[seqfan] Re: what are "sorted tuples" called?

[Richard J. Mathar]

wm> Date: Tue, 5 May 2020 19:04:21 +0200
wm> From: "Wouter Meeussen" <wouter.meeussen at telenet.be>
wm> To: "seqfan" <seqfan at list.seqfan.eu>
wm> Subject: [seqfan] what are "sorted tuples" called?
wm> ...
wm> I had a lot of fun trying to count the ?sorted tuples? of length n from an alphabet of k letters (1<=k<=n) that are neighbour-free.
wm> 
wm> Example of ?sorted tuples? of length n=3 chosen from a k=4 letter alphabet :
wm> 
wm> {1, 1, 1}, {1, 1, 2}, {1, 1, 3}, {1, 1, 4}, {1, 2, 2},
wm> {1, 2, 3}, {1, 2, 4}, {1, 3, 3}, {1, 3, 4}, {1, 4, 4},
wm> {2, 2, 2}, {2, 2, 3}, {2, 2, 4}, {2, 3, 3}, {2, 3, 4},
wm> {2, 4, 4}, {3, 3, 3}, {3, 3, 4}, {3, 4, 4}, {4, 4, 4}
wm> 
wm> I?m sure these constructs have a proper name. Any hints?

I don't understand what is  meant by "neighbour-free" (?).
As far as I can see the example counts the non-decreasing
compositions with n parts  with all parts <=k.