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A096597
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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.
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2
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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
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OFFSET
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1,3
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COMMENTS
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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
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LINKS
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FORMULA
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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
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)
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EXAMPLE
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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}}.
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MATHEMATICA
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(* see A089924 for "planepartitions[]" *) Table[Rest@CoefficientList[Plus@@(x ^ Max[Flatten[ # ], Length[ # ], Max[Length/@# ]]&/@ planepartitions[n]), x], {n, 19}]
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PROG
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(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
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CROSSREFS
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KEYWORD
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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