Abstract: Using ideas from 3-manifolds, Hatcher-Wahl defined a notion of automorphism groups of free groups with boundary. We study their Torelli subgroups, adapting ideas introduced by Putman for surface mapping class groups. Our main results show that these groups are finitely generated, and also that they satisfy an appropriate version of the Birman exact sequence.

Abstract: We consider a generalization of Eulerian numbers which count the number of placements of $cn$ “rooks” on an $n \times n$ board where there are exactly $c$ rooks in each row and each column, and exactly $k$ rooks below the main diagonal. The standard Eulerian numbers correspond to the case $c = 1$. We show that for any $c$ the resulting numbers are symmetric and give generating functions of these numbers for small values of $k$.

Abstract: Juggling patterns can be described by a closed walk in a (directed) state graph, where each vertex (or state) is a landing pattern for the balls and directed edges connect states that can occur consecutively. The number of such patterns of length $n$ is well known, but a long-standing problem is to count the number of prime juggling patterns (those juggling patterns corresponding to cycles in the state graph). For the case of $b = 2$ balls we give an expression for the number of prime juggling patterns of length $n$ by establishing a connection with partitions of $n$ into distinct parts. From this we show the number of two-ball prime juggling patterns of length $n$ is $(\gamma−o(1)) 2^n$ where $\gamma = 1.32963879259\ldots$. For larger $b$ we show there are at least $bn−1$ prime cycles of length $n$.