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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 (list; graph; refs; listen; history; text; internal format)
 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COMMENTS Irreducible representations of S_n contained in the special linear group were first considered by L. Solomon (unpublished).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 = 0; b[n_] := Module[{bb, e, pos, k, r}, bb = Reverse[IntegerDigits[n, 2]]; e = bb[]; 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) + Sum[2^((v + 1) (k - 2) - v (v - 1)/2), {v, 1, k - 1}] + e 2^(k (k - 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: A064288 A054085 A021443 * A306238 A318086 A244049 Adjacent sequences:  A045920 A045921 A045922 * A045924 A045925 A045926 KEYWORD nonn,nice AUTHOR EXTENSIONS a(31)-a(48) from Amritanshu Prasad, May 11 2016 STATUS approved

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Last modified February 27 04:56 EST 2020. Contains 332299 sequences. (Running on oeis4.)