There are two possible solutions here, one using the Pólya Enumeration
Theorem and another one using Burnside's lemma and Stirling numbers of
the second kind. The second solution is much more efficient than the
first one. The answers are $126$ for the action of the symmetric group
and $1643544$ for the cyclic group.
For both solutions we need the cycle indices of the respective groups
which are
$$Z(S_{10}) =
1/9\,a_{{9}}a_{{1}}+1/16\,a_{{8}}{a_{{1}}}^{2}+1/42\,a_{{7}}{a_{
{1}}}^{3}+{\frac {a_{{6}}{a_{{1}}}^{4}}{144}}+{\frac {a_{{2}}{a_
{{1}}}^{8}}{80640}}\\+{\frac {a_{{3}}{a_{{1}}}^{7}}{15120}}+{
\frac {{a_{{1}}}^{6}{a_{{2}}}^{2}}{5760}}+{\frac {a_{{4}}{a_{{1}
}}^{6}}{2880}}+{\frac {a_{{5}}{a_{{1}}}^{5}}{600}}+{\frac {{a_{{
1}}}^{4}{a_{{2}}}^{3}}{1152}}+{\frac {{a_{{1}}}^{4}{a_{{3}}}^{2}
}{432}}+{\frac {{a_{{1}}}^{2}{a_{{4}}}^{2}}{64}}\\+{\frac {{a_{{1}
}}^{2}{a_{{2}}}^{4}}{768}}+{\frac {a_{{1}}{a_{{3}}}^{3}}{162}}+{
\frac {{a_{{2}}}^{3}a_{{4}}}{192}}+{\frac {{a_{{2}}}^{2}{a_{{3}}
}^{2}}{144}}+1/48\,{a_{{2}}}^{2}a_{{6}}+{\frac {a_{{2}}{a_{{4}}}
^{2}}{64}}\\+1/16\,a_{{2}}a_{{8}}+{\frac {{a_{{3}}}^{2}a_{{4}}}{72
}}+1/21\,a_{{3}}a_{{7}}+1/24\,a_{{4}}a_{{6}}+{\frac {1}{50}}\,{a
_{{5}}}^{2}+1/10\,a_{{10}}\\+{\frac {{a_{{1}}}^{10}}{3628800}}+{
\frac {{a_{{2}}}^{5}}{3840}}+1/24\,a_{{1}}a_{{2}}a_{{3}}a_{{4}}+
{\frac {a_{{1}}{a_{{2}}}^{3}a_{{3}}}{144}}+1/40\,a_{{1}}{a_{{2}}
}^{2}a_{{5}}\\+1/14\,a_{{1}}a_{{2}}a_{{7}}+1/18\,a_{{1}}a_{{3}}a_{
{6}}+1/20\,a_{{1}}a_{{4}}a_{{5}}+{\frac {{a_{{1}}}^{2}{a_{{2}}}^
{2}a_{{4}}}{64}}+{\frac {{a_{{1}}}^{2}a_{{2}}{a_{{3}}}^{2}}{72}}
\\+1/24\,{a_{{1}}}^{2}a_{{2}}a_{{6}}+1/30\,{a_{{1}}}^{2}a_{{3}}a_{
{5}}+{\frac {{a_{{1}}}^{3}{a_{{2}}}^{2}a_{{3}}}{144}}+{\frac {{a
_{{1}}}^{3}a_{{2}}a_{{5}}}{60}}+{\frac {{a_{{1}}}^{3}a_{{3}}a_{{
4}}}{72}}\\+{\frac {{a_{{1}}}^{5}a_{{2}}a_{{3}}}{720}}+{\frac {{a_
{{1}}}^{4}a_{{2}}a_{{4}}}{192}}+1/30\,a_{{2}}a_{{3}}a_{{5}}$$
and
$$Z(C_{10}) =
1/10\,{a_{{1}}}^{10}+1/10\,{a_{{2}}}^{5}+2/5\,{a_{{5}}}^{2}+2/5
\,a_{{10}}.$$
Using a CAS like Maple we can then perform coefficient extraction of
all terms in
$$Z(S_{10})(B_1+B_2+\cdots+B_6)$$
and
$$Z(C_{10})(B_1+B_2+\cdots+B_6)$$
that contain all six variables, e.g. from
$$[B_1^5 B_2 \cdots B_6] Z(C_{10})(B_1+B_2+\cdots+B_6).$$
We then sum these and set the six variables to one to get the count we
are interested in. The substitution follows the standard rule of $a_q
= B_1^q + B_2^q + \cdots + B_6^q.$ An excerpt from $Z(C_{10})$ looks
like this:
$$\cdots +504\,B_{{3}}B_{{4}}{B_{{5}}}^{3}{B_{{6}}}^
{5}+252\,B_{{3}}B_{{4}}{B_{{5}}}^{2}{B_{{6}}}^{6}+72\,B_{{3}}B_{{4}
}B_{{5}}{B_{{6}}}^{7}\\+9\,B_{{3}}B_{{4}}{B_{{6}}}^{8}+B_{{3}}{B_{{5}
}}^{9}+9\,B_{{3}}{B_{{5}}}^{8}B_{{6}}+36\,B_{{3}}{B_{{5}}}^{7}{B_{{
6}}}^{2}+84\,B_{{3}}{B_{{5}}}^{6}{B_{{6}}}^{3}\\+126\,B_{{3}}{B_{{5}}
}^{5}{B_{{6}}}^{4}+126\,B_{{3}}{B_{{5}}}^{4}{B_{{6}}}^{5}+84\,B_{{3
}}{B_{{5}}}^{3}{B_{{6}}}^{6}\\+36\,B_{{3}}{B_{{5}}}^{2}{B_{{6}}}^{7}+
9\,B_{{3}}B_{{5}}{B_{{6}}}^{8}+B_{{3}}{B_{{6}}}^{9}\\+{B_{{4}}}^{10}+
{B_{{4}}}^{9}B_{{5}}+{B_{{4}}}^{9}B_{{6}}+5\,{B_{{4}}}^{8}{B_{{5}}}
^{2}+9\,{B_{{4}}}^{8}B_{{5}}B_{{6}}+\cdots$$
This is what the procedures V1 and W1 do. This method is very
costly as we must carry out expensive polynomial exponentiations to
get the coefficients and coefficients are being computed that do not
contribute to the desired result.
A much more efficient approach is to use the Burnside lemma
directly. Given a term from the cycle index $Z(S_{10})$ or $Z(C_{10})$
the admissible assignments must be constant on each cycle and in the
context of this particular problem all six colors must be
present. This means we have a set partition of the cycles into six
non-empty sets where each set receives one color. Set partitions are
counted by the Stirling numbers of the second kind. Before we conclude
we need to take into account that all $720$ permutations of the six
colors correspond to a valid assignment. This gives a very efficient
algorithm with good time and space complexity parameters (number of
operations proportional to the number of terms in the cycle
index). The algorithm is implemented in **V2** and **W2**.
The Maple code for these was as follows:
with(combinat);
with(numtheory);
pet_varinto_cind :=
proc(poly, ind)
local subs1, subs2, polyvars, indvars, v, pot, res;
res := ind;
polyvars := indets(poly);
indvars := indets(ind);
for v in indvars do
pot := op(1, v);
subs1 :=
[seq(polyvars[k]=polyvars[k]^pot,
k=1..nops(polyvars))];
subs2 := [v=subs(subs1, poly)];
res := subs(subs2, res);
od;
res;
end;
pet_cycleind_cyclic :=
proc(n)
local d, s;
s := 0;
for d in divisors(n) do
s := s + phi(d)*a[d]^(n/d);
od;
s/n;
end;
pet_cycleind_symm :=
proc(n)
local p, s;
option remember;
if n=0 then return 1; fi;
expand(1/nadd(a[l]pet_cycleind_symm(n-l), l=1..n));
end;
pet_flatten_term :=
proc(varp)
local terml, d, cf, v;
terml := [];
cf := varp;
for v in indets(varp) do
d := degree(varp, v);
terml := [op(terml), seq(v, k=1..d)];
cf := cf/v^d;
od;
[cf, terml];
end;
V1 :=
proc()
local sind, res, term;
sind :=
pet_varinto_cind(add(cat('B', q), q=1..6),
pet_cycleind_symm(10));
res := 0;
for term in expand(sind) do
if nops(indets(term)) = 6 then
res := res + term;
fi;
od;
subs([seq(cat('B', q)=1, q=1..6)], res);
end;
W1 :=
proc()
local sind, res, term;
sind :=
pet_varinto_cind(add(cat('B', q), q=1..6),
pet_cycleind_cyclic(10));
res := 0;
for term in expand(sind) do
if nops(indets(term)) = 6 then
res := res + term;
fi;
od;
subs([seq(cat('B', q)=1, q=1..6)], res);
end;
V2 :=
proc()
local ind, term, flat, res;
ind := pet_cycleind_symm(10);
res := 0;
for term in ind do
flat := pet_flatten_term(term);
res := res + flat[1]*
6!*stirling2(nops(flat[2]), 6);
od;
res;
end;
W2 :=
proc()
local ind, term, flat, res;
ind := pet_cycleind_cyclic(10);
res := 0;
for term in ind do
flat := pet_flatten_term(term);
res := res + flat[1]*
6!*stirling2(nops(flat[2]), 6);
od;
res;
end;
Addendum 02 Jun 2016. The case of the symmetric group has a simple closed form as seen at this MSE link. In this particular instance we obtain
$$6! \times\frac{1}{10!} \sum_{q=1}^{10} \left[10\atop q\right] {q\brace 6} = 126.$$
Using the Stirling numbers of the first kind to count terms in the cycle index of the symmetric group that consist of $q$ cycles is of course much better than computing all terms of the cycle index.
The closed form for the cyclic group is
$$6! \times \frac{1}{10}\sum_{d|10} \varphi(d) {10/d\brace 6}
= 1643544.$$
The case of cyclic and dihedral symmetry including Maple code is documented at OEIS A087854 and OEIS A273891.
