Problem Archive

Consider the following algorithm for sorting a list:

• 1. Starting from the beginning of the list, check each pair of adjacent elements in turn.
• 2. If the elements are out of order:
  • a. Move the smallest element of the pair at the beginning of the list.
  • b. Restart the process from step 1.
• 3. If all pairs are in order, stop.

For example, the list $\{\,4\,1\,3\,2\,\}$ is sorted as follows:

• $\underline{4\,1}\,3\,2$ ($4$ and $1$ are out of order so move $1$ to the front of the list)
• $1\,\underline{4\,3}\,2$ ($4$ and $3$ are out of order so move $3$ to the front of the list)
• $\underline{3\,1}\,4\,2$ ($3$ and $1$ are out of order so move $1$ to the front of the list)
• $1\,3\,\underline{4\,2}$ ($4$ and $2$ are out of order so move $2$ to the front of the list)
• $\underline{2\,1}\,3\,4$ ($2$ and $1$ are out of order so move $1$ to the front of the list)
• $1\,2\,3\,4$ (The list is now sorted)

Let $F(L)$ be the number of times step 2a is executed to sort list $L$. For example, $F(\{\,4\,1\,3\,2\,\}) = 5$.

We can list all permutations $P$ of the integers $\{1, 2, \dots, n\}$ in lexicographical order, and assign to each permutation an index $I_n(P)$ from $1$ to $n!$ corresponding to its position in the list.

Let $Q(n, k) = \min(I_n(P))$ for $F(P) = k$, the index of the first permutation requiring exactly $k$ steps to sort with First Sort. If there is no permutation for which $F(P) = k$, then $Q(n, k)$ is undefined.

For $n = 4$ we have:

P	I4(P)	F(P)
{1, 2, 3, 4}	1	0	Q(4, 0) = 1
{1, 2, 4, 3}	2	4	Q(4, 4) = 2
{1, 3, 2, 4}	3	2	Q(4, 2) = 3
{1, 3, 4, 2}	4	2
{1, 4, 2, 3}	5	6	Q(4, 6) = 5
{1, 4, 3, 2}	6	4
{2, 1, 3, 4}	7	1	Q(4, 1) = 7
{2, 1, 4, 3}	8	5	Q(4, 5) = 8
{2, 3, 1, 4}	9	1
{2, 3, 4, 1}	10	1
{2, 4, 1, 3}	11	5
{2, 4, 3, 1}	12	3	Q(4, 3) = 12
{3, 1, 2, 4}	13	3
{3, 1, 4, 2}	14	3
{3, 2, 1, 4}	15	2
{3, 2, 4, 1}	16	2
{3, 4, 1, 2}	17	3
{3, 4, 2, 1}	18	2
{4, 1, 2, 3}	19	7	Q(4, 7) = 19
{4, 1, 3, 2}	20	5
{4, 2, 1, 3}	21	6
{4, 2, 3, 1}	22	4
{4, 3, 1, 2}	23	4
{4, 3, 2, 1}	24	3

Let $R(k) = \min(Q(n, k))$ over all $n$ for which $Q(n, k)$ is defined.

Find $R(12^{12})$.