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A007716
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Number of polynomial symmetric functions of matrix of order n under separate row and column permutations.
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4
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1, 4, 10, 33, 91, 298, 910, 3017, 9945, 34207, 119369, 429250, 1574224, 5916148, 22699830, 89003059, 356058540, 1453080087, 6044132794, 25612598436, 110503627621, 485161348047, 2166488899642, 9835209912767, 45370059225318
(list; graph; refs; listen; history; internal format)
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OFFSET
| 1,2
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COMMENTS
| Also number of nonnegative integer n X n matrices with sum of elements equal to n, under row and column permutations.
This is a two-dimensional generalization of the partition function (A000041), which equals the number of length n vectors of nonnegative integers with sum n, equivalent under permutations. - Franklin T. Adams-Watters, Sep 19 2011.
Also number of non-isomorphic multiset partitions of weight n. [Gus Wiseman (gus(AT)nafindix.com), Sep 19 2011]
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FORMULA
| a(n) is coefficient of x^n in cycle index Z(S_nXS_n; x_1, x_2, ...) if we replace x_i by 1+x^i+x^(2*i)+x^(3*i)+x^(4*i)+..., where S_nXS_n is cartesian product of symmetric groups S_n of degree n - Vladeta Jovovic (vladeta(AT)eunet.rs), Mar 09 2000
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EXAMPLE
| The 10 non-isomorphic multiset partitions of weight 3 are: {{1, 1, 1}}, {{1, 1, 2}}, {{1, 2, 3}}, {{1}, {1, 1}}, {{1}, {1, 2}}, {{1}, {2,
2}}, {{1}, {2, 3}}, {{1}, {1}, {1}}, {{1}, {1}, {2}}, {{1}, {2}, {3}}.
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CROSSREFS
| Cf. A053307, A052365, A052366, A052367, A052372, A052373, A049311, A054688, A000041.
Sequence in context: A052367 A052372 A052373 * A122948 A149171 A149172
Adjacent sequences: A007713 A007714 A007715 * A007717 A007718 A007719
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KEYWORD
| nice,nonn
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AUTHOR
| C. L. Mallows (colinm(AT)research.avayalabs.com)
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EXTENSIONS
| More terms from Vladeta Jovovic (vladeta(AT)eunet.rs), Jun 28 2000
Terms a(19) and up from Max Alekseyev (maxale(AT)gmail.com), Jan 22 2010
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