Hopkins B.Tangboonduangjit A.Mahidol University2025-09-232025-09-232025-05-01Fibonacci Quarterly Vol.63 No.2 (2025) , 400-41100150517https://repository.li.mahidol.ac.th/handle/123456789/112228Mansour and Shattuck introduced the notion of water cells for integer compositions in 2018. We focus on compositions with parts restricted to 1 and 2 and consider the array of counts for such compositions of n with k water cells, establishing generating functions for the columns and diagonal sums, recurrences within the array in the spirit of Pascal’s lemma, and connections to other restricted compositions. Most of our proofs are combinatorial, but we also make use of Riordan arrays.MathematicsArticleSCOPUS10.1080/00150517.2025.25133352-s2.0-105016116705