A091298 -id:A091298 - cp's OEIS frontend

A306096 Number of plane partitions of n where parts are colored in (at most) 6 colors.

1, 6, 78, 726, 7278, 62574, 586878, 4889166, 42892710, 354335982, 2976581670, 23990771094, 197564663094, 1565310230790, 12548473437822, 98526949264374, 776195574339102, 6008457242324814, 46729763436714126, 357901583160822990, 2748384845416097718

Offset: 0

M. F. Hasler, Oct 16 2018

nonn

a(0) = 1 corresponds to the empty sum, in which all terms are colored in one among six given colors, since there is no term at all.

			For n = 1, there is only the partition [1], which can be colored in any of the six colors, whence a(1) = 6.
For n = 2, there are the partitions [2], [1,1] and [1;1]. Adding colors, this yields a(2) = 6 + 36 + 36 = 78 distinct possibilities.

M. F. Hasler, Table of n, a(n) for n = 0..50

Cf. A091298, A208447.

Column 6 of A306100 and A306101. See A306099, A306093, A306094, A306095 for columns 2, 3, 4 and 5.

PARI
```
a(n)=sum(k=1,n,A091298(n,k)*6^k,!n)
```

a(n) = Sum_{k=1..n} A091298(n,k)*6^k, for n > 0.

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.

Original entry on oeis.org

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

Wouter Meeussen, Aug 14 2004

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

			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}}.

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.

Cf. A096272, A091360.

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

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

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)

Edited by M. F. Hasler, Sep 24 2018

A306098 Number of equivalence classes, modulo transposition, of non-symmetric plane partitions of n.

Original entry on oeis.org

0, 0, 1, 2, 5, 10, 21, 39, 74, 133, 239, 415, 719, 1216, 2048, 3393, 5586, 9087, 14695, 23530, 37462, 59172, 92947, 145024, 225123, 347421, 533614, 815378, 1240410, 1878302, 2832586, 4253800, 6363760, 9483831, 14083418, 20839900, 30735490, 45181303, 66210373, 96730731

Offset: 0

M. F. Hasler, Sep 26 2018

nonn

A plane partition of n is a matrix of nonnegative integers that sum up to n, and such that A[i,j] >= A[i+1,j], A[i,j] >= A[i,j+1] for all i,j. We can consider A of infinite size but there are at most n nonzero rows and columns and we can ignore empty rows or columns. It is symmetric iff A = transpose(A), or A[i,j] = A[j,i] for all i,j.

For any n, we have the total number of plane partitions of n, A000219(n) = A005987(n) + 2*a(n), where A005987 is the number of symmetric plane partitions. For any of the non-symmetric plane partitions, its transpose is a different plane partition of n. So the difference A000219 - A005987 is always even, equal to twice a(n).

			The only plane partition of n = 0 is the empty partition []; by convention we do consider it to be symmetric (like a 0 X 0 matrix), so there is no non-symmetric plane partition of 0: a(0) = 0.
The only plane partition of n = 1 is the partition [1] which is symmetric, so there's again no non-symmetric plane partition of 1: a(1) = 0.
For n = 2 we have the partitions [2], [1 1] and [1; 1] (where ; denotes the end of a row). The first one is symmetric, the two others aren't, but are the transpose of each other, so a(2) = 1.
For n = 3 we have the partitions [3], [2 1], [2; 1], [1 1; 1 0], [1 1 1], [1; 1; 1]. The first and the fourth are symmetric, second and third, and fifth and sixth are non-symmetric, and pairwise the transpose of each other, so a(3) = 2.

Cf. A000219, A005987, A091298.

a(n)=#select(t->(t=matconcat(t~))~!=t,PlanePartitions(n))/2 \\ For illustrative purpose: remove "#" to see the list. See A091298 for PlanePartitions(). More efficiently: A306098(n)=(A000219(n)-A005987(n))/2

a(n) = (A000219(n) - A005987(n))/2.

A319648 Total number of parts in all plane partitions of n.

Original entry on oeis.org

0, 1, 5, 14, 38, 85, 196, 401, 830, 1615, 3119, 5802, 10718, 19246, 34276, 59889, 103656, 176801, 299025, 499732, 828638, 1360696, 2218128, 3586194, 5759839, 9184715, 14557974, 22929745, 35916469, 55942850, 86695329, 133671740, 205144324, 313380895, 476667370

Offset: 0

Alois P. Heinz, Sep 25 2018

nonn

			The plane partitions of 2 are [2], [1 1] and [1; 1]. There is a total of a(2) = 5 parts. - _M. F. Hasler_, Sep 27 2018

Alois P. Heinz, Table of n, a(n) for n = 0..50

Row sums of A092288.

Cf. A000219.

A319648(n)={vecsum(apply(pp->vecsum(apply(p->#p,pp)),PlanePartitions(n)))} \\ See A091298 for PlanePartitions(). For illustration mainly, becomes slow for n > 15. - M. F. Hasler, Sep 27 2018

M319648=[]; A319648(n,L=0,s)={if(L, n>1||return([1,1]); #L>2||(s=setsearch(M319648,[[n,L],[]],1))>#M319648|| M319648[s][1]!=[n,L]|| return(M319648[s][2]); my(S=[1,n]); for(m=2,n, forpart(P=m, vecmin(L-Vecrev(P,#L))<0&&next; S+=if(mA319648(n-m,Vecrev(P))*[1,#P;0,1],[1,#P]),L[1],#L)); #L>2|| M319648=setunion(M319648,[[[n,L],S]]); S, my(S=n); n>1&& forpart(P=n,S+=#P); for(m=2,n-1,forpart(P=m,S+=A319648(n-m,Vecrev(P))*[#P,1]~));S)} \\ M. F. Hasler, Sep 30 2018

a(n) = Sum_{k=1..n} k*A091298(n,k). - M. F. Hasler, Sep 27 2018

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A306096 Number of plane partitions of n where parts are colored in (at most) 6 colors.

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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.

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A306098 Number of equivalence classes, modulo transposition, of non-symmetric plane partitions of n.

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A319648 Total number of parts in all plane partitions of n.

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