9
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I need to determine how many strictly increasing sequences of a certain length (e.g. 5) are contained in a list of integers. I would be very happy if someone could help me. A sequence would be considered strictly increasing if a<b<c.

list=BlockRandom[RandomInteger[11,{11111}],RandomSeeding->1234];
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9
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length = 5;

Count[And @@@ Positive[Partition[Differences[list], length-1, 1]], True]
42

You can also use:

Count[Positive[Partition[Differences[list], length-1, 1]], {True ..}]

Count[Partition[Unitize @ Ramp @ Differences[list], length-1, 1], {1 ..}]

Count[Subsequences[Unitize @ Ramp @ Differences[list], {length-1}], {1 ..}]

Total[Times @@@ Subsequences[Unitize @ Ramp @ Differences @ list, {length-1}]]

Total @ BlockMap[Apply[Times], Unitize @ Ramp @ Differences @ list, length-1, 1]

Count[Less @@@ Partition[list, length, 1], True]

Count[Subsequences[list, {length}], _?(Apply[Less])]

Count[BlockMap[Apply[Less], list, length, 1], True]

Total @ Boole[Less @@@ Subsequences[list, {length}]]

Total @ Boole[BlockMap[Apply[Less], list, length, 1]]

Length @ Select[Partition[list, length, 1], Apply[Less]]

Total[Boole[Less @@@ Partition[list, length, 1]]]
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