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Polycubes with Small Perimeter Defect

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Abstract

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.

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Notes

  1. It is easy to observe [11] that polycubes of size n can span at most \(n{-}1\) dimensions, hence, \(\text {DX}(n,d) = 0\) for any \(d \ge n\). Therefore, Lunnon’s formula can be rewritten as \(A_d(n) = \sum _{i=0}^{\min (d,n{-}1)} \left( {\begin{array}{c}d\\ i\end{array}}\right) \text {DX}(n,i)\).

  2. The sporadic values are 0 for \(1 \le n \le 3\); 9 for \(n = 4\); and 28 for \(n = 5\). In the sequel, we will not provide sporadic values due to lack of interest.

  3. A quasi-polynomial is a generalization of a polynomial, in which coefficients are periodic functions with an integral period.

  4. The nth cyclotomic polynomial \(\Phi _n(x)\) is the (unique) irreducible polynomial with integer coefficients that divides \(x^n-1\), and does not divide \(x^k-1\) for any \(k<n\). The first cyclotomic polynomials are \(\Phi _1(x)=x-1\), \(\Phi _2(x)=x+1\), \(\Phi _3(x)=x^2+x+1\).

  5. That is, in a polycube of size at least 2, any cell Z is an L-cell unless it has degree 1, or its degree is 2 and both its neighbors are attached to opposite faces of Z.

  6. To see a simpler example for this phenomenon, one can consider the case shown in Fig. 3(b) for \(k=2\), split it into three subcases depending on whether the left vertical bar is shorter, longer, or equal in length to the right one, and calculate generating functions for each of them. We leave the details to the reader.

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Author notes

  1. Yufei Zheng

    Present address: Department of Computer Science, Princeton University, Princeton, NJ, 08540, USA

Authors and Affiliations

  1. Institute of Discrete Mathematics and Geometry, TU Wien—Vienna University of Technology, 1040, Vienna, Austria

    Andrei Asinowski

  2. Department of Mathematics, University of Klagenfurt, 9020, Klagenfurt, Austria

    Andrei Asinowski

  3. Department of Computer Science, Technion—Israel Institute of Technology, Haifa, 3200003, Israel

    Gill Barequet & Yufei Zheng

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Correspondence to Gill Barequet.

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On behalf of all authors, the corresponding author states that there is no conflict of interest. Data sharing is not applicable to this article as no datasets were generated or analyzed during the current study.

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Communicated by Frédérique Bassino.

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Research by the first author has been supported by FWF Grants SFB F50-03 and P28466-N35. Research by the second and third authors has been supported in part by Grant 575/15 from the Israel Science Foundation (ISF) and by Grant 2017684 from the Binational Science Foundation (BSF). Preliminary versions of this paper appeared in Ref. [5] (2D only) and Ref. [6]

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Asinowski, A., Barequet, G. & Zheng, Y. Polycubes with Small Perimeter Defect. Ann. Comb. 26, 997–1020 (2022). https://doi.org/10.1007/s00026-022-00601-7

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