# How Shuffled is this Sequence?

Given a sequence $$\S\$$ with length $$\n\$$ of exactly one each of whole numbers from $$\1\$$ to $$\n\$$, your task is to return a number $$\\sigma\$$ indicating how shuffled it is.

# Definition of shuffledness

Humans perceive shuffling randomness locally [Citation Needed], so the shuffledness of a sequence is based on the ability to find various patterns within each sliding window $$\W\$$ of 5 consecutive elements of the sequence ($$\W = \{S_x, S_{x+1}, S_{x+2}, S_{x+3}, S_{x+4}\}\$$ for some $$\x\$$- these will overlap, as each $$\x\$$ that is at least $$\4\$$ less than $$\n\$$ is considered). For each window of 5 elements, add a point for each test of the following that fails: (Note the 1-indexing due to the use of mathematical notation)

• Apply the following tests to the whole window:
1. $$\W\$$ is monotonic
2. $$\W\$$ is in ascending order
3. Some permutation of $$\W\$$ is a run of sequential integers (e.g. 1, 2, 3, 4, 5)
4. For all $$\W_i\$$ where $$\i>1\$$, $$\W_i = W_{i-1}+k\$$ for some integer $$\k\$$.
• This corresponds to regular linear intervals
5. For all $$\W_i\$$ where $$\i>1\$$, $$\W_i = W_{i-1}+ax+k\$$ for some integers $$\k, a\$$, where $$\x=i\$$.
• This corresponds to regular quadratic intervals (which could also be linear)
• Apply the same tests for subwindows where 1 or 2 elements have been removed (if any one of those reduced windows passes a particular test, this step passes. Only count one failure when a test fails for all reduced windows.
• Apply the same tests for windows with some pair of adjacent elements optionally swapped (this step will pass if all tests passed for the whole unmodified window) Replace test 3 with the following:
• $$\W\$$ is an ascending or descending sequence of sequential integers (e.g. 5, 4, 3, 2, 1)

### Notes

• Ascending sequential sequences will pass every test for all windows and therefore be scored as $$\0\$$
• Descending sequential sequences will pass all tests except the "ascending" one for all windows and will therefore be scored as $$\3(n-4)\$$
• The naive maximum shuffledness is $$\15(n-4)\$$

# Examples

• [1, 5, 3, 2, 4]
• All elements:
• Passes test 3
• Fails all other tests (+4)
• Elements removed:
• Passes tests 1, 2, 5 with [1, 2, 4] (remove 5 and 3)
• Passes test 3 with [5, 3, 2, 4] (remove 1)
• Fails test 4 (+1)
• Elements swapped:
• Fails all tests (+5)
• Test 3 fails because it requires the elements to be in order for this case only
• $$\\therefore \sigma = 10\$$
• [1, 2, 4, 3, 5]
• All elements:
• Passes test 3
• Fails all other tests (+4)
• Elements removed:
• All tests pass with [1, 2, 3] (remove 4, 5)
• Elements swapped:
• Passes all tests (swap 4, 3)
• $$\\therefore \sigma = 4\$$
• [3, 6, 9, 12, 15] (a sliding window)
• All elements:
• Passes tests 1, 2, 4, 5
• Fails test 3 (+1)
• Elements removed: same (+1)
• Elements swapped: same (+1) (no swap)
• $$\\therefore \sigma = 3\$$
• [8, 12, 13, 11, 6] (a sliding window)
• All elements:
• Passes test 5 ($$\a=-3, k=4\$$)
• Fails all other tests (+4)
• Elements removed:
• Passes tests 1, 2, 5 with [8, 12, 13] (remove 11, 6)
• Passes test 3 with [12, 13, 11] (remove 8, 6)
• Fails test 4 (+1)
• Elements swapped: same as all elements (+4) (no swap)
• $$\\therefore \sigma = 9\$$
• [2, 4, 6, 1, 3, 5] (whole sequence)
• First window: ([2, 4, 6, 1, 3])
• All elements: Fails all tests (+5)
• Elements removed:
• Passes tests 1, 2, 4, 5 with [2, 4, 6]
• Passes test 3 with [2, 1, 3]
• Elements swapped: Fails all tests (+5)
• Second window: ([4, 6, 1, 3, 5])
• All elements: Fails all tests (+5)
• Elements removed:
• Passes tests 1, 2, 4, 5 with [1, 3, 5]
• Passes test 3 with [4, 6, 5]
• Elements swapped: Fails all tests (+5)
• $$\\therefore \sigma = 20\$$

# Rules and such

• Standard IO rules, loopholes, etc
• You must handle sequences of up to 1000 elements long
• You may assume $$\S\$$ is at least 5 elements long
• No validation necessary. You may assume that all inputs contain exactly one each of every whole number from $$\1\$$ to $$\n\$$
• Shortest code wins!
• In the first example, how can it fail all tests for the elements swapped bit? – Nick Kennedy Dec 28 '19 at 0:22
• How is $W_x$ defined? – flawr Dec 28 '19 at 11:17
• @NickKennedy The swapped elements test requires the run to actually be in order – Beefster Dec 30 '19 at 4:10
• @flawr $W_x$ is the $x$th element of $W$ – Beefster Dec 30 '19 at 4:11
• @Beefster I'd suggest adding that in a small comment, or writing something like $W = (W_1,W_2,\ldots)$. (It was confusing for me as $x$ is an unusual variable to use for indexing.) – flawr Dec 30 '19 at 14:44

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