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A045923
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Number of irreducible representations of symmetric group S_n for which every matrix has determinant 1.
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2
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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
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1,4
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COMMENTS
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Irreducible representations of S_n contained in the special linear group were first considered by L. Solomon (unpublished).
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REFERENCES
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R. P. Stanley, Enumerative Combinatorics, vol. 2, Cambridge University Press, Cambridge and New York, 1999, Exercise 7.55.
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LINKS
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FORMULA
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EXAMPLE
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a(5)=2, since only the irreducible representations indexed by the partitions (5) and (3,2) are contained in the special linear group.
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MATHEMATICA
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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];
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CROSSREFS
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KEYWORD
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nonn,nice
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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