A045923 Number of irreducible representations of symmetric group S_n for which every matrix has determinant 1.

1, 1, 1, 2, 2, 7, 7, 10, 10, 34, 40, 53, 61, 103, 112, 143, 145, 369, 458, 579, 712, 938, 1127, 1383, 1638, 2308, 2754, 3334, 3925, 5092, 5818, 6989, 7759, 12278, 14819, 17881, 21477, 25887, 30929, 36954, 43943, 52918, 62749, 74407, 87854, 104534, 122706, 144457

Offset: 1

Author

Richard Stanley

Keywords

• nonn
• nice

Comments

Irreducible representations of S_n contained in the special linear group were first considered by L. Solomon (unpublished).

Examples

			a(5)=2, since only the irreducible representations indexed by the partitions (5) and (3,2) are contained in the special linear group.
		

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).

Crossrefs

Cf. A000041, A272090.

Programs

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 *)

Formula

a(n) = A000041(n) - A272090(n). - Amritanshu Prasad, May 11 2016

Extensions

a(31)-a(48) from Amritanshu Prasad, May 11 2016