OFFSET
0,4
COMMENTS
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).
EXAMPLE
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.
PROG
CROSSREFS
KEYWORD
nonn
AUTHOR
M. F. Hasler, Sep 26 2018
STATUS
approved