(pdf) (abstract) 

E. Banaian, S. Butler, C. Cox, J. Davis, J. Landgraf, S. Ponce
A generalization of Eulerian numbers via rook placements
Involve 10 (2017), no. 4, 691705.
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 $k$.

(pdf) (abstract) 

E. Banaian, S. Butler, C. Cox, J. Davis, J. Landgraf, S. Ponce
Counting prime juggling patterns
Graphs Combin. 32 (2016), no. 5, 16751688.
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 longstanding 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 twoball 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 $n$.
