Ascendant gap chain of a prime constellation

Question

Given a prime constellation, define its "ascendant gap chain" as the sequence of prime gaps appearing therein sorted in increasing order.

Do all prime constellations of length $k$ share the same ascendant gap chain?

Answer 1

No. According to Tony Forbes's list here, there are six possible patterns of prime 13-tuplets with the minimal width (48):

13  48 : 0  6  12  16  18  22  28  30  36  40  42  46  48 
13  48 : 0  4  6  10  16  18  24  28  30  34  40  46  48 
13  48 : 0  4  6  10  16  18  24  28  30  34  36  46  48 
13  48 : 0  2  6  8  12  18  20  26  30  32  36  42  48 
13  48 : 0  2  8  14  18  20  24  30  32  38  42  44  48 
13  48 : 0  2  12  14  18  20  24  30  32  38  42  44  48 

The third and last of these contain gaps of length 10, but the others don't. You can find explicit examples of prime 13-tuplets both with and without gaps of length 10 here; for example:

10527733922579 + d, d = 0, 2, 12, 14, 18, 20, 24, 30, 32, 38, 42, 44, 48
186460616596321 + d, d = 0, 6, 12, 16, 18, 22, 28, 30, 36, 40, 42, 46, 48