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

Abstract

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 $(L_1,L_2,\dots )$ may tend to a limit shape $\varphi$. We show that every convex $\varphi$ 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 $L_n=\{1,1+\lfloor 1/q_n\rfloor ,1+\lfloor 2/q_n\rfloor ,\dots \}$, 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 $n{q}_n^2\to 0$), exponential (in case $n{q}_n^2\to \mathrm{\infty }$), or interpolating between these shapes (in case $n{q}_n^2\to C>0$).