What is the connection between these quantities? $S = 6,15,20,15,6,1$ 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:
http://onlinelibrary.wiley.com/doi/10.1002/bip.360220812/abstract;jsessionid=3DB31A510C70A7C72FDA23FAB9FFB3E9.f04t04
 A: See:
OEIS A135278: T(n,m) is the number of m-faces of a regular n-simplex
https://oeis.org/A135278
and in particular:
a) this comment by Tom Copeland (24 July 2017) on the relationship between the combinatorics of the n-simplex and states of a quantum system: "For a correlation between the states of a quantum system and the combinatorics of the n-simplex, see Boya and Dixit";
b) the paper by Boya and Dixit referenced in his comment:
https://arxiv.org/pdf/0808.1930.pdf 
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 {tcag0}
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.)
This deliberate omission MAY be why we're not seeing 512 cases.
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

A: This one is a sixth tragmic-power of a point.  I imagine we should look for a power-product.
