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$S = 6,15,20,15,6,1$ 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

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

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  • $\begingroup$ @WendyKrieger - thanks as always - I have five more cases to do out of six to see if the same numbers arise. Also, note that in my team's empirical research, the numbers arise multiplied by 4096, as: 24576, 61440, 81920, 61440, 24576, 4096 $\endgroup$ – David Halitsky Dec 29 '17 at 13:13
  • $\begingroup$ From the sample, there seems to be an impost in the first value, of 512. Then it gives the row of the triangle, less 512,0,0,0,.... Curiouser and curiouser. $\endgroup$ – wendy.krieger Dec 30 '17 at 10:18
  • $\begingroup$ @WendyKrieger - I've added an example to the post here, and also sent you an email with the complete set of 261632 cases underlying row 1 of ht PDF previously sent. I added the example here so that peolple can see the nature of the underlying biomolecular data which led to the numbers similar to the OEIS sequence. $\endgroup$ – David Halitsky Dec 30 '17 at 14:37
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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
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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

Case 2 rrryryrrr vs yryrrryry

                N     Diff        N       N     Diff      Sum of   Sum of N's
Mismatches   Less     Less     Zero    More     More       N's      / 32768
    6       16348    -166.618    60   16360   +166.500   32768        1      
    7       49285    -210.190    93   48926   +211.733   98304        3
    8       49403    -246.811    86   48815   +249.784   98304        3  
    9       16489    -279.088    54   16225   +283.629   32768        1

Case 5 ryryyyryr vs yyyryryyy 

                N      Diff      N       N      Diff     Sum of   Sum of N's
Mismatches   Less      Less    Zero    More     More       N's     / 32768
    6       16360   -166.500     60   16348    +166.618  32768        1      
    7       48926   -211.733     93   49285    +210.190  98304        3
    8       48815   -249.784     86   49403    +246.811  98304        3
    9       16225   -283.629     54   16489    +279.088  32768        1

Case 1 rrrryrrrr vs ryrrrrryr 

                    N     Diff        N       N     Diff      Sum of   Sum of N's
    Mismatches   Less     Less     Zero    More     More       N's      / 4096
        3        2071   -160.563    22     2003   +166.014    4096        1
        4       12269   -191.979    36    12271   +191.948   24576        6
        5       30782   -216.581    77    30581   +218.004   61440       15
        6       41153   -238.724   122    40645   +241.707   81920       20
        7       30841   -259.855    50    30549   +262.338   61440       15
        8       12305   -279.920    15    12256   +281.039   24576        6
        9        2052   -297.321    16     2028   +300.840    4096        1

Case 6 yryyyyyry vs yyyyryyyy 

                    N      Diff      N       N      Diff     Sum of   Sum of N's
    Mismatches   Less      Less    Zero    More     More       N's     / 4096
        3        2003   -166.014    22     2071   +160.563    4096        1
        4       12271   -191.948    36    12269   +191.979   24576        6
        5       30581   -218.004    77    30782   +216.581   61440       15
        6       40645   -241.707   122    41153   +238.724   81920       20
        7       30549   -262.338    50    30841   +259.855   61440       15
        8       12256   -281.039    15    12305   +279.920   24576        6
        9        2028   -300.840    16     2052   +297.321    4096        1
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