Limit shapes of stable and recurrent configurations of a generalized Bulgarian solitaire

Bulgarian solitaire is played on \( n \) cards divided into several piles; a move consists of picking one card from each pile to form a new pile. This can be seen as a process on the set of integer partitions of \( n \): if sorted configurations are represented by Young diagrams, a move in the solitaire consists of picking all cards in the bottom layer of the diagram and inserting the picked cards as a new column. Here we consider a generalization, \( L \)-solitaire, wherein a fixed set of layers \( L \) (that includes the bottom layer) are picked to form a new column.

\( L \)-solitaire has the property that if a stable configuration of \( n \) cards exists it is unique. Moreover, the Young diagram of a configuration is convex if and only if it is a stable (fixpoint) configuration of some \( L \)-solitaire. If the Young diagrams representing card configurations are scaled down to have unit area, the stable configurations corresponding to an infinite sequence of pick-layer sets \( (L_1, L_2, \ldots) \) may tend to a limit shape \( \phi \). We show that every convex \( \phi \) with certain properties can arise as the limit shape of some sequence of \( L_n \). We conjecture that recurrent configurations have the same limit shapes as stable configurations.

For the special case \( L_n = \{1, 1 + \lfloor 1/q_n \rfloor, 1 + \lfloor 2/q_n \rfloor, \ldots\} \), where the pick layers are approximately equidistant with average distance \( 1/q_n \) for some \( q_n \in (0,1] \), these limit shapes are linear (in case \( nq_n^2 \to 0 \)), exponential (in case \( nq_n^2 \to \infty \)), or interpolating between these shapes (in case \( nq_n^2 \to C > 0 \)).