For positive integers and , consider a multiset of nonempty subsets such that there is unique partition these into partitions . We study the maximum possible size multiset. focus on regime show When for any this lower bound simplifies to we matching upper optimal up factor also compute exactly when

Given a sequence of integers aj, j ≥ 1, a multiset is a combinatorial object composed of unordered components, such that there are exactly aj one-component multisets of size j. When aj ≍ jr−1yj for some r > 0, y ≥ 1, then the multiset is called expansive. Let cn be the number of multisets of total size n. Using a probabilistic approach, we prove for expansive multisets that cn/cn+1 → 1 and that...

Gamma is a programming model where computation can be seen as chemical reactions between data represented as molecules floating in a chemical solution. This model can be formalized as associative, commutative, conditional rewritings of multisets where rewrite rules and multisets represent chemical reactions and solutions, respectively. In this article, we generalize the notion of multiset used ...

CP-nets and their variants constitute one of the main AI approaches for specifying and reasoning about preferences. CI-nets, in particular, are a CP-inspired formalism for representing ordinal preferences over sets of goods, which are typically monotonic. Considering also that goods often come in multisets rather than sets, a natural question is whether CI-nets can be used more or less directly...

Current methods which compress multisets at an optimal rate have computational complexity that scales linearly with alphabet size, making them too slow to be practical in many real-world settings. We show how convert a compression algorithm for sequences into one multisets, exchange additional term is quasi-linear sequence length. This allows us of exchangeable symbols rate, decoupled from the ...

Scheduling The problem of all optimizations is to decide if their modifications to the reservation vector are valid or not. I.e. if the constrains for scheduling instructions remain exactly the same before and after the (possibly illegal) optimization. The problem can be solved by converting the reservation vector back into the unambiguous set of relevant issue-multisets. If – after modifying t...