$S = 6,15,20,15,6,1$ is a row of Pascal's triangle with its left-hand edge removed.
What is the relationship between $S$, the number of m-faces of a regular n-simplex, and states of a finite quantum system?
Edited to add: Some MSE contributors who also read at MO may recall this MO post and the very mixed reactions to it.
Edited 12/30/2017: I'm adding this so people can see the nature of the biomolecular data which gave rise to the posted question:
One example of 261632 cases over the DNA alphabet {tcag} instantiating the palindrome pair rrrryrrrr and ryrrrrryr over the 2-letter {y,r} alphabet. (The {y,r}-alphabet reduces t,c to y and a,g to r, and the {w,s} alphabet reduces t,a to w and c,g to s.)
ggggcgggg 9-tuple 1 over {tcag}
gcgggggca 9-tuple 2 over {tcag}
rrrryrrrr 9-tuple 1 over {y,r} (palindromic)
ryrrrrryr 9-tuple 2 over {y,r} (palindromic)
sssssssss 9-tuple 2 over {w,s}
ssssssssw 9-tuple 2 over {w,s}
4 count of positions in which tuples 1 and 2 differ
0 difference indicator for position 1
2 difference indicator for position 2
0 difference indicator for position 3
0 difference indicator for position 4
5 difference indicator for position 5
0 difference indicator for position 6
0 difference indicator for position 7
8 difference indicator for position 8
9 difference indicator for position 9
Note that the 261632 cases deliberately includes NO case in which tuples 1 and 2 have the SAME representation over {w,s}. (This is because energetically speaking, they are basically alike.)
See this post for background discussion of the "reduced" {y,r} and {w,s} alphabets:
Cases where ANY 2 of 3 +/- choices select one of four possible elements
Note also that 2-tuples over the DNA {tcag} alphabet or the RNA {ucag} alphabet have associated "relative-delta-H enthalpies" as follows. (These indicate relative strength of complementary binding of these 2-tuples across the two strands in duplex ("double-helix") DNA or RNA - the values below are for RNA, not DNA).
aa 2.80
ga 1.41
ua 2.07
ca 1.16
ag 1.52
au 2.86
gg 0.27
ac 1.91
ug 1.16
cg 0.00
gu 1.91
gc 0.95
uu 2.80
cu 1.52
uc 1.41
cc 0.27
The paper reporting these values can be found here:
Here is the way the rows of Pascal's triangle for n = 3, 6 actually occur in my team's energetic data. The two 1-3-3-1 cases probably represent the initial energetic scaffold for an early version of the genetic code and the two 1-6-15-20-15-6-1 cases probably represent a later energetic scaffold underlying a more complex exfoliation of the original code. So, the relevant OEIS sequence is actually A135278
https://oeis.org/A135278
See in particular the Boya and Dixit paper (referenced by Tom Copeland) regarding m-faces of n-simplices in relation to states of a finite quantum system.
https://arxiv.org/abs/0808.1930
In our case, the energetics derive from the relative-delta-H enthalpies of dinucleotides bound to complementary dinucleotides across the two strands of a DNA or RNA double helix, as calculated by Ornstein and Fresco in these two papers:
https://www.ncbi.nlm.nih.gov/pubmed/6616016
http://onlinelibrary.wiley.com/doi/10.1002/bip.360220812/abstract