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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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.