Bargraph statistics on words and set partitions

In this paper, we consider statistics on partitions of an n-element set represented as a subset of the bargraphs that have n horizontal steps. More precisely, we find the joint distribution of the area and up step statistics on the latter subset of bargraphs, thereby obtaining new refined counts on partitions having a fixed number of blocks. Furthermore, we give explicit formulas in terms of the Stirling numbers for the total area and number of up steps in bargraphs corresponding to partitions, providing both algebraic and combinatorial proofs. Finally, we find asymptotic estimates for the average and total values of these statistics and as a consequence obtain some new identities for the Stirling and Bell numbers.