2017/03/21 by Eriksson, Kimmo, Jonsson, Markus, Sjöstrand, Jonas
#Combinatorics (math.CO) #FOS: Mathematics
paper · doi:10.48550/arxiv.1703.07099
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)≤ 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 (λ1 ≥ λ2≥ …) 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 (σ1, σ2, …) 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 σn. For the special case when σn(h)=\lceil qn h \rceil for 0 < qn ≤ 1, these limit shapes are triangular (in case qn2 n→ 0), or exponential (in case qn2 n→ ∞), or interpolating between these shapes (in case qn2 n→ C>0).