Enumeration of tilings of diamonds and hexagons with defects

We show how to count tilings of Aztec diamonds and hexagons with defects using determinants. In several cases these determinants can be evaluated in closed form. In particular, we obtain solutions to problems 1, 2, and 10 in James Propp's list of problems on enumeration of matchings

A Combinatorial Interpretation of the Free Fermion Condition of the Six-Vertex Model

The free fermion condition of the six-vertex model provides a 5 parameter sub-manifold on which the Bethe Ansatz equations for the wavenumbers that enter into the eigenfunctions of the transfer matrices of the model decouple, hence allowing explicit solutions. Such conditions arose originally in early field-theoretic S-matrix approaches. Here we provide a combinatorial explanation for the condition in terms of a generalised Gessel-Viennot involution. By doing so we extend the use of the Gessel-Viennot theorem, originally devised for non-intersecting walks only, to a special weighted type of \emph{intersecting} walk, and hence express the partition function of $N$ such walks starting and finishing at fixed endpoints in terms of the single walk partition functions

Three osculating walkers

We consider three directed walkers on the square lattice, which move simultaneously at each tick of a clock and never cross. Their trajectories form a non-crossing configuration of walks. This configuration is said to be osculating if the walkers never share an edge, and vicious (or: non-intersecting) if they never meet. We give a closed form expression for the generating function of osculating configurations starting from prescribed points. This generating function turns out to be algebraic. We also relate the enumeration of osculating configurations with prescribed starting and ending points to the (better understood) enumeration of non-intersecting configurations. Our method is based on a step by step decomposition of osculating configurations, and on the solution of the functional equation provided by this decomposition

Epinephrine and clonidine do not improve intrathecal sufentanil analgesia after total hip replacement†

Background. We compared analgesia after intrathecal sufentanil alone, sufentanil with epinephrine 200µg and sufentanil with clonidine 30 µg in patients after total hip replacement, the endpoints being onset and duration of action. Methods. We performed a randomized double‐blind study of 45 patients for elective total hip arthroplasty using continuous spinal anaesthesia. As soon as a pain score higher than 3 on a 10 cm visual analogue scale was reported, sufentanil 7.5 µg alone, sufentanil 7.5 µg + epinephrine 200 µg or sufentanil 7.5 µg + clonidine 30 µg in 2 ml normal saline was given intrathecally. Pain scores, rescue analgesia (diclofenac and morphine) and adverse effects (respiratory depression, postoperative nausea and vomiting, itching) were observed for 24h after surgery. Results. Time to a pain score of 3 [281 (36) vs 288 (23) vs 305 (30) min] were similar in all three groups. Adverse effects and analgesic requirements during the first 24h were also similar. Conclusion. After total hip replacement, all three analgesic regimens gave good analgesia with comparable onset and duration of action, and minor adverse effects. Br J Anaesth 2002; 89: 562-

Geestelijke begeleiding in transitie. Een exploratief onderzoek naar good practices van het vak humanistisch geestelijke begeleiding in het werkveld van de zorg

Dit onderzoek richt zich op het vak geestelijke begeleiding in het werkveld van de zorg. Door maatschappelijke veranderingen en de aard van het beroep is het vak intramuraal steeds meer onder druk komen te staan. Het vak ontwikkelt zich en er wordt gezocht naar mogelijkheden voor de ontwikkeling ervan in nieuwe werkvelden. In een kwalitatief empirische studie is onderzocht wat wordt gezien als good practices van het vak geestelijke begeleiding. Zes stakeholders en vier humanistisch geestelijk begeleiders zijn geïnterviewd. De onderzoeksvraag is: Wat zijn volgens stakeholders en humanistisch geestelijk begeleiders good practices van geestelijke begeleiding binnen het werkveld van de zorg, en wat is de waarde ervan voor de ontwikkeling van het vak? Er is een verkennend onderzoek verricht naar wat good practices zijn van het vak geestelijke begeleiding, welke criteria hiervoor worden gehanteerd en wat succesfactoren zijn om hiertoe te komen. Hierbij is zowel gekeken naar traditionele werkvelden (intramuraal in loondienst) als nieuwe werkvelden (extramuraal en vrijgevestigd werken) binnen de zorg

Random Words, Toeplitz Determinants and Integrable Systems. I

It is proved that the limiting distribution of the length of the longest weakly increasing subsequence in an inhomogeneous random word is related to the distribution function for the eigenvalues of a certain direct sum of Gaussian unitary ensembles subject to an overall constraint that the eigenvalues lie in a hyperplane.Comment: 15 pages, no figure

Thomson, Raman and Rayleigh scattering on atmospheric plasma

Abstract only

Counting matrices over finite fields with support on skew Young diagrams and complements of Rothe diagrams

We consider the problem of finding the number of matrices over a finite field with a certain rank and with support that avoids a subset of the entries. These matrices are a q-analogue of permutations with restricted positions (i.e., rook placements). For general sets of entries these numbers of matrices are not polynomials in q (Stembridge 98); however, when the set of entries is a Young diagram, the numbers, up to a power of q-1, are polynomials with nonnegative coefficients (Haglund 98). In this paper, we give a number of conditions under which these numbers are polynomials in q, or even polynomials with nonnegative integer coefficients. We extend Haglund's result to complements of skew Young diagrams, and we apply this result to the case when the set of entries is the Rothe diagram of a permutation. In particular, we give a necessary and sufficient condition on the permutation for its Rothe diagram to be the complement of a skew Young diagram up to rearrangement of rows and columns. We end by giving conjectures connecting invertible matrices whose support avoids a Rothe diagram and Poincar\'e polynomials of the strong Bruhat order.Comment: 24 pages, 9 figures, 1 tabl