The Cycling of Partitions and Compositions under Repeated Shifts

Abstract

In “Bulgarian Solitaire,” a player divides a deck ofncards 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 ofn. Brandt proved that forn=1+2+···+k, the cycle is just the single partition into piles of distinct sizes 1,2,…,k. LetDB(n) denote the maximum number of moves required to reach a cycle. Igusa and Etienne proved thatDB(n)≤k2−kwhenevern≤1+2+···+k, and equality holds whenn=1+2+···+k. We present a simple new derivation of these facts. We improve the bound toDB(n)≤k2−2k−1, whenevern<1+2+···+kwithk≥4. We present a lower bound forDB(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.