Construction of Balanced n-ary Designs via Complete Multiset Spaces

We study balanced n-ary block designs that allow repeated treatments within blocks, motivated by applications in dose-response and repeated-measure experiments. We introduce a master (k + 1)-ary design with explicit replication and pairwise concurrence parameters. Systematic symmetry-preserving deletions generate infinite families of lower-arity balanced designs, for which explicit formulas for replication numbers, structured expressions for pairwise concurrences, and the resulting information matrix under the fixed-effects model are obtained. These results establish a rigorous link between combinatorial S_v -symmetry and statistical optimality. Simulation studies show that the proposed multiset-based designs, by maintaining equireplication, yield lower contrast variances than random multiset arrangements with unequal replication numbers. Potential applications include doseresponse studies, resampling-based simulations, and repeated-measures experiments.