A045923 - OEIS

%I #33 Aug 09 2018 13:26:25

%S 1,1,1,2,2,7,7,10,10,34,40,53,61,103,112,143,145,369,458,579,712,938,

%T 1127,1383,1638,2308,2754,3334,3925,5092,5818,6989,7759,12278,14819,

%U 17881,21477,25887,30929,36954,43943,52918,62749,74407,87854,104534,122706,144457

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

%C Irreducible representations of S_n contained in the special linear group were first considered by L. Solomon (unpublished).

%D R. P. Stanley, Enumerative Combinatorics, vol. 2, Cambridge University Press, Cambridge and New York, 1999, Exercise 7.55.

%H Amritanshu Prasad, <a href="/A045923/b045923.txt">Table of n, a(n) for n = 1..999</a>

%H 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 a(n) = A000041(n) - A272090(n). - _Amritanshu Prasad_, May 11 2016

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

%t b[1] = 0;

%t b[n_] := Module[{bb, e, pos, k, r},

%t bb = Reverse[IntegerDigits[n, 2]];

%t e = bb[[1]];

%t pos = DeleteCases[Flatten[Position[bb, 1]], 1] - 1;

%t r = Length[pos];

%t Do[k[i] = pos[[i]], {i, 1, r}];

%t 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 ];

%t a[n_] := PartitionsP[n] - b[n];

%t Array[a, 50] (* _Jean-François Alcover_, Aug 09 2018, after _Amritanshu Prasad_ *)

%Y Cf. A000041, A272090.

%K nonn,nice

%O 1,4

%A _Richard Stanley_

%E a(31)-a(48) from _Amritanshu Prasad_, May 11 2016