The Band-Hit Statistic for Toroidal Circulant Bands

We study the number of hits of a uniformly random permutation matrix on a circulant band consisting of k consecutive cyclic diagonals in an n × n toroidal board. The case of n hits is counted by the permanent of the corresponding indicator matrix, while the full distribution is naturally related to the classical hit-number framework of rook theory. Let X(n,k) denote the number of hits. Using an exact finite-state dynamic programming method, we compute its higher-order moments over a broad range of n, k, and m. The numerical data reveal a stable region associated with floor(m/2)(k−1) + 1 ≤ n. Within the scope of the present calculations, we confirmed that, in this region, the higher-order moment can be expressed in terms of the mth Touchard polynomial T_m(k) and an explicit finite correction term involving Stirling numbers of the second kind. The complete formula is given in the accompanying paper. Here, T_m(k) is equivalently the mth moment of a Poisson random variable with mean k. Furthermore, based on these results, we present several conjectures regarding the higher-order moment. This record also contains the computational materials accompanying the paper, including a Jupyter notebook, C source code, precomputed CSV data, and reproduction instructions.