Sierpinski Matrices: Structure & Applications

• Sierpinski matrices are structured matrices with binary (or generalized) entries reflecting the fractal geometry of the Sierpinski triangle.
• They are generated recursively via Kronecker products, encoding combinatorial properties such as Lucas's theorem and carry-free digit sums.
• Applications span discrete mathematics, spectral theory, and harmonic analysis on fractals, offering insights into matrix groups and integer sequences.

A Sierpinski matrix is a canonical family of structured matrices—typically with entries in $\{0,1\}$, or in more general arithmetics—whose sparsity and recursive self-similarity encode the combinatorial geometry of the Sierpinski gasket or triangle. These matrices appear across discrete mathematics, algebra, combinatorics, theoretical computer science, and harmonic analysis on fractals, and they serve as fundamental objects in binomial recurrences, digital expansions, and algebraic theory. Modern research encompasses various Sierpinski matrices: classical mod 2 forms, parameterized and generalized versions, two-dimensional fractal matrices, Weyr and Jordan canonical matrices, and matrices arising from graph energy and self-similar Dirichlet forms.

1. Classical Sierpinski Matrix and Modulo-2 Structures

The prototypical Sierpinski matrix $S$ is the infinite lower-triangular binary matrix with entries

$$S_{n,k} = \binom{n-1}{k-1} \bmod 2, \qquad n, k \ge 1.$$

The nonzero pattern of $S$ (or any finite $n\times n$ principal submatrix) is precisely the left-justified Sierpinski triangle: a fractal pattern arising from Pascal’s triangle reduced modulo 2. The pattern reflects Lucas’s theorem, and the support of $1$'s corresponds to pairs $(n, k)$ such that adding $k-1$ and $n-k$ in base 2 involves no carries (Beck et al., 2021).

Variants include the right-justified Sierpinski matrix $M_{n,k} = \binom{k-1}{n-k} \bmod 2$, which displays the same pattern in reflected orientation. The structure of these matrices encodes combinatorial properties of binary expansions and recurrences.

2. Recursive and Fractal Constructions

Sierpinski matrices admit elegant recursive and Kronecker (tensor) factorizations:

• For $n\geq1$, the classical Sierpinski matrix $S_n$ of size $2^n\times2^n$ satisfies

$$S_{n+1} = \begin{pmatrix} S_n & S_n \ 0 & S_n \end{pmatrix}$$

• More generally, the parameterized family $S_n(x)$—with $x$ an indeterminate—follows the Kronecker recurrence

$$S_1(x) = \begin{pmatrix} 1 & x \ 0 & 1 \end{pmatrix}, \qquad S_{n+1}(x) = S_1(x) \otimes S_n(x)$$

and admits the explicit entrywise formula,

$$(S_n(x))_{j,k} = \begin{cases} x^{s(j-k)} & 0 \leq k \leq j,\ (k, j-k) \text{ carry-free} \ 0 & \text{otherwise} \end{cases}$$

where $s(m)$ denotes the sum of binary digits of $m$ (Nguyen, 2014).

The Kronecker structure imparts the matrix with a self-similar, fractal geometry. Entries are nonzero precisely on patterns matching discrete Sierpinski gaskets, and blockwise recurrences dictate the branching and scaling of the support across digit levels.

3. Generalizations: Digital, $b$-ary, and Higher-Dimensional Matrices

Generalized Sierpinski matrices extend this structure to arbitrary prime bases $b$ and parameterized recursions:

• The generalized Sierpinski matrices $S_{b,N}(x)$ are defined recursively by

$$S_{b,1}(x) = [\binom{x + (j-k) - 1}{j-k}]_{0 \leq j, k < b}, \qquad S_{b,N+1}(x) = S_{b,1}(x) \otimes S_{b,N}(x),$$

yielding lower-triangular $b^N \times b^N$ matrices (Nguyen, 2015).

• For $M(i,j)$ defined recursively as

$$M[0,0]=1,\quad M[0,j]=a^j,\ M[i,0]=c^i,\quad M[i,j]=aM[i,j-1]+bM[i-1,j-1]+cM[i-1,j],$$

one obtains two-dimensional Sierpinski matrices whose mod $p$ reductions for primes $p$ generate the Sierpinski carpet and related discrete fractals (0901.3189).

• Fractal self-similarity is formalized numerically: for $p$ a prime and $M$ as above, $M$ exhibits $p$-self-similarity if

$M[sp^k+i, tp^k+j] \equiv M[s,t]\,M[i,j] \pmod p.$

This family encompasses the Sierpinski triangle ($a=b=c=1,\,p=2$), the Sierpinski carpet ($a=b=c=1,\,p=3$), and an infinite range of combinatorial patterns determined by recursion parameters.

4. Canonical Forms, Spectral Theory, and Weyr Structures

Sierpinski matrices over fields of characteristic 0 (or sufficiently large prime) have a unique Weyr structure characterized by binomial coefficients. In the recursive form

$$B_{n} = \begin{pmatrix} B_{n-1} & B_{n-1} \ 0 & B_{n-1} \end{pmatrix},\quad B_0 = [1],$$

the Weyr structure of $B_n$ is the multiset

$$\left\{ \binom{n}{\lfloor n/2 \rfloor}, \binom{n}{\lfloor n/2 \rfloor - 1}, \ldots, \binom{n}{0} \right\}$$

arranged in non-increasing order. Each block’s size corresponds to the combinatorics of binary expansions (O'Meara et al., 2017).

The spectral decomposition thus directly encodes classical combinatorial identities; the same binomial coefficients describe the sizes of Jordan blocks in the Jordan canonical form, though early proofs of the Jordan analogue invoked deep results from algebraic geometry.

5. Invertibility, Harmonic Extensions, and Analysis on Fractals

In the context of harmonic structures on the Sierpinski gasket $SG_k$, the associated harmonic-extension (“splitting”) matrices $A_i$ are $3\times3$ rational matrices that map boundary Dirichlet data to values on lower-dimensional cells. Tsougkas’ theorem asserts that these matrices are invertible for all $k\geq 2$: $$\det A_i = \frac{(\alpha_k - \beta_k)^2(\alpha_k + 2\beta_k)}{D_k^3} > 0,$$ where $\alpha_k, \beta_k$ are explicit integers and $D_k$ is a positive integer depending on $k$ (Tsougkas, 2016).

Non-degeneracy of all $A_i$ is critical for analysis on post-critically finite self-similar sets: it guarantees finite-dimensionality and uniqueness of harmonic functions, well-posedness of the Laplacian, absolute continuity of energy measures, and robust spectral properties, and can be characterized using Tutte’s barycentric embedding and graph connectivity criteria.

6. Group Properties, Digital Binomial Theorems, and Algebraic Applications

Sierpinski matrices in the parameterized family satisfy the matrix group law

$S(x) S(y) = S(x+y),$

implying that $\{S(x) : x \in \mathbb{R}\}$ forms a one-parameter abelian matrix group under multiplication (Nguyen, 2014, Nguyen, 2015). This leads to a digital (carry-free) version of the binomial theorem, expressing

$$(x+y)^{s(m)} = \sum_{\substack{0\leq k \leq m \ (k, m-k)\,\text{carry-free}}} x^{s(k)} y^{s(m-k)},$$

where $s(\cdot)$ is the binary sum-of-digits function and the summation is over carry-free decompositions. Generalizations to base $b$ follow analogously.

Applications extend to commutative Artinian algebras, where the multiplication operator by $g = \prod_{i=1}^n (1+x_i)$ on $A = \mathbb{F}[x_1,\dots,x_n]/(x_1^2,\dots,x_n^2)$ has the classical Sierpinski matrix as its representing matrix, implying strong Lefschetz properties and bijectivity between graded pieces (O'Meara et al., 2017). Group actions, braid relations, and connections to Prouhet-Thue-Morse and PTM polynomials arise through detailed matrix and generating function methods (Nguyen, 2015).

7. Connections to Integer Sequences and Additional Identities

Antidiagonal, row, and column sums of Sierpinski-type matrices, and their related Pascal matrices, yield classic integer sequences:

Matrix	Row Sums	Column Sums	Antidiagonal Sums
$P$	Fibonacci $F_n$	Powers of $2$	Padovan $P_{n+2}$
$P^{-1}$	$(-1)^n F_{n-2}$	—	Fine numbers
$M$	Stern sequence $s(n)$	$2^{d(k-1)}$ (odd entries in Pascal)	Recursion of $s(n)$
$M^{-1}$	$1$ if $n$ is power of $2$, else $0$	—	Thue–Morse–twisted recursion
$S, S^{-1}$	Powers of $2$, Stern numbers	—	Signs: periodic or 3-cycle

Here, $d(k-1)$ denotes the binary digit sum of $k-1$. Summation identities follow from generating function arguments and reflect the deep interplay between Sierpinski matrices, binary combinatorics, and sequence theory (Beck et al., 2021).

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Sierpinski matrices stand at the crossroads of combinatorics, spectral theory, group algebra, and fractal analysis, with recurring roles in spectral decimation, self-similar tilings, matrix factorizations, and advanced algebraic constructions. Current research continues to expand their reach into generalized binomial bases, group-theoretic structures, and analysis on non-classical fractals.