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

1,4

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Irreducible representations of S_n contained in the special linear group were first considered by L. Solomon (unpublished).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

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

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

Cf. A000041, A272090.

Sequence in context: A054085 A357412 A021443 * A306238 A318086 A244049

Adjacent sequences: A045920 A045921 A045922 * A045924 A045925 A045926

Keyword

nonn,nice

Author

Richard Stanley

Extensions

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

Status

approved