A089924 -id:A089924 - cp's OEIS frontend

A094504 T(n,m) equals number of solid partitions of n containing m plane partitions.

1, 3, 1, 6, 3, 1, 13, 9, 3, 1, 24, 22, 9, 3, 1, 48, 54, 25, 9, 3, 1, 86, 120, 63, 25, 9, 3, 1, 160, 267, 153, 66, 25, 9, 3, 1, 282, 559, 357, 162, 66, 25, 9, 3, 1, 500, 1158, 805, 390, 165, 66, 25, 9, 3, 1, 859, 2314, 1761, 898, 399, 165, 66, 25, 9, 3, 1, 1479, 4559, 3761, 2025, 931, 402, 165, 66, 25, 9, 3, 1

Offset: 1

Wouter Meeussen, Jun 05 2004

First column equals the number of plane partitions of n, corresponding to the 'single layer' solid partitions.

Rows read backward tend to limiting sequence 1, 3, 9, 25, 66, 165, 402, ... A096322.

			T(5,3) = 9 since these 9 solid partitions are [{{3}},{{1}},{{1}}], [{{2,1}},{{1}},{{1}}], [{{1,1,1}},{{1}},{{1}}], [{{2},{1}},{{1}},{{1}}], [{{1,1},{1}},{{1}},{{1}}], [{{1},{1},{1}},{{1}},{{1}}], [{{2}},{{2}},{{1}}], [{{1,1}},{{1,1}},{{1}}], [{{1},{1}},{{1},{1}},{{1}}].
Triangle begins:
   1;
   3,  1;
   6,  3,  1;
  13,  9,  3, 1;
  24, 22,  9, 3, 1;
  48, 54, 25, 9, 3, 1;
  ...

Wouter Meeussen, Table of n, a(n) for n = 1..136 (16 rows)
Wouter Meeussen, Mma functions for plane and solid partitions
Wouter Meeussen, Mma functions for plane and solid partitions

Cf. A000293, A090984, A089924, A096322.

(* uses "Mma functions for plane and solid partitions" also used in A090984, A089924 *)
 Table[Length/@Split[Sort[Length/@Flatten[solidformBTK/@Partitions[n]]]], {n, 16}]

Finding a G.f. for the solid partitions is an open problem.

Renewed linked Mma program file.Wouter Meeussen, Feb 20 2025

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

Original entry on oeis.org

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

Wouter Meeussen, Feb 24 2004

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

			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;
  ...

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.

Row sums give A000219.
Cf. A090539, A092288, A089924.
Column 1 is A000012. Column 2 is A052928. Diagonal and subdiagonal are A000041.

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

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

A094508 Triangle read by rows: T[n,m] = number of solid partitions of n with trace m, where the trace of a solid partitions is defined as the sum of the traces of the constituent plane partitions.

Original entry on oeis.org

1, 2, 2, 3, 4, 3, 4, 11, 6, 5, 5, 18, 19, 10, 7, 6, 33, 42, 34, 14, 11, 7, 48, 85, 80, 50, 22, 15, 8, 74, 156, 186, 128, 80, 30, 22, 9, 100, 275, 368, 318, 208, 112, 44, 30, 10, 140, 446, 725, 696, 534, 304, 165, 60, 42, 11, 180, 705, 1300, 1464, 1214, 808, 450, 228, 84, 56

Offset: 1

Wouter Meeussen, Jun 05 2004

Last column equals the partition numbers, corresponding to the 'single column' solid partitions.

			Table starts {1}, {2,2},{3,4,3},{4,11,6,5},..
T[4,3]=6 since these 6 solid partitions with trace 3 are:
[{{3,1}}], [{{3},{1}}], [{{2,1}},{{1}}], [{{2},{1}},{{1}}], [{{1,1}},{{1}},{{1}}], [{{1},{1}},{{1}},{{1}}]

Cf. A000293, A090984, A089924.

uses functions defined in A090984, A089924. solidform[q_?PartitionQ]:=Module[{}, Select[Flatten[Outer[z, Sequence@@(planepartitions/@q), 1]], And@@Apply[coversplaneQ, Partition[ #/.z->List, 2, 1], {1}]&]];tomatrix[par_]:=Block[{l=Max[Length/@ par]}, Map[PadRight[ #, l]&, par]]; Table[Length/@Split[Sort[Plus@@@Map[Tr[tomatrix[ # ]]&, Flatten[solidform/ @Partitions[n]], {2}]]], {n, 12}]

Finding a GF for the solid partitions is an open problem.

A091357 Number of planar partitions of n with exactly 3 rows.

Original entry on oeis.org

1, 2, 5, 11, 22, 42, 78, 138, 239, 405, 669, 1088, 1741, 2744, 4267, 6564, 9975, 15019, 22394, 33111, 48549, 70678, 102127, 146636, 209186, 296697, 418401, 586985, 819218, 1137962, 1573336, 2165888, 2968914, 4053563, 5512820, 7469989

Offset: 3

Christian G. Bower, Jan 02 2004

nonn

Column 3 of A091355.

Cf. A000219, A089924, A091356-A091360.

a(n) = A000991(n)-A000990(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.

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

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A094504 T(n,m) equals number of solid partitions of n containing m plane partitions.

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

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A094508 Triangle read by rows: T[n,m] = number of solid partitions of n with trace m, where the trace of a solid partitions is defined as the sum of the traces of the constituent plane partitions.

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A091357 Number of planar partitions of n with exactly 3 rows.

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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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