A306098 Number of equivalence classes, modulo transposition, of non-symmetric plane partitions of n.

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

Offset: 0

Author

M. F. Hasler, Sep 26 2018

Keywords

• nonn

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).

Examples

			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.
		

Crossrefs

Cf. A000219, A005987, A091298.

Programs

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

Formula

a(n) = (A000219(n) - A005987(n))/2.