{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,5,13]],"date-time":"2025-05-13T21:58:55Z","timestamp":1747173535220,"version":"3.40.5"},"reference-count":40,"publisher":"Cambridge University Press (CUP)","issue":"2","license":[{"start":{"date-parts":[[2023,11,16]],"date-time":"2023-11-16T00:00:00Z","timestamp":1700092800000},"content-version":"unspecified","delay-in-days":0,"URL":"https:\/\/www.cambridge.org\/core\/terms"}],"content-domain":{"domain":["cambridge.org"],"crossmark-restriction":true},"short-container-title":["Combinator. Probab. Comp."],"published-print":{"date-parts":[[2024,3]]},"abstract":"Abstract<\/jats:title>The factorially normalized Bernoulli polynomials$b_n(x) = B_n(x)\/n!$<\/jats:tex-math><\/jats:alternatives><\/jats:inline-formula>are known to be characterized by$b_0(x) = 1$<\/jats:tex-math><\/jats:alternatives><\/jats:inline-formula>and$b_n(x)$<\/jats:tex-math><\/jats:alternatives><\/jats:inline-formula>for$n \\gt 0$<\/jats:tex-math><\/jats:alternatives><\/jats:inline-formula>is the anti-derivative of$b_{n-1}(x)$<\/jats:tex-math><\/jats:alternatives><\/jats:inline-formula>subject to$\\int _0^1 b_n(x) dx = 0$<\/jats:tex-math><\/jats:alternatives><\/jats:inline-formula>. We offer a related characterization:$b_1(x) = x - 1\/2$<\/jats:tex-math><\/jats:alternatives><\/jats:inline-formula>and$({-}1)^{n-1} b_n(x)$<\/jats:tex-math><\/jats:alternatives><\/jats:inline-formula>for$n \\gt 0$<\/jats:tex-math><\/jats:alternatives><\/jats:inline-formula>is the$n$<\/jats:tex-math><\/jats:alternatives><\/jats:inline-formula>-fold circular convolution of$b_1(x)$<\/jats:tex-math><\/jats:alternatives><\/jats:inline-formula>with itself. Equivalently,$1 - 2^n b_n(x)$<\/jats:tex-math><\/jats:alternatives><\/jats:inline-formula>is the probability density at$x \\in (0,1)$<\/jats:tex-math><\/jats:alternatives><\/jats:inline-formula>of the fractional part of a sum of$n$<\/jats:tex-math><\/jats:alternatives><\/jats:inline-formula>independent random variables, each with the beta$(1,2)$<\/jats:tex-math><\/jats:alternatives><\/jats:inline-formula>probability density$2(1-x)$<\/jats:tex-math><\/jats:alternatives><\/jats:inline-formula>at$x \\in (0,1)$<\/jats:tex-math><\/jats:alternatives><\/jats:inline-formula>. This result has a novel combinatorial analog, theBernoulli clock<\/jats:italic>: mark the hours of a$2 n$<\/jats:tex-math><\/jats:alternatives><\/jats:inline-formula>hour clock by a uniformly random permutation of the multiset$\\{1,1, 2,2, \\ldots, n,n\\}$<\/jats:tex-math><\/jats:alternatives><\/jats:inline-formula>, meaning pick two different hours uniformly at random from the$2 n$<\/jats:tex-math><\/jats:alternatives><\/jats:inline-formula>hours and mark them$1$<\/jats:tex-math><\/jats:alternatives><\/jats:inline-formula>, then pick two different hours uniformly at random from the remaining$2 n - 2$<\/jats:tex-math><\/jats:alternatives><\/jats:inline-formula>hours and mark them$2$<\/jats:tex-math><\/jats:alternatives><\/jats:inline-formula>, and so on. Starting from hour$0 = 2n$<\/jats:tex-math><\/jats:alternatives><\/jats:inline-formula>, move clockwise to the first hour marked$1$<\/jats:tex-math><\/jats:alternatives><\/jats:inline-formula>, continue clockwise to the first hour marked$2$<\/jats:tex-math><\/jats:alternatives><\/jats:inline-formula>, and so on, continuing clockwise around the Bernoulli clock until the first of the two hours marked$n$<\/jats:tex-math><\/jats:alternatives><\/jats:inline-formula>is encountered, at a random hour$I_n$<\/jats:tex-math><\/jats:alternatives><\/jats:inline-formula>between$1$<\/jats:tex-math><\/jats:alternatives><\/jats:inline-formula>and$2n$<\/jats:tex-math><\/jats:alternatives><\/jats:inline-formula>. We show that for each positive integer$n$<\/jats:tex-math><\/jats:alternatives><\/jats:inline-formula>, the event$( I_n = 1)$<\/jats:tex-math><\/jats:alternatives><\/jats:inline-formula>has probability$(1 - 2^n b_n(0))\/(2n)$<\/jats:tex-math><\/jats:alternatives><\/jats:inline-formula>, where$n! b_n(0) = B_n(0)$<\/jats:tex-math><\/jats:alternatives><\/jats:inline-formula>is the$n$<\/jats:tex-math><\/jats:alternatives><\/jats:inline-formula>th Bernoulli number. For$ 1 \\le k \\le 2 n$<\/jats:tex-math><\/jats:alternatives><\/jats:inline-formula>, the difference$\\delta _n(k)\\,:\\!=\\, 1\/(2n) -{\\mathbb{P}}( I_n = k)$<\/jats:tex-math><\/jats:alternatives><\/jats:inline-formula>is a polynomial function of$k$<\/jats:tex-math><\/jats:alternatives><\/jats:inline-formula>with the surprising symmetry$\\delta _n( 2 n + 1 - k) = ({-}1)^n \\delta _n(k)$<\/jats:tex-math><\/jats:alternatives><\/jats:inline-formula>, which is a combinatorial analog of the well-known symmetry of Bernoulli polynomials$b_n(1-x) = ({-}1)^n b_n(x)$<\/jats:tex-math><\/jats:alternatives><\/jats:inline-formula>.<\/jats:p>","DOI":"10.1017\/s0963548323000421","type":"journal-article","created":{"date-parts":[[2023,11,16]],"date-time":"2023-11-16T05:52:52Z","timestamp":1700113972000},"page":"210-237","update-policy":"https:\/\/doi.org\/10.1017\/policypage","source":"Crossref","is-referenced-by-count":0,"title":["The Bernoulli clock: probabilistic and combinatorial interpretations of the Bernoulli polynomials by circular convolution"],"prefix":"10.1017","volume":"33","author":[{"ORCID":"https:\/\/orcid.org\/0000-0003-3788-9280","authenticated-orcid":false,"given":"Yassine","family":"El Maazouz","sequence":"first","affiliation":[]},{"given":"Jim","family":"Pitman","sequence":"additional","affiliation":[]}],"member":"56","published-online":{"date-parts":[[2023,11,16]]},"reference":[{"key":"S0963548323000421_ref2","doi-asserted-by":"publisher","DOI":"10.1007\/978-4-431-54919-2"},{"key":"S0963548323000421_ref7","doi-asserted-by":"publisher","DOI":"10.1515\/forum-2015-0257"},{"key":"S0963548323000421_ref5","doi-asserted-by":"publisher","DOI":"10.1090\/S0273-0979-01-00912-0"},{"key":"S0963548323000421_ref10","first-page":"1","article-title":"A new approach to Bernoulli polynomials","volume":"26","author":"Costabile","year":"2006","journal-title":"Rend. 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