GRAPHENE NANOCONES and PASCAL MATRICES
– some determinantal conjectures –

Physics often entails intriguing mathematics, and Carbon Nanocones are no exception. They are conical honeycomb lattices of Carbon atoms, with a cap of Carbon rings causing different cone apertures [5]. The ones with smallest angle, near $19^{\circ}$, were sinthetized by Ge and Sattler in 1994, in the hot vapour phase of carbon; the others were found by accident, in pyrolysis of heavy oil [17].

The simplest description for such structures is through the adjacency matrix. Chemists give weight $-\beta$ to lattice links among atoms, and name it Hückel matrix. Because of the $C_{n}$ symmetry of nanocones, the Hückel matrix can be restricted to a single honeycomb triangle, with Bloch boundary conditions $e^{\pm i2\pi/n}$ on two sides. Properties of the spectrum, based on graph theory and symmetries, were studied in [15]. Recently, based on numerics, new rules were established for the filling of levels, that differ from the ordinary Hückel rules [4].
This paper originates from a question by an author in [4]: what is the determinant of the Huckel matrix of a nanocone? The answer here conjectured opens a nice connection with Pascal matrices and combinatorics.

It is convenient to replace the Bloch factors by free parameters $x$ and $y$, or even by parameters $x_{i}$ and $y_{i}$ for each row.
The size of a Hückel matrix with $n$ rows is $(n+1)^{2}$, which is the number of atoms in the triangle.
$\displaystyle\small{H_{2}=\left[\begin{array}[]{c|ccc|ccccc}S_{0}&0&1&&&&&&\\ \hline\cr 0&0&1&y_{1}&0&1&&&\\ 1&1&0&1&0&&0&&\\ &x_{1}&1&0&0&&&1&\\ \hline\cr&0&0&0&0&1&&&y_{2}\\ &1&&&1&0&1&&\\ &&0&&&1&0&1&\\ &&&1&&&1&0&1\\ &&&&x_{2}&&&1&0\end{array}\right]}$


Example. A triangle with one hexagon and its Hückel matrix. In this representation, the diagonal blocks are the rows of the graph, with 1,3,5 atoms. Edges are 1s in the matrix. Dots connect row extrema with weights $x_{i}=y_{i}=1$ ($S_{0}=x_{0}+y_{0}$).

The determinants of small Hückel matrices are evaluated:

$\bullet$ The determinants are homogeneous palindromic polynomials with positive coefficients.
$\bullet$ If the parameters are $x_{1},x_{2},...,y_{n}$ are kept different, the determinants are homogeneous polynomial of degree $n+1$ in the variables, with coefficients that are perfect squares of integers (this feature was noticed to me by Andrea Sportiello).

Their evaluation for small size $n$ with Schur’s formula for block matrices showed that the determinant of a Hückel matrix of size $(n+1)^{2}$ eventually reduces to the determinant of a matrix of binomials of size $(n+1)$ (a step is illustrated in section 8). Binomials occur in many counting problems of benzenoids [6, 9].
After discovering the beautiful Pascal matrices in Lunnon’s paper [19], I found with Mathematica that the polynomial coefficients in $\det H_{n}$ for accessible values of $n$, match the sequences A045912 in OEIS [22] for the characteristic polynomials of Pascal matrices. This led me to the first conjecture; the others then came out.

Conjectures 1 and 2 imply symmetries of the eigenvalues of Hückel matrices of nanocones and the larger family of graphannulenes, that support the generalized Hückel rules for the electronic structure proposed by Evangelisti et al. in ref.[4].

Before stating the conjectures, let’s introduce Pascal matrices.

There are various forms of Pascal matrices, and much literature [19, 7, 24, 10]. The lower triangular Pascal matrix $P_{n}$ has size $n+1$ and elements that are the binomial coefficients of the Pascal triangle. The inverse matrix also has binomial coefficients, but with alternating signs. For example:

The product $Q_{n}=P_{n}P_{n}^{T}$ is the symmetric positive Pascal matrix. It has unit determinant and matrix elements $(Q_{n})_{ij}=\binom{i+j}{j}$ $i,j=0,...,n$:

Since the eigenvalues are strictly positive, the coefficients of the polynomial $\det(z+Q_{n})$ are positive. Moreover the eigenvalues come in pairs: $q_{n+1-j}=1/q_{j}$; if the matrix size is odd ($n$ even), an eigenvalue of $Q_{n}$ is 1.
Note that the entry $(i,j)$ (independent of $n$) counts the number of paths with $i+j$ steps $\rightarrow$ or $\downarrow$ that connect the corner ($0,0$) with ($i,j$). For example, ($3,4$) can be reached in $35=\binom{7}{3}$ different ways from ($0,0$).

With all $x_{k}=x$ and $y_{k}=y$, we obtain the interesting identity:

In the next section I report the very few known analytic values of $\det H_{n}$, given by eq.(8) for special values of the ratio $\omega=y/x=e^{2i\theta}$. They result from amazing combinatorial problems, that are here reported for the delight of the reader.

The characteristic polynomial of the symmetric Pascal matrix appears in diverse combinatorial problems with the following generalization:

George Andrews [2] in 1979, in proving the weak Macdonald conjecture for certain plane partitions, obtained the closed expression of (9) with $\omega=1$. He exploited identities for hypergeometric series. For the Pascal matrix ($m=0$) it greatly simplifies:

This number is also the determinant of the periodic Hückel matrix, $\det H_{n}(1,1)$.

A plane partition $\pi$ of $N$ of shape $(a,b,c)$ is an array $a\times b$ of numbers $c\geq\pi_{ij}\geq 0$ ($i=1,..,a$, $j=1,..,b$) with the property that all rows and columns are weakly decreasing, and the sum of all $\pi_{ij}$ is $N$.
These are two plane partitions of 38 in $(3,5,7)$:

A plane partition may be represented by stacking $\pi_{ij}$ cubes on each square cell $(i,j)$ in the rectangle $a\times b$. A stacking of cubes defines ascending staircases that correspond to sets of non-intersecting walks in a lattice. The beautiful Gessel-Viennot theorem [11] counts them as determinants of certain binomial matrices¹¹1see also the nice presentation by Aigner in [1].

It was later discovered [13] that the projection of the bounding box $a\times b\times c$ on a oblique plane normal to (1,1,1) is an hexagon $H(a,b,c)$ with side-lengths $a,b,c,a,b,c$ (in cyclic order) and angles $\pi/3$ in a triangular lattice. Each stacking is one-to-one with a lozenge tiling of the hexagon. A lozenge is the union of two triangular unit cells of the foreground triangular lattice, and has angles $\pi/3$ and $2\pi/3$.

Plane partitions were classified by Stanley and others into 10 symmetry classes and enumerated ([18] is a nice review by Christian Krattenthaler).
Class 1 are the unrestricted partitions. They were enumerated by Percy MacMahon (1896): the number of all plane partitions in $(a,b,c)$, i.e. lozenge tilings of $H(a,b,c)$, is

where $H(k)=1!2!...(k-1)!$.
The other extremum is Class 10, with totally symmetric self-complementary plane partition in $(2n,2n,2n)$. Viewed as a stack of cubes in the cubic box of side $2n$, a TSSC partition has the property of being equal to its complement (the void in the box, see fig.3) and, viewed as a tiling, it is invariant under rotations by $\pi/3$ and reflections in the main diagonals of the hexagon. The enumeration was finally obtained by Andrews [3]:

$A(n)$ is also the number of $n\times n$ matrices $\{0,\pm 1\}$ whose row-sums and column-sums are all 1, and in every row and column the non-zero entries alternate in sign [26].

With the aid of the identity

the numbers (10) can be rewritten as follows:

Ciucu, Eisenköbl, Krattenthaler and Zare evaluated weighted sums over the tilings of the hexagon $H(a,b+m,c,a+m,b,c+m)$ with a central triangular hole of sides $m,m,m$. With $a=b=c=n+1$ the counts are given by the determinant (9) with weights $\omega=\pm 1,e^{i2\pi/3},e^{i\pi/3}$ (equations 3.3, 3.4, 3.5 in [8]).
With $m=0$ they simplify to eqs.(13), (14) and (15) given below. They are the weighted counts $\sum_{\pi}\omega^{\#\pi}$ ($\#\pi$ is the number of cubes on the main diagonal) over the cyclically symmetric partitions $\pi$ with shape $(n+1)^{3}$ (Corollary 8 of [8]). In other words, the determinants give the weighted sums over the tilings of $H(n+1,n+1,n+1)$ that are invariant under rotations by $\pi/3$ around the center of the hexagon (Class 3). The different weights $\omega$ give:

Up to sign, it coincides with $\det H_{2n+1}(1,-1)$. For an odd size $\det(Q_{2n}-I_{2n+1})=0$. Some numbers are listed by Stembridge (1998), who first studied this case²²2page 6 in http://www.math.lsa.umich.edu/ jrs/papers/strange.pdf.
Eqs. 3.4 and 3.5 with $m=0$ have simpler expressions given in Table 1 of [21]:

The numbers $A_{HT}(n)$ enumerate the half-turn symmetric alternating sign $n\times n$ matrices³³3It is the subclass of alternating sign $n\times n$ matrices such that $A_{ij}=A_{n+1-i,n+1-j}$. They are related to the ice model of statistical mechanics. [23]:

The generalized Pascal matrices also occur in the works by Mitra and Nienhuis [20, 21]. On an infinite cylindric square lattice of even circumference $L$, they consider coverings of closed paths that at each vertex make a left or right turn with equal probability. Two paths can have a vertex in common, but do not intersect.
The probability $P(L,m)$ that a point (not a vertex) of the cylinder is surrounded by $m$ loops is guessed as $Q(L,m)/A_{HT}(L)^{2}$, where $Q(L,m)$ is expressed with the polynomial coefficients of the Pascal matrix. At the same time, $P(L,m)$ is estimated by Coulomb gas techniques. The following asymptotics is obtained:

The large $n$ (i.e. $L+1$) expansion of (8) for any value $|\omega|=1$ is discussed by Mitra [21].

The Hückel matrices $H_{n}(\theta)=H_{n}(e^{-i\theta},e^{i\theta})$ are Hermitian and their determinants can be written as polynomials of $\cos\theta$. The known results for the Pascal matrix give:

The following table is a check. It shows the first values of $A(n)$, $A_{HT}(n)$ and $\det H_{n}(\theta)$, that coincide with the values of the determinants listed in introduction. The value for $\theta=\pi/4$ is compared with the expansion (17) by Mitra; the values are very close.

If rows $0,1,...,k-1$ are removed from a honeycomb triangle, a trapezium results, with rows of lengths $2k+1,...,2n+1$. The corresponding Hückel matrix $H_{k,n}({\bf x},{\bf y})$ is obtained by deleting the first $k^{2}$ rows and columns of $H_{n}({\bf x},{\bf y})$. Its size is $(n+1)^{2}-k^{2}=(2k+1)+...+(2n+1)$.

For example, the graph and the matrix $H_{6,7}$ (size $28$) are:
$\displaystyle\left[\begin{array}[]{cc}T_{6}&R_{7}^{T}\\ R_{7}&T_{7}\end{array}\right]$

Conjecture 2: The determinant of the Hückel matrix $H_{k,n}({\bf x},{\bf y})$ of a honeycomb trapezium with rows with $2k+1$, $2k+3$, … , $2n+1$ sites, size $(n+1)^{2}-k^{2}$, is equal to the determinant of the following matrix of size $n+1-k$:

In these examples the determinants of Hückel matrices of size $28$, $51$ and $64$ are evaluated as matrices of size $2$, $3$ and $4$ ($S_{n}=x_{n}+y_{n}$):

It appears that all coefficients are perfect squares.

The equality of the permanent and the determinant of a matrix is a rare circumstance [12, 25]. Numerical checks on Hückel matrices for triangles and trapezia of small sizes, support the conjecture:

Conjecture 3. The permanent and the determinant of $H_{n}({\bf x},{\bf y})$ coincide. The same is true for $H_{k,n}({\bf x},{\bf y})$.

Accordingly, $\det H_{n}={\sum}_{\sigma}H_{1,\sigma_{1}}H_{2,\sigma_{2}}...H_{N,\sigma_{N}}$ where $\sigma$ are even permutations of $1,...,N$, with $N=(n+1)^{2}$. Each nonzero term contains $n(n+1)$ unit factors and $n+1$ factors $x_{j}$ or $y_{k}$ and is visualized as $N$ arrows $1\to\sigma_{1}$ … $N\to\sigma_{N}$ along the edges of the graph. All vertices of the graph participate with one outgoing and one incoming arrow that make closed oriented self-avoiding loops and dimers (two opposite arrows on the same edge). Such a configuration is a “loop covering” $G$ of the graph.

Theorem (Harari [14]): Let $G$ be a covering of the graph with dimers and oriented loops. If $\epsilon_{G}$ is the number of dimers and oriented loops of even length in $G$, and $p_{G}$ is the product of the values of the edges in $G$, then:

For honeycomb triangles and trapezia only even permutations contribute to the determinant, $\epsilon_{G}=1$.

Some facts about determinants of Hückel matrices of graphene triangles and trapezia are now proven.

As the result (20) only depends on the positions of the non-zero matrix elements and not on their values, every single product $H_{1,\sigma_{1}}...H_{N,\sigma_{N}}$ that does not contain a monomial of degree $n+1$ is identically zero.

The graph of the triangle of graphene with $n+1$ rows has $b=2n+1$ blue dots at the end of the rows, $b^{\prime}=\tfrac{1}{2}n(n-1)$ blue dots in bulk, $r=\tfrac{1}{2}n(n+1)$ red dots. The total number of vertices is $b+b^{\prime}+r=(n+1)^{2}$.
Depending on the numbering of vertices, one has different matrix representations of the graph.