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

Abstract

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 $n>0$ is the antiderivative 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 $n>0$ 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 {\em Bernoulli clock}: mark the hours of a $2n$ hour clock by a uniform random permutation of the multiset $\{1,1,2,2,\dots ,n,n\}$, meaning pick two different hours uniformly at random from the $2n$ hours and mark them $1$, then pick two different hours uniformly at random from the remaining $2n-2$ 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 $1\le k\le 2n$, the difference ${\delta }_n(k):=1/(2n)-\P (I_n=k)$ is a polynomial function of $k$ with the surprising symmetry ${\delta }_n(2n+1-k)=(-1{)}^n{\delta }_n(k)$, which is a combinatorial analog of the well known symmetry of Bernoulli polynomials $b_n(1-x)=(-1{)}^n{b}_n(x)$.