A096597 Triangle read by rows: T[n,m] = number of plane partitions of n whose 3-dimensional Ferrers plot just fits inside an m X m X m box, i.e., with Max[parts, rows, columns] = m.

1, 0, 3, 0, 3, 3, 0, 4, 6, 3, 0, 3, 12, 6, 3, 0, 3, 21, 15, 6, 3, 0, 1, 31, 30, 15, 6, 3, 0, 1, 42, 60, 33, 15, 6, 3, 0, 0, 54, 102, 69, 33, 15, 6, 3, 0, 0, 64, 175, 132, 72, 33, 15, 6, 3, 0, 0, 73, 270, 246, 141, 72, 33, 15, 6, 3, 0, 0, 81, 417, 432, 276, 144, 72, 33, 15, 6, 3, 0, 0, 83

Offset: 1

Author

Wouter Meeussen, Aug 14 2004

Keywords

• nonn
• tabl

Comments

Row sums equal A000219 (plane partitions).

Conjecture: the last (floor(n/2)) terms of each row read backwards are 3*A091360 (partial sums of A000219).

Björner & Stanley (2010) give in eq.(3.7) MacMahon's generating function pp(r,s,t) for the number of plane partitions with rows <= r, columns <= s, parts <= t. For r = s = t = m, it simplifies to the g.f. f(m) given in formula. A g.f. for column m of this table is then f(m) - f(m-1). - M. F. Hasler, Sep 26 2018

Examples

			The table starts:
  n : T[n,1..n]
  1 : [1]
  2 : [0, 3]
  3 : [0, 3,  3]
  4 : [0, 4,  6,   3]
  5 : [0, 3, 12,   6,  3]
  6 : [0, 3, 21,  15,  6,  3]
  7 : [0, 1, 31,  30, 15,  6,  3]
  8 : [0, 1, 42,  60, 33, 15,  6, 3]
  9 : [0, 0, 54, 102, 69, 33, 15, 6, 3]
etc.
T[5,2] = 3 counts the plane partitions {{2,1},{2}}, {{2,1},{1,1}} and {{2,2},{1}}.
		

Links

• George E. Andrews, On a Partition Function of Richard Stanley, The Electronic Journal of Combinatorics, Volume 11, Issue 2 (2004-6) (The Stanley Festschrift volume), #R1.
• A. Björner and R. P. Stanley, with A combinatorial miscellany, L'Enseignement Math., Monograph No. 42, 2010.
• M. F. Hasler, A096597, rows 1..50, Sep 26 2018.

Crossrefs

Cf. A096272, A091360.

Programs

Mathematica

(* see A089924 for "planepartitions[]" *) Table[Rest@CoefficientList[Plus@@(x ^ Max[Flatten[ # ], Length[ # ], Max[Length/@# ]]&/@ planepartitions[n]), x], {n, 19}]

PARI

A096597_row(n,c=vector(n))={for(i=1,#n=PlanePartitions(n),c[vecmax([#n[i], #n[i][1], n[i][1][1]])]++);c} \\ See A091298 for PlanePartitions().
{A096597(n,m,x=(O('x^n)+1)*'x,f(r)=prod(k=1,2*r-1,((1-x^(k+r))/(1-x^k))^min(k,2*r-k)))=polcoeff(f(m)-f(m-1),n)} \\ Replace "polcoeff(...,n)" by "Vec(...)" to get the whole column m up to row n (for "Vec(...,-n)", padded with leading 0's). - M. F. Hasler, Sep 26 2018

Formula

k-th column is CoefficientList[Series[qMacMahon[k]-qMacMahon[k-1], {q, 0, 3^k}], q] with qMacMahon[n_Integer]:=Product[qan[i+j+k-1]/qan[i+j+k-2], {i, n}, {j, n}, {k, n}] and qan[n_]:=(q^n-1)/(q-1). - Wouter Meeussen, Aug 28 2004

From M. F. Hasler, Sep 26 2018: (Start)

G.f. of column m: f(m)-f(m-1), where f(m) = Product_{k=1..2*m-1} ((1-X^(k+m))/(1-X^k))^min(k,2*m-k).

From the definition, we have T[n,m] = 0 if n > m^3.

Columns and reversed rows converge to 3*A091360: T[m+k,m] = T[2m,2m-k] = 3*A091360(k) for 0 <= k < m-1. (End)

Extensions

Edited by M. F. Hasler, Sep 24 2018