A096272 -id:A096272 - cp's OEIS frontend

A096771 Triangle read by rows: T(n,m) is the number of partitions of n that (just) fit inside an m X m box, but not in an (m-1) X (m-1) box. Partitions of n with Max(max part, length) = m.

1, 0, 2, 0, 1, 2, 0, 1, 2, 2, 0, 0, 3, 2, 2, 0, 0, 3, 4, 2, 2, 0, 0, 2, 5, 4, 2, 2, 0, 0, 1, 7, 6, 4, 2, 2, 0, 0, 1, 6, 9, 6, 4, 2, 2, 0, 0, 0, 7, 11, 10, 6, 4, 2, 2, 0, 0, 0, 5, 14, 13, 10, 6, 4, 2, 2, 0, 0, 0, 5, 15, 19, 14, 10, 6, 4, 2, 2, 0, 0, 0, 3, 17, 22, 21, 14, 10, 6, 4, 2, 2, 0, 0, 0, 2, 17, 29

Offset: 1

Wouter Meeussen, Aug 21 2004

Row sums are A000041. Columns are finite and sum to A051924. The final floor(n/2) terms of each row are the reverse of the initial terms of 2*A000041.

			T(5,3)=3, counting 32, 311 and 221.
From _Gus Wiseman_, Apr 12 2019: (Start)
Triangle begins:
  1
  0  2
  0  1  2
  0  1  2  2
  0  0  3  2  2
  0  0  3  4  2  2
  0  0  2  5  4  2  2
  0  0  1  7  6  4  2  2
  0  0  1  6  9  6  4  2  2
  0  0  0  7 11 10  6  4  2  2
  0  0  0  5 14 13 10  6  4  2  2
  0  0  0  5 15 19 14 10  6  4  2  2
  0  0  0  3 17 22 21 14 10  6  4  2  2
  0  0  0  2 17 29 27 22 14 10  6  4  2  2
  0  0  0  1 17 33 36 29 22 14 10  6  4  2  2
  0  0  0  1 15 39 45 41 30 22 14 10  6  4  2  2
  0  0  0  0 14 41 57 52 43 30 22 14 10  6  4  2  2
  0  0  0  0 11 47 67 69 57 44 30 22 14 10  6  4  2  2
  0  0  0  0  9 46 81 85 76 59 44 30 22 14 10  6  4  2  2
(End)

Andrew Howroyd, Table of n, a(n) for n = 1..1275 (rows 1..50)
Findstat, FindStat - St000784: The maximum of the length and the largest part of the integer partition

A version with reflected rows is A338621.
Cf. A051924, A096272, A096597, A115994, A252464, A263297, A325193, A368985.
Related triangles are A115720, A325188, A325189, A325192, A325200, with Heinz-encoded versions A257990, A325169, A065770, A325178, A325195.

Table[Count[Partitions[n], q_List /; Max[Length[q], Max[q]]===k], {n, 16}, {k, n}]

row(n)={my(r=vector(n)); forpart(p=n, r[max(#p,p[#p])]++); r} \\ Andrew Howroyd, Jan 12 2024

Sum_{k>=1} k*T(n,k) = A368985(n). - Andrew Howroyd, Jan 12 2024

A096573 Number of fixed points of mirroring operation on solid partitions.

Original entry on oeis.org

1, 2, 4, 8, 13, 24, 39, 68, 110, 182, 288, 468, 728, 1150, 1770, 2751, 4175, 6388, 9597, 14495, 21571, 32200, 47498

Offset: 1

Wouter Meeussen, Jun 27 2004

Uses function "solidformBTK" from link below.

			Solid partition [{{3, 1, 1, 1}, {3}}, {{2, 1}}, {{1}}, {{1}}, {{1}}] mirrors into [{{3, 3}, {1}, {1}, {1}}, {{2}, {1}}, {{1}}, {{1}}, {{1}}] by mirroring each layer as a plane partition.

Wouter Meeussen, Mma functions for plane and solid partitions
Luigi Togliani, Forms of Crossed and Simple Polygons, Science & Philosophy (2019) Vol. 7, No. 2, 71-82.

Cf. A000293, A094504, A094508, A096272, A096574, A096575, A096576, A096577, A096578, A096579, A096580, A096581, A119266.

Tr/@Table[Count[solidformBTK[par],arg_z/;flip[arg]==arg],{n,20},{par,IntegerPartitions[n]}] (* Wouter Meeussen, Feb 05 2025  *)

a(16)-a(23) from Wouter Meeussen, Feb 05 2025

A096574 Number of asymmetric solid partitions under mirroring operation.

Original entry on oeis.org

0, 1, 3, 9, 23, 58, 134, 308, 677, 1470, 3106, 6479, 13260, 26827, 53516

Offset: 1

Wouter Meeussen, Jun 27 2004

Uses function "solidformBTK" from link above.

			Solid partition [{{3, 1, 1, 1}, {3}}, {{2, 1}}, {{1}}, {{1}}, {{1}}] mirrors into [{{3, 3}, {1}, {1}, {1}}, {{2}, {1}}, {{1}}, {{1}}, {{1}}] by mirroring each layer as a plane partition.

Wouter Meeussen, Mma functions for plane and solid partitions

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

a(n) = (A000293(n) - A096573(n))/2.

A096575 Number of fixed points of solid partitions under rotation operation.

Original entry on oeis.org

1, 1, 1, 2, 2, 2, 4, 6, 6, 8, 11, 13, 17, 24, 28, 36, 47, 56, 69, 94, 114, 138, 177, 218, 262

Offset: 1

Wouter Meeussen, Jun 27 2004

Rotation has permutation cycle length 1 or 3. Uses function "solidformBTK" from link below.
Is this the same sequence as A002722? - R. J. Mathar, Sep 04 2008 [This still seems to be true even after 20 terms. - N. J. A. Sloane, Feb 05 2025]
Rotation of each of the plane partitions in a solid partition appears to lead to the same count of fixed points as rotating the 3D-partition as a whole. - Wouter Meeussen, Feb 05 2025

			Solid partition [{{3, 1, 1, 1}, {3}}, {{2, 1}}, {{1}}, {{1}}, {{1}}] rotates into [{{4, 1}, {1, 1}, {1, 1}}, {{2}, {1}}, {{1}}, {{1}}, {{1}}] by rotating each layer as a plane partition.

Wouter Meeussen, Mma functions for plane and solid partitions

Cf. A000293, A094504, A094508, A096272, A096573, A096574, A096576, A096577, A096578, A096579, A096580, A096581, A119266.

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

a(16)-a(23) from Wouter Meeussen, Feb 05 2025

a(24)-a(25) from Wouter Meeussen, Jul 27 2025

A096577 Number of fixed points of solid partitions under 'time-lapse' operation.

Original entry on oeis.org

1, 0, 0, 0, 1, 0, 1, 0, 2, 0, 2, 0, 2, 0, 4, 1, 5, 0, 7, 1, 7, 0, 14

Offset: 1

Wouter Meeussen, Jun 27 2004

Operation 'time lapse', or 'lapse', L, operates on a solid partition by creating a new one, layer by layer. Layer k is defined by its 3-dimensional-Ferrers plot, equal to the (existence of) elements of the solid partition with value >= k. As if taking a time-lapse picture of the solid partition, filtering out elements less than k and projecting the resulting structure (filled with ones) to the base plane. Given there are three planes to project into, together with the starting solid partition, that makes four 'isomers'.

			Solid partition [{{3,1,1,1},{3}},{{2,1}},{{1}},{{1}},{{1}}] lapses (L) into
[{{4,1},{2},{1},{1},{1}},{{1,1},{1}},{{1,1}}], then into
[{{2,1,1,1,1},{2,1},{2}},{{1,1}},{{1}},{{1}}], further into
[{{5,2,1},{2},{1},{1}},{{1,1,1}}] and returns after L^4 to
[{{3,1,1,1},{3}},{{2,1}},{{1}},{{1}},{{1}}].

Wouter Meeussen, Mma functions for plane and solid partitions

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

(* See link above. *)
Tr/@Table[Count[solidformBTK[par],arg_z/;lapse[arg]==arg],{n,20},{par,IntegerPartitions[n]}] (* Wouter Meeussen, Feb 05 2025 *)

a(16)-a(23) from Wouter Meeussen, Mar 19 2025

A096576 Number of solid partitions asymmetric under rotation operation.

Original entry on oeis.org

0, 1, 3, 8, 19, 46, 101, 226, 486, 1038, 2163, 4471, 9077, 18260, 36258

Offset: 1

Wouter Meeussen, Jun 27 2004

Rotation has permutation cycle length 1 or 3. Uses function "solidformBTK" from link above.

			Solid partition [{{3, 1, 1, 1}, {3}}, {{2, 1}}, {{1}}, {{1}}, {{1}}] rotates into [{{4, 1}, {1, 1}, {1, 1}}, {{2}, {1}}, {{1}}, {{1}}, {{1}}] by rotating each layer as a plane partition.

Wouter Meeussen, Mma functions for plane and solid partitions

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

Mathematica
```
(* See A096575. *)
```

a(n) = (A000293(n) - A096575(n))/3.

A096578 Number of solid partitions with period (cycle length) two under 'time-lapse' operation.

Original entry on oeis.org

0, 0, 1, 1, 1, 2, 3, 6, 7, 11, 15, 25, 33, 48, 65

Offset: 1

Wouter Meeussen, Jun 27 2004

Operation 'time lapse', or 'lapse', L, operates on a solid partition by creating a new one, layer by layer. Layer k is defined by its 3-dimensional-Ferrers plot, equal to the (existence of) elements of the solid partition with value >= k. As if taking a time-lapse picture of the solid partition, filtering out elements less than k and projecting the resulting structure (filled with ones) to the base plane. Given there are three plane to project into, together with the starting solid partition, that makezs four 'isomers'.

			Solid partition [{{3,1,1,1},{3}},{{2,1}},{{1}},{{1}},{{1}}] lapses (L) into
[{{4,1},{2},{1},{1},{1}},{{1,1},{1}},{{1,1}}], then into
[{{2,1,1,1,1},{2,1},{2}},{{1,1}},{{1}},{{1}}], further into
[{{5,2,1},{2},{1},{1}},{{1,1,1}}] and returns after L^4 to
[{{3,1,1,1},{3}},{{2,1}},{{1}},{{1}},{{1}}]

Wouter Meeussen, Mma functions for plane and solid partitions

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

Mathematica
```
(* See link above. *)
```

A096581 Number of solid partitions non-symmetric under L^2 (L= 'time-lapse' symmetry operation) on solid partitions.

Original entry on oeis.org

0, 2, 4, 12, 28, 68, 150, 336, 724, 1550, 3234, 6688, 13590, 27354, 54334

Offset: 1

Wouter Meeussen, Jun 27 2004

			Solid partition [{{3,1,1,1},{3}},{{2,1}},{{1}},{{1}},{{1}}] lapses (L) into
[{{4,1},{2},{1},{1},{1}},{{1,1},{1}},{{1,1}}], then into
[{{2,1,1,1,1},{2,1},{2}},{{1,1}},{{1}},{{1}}], further into
[{{5,2,1},{2},{1},{1}},{{1,1,1}}] and returns after L^4 to
[{{3,1,1,1},{3}},{{2,1}},{{1}},{{1}},{{1}}]

Wouter Meeussen, Mma functions for plane and solid partitions

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

Mathematica
```
(* See link above. *)
```

By definition, A000293(n) = A096580(n) + 2*a(n).

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

Original entry on oeis.org

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

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

cp's OEIS Frontend

A096771 Triangle read by rows: T(n,m) is the number of partitions of n that (just) fit inside an m X m box, but not in an (m-1) X (m-1) box. Partitions of n with Max(max part, length) = m.

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A096573 Number of fixed points of mirroring operation on solid partitions.

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A096574 Number of asymmetric solid partitions under mirroring operation.

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A096575 Number of fixed points of solid partitions under rotation operation.

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A096577 Number of fixed points of solid partitions under 'time-lapse' operation.

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A096576 Number of solid partitions asymmetric under rotation operation.

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A096578 Number of solid partitions with period (cycle length) two under 'time-lapse' operation.

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A096581 Number of solid partitions non-symmetric under L^2 (L= 'time-lapse' symmetry operation) on solid partitions.

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