetics. In the standard-of-care plus rituximab group, median age was 46 years (IQR 23-65), 159 (55%) of 294 participants were male, 130 (45%) were female, and 140 (60%) of 235 participants had high-risk cytogenetics. After a median follow-up of 53·7 months (IQR 40·3-70·4), 3-year event-free survival was 43·7% (95% CI 37·8-49·5) for standard of care versus 51·4% (45·4-57·1) for standard of care plus rituximab (hazard ratio [HR] 0·85 [95% CI 0·69-1·06]; p=0·14). The most common adverse events were infections and cytopenias, with no difference between the groups in the rates of adverse events. There were 11 (4%) fatal (grade 5) events in induction phases 1 and 2 in the standard-of-care group and 13 (5%) events in the standard-of-care plus rituximab group). 3-year non-relapse mortality was 23·7% (95% CI 19·0-29·4) in the standard-of-care group versus 20·6% (16·2-25·9) in the standard-of-care plus rituximab group (HR 0·88 [95% CI 0·62-1·26]; p=0·49). INTERPRETATION: Standard of care plus four doses of rituximab did not significantly improve event-free survival over standard of care. Rituximab is beneficial in acute lymphoblastic leukaemia but four doses during induction is likely to be insufficient. FUNDING: Cancer Research UK and Blood Cancer UK