On equal values of Stirling numbers of the second kind

S-k(n) denote the Stirling number of the second kind with parameters k and n, i. e. S-k(n) the number of the partition of n elements into k non-empty sets. We formulate the following conjecture concerning the common values of Stirling numbers: Let 1 < a < b be fixed integers. Then all the solutions of the equation S-a(x) = S-b(y) with x > a, y > b are S-5(6) = S-2(5) = 15 and S-90(91) = S-2(13) = 4095. In this note our conjecture is proved for max(a, b) <= 100 and log b/log a is not an element of Q by using some powerful tools from the modern theory of Diophantine equations. (C) 2011 Elsevier Inc. All rights reserved.