A000293 -id:A000293 - cp's OEIS frontend

A096322 Limiting sequence formed by rows of A094504 read backwards: rightmost floor(n/2)+1 terms of row n in table A094504.

1, 3, 9, 25, 66, 165, 402, 943, 2163, 4835, 10598, 22785, 48215, 100470, 206620, 419662, 842928, 1675487, 3298688, 6436210, 12453352, 23905923, 45550529, 86180937, 161964145, 302447657

Offset: 1

Wouter Meeussen, Jun 27 2004

Same sequence, multiplied by four, occurs in A096272.

a(n) is the number of solid partitions with layer structure an integer partition of (2n-2) in exactly (n-1) parts. - Wouter Meeussen, Mar 12 2025

			For n=3 the a(3)= 9 solid partitions are generated by the integer partitions of (2n-2) in exactly (n-1) parts with parts =1 and duplicate parts deleted, so just {3} and {2} :
 z[{{3}}], z[{{2,1}}], z[{{1,1,1}}], z[{{2},{1}}], z[{{1,1},{1}}], z[{{1},{1},{1}}] and  z[{{2}}], z[{{1,1}}], z[{{1},{1}}]

Wouter Meeussen, Mma functions for plane and solid partitions

Cf. A000293, A094504, A094508, A096272, A096573, A096574, A096575, A096576, A096577, A096578, A096579, A096580, A096581.

Extended to n=26, Wouter Meeussen, May 23 2025

A379277 Number of solid partitions with multiplicities of parts matching the n-th composition in standard order.

Original entry on oeis.org

1, 3, 3, 6, 9, 6, 9, 13, 21, 24, 33, 13, 21, 24, 33, 24, 48, 57, 84, 51, 93, 90, 135, 24, 48, 57, 84, 51, 93, 90, 135, 48, 102, 144, 213, 138, 258, 252, 387, 111, 228, 282, 426, 219, 417, 408, 633, 48, 102, 144, 213, 138, 258, 252, 387, 111, 228, 282, 426, 219

Offset: 1

John Tyler Rascoe, Dec 19 2024

nonn

			The 5th composition in standard order, (2,1) corresponds to a solid partition with 3 parts (a,b,c) with a = b and a > c. There are 9 ways to arrange these parts into valid a solid partition giving a(5) = 9.

John Tyler Rascoe, Table of n, a(n) for n = 1..1024
John Tyler Rascoe, Python program.

Cf. A000041, A000219, A000293, A066099, A207542.

Python
```
# see links
```

a(2^k) = A000219(k+1).

a(2^k-1) = A207542(k) for k > 0.

A096653 Lower triangular matrix T, read by rows, such that the row sums of T^n form the (3n)-dimensional partition numbers.

Original entry on oeis.org

1, 0, 1, 0, 3, 1, 0, 6, 3, 1, 0, 13, 9, 3, 1, 0, 24, 19, 12, 3, 1, 0, 48, 48, 25, 15, 3, 1, 0, 86, 84, 84, 31, 18, 3, 1, 0, 160, 228, 99, 135, 37, 21, 3, 1, 0, 282, 129, 721, 57, 204, 43, 24, 3, 1, 0, 500, 2521, -2267, 2087, -93, 294, 49, 27, 3, 1, 0, 859, -16291, 29876, -13253, 5229, -417, 408, 55, 30, 3, 1, 0, 1479, 199621, -317919

Offset: 0

Paul D. Hanna, Jul 06 2004

Row sums of T form A000293 (solid partitions); row sums of T^2 form A000416(6-D).

			Triangle T begins:
{1},
{0,1},
{0,3,1},
{0,6,3,1},
{0,13,9,3,1},
{0,24,19,12,3,1},
{0,48,48,25,15,3,1},
{0,86,84,84,31,18,3,1},
{0,160,228,99,135,37,21,3,1},
{0,282,129,721,57,204,43,24,3,1},
{0,500,2521,-2267,2087,-93,294,49,27,3,1},
{0,859,-16291,29876,-13253,5229,-417,408,55,30,3,1},
{0,1479,199621,-317919,165456,-46401,11539,-996,549,61,33,3,1},
{0,2485,-2547804,4150781,-2100853,627628,-126896,23006,-1926,720,67,36,3,1},...
Row sums are: {1,1,4,10,26,59,140,307,684,1464,3122,6500,...} (A000293).
T^2 begins:
{1},
{0,1},
{0,6,1},
{0,21,6,1},
{0,71,27,6,1},
{0,216,101,33,6,1},
{0,657,363,131,39,6,1},
{0,1907,1185,552,161,45,6,1},
{0,5507,3931,1824,789,191,51,6,1},
{0,15522,11574,7449,2520,1080,221,57,6,1},...
with row sums: {1,1,7,28,105,357,1197,3857,12300,38430,...} (A000416).

Cf. A096651, A096652, A000293, A000416.

Matrix cube of triangle A096651.

A098052 T(n,k) counts the solid partitions of n that can be extended to a solid partition of n+1 in exactly (k+3) ways. Equivalently, the number of solid partitions of n that have exactly k+3 partitions of n+1 majoring them.

Original entry on oeis.org

1, 4, 4, 6, 10, 12, 0, 4, 4, 30, 12, 12, 0, 0, 1, 16, 48, 18, 48, 0, 6, 4, 4, 70, 72, 100, 27, 12, 22, 20, 102, 114, 232, 76, 66, 68, 6, 10, 114, 231, 448, 232, 180, 201, 48, 16, 204, 330, 728, 628, 462, 546, 184, 24

Offset: 4

Wouter Meeussen, Sep 11 2004

Row sums are A000293 (solid partitions) by definition.
First column is conjectured to be A007426 = tau_4(n).
All solid partitions can be extended in at least 4 ways (hence the offset 4).

			T(5,7)=1 because there is only 1 solid partition of 5 [{{2, 1}, {1}}, {{1}}] that can be extended to a solid partition of 6 in exactly (7+3 =10) ways:
  [{{2,1},{2}},{{1}}], [{{2,1},{1,1}},{{1}}], [{{2,2},{1}},{{1}}],
  [{{3,1},{1}},{{1}}], [{{2,1,1},{1}},{{1}}], [{{2,1},{1},{1}},{{1}}],
  [{{2,1},{1}},{{2}}], [{{2,1},{1}},{{1,1}}], [{{2,1},{1}},{{1},{1}}],
  [{{2,1},{1}},{{1}},{{1}}].
Table starts
  1;
  4;
  4,6;
  10,12,0,4;
  4,30,12,12,0,0,1;
  16,48,18,48,0,6,4;
  4,70,72,100,27,12,22;
  20,102,114,232,76,66,68,6;
  ...

Wouter Meeussen, Mma functions for plane and solid partitions

Cf. A000029, A007426, A097994.

(* functions 'solidform' and 'coversplaneQ', see A096574 *)
Table[ Rest@BinCounts[Count[Flatten[solidformBTK/@IntegerPartitions[n+1]],q_/;coverssolidQ[q,#]]&/@Flatten[solidformBTK/@IntegerPartitions[n]]] ,{n,1,8}] (* Wouter Meeussen, Feb 03 2025 *)

A379278 Number of solid partitions of n such that all parts occur with the same multiplicity.

Original entry on oeis.org

1, 1, 4, 10, 20, 31, 97, 105, 228, 466, 657, 953, 2958, 2675, 4884, 11635, 13485, 19136, 58099, 48816, 89138, 219474, 197247, 296097, 1026590, 713425, 1099311, 3386891, 2744274, 3788578, 15225795, 8562311, 13588731, 47251379, 28547765, 43887961, 200572890, 90616026

Offset: 0

John Tyler Rascoe, Dec 19 2024

nonn

			For a(3) = 10 there are 6 arrangements of parts (1,1,1), 3 arrangements of parts (2,1), and 1 arrangement of (3).

John Tyler Rascoe, Python program.

Cf. A000041, A000219, A000293, A207542, A323657.

Python
```
# see links
```

a(27)-a(37) from Bert Dobbelaere, Apr 24 2025

A382247 Number of fixed points of solid partitions under twice the 'time-lapse' operation.

Original entry on oeis.org

1, 0, 2, 2, 3, 4, 7, 12, 16, 22, 32, 50, 68, 96, 134, 195, 261, 364, 497, 701, 941, 1288, 1738

Offset: 1

Wouter Meeussen, Mar 19 2025

Permutes the 4 axes of the 4D-Ferrers plot of the solid partitions as 2143.

			z[{{2},{2}}] -> z[{{1,1}},{{1,1}}] -> z[{{2},{2}}] under the 'lapse' operation.

Wouter Meeussen, Mma functions for plane and solid partitions

Cf. A000293, A094504, A094508, A096272, A096573, A096574, A096575, A096576, A096578, A096579, A096580, A096581, A119266.

Tr/@Table[Count[solidformBTK[par], arg_z/; Nest[lapse,arg,2]===arg], {n, 20}, {par, IntegerPartitions[n]}]

A385373 Number of solid partitions with multiplicities (1, ..., n).

Original entry on oeis.org

1, 1, 6, 138, 14049, 6851919

Offset: 0

John Tyler Rascoe, Jun 27 2025

A solid partition with d distinct parts (p_1^(k_1) > p_2^(k_2) > ... > p_d^(k_d)) has the multiset of multiplicities (k_1, k_2, ..., k_d).

Alternatively, a(n) is the number of chains of plane partitions ordered by inclusion, comprised of n consecutive triangular numbers starting with 1.

			For n = 2 a solid partition having multiplicities (1,2) has two distinct parts (a,b^2) with a < b, and there are 6 ways to arrange these parts.

John Tyler Rascoe, Python program.

Cf. A000217, A000219, A000293, A164894, A379277.

Python
```
# see Links
```

a(n) = A379277(A164894(n)) for n > 0.

A098530 T(n,k) counts solid partitions of n+1 that can be 'shrunk' in k ways to a solid partition of n by removing 1 element from it. Equivalently, it counts how many solid partitions of n+1 have k different solid partitions of n it just covers.

Original entry on oeis.org

4, 4, 6, 10, 12, 4, 4, 42, 12, 1, 16, 60, 60, 4, 4, 105, 164, 34, 20, 162, 316, 180, 6, 10, 202, 672, 484, 96

Offset: 1

Wouter Meeussen, Sep 12 2004

Sequence starts 4; 4,6; 10,12,4; 4,42,12,1; 16,60,60,4; 4,105,164,34; Row sums are A000293= the solid partitions of n+1 apart from offset. First column conjectured to be the (beheaded) A007426.

			T(3,3)=4 because the only solid partitions of 3+1=4 that can be shrunk in exactly 3 ways to plane partitions of 3 are
[{{2,1},{1}}], [{{2,1}},{{1}}], [{{2},{1}},{{1}}] and [{{1,1},{1}},{{1}}].

Cf. A000293, A007426, A098529.

(* functions 'solidform' and 'coverssolidQ', see A098052 *) Table[Frequencies[Count[Flatten[solidform / @ Partitions[n+1]], q_/;coverssolidQ[q, # ]]&/ @ Flatten[solidform / @ Partitions[n]]], {n, 1, 8}]

A210392 10-dimensional partition numbers.

Original entry on oeis.org

1, 11, 66, 341, 1606, 7238, 31548, 134728, 565983, 2350183, 9661465, 39401792, 159527302, 641733862, 2565774277, 10198601886, 40305279454, 158376907546, 618742851276, 2403142436321

Offset: 1

Jonathan Vos Post, Mar 20 2012

nonn

d=10 column of Table 1, p. 30, of Govindarajan. The d=1 column is A000041(n) for n > 0, d=2 column is A000219(n) for n > 0, d=3 column is A000293(n) for n > 0, etc.

Suresh Govindarajan, Notes on higher-dimensional partitions, arXiv:1203.4419v1 [math.CO], Mar 20, 2012.

Cf. A000041, A000219, A000293.

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

A096322 Limiting sequence formed by rows of A094504 read backwards: rightmost floor(n/2)+1 terms of row n in table A094504.

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A379277 Number of solid partitions with multiplicities of parts matching the n-th composition in standard order.

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A096653 Lower triangular matrix T, read by rows, such that the row sums of T^n form the (3n)-dimensional partition numbers.

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Formula

A098052 T(n,k) counts the solid partitions of n that can be extended to a solid partition of n+1 in exactly (k+3) ways. Equivalently, the number of solid partitions of n that have exactly k+3 partitions of n+1 majoring them.

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Mathematica

A379278 Number of solid partitions of n such that all parts occur with the same multiplicity.

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Python

Extensions

A382247 Number of fixed points of solid partitions under twice the 'time-lapse' operation.

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Programs

Mathematica

A385373 Number of solid partitions with multiplicities (1, ..., n).

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Python

Formula

A098530 T(n,k) counts solid partitions of n+1 that can be 'shrunk' in k ways to a solid partition of n by removing 1 element from it. Equivalently, it counts how many solid partitions of n+1 have k different solid partitions of n it just covers.

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Author

Keywords

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Programs

Mathematica

A210392 10-dimensional partition numbers.

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