

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
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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Irreducible representations of S_n contained in the special linear group were first considered by L. Solomon (unpublished).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 nontrivial 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] (* JeanFrançois Alcover, Aug 09 2018, after Amritanshu Prasad *)


CROSSREFS

Cf. A000041, A272090.
Sequence in context: A064288 A054085 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



