OFFSET
0,3
COMMENTS
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
REFERENCES
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).
LINKS
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.
FORMULA
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)
EXAMPLE
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)
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. - Geoffrey Critzer, Nov 29 2014
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 + ...
MAPLE
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
MATHEMATICA
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 *)
PROG
(PARI) {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 */
(PARI) {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 */
(PARI) my(N=66, x='x+O('x^N)); Vec( prod(n=1, N, (1-x^n)^-n) ) \\ Joerg Arndt, Mar 25 2014
(PARI) A000219(n)=#PlanePartitions(n) \\ See A091298 for PlanePartitions(). For illustrative use: much slower than the above. - M. F. Hasler, Sep 24 2018
(Python)
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
(Julia)
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
(SageMath) # uses[EulerTransform from A166861]
b = EulerTransform(lambda n: n)
print([b(n) for n in range(37)]) # Peter Luschny, Nov 11 2020
CROSSREFS
KEYWORD
nonn,nice,easy,core
AUTHOR
EXTENSIONS
Corrected by N. J. A. Sloane, Jul 29 2006
Minor edits by Vaclav Kotesovec, Oct 27 2014
STATUS
approved