PPubMed11 May 2023Cited 3×

On the combinatorics of crystal structures. II. Number of Wyckoff sequences of a given subdivision complexity On the combinatorics of crystal structures. II

Wolfgang Hornfeck, Kamil Červený, M. I. Aroyo

Abstract

The number of Wyckoff sequences of a given subdivision complexity is calculated by means of a generating polynomial approach and a dynamic programming approach. The result depends on the choice of space-group symmetry (which is obligatory) and Wyckoff sequence length (which is optional). It also takes into account specified values for the total number of combinatorial and coordinational degrees of freedom, thereby representing crystal structures of invariant subdivision complexity.

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The paper introduces generating‑polynomial and dynamic‑programming methods to count Wyckoff sequences for a given space group and degrees of freedom, showing their use on real crystal data.

What’s new

i.Developed a generating polynomial approach to count Wyckoff sequences.
ii.Developed a dynamic programming algorithm for the same counting problem.
iii.Applied both methods to real crystal structure data for several space groups.
iv.Connected the combinatorial counts to subdivision complexity and Shannon entropy measures.
v.Provided open‑source Python and Mathematica implementations of the algorithms.

How they did it

The authors formulate the counting problem as a restricted partition problem and derive a generating polynomial whose coefficients give the number of Wyckoff sequences. They also design a dynamic programming algorithm that solves the same problem more efficiently. Both approaches are implemented in Python and Mathematica and tested on example space groups.

What they found

For space group P421*m (No.113) 44 possible Wyckoff letter sequences were found, with a search space of 11 648 multiplier choices.
The Pearson database contains 759 Wyckoff sequences for this group, reduced to 63 unique sequences after removing duplicates.
For space group Fm3m (No.225) 17 matching sequences were identified, of which 7 are realized in known crystal structures; restricting length k yields 4 solutions.
For space group Cmcm (No.63) 67 matching sequences were identified, with 11 realized in nature.
The dynamic programming algorithm outperforms the generating‑polynomial method in computational speed.

Caveats

✕The generating‑polynomial approach is non‑constructive and does not produce the actual sequences.
✕Exhaustive enumeration becomes infeasible for very large search spaces, limiting practical tractability.
✕Methods require a fixed space‑group symmetry and specified total degrees of freedom, restricting generality.
✕Wyckoff sequences ignore unit‑cell parameters and atomic coordinates, so they cannot capture full structural detail.

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