A thought about pancakes

The pancake sorting problem is to determine the diameter of the pancake graph, the Cayley graph of {S_n} with respect to the generating set {P_n = \{p_1, \dots, p_n\}}, where {p_k} is defined (in two-line notation) by

\displaystyle  p_k = \left(\begin{array}{ccccccc} 1 & 2 & \dots & k & k+1 & \cdots & n \\ k & k-1 & \dots & 1 & k+1 & \cdots & n \end{array}\right)

(i.e., {p_k} flips the top {k} pancakes). It is not hard to see that the diameter is between {n} and {2n}. Both lower and upper bound have been sharpened by a constant factor, and a natural conjecture is that {\mathrm{diam}~\mathrm{Cay}(S_n, P_n) / n} converges to some intermediate constant (to be determined, ideally).

Here is a natural model problem: estimate {|P^m|} for each fixed {m} as {n \to \infty}. Clearly {|P^m| \leq n^m}, and we should be able to detect some fall-off from this trivial upper bound.

Proposition 1 There is an integer sequence {(a_m)}, beginning

\displaystyle  1, 2, 6, 22, 90, 486, 3014, 22185, 182229, \dots

(not in the OEIS) such that {|P^m| \sim (a_m/m!) n^m} as {n \to \infty} with {m} fixed.

To define {(a_m)} it is useful to introduce the hyperoctahedral group {B_m = \{\pm1\} \wr S_m} (cf. burnt pancake sorting). You can think of {B_m} as the group of “signed” or “antipodal” permutations: permutations {\sigma} of {\{\pm1, \dots, \pm m\}} such that {\sigma(-x) = -\sigma(x)} for all {x}. There is a natural cover {B_m \to S_m} defined by forgetting signs. We lift {P_m} to {B_m} in a particular way: define {Q_m = \{q_1, \dots, q_m\}}, where

\displaystyle  q_k = \left(\begin{array}{ccccccc} 1 & 2 & \dots & k & k+1 & \cdots & n \\ -k & -(k-1) & \dots & -1 & k+1 & \cdots & n \end{array}\right).

Now define

\displaystyle  a_m = \# \{ q_{\sigma(1)} \cdots q_{\sigma(m)} : \sigma \in S_m \}.

I computed the initial terms in the sequence using this simple formula. The first nontrivial relations are

\displaystyle  q_1 q_2 q_4 q_3 = q_2 q_4 q_3 q_1

and its inverse (to verify I recommend using four asymmetric cards from a deck of cards), which together account for {a_4 = 22 < 24}.

To sketch the proof of Proposition 1, consider composing {m} random elements of {P_n}. You can think of this as a two-step process: randomly divide the stack {\{1, \dots, n\}} in {m} places, and then rearrange the resulting {m+1} pieces, possibly with some reversals. With high probability the points of division are all distinct, so they can be inferred from the result (this is true for {m} up to about {n^{1/2}}). The only question is whether the order can be inferred, and this is precisely what is defined by {a_m}.

It would be informative for the pancake problem to know more about the sequence {(a_m)}. I don’t know whether {a_m} can be computed more efficiently than the naive way above, e.g., whether there is a recurrence: if so that would really help a lot.

There is at least a submultiplicativity property that follows from the proposition:

Corollary 2 For {m, k\geq 1}, {a_{m+k} / (m+k)! \leq a_m / m! \cdot a_k / k!}. Hence there is a constant {c_1 > 0} such that

\displaystyle  a_m/m! < e^{-c_1m + o(m)}.

Proof: This follows from {|P^{m+k}| \leq |P^m| |P^k|}, dividing by {n^{m+k}}, and taking the limit. The second part follows from the subadditivity lemma and the data point {a_4 = 22 < 24}. \Box

I think a stronger bound should be true: there should be a constant {c_2 > 0} such that

\displaystyle  a_m < m!^{1-c_2 + o(1)}.

(One might then contemplate whether {n/(1 - c_2) + o(n)} is the diameter of the pancake graph, but that seems a little bold at this stage.)