On Dimensions of a Random Solid Diagram

Authors:
Boris Pittel
Affiliations:
Department of Mathematics, Ohio State University, 231 West 18th Avenue, Columbus, Ohio, OH 43210-1174, USA (e-mail: bgp@math.ohio-state.edu)
Venue:
Combinatorics, Probability and Computing
Year:
2005

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Abstract

A solid diagram of volume n is a packing of n unit cubes into a corner so that the heights of vertical stacks of cubes do not increase in either of two horizontal directions away from the corner. An asymptotic distribution of the dimensions – heights, depths, and widths – of the diagram chosen uniformly at random among all such diagrams is studied. For each k, the planar base of k tallest stacks is shown to be Plancherel distributed in the limit $n\to\infty$.