Treffer: 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, theBernoulli 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)$.