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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.

%I A096597 #30 Oct 28 2021 06:30:27
%S A096597 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,
%T A096597 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,
%U A096597 73,270,246,141,72,33,15,6,3,0,0,81,417,432,276,144,72,33,15,6,3,0,0,83
%N 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.
%C A096597 Row sums equal A000219 (plane partitions).
%C A096597 Conjecture: the last (floor(n/2)) terms of each row read backwards are 3*A091360 (partial sums of A000219).
%C A096597 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
%H A096597 George E. Andrews, <a href="https://doi.org/10.37236/1858">On a Partition Function of Richard Stanley</a>, The Electronic Journal of Combinatorics, Volume 11, Issue 2 (2004-6) (The Stanley Festschrift volume), #R1.
%H A096597 A. Björner and R. P. Stanley, with  <a href="http://www-math.mit.edu/~rstan/papers/comb.pdf">A combinatorial miscellany</a>, L'Enseignement Math., Monograph No. 42, 2010.
%H A096597 M. F. Hasler, <a href="/A096597/a096597_3.txt">A096597, rows 1..50</a>, Sep 26 2018.
%F A096597 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
%F A096597 From _M. F. Hasler_, Sep 26 2018: (Start)
%F A096597 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).
%F A096597 From the definition, we have T[n,m] = 0 if n > m^3.
%F A096597 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)
%e A096597 The table starts:
%e A096597   n : T[n,1..n]
%e A096597   1 : [1]
%e A096597   2 : [0, 3]
%e A096597   3 : [0, 3,  3]
%e A096597   4 : [0, 4,  6,   3]
%e A096597   5 : [0, 3, 12,   6,  3]
%e A096597   6 : [0, 3, 21,  15,  6,  3]
%e A096597   7 : [0, 1, 31,  30, 15,  6,  3]
%e A096597   8 : [0, 1, 42,  60, 33, 15,  6, 3]
%e A096597   9 : [0, 0, 54, 102, 69, 33, 15, 6, 3]
%e A096597 etc.
%e A096597 T[5,2] = 3 counts the plane partitions {{2,1},{2}}, {{2,1},{1,1}} and {{2,2},{1}}.
%t A096597 (* see A089924 for "planepartitions[]" *) Table[Rest@CoefficientList[Plus@@(x ^ Max[Flatten[ # ], Length[ # ], Max[Length/@# ]]&/@ planepartitions[n]), x], {n, 19}]
%o A096597 (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().
%o A096597 {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
%Y A096597 Cf. A096272, A091360.
%K A096597 nonn,tabl
%O A096597 1,3
%A A096597 _Wouter Meeussen_, Aug 14 2004
%E A096597 Edited by _M. F. Hasler_, Sep 24 2018