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524. First Sort II

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:

• 4 1 3 2 (4 and 1 are out of order so move 1 to the front of the list)
• 1 4 3 2 (4 and 3 are out of order so move 3 to the front of the list)
• 3 1 4 2 (3 and 1 are out of order so move 1 to the front of the list)
• 1 3 4 2 (4 and 2 are out of order so move 2 to the front of the list)
• 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, ..., 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(12¹²).