# Partitioning with overlap

I want to explain the following procedure for th number 1 to 10.

I want to partition the numbers to the group of 3 like this:

1 2 3 : 2 3 4: 3 4 5 : 4 5 6 : 5 6 7: 6 7 8: 7 8 9: 8 9 10.

How can I explain it?

In real data set the partition size is 100 and I have more than 5000 numbers.

I'd try saying:

Eight triplets of consecutive numbers from (1,2,3) through (8,9,10).

Although, in character count, your complete series is shorter… ;-)

There are many ways to describe it. I might say,

A set of all possible three consecutive integers (chosen from 1 to 10).

or

An ordered list of [n,n+1,n+2], where n is an integer from 1 to 8.

• Or: All triples (x, y, z), where y = x + 1, z = y + 1, x ≥ 1, and z ≤ 10. (You're right; there are many possible ways to describe it.)
– J.R.
Feb 14, 2014 at 2:34

Partitioning of a set with overlapping elements is mathematically called a covering. If the target audience is mathematically inclined, you could define this term and use it as follows:

Coverings of the set of integers from 1 to 10, consisting of 3 consecutive numbers.

• I'm not sure about this usage of "covering". Could you give me some link so I can read more? I thought that in combinatorics, a covering would mean the entire set itself (a covering = something that covers the given structure), not each element (or a partition) in the set. Feb 13, 2014 at 23:05
• @DamkerngT. I read about it in a Discrete Mathematics textbook. en.wikipedia.org/wiki/Set_cover_problem. Perhaps my statement was ambiguous, by "A partition", I did not mean one member of such a partition, but the partition itself. Feb 14, 2014 at 10:28
• Thank you for the link. I'm still unsure about the use of "covering". And if you meant to say "Coverings" to mean "A partition", I think it might be better to say "A covering" or even better, "A cover". If we followed the definition given in your link, e.g. U = {1,2,3,4,5} and the set of sets S = {{1,2,3},{2,4},{3,4},{4,5}} so that the union of S is U. A cover C would be a subset of S whose union is U. Feb 14, 2014 at 11:07
• Uhm, yeah, but the question isn't asking for one partition, but the collection of the 8 partition. Feb 14, 2014 at 11:42
• And I would call the 8-partition collection a cover, not covers, at least according to the definition in Wikipedia. But it's up to you to choose your terms though. :-) Feb 14, 2014 at 11:53

You could say, "All partitions of the numbers from 1 to 10 consisting of three consecutive numbers, for example [1, 2, 3], and [2, 3, 4].".