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

Wouter Meeussen wouter.meeussen at telenet.be
Wed May 6 17:31:44 CEST 2020

```Richard,

the example was just to clarify what I meant by 'sorted tuples'.
The concept of 'neighbourfree' means that part i cannot be followed by i+1.
With the same example that leaves only these 10:

{1, 1, 1},               {1, 1, 3}, {1, 1, 4},
{1, 3, 3},               {1, 4, 4},
{2, 2, 2},               {2, 2, 4},
{2, 4, 4}, {3, 3, 3},                              {4, 4, 4}

the tabular counting gives (up to 7)

1    1    1    1    1    1    1
2    2    2    2    2    2    2
3    4    5    6    7    8    9
4    7    10    13    16    19    22
5    11    18    26    35    45    56
6    16    30    48    70    96    126
7    22    47    83    131    192    267

Wouter

-----Original Message-----
From: Richard J. Mathar
Sent: Wednesday, May 06, 2020 4:58 PM
To: seqfan at list.seqfan.eu
Subject: [seqfan] Re: what are "sorted tuples" called?

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.

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