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A069999
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Number of possible dimensions for commutators of n X n matrices; it is independent of the field. Or, given a partition P = (p_1, p_2, ..., p_m) of n with p_1 >= p_2 >= ... >= p_m, let S(P) = sum_j (2j-1)p_j; then a(n) = number of integers that are an S(P) for some partition.
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1
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1, 2, 3, 5, 7, 9, 13, 18, 21, 27, 34, 39, 46, 54, 61, 72, 83, 92, 106, 118, 130, 145, 162, 176, 193, 209, 226, 246, 265, 284, 308, 330, 352, 375, 402, 426, 453, 480, 508, 538, 570, 598, 631, 661, 694, 730, 765, 800, 835, 872, 911, 951, 992, 1030, 1071, 1115
(list; graph; refs; listen; history; internal format)
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OFFSET
| 1,2
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COMMENTS
| Or, given such a partition P of n, let T(P) = sum_i p_i^2; then a(n) = number of integers that are a T(P) for some P. While T(P) need not equal S(P) for a given partition, the two sets of integers are equal. Or, expand the infinite product prod_k 1/(1-x^{k^2}y^k) as a power series; then a(n) = number of terms of the form x^my^n having a nonzero coefficient.
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REFERENCES
| Zachary Albertson and Evan Willett, "Possible Dimensions of Commutators of Matrices", Senior Thesis, Wake Forest University, May 09, 2002.
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LINKS
| David Savitt and R. P. Stanley, A Note on the Symmetric Powers of the Standard Representation of S_n, Electronic J. Combinat., 7 (2000) #R6.
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FORMULA
| No generating function is known.
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CROSSREFS
| Sequence in context: A028870 A057886 A200672 * A035563 A028378 A143587
Adjacent sequences: A069996 A069997 A069998 * A070000 A070001 A070002
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KEYWORD
| easy,nonn,nice
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AUTHOR
| Jim Kuzmanovich (kuz(AT)wfu.edu), Apr 26 2002
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EXTENSIONS
| More terms from Robert Gerbicz (gerbicz(AT)freemail.hu), Aug 27 2002
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