The Cycling of Partitions and Compositions under Repeated Shifts

In “Bulgarian Solitaire”, a player divides a deck of n cards into piles. Each move consists of taking a card from each pile to form a single new pile. One is concerned only with how many piles there are of each size. Starting from any division into piles, one always reaches some cycle of partitions of n. Brandt proved that for n = 1+2+ · · ·+k, the cycle is just the single partition into piles of distinct sizes 1, 2, . . . , k. Let DB(n) denote the maximum number of moves required to reach a cycle. Igusa and Etienne proved that DB(n) ≤ k − k whenever n ≤ 1 + 2 + · · ·+ k, and equality holds when n = 1 + 2 + · · ·+ k. We present a simple new derivation of these facts. We improve the bound to DB(n) ≤ k − 2k − 1, whenever n < 1 + 2 + · · ·+ k with k ≥ 4. We present a lower bound for DB(n) that is likely to be the actual value. We introduce a new version of the game, Carolina Solitaire, in which the piles are kept in order, so we work with compositions rather than partitions. Many analogous results can be obtained. Running head: Cycling Under Repeated Shifts 1 Research supported in part by grant NSA/MSP MDA904–95H1024. 2 Research supported in part by grant NSF DMS–9701211. Cycling Under Repeated Shifts Section