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A045923
Number of irreducible representations of symmetric group S_n for which every matrix has determinant 1.
2
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Irreducible representations of S_n contained in the special linear group were first considered by L. Solomon (unpublished).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
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
Sequence in context: A054085 A357412 A021443 * A306238 A318086 A244049
KEYWORD
nonn,nice
EXTENSIONS
a(31)-a(48) from Amritanshu Prasad, May 11 2016
STATUS
approved