Rank is a fundamental concept in linear algebra and has numerous applications in diverse areas of discrete mathematics and theoretical computer science, such as algebraic complexity , communication complexity , and extremal combinatorics , to name only a few. A common phenomenon is that low rank often helps in proving combinatorial upper bounds or designing algorithms, e.g., through representative sets [7, 13, 21] or the polynomial method (which ultimately relies on fast rectangular matrix multiplication, enabled through low-rank factorizations of problem-related matrices ). In particular, rank has recently found applications in fine-grained complexity (see  and the references therein) and parameterized complexity. In the latter, several influential results, such as algorithms for kernelization , the longest path problem , and connectivity problems parameterized by treewidth [12, 11, 7], rely crucially on low-rank factorizations.
In view of the utility of low rank in proving upper bounds, it is natural to ask, conversely, whether high rank translates into lower bounds. Indeed, examples for this connection can be found in communication complexity [22, Section 1.4] and circuit complexity . In the present paper, we find such applications also in fine-grained and parameterized complexity: We develop a technique that allows us to transform rank lower bounds into conditional lower bounds for the problem #HC of counting Hamiltonian cycles. The decision version HC of #HC, which asks for the existence of a Hamiltonian cycle, is a classical subject of algorithmic research. For decades, the well-known time dynamic programming algorithm  was essentially the fastest known algorithm for HC, until a breakthrough result  showed that HC can actually be solved in randomized time. This result spawned several novel algorithmic insights into HC, but also showed that we still do not understand this problem in a satisfactory way: No deterministic time algorithm for HC is known, and even no randomized time algorithms are known for the more general traveling salesman problem, the directed Hamiltonian cycle problem, or the counting version #HC.
One of the novel algorithmic techniques for HC following in the wake of  is closely tied to the rank of the so-called matchings connectivity matrix . For even , the matchings connectivity matrix is indexed by the perfect matchings of the complete graph , and the entry for perfect matchings and is defined as if the union is a single cycle, and otherwise. See Figure 1 for an example. The matchings connectivity matrix can be seen as a description of the behavior of Hamiltonian cycles under graph separators, an interpretation that proved useful for algorithmic applications. For instance, the authors of  show that the rank of over is precisely and use this surprisingly low rank to count Hamiltonian cycles modulo in bipartite directed graphs in time, which was recently improved to time in . A randomized algorithm for the decision version follows from witness isolation.
The low rank of also enabled an time algorithm for HC on graphs with a given path decomposition of width . For this problem, the standard dynamic programming approach would require to keep track of all partitions of separators, resulting in a running time of ; it is thus somewhat remarkable that the single-exponential running time of can be achieved. Even more surprisingly, the base appears to be optimal, as it is known that any time algorithm would violate the Strong Exponential Time Hypothesis (SETH) . This was proven by combining a general reduction technique for SETH-based lower bounds from  with a special property of , namely, that contains a principal minor of size that is a permutation matrix. In other words, there is a collection of perfect matchings such that every perfect matching in this collection can be extended to a Hamiltonian cycle by precisely one other member.
The general technique for SETH-based lower bounds from  was successfully applied to various problems parameterized by pathwidth: As a result, the optimal base in the exponential dependence on the pathwidth has been identified for many problems, assuming SETH. However, there are still natural open problems left, such as #HC: It has been shown that this problem can be solved in time , later extended to time when parameterized by treewidth , where denotes the matrix multiplication constant. A tight lower bound however remained elusive, and this might justify optimism towards improved algorithms: For example, if we could lift the time algorithm for #HC modulo to an time algorithm for #HC, we could solve #HC on bipartite graphs in time, since bipartite graphs have pathwidth at most .
Our main results
We strike out the route towards faster algorithms for #HC sketched above: We show that the current pathwidth (and, assuming , treewidth) based algorithms are optimal assuming SETH.
Assuming SETH, there is no such that #HC can be solved in time on graphs with a given path decomposition of width .
This theorem gives a natural example for an NP-hard problem whose decision version (with base ) and counting version (with base ) differ provably under SETH.
We prove Theorem 1.1 by starting from the general reduction technique in , augmented with a novel idea: We extend the technique in such a way that it can exploit arbitrary lower bounds on the matrix rank of
, without further insights into the particular structure of basis vectors. That is, we derive Theorem1.1 as a consequence of the following more general “black-box” connection between the rank of and the running time for #HC: If the exponential base of the rank can be lower-bounded by , then we do not expect time algorithms.
Let be such that , where , as even tends to infinity.111This implies that can be lower-bounded by up to sub-exponential factors. Assuming SETH, there is no such that #HC can be solved in time on graphs with a given path decomposition of width .
For prime numbers , the same applies to #HC modulo when replacing by , which is defined analogously to by taking over .
To prove Theorem 1.1, we then combine Theorem 1.2 with our second main contribution: We determine the rank of over up to polynomial factors, and for primes , we additionally give lower bounds on the rank over that are higher than the rank over .
The rank of over the rational numbers is at least . For any prime , the rank of over is at least , and for prime , it is at least . See Theorem 3.7 for a full list.
The bound over is obtained by a novel application of representation theory, inspired by a previous approach from , where the rank of a bipartite version of over was found to be up to polynomial factors. In the bipartite version of , only perfect matchings contained in the complete bipartite graph are considered. In our non-bipartite version, any perfect matching in the complete graph is allowed; this is more appropriate for algorithmic applications, and our new bound over shows that going to the non-bipartite setting increases the rank significantly.
Combined with Theorem 1.2, our bound over suggests that #HC modulo prime is harder than modulo : We can solve #HC modulo in time, where , but we cannot solve #HC modulo in time unless SETH fails. This connects to recent results [5, 6], which show that the counting Hamiltonian cycles modulo (not parameterized by pathwidth) can be solved in time , where depends on the constant .
Connection matrices and fingerprints
The matchings connectivity matrix fits into a bigger picture of so-called connection matrices for graph parameters, and our bounds on the rank of translate into rank bounds in this framework.
The connection matrices of a graph parameter are a sequence of matrices , for , which describe the behavior of under graph separators of size . To define these matrices, say that a -boundaried graph, for , is a simple graph with distinguished vertices that are labeled . Two -boundaried graphs and can be glued together, yielding a graph , by taking the disjoint union of and and identifying vertices with the same label. The -th connection matrix of then is an infinite matrix whose rows and columns are indexed by -boundaried graphs such that the entry is .
The ranks of connection matrices are closely related to graph-theoretic, algorithmic, and model-theoretic properties of graph parameters [20, 26, 25]. In particular, the connection matrices for the number of Hamiltonian cycles were studied in [25, 26], where their rank was upper-bounded by . As a consequence of Theorem 1.3, we can improve upon this and obtain the following essentially tight bounds.
For , the rank of the connection matrix for the number of Hamiltonian cycles is , up to polynomial factors.
To prove this theorem, we use a third matrix, the fingerprint matrix for Hamiltonian cycles, which will also play an important role in our main reduction.222To avoid (or add) confusion, let us stress that we consider three matrices related to the Hamiltonian cycle problem: The matchings connectivity matrix , the connection matrix for the number of Hamiltonian cycles, and the fingerprint matrix for Hamiltonian cycles. We will revisit their differences in Section 2. While these matrices are closely related, our arguments benefit from using different matrices for different proofs. A fingerprint of a -boundaried graph is a pair , where assigns , or to each boundary vertex, and is a perfect matching on the boundary vertices to which assigns . Fingerprints are essentially the states one would use in the natural dynamic programming routine for counting Hamiltonian cycles parameterized by pathwidth; they describe the behavior of a Hamiltonian cycle on a given side of a separation. A pair of fingerprints and on combines if for every and additionally forms a single cycle. The fingerprint matrix is a binary matrix, indexed by fingerprints, and the value at a pair of fingerprints is iff the two fingerprints combine.
Theorem 1.3: Rank of the matchings connectivity matrix
To prove Theorem 1.3, we give two different lower bounds on the rank of : One is relatively simple and contained in Section 3. For this bound, we first explicitly compute the rank of small matching connectivity matrices and then use a product construction to give lower bounds for larger orders. While the resulting bound is loose, it also applies to the rank of over for prime , whereas our more sophisticated main bound does not. In particular, we can use the bound to show that the rank of over and other primes is asymptotically larger than the rank over .
Our main result however concerns the rank of over , which we establish to be up to polynomial factors in Section 4. To this end, we build upon representation-theoretic techniques that were also used in Raz and Spieker’s bound  for the bipartite version of , and which we first survey briefly: A hook partition of some number is a number partition with the particular form for some . One can view as a Ferrers diagram, which is a left-adjusted diagram made of cells, as shown in Figure 3(a). A standard Young tableau of shape is a labeling of this diagram with numbers from such that the numbers are strictly increasing in each row and each column, see Figure 3(b). Raz and Spieker showed that the rank of the bipartite variant of can be expressed as a weighted sum over all hook partitions of , where each is weighted by the squared number of Young tableaux of shape . This sum simplifies to the central binomial coefficient , showing that the bipartite variant of has rank , up to polynomial factors.
To address the non-bipartite setting, we found ourselves in need of additional techniques from algebraic combinatorics that were not present in Raz and Spieker’s original bound, such as the perfect matching association scheme  and zonal spherical functions . With these at hand, we prove that the rank of can be lower-bounded by a similar sum over number partitions as in the bipartite case, this time however ranging over domino hook partitions , which have the form for some . As in the bipartite case, we then observe that this sum simplifies significantly, this time however to (essentially) a product of two consecutive Catalan numbers. This entails a lower bound of for the rank of , up to polynomial factors. It then follows easily from the upper bound in the bipartite setting that this bound is tight up to polynomial factors.
Theorem 1.2: SETH-hardness via assignment propagation
To describe how we turn lower bounds on the rank of into algorithmic intractability results for #HC under SETH, let us first survey the general construction from , which we dub a block propagation scheme: Given a CNF-formula with variables, such a scheme produces an equivalent instance of the target problem with parameter value for some constant . An algorithm with running time for the target problem would then refute SETH, as it would imply a time algorithm for CNF-SAT.
The constructed target instance has the outline sketched in Figure 2:
The variables of are grouped into blocks of constant size , where depends only on the in the running time we wish to rule out. The variable blocks are represented as rows, each propagating an assignment of type using a thin graph of pathwidth . Specifically, an assignment is represented as the type of a partial solution for the target problem on a -boundaried graph (e.g., as a partial coloring of the boundary, or in our case, as a fingerprint of a Hamiltonian cycle.) The relationship between and is important in this construction: Intuitively, if we can choose small for large , then the target problem has a large “combinatorial capacity” in the sense that it allows us to pack assignments to large blocks into thin wires.
In a block propagation scheme, the clauses of are then represented as columns; the column corresponding to clause checks whether the overall assignment of type propagated by the rows satisfies . To this end, one can use cell gadgets, which are graphs with left and right interface vertices, and additional top/bottom interface vertices, where depends only on the target problem. The cell gadget is placed at the intersection of a row and a column, and it needs to “decode” an assignment from the state of the left interface vertices, decide whether satisfies the clause, and “encode” back into the state of right vertices. The top and bottom interface is used to propagate, from the top of a column downwards, whether the respective clause is already satisfied by the partial assignments to the blocks above a given cell. Due to the grid-like construction, the overall pathwidth of the instance is usually easily seen to be bounded by , where the additive constant accounts for the size of cell gadgets.
The main technical effort in these reductions lies in constructing the cell gadget, and this usually subsumes constructing a “state tester”, a gadget that tests whether, in a solution to , the left/right interface vertices are in a particular state (say, a particular partial coloring, or a particular fingerprint of a Hamiltonian cycle). This requires constructing a graph that can be extended to a solution of the target instance iff the relevant vertices are in state , and for various problems, such constructions can be achieved with some effort. In the case of #HC, we face the situation that testers for fingerprints of Hamiltonian cycles do not exist: There are fingerprints such that any graph that extends to a Hamiltonian cycle also extends some unwanted fingerprints . Our main insight here is that this problem can be solved by firstly restricting to a set of good fingerprints that induce a full-rank submatrix of the fingerprint matrix , and secondly simulating a “linear combination” of testers, with coefficients obtained from the inverse of . In the fingerprint tester for , other fingerprints will have extensions of non-zero weight, but the weights of these extensions are chosen in such a way (depending on ) that extensions of cancel out. (A similar idea was used before to obtain conditional lower bounds for the complexity of permanents .) This allows us to simulate a state tester for fingerprints, and the use of cancellations also shows why this lower bound works only for #HC and not for HC—which is fortunate, since HC does admit an time algorithm.
2 Preliminaries and notation
If is a CNF-formula, and is a (partial) assignment to its variables, we write to denote that satisfies . For integers , we let . All graphs in this paper will be undirected. If is a graph, is a vertex and is an edge set, we let denote the number of edges in that are incident to . We write for the set of all perfect matchings of the complete graph .
Strong Exponential Time Hypothesis:
As formulated by Impagliazzo and Paturi , the complexity assumption SETH states that for every , there is a constant such that -CNF-SAT (the satisfiability problem for -CNF formulas on variables) cannot be solved in time . It is common to state this hypothesis as ruling out even randomized algorithms, making it slightly stronger. A result from Calabro et al. [9, Theorem 1] gives a randomized reduction from -CNF-SAT to the problem UNIQUE--CNF-SAT, where the -CNF formula is guaranteed to have at most one satisfying assignment. This allows us to assume that the -CNF formulas in the statement of (randomized) SETH have at most one satisfying assignment.
A path decomposition of a graph is a path in which each node has an associated set of vertices (called a bag) such that and the following holds:
For each edge there is a node in such that .
If then for all nodes on the (unique) path from to in .
The width of is the size of the largest bag minus one, and the pathwidth of a graph is the minimum width over all possible path decompositions of . A path decomposition starts in if the first bag contains and ends in if the last bag contains .
Since our focus here is on path decompositions, we only mention in passing that the related notion of treewidth can be defined in the same way, except for letting the nodes of the decomposition form a tree instead of a path.
Given a field and two matrices and , the Kronecker product is a matrix in . Its rows can be indexed by pairs , and similarly for columns. The entry of at row and column is defined as . For , the -th Kronecker power of is the -fold product , and we consider its rows and columns to be indexed by and respectively.
If and each have full rank over , then so does . Note that this requires to be a field; it would fail if contained zero divisors. For our purposes of computing the rank of matrices over , this means we require to be prime.
The HC-fingerprints (which we often abbreviate as fingerprints) capture the states of the natural dynamic program for Hamiltonian cycles:
Definition 2.1 (Fingerprint, Partial Solutions).
Let be a graph and let , where is the set of ‘boundary vertices’. A fingerprint on is a pair where and is a perfect matching on . A partial solution in for is an edge set such that (i) for every , (ii) for every , and (iii) if then and are the endpoints of the same path of .
Two fingerprints and on combine (or match) if for every and forms a single cycle or is empty.
Variants of connection matrices:
Our paper studies three related matrices that describe the behavior of Hamiltonian cycles under separators. We recall their definitions here for reference:
This binary matrix is indexed by perfect matchings , and is iff is a single cycle. This matrix appears naturally in our rank lower bounds.
This binary matrix is indexed by all fingerprints on a fixed set of size . An entry equals iff and combine. This matrix will be crucially used in the algorithmic lower bound.
This integer-valued matrix is indexed by all -boundaried graphs, and hence infinite. The entry counts the Hamiltonian cycles in the graph . (If an edge between boundary vertices is present in both and , we count it twice in .) This matrix will not be used in later sections, but we still mentioned it to connect to the established literature on connection matrices.
The subscript is omitted when clear from the context. As we show below, we can easily transform rank bounds for into bounds for and , thus justifying our focus on the rank of :
If is a polynomial such that , then for some polynomial .
Subject to proper indexing, the matrix is a block-antidiagonal matrix that has a block for every vector , since fingerprints with degree functions not satisfying for all cannot match. Therefore, we obtain:
where is a polynomial satisfying for , and the last inequality follows from the binomial theorem. The upper bound follows similarly. ∎
Furthermore, a simple argument shows that the fingerprint matrix and the connection matrix actually have the same rank.
For every , the matrices and have the same rank.
We first show that by finding as a submatrix of . To this end, we construct a -boundaried graph for every -fingerprint and then find as the submatrix induced by these graphs. Given , the graph is constructed as follows: At first, it contains only the boundary vertices . Then we add an arbitrary partial solution for to . For instance, if and is non-empty, pick the lexicographically first edge of the matching , say , and connect to in with a path that passes through all vertices in in an arbitrary order. Then add all edges in to as edges. If is empty, add a Hamiltonian cycle on . Finally, subdivide all edges of the graph; this adds some number of subdivision vertices to , which we consider not to be part of the boundary. Note that the degree of boundary vertex in is precisely .
Given two -fingerprints , we observe that any Hamiltonian cycle in uses all edges of the graph, as every edge is incident to a (subdivision) vertex of degree . This implies, firstly, that the number of Hamiltonian cycles in is either or . Secondly, it implies that needs to be the constant -function for to have a Hamiltonian cycle. If this condition is fulfilled, then by construction, has a Hamiltonian cycle iff forms a single cycle. Summarizing, we have that and that iff and match. This shows that the set of -boundaried graphs , for -fingerprints , induce the fingerprint matrix as a submatrix in , and the lower bound on the rank of follows.
For the upper bound of , we find a matrix such that . The rows of are indexed by -boundaried graphs , the columns are indexed by fingerprints , and we define to count the partial solutions in for the fingerprint .
Given two -boundaried graphs and , every Hamiltonian cycle in induces a partial solution in each of and , for fingerprints and , respectively. The pair of fingerprints can be determined uniquely from the partial solutions of , and since is a Hamiltonian cycle, it follows that and match. Given a matching pair of fingerprints , the number of Hamiltonian cycles of with is precisely , as the extensions in each of and can be chosen independently, provided they agree with and respectively. We conclude that the number of Hamiltonian cycles in can be expressed as
It follows that as claimed, establishing the upper bound on the rank. ∎
3 A simple rank lower bound
Let be a fixed prime. To obtain the lower bound on the rank of over , we proceed in two steps: First, we use a computer program to compute, for a small constant , the rank of over . Then we use a product construction to amplify this initial rank to a lower bound on the rank of for .
3.1 The initial matrix
We choose maximally such that can still be computed, e.g., by the MATLAB script provided in the ancillary files. If the rank of over is , then the symmetry of implies the existence of a set of perfect matchings in such that the submatrix has full rank over . Our computations enable the following choices:
For any prime , the matrix has (full) rank over . Furthermore, the matrix has rank over , rank over , and rank over for .
The dimensions of are . Our calculations show that the determinant of is non-zero and contains only the prime factors and . It follows that has full rank over for any prime . Over , the rank of is found to be , but we will obtain a better bound by going to , a matrix of dimensions , where the claimed rank bounds can be obtained by calculation. ∎
For concreteness, we illustrate our approach in the next subsections with as initial matrix, see Figure 1, and revisit the better choices provided in Lemma 3.1 at the end of the proof. Calculation shows that , so has full rank over for primes , and we can choose a set of size to get a full-rank matrix . Already gives a lower bound on the rank of over with that is higher than the rank over .
3.2 Amplification via Kronecker products
After having obtained
, we then “tensor up” this matrix to obtain rank lower bounds onfor : To this end, we find the Kronecker power , which is a full-rank matrix of dimensions , as a submatrix of . It follows that whenever is divisible by . For , this yields over when .
To proceed, it will be useful to define a particular graph on vertices for each , which can be viewed as a subgraph of . Only perfect matchings contained in will be relevant. The graph consists of disjoint copies of and “patch edges” between adjacent -copies that will be used to combine solutions of the individual -copies to a global solution.
Let be fixed. For , let be obtained as follows, see also Figure 3:
Take disjoint copies of and denote the vertices of copy by .
For each and , add an edge from to , interpreting as . These are the patch edges.
The perfect matchings of contain a particular subset of size that is essentially the -th power of ; this set will be the row set of the full-rank submatrix we wish to find in . The elements of are disjoint unions of perfect matchings, one for each -copy.
Given a tuple , for , we define a perfect matching of the graph by
We write for the perfect matchings that can be obtained from this way.
It remains to find an appropriate column set of perfect matchings. Note that we cannot reuse for this purpose: If , the union of any two perfect matchings in is disconnected, and therefore contains only zeroes.
We do however obtain a suitable column set, which we denote by , by using the patch edges of : Each perfect matching in is obtained from some by deleting one particular edge from each -copy, and patching the resulting isolated vertices to adjacent -copies.
Given a tuple , for , we define a perfect matching of the graph :
Start with the perfect matching .
For , let denote the neighbor of in . Delete the edge from to in , rendering these two vertices isolated.
For , include the patch edge from to . (Consider here.)
We then define .
It is easily seen that is indeed a perfect matching of the graph , for each : We started with the perfect matching , then reduced the degree of and to for all , and then increased these degrees back to in the third step. No other degrees were affected.
Having defined our rows and columns , we proceed to study the submatrix . Note that both and correspond bijectively to , so the indexing of already puts this matrix close to the -th Kronecker power of . Its content also does not fail us:
Identifying and each with in the natural way, we have
Given with and , let be the union of its corresponding perfect matchings. We observe that is a Hamiltonian cycle (in ) if and only if is a Hamiltonian cycle (in ) for each :
In the “if” direction, note that is the result of deleting one edge each from Hamiltonian cycles, then adding edges between the endpoints of the resulting Hamiltonian paths so as to obtain a Hamiltonian cycle in .
In the “only if” direction, note that the restriction of to the -th -copy for is a Hamiltonian path between the and some neighbor. By adding back the edge between and its neighbor and deleting the patch edges, we obtain a Hamiltonian cycle in each copy of .
The claim then follows from the definition of and the Kronecker product. ∎
Since has full rank over and was required to be prime, the Kronecker power also has full rank, so we obtain:
The matrix has full rank over . Consequently, the rank of over is at least .
In conclusion, by using , we obtain that, for prime , the rank of over is at least . Using the larger initial matrices provided by Lemma 3.1, we obtain the following stronger bounds:
For prime , the rank of over is at least
The bounds can be improved by using larger initial matrices , but we hit our computational limit with the matrix . For this matrix, we could no longer compute determinants of the relevant submatrices to determine their prime factors, but we could still compute the rank of for primes up to , thus obtaining the last three entries in Theorem 3.7.
4 The rank of the matchings connectivity matrix over the rational numbers
In this section we establish the first part of Theorem 1.3. For this we need some basics on the representation theory of the symmetric group which we first briefly outline.
4.1 The Representation Theory of the Symmetric Group
The representation theory of the symmetric group is remarkable, as much of it may be explained via the combinatorics of integer partitions and tableaux. We outline the relevant combinatorial aspects of the theory, leaving the algebraic basics of finite group representation theory for Appendix B. The reader is referred to  for a gentle but more thorough introduction.
Let such that and denote an integer partition of . If parts of the integer partition have the same size , then we express them by the shorthand . Let denote the number of integer partitions of . It is well-known that there is a one-to-one correspondence between the irreducible representations of and the integer partitions of . We let denote the irreducible representation of corresponding to .
For an integer partition , the Ferrers diagram of is an associated left-justified tableau that has cells in the th row. Abusing notation, we let also refer the Ferrers diagram of . In Figure 3(a) the Ferrers diagram for is illustrated.
We obtain a standard Young tableau from a Ferrers diagram by labeling its cells with numbers such that the numbers along each row are strictly increasing, and the numbers along each column are strictly increasing. In Figure 3(b) a standard Young tableau of shape is shown.
Let denote the number of standard Young tableaux of shape . There is an elegant combinatorial formula for expressing .
We say that a tableau covers a tableau if the cells of are contained in the cells of . A hook is a tableau of shape , equivalently, a tableau that does not cover the shape . The partition is not a hook, as it covers , illustrated in Figure 3(c).
For each cell of a Ferrers diagram, say at row and column , if we take along with all cells in row to the right of , and all cells in column that lie below , we obtain a hook for some . Let denote the number of cells in this hook, the so-called hook length. In Figure 3(d), we have annotated each cell with its corresponding hook length.
The following result connects hook lengths with enumerating standard Young tableaux.
Theorem 4.1 (Hook Theorem ).
For instance, it is easy to see using the hook formula that for any hook . A classic result in the representation theory of the symmetric group is that equals the dimension of the irreducible corresponding to .
Proposition 4.2 (Dimensions of Irreducibles of ).
4.2 Relating rank to the number of Young tableaux
We proceed by studying where we let . Let be a partition of the vertices of into two parts of size . Consider the sub-matrix of induced by the perfect matchings of that are also bipartite perfect matchings with respect to the bipartition . In 
, Raz and Spieker show that the eigenspaces ofare in fact irreducible representations of
, and that the eigenspaces corresponding to nonzero eigenvalues ofcorrespond to the hooks of length . This result paired with some elementary combinatorics implies the following theorem.
Theorem 4.3 (Raz & Spieker ).
Since there are ways to partition into two parts of size , this already gives an upper bound of on the rank of . We will show this is almost tight.333Moreover, it can be verified experimentally that for some constant the rank of is strictly smaller than this bound, but lower order terms will not be relevant for us. One of our key technical theorems is the following exact formula for the rank of .
For any , let us write . Then
This result can be seen as the non-bipartite analogue of Theorem 4.3. To prove it, we determine the nonzero eigenvalues of ; however, this will require a fair amount of algebraic combinatorics, which we now develop.
Let denote the hyperoctahedral group of order , equivalently, the group of permutations such that
It is well-known that the set of perfect matchings of can be written as . Even though is not a group, these cosets possess a remarkable amount of algebraic structure.
Let be the vector space of real-valued functions over perfect matchings, equivalently, the space of real-valued functions over that are -invariant, that is, . In , Thrall showed this vector space admits the following decomposition into irreducible representations of .
Theorem 4.5 (Thrall ’42).
A consequence of Thrall’s result is that admits a symmetric association scheme, the so-called perfect matching association scheme [14, Section 15.4].
Definition 4.6 (Symmetric Association Scheme).
A collection of binary matrices is a symmetric association scheme if the following axioms are satisfied.
is the identity matrix.
where is the all-ones matrix.
for all .
is a linear combination of for all .
for all .
Recall that the union of any two perfect matchings is a disjoint union of cycles, which can be represented by an integer partition of the form where for some . The perfect matching association scheme is simply the collection of matrices defined such that if , and otherwise.
Since is a symmetric association scheme, the eigenspaces of the ’s coincide, and are precisely the irreducibles in the decomposition given by Thrall [14, Section 15.4]. In light of this, we can take the distinct eigenvalues of these matrices as column vectors and collect them in a matrix . For example, when , we have , and
For any , we call the -sphere, where
The following lemma gives a simple way to determine their size.
Lemma 4.8 ().
Let denote the number of parts of , denote the number of parts of that equal , and set . Then
Note that the first row of lists the sizes of the respective spheres. This is no coincidence, as each has constant row sum , and so its largest eigenvalue is respectively . It is known that the entries of are determined by the zonal spherical functions [29, Chapter VII], which can be thought of as an analogue of irreducible characters in our association scheme setting.
Theorem 4.9 ().