Biorthogonal eigenvectors of the Holte carry matrix and cascade-free enumeration

Abstract: For $k$-summand base-$N$ addition, the carry process is a Markov chain on ${0,\ldots,k-1}$ whose transition matrix--the Holte matrix $T$--has eigenvalues ${N<sup>{-j}}_{j=0}<sup>{k-1}$, all simple and independent of $N$. We give the complete biorthogonal eigenvector system. The left eigenvectors factor as $$\sum_i u_j[i] xⁱ = c_{k,j} (x-1)^j A_{k-j}(x)$$, where $c_{k,j} = |s(k,k-j)|/k!$ involves unsigned Stirling numbers and $A_n(x)$ is the Eulerian polynomial. The right eigenvectors satisfy $$\sum_i \binom{k-1}{i} v_j[i] xⁱ = (1+x)^{k-1-j} Q_j(x)$$, where the quotient polynomials $Q_j$ have palindrome symmetry $x^j Q_j(1/x) = (-1)^j Q_j(x)$ and converge to $(1-x)<sup>j$ as $k \to \infty$; for $j \le 3$, we give explicit closed forms in terms of $k$. The cascade-free avoidance count satisfies $a(L) = (\sqrt{d})^L U_L(x)$ (Chebyshev polynomial of the second kind) whenever the restricted transfer matrix has dimension $d \le 2$; we prove this is sharp: for $k$-summand addition, Chebyshev form holds for $k = 3$ and fails for $k \ge 4$. The proof uses oscillatory matrix theory to establish non-vanishing of all spectral residues. The characteristic polynomial of the restricted transfer matrix is determined in closed form by a Stirling-weighted Lagrange interpolation at the Holte eigenvalues. Two systems with binary carry state spaces are shadow-equivalent if and only if they share the pair $(N, d)$. The general classification for $k$-state systems reduces to the characteristic polynomial of $T$.

• The paper establishes an explicit biorthogonal system for the Holte carry matrix, revealing its eigenvalues and eigenvectors independent of base N.
• It provides closed-form generating functions for left and right eigenvectors, linking them to classical combinatorial constructs such as Eulerian numbers and Foulkes characters.
• The work identifies strict conditions under which cascade-free enumeration via Chebyshev parametrization is possible, delineating a threshold for k values.

Biorthogonal Eigenvectors of the Holte Carry Matrix and Cascade-Free Enumeration

Spectral Theory of the Holte Carry Matrix

The Holte carry matrix $T$, associated with the Markovian process of carries in base-$N$ addition of $k$ summands, exhibits a fully computable, algebraic spectral structure. Holte’s original result establishes that the eigenvalues of $T$ are exactly $\{N^{-j}\}_{j=0}^{k-1}$, all algebraically simple and independent of the base $N$. This spectrum is realized as part of a lower-triangular action of $T$ on the monomial filtration in the state space of carries, which are naturally elements of $\{0,\ldots,k-1\}$.

Crucially, the eigenvectors of $T$ are $N$-independent, reflecting deep connections with the symmetric group $N$0 and its representation theory. The stationary distribution aligns with the Eulerian numbers $N$1, quantifying permutations with a prescribed number of descents, and the eigenvectors themselves relate to classical combinatorial constructs: Foulkes characters and Eulerian idempotents.

Explicit Construction of the Biorthogonal System

The paper rigorously derives and packages the complete biorthogonal system of left and right eigenvectors for $N$2 in closed form, advancing prior results by Holte, Diaconis, Fulman, and others. The left eigenvectors $N$3 possess a generating function

$N$4

with normalization constant $N$5, where $N$6 is the unsigned Stirling number of the first kind and $N$7 is the Eulerian polynomial.

The right eigenvectors admit a binomial transform characterization: $N$8 where $N$9 is a palindromic (for even $k$0) or anti-palindromic (for odd $k$1) quotient polynomial of degree $k$2, structurally constrained by $k$3 and satisfying $k$4 as $k$5. Explicit closed forms for $k$6 and $k$7 are computed, establishing the rate of stabilization to the canonical binomial.

This biorthogonal system, uniquely determined up to scaling, induces a biorthogonal basis with the usual normalization $k$8, facilitating spectral expansions for all quantities associated with the carry process.

Cascade-Free Enumeration and the Chebyshev Threshold

A central application is to the avoidance problem: enumerate length-$k$9 digit sequences in which the carry state never reaches the absorbing upper bound ($T$0). For the dynamical system restricted to non-forbidden states, the evolution is governed by the matrix $T$1. The Chebyshev polynomial $T$2 and its parametrizations arise classically in combinatorics for second-order recurrences, and the paper gives a sharp threshold for their universality.

It is proved that a Chebyshev-parametrization of the cascade-free sequence count $T$3 is possible if and only if the restricted matrix has dimension $T$4: for $T$5, $T$6 is geometric, and for $T$7, a Chebyshev-$T$8 parametrization in terms of the discriminant holds. For $T$9, no such form exists; the spectrum is strictly higher-order and the minimal polynomial for $\{N^{-j}\}_{j=0}^{k-1}$0 is irreducible of degree $\{N^{-j}\}_{j=0}^{k-1}$1. This threshold is established by combining oscillatory matrix theory (guaranteeing non-vanishing spectral residues upon eigenvalue separation) and explicit computation of the characteristic polynomial of $\{N^{-j}\}_{j=0}^{k-1}$2 via Stirling-weighted Lagrange interpolation at the Holte eigenvalues, parameterized by Foulkes character values.

In the positive Chebyshev regime ($\{N^{-j}\}_{j=0}^{k-1}$3), the closed form produces the Fibonacci bisection $\{N^{-j}\}_{j=0}^{k-1}$4 for base-3 addition. Contradictory to any naive generalization, if $\{N^{-j}\}_{j=0}^{k-1}$5, all Chebyshev-parametrized formulas fail, and recurrence relations necessarily exhibit degree at least three (or higher). The result is both a classification and a limitation theorem for the explicit combinatorics of digit-sequence constraints dictated by carry processes.

Structural Classification and Moduli Spaces

The conjugacy class of the Holte matrix is shown to be completely determined by its spectrum. More generally, for binary carry processes ($\{N^{-j}\}_{j=0}^{k-1}$6), systems are shadow-equivalent if and only if they share the same trace and determinant (i.e., parameters $\{N^{-j}\}_{j=0}^{k-1}$7). The moduli space of equivalence classes is fully characterized: pairs $\{N^{-j}\}_{j=0}^{k-1}$8 such that $\{N^{-j}\}_{j=0}^{k-1}$9 has an integer factorization with summands bounded by $N$0. For general $N$1 and simple spectrum, the characteristic polynomial uniquely determines the equivalence class.

These results imply that additive carry propagation (in positional representations as well as more general Markovian digit operations) is rigidly determined by spectral invariants, and shadow equivalence classes are parameterized by the roots of the characteristic polynomial of their (possibly restricted) transfer matrices.

Theoretical and Practical Implications

This work provides a definitive algebraic framework for the spectral analysis of carry matrices. The explicit relationship between eigenvectors and classical symmetric group invariants enriches the combinatorial and representation-theoretic understanding of carry phenomena. In practical terms, the results delimit the class of constraints (e.g., cascade-freeness) that admit explicit, low-complexity enumeration formulas, identifying precisely the regimes where Chebyshev (or other orthogonal polynomial) parametrizations are admissible. For arbitrary $N$2, the Stirling–Lagrange interpolation formula enables computable spectral invariants governing enumeration and classification, potentially impacting both algorithmic implementations for combinatorial generation and the analysis of digit-wise algorithms in number systems.

On the theoretical side, this spectral rigidity and its representation-theoretic underpinnings suggest new connections to symmetric function theory, descent algebras, and Markovian random walks on state spaces parameterized by grouping structures on permutations, opening further directions for investigation.

Conclusion

The paper achieves a complete and explicit algebraic-combinatorial description of the eigenstructure of the Holte carry matrix, with a new generating-function factorization for the left eigenvectors and a palindromic quotient characterization for the right eigenvectors. The Chebyshev threshold for cascade-free sequence enumeration is sharply determined, and the family of systems admitting such formulas is classified. Both the structural moduli of carry matrices and their enumeration-theoretic consequences are reduced to spectral data and representation-theoretic quantities, resolving several open cases on the algebraic nature of constrained digit processes (2604.04591).