The Bernoulli clock: probabilistic and combinatorial interpretations of the Bernoulli polynomials by circular convolution

The factorially normalized Bernoulli polynomials $b_n(x) = B_n(x)/n!$ are known to be characterized by $b_0(x) = 1$ and $b_n(x)$ for $n \gt 0$ is the anti-derivative of $b_{n-1}(x)$ subject to $\int _0^1 b_n(x) dx = 0$ . We offer a related characterization: $b_1(x) = x - 1/2$ and $({-}1)^{n-1} b_n(x)$ for $n \gt 0$ is the $n$ -fold circular convolution of $b_1(x)$ with itself. Equivalently, $1 - 2^n b_n(x)$ is the probability density at $x \in (0,1)$ of the fractional part of a sum of $n$ independent random variables, each with the beta $(1,2)$ probability density $2(1-x)$ at $x \in (0,1)$ . This result has a novel combinatorial analog, the Bernoulli clock: mark the hours of a $2 n$ hour clock by a uniformly random permutation of the multiset $\{1,1, 2,2, \ldots, n,n\}$ , meaning pick two different hours uniformly at random from the $2 n$ hours and mark them $1$ , then pick two different hours uniformly at random from the remaining $2 n - 2$ hours and mark them $2$ , and so on. Starting from hour $0 = 2n$ , move clockwise to the first hour marked $1$ , continue clockwise to the first hour marked $2$ , and so on, continuing clockwise around the Bernoulli clock until the first of the two hours marked $n$ is encountered, at a random hour $I_n$ between $1$ and $2n$ . We show that for each positive integer $n$ , the event $( I_n = 1)$ has probability $(1 - 2^n b_n(0))/(2n)$ , where $n! b_n(0) = B_n(0)$ is the $n$ th Bernoulli number. For $ 1 \le k \le 2 n$ , the difference $\delta _n(k)\,:\!=\, 1/(2n) -{\mathbb{P}}( I_n = k)$ is a polynomial function of $k$ with the surprising symmetry $\delta _n( 2 n + 1 - k) = ({-}1)^n \delta _n(k)$ , which is a combinatorial analog of the well-known symmetry of Bernoulli polynomials $b_n(1-x) = ({-}1)^n b_n(x)$ .