150 research outputs found
The computational challenge of enumerating high-dimensional rook walks
We provide guessed recurrence equations for the counting sequences of rook
paths on d-dimensional chess boards starting at (0..0) and ending at (n..n),
where d=2,3,...,12. Our recurrences suggest refined asymptotic formulas of
these sequences. Rigorous proofs of the guessed recurrences as well as the
suggested asymptotic forms are posed as challenges to the reader
Automatic Classification of Restricted Lattice Walks
We propose an experimental mathematics approach leading to the
computer-driven discovery of various structural properties of general counting
functions coming from enumeration of walks
Constructing minimal telescopers for rational functions in three discrete variables
We present a new algorithm for constructing minimal telescopers for rational
functions in three discrete variables. This is the first discrete
reduction-based algorithm that goes beyond the bivariate case. The termination
of the algorithm is guaranteed by a known existence criterion of telescopers.
Our approach has the important feature that it avoids the potentially costly
computation of certificates. Computational experiments are also provided so as
to illustrate the efficiency of our approach
Explicit formula for the generating series of diagonal 3D rook paths
Let denote the number of ways in which a chess rook can move from a
corner cell to the opposite corner cell of an
three-dimensional chessboard, assuming that the piece moves closer to the goal
cell at each step. We describe the computer-driven \emph{discovery and proof}
of the fact that the generating series admits
the following explicit expression in terms of a Gaussian hypergeometric
function: G(x) = 1 + 6 \cdot \int_0^x \frac{\,\pFq21{1/3}{2/3}{2} {\frac{27
w(2-3w)}{(1-4w)^3}}}{(1-4w)(1-64w)} \, dw.Comment: To appear in "S\'eminaire Lotharingien de Combinatoire
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