Riordan Paths and Derangements

Riordan paths are Motzkin paths without horizontal steps on the x-axis. We establish a correspondence between Riordan paths and $(321,3\bar{1}42)$-avoiding derangements. We also present a combinatorial proof of a recurrence relation for the Riordan numbers in the spirit of the Foata-Zeilberger proof of a recurrence relation on the Schr\"oder numbers.Comment: 9 pages, 2 figure

Rank Minimization over Finite Fields: Fundamental Limits and Coding-Theoretic Interpretations

This paper establishes information-theoretic limits in estimating a finite field low-rank matrix given random linear measurements of it. These linear measurements are obtained by taking inner products of the low-rank matrix with random sensing matrices. Necessary and sufficient conditions on the number of measurements required are provided. It is shown that these conditions are sharp and the minimum-rank decoder is asymptotically optimal. The reliability function of this decoder is also derived by appealing to de Caen's lower bound on the probability of a union. The sufficient condition also holds when the sensing matrices are sparse - a scenario that may be amenable to efficient decoding. More precisely, it is shown that if the n\times n-sensing matrices contain, on average, \Omega(nlog n) entries, the number of measurements required is the same as that when the sensing matrices are dense and contain entries drawn uniformly at random from the field. Analogies are drawn between the above results and rank-metric codes in the coding theory literature. In fact, we are also strongly motivated by understanding when minimum rank distance decoding of random rank-metric codes succeeds. To this end, we derive distance properties of equiprobable and sparse rank-metric codes. These distance properties provide a precise geometric interpretation of the fact that the sparse ensemble requires as few measurements as the dense one. Finally, we provide a non-exhaustive procedure to search for the unknown low-rank matrix.Comment: Accepted to the IEEE Transactions on Information Theory; Presented at IEEE International Symposium on Information Theory (ISIT) 201

Parity Reversing Involutions on Plane Trees and 2-Motzkin Paths

The problem of counting plane trees with $n$ edges and an even or an odd number of leaves was studied by Eu, Liu and Yeh, in connection with an identity on coloring nets due to Stanley. This identity was also obtained by Bonin, Shapiro and Simion in their study of Schr\"oder paths, and it was recently derived by Coker using the Lagrange inversion formula. An equivalent problem for partitions was independently studied by Klazar. We present three parity reversing involutions, one for unlabelled plane trees, the other for labelled plane trees and one for 2-Motzkin paths which are in one-to-one correspondence with Dyck paths.Comment: 8 pages, 4 figure

Gender in Engineering Departments: Are There Gender Differences in Interruptions of Academic Job Talks?

We use a case study of job talks in five engineering departments to analyze the under-studied area of gendered barriers to finalists for faculty positions. We focus on one segment of the interview day of short-listed candidates invited to campus: the “job talk”, when candidates present their original research to the academic department. We analyze video recordings of 119 job talks across five engineering departments at two Research 1 universities. Specifically, we analyze whether there are differences by gender or by years of post-Ph.D. experience in the number of interruptions, follow-up questions, and total questions that job candidates receive. We find that, compared to men, women receive more follow-up questions and more total questions. Moreover, a higher proportion of women’s talk time is taken up by the audience asking questions. Further, the number of questions is correlated with the job candidate’s statements and actions that reveal he or she is rushing to present their slides and complete the talk. We argue that women candidates face more interruptions and often have less time to bring their talk to a compelling conclusion, which is connected to the phenomenon of “stricter standards” of competence demanded by evaluators of short-listed women applying for a masculine-typed job. We conclude with policy recommendations

Mifepristone Antagonization With Progesterone to Prevent Medical Abortion: A Randomized Controlled Trial.

ObjectiveTo estimate the efficacy and safety of mifepristone antagonization with high-dose oral progesterone.MethodsWe planned to enroll 40 patients in a double-blind, placebo-controlled, randomized trial. We enrolled patients at 44-63 days of gestation with ultrasound-confirmed gestational cardiac activity who were planning surgical abortion. Participants ingested mifepristone 200 mg and initiated oral progesterone 400 mg or placebo 24 hours later twice daily for 3 days, then once daily until their planned surgical abortion 14-16 days after enrollment. Follow-up visits were scheduled 3±1, 7±1, and 15±1 days after mifepristone intake with ultrasonography and blood testing for human chorionic gonadotropin and progesterone. Participants exited from the study when they had their surgical abortion or earlier for gestational cardiac activity absence, gestational sac expulsion, or medically indicated suction aspiration. We assessed the primary outcome of continued gestational cardiac activity at approximately 2 weeks (15±1 day), side effects after drug ingestion, and safety outcomes including hemorrhage and emergent treatment.ResultsWe enrolled participants from February to July 2019 and stopped enrollment after 12 patients for safety concerns. Mean gestational age was 52.5 days. Two (one per group) voluntarily discontinued 3 days after mifepristone ingestion for subjective symptoms (nausea and vomiting, bleeding). Among the remaining 10 patients (five per group), gestational cardiac activity continued for 2 weeks in four in the progesterone group and two in the placebo group. One patient in the placebo group had no gestational cardiac activity 3 days after mifepristone use. Severe hemorrhage requiring ambulance transport to hospital occurred in three patients; one received progesterone (complete expulsion, no aspiration) and two received placebo (aspiration for both, one required transfusion). We halted enrollment after the third hemorrhage. No other significant side effects were reported.ConclusionWe could not estimate the efficacy of progesterone for mifepristone antagonization due to safety concerns when mifepristone is administered without subsequent prostaglandin analogue treatment. Patients in early pregnancy who use only mifepristone may be at high risk of significant hemorrhage.Clinical trial registrationClinicalTrials.gov, NCT03774745