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A091298 Triangle read by rows: T(n,k) is the number of plane partitions of n containing exactly k parts.

%I A091298 #38 Feb 16 2025 08:32:52
%S A091298 1,1,2,1,2,3,1,4,3,5,1,4,7,5,7,1,6,10,13,7,11,1,6,14,20,19,11,15,1,8,
%T A091298 18,33,32,31,15,22,1,8,25,43,56,54,43,22,30,1,10,29,66,81,99,78,64,30,
%U A091298 42,1,10,37,83,126,150,148,118,88,42,56,1,12,44,114,174,246,235,230,166,124,56,77
%N A091298 Triangle read by rows: T(n,k) is the number of plane partitions of n containing exactly k parts.
%C A091298 First column is 1, representing the single-part {{n}}, last column is P(n), since the all-ones plane partitions form the Ferrers-Young plots of the (linear) partitions of n.
%C A091298 A plane partition of n is a two-dimensional table (or matrix) with nonnegative elements summing up to n, and nonincreasing rows and columns. (Zero rows and columns are ignored.) - _M. F. Hasler_, Sep 22 2018
%H A091298 Alois P. Heinz, <a href="/A091298/b091298.txt">Rows n = 1..50</a>
%H A091298 A. Rovenchak, <a href="http://arxiv.org/abs/1401.4367">Enumeration of plane partitions with a restricted number of parts</a>, arXiv preprint arXiv:1401.4367 [math-ph], 2014.
%H A091298 E. W. Weisstein, <a href="https://mathworld.wolfram.com/PlanePartition.html">Plane partition</a>.
%H A091298 Wikipedia, <a href="https://en.wikipedia.org/wiki/Plane_partition">Plane partition</a>.
%e A091298 This plane partition of n=7: {{3,1,1},{2}} contains 4 parts: 3,1,1,2.
%e A091298 Triangle T(n,k) begins:
%e A091298   1;
%e A091298   1,  2;
%e A091298   1,  2,  3;
%e A091298   1,  4,  3,  5;
%e A091298   1,  4,  7,  5,  7;
%e A091298   1,  6, 10, 13,  7, 11;
%e A091298   1,  6, 14, 20, 19, 11, 15;
%e A091298   1,  8, 18, 33, 32, 31, 15, 22;
%e A091298   1,  8, 25, 43, 56, 54, 43, 22, 30;
%e A091298   1, 10, 29, 66, 81, 99, 78, 64, 30, 42;
%e A091298   ...
%t A091298 (* see A089924 for "planepartition" *) Table[Length /@ Split[Sort[Length /@ Flatten /@ planepartitions[n]]], {n, 16}]
%o A091298 (PARI) A091298(n,k)=sum(i=1,#n=PlanePartitions(n),sum(j=1,#n[i],#n[i][j])==k)
%o A091298 PlanePartitions(n,L=0,PP=List())={ n<2&&return([if(n,[[1]],[])]); for(N=1,n, my(P=apply(Vecrev, if(L, select(p->vecmin(L-Vecrev(p,#L))>=0, partitions(N,L[1],#L)), partitions(N)))); if(N<n, for(i=1, #P, my(pp = PlanePartitions(n-N,P[i])); for(j=1, #pp, listput(PP, concat([P[i]], pp[j])))), for(i=1,#P, listput(PP, [P[i]])))); Set(PP)} \\ _M. F. Hasler_, Sep 24 2018
%Y A091298 Row sums give A000219.
%Y A091298 Cf. A090539, A092288, A089924.
%Y A091298 Column 1 is A000012. Column 2 is A052928. Diagonal and subdiagonal are A000041.
%K A091298 nonn,tabl
%O A091298 1,3
%A A091298 _Wouter Meeussen_, Feb 24 2004