Human and constructive proof of combinatorial identities: an example from Romik

International audienceIt has become customary to prove binomial identities by means of the method for automated proofs as developed by Petkovšek, Wilf and Zeilberger. In this paper, we wish to emphasize the role of "human'' and constructive proofs in contrast with the somewhat lazy attitude of relaying on "automated'' proofs. As a meaningful example, we consider the four formulas by Romik, related to Motzkin and central trinomial numbers. We show that a proof of these identities can be obtained by using the method of coefficients, a human method only requiring hand computations

A strip-like tiling algorithm

AbstractWe extend our previous results on the connection between strip tiling problems and regular grammars by showing that an analogous algorithm is applicable to other tiling problems, not necessarily related to rectangular strips. We find generating functions for monomer and dimer tilings of T- and L-shaped figures, holed and slotted strips, diagonal strips and combinations of them, and show how analogous results can be obtained by using different pieces

Complementary Riordan arrays

Abstract Recently, the concept of the complementary array of a Riordan array (or recursive matrix) has been introduced. Here we generalize the concept and distinguish between dual and complementary arrays. We show a number of properties of these arrays, how they are computed and their relation with inversion. Finally, we use them to find explicit formulas for the elements of many recursive matrices

Acute Delta Hepatitis in Italy spanning three decades (1991–2019): Evidence for the effectiveness of the hepatitis B vaccination campaign

Updated incidence data of acute Delta virus hepatitis (HDV) are lacking worldwide. Our aim was to evaluate incidence of and risk factors for acute HDV in Italy after the introduction of the compulsory vaccination against hepatitis B virus (HBV) in 1991. Data were obtained from the National Surveillance System of acute viral hepatitis (SEIEVA). Independent predictors of HDV were assessed by logistic-regression analysis. The incidence of acute HDV per 1-million population declined from 3.2 cases in 1987 to 0.04 in 2019, parallel to that of acute HBV per 100,000 from 10.0 to 0.39 cases during the same period. The median age of cases increased from 27 years in the decade 1991-1999 to 44 years in the decade 2010-2019 (p < .001). Over the same period, the male/female ratio decreased from 3.8 to 2.1, the proportion of coinfections increased from 55% to 75% (p = .003) and that of HBsAg positive acute hepatitis tested for by IgM anti-HDV linearly decreased from 50.1% to 34.1% (p < .001). People born abroad accounted for 24.6% of cases in 2004-2010 and 32.1% in 2011-2019. In the period 2010-2019, risky sexual behaviour (O.R. 4.2; 95%CI: 1.4-12.8) was the sole independent predictor of acute HDV; conversely intravenous drug use was no longer associated (O.R. 1.25; 95%CI: 0.15-10.22) with this. In conclusion, HBV vaccination was an effective measure to control acute HDV. Intravenous drug use is no longer an efficient mode of HDV spread. Testing for IgM-anti HDV is a grey area requiring alert. Acute HDV in foreigners should be monitored in the years to come

Global disparities in surgeons’ workloads, academic engagement and rest periods: the on-calL shIft fOr geNEral SurgeonS (LIONESS) study

: The workload of general surgeons is multifaceted, encompassing not only surgical procedures but also a myriad of other responsibilities. From April to May 2023, we conducted a CHERRIES-compliant internet-based survey analyzing clinical practice, academic engagement, and post-on-call rest. The questionnaire featured six sections with 35 questions. Statistical analysis used Chi-square tests, ANOVA, and logistic regression (SPSS® v. 28). The survey received a total of 1.046 responses (65.4%). Over 78.0% of responders came from Europe, 65.1% came from a general surgery unit; 92.8% of European and 87.5% of North American respondents were involved in research, compared to 71.7% in Africa. Europe led in publishing research studies (6.6 ± 8.6 yearly). Teaching involvement was high in North America (100%) and Africa (91.7%). Surgeons reported an average of 6.7 ± 4.9 on-call shifts per month, with European and North American surgeons experiencing 6.5 ± 4.9 and 7.8 ± 4.1 on-calls monthly, respectively. African surgeons had the highest on-call frequency (8.7 ± 6.1). Post-on-call, only 35.1% of respondents received a day off. Europeans were most likely (40%) to have a day off, while African surgeons were least likely (6.7%). On the adjusted multivariable analysis HDI (Human Development Index) (aOR 1.993) hospital capacity > 400 beds (aOR 2.423), working in a specialty surgery unit (aOR 2.087), and making the on-call in-house (aOR 5.446), significantly predicted the likelihood of having a day off after an on-call shift. Our study revealed critical insights into the disparities in workload, access to research, and professional opportunities for surgeons across different continents, underscored by the HDI

Analysis of algorithms as a teaching experience

Teaching analysis of algorithms to students in Computer Science degrees, using the approach popularized by Knuth in his series of books ``The Art of Computer Programming" and later by Sedgewick and Flajolet in the book ``An Introduction to the Analysis of Algorithms", is not a simple task since, in general, these students are more interested in the implementation of an algorithm than in the corresponding theoretical aspects. This approach concentrates on precisely characterizing the performance of algorithms by determining their best, worst and average case performance using a methodology based on symbolic tools such as recurrence relations and generating functions.The most difficult aspect is to understand the average case since this corresponds to studying the algorithm as its possible inputs vary: this represents the most important goal since generally students have no difficulty in understanding the best and worst cases, corresponding to particular input configurations.A compromise that has been successful over the years consists in teaching students the analytical aspects of the problem and then organize a simulation of the algorithm with a system of symbolic computation in order to precisely check the theoretical results