doi:10.1007/s00025-021-01393-7engAdell J.A.Cárdenas-Morales D.On the Uniqueness Conjecture for the Maximum Stirling Numbers of the Second KindART-2021-126629The Stirling numbers of the second kind S(n,  k) satisfy S(n, 0)<¿<S(n, kn)=S(n, kn+1)>¿>S(n, n).A long standing conjecture asserts that there exists no n= 3 such that S(n, kn) = S(n, kn+ 1). In this note, we give a characterization of this conjecture in terms of multinomial probabilities, as well as sufficient conditions on n ensuring that S(n, kn) > S(n, kn+ 1). © 2021, The Author(s), under exclusive licence to Springer Nature Switzerland AG.2021http://zaguan.unizar.es/record/11813810.1007/s00025-021-01393-7http://zaguan.unizar.es/record/118138oai:zaguan.unizar.es:118138Results in Mathematics 76, 2 (2021), [8 pp]All rights reservedhttp://www.europeana.eu/rights/rr-f/info:eu-repo/semantics/openAccess