Finding permutations (with repetition) that adhere to precedence rules

[CheddarCheeses]

I am looking for an equation to find a number of "viable permutations" that adhere to a rule of precedence. Lets say given set ? with (1, 2, 3, 4, 5, 6, 7), how many permutations with ? trials would have a 7 come after a 6, which comes after a 5, which comes after a 4, and so on from 1−7. They do not need to be immediately following each other, so long as the numbers appear in that order anywhere in the permutation. For example if ?=15, the total possible permutations would be 7^15=4.7475615?+12. I need to know how many of these have a 1,2,3,4,5,6,7 anywhere in the permutation in order, but there can be any random numbers in between them

 

[pka]

I am looking for an equation to find a number of "viable permutations" that adhere to a rule of precedence. Lets say given set ? with (1, 2, 3, 4, 5, 6, 7), how many permutations with ? trials would have a 7 come after a 6, which comes after a 5, which comes after a 4, and so on from 1−7. They do not need to be immediately following each other, so long as the numbers appear in that order anywhere in the permutation. For example if ?=15, the total possible permutations would be 7^15=4.7475615?+12. I need to know how many of these have a 1,2,3,4,5,6,7 anywhere in the permutation in order, but there can be any random numbers in between them

As I understand the question: we are asked for the number of ways that the string [imath]1,\;2,\;3\;,4\;,5\;,6\;,7\;,8\;,9\;,10\;,11\;,12\;,13\;,14\;,15[/imath] can be arranged so that the numbers [imath]1,\;2,\;3\;,4\;,5\;,6\;,7[/imath] appear in that order no matter how they are spread out.
The string [imath]1,\;13,\;15,\;11,\;2,\;10,\;3,\;4,\;8,\; 14\;,5,\;9\;,6\;,7,\;12[/imath] works
How many ways can the string [imath]|,\;|,\;|\;,|\;,|\;,|\;,|\;,8\;,9\;,10\;,11\;,12\;,13\;,14\;,15[/imath] can be rearranged?
Answer: [imath]\dfrac{15!}{7!}=259459200[/imath] ways.
Rearrange the string and then replace each | with a digit [imath]1,2,3,4,5,6,7[/imath] in that order.

[imath][/imath] [imath][/imath] [imath][/imath] [imath][/imath]