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A306098
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Number of equivalence classes, modulo transposition, of non-symmetric plane partitions of n.
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2
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0, 0, 1, 2, 5, 10, 21, 39, 74, 133, 239, 415, 719, 1216, 2048, 3393, 5586, 9087, 14695, 23530, 37462, 59172, 92947, 145024, 225123, 347421, 533614, 815378, 1240410, 1878302, 2832586, 4253800, 6363760, 9483831, 14083418, 20839900, 30735490, 45181303, 66210373, 96730731
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OFFSET
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0,4
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COMMENTS
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A plane partition of n is a matrix of nonnegative integers that sum up to n, and such that A[i,j] >= A[i+1,j], A[i,j] >= A[i,j+1] for all i,j. We can consider A of infinite size but there are at most n nonzero rows and columns and we can ignore empty rows or columns. It is symmetric iff A = transpose(A), or A[i,j] = A[j,i] for all i,j.
For any n, we have the total number of plane partitions of n, A000219(n) = A005987(n) + 2*a(n), where A005987 is the number of symmetric plane partitions. For any of the non-symmetric plane partitions, its transpose is a different plane partition of n. So the difference A000219 - A005987 is always even, equal to twice a(n).
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LINKS
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FORMULA
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EXAMPLE
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The only plane partition of n = 0 is the empty partition []; by convention we do consider it to be symmetric (like a 0 X 0 matrix), so there is no non-symmetric plane partition of 0: a(0) = 0.
The only plane partition of n = 1 is the partition [1] which is symmetric, so there's again no non-symmetric plane partition of 1: a(1) = 0.
For n = 2 we have the partitions [2], [1 1] and [1; 1] (where ; denotes the end of a row). The first one is symmetric, the two others aren't, but are the transpose of each other, so a(2) = 1.
For n = 3 we have the partitions [3], [2 1], [2; 1], [1 1; 1 0], [1 1 1], [1; 1; 1]. The first and the fourth are symmetric, second and third, and fifth and sixth are non-symmetric, and pairwise the transpose of each other, so a(3) = 2.
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PROG
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(PARI) a(n)=#select(t->(t=matconcat(t~))~!=t, PlanePartitions(n))/2 \\ For illustrative purpose: remove "#" to see the list. See A091298 for PlanePartitions(). More efficiently: A306098(n)=(A000219(n)-A005987(n))/2
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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