Bernoulli Polynomials as Permanent of Matrices

Bernoulli polynomials and Bernoulli numbers are fundamental sequences in mathematics with important applications in number theory, combinatorics, and analysis. Classical treatments are typically based on generating functions, recurrence relations, and analytic representations. In this paper, we introduce an alternative algebraic formulation based on matrix permanents. We prove that the $n$th Bernoulli polynomial $B_n(x)$ can be expressed as $\displaystyle \frac{1}{(n-1)!}$ times the permanent of a structured $(n+1)\times(n+1)$ matrix. As a special case, Bernoulli numbers are obtained by evaluating these polynomial representations at zero, yielding a permanent representation for $B_n$. The construction requires only elementary linear algebra and reveals a structural connection between binomial coefficients, permanent expansions, and the classical Bernoulli recurrence. This approach provides a new combinatorial and algebraic perspective on Bernoulli sequences and establishes a link between matrix theory and special polynomial families.