OFFSET
1,4
COMMENTS
Irreducible representations of S_n contained in the special linear group were first considered by L. Solomon (unpublished).
REFERENCES
R. P. Stanley, Enumerative Combinatorics, vol. 2, Cambridge University Press, Cambridge and New York, 1999, Exercise 7.55.
LINKS
Amritanshu Prasad, Table of n, a(n) for n = 1..999
A. Ayyer, A. Prasad and S. Spallone, Representations of symmetric groups with non-trivial determinant, arXiv:1604.08837 [math.RT] (2016).
FORMULA
EXAMPLE
a(5)=2, since only the irreducible representations indexed by the partitions (5) and (3,2) are contained in the special linear group.
MATHEMATICA
b[1] = 0;
b[n_] := Module[{bb, e, pos, k, r},
bb = Reverse[IntegerDigits[n, 2]];
e = bb[[1]];
pos = DeleteCases[Flatten[Position[bb, 1]], 1] - 1;
r = Length[pos];
Do[k[i] = pos[[i]], {i, 1, r}];
2^Sum[k[i], {i, 2, r}] (2^(k[1] - 1) + Sum[2^((v + 1) (k[1] - 2) - v (v - 1)/2), {v, 1, k[1] - 1}] + e 2^(k[1] (k[1] - 1)/2))
];
a[n_] := PartitionsP[n] - b[n];
Array[a, 50] (* Jean-François Alcover, Aug 09 2018, after Amritanshu Prasad *)
CROSSREFS
KEYWORD
nonn,nice
AUTHOR
EXTENSIONS
a(31)-a(48) from Amritanshu Prasad, May 11 2016
STATUS
approved