Notes on Number Theory and Discrete Mathematics Print ISSN 1310–5132, Online ISSN 2367–8275 2022, Volume 28, Number 1, Pages 81–91 DOI: 10.7546/nntdm.2022.28.1.81-91 Set partitions with isolated successions Toufik Mansour 1 and Augustine O. Munagi 2 1 Department of Mathematics, University of Haifa 3498838 Haifa, Israel e-mail: tmansour@univ.haifa.ac.il 2 School of Mathematics, University of the Witwatersrand Johannesburg 2050, South Africa e-mail: augustine.munagi@wits.ac.za Received: 24 March 2021 Revised: 14 February 2022 Accepted: 15 February 2022 Online First: 15 February 2022 Abstract: We enumerate partitions of the set {1,...,n} according to occurrences of isolated successions, that is, integer strings a, a +1,...,b in a block when neither a − 1 nor b +1 lies in the same block. Our results include explicit formulas and generating functions for the number of partitions containing isolated successions of a given length. We also consider a corresponding analog of the associated Stirling numbers of the second kind. Keywords: Partition, Isolated succession, Recurrence, Generating function. 2020 Mathematics Subject Classification: 05A18, 05A15, 05A19. 1 Introduction A partition of [n]= {1, 2,...,n} is a decomposition of [n] into nonempty subsets called blocks. A partition into k-blocks is also called a k-partition and denoted by B 1 /B 2 /.../B k , where the blocks are arranged in standard order: min(B 1 ) < ··· < min(B k ) (see [4]). The number of k-partitions of [n] is the Stirling number of the second kind S (n, k) which satisfies the recurrence relation: S (n, k)= S (n − 1,k − 1) + kS (n − 1,k), (1) where S (0, 0) = 1,S (n, 0) = S (0,n)=0 for n> 0. 81