fromstirarrangement

Fromstir arrangement is a term used in theoretical combinatorics to describe a class of sequences formed by combining a from operation with a stir operation on a finite multiset or sequence. In this framework, the from operation selects elements from an input pool according to a prescribed rule, while the stir operation partitions the selected elements into blocks and then orders elements within and between blocks according to additional constraints. The result is a linear arrangement whose structure reflects both the selection rule and the inter-block organization.

Formally, start with a set of n elements and a partition into k nonempty blocks. A fromstir arrangement is obtained by choosing a from-rule that assigns each element to a block and a stir-rule that orders the blocks and the elements inside each block. The final sequence is the concatenation of the blocks in a specified order, with internal orderings determined by the stir-rule. Different choices of from- and stir-rules yield distinct families of arrangements. In many treatments, the counting of fromstir arrangements connects to generalized Stirling numbers or products of partition numbers with factorial factors, depending on the exact rules used.

Variants of the concept impose additional constraints, such as fixed block sizes, avoidance of certain adjacent patterns, or symmetry requirements on block order. These variants affect both the combinatorial count and the structural properties of the resulting arrangements, making fromstir arrangements a flexible tool for studying layered or hierarchical orderings.

Examples of the idea are often illustrated with small sets, such as a three-element set partitioned into two blocks, where the singleton and pair configurations yield different valid sequences under a chosen stir-rule. The concept is primarily of theoretical interest, with potential applications in enumerative combinatorics, randomized generation of structured sequences, and data organization methods that benefit from layered ordering.