The Holte matrix has biorthogonal eigenvectors expressed via Stirling numbers, Eulerian polynomials, and symmetric quotient polynomials, with cascade-free avoidance counts being Chebyshev polynomials exactly for k=3 but not for k>=4.
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Cascade-free sequence counts for GEN/PROP/KILL digit operations satisfy a(L) = N a(L-1) - d a(L-2) and equal scaled Chebyshev polynomials, with base-3 doubling equaling the Fibonacci bisection F(2L+2).
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Biorthogonal eigenvectors of the Holte carry matrix and cascade-free enumeration
The Holte matrix has biorthogonal eigenvectors expressed via Stirling numbers, Eulerian polynomials, and symmetric quotient polynomials, with cascade-free avoidance counts being Chebyshev polynomials exactly for k=3 but not for k>=4.
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Cascade-free sequences, dispersion index, and state avoidance for stateful digit-wise operations
Cascade-free sequence counts for GEN/PROP/KILL digit operations satisfy a(L) = N a(L-1) - d a(L-2) and equal scaled Chebyshev polynomials, with base-3 doubling equaling the Fibonacci bisection F(2L+2).