A090984 -id:A090984 - 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

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.

A098529 Triangle read by rows: T(n,k) counts plane partitions of n+1 that can be 'shrunk' in k ways to a plane partition of n by removing 1 element from it. Equivalently, it counts how many partitions of n+1 have k different partitions of n it just covers.

Original entry on oeis.org

1, 3, 3, 3, 6, 6, 1, 3, 18, 3, 9, 24, 15, 3, 42, 38, 3, 10, 60, 69, 21, 6, 72, 153, 45, 6, 9, 114, 220, 141, 15, 1, 3, 120, 399, 274, 60, 3, 18, 159, 558, 570, 162, 12, 3, 174, 834, 1029, 399, 46, 9, 267, 1080, 1749, 921, 138, 3

Offset: 0

Wouter Meeussen, Sep 12 2004

Sequence starts 1; 3; 3,3; 6,6,1; 3,18,3; 9,24,15; 3,42,38,3; Row sums are A000219= the plane partitions of n+1 apart from offset. Sum(all k, k * T(n,k) ) = A090984(n) by definition. First column is A007425. Row lengths are A120565. - Franklin T. Adams-Watters, Jun 14 2006

			T(4,1)=2 because the only plane partitions of 4+1=5 that can be shrunk in only 1 way to plane partitions of 4 are {{5}} and {{1},{1},{1},{1},{1}}, producing {{4}} and {{1},{1},{1},{1}} respectively.
T(4,1)=3 because the only plane partitions of 4+1=5 that can be shrunk in only 1 way to plane partitions of 4 are {{5}},{{1,1,1,1,1}} and {{1},{1},{1},{1},{1}}, producing {{4}},{{1,1,1,1}} and {{1},{1},{1},{1}} respectively.

Cf. A000219, A090984, A007425, A120565.

(* functions 'planepartitions' and 'coversplaneQ', see A096574 *) Table[Frequencies[Count[planepartitions[n], q_/; coversplaneQ[ #, q]]&/@ planepartitions[n+1]], {n, 1, 12}]

Corrected and extended by Franklin T. Adams-Watters, Jun 14 2006

More terms from Wouter Meeussen, May 05 2007

A097994 T(n,k) counts plane partitions of n that can be 'extended' in (k+2) ways to a plane partition of n+1 by adding 1 element to it. Equivalently, it counts how many partitions of n have (k+2) different partitions of n+1 just covering it.

Original entry on oeis.org

1, 3, 0, 3, 3, 0, 6, 6, 0, 1, 3, 15, 3, 3, 0, 9, 21, 6, 12, 0, 0, 3, 34, 21, 25, 3, 0, 0, 10, 45, 36, 54, 15, 0, 0, 0, 6, 54, 72, 108, 36, 6, 0, 0, 0, 9, 84, 102, 172, 117, 15, 0, 1, 0, 0, 3, 84, 174, 306, 228, 54, 7, 3, 0, 0, 0, 18, 114, 225, 483, 447, 162, 18, 12, 0, 0, 0, 0, 3, 114

Offset: 1

Wouter Meeussen, Sep 07 2004

The first column starts a t k=3 since all plane partitions can be extended in at least 3 ways. Row sums are A000219 by definition. Sum T(n,k) (k+2) =A090984.

			T(4,4)=1 because {{2,1},{1}} is the only plane partition of 4 that can be extended in 4+2 = 6 ways to a plane partition of 5.

Cf. A000219, A090984.

(* functions 'planepartitions' and 'coversplaneQ', see A096574 *) Table[Frequencies[Count[planepartitions[n+1], q_/; coversplaneQ[q, # ]]&/@ planepartitions[n]], {n, 1, 16}]

A112669 Triangle read by rows: T(n,k) = number of plane partitions of n that can be extended in k ways to a plane partition of n+1 by adding 1 element to it.

Original entry on oeis.org

1, 3, 3, 3, 6, 6, 0, 1, 3, 15, 3, 3, 9, 21, 6, 12, 3, 34, 21, 25, 3, 10, 45, 36, 54, 15, 6, 54, 72, 108, 36, 6, 9, 84, 102, 172, 117, 15, 0, 1, 3, 84, 174, 306, 228, 54, 7, 3, 18, 114, 225, 483, 447, 162, 18, 12, 3, 114, 348, 724, 824, 369, 66, 37, 9, 171, 453

Offset: 1

Wouter Meeussen, Sep 07 2004

In other words, it shows how many partitions of n have k different partitions of n+1 just covering it.

			As an irregular triangle:
1
3
3 3
6 6 0 1
3 15 3 3
9 21 6 12
3 34 21 25 3
10 45 36 54 15
6 54 72 108 36 6
As a table:
k:=1 k:=2 k:=3 k:=4 k:=5 k:=6 k:=7 k:=8 k:=9 k:=10 k:=11 k:=12
n:=1 0 0 1 0 0 0 0 0 0 0 0 0
n:=2 0 0 3 0 0 0 0 0 0 0 0 0
n:=3 0 0 3 3 0 0 0 0 0 0 0 0
n:=4 0 0 6 6 0 1 0 0 0 0 0 0
n:=5 0 0 3 15 3 3 0 0 0 0 0 0
n:=6 0 0 9 21 6 12 0 0 0 0 0 0
n:=7 0 0 3 34 21 25 3 0 0 0 0 0
n:=8 0 0 10 45 36 54 15 0 0 0 0 0
n:=9 0 0 6 54 72 108 36 6 0 0 0 0

Row sums are A000219; the weighted products (dot product with the k's) is A090984.

A244252 Total number of incoming edges at depth n in the solid partitions graph.

Original entry on oeis.org

1, 4, 16, 46, 128, 332, 842, 2042, 4846, 11146, 25114, 55310, 119662, 254354, 532784, 1100411, 2245118, 4528212, 9038898, 17868025, 35006932, 68008606, 131083778, 250774482, 476372848, 898837825, 1685107392, 3139812791, 5816015908, 10712596279, 19625001436, 35765137033, 64853219808, 117031972499, 210211082354, 375886565558, 669232663688, 1186538314110, 2095236499224, 3685445929502

Offset: 1

Suresh Govindarajan, Jun 23 2014

The solid partition graph is constructed as a directed graph whose vertices are solid partitions. The root vertex of the graph is the unique solid partition with one node. Given a solid partition, draw on outward directed edge to all solid partitions that can be obtained by the addition of a single node to the solid partition. The depth of a given vertex is given by the number of its nodes.

			a(2) = 4 as all four solid partitions of 2 are connected to the root vertex.

N. Destainville and S. Govindarajan, Estimating the asymptotics of solid partitions, arXiv:1406.5605 [cond-mat.stat-mech], 2014.

Cf. A000293, A090984, A000070.

cp's OEIS Frontend

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

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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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A098529 Triangle read by rows: T(n,k) counts plane partitions of n+1 that can be 'shrunk' in k ways to a plane partition of n by removing 1 element from it. Equivalently, it counts how many partitions of n+1 have k different partitions of n it just covers.

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A097994 T(n,k) counts plane partitions of n that can be 'extended' in (k+2) ways to a plane partition of n+1 by adding 1 element to it. Equivalently, it counts how many partitions of n have (k+2) different partitions of n+1 just covering it.

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A112669 Triangle read by rows: T(n,k) = number of plane partitions of n that can be extended in k ways to a plane partition of n+1 by adding 1 element to it.

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A244252 Total number of incoming edges at depth n in the solid partitions graph.

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