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

%I A045923 #33 Aug 09 2018 13:26:25
%S A045923 1,1,1,2,2,7,7,10,10,34,40,53,61,103,112,143,145,369,458,579,712,938,
%T A045923 1127,1383,1638,2308,2754,3334,3925,5092,5818,6989,7759,12278,14819,
%U A045923 17881,21477,25887,30929,36954,43943,52918,62749,74407,87854,104534,122706,144457
%N A045923 Number of irreducible representations of symmetric group S_n for which every matrix has determinant 1.
%C A045923 Irreducible representations of S_n contained in the special linear group were first considered by L. Solomon (unpublished).
%D A045923 R. P. Stanley, Enumerative Combinatorics, vol. 2, Cambridge University Press, Cambridge and New York, 1999, Exercise 7.55.
%H A045923 Amritanshu Prasad, <a href="/A045923/b045923.txt">Table of n, a(n) for n = 1..999</a>
%H A045923 A. Ayyer, A. Prasad and S. Spallone, <a href="https://arxiv.org/abs/1604.08837">Representations of symmetric groups with non-trivial determinant</a>, arXiv:1604.08837 [math.RT] (2016).
%F A045923 a(n) = A000041(n) - A272090(n). - _Amritanshu Prasad_, May 11 2016
%e A045923 a(5)=2, since only the irreducible representations indexed by the partitions (5) and (3,2) are contained in the special linear group.
%t A045923 b[1] = 0;
%t A045923 b[n_] := Module[{bb, e, pos, k, r},
%t A045923 bb = Reverse[IntegerDigits[n, 2]];
%t A045923 e = bb[[1]];
%t A045923 pos = DeleteCases[Flatten[Position[bb, 1]], 1] - 1;
%t A045923 r = Length[pos];
%t A045923 Do[k[i] = pos[[i]], {i, 1, r}];
%t A045923 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))
%t A045923 ];
%t A045923 a[n_] := PartitionsP[n] - b[n];
%t A045923 Array[a, 50] (* _Jean-François Alcover_, Aug 09 2018, after _Amritanshu Prasad_ *)
%Y A045923 Cf. A000041, A272090.
%K A045923 nonn,nice
%O A045923 1,4
%A A045923 _Richard Stanley_
%E A045923 a(31)-a(48) from _Amritanshu Prasad_, May 11 2016