cp's OEIS Frontend

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

%I A306098 #17 Oct 08 2018 17:31:07
%S A306098 0,0,1,2,5,10,21,39,74,133,239,415,719,1216,2048,3393,5586,9087,14695,
%T A306098 23530,37462,59172,92947,145024,225123,347421,533614,815378,1240410,
%U A306098 1878302,2832586,4253800,6363760,9483831,14083418,20839900,30735490,45181303,66210373,96730731
%N A306098 Number of equivalence classes, modulo transposition, of non-symmetric plane partitions of n.
%C A306098 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.
%C A306098 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).
%F A306098 a(n) = (A000219(n) - A005987(n))/2.
%e A306098 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.
%e A306098 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.
%e A306098 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.
%e A306098 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.
%o A306098 (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
%Y A306098 Cf. A000219, A005987, A091298.
%K A306098 nonn
%O A306098 0,4
%A A306098 _M. F. Hasler_, Sep 26 2018