Journal of Integer Sequences, Vol. 20 (2017), Article 17.9.5

The Perimeter of Proper Polycubes


Sebastian Luther
Institut für Theoretische Physik
Otto-von-Guericke Universität Magdeburg
Postfach 4120
39016 Magdeburg
Germany

Stephan Mertens
Institut für Theoretische Physik
Otto-von-Guericke Universität
Magdeburg Postfach 4120
39016 Magdeburg
Germany
and
Santa Fe Institute
1399 Hyde Park Rd.
Santa Fe, NM 87501
USA

Abstract:

We derive formulas for the number of polycubes of size n and perimeter t that are proper in n - 1 and n - 2 dimensions. These formulas complement computer based enumerations of perimeter polynomials in percolation problems. We demonstrate this by computing the perimeter polynomial for n = 12 in arbitrary dimension d.

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(Concerned with sequences A000041 A001931 A127670 A151830 A151831 A151832 A151833 A151834 A151835 A171860 A191092.)

Received July 7 2017; revised versions received October 22 2017; October 25 2017. Published in Journal of Integer Sequences, October 27 2017.

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