Limit shapes of stable 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. In a recent generalization, $σ$-Bulgarian solitaire, the number of cards you pick from a pile is some function $σ$ of the pile size, such that you pick $σ(h)\le h$ cards from a pile of size $h$. Here we consider a special class of such functions. Let us call $σ$ well-behaved if $σ(1)=1$ and if both $σ(h)$ and $h-σ(h)$ are non-decreasing functions of $h$. Well-behaved $σ$-Bulgarian solitaire has a geometric interpretation in terms of layers at certain levels being picked in each move. It also satisfies that if a stable configuration of $n$ cards exists it is unique. Moreover, if piles are sorted in order of decreasing size ($λ_1 \ge λ_2\ge \dots$) then a configuration is convex if and only if it is a stable configuration of some well-behaved $σ$-Bulgarian solitaire. If sorted configurations are represented by Young diagrams and scaled down to have unit height and unit area, the stable configurations corresponding to an infinite sequence of well-behaved functions ($σ_1, σ_2, \dots$) may tend to a limit shape $ϕ$. We show that every convex $ϕ$ with certain properties can arise as the limit shape of some sequence of well-behaved $σ_n$. For the special case when $σ_n(h)=\lceil q_n h \rceil$ for $0 < q_n \le 1$, these limit shapes are triangular (in case $q_n^2 n\rightarrow 0$), or exponential (in case $q_n^2 n\rightarrow \infty$), or interpolating between these shapes (in case $q_n^2 n\rightarrow C>0$).