A091298 -id:A091298 - cp's OEIS frontend

A000219 Number of plane partitions (or planar partitions) of n.

1, 1, 3, 6, 13, 24, 48, 86, 160, 282, 500, 859, 1479, 2485, 4167, 6879, 11297, 18334, 29601, 47330, 75278, 118794, 186475, 290783, 451194, 696033, 1068745, 1632658, 2483234, 3759612, 5668963, 8512309, 12733429, 18974973, 28175955, 41691046, 61484961, 90379784, 132441995, 193487501, 281846923

Offset: 0

N. J. A. Sloane

Two-dimensional partitions of n in which no row or column is longer than the one before it (compare A001970). E.g., a(4) = 13:
4.31.3.22.2.211.21..2.1111.111.11.11.1 but not 2
.....1....2.....1...1......1...11.1..1........ 11
....................1.............1..1
.....................................1
In the above, one also must require that rows & columns are nondecreasing, e.g., [1,1; 2] is also forbidden (which implies that row and column lengths are nondecreasing, if empty cells are identified with cells filled with 0's). - M. F. Hasler, Sep 22 2018
Can also be regarded as number of "safe pilings" of cubes in the corner of a room: the height should not increase away from the corner. - Wouter Meeussen
Also number of partitions of n objects of 2 colors, each part containing at least one black object; see example. - Christian G. Bower, Jan 08 2004
Number of partitions of n into 1 type of part 1, 2 types of part 2, ..., k types of part k. E.g., n=3 gives 111, 12, 12', 3, 3', 3''. - Jon Perry, May 27 2004
The bijection between the partitions in the two preceding comments goes by identifying a part with k black objects with a part of type k. - David Scambler and Joerg Arndt, May 01 2013
Can also be regarded as the number of Jordan canonical forms for an n X n matrix. (I.e., a 5 X 5 matrix has 24 distinct Jordan canonical forms, dependent on the algebraic and geometric multiplicity of each eigenvalue.) - Aaron Gable (agable(AT)hmc.edu), May 26 2009
(1/n) * convolution product of n terms * A001157 (sum of squares of divisors of n): (1, 5, 10, 21, 26, 50, 50, 85, ...) = a(n). As shown by [Bressoud, p. 12]: 1/6 * [1*24 + 5*13 + 10*6 + 21*3 + 26*1 + 50*1] = 288/6 = 48. - Gary W. Adamson, Jun 13 2009
Convolved with the aerated version (1, 0, 1, 0, 3, 0, 6, 0, 13, ...) = A026007: (1, 1, 2, 5, 8, 16, 28, 49, 83, ...). - Gary W. Adamson, Jun 13 2009
Starting with offset 1 = row sums of triangle A162453. - Gary W. Adamson, Jul 03 2009
Unfortunately, Wright's formula is also incomplete in the paper by G. Almkvist: "Asymptotic formulas and generalized Dedekind sums", p. 344, (the denominator should have sqrt(3*Pi) not sqrt(Pi)). This error was already corrected in the paper by Steven Finch: "Integer Partitions". - Vaclav Kotesovec, Aug 17 2015
Also the number of non-isomorphic weight-n chains of multisets whose dual is also a chain of multisets. The dual of a multiset partition has, for each vertex, one block consisting of the indices (or positions) of the blocks containing that vertex, counted with multiplicity. The weight of a multiset partition is the sum of sizes of its parts. - Gus Wiseman, Sep 25 2018

			A planar partition of 13:
  4 3 1 1
  2 1
  1
a(5) = (1/5!)*(sigma_2(1)^5+10*sigma_2(2)*sigma_2(1)^3+20*sigma_2(3)*sigma_2(1)^2+ 15*sigma_2(1)*sigma_2(2)^2+30*sigma_2(4)*sigma_2(1)+20*sigma_2(2)*sigma_2(3)+24*sigma_2(5)) = 24. - _Vladeta Jovovic_, Jan 10 2003
From _David Scambler_ and _Joerg Arndt_, May 01 2013: (Start)
There are a(4) = 13 partitions of 4 objects of 2 colors ('b' and 'w'), each part containing at least one black object:
1 black part:
  [ bwww ]
2 black parts:
  [ bbww ]
  [ bww, b ]
  [ bw, bw ]
3 black parts:
  [ bbbw ]
  [ bbw, b ]
  [ bb, bw ]
(but not: [bw, bb ] )
  [ bw, b, b ]
4 black parts:
  [ bbbb ]
  [ bbb, b ]
  [ bb, bb ]
  [ bb, b, b ]
  [ b, b, b, b ]
(End)
From _Geoffrey Critzer_, Nov 29 2014: (Start)
The corresponding partitions of the integer 4 are:
  4'''
  4''
  3'' + 1
  2' + 2'
  4'
  3' + 1
  2 + 2'
  2' + 1 + 1
  4
  3 + 1
  2 + 2
  2 + 1 + 1
  1 + 1 + 1 + 1.
(End)
From _Gus Wiseman_, Sep 25 2018: (Start)
Non-isomorphic representatives of the a(4) = 13 chains of multisets whose dual is also a chain of multisets:
  {{1,1,1,1}}
  {{1,1,2,2}}
  {{1,2,2,2}}
  {{1,2,3,3}}
  {{1,2,3,4}}
  {{1},{1,1,1}}
  {{2},{1,2,2}}
  {{3},{1,2,3}}
  {{1,1},{1,1}}
  {{1,2},{1,2}}
  {{1},{1},{1,1}}
  {{2},{2},{1,2}}
  {{1},{1},{1},{1}}
(End)
G.f. = 1 + x + 3*x^2 + 6*x^3 + 13*x^4 + 24*x^5 + 48*x^6 + 86*x^7 + 160*x^8 + ...

G. Almkvist, The differences of the number of plane partitions, Manuscript, circa 1991.
G. E. Andrews, The Theory of Partitions, Addison-Wesley, 1976, p. 241.
D. M. Bressoud, Proofs and Confirmations, Camb. Univ. Press, 1999; pp(n) on p. 10.
Miklos Bona, editor, Handbook of Enumerative Combinatorics, CRC Press, 2015, page 575.
L. Carlitz, Generating functions and partition problems, pp. 144-169 of A. L. Whiteman, ed., Theory of Numbers, Proc. Sympos. Pure Math., 8 (1965). Amer. Math. Soc., see p. 145, eq. (1.6).
I. P. Goulden and D. M. Jackson, Combinatorial Enumeration, Wiley, N.Y., 1983, (5.4.5).
P. A. MacMahon, Memoir on the theory of partitions of numbers - Part VI, Phil. Trans. Royal Soc., 211 (1912), 345-373.
P. A. MacMahon, Combinatory Analysis. Cambridge Univ. Press, London and New York, Vol. 1, 1915 and Vol. 2, 1916; see vol. 2, p 332.
P. A. MacMahon, The connexion between the sum of the squares of the divisors and the number of partitions of a given number, Messenger Math., 54 (1924), 113-116. Collected Papers, MIT Press, 1978, Vol. I, pp. 1364-1367. See Table II. - N. J. A. Sloane, May 21 2014
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Suresh Govindarajan, Table of n, a(n) for n = 0..6500 (first 401 terms from T. D. Noe)
G. Almkvist, Asymptotic formulas and generalized Dedekind sums, Exper. Math., 7 (No. 4, 1998), pp. 343-359.
G. E. Andrews and P. Paule, MacMahon's partition analysis XII: Plane Partitions, J. Lond. Math. Soc., 76 (2007), 647-666.
A. O. L. Atkin, P. Bratley, I. G. McDonald and J. K. S. McKay, Some computations for m-dimensional partitions, Proc. Camb. Phil. Soc., 63 (1967), 1097-1100.
A. O. L. Atkin, P. Bratley, I. G. McDonald and J. K. S. McKay, Some computations for m-dimensional partitions, Proc. Camb. Phil. Soc., 63 (1967), 1097-1100. [Annotated scanned copy]
Michael Beeler, R. William Gosper and Richard C. Schroeppel, HAKMEM, ITEM 18, Memo AIM-239, Artificial Intelligence Laboratory, Massachusetts Institute of Technology, Cambridge, Mass., 1972.
Edward A. Bender, Asymptotic methods in enumeration, SIAM Review 16 (1974), no. 4, p. 509.
E. A. Bender and D. E. Knuth, Enumeration of Plane Partitions, J. Combin. Theory A. 13, 40-54, 1972.
S. Benvenuti, B. Feng, A. Hanany and Y. H. He, Counting BPS operators in gauge theories: Quivers, syzygies and plethystics, arXiv:hep-th/0608050, p. 41-42.
Henry Bottomley, Illustration of initial terms
D. M. Bressoud and J. Propp, How the alternating sign matrix conjecture was solved, Notices Amer. Math. Soc., 46 (No. 6, 1999), 637-646.
Shouvik Datta, M. R. Gaberdiel, W. Li, and C. Peng, Twisted sectors from plane partitions, arXiv preprint arXiv:1606.07070 [hep-th], 2016.
Wenjie Fang, Hsien-Kuei Hwang, and Mihyun Kang, Phase transitions from exp(n^(1/2)) to exp(n^(2/3)) in the asymptotics of banded plane partitions, arXiv:2004.08901 [math.CO], 2020.
Steven Finch, Integer Partitions, September 22, 2004. [Cached copy, with permission of the author]
P. Flajolet and R. Sedgewick, Analytic Combinatorics, 2009; see page 580.
Bernhard Heim, Markus Neuhauser and Robert Tröger, Inequalities for Plane Partitions, arXiv:2109.15145 [math.CO], 2021.
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 141
Vaclav Kotesovec, A method of finding the asymptotics of q-series based on the convolution of generating functions, arXiv:1509.08708 [math.CO], 2015-2016, p. 18.
Vaclav Kotesovec, Graphs - The asymptotic ratio (250000 terms)
D. E. Knuth, A Note on Solid Partitions, Math. Comp. 24, 955-961, 1970.
Oleg Lazarev, Matt Mizuhara and Ben Reid, Some Results in Partitions, Plane Partitions, and Multipartitions, 13 August 2010.
P. A. MacMahon, Combinatory analysis.
J. Mangual, McMahon's Formula via Free Fermions, arXiv preprint arXiv:1210.7109 [math.CO], 2012. - From _N. J. A. Sloane_, Jan 01 2013
Ville Mustonen and R. Rajesh, Numerical Estimation of the Asymptotic Behaviour of Solid Partitions ..., arXiv:cond-mat/0303607 [cond-mat.stat-mech], 2003.
L. Mutafchiev and E. Kamenov, On The Asymptotic Formula for the Number of Plane Partitions..., arXiv:math/0601253 [math.CO], 2006; C. R. Acad. Bulgare Sci. 59(2006), No. 4, 361-366.
Ken Ono, Sudhir Pujahari and Larry Rolen, Turán inequalities for the plane partition function, arXiv:2201.01352 [math.NT], 2022.
I. Pak, Partition bijections, a survey, Ramanujan J. 12 (2006) 5-75.
A. Rovenchak, Enumeration of plane partitions with a restricted number of parts, arXiv preprint arXiv:1401.4367 [math-ph], 2014.
Raphael Schumacher, The self-counting identity, Fib. Quart., 55 (No. 2 2017), 157-167.
N. J. A. Sloane, Transforms
J. Stienstra, Mahler measure, Eisenstein series and dimers, arXiv:math/0502197 [math.NT], 2005.
Balázs Szendrői, Non-commutative Donaldson-Thomas invariants and the conifold, Geometry & Topology 12.2 (2008): 1171-1202.
Eric Weisstein's World of Mathematics, Plane Partition
E. M. Wright, Rotatable partitions, J. London Math. Soc., 43 (1968), 501-505.
Index entries for "core" sequences

Cf. A000784, A000785, A000786, A005380, A005987, A048141, A048142, A089300.
Cf. A023871-A023878, A026007, A001157, A162453, A285216.
Differences: A191659, A191660, A191661.
Row sums of A089353 and A091438 and A091298.
Column k=1 of A144048. - Alois P. Heinz, Nov 02 2012
Sequences "number of r-line partitions": A000041 (r=1), A000990 (r=2), A000991 (r=3), A002799 (r=4), A001452 (r=5), A225196 (r=6), A225197 (r=7), A225198 (r=8), A225199 (r=9).
Cf. A249386, A249387.
Cf. A161870, A255610, A255611, A255612, A255613, A255614, A193427.

using Nemo, Memoize
@memoize function a(n)
    if n == 0 return 1 end
    s = sum(a(n - j) * divisor_sigma(j, 2) for j in 1:n)
    return div(s, n)
end
[a(n) for n in 0:20] # Peter Luschny, May 03 2020

series(mul((1-x^k)^(-k),k=1..64),x,63);
# second Maple program:
a:= proc(n) option remember; `if`(n=0, 1, add(
      a(n-j)*numtheory[sigma][2](j), j=1..n)/n)
    end:
seq(a(n), n=0..50);  # Alois P. Heinz, Aug 17 2015

CoefficientList[Series[Product[(1 - x^k)^-k, {k, 64}], {x, 0, 64}], x]
Zeta[3]^(7/36)/2^(11/36)/Sqrt[3 Pi]/Glaisher E^(3 Zeta[3]^(1/3) (n/2)^(2/3) + 1/12)/n^(25/36) (* asymptotic formula after Wright; Vaclav Kotesovec, Jun 23 2014 *)
a[0] = 1; a[n_] := a[n] = Sum[a[n - j] DivisorSigma[2, j], {j, n}]/n; Table[a[n], {n, 0, 50}] (* Jean-François Alcover, Sep 21 2015, after Alois P. Heinz *)
CoefficientList[Series[Exp[Sum[DivisorSigma[2, n] x^n/n, {n, 50}]], {x, 0, 50}], x] (* Eric W. Weisstein, Feb 01 2018 *)

{a(n) = if( n<0, 0, polcoeff( exp( sum( k=1, n, x^k / (1 - x^k)^2 / k, x * O(x^n))), n))}; /* Michael Somos, Jan 29 2005 */

{a(n) = if( n<0, 0, polcoeff( prod( k=1, n, (1 - x^k + x * O(x^n))^-k), n))}; /* Michael Somos, Jan 29 2005 */

my(N=66, x='x+O('x^N)); Vec( prod(n=1,N, (1-x^n)^-n) ) \\ Joerg Arndt, Mar 25 2014

A000219(n)=#PlanePartitions(n) \\ See A091298 for PlanePartitions(). For illustrative use: much slower than the above. - M. F. Hasler, Sep 24 2018

from sympy import cacheit
from sympy.ntheory import divisor_sigma
@cacheit
def A000219(n):
    if n <= 1:
        return 1
    return sum(A000219(n - k) * divisor_sigma(k, 2) for k in range(1, n + 1)) // n
print([A000219(n) for n in range(20)])
# R. J. Mathar, Oct 18 2009

# uses[EulerTransform from A166861]
b = EulerTransform(lambda n: n)
print([b(n) for n in range(37)]) # Peter Luschny, Nov 11 2020

G.f.: Product_{k >= 1} 1/(1 - x^k)^k. - MacMahon, 1912.
Euler transform of sequence [1, 2, 3, ...].
a(n) ~ (c_2 / n^(25/36)) * exp( c_1 * n^(2/3) ), where c_1 = A249387 = 2.00945... and c_2 = A249386 = 0.23151... - Wright, 1931. Corrected Jun 01 2010 by Rod Canfield - see Mutafchiev and Kamenov. The exact value of c_2 is e^(2c)*2^(-11/36)*zeta(3)^(7/36)*(3*Pi)^(-1/2), where c = Integral_{y=0..inf} (y*log(y)/(e^(2*Pi*y)-1))dy = (1/2)*zeta'(-1).
The exact value of c_1 is 3*2^(-2/3)*Zeta(3)^(1/3) = 2.0094456608770137530649... - Vaclav Kotesovec, Sep 14 2014
a(n) = (1/n) * Sum_{k=1..n} a(n-k)*sigma_2(k), n > 0, a(0)=1, where sigma_2(n) = A001157(n) = sum of squares of divisors of n. - Vladeta Jovovic, Jan 20 2002
G.f.: exp(Sum_{n>0} sigma_2(n)*x^n/n). a(n) = Sum_{pi} Product_{i=1..n} binomial(k(i)+i-1, k(i)) where pi runs through all nonnegative solutions of k(1)+2*k(2)+..+n*k(n)=n. - Vladeta Jovovic, Jan 10 2003
From Vaclav Kotesovec, Nov 07 2016: (Start)
More precise asymptotics: a(n) ~ Zeta(3)^(7/36) * exp(3 * Zeta(3)^(1/3) * (n/2)^(2/3) + 1/12) / (A * sqrt(3*Pi) * 2^(11/36) * n^(25/36))
* (1 + c1/n^(2/3) + c2/n^(4/3) + c3/n^2), where
c1 = -0.23994424421250649114273759... = -277/(864*(2*Zeta(3))^(1/3)) - Zeta(3)^(2/3)/(1440*2^(1/3))
c2 = -0.02576771365117401620018082... = 353*Zeta(3)^(1/3)/(248832*2^(2/3)) - 17*Zeta(3)^(4/3)/(3225600*2^(2/3)) - 71575/(1492992*(2*Zeta(3))^(2/3))
c3 = -0.00533195302658826100834286... = -629557/859963392 - 42944125/(7739670528*Zeta(3)) + 14977*Zeta(3)/1114767360 - 22567*Zeta(3)^2/250822656000
and A = A074962 is the Glaisher-Kinkelin constant.
(End)

Corrected by N. J. A. Sloane, Jul 29 2006

Minor edits by Vaclav Kotesovec, Oct 27 2014

A005987 Number of symmetric plane partitions of n.

Original entry on oeis.org

1, 1, 1, 2, 3, 4, 6, 8, 12, 16, 22, 29, 41, 53, 71, 93, 125, 160, 211, 270, 354, 450, 581, 735, 948, 1191, 1517, 1902, 2414, 3008, 3791, 4709, 5909, 7311, 9119, 11246, 13981, 17178, 21249, 26039, 32105, 39213, 48159, 58669, 71831, 87269

Offset: 0

N. J. A. Sloane

From M. F. Hasler, Sep 26 2018: (Start)
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 ignore empty rows or columns. It is symmetric iff A = transpose(A), i.e., A[i,j] = A[j,i] for all i,j.
For any n, we have A000219(n) = a(n) + 2*A306098(n) where A306098(n) is the number of equivalence classes, modulo transposition, of non-symmetric plane partitions. (For any of these, its transpose is a different plane partition of n.) (End)

			From _M. F. Hasler_, Sep 26 2018: (Start)
The only plane partition of n = 0 is the empty partition []; we consider it to be symmetric (as a 0 X 0 matrix), so a(0) = 1.
The only plane partition of n = 1 is the partition [1] which is symmetric, so a(1) = 1.
For n = 2 we have the partitions [2], [1 1] and [1; 1] (where ; denotes the end of a row). Only the first one is symmetric, 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, so a(3) = 2. (End)

D. M. Bressoud, Proofs and Confirmations, Camb. Univ. Press, 1999; p. 134.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
R. P. Stanley, Enumerative Combinatorics, Cambridge, Vol. 2, 1999; see Corollary 7.20.5

T. D. Noe, Table of n, a(n) for n = 0..1000
A. Björner and R. P. Stanley, with A combinatorial miscellany, L'Enseignement Math., Monograph No. 42, 2010.
R. P. Stanley, Theory and application of plane partitions II, Studies in Appl. Math., 50 (1971), 259-279. DOI:10.1002/sapm1971503259. [Scan on author's personal web page].

Cf. A000784, A000785, A000786, A000219, A048142.

terms = 46; s = Product[1/(1 - x^(2i-1))/(1 - x^(2i))^Floor[i/2], {i, 1, Ceiling[terms/2]}] + O[x]^terms; CoefficientList[s, x] (* Jean-François Alcover, Jul 10 2017 *)

a(n)=polcoeff(prod(k=1,n,(1-x^k)^-if(k%2,1,k\4),1+x*O(x^n)), n) \\ Michael Somos, May 19 2000

show(n)=select(t->(t=matconcat(t~))~==t, PlanePartitions(n)) \\ Using PlanePartitions() given in A091298, this selects and returns the list of symmetric plane partitions of n. - M. F. Hasler, Sep 26 2018

G.f.: Product_{i=1..oo} 1/(1-x^(2i-1))/(1-x^(2i))^floor(i/2). (Stanley 1971, Prop.14.3; Björner & Stanley 2010, p. 33).

a(n) ~ exp(3 * Zeta(3)^(1/3) * n^(2/3) / 2^(5/3) + Pi^2 * n^(1/3) / (2^(10/3) * Zeta(3)^(1/3)) - Pi^4 / (384*Zeta(3)) + 1/24) * Zeta(3)^(13/72) / (2^(77/72) * sqrt(3*Pi*A) * n^(49/72)), where A is the Glaisher-Kinkelin constant A074962. - Vaclav Kotesovec, May 05 2018

More terms from Wouter Meeussen, Dec 11 1999

Edited by M. F. Hasler, Sep 26 2018

A306100 Square array T(n,k) = number of plane partitions of n with parts colored in (at most) k colors; n >= 0, k >= 0; read by antidiagonals.

Original entry on oeis.org

1, 1, 0, 1, 1, 0, 1, 2, 3, 0, 1, 3, 10, 6, 0, 1, 4, 21, 34, 13, 0, 1, 5, 36, 102, 122, 24, 0, 1, 6, 55, 228, 525, 378, 48, 0, 1, 7, 78, 430, 1540, 2334, 1242, 86, 0, 1, 8, 105, 726, 3605, 8964, 11100, 3690, 160, 0, 1, 9, 136, 1134, 7278, 25980, 56292, 47496, 11266, 282, 0

Offset: 0

M. F. Hasler, Sep 22 2018

			The array starts:
  [1  1    1     1     1      1 ...] = A000012
  [0  1    2     3     4      5 ...] = A001477
  [0  3   10    21    36     55 ...] = A014105
  [0  6   34   102   228    430 ...] = A067389
  [0 13  122   525  1540   3605 ...]
  [0 24  378  2334  8964  25980 ...]
  [0 48 1242 11100 56292 203280 ...]

Alois P. Heinz, Antidiagonals n = 0..50, flattened
OEIS wiki, Plane partitions.
Wikipedia, Plane partition.

Columns k=0-5 give: A000007, A000219, A306099, A306093, A306094, A306095.
See A306101 for a variant.
Cf. A091298, A208447, A001477, A014105, A067389.

A306100(n,k)=sum(j=1,n,A091298(n,j)*k^j)

T(n,k) = Sum_{j=0..n} A091298(n,j)*k^j, assuming A091298(n,0) = A000007(n).

T(n,k) = Sum_{i=0..k} C(k,i) * A319600(n,i). - Alois P. Heinz, Sep 28 2018

Edited by Alois P. Heinz, Sep 26 2018

A306093 Number of plane partitions of n where parts are colored in 3 colors.

Original entry on oeis.org

1, 3, 21, 102, 525, 2334, 11100, 47496, 210756, 886080, 3759114, 15378051, 63685767, 255417357, 1030081827, 4078689249, 16150234665, 62991117084, 245948154087, 947944122906, 3653360869998, 13946363438502, 53149517598207, 200994216333375, 759191650345380

Offset: 0

M. F. Hasler, Sep 22 2018

nonn

a(0) = 1 corresponds to the empty sum, in which all terms are colored in one among three 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 three colors, whence a(1) = 3.
For n = 2, there are the partitions [2], [1,1] and [1;1]. Adding colors, this yields a(2) = 3 + 9 + 9 = 21 distinct possibilities.

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

Cf. A091298, A208447.

Column 3 of A306100 and A306101. See A306099 for column 2, A306094 .. A306096 for columns 4 .. 6.

PARI
```
a(n)=sum(k=1,n,A091298(n,k)*3^k,!n)
```

a(n) = Sum_{k=1..n} A091298(n,k)*3^k.

a(12) corrected and a(13)-a(24) added by Alois P. Heinz, Sep 24 2018

A306099 Number of plane partitions of n where parts are colored in 2 colors.

Original entry on oeis.org

1, 2, 10, 34, 122, 378, 1242, 3690, 11266, 32666, 94994, 267202, 754546, 2072578, 5691514, 15364290, 41321962, 109634586, 290048746, 758630698, 1977954706, 5111900410, 13161995010, 33645284962, 85727394018, 217042978882, 547750831210, 1375147078146, 3441516792442

Offset: 0

M. F. Hasler and Rick L. Shepherd, following an idea from David S. Newman, Sep 22 2018

nonn

a(0) = 1 corresponds to the empty sum, in which all terms are colored in one among two 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 two colors, whence a(1) = 2.
For n = 2, there are the partitions [2], [1,1] and [1;1]. Adding colors, this yields a(2) = 2 + 4 + 4 = 10 distinct possibilities.

Alois P. Heinz, Table of n, a(n) for n = 0..50
OEIS wiki: Plane partitions
Wikipedia, Plane partition

Cf. A091298, A208447.

Column 2 of A306100 and A306101. See A306093 .. A306096 for columns 3 .. 6.

PARI
```
a(n)=!n+sum(k=1,n,A091298(n,k)<
				
```

a(n) = Sum_{k=1..n} A091298(n,k)*2^k.

a(12) corrected and a(13)-a(28) added by Alois P. Heinz, Sep 24 2018

A306101 Square array T(n,k) = number of plane partitions of n with parts colored in (at most) k colors; n, k >= 1; read by antidiagonals.

Original entry on oeis.org

1, 2, 3, 3, 10, 6, 4, 21, 34, 13, 5, 36, 102, 122, 24, 6, 55, 228, 525, 378, 48, 7, 78, 430, 1540, 2334, 1242, 86, 8, 105, 726, 3605, 8964, 11100, 3690, 160, 9, 136, 1134, 7278, 25980, 56292, 47496, 11266, 282, 10, 171, 1672, 13237, 62574, 203280, 316388, 210756, 32666, 500, 11, 210, 2358, 22280, 132258, 586878, 1417530

Offset: 1

M. F. Hasler, Sep 22 2018

One could have included a row 0 with all 1's, since there is exactly one partition of n = 0, the empty sum, for which all terms (since there are none) are colored in one among k colors.

			The array starts:
  [      1       2       3       4       5 ...] = A000027
  [      3      10      21      36      55 ...] = A014105
  [      6      34     102     228     430 ...] = A067389
  [     13     122     525    1540    3605 ...]
  [     24     378    2334    8964   25980 ...]
  [     48    1242   11100   56292  203280 ...]
   A000219 A306099 A306093 A306094 A306094
For concrete examples, see A306099 and A306093.

Alois P. Heinz, Antidiagonals n = 1..50, flattened

Cf. A091298, A208447, A000219, A000027, A014105, A067389.
See A306100 for a variant.
Cf. A000219, A306099, A306093, A306094, A306095 for columns 1..5.

A306101(n,k)=sum(j=1,n,A091298(n,j)*k^j)

T(n,k) = Sum_{j=1..n} A091298(n,j)*k^j.

A306094 Number of plane partitions of n where parts are colored in (at most) 4 colors.

Original entry on oeis.org

1, 4, 36, 228, 1540, 8964, 56292, 316388, 1857028, 10301892, 57884132, 312915172, 1720407492, 9132560068, 48898964964, 256790538660, 1350883911620, 6992031608260, 36296271612324, 185785685287076, 952221494828996, 4831039856692356, 24489621255994276

Offset: 0

M. F. Hasler, Sep 22 2018

nonn

a(0) = 1 corresponds to the empty sum, in which all terms are colored in one among four 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 four colors, whence a(1) = 4.
For n = 2, there are the partitions [2], [1,1] and [1;1]. Adding colors, this yields a(2) = 4 + 16 + 16 = 36 distinct possibilities.

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

Cf. A091298, A208447.

Column 4 of A306100 and A306101. See A306099 and A306093 for columns 2 and 3.

PARI
```
a(n)=!n+sum(k=1,n,A091298(n,k)*4^k)
```

a(n) = Sum_{k=1..n} A091298(n,k)*4^k.

a(12) corrected and a(13)-a(22) added by Alois P. Heinz, Sep 24 2018

A092288 Triangle read by rows: T(n,k) = count of parts k in all plane partitions of n.

Original entry on oeis.org

1, 4, 1, 11, 2, 1, 28, 7, 2, 1, 62, 15, 5, 2, 1, 137, 38, 13, 5, 2, 1, 278, 76, 28, 11, 5, 2, 1, 561, 164, 60, 26, 11, 5, 2, 1, 1080, 316, 124, 52, 24, 11, 5, 2, 1, 2051, 623, 244, 108, 50, 24, 11, 5, 2, 1, 3778, 1156, 469, 208, 100, 48, 24, 11, 5, 2, 1, 6885, 2160, 886, 404, 194, 98, 48, 24, 11, 5, 2, 1

Offset: 1

Wouter Meeussen, Feb 01 2004

For large n the rows end in A091360 = partial sums of A000219 (count of plane partitions).

			Triangle begins:
    1;
    4,  1;
   11,  2,  1;
   28,  7,  2,  1;
   62, 15,  5,  2,  1;
  137, 38, 13,  5,  2,  1;
  ...

Alois P. Heinz, Rows n = 1..50, flattened
M. F. Hasler, A092288, rows 1 - 18.

Column k=1 gives A090539.
Row sums give A319648.
T(2n+1,n+1) gives A091360.
Cf. A000219, A066633.

Table[Length /@ Split[Sort[Flatten[planepartitions[k]]]], {k, 12}]

A092288_row(n, c=vector(n), m, k)={for(i=1, #n=PlanePartitions(n), for(j=1,#m=n[i], for(i=1,#k=m[j], c[k[i]]++))); c} \\ See A091298 for PlanePartitions(). See below for more efficient code.
M92288=[]; A092288(n,k,L=0)={n>1||return(if(L,[n,n==k],n==k)); if(#L&& #L<3, my(j=setsearch(M92288,[[n,k,L],[]],1)); j<=#M92288&& M92288[j][1]==[n,k,L]&& return(M92288[j][2])); my(c(p)=sum(i=1,#p,p[i]==k),S=[0,0],t); for(m=1,n,my(P=if(L,select(p->vecmin(L-Vecrev(p,#L))>=0, partitions(m,L[1],#L)), partitions(m))); if(mA092288(n-m,k,Vecrev(P[i])); S+=[t[1], t[1]*c(P[i])+t[2]], S+=[#P,vecsum(apply(c,P))])); if(L, #L<3&& M92288= setunion(M92288,[[[n,k,L],S]]);S,S[2])} \\ M. F. Hasler, Sep 26 2018

A090539 Total number of parts equal to 1 in all plane partitions of n.

Original entry on oeis.org

0, 1, 4, 11, 28, 62, 137, 278, 561, 1080, 2051, 3778, 6885, 12255, 21589, 37409, 64160, 108612, 182234, 302461, 497997, 812579, 1316225, 2115608, 3378239, 5357855, 8447086, 13237631, 20632630, 31985504, 49339795, 75738099, 115731636, 176055280, 266697522

Offset: 0

Wouter Meeussen, Feb 01 2004

nonn

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

Column k=1 of A092288.

Cf. A066633, A000219, A091298.

Table[Count[Flatten[planepartitions[k]], 1], {k, 17}]

A090539(n)=A092288(n,1) \\ M. F. Hasler, Sep 26 2018

a(18)-a(27) from Vaclav Kotesovec, May 05 2018

a(28)-a(34) from Alois P. Heinz, Sep 24 2018

A306095 Number of plane partitions of n where parts are colored in (at most) 5 colors.

Original entry on oeis.org

1, 5, 55, 430, 3605, 25980, 203280, 1417530, 10373080, 71595830, 501688880, 3376856755, 23181027055, 153326091805, 1024829902855, 6713038952355, 44092634675905, 284723995000530, 1845944380173205, 11791816763005330, 75485171060740630, 478105767714603130

Offset: 0

M. F. Hasler, Sep 22 2018

nonn

a(0) = 1 corresponds to the empty sum, in which all terms are colored in one among five 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 five colors, whence a(1) = 5.
For n = 2, there are the partitions [2], [1,1] and [1;1]. Adding colors, this yields a(2) = 5 + 25 + 25 = 55 distinct possibilities.

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

Cf. A091298, A208447.

Column 5 of A306100 and A306101. See A306099, A306093, A306094, A306096 for columns 2, 3, 4 and 6.

PARI
```
a(n)=!n+sum(k=1,n,A091298(n,k)*5^k)
```

a(n) = Sum_{k=1..n} A091298(n,k)*5^k.

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A000219 Number of plane partitions (or planar partitions) of n.

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A005987 Number of symmetric plane partitions of n.

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A306100 Square array T(n,k) = number of plane partitions of n with parts colored in (at most) k colors; n >= 0, k >= 0; read by antidiagonals.

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A306093 Number of plane partitions of n where parts are colored in 3 colors.

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A306099 Number of plane partitions of n where parts are colored in 2 colors.

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A306101 Square array T(n,k) = number of plane partitions of n with parts colored in (at most) k colors; n, k >= 1; read by antidiagonals.

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