Polycubes with Small Perimeter Defect

Andrei Asinowski, Gill Barequet, Yufei Zheng

Research output: Contribution to journalArticlepeer-review


In this paper, we consider enumeration of d-dimensional polycubes, whose perimeter (defined as the number of empty cells neighboring the polycube) has a fixed deviation from the maximum possible value. We provide a general framework for deriving such formulae, as well as several explicit formulae. In particular, we prove that for any fixed dimension d, the generating function that enumerates polycubes with a fixed defect (with respect to their volume) is rational. Moreover, its denominator is a product of cyclotomic polynomials.

Original languageEnglish
Pages (from-to)997-1020
Number of pages24
JournalAnnals of Combinatorics
Issue number4
StatePublished - Dec 2022


  • Cyclotomic polynomials
  • Linear recurrence
  • Polycubes
  • Rational generating function

All Science Journal Classification (ASJC) codes

  • Discrete Mathematics and Combinatorics


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