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Bulgarian Solitaire is an interesting self-map on the set of integer
partitions of a fixed number $n$. As a finite dynamical system, its long-term
behavior is well-understood, having recurrent orbits parametrized by necklaces
of beads with two colors black $B$ and white $W$. However, the behavior of the
transient elements within each orbit is much less understood.

Recent work of Pham considered the orbits corresponding to a family of
necklaces $P^\ell$ that are concatenations of $\ell$ copies of a fixed
primitive necklace $P$. She proved striking limiting behavior as $\ell$ goes to
infinity: the level statistic for the orbit, counting how many steps it takes a
partition to reach the recurrent cycle, has a limiting distribution, whose
generating function $H_p(x)$ is rational. Pham also conjectured that $H_P(x),
H_{P^*}(x)$ share the same denominator whenever $P^*$ is obtained from $P$ by
reading it backwards and swapping $B$ for $W$.

Here we introduce a new representation of Bulgarian Solitaire that is
convenient for the study of these generating functions. We then use it to prove
two instances of Pham's conjecture, showing that

$$H_{BWBWB \cdots WB}(x)=H_{WBWBW \cdots BW}(x)$$ and that $H_{BWWW\cdots
W}(x),H_{WBBB\cdots B}(x)$ share the same denominator.