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In 2007, Andrews studied the odd Durfee symbols and their odd ranks. Let $N^0(m,k,n)$ denote the number of odd Durfee symbols of $n$ with odd rank congruent to $m$ modulo $k$. Motivated by Andrews' work, many authors obtained generating functions of $N^0(m,k,n)$ from which relations between odd ranks are proved. In this paper, we establish a family of congruences for odd ranks modulo powers of $5$.

It is still an open and challenging problem for mobile robots navigating along time-efficient and collision-free paths in a crowd. The main challenge comes from the complex and sophisticated interaction mechanism, which requires the robot to understand the crowd and perform proactive and foresighted behaviors. Deep reinforcement learning is a promising solution to this problem. However, most previous learning methods incur a tremendous computational burden. To address these problems, we propose a graph-based deep reinforcement learning method, SG-DQN, that (i) introduces a social attention mechanism to extract an efficient graph representation for the crowd-robot state; (ii) directly evaluates the coarse q-values of the raw state with a learned dueling deep Q network(DQN); and then (iii) refines the coarse q-values via online planning on possible future trajectories. The experimental results indicate that our model can help the robot better understand the crowd and achieve a high success rate of more than 0.99 in the crowd navigation task. Compared against previous state-of-the-art algorithms, our algorithm achieves an equivalent, if not better, performance while requiring less than half of the computational cost., Comment: 8 pages, 7 figures, and two tables. The paper is in submission