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

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, 18, 33, 32, 31, 15, 22, 1, 8, 25, 43, 56, 54, 43, 22, 30, 1, 10, 29, 66, 81, 99, 78, 64, 30, 42, 1, 10, 37, 83, 126, 150, 148, 118, 88, 42, 56, 1, 12, 44, 114, 174, 246, 235, 230, 166, 124, 56, 77

Offset: 1

Author

Wouter Meeussen, Feb 24 2004

Keywords

• nonn
• tabl

Comments

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.

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

Examples

			This plane partition of n=7: {{3,1,1},{2}} contains 4 parts: 3,1,1,2.
Triangle T(n,k) begins:
  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, 18, 33, 32, 31, 15, 22;
  1,  8, 25, 43, 56, 54, 43, 22, 30;
  1, 10, 29, 66, 81, 99, 78, 64, 30, 42;
  ...
		

Links

• Alois P. Heinz, Rows n = 1..50
• A. Rovenchak, Enumeration of plane partitions with a restricted number of parts, arXiv preprint arXiv:1401.4367 [math-ph], 2014.
• E. W. Weisstein, Plane partition.
• Wikipedia, Plane partition.

Crossrefs

Row sums give A000219.

Cf. A090539, A092288, A089924.

Column 1 is A000012. Column 2 is A052928. Diagonal and subdiagonal are A000041.

Programs

Mathematica

(* see A089924 for "planepartition" *) Table[Length /@ Split[Sort[Length /@ Flatten /@ planepartitions[n]]], {n, 16}]

PARI

A091298(n,k)=sum(i=1,#n=PlanePartitions(n),sum(j=1,#n[i],#n[i][j])==k)
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(NM. F. Hasler, Sep 24 2018