Carbohydrate conformations I have to questions regarding the conformations of carbohydrates.


There are 38 different conformations for an aldohexose or ketohexose, including $\alpha$ and $\beta$ anomers, namely 2 chairs, 6 boats, 6 skew-boats, 12 half-chairs and 12 envelopes.

Each of these 38 conformations can have 729 different isomers ($5~\ce{OH}$ and $1~\ce{CH2OH}$) because of dihedral and tetrahedral angle ($3^6$ isomers, 3 (gg, gt, tg for each dihedral angle)).

For example, glucose has $2~(\alpha~\text{and}~\beta~\text{anomers}) \times 38 \times 3^6 = 55404$ conformations.

Is this calculation correct?  


The nomenclature of the 38 different conformations is still not clear to me, which leads me to my second question.


For chair form, the literature uses $\ce{^4C_1}$ and so on. But I'm not sure, which 4 carbon constitutes the reference plane.

Similarly other used terms are $\ce{^{3,O}B}$ for boats, $\ce{^3S_5}$ for skew conformations.

Can anyone please explain how to interpret these symbols?
After going through several literature, finally I am able to answer these questions. It might be important starting point for someone wanting to know carbohydrate chemistry, that's why I'm posting my answer.  


Yes, I'm right. For α- and β-glucose 27702 configuration are possible. So, in total $27702 \times 2 = 55404$ possible structures for glucose [1]. 


  For xylose, 81 rotamers were optimized per puckering conformation for a total of 3078 conformations. For α-glucose, β-glucose, and β-mannose, 729 rotamers were generated for each pucker, resulting in 27 702 conformations for each of these six-carbon monosaccharides. For GlcNAc, 972 conformations per ring pucker were generated for a total of 36 936 geometries. Thus, the initial set of conformations considered for all molecules in this study comprised 123 201 geometries.

Actually these nomenclature is known as Cremer-Pople (Dr. Pople is the Noble prize winner chemist for his outstanding contribution to formulate DFT) nomenclature. They defined puckering parameters and generalized ring structure conformations for 3/4/5/6 or more carbon species. That's a real interesting world!
Courtesy: Shinya Fushinobu, University of Tokyo
It might not be clear from this picture. So I will try to explain them separately.  


Chair: Two chair conformation possible. (O-C2-C3-C5 forms the reference plane). If C-4 is on top then $^4C_1$ and vice versa. 

Boats: Boats forms when two plane intersects each other. The bottom two corner atoms are used to distinguish. If two bottom corners are upside down then they are used as superscript on the left and vice versa.

Skew-Boats: Same as chair. Reference plane is formed by (O-C2-C3-C5). If C5 is on the top and C1 is in bottom then it's named as $^5S_1$ 



Reference


Mayes, H. B.; Broadbelt, L. J.; Beckham, G. T. Journal of the American Chemical Society 2014, 136 (3), 1008–1022 DOI: 10.1021/ja410264d.