(1) As for the number of alkanes ($\ce{C_nH_{2n+2}}$), Table 1, which is extracted from the data reported in
S. Fujita, MATCH Commun. Math. Comput. Chem., 57,
299--340 (2007) (access free),
shows the comparison between two enumerations based on
Polya's theorem and on Fujita's proligand method.
The number of alkanes ($\ce{C_nH_{2n+2}}$)
as constitutional isomers (structural isomers) and as steric isomers
is calculated by Polya's theorem (G. Polya and R. C. Read,
Combinatorial Enumeration of Groups, Graphs, and Chemical Compounds, Springer (1987)).
In the process of calculating constitutional isomers,
one 2D structure (graph or constitution) is counted just once.
In the process of calculating steric isomers, one achiral molecule or each chiral molecule
of an enantiomeric pair is counted just once, where achiral molecules and chiral molecules
are not differentiated from each other.
On the other hand,
the number of alkanes ($\ce{C_nH_{2n+2}}$)
as three-dimensional (3D) structural isomers
and as steric isomers is
calculated by Fujita's proligand method
(S. Fujita,
Combinatorial Enumeration of Graphs, Tree-Dimensional Structures, and
Chemical Compounds, Unversity of Kragujevac (2013)).
In the process of calculating 3D structural isomers,
one achiral molecule or one pair of enantiomers is counted just once,
where achiral molecules and chiral molecules (enantiomeric pairs)
are differentiated from each other.
For more information, see an account article entitled
"Numbers of Alkanes and Monosubstituted Alkanes.
A Long-Standing Interdisciplinary Problem over 130 Years"
(
S. Fujita, Bull. Chem. Soc. Japan, 83, 1--18 (2010),
access free). This account article has discussed the difference between graph enumeration (Polya's theorem) and 3D structural enmeration (Fujita's proligand method) during recursive calculation.
It should be emphasized that graph-theoretical enumerations of chemical compounds
as constitutional isomers (structural isomers) and as steric isomers (based on asymmetry)
should be differentiated from stereochemical enumerations of chemical compounds
as 3D structural isomers and as steric isomers (based on chirality).
Although steric isomers based on asymmetry (graphs governed by permutation groups) and
steric isomers based on chirality (3D structures governed by point groups) give identical
enumeration results, they are conceptually different entities.
This point of view stems from Fujita's stereoisogram approach,
which is described in a recent book
(S. Fujita
Mathematical Stereochemistry, De Gruyter (2015)).
(2) Enumeration of achiral and chiral alkanes of a given carbon content has been
conducted by considering internal branching
(
S. Fujita, Bull. Chem. Soc. Jpn., 81, 1423--1453 (2008)).
Figure 3 of this report is cited below.
The symbol [q, t, s, p] means
the presence of q quaternary carbons, t tertiary carbons, s secondary carbons,
and p primary carbons. Alkanes are categorized into
centroidal and bicentroidal alkanes, which are the 3D extension of
centroidal and bicentroidal trees of graph theory.