Skew Pieri Rules for Hall-Littlewood Functions

International audience We produce skew Pieri Rules for Hall–Littlewood functions in the spirit of Assaf and McNamara (FPSAC, 2010). The first two were conjectured by the first author (FPSAC, 2011). The key ingredients in the proofs are a q-binomial identity for skew partitions that are horizontal strips and a Hopf algebraic identity that expands products of skew elements in terms of the coproduct and antipode. Nous produisons quelques règles dissymètrique de Pieri pour les fonctions Hall–Littlewood au sens de Assaf et McNamara (FPSAC, 2010). Les premières deux règles ont ètè conjecturèe par le premier auteur (FPSAC, 2011). Les principaux ingrèdients dans les preuves sont une identitè q-binomiale pour les partitions dissymètrique qui sont bandes horizontales et une identitè de Hopf qui exprime les produits d'èlèments dissymètrique en termes du coproduit et de l'antipode.

Kinship and differentialities: alternatives to identity and to ethnic frontiers in the study of migrations

Based on work undertaken by the Laboratory of Migratory Studies, focused on research in the Valadares region of Minas Gerais State and on studies about the Japanese presence in Brazil, I sought to discuss alternatives to approaches that focus on the binomial identity-ethnic borders. The development of another perspective, critical of the concept of identity, arose at the interlacing of the relationships between migration and kinship with a perspective influenced by the work of Tim Ingold.

Counting Defective Parking Functions

Suppose that $m$ drivers each choose a preferred parking space in a linear car park with $n$ spaces. Each driver goes to the chosen space and parks there if it is free, and otherwise takes the first available space with a larger number (if any). If all drivers park successfully, the sequence of choices is called a parking function. In general, if $k$ drivers fail to park, we have a defective parking function of defect $k$. Let ${\rm cp}(n,m,k)$ be the number of such functions. In this paper, we establish a recurrence relation for the numbers ${\rm cp}(n,m,k)$, and express this as an equation for a three-variable generating function. We solve this equation using the kernel method, and extract the coefficients explicitly: it turns out that the cumulative totals are partial sums in Abel's binomial identity. Finally, we compute the asymptotics of ${\rm cp}(n,m,k)$. In particular, for the case $m=n$, if choices are made independently at random, the limiting distribution of the defect (the number of drivers who fail to park), scaled by the square root of $n$, is the Rayleigh distribution. On the other hand, in the case $m=\omega(n)$, the probability that all spaces are occupied tends asymptotically to one.