crystal lattices
On the combinatorics of crystal structures. II. Number of Wyckoff sequences of a given subdivision complexity
^{a}FZU – Institute of Physics of the Czech Academy of Sciences, Na Slovance 1999/2, 182 00 Prague 8, Czechia
^{*}Correspondence email: hornfeck@fzu.cz
Wyckoff sequences are a way of encoding combinatorial information about crystal structures of a given symmetry. In particular, they offer an easy access to the calculation of a crystal structure's combinatorial, coordinational and configurational complexity, taking into account the individual multiplicities (combinatorial degrees of freedom) and arities (coordinational degrees of freedom) associated with each
However, distinct Wyckoff sequences can yield the same total numbers of combinatorial and coordinational In this case, they share the same value for their Shannon based subdivision complexity. The enumeration of Wyckoff sequences with this property is a combinatorial problem solved in this work, first in the general case of fixed subdivision complexity but nonspecified Wyckoff sequence length, and second for the restricted case of Wyckoff sequences of both fixed subdivision complexity and fixed Wyckoff sequence length. The combinatorial results are accompanied by calculations of the combinatorial, coordinational, configurational and subdivision complexities, performed on Wyckoff sequences representing actual crystal structures.Keywords: Wyckoff sequences; combinatorics; Shannon entropy; structural complexity.
1. Introduction
Any standardized Wyckoff sequence: a string composed of the type number (sometimes the Hermann–Mauguin symbol is used instead) and followed by all the Wyckoff letters for each partially or fully occupied in the the letters are put in reverse alphabetic order and augmented by their superscripted frequency of occurrence, in case a certain nonfixed is occupied multiple times.
can be conveniently related to a descriptor uniquely encoding its combinatorial properties, namely itsStandardization is necessary because many crystal structures do have equivalent descriptions in terms of their ), ch. 8]. A comprehensive scheme for standardization has been theoretically developed by Parthé and coworkers (TYPIX) and implemented into the software STRUCTURE TIDY (Parthé & Gelato, 1984, 1985; Gelato & Parthé, 1987; Parthé et al., 1993a,b). The uniqueness of the Wyckoff sequence thus depends on and follows from the uniqueness of the standardization.
and atomic coordinates, depending either on matters of possible unitcell choices or the symmetry properties of space groups [keywords: symmetry of symmetry, Cheshire groups, Euclidean normalizers; see Müller (2013The Wyckoff positions of a X for which the sitesymmetry groups are conjugate subgroups of (see Hahn, 2005, ch. 8.3.2, p. 733; Aroyo, 2016, ch. 1.4.4.2, p. 62).
encompass all possible distinct sets of symmetryequivalent sites within a Or, to put it in rigorous mathematical terms: a of a consists of all pointsIt is important to distinguish the notions of et al., 2018). This is of special importance in the study of group–subgroup relations, in which a group and one of its proper subgroups can be of the same spacegroup type, but not of the same Another consequence is that the number of space groups is infinite, whereas the number of spacegroup types is finite: 230 distinct ones in three dimensions, including 11 enantiomorphous types. In particular, the Hermann–Mauguin symbol represents the spacegroup type, while concepts related to the Wyckoff positions, such as the Wyckoff sequences studied in this work, are related to the in the sense that they are inseparably connected with the coordinate description of crystal structures. However, the number of distinct Wyckoff positions is considered to be finite, with a total of 1731 positions for all space groups, thus allowing for the combinatorial analysis to follow.
and spacegroup type. A encompasses all of the symmetry of an actual including its translation symmetry, specified by its lattice parameters. A spacegroup type is an abstract notion comprising all those, otherwise distinct, space groups which share the same representation by matrixcolumn pairs of symmetry operations (NespoloThe Wyckoff positions are labelled by up to 27 possible Wyckoff letters (a to z and α), being devoid of any further meaning, but depending on the choice of Since the Wyckoff sequence composed of these letters does not include any specific information about either the unitcell parameters or the atomic coordinates, it is a more abstract notion as well, and useful as such for purposes of classification and systematics (see, e.g., Allmann & Hinek, 2007). Note that this abstraction means that actual crystal structures can share the same Wyckoff sequence, while being distinct with respect to the specific values of the free parameter(s) of their nonfixed Wyckoff position(s). Such crystal structures are called isopointal (LimadeFaria et al., 1990).
Yet, Wyckoff sequences can be studied without referring to specific geometric crystal structures. Thus, any Wyckoff sequence really describes an infinite family of crystal structures, sharing a common parametrization in terms of their geometric et al., 2022).
according to the observed partial or full occupancy of their general position(s) and, if present, special position(s) of their corresponding spacegroup type. Indeed, the Wyckoff sequence has been used in the aforementioned sense, as a coordinatefree representation of crystal structures in modern machine learning approaches to materials discovery (GoodallOne particular advantage of this abstract, combinatorial point of view is due to the fact that the number of Wyckoff sequences of given length k is finite (Hornfeck, 2022a), making an exhaustive study possible for small values of k. In fact, most of the actual crystal structures found so far in nature have Wyckoff sequences of length below k = 50. For the spacegroup type Pmmm (No. 47) with ν = 19 nonfixed sites and φ = 8 fixed ones, constituting the case with the highest number of possible sequences, this would correspond to about 3.5×10^{18} distinct sequences to consider in total, up to the length of k = 50 inclusive (see Appendix A for the computation and the exact result).
Another advantage, and one focus of this work, is due to the fact that the Wyckoff sequence translates into the information about a crystal structure's combinatorial (M) and coordinational (A) collective associated with a weighted sum of each Wyckoff position's individual multiplicity ( M_{i}) and arity ( A_{i}), respectively, the latter being the number of independent coordinate parameters required to be specified in a standardized description of an actual crystal structure.
Both the combinatorial and coordinational , 2014) and extended by Hornfeck (2020, 2022b) based on the utilization of the Shannon as a complexity measure.
can be used to assess a crystal structure's combinatorial and coordinational complexity by means of an approach pioneered by Krivovichev (2012Taking a general point of view, the collective M+A, result as the sum of the individual combinatorial (M) and coordinational (A) In a similar way, on the level, the collective combinatorial and coordinational M or A, result as the sum of the individual combinatorial and coordinational M_{i} or A_{i}, respectively.
represent a certain system (a macrostate), while the individual each represent a certain subsystem (a microstate). This is true on different levels of hierarchy. For instance, on the level, the collective configurationalIn combinatorial terms this splitting of a system into subsystems corresponds to an integer partition. However, in their crystallographic application, the number of partitions is not simply given by the numbertheoretic i.e. modulo centring translations, which, in the following, will be the preferred choice of description), while the possible values of the Wyckoff arities are restricted to the set . Moreover, for a given choice of spacegroup type, the values of either set that can occur are restricted further (although not their frequencies of occurrence, except for fixed positions, which can occur only once), in accordance with the existing Wyckoff positions. A final restriction is imposed by each introducing a coupling of values from both sets. The combinatorial problem under consideration thus is one of counting the number of restricted, coupled partitions.
yet restricted by and the coupling of combinatorial and coordinational degrees of freedoms as found within each In particular, the possible values of the Wyckoff multiplicities considering all spacegroup types are restricted to the set (assuming primitive unit cells,Thus, due to these particular restrictions, it is a nontrivial task to describe a macrostate defined by the collective (M,A) by the composition from or the subdivision into its associated microstates (M_{i},A_{i}). Foremost, it is a natural question to ask, given one macrostate, what is the number of microstates corresponding to it?
The following section will give the answer to this question, with the main results being a generating polynomial approach (Section 2.4, theory; Section 2.7, algorithm) and a dynamic programming approach (Section 2.8, theory and algorithm).
2. Combinatorics of Wyckoff sequences
To state the problem succinctly: how many distinct Wyckoff sequences (of optionally fixed length k) exist, given a pair of combinatorial and coordinational (M,A) together with a choice of space group? (Note that a has to be fixed, since this determines the alphabet of Wyckoff letters which can appear in the Wyckoff sequence.)
2.1. A problem of crystals – exposition
The problem statement can be translated rather straightforwardly into some algebraic form. Note that two integer values exist for each individual i, its multiplicity M_{i} and its arity A_{i} (coordinational degree of freedom). Both have to be considered in a coupled way, which will be done in the notation by the use of column vectors. Then, the total multiplicity and arity, M and A, respectively, corresponding to a certain set of Wyckoff sequences of crystal structures, are given as
Here, the sum index i runs over all the n existing Wyckoff positions for a given with the integer multipliers denoting the frequency of occurrence of a given site in the sum of individual multiplicities M_{i} and arities A_{i} of the combined sites. In one general point of view, as mentioned in the Introduction, one can call the pair (M,A) the system variables (describing the macrostate), while the M_{i} and arities A_{i} would be the subsystem variables (describing the microstates).
Information about the Wyckoff positions of the 230 threedimensional spacegroup types is compiled in Vol. A of the International Tables for Crystallography (Aroyo, 2016), which is the authoritative source. Alternatively, it can also be retrieved online from the Bilbao Crystallographic Server (https://www.cryst.ehu.es/) using the routine WYCKPOS (Aroyo et al., 2006a,b, 2011).
The values of the Wyckoff multiplicities M_{i} are explicitly stated as the numeral part of the Wyckoff symbol assigned to each (the nonnumeral part is given by the Wyckoff letter). Note, however, that these values are given for the centred unit cells, in which case M should also be specified with respect to the unitcell content of the centred in order to maintain the correct correspondence. Alternatively, the values of the Wyckoff multiplicities can be reduced by the division of an integer factor depending on the centring type (2 for C, A and I centring; 3 for R centring; 4 for F centring), if the primitive and the number of atoms it contains, is taken as a reference. Indeed, choosing the primitive as a reference is strongly recommended in the context of crystallographic complexity calculations (cf. Section 3), in order to make the results of these calculations comparable for crystal structures differing in their centring type, since any existing centring translation just repeats parts of a inside a thereby contributing no additional information to its description, and, accordingly, its complexity.
The values of the Wyckoff arities A_{i}, although not explicitly stated in the aforementioned sources, can be deduced from them, namely by means of visual inspection with respect to the number of positional variables x, y and z occurring in the listing of the general coordinates as provided for each Wyckoff position.
In general, this information can be obtained for any crystallographic et al. (1978) for the case of four dimensions].
including higherdimensional ones, namely by inspecting the number of symmetryequivalent positions modulo translations, to obtain the multiplicities, as well as by elucidating the dependency of their general coordinates with respect to symmetry, to obtain the arities. In fact, all this information can be obtained in a purely algorithmic manner [see BrownIt should be noted that the problem at hand can be given different mathematical interpretations and representations. Equation (1) already suggests a geometrical interpretation as twodimensional vectors, which happen to live on the twodimensional integer lattice (square lattice). The problem then appears as a special case of a lattice path problem (Fig. 1), in which the vectors associated with the Wyckoff positions of a define the set of steps.
Alternatively, the twodimensional plane can be identified with the complex plane of Wessel, Argand and Gauß suggesting a change of notation, in which i is the imaginary unit ():
While these interpretations and representations are mathematically equivalent, and thus do not seem to make any difference per se, the knowledge of these alternatives can be of importance when it comes to searching for subfields of mathematics discussing already existing solutions to a given problem, or general methods to find them, and also in the case of implementation into computer code.
2.2. Conditions on the multipliers
For any given site the multipliers are restricted to discrete intervals
of potential integer values, which are denoted in shorthand as in the following. Naturally, by definition, the frequencies of occurrence are bounded from below, likewise for all sites, by all minimal multipliers , meaning that a site is absent in this case. The variable upper bounds, the maximal multipliers, are determined according to the case distinction
differentiating between nonfixed and fixed sites. Collecting the terms for the nonfixed and fixed sites separately, the summation of equation (1) can be split into separate parts,
in which ν and φ now denote the total number of nonfixed and fixed sites, respectively (compare Hornfeck, 2022a).
In any case, the repetition r_{i} is given as
with denoting the floor function, and the smaller ratio restricting the number of possible occurrences of the corresponding site from above. Stated in a different way, each multiplier for any nonfixed site has to fulfil both of the conditions
simultaneously, with being defined as the largest integer doing so.
These conditions originate from the fact that the Wyckoff multiplicities and arities are nonnegative integers; thus any surpassing of either one of the limits M or A cannot be balanced by the addition of another hence cannot be a part of the solution, and thus signifies an end to the Wyckoff sequence construction process. This guarantees the existence of maximal values and fixes the size of the search space of which the solution space is a subspace.
While the determination of the size of the solution space, the number of Wyckoff sequences existing for a given choice of (M,A), is our main task, the determination of the size of the search space appears as a first step towards a result, in that it gives a numerical upper bound for the size of the solution space, a combinatorial overview about the Wyckoff sequences to expect, with respect to their length, as well as an estimate regarding the computational tractability of their actual construction based on an exhaustive exploration of the search space.
2.3. Size of the search space
For nonfixed sites the multiplier intervals are given as , with variable r_{i}, while for the fixed sites the multiplier intervals are given as [[0,1]] = {0,1}, invariably. The over all interval sets of multipliers of either type determines the search space:
This corresponds to the set of all possible Wyckoff letter sequences, being Wyckoff sequences without the spacegroup type symbol/number prefix explicitly stated, as constructed from the multiset , in which the denote a general Wyckoff letter out of n possible letters for a given Note that the choice of while defining the alphabet of Wyckoff letters and limiting the number of terms to consider in the does not determine the size of the search space by itself. This size, the cardinality
of the search space, in which individual solutions have to be found, if they exist, is determined by the values of the maximal multipliers. Thus, the size of the search space is determined only if both the M and A are fixed. Then, the search space size gives an absolute upper bound on the potential number of solutions, yet usually, and in anticipation of our following results, the actual number of solutions will be much lower or even zero. This difference in the number of solutions is due to the construction of the search space by means of the namely because the restrictions imposed by the values for individual Wyckoff positions do not take into account their cumulative, conditional interactions.
and the total numbers ofAs is often the case in combinatorial problems, the same cardinality can be obtained by an alternative counting method, namely as the result of the summation of all coefficients in the expansion of the univariate polynomial defined by
This is the generating polynomial for a multiset with finite multiplicities [compare its description and, in particular, equation (4) in Hornfeck (2022a)]. Upon its expansion the coefficients c_{k} for each term x^{k} of this polynomial count the number of sequences of length k, thereby representing a more differentiated view of the search space's contents. Eventually,
in which is the multiset's cardinality.
The size of the search space gains some importance due to the fact that the combinatorial approach we will describe in the remainder of this work is nonconstructive, as is commonly the case for such combinatorial questions, since it gives only the number of potential Wyckoff sequences matching with a given parameter pair (M,A), but does not reveal the Wyckoff sequences themselves; these can be discovered by an exhaustive check of all admissible multiplier combinations existing within the combined multiplier intervals.
We envision that the search space size can be reduced to some degree by applying effective intermediate checks for multiplier combinations already violating the upper limit as imposed by the choice of the parameters (M,A), overshooting either one parameter at a time or both simultaneously, possibly in combination with the use of a clever data structure such as pruned trees. However, search space sizes below are easily tractable on a standard desktop personal computer, which should encompass most tasks related to the comparison of the potential combinatorial solutions with Wyckoff sequences representing actual crystal structures.
2.4. A generating polynomial approach to find solutions
Our combinatorial problem stated above is solved in two steps: first, by reducing it, conceptually, to an analogous classical problem of combinatorics, the coin change problem, as can be found in many textbooks on the topic (for instance, Marcus, 1998, p. 89), and second, by adapting the classical problem to the crystallographic one.
The classical coin change problem is stated in Appendix B, and can be seen as an illustration of the use of generating polynomials, defined in some abstract variable. Notably, the variable is an indeterminate symbol only, entailing no specific meaning other than to allow algebraic operations performed on it; hence it merely acts as a placeholder and bookkeeping device, yet is the decisive one, in order to systematically find a solution.
This use of generating polynomials has already been described in the first entry of this series (Hornfeck, 2022a) to which the reader, interested in more detailed information, is referred. An introduction to the wider field of generating functions is given by Graham et al. (1994), and a more detailed exposition of the main ideas involved is given by Wilf (2006).
Now, adapting the classical coin change problem to our crystallographic one is carried out in three steps: (i) matching the number of distinct types of coins with the number of distinct Wyckoff positions; (ii) identifying the values of distinct types of coins with the pair of values of the Wyckoff position's multiplicity M_{i} and arity A_{i}; and (iii) treating the pair of values ( M_{i}, A_{i}) in a coupled way, by introducing two abstract variables (x,y) in the generating polynomials, instead of one.
Thus, in some way, the crystallographic problem is a coin change problem with a twist, based on imaginary coins with denominations on both their front and back sides and a pair of target values to reach upon summation. Similar combinatorial problems arise for the case of real cards, coupling values denoted by digits or letters with symbolic ones such as diamonds, hearts, spades and clubs.
In particular, the third adaptation step is crucial for the correct enumeration. As a consequence of it, generating polynomials of the kind
are assigned to each n Wyckoff positions,
with their product over allyielding the solution, namely by the value of the coefficient of x^{M}y^{A} in the expanded form of the polynomial .
As an aside, one can note that the product over all n sites can be split,
according to the contributions of ν nonfixed and φ fixed sites [compare equation (5)]. This splitting reduces the problem for the fixed sites to a univariate one.
It should be noted that this approach can be further generalized, in principle, by taking into account chemical )] in terms of atomic numbers of atoms occupying a given as well. Then, one would have to introduce a third variable z into the respective polynomials with all other procedures considered to be analogously performed. However, there is a difference, in that the atomic numbers are not restricted in any way, that is to say there exists no natural coupling between them and the Wyckoff multiplicities and arities – they are a pure matter of choice. In contrast, the Wyckoff multiplicities and arities are coupled in their values for each of a Thus, regarding this relative arbitrariness of choice and the relative unimportance of this general case, we refrain from expanding in this direction for the moment. However, our opinion on this topic might change in the future, if there should be an interesting application for fixing the total chemical that is the total electron count within a reduced to a set value, together with the other and asking for the number of crystal structures fulfilling this condition. In this case, any generalization required can be obtained in a straightforward manner by following the same extension procedure as described in the following.
[as introduced by Hornfeck (20202.5. A generalization including the Wyckoff sequence length
Another and considerably more useful generalization is made by taking into account the length k of the Wyckoff sequence as a further restriction, which can be seen as an extension and a to a previous result in the combinatorics of Wyckoff sequences (Hornfeck, 2022a). This can be achieved by adjusting equation (1) according to
in addition to
used for the determination of the for each individual site i. Trivially, each site itself is of length k_{i} = 1 (thus, ), which gives the total length of the Wyckoff sequence as the sum of the multipliers : . Again, the final result is given as the value of the coefficient of x^{M}y^{A}z^{k} in the expanded polynomial
in which
defines the individual terms.
As was the case before, the product over all n sites
can be split according to the contributions of ν nonfixed and φ fixed sites.
Finally, it should be noted that there exist general methods for obtaining explicit formulas for the coefficients of generating functions, for instance for the powers of bivariate generating functions (Kruchinin et al., 2021), as this is an active field of mathematical research.
2.6. A problem of crystals – exemplification
In the following, an illustrative example is discussed in full calculational detail.
The tetragonal spacegroup type (No. 113) encompasses a total of six distinct Wyckoff positions,
here written in a more convenient inline notation, indexed according to their Wyckoff letters, and with their corresponding multiplicities and arities stated:
The chosen A_{i} = 0,1,2,3 are present. The six Wyckoff positions comprise a total of three and four distinct values for the Wyckoff multiplicity and arity, respectively, thereby allowing us to illustrate the combinatorial calculation in some detail.
is the one with the smallest number of Wyckoff positions for which all possible arity valuesNow, with some arbitrary chosen M = 12 and A = 3 given for this particular example, this results in the following maximal multipliers:
for each
and consequently in the following generating polynomials:Note how the two fixed sites with Wyckoff letters a and b give rise to exactly the same polynomial factor in one variable x only (namely the one used for the Wyckoff multiplicities), and how the polynomials corresponding to the nonfixed sites are restricted in terms of the monomials of highest degree in either x or y by the given values of either M or A or both.
Note also that the number of variables in each polynomial corresponds to the splitting of all n sites into ν contributions from the nonfixed sites (bivariate case) and φ contributions from the fixed ones (univariate case) [compare equation (5)].
The expanded polynomial consisting of a total of 60 terms is given by
The decisive term x^{M}y^{A} = x^{12}y^{3} has 6 as its coefficient, which thus corresponds to the six existing solutions, looked for in the search space; they can be stated in terms of their Wyckoff letter sequence, with multiple occurrences of the same letter marked by superscripts:
Note, in particular, the cases d^{2}cb and d^{2}ca which differ only in the exact choice of the fixed site, being otherwise degenerate in their column vector summation [compare equation (1)].
In fact, the bivariate polynomial given in equation (24) contains the information about all possible solutions (M,A) which can be constructed from the multiset of Wyckoff letters [a,b,3c,3d,e,f] with restricted multiset multiplicities, as determined by the maximal multipliers. Since the multiplier intervals are [[0,3]] for both the Wyckoff positions c and d, as well as [[0,1]] for all other sites, the search space has a size of 4^{2}×2^{4} = 256 cases, whose distribution according to the length k of the sequence can be read off from the expansion (11 terms; not shown) of the generating polynomial
since for four out of the six Wyckoff sites and for the remaining two. The observed six solutions with the property (M,A) = (12,3) constitute only a tiny fraction of the search space, which is generally true, the size of the search space typically being overwhelmingly larger than the number of solutions.
In particular, the full set of solutions contains the singular occurrence of the empty sequence, { }, corresponding to the trivial monomial x^{0}y^{0} = 1, as well as that of the maximal length sequence, abcccdddef, corresponding to the monomial x^{34}y^{11} of highest degree, for which M = 34 and A = 11. Note how one can instantly check that there exists no solution for, say, the case M = 7 and A = 1, because no monomial of the form x^{7}y appears in , or, to put it another way, the coefficient of this monomial in the expansion of is equal to zero. Note also how one specific solution, namely solution d^{3}, corresponds to the occurrence of the monomial x^{12}y^{3}, the j = 3 case of the basic monomial x^{j 4}y^{j 1} representing the in the generating polynomial P_{d}(x,y) of equation (23).
Taking into account the length k as an additional parameter, one has to adjust the individual polynomial terms according to
Expansion of their product yields a polynomial in 142 terms (not shown), with, for instance, the value of the coefficient for x^{M}y^{A}z^{k} = x^{12}y^{3}z^{4} being equal to 3, corresponding to the Wyckoff letter sequences
which form a subset of the six aforementioned solutions.
2.7. Summary of the generation function approach
To summarize the aforementioned results in an algorithmic form, using the generating polynomial approach one has to perform the following steps to obtain a solution (with the first two items in the enumeration defining the input to the algorithm and the last one its output):
(i) Fix a
and thereby the set of Wyckoff positions and the potential alphabet of Wyckoff letters from which a Wyckoff sequence can be formed.(ii) Fix the values for the total Wyckoff multiplicity M and the total Wyckoff arity A and (optionally) the Wyckoff sequence length k, all of which are expected to be integers.
(iii) Retrieve all the individual values for the Wyckoff multiplicities and arities .
(iv) From the individual Wyckoff multiplicities and arities compute the individual maximal multipliers .
(v) From the individual maximal multipliers construct the individual generating polynomials, P_{i}(x,y) or P_{i}(x,y,z).
(vi) From the individual generating polynomials build their product, or , and expand it.
(vii) From the expansion read off the coefficient for the monomial of the form x^{M}y^{A} or x^{M}y^{A}z^{k}; this is the solution.
2.8. A dynamic programming approach to find solutions
Finally, after illustrating the aforementioned combinatorial method on an explicit example, we want to highlight the possibility of an alternative algorithmic way of calculation which turns out to be very efficient, by making optimal use of the recursive nature and overlapping B. In the following, we show the adapted pseudocode for the bivariate and finite coin change problem in its crystallographic application, thus taking into account the case distinction for the multipliers of nonfixed and fixed Wyckoff sites (Fig. 2). The adaptation to the trivariate and finite case including the length k as a parameter follows a straightforward procedure (Fig. 3), as would be the case for any future multivariate generalization, for instance also taking into account chemical degrees of freedom.
of the problem, in such a way that subsolutions are only ever computed once and retrieved as needed for the calculation of the solution. This algorithmic way is known under the name dynamic programming. A simple example for the classical univariate and infinite coin change problem is given in AppendixIn either case, the approach starts with the trivial base case (there is always one possibility for the null tuple), from which it proceeds bottomup making use of a tabulation implementation of subsolution values cached into either a matrix (bivariate case) or a tensor (trivariate case), which is updated iteratively until the target case is reached and the solution is returned. Equation (29) shows the solution matrix for the case defined by the target tuple (M,A) = (12,3) and the multiset of tuples in which the values for M_{i} and A_{i} increase along the rows and the columns, respectively. Note that for a more economic display, the matrix is shown in its transposed form, with columns and rows interchanged, such that its upperleft corner represents the matrix element and the lowerright corner represents the matrix element , from which the solution, , can be read off:
A direct comparison shows that all nonzero entries represent the nonvanishing coefficients as occurring in the polynomial expansion given in equation (24), yet with terms only up to the solution x^{12}y^{3} monomial inclusive. Note that the algebraic structure of the Wyckoff positions is reflected in the matrix as well, since the reachable positions are determined by the smallest increments and their parity as observed for the Wyckoff multiplicities and arities. For instance, since all the Wyckoff multiplicities are even numbers, all odd columns of the above matrix are given by the null vector.
A Python implementation of the algorithm for the bivariate case is given in Appendix C, which also contains a Mathematica implementation of the generating polynomial approach.
3. Complexity of Wyckoff sequences
The aforementioned problem of crystals arose in the context of calculating Shannon ). Apart from its chemical – its decoration (colouring) of Wyckoff sites with atoms – a is geometrically defined by its combined combinatorial and coordinational the collective multiplicities M and arities A of its occupied Wyckoff positions.
based complexity measures for crystal structures, taking into account a crystal structure's fundamental chemical, combinatorial and coordinational (Hornfeck, 20203.1. Shannon based complexity measures
In general, a given multiset of n individual X_{i} defines a discrete probability distribution
where denotes the collective number of
as obtained from the partition of the individual numbers of From this a Shannoncan be obtained. Note that L(0) = 0, by definition.
As mentioned before, the general interpretation is that of a system with X on a higher level of structural hierarchy being subdivided into n subsystems of X_{i} each on a lower level of structural hierarchy. For more details of the general theory, the reader is referred to previous work done by the author (Hornfeck, 2020, 2022b).
In particular, a crystal structure's M collective combinatorial are associated with the multiset of individual Wyckoff multiplicities M_{i}, thereby yielding a fundamental combinatorial Shannon entropy:
Note that in order to ensure the comparability of combinatorial complexity values between different crystal structures, the values of the Wyckoff multiplicities M_{i} have to refer to a primitive [although this is really important only for derived complexity values such as the maximal complexity, , the normal complexity, , or the total complexity, , which have been described by Hornfeck (2020)].
Now, in the same way, a crystal structure's A collective coordinational are associated with the multiset of individual Wyckoff arities A_{i}, thereby yielding a fundamental coordinational Shannon
Now, proceeding in a completely analogous manner, a crystal structure's F = M+A collective configurational are associated with the combined multiset
of individual Wyckoff multiplicities M_{i} and individual Wyckoff arities A_{i}, corresponding to the individual configurational F_{i}, thereby yielding a composite configurational Shannon
While the association with the combined multiset is natural, it is noteworthy to mention that this does not mean that the corresponding entropies are (simply) additive; on the contrary
Most importantly, however, by means of the strong additivity property of the Shannon
the configurational is equivalent to the general expressionincluding appropriate weighting factors w_{M} = M/(M+A) and w_{A} = A/(M+A) and an additional subdivision (mixing) complexity based on them,
This equivalence now allows an assessment of the relative complexity related to the splitting of a given collective number of M+A, for the composite system, into individual numbers of M and A, for the fundamental (sub)systems.
The subdivision complexity is a Shannon and for all the combinatorially enumerated cases it is an invariant characterizing the subdivision step on the higher level of hierarchy. It can be seen as connecting the collective treatment of within the combined configurational complexity , with the individual treatment of within the separated combinatorial and coordinational complexities, and , respectively. This allows for a complexity partition analysis (Hornfeck, 2022b). This central importance of the subdivision complexity and its relationship to the other complexity measures is graphically depicted in Fig. 4.
of the weighting factors occurring in equation (37)In the same manner, other subdivision complexities exist on the lower X denoting either M or A.
level of hierarchy, characterizing the subdivision steps , withFurthermore, it should be noted that the strong additivity property also holds for the case of maximal Shannon entropies
where takes on the same value as in equation (37). The maximal entropies are reached in the case of maximal subdivision and thus perfect equidistribution of (corresponding to maximally expanded partitions of variable length equal to the number of degrees of freedom). This makes their calculation particularly simple, resulting eventually in , = log_{2}A and .
Each maximal
can also be used in turn to define its corresponding nonmaximal for instanceby means of taking the difference between the maximal entropies for the collective (here, F) and individual (here, F_{i}) the latter ones attributed with their appropriate weighting factors F_{i}/F.
All the aforementioned interrelations are depicted in Fig. 5.
 Figure 5 level, thus highlighting the applicability of the strong additivity property on both levels of hierarchy (in this case the common weighting factors 
4. Combining the combinatorics and complexity of Wyckoff sequences
Some elementary statistical results shall be stated about the magnitude of the collective ).
to expect, in extreme and on average, and with respect to actual data as retrieved from the 20 040 unique Wyckoff sequences compiled in the Pearson's Crystal Data Database for Inorganic Compounds (Villars & Cenzual, 2020The minimal observed combinatorial and coordinational M_{min} = 1, A_{min} = 0, the maximal are M_{max} = 5926, A_{max} = 1476, the mean values are 94.3]" class="img_align_bottom mathimage" style="position: relative; display: inline; paddingtop: 0px; paddingbottom: 0px; verticalalign: 2.465474pt;" height="14" width="227" />, , the median values are , , and the mode values are , , respectively.
are, as determined for the individual, independent distributions,Our combinatorial result now tells us how many of these partitions, being solutions to a combinatorial problem restricted by (M,A) and, potentially, the length k of the Wyckoff sequence. In the following we will illustrate this by performing explicit calculations for three examples.
can be realized for a given given the fixed number ofIt should be noted that the examples given here were chosen such that the discussion of the results could be made explicit, with the full set of potential Wyckoff sequence solutions being listed, and with some actual representatives being found among the known crystal structures. In general, the number of potential solutions might exhibit a rapid growth (combinatorial explosion), while the number of actual representatives lags behind greatly.
4.1. Example 1: spacegroup type P42_{1}m (No. 113) revisited
As our first example, we revisit the case of spacegroup type (No. 113). Table 1 compiles some values of specific Shannon entropies for the Wyckoff sequences given in this example, taking into account the combinatorial, coordinational and configurational highlighting the constant term , with the other Shannon entropies related according to their strong additive sum:

A search in the Pearson's Crystal Data ) for the spacegroup type (No. 113) reveals a total of 759 Wyckoff sequences for which prototypes have been assigned. Reducing for multiple entries arising from multiple determinations sharing the same prototype yields 79 entries with a unique Wyckoff sequence/prototype combination. (Notably, 538 entries share the same Wyckoff sequence 113 fe^{3}ca, with seven distinct prototypes associated to it, of which the tP24–Ca_{2}MgSi_{2}O_{7} prototype alone occurs 307 times, thereby explaining the heavy degree of reduction observed.) Reducing again for distinct prototypes sharing the same Wyckoff sequence yields 63 unique Wyckoff sequences.
Database for Inorganic Compounds (Villars & Cenzual, 2020Of these, 55 correspond to a unique pair (M,A) of combinatorial and coordinational with six pairs occurring two times, and one pair, (28,12), occurring three times. The associated Wyckoff sequences and prototypes are: fe^{4}cb (Th_{2}Se_{5} prototype), fe^{4}ca [(Ca_{0.25}La_{0.75})_{2}Ga_{3}O_{7.25} prototype] and e^{4}dc^{3}a (Ce_{2}CoAl_{7}Ge_{4}/LiSmTiO_{4} prototypes).
On the basis of this work, one obtains a larger number of 44 possible Wyckoff letter sequence solutions [based on the Wyckoff multiplicities and arities, compare equation (21)], which can be constructed from an exhaustive exploration of a search space of much larger size of 11 648 multiplier choices (representatives with prototypes marked by an asterisk):
While the combinatorial approach is nonconstructive, it might often suffice to only know the number of possible solutions.
4.2. Example 2: spacegroup type Fm3m (No. 225)
As another example, we consider the cubic spacegroup type (No. 225) encompassing a total of 12 distinct Wyckoff positions of the following multiplicities and arities:
Note that the multiplicities stated here are those for a reduced (primitive) Fcentred would be four times larger. Again, all of the four possible values for the Wyckoff arity are present in this example, in addition to seven distinct values for the Wyckoff multiplicity. This example was chosen because it partly corresponds to actual crystal structures.
while the multiplicities for anNow, for the given choice of (M,A) = (28,3) one finds 17 matching Wyckoff letter sequences:
Here, the seven sequences preceded by an asterisk are realized as crystal structures. A compilation of their complexity values, which are interrelated according to
is given in Table 2. Here, only the combinatorial contribution is responsible for the variable amount of configurational complexity , the other terms, in particular the value for , being constant.
If one performs the same calculation with the additional restriction of the Wyckoff sequence length to the value k = 5 one obtains the number of four solutions, which the reader can easily check on the list given above.
4.3. Example 3: spacegroup type Cmcm (No. 63)
As a final example, we consider the orthorhombic spacegroup type Cmcm (No. 63) encompassing a total of eight distinct Wyckoff positions of the following multiplicities and arities:
Note that the multiplicities stated here are those for a reduced (primitive) Ccentred would be two times larger. Again, all of the four possible values for the Wyckoff arity are present in this example, in addition to three distinct values for the Wyckoff multiplicity.
while the multiplicities for aNow, for the given choice of (M,A) = (22,10) one finds 67 matching Wyckoff letter sequences:
Here, the 11 sequences preceded by an asterisk are realized as crystal structures. A compilation of their complexity values, which are interrelated according to
is given in Table 3. Here, both the combinatorial and coordinational are responsible for the variable amount of configurational complexity .

5. Conclusion
As a contribution to the combinatorics of Wyckoff sequences, we have presented two methods to calculate their number for a fixed and 2.7 for the key results), while the second makes use of a dynamic programming algorithm (Section 2.8). While the generating polynomial approach appears to be conceptually easier to understand, the dynamic programming algorithm is considerably better in its computational performance. The methods have been exemplified on cases of ideal and actual crystal structures with invariant subdivision complexity and variable configurational complexity in the sense of Hornfeck (2020), thus relating the combinatorics of Wyckoff sequences to the complexities of crystal structures.
given a pair of combinatorial and coordinational total and, optionally, their length. The first method is based on a generating polynomial approach (see Sections 2.4APPENDIX A
Number of Wyckoff sequences of length k
The number of Wyckoff sequences of length k is given as
in which ν and φ, respectively, denote the number of nonfixed and fixed Wyckoff positions of a given [equation (7) in Hornfeck (2022a)]. In the worst case and , thus
gives the number of distinct Wyckoff sequences up to k = 50 inclusive.
APPENDIX B
The classical coin change problem
Assume you are in the possession of the following multiset of coins: three coins of value v = 1, four coins of value v = 2, two coins of value v = 5. How many ways exist to pay a total price of value V = 12?
Let S_{v} denote the individual sum of coins of value v, hence
The values which each individual sum may take are determined and limited in number by the number of available coins of each nominal value. Thus, one gets
as all possible values. Now, the trick is that one identifies these values with the exponents of a couple of generating polynomials, one for each type of coin, namely
in which x^{0} = 1 by definition. From these individual polynomials one forms their product
and analyses its expansion
The coefficient of x^{12} in gives the desired solution, here it is equal to four, describing the four cases (i) 2×1+2×5, (ii) 1×2+2×5, (iii) 3×1+2×2+1×5 and (iv) 1×1+3×2+1×5, all amounting to 12.
A generalization is possible, in case the number of coins available for all given values v is assumed to be infinite. Then, the generating polynomials are replaced by the respective generating functions of a geometric series,
and the solution to the problem is given by their product
in the same way as before, namely as the value of the coefficient [x^{V}] in , in which V denotes the total price. In the infinite case the result is , necessarily containing at least as many or even more solutions than were found in the finite case. In this general form the problem has been addressed by Pólya (1956). A closed form solution for the univariate infinite case has been described by Graham et al. (1994, pp. 327–330 and pp. 344–346).
Note that
is the generating function of the numbertheoretic integer , p. 71), and our problem asks for the number of restricted partitions into specified, possibly repeated parts (a finite or infinite number of coins with a finite set of distinct denominations). In this context the number of nonnegative integer solutions is known as Sylvester's denumerant (Sylvester, 1857) which has a rich history in number theory, partly due to the difficulty of obtaining exact results (Alfonsín, 2005, ch. 4).
(Alfonsín, 2005Finally, it should be emphasized that an elegant method of solving this problem in practice is given by the concept of dynamical programming. Since the problem again and again reduces to simpler subproblems of the same kind, the solutions of these subproblems can be stored whenever they occur for the first time. Moreover, using a bottomup approach (topdown would be equally feasible), the solutions for subproblems can be systematically generated from the already known solutions. For this purpose the function
here given in a Python implementation is initialized with a result vector of length n+1 of the form . This result vector is subsequently updated in a separate loop for every coin value. The final value at position n of the result vector then is the solution to the (infinite) coin problem described before (for n = 12 the result is equal to 13).
APPENDIX C
Code examples
Figs. 6 and 7 give working examples for the calculation of the number of Wyckoff sequences with the same subdivision complexity by means of the generating polynomial and dynamic programming approaches. The input values are taken from the example described in Section 2.6, with the output being equal to six.
Acknowledgements
The author thanks Philipp Kuhn for discussions, one reviewer for helpful comments which greatly improved the manuscript, Michal Dušek for his kind endorsement, as well as the organizers of the conference Mathematics and Computer Science for Materials Innovation (MACSMIN), Vitaliy Kurlin and Mois Aroyo, for the invitation to present this work at MACSMIN2022 (Crystal Lattice Classifications) on location at the Materials Innovation Factory, Liverpool, United Kingdom.
Funding information
This work was supported by the Czech Science Foundation, project No. 2105926X. We also acknowledge CzechNanoLab Research Infrastructure supported by MEYS CR (LM2018110).
References
Alfonsín, J. L. R. (2005). The Diophantine Frobenius Problem. Oxford: Oxford University Press. Google Scholar
Allmann, R. & Hinek, R. (2007). Acta Cryst. A63, 412–417. Web of Science CrossRef IUCr Journals Google Scholar
Aroyo, M. I. (2016). Editor. International Tables for Crystallography, Vol. A, Spacegroup Symmetry, 6th ed. Chichester: Wiley. Google Scholar
Aroyo, M. I., Kirov, A., Capillas, C., PerezMato, J. M. & Wondratschek, H. (2006a). Acta Cryst. A62, 115–128. Web of Science CrossRef CAS IUCr Journals Google Scholar
Aroyo, M. I., PerezMato, J. M., Capillas, C., Kroumova, E., Ivantchev, S., Madariaga, G., Kirov, A. & Wondratschek, H. (2006b). Z. Kristallogr. 221, 15–27. Web of Science CrossRef CAS Google Scholar
Aroyo, M. I., PerezMato, J. M., Orobengoa, D., Tasci, E., de la Flor, G. & Kirov, A. (2011). Bulg. Chem. Commun. 43, 183–197. CAS Google Scholar
Brown, H., Bülow, R., Neubüser, J., Wondratschek, H. & Zassenhaus, H. (1978). Crystallographic Groups of Four Dimensional Space. New York: Wiley. Google Scholar
Gelato, L. M. & Parthé, E. (1987). J. Appl. Cryst. 20, 139–143. CrossRef Web of Science IUCr Journals Google Scholar
Goodall, R. E. A., Parackal, A. S., Faber, F. A., Armiento, R. & Lee, A. A. (2022). Sci. Adv. 8, eabn4117. CrossRef PubMed Google Scholar
Graham, R. L., Knuth, D. L. & Patashnik, O. (1994). Concrete Mathematics – a Foundation for Computer Science, 2nd ed. New York: AddisonWesley. Google Scholar
Hahn, T. (2005). Editor. International Tables for Crystallography, Vol. A, Spacegroup symmetry. 5th ed. Heidelberg: Springer. Google Scholar
Hornfeck, W. (2020). Acta Cryst. A76, 534–548. Web of Science CrossRef IUCr Journals Google Scholar
Hornfeck, W. (2022a). Acta Cryst. A78, 149–154. CrossRef IUCr Journals Google Scholar
Hornfeck, W. (2022b). Z. Kristallogr. 237, 127–134. CrossRef CAS Google Scholar
Krivovichev, S. (2012). Acta Cryst. A68, 393–398. Web of Science CrossRef IUCr Journals Google Scholar
Krivovichev, S. V. (2014). Angew. Chem. Int. Ed. 53, 654–661. Web of Science CrossRef CAS Google Scholar
Kruchinin, D., Kruchinin, V. & Shablya, Y. (2021). Mathematics, 9, 428. CrossRef Google Scholar
LimadeFaria, J., Hellner, E., Liebau, F., Makovicky, E. & Parthé, E. (1990). Acta Cryst. A46, 1–11. CrossRef CAS IUCr Journals Google Scholar
Marcus, D. A. (1998). Combinatorics – a Problem Oriented Approach. Washington, D. C.: The Mathematical Association of America. Google Scholar
Müller, U. (2013). Symmetry Relationships between Crystal Structures – Applications of Crystallographic Group Theory in Crystal Chemistry. Oxford: Oxford University Press. Google Scholar
Nespolo, M., Aroyo, M. I. & Souvignier, B. (2018). J. Appl. Cryst. 51, 1481–1491. Web of Science CrossRef CAS IUCr Journals Google Scholar
Parthé, E., Cenzual, K. & Gladyshevskii, R. E. (1993a). J. Alloys Compd. 197, 291–301. Google Scholar
Parthé, E., Gelato, L., Chabot, B., Penzo, M., Cenzual, K. & Gladyshevskii, R. (1993b). TYPIX – Standardized Data and Crystal Chemical Characterization of Inorganic Structure Types (Vol. 1). In Gmelin Handbook of Inorganic and Organometallic Chemistry, 8th ed. Berlin: Springer. Google Scholar
Parthé, E. & Gelato, L. M. (1984). Acta Cryst. A40, 169–183. CrossRef Web of Science IUCr Journals Google Scholar
Parthé, E. & Gelato, L. M. (1985). Acta Cryst. A41, 142–151. CrossRef Web of Science IUCr Journals Google Scholar
Pólya, G. (1956). Am. Math. Mon. 63, 689–697. Google Scholar
Sylvester, J. J. (1857). Q. J. Pure Appl. Math. 1, 141–152. Google Scholar
Villars, P. & Cenzual, K. (2020). Pearson's Crystal Data Crystal Structure Database for Inorganic Compounds. ASM International, Materials Park, Ohio, USA. https://www.crystalimpact.com/pcd/Default.htm. Google Scholar
Wilf, H. S. (2006). Generatingfunctionology, 3rd ed. Wellesley: A. K. Peters. Google Scholar
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