604 research outputs found
Proof of George Andrews's and David Robbins's q-TSPP Conjecture
The conjecture that the orbit-counting generating function for totally
symmetric plane partitions can be written as an explicit product formula, has
been stated independently by George Andrews and David Robbins around 1983. We
present a proof of this long-standing conjecture
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
A polytope related to empirical distributions, plane trees, parking functions, and the associahedron
We define an n-dimensional polytope Pi_n(x), depending on parameters x_i>0,
whose combinatorial properties are closely connected with empirical
distributions, plane trees, plane partitions, parking functions, and the
associahedron. In particular, we give explicit formulas for the volume of
Pi_n(x) and, when the x_i's are integers, the number of integer points in
Pi_n(x). We give two polyhedral decompositions of Pi_n(x), one related to order
cones of posets and the other to the associahedron.Comment: 41 page
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