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%I M2566 N1016 #260 Oct 27 2023 18:21:20
%S 1,1,3,6,13,24,48,86,160,282,500,859,1479,2485,4167,6879,11297,18334,
%T 29601,47330,75278,118794,186475,290783,451194,696033,1068745,1632658,
%U 2483234,3759612,5668963,8512309,12733429,18974973,28175955,41691046,61484961,90379784,132441995,193487501,281846923
%N Number of planar partitions (or plane partitions) of n.
%C 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:
%C 4.31.3.22.2.211.21..2.1111.111.11.11.1 but not 2
%C .....1....2.....1...1......1...11.1..1........ 11
%C ....................1.............1..1
%C .....................................1
%C 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
%C 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_
%C 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
%C 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
%C 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
%C 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
%C (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
%C 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
%C Starting with offset 1 = row sums of triangle A162453. - _Gary W. Adamson_, Jul 03 2009
%C 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
%C 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
%D G. Almkvist, The differences of the number of plane partitions, Manuscript, circa 1991.
%D G. E. Andrews, The Theory of Partitions, Addison-Wesley, 1976, p. 241.
%D D. M. Bressoud, Proofs and Confirmations, Camb. Univ. Press, 1999; pp(n) on p. 10.
%D Miklos Bona, editor, Handbook of Enumerative Combinatorics, CRC Press, 2015, page 575.
%D 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).
%D I. P. Goulden and D. M. Jackson, Combinatorial Enumeration, Wiley, N.Y., 1983, (5.4.5).
%D P. A. MacMahon, Memoir on the theory of partitions of numbers - Part VI, Phil. Trans. Royal Soc., 211 (1912), 345-373.
%D P. A. MacMahon, Combinatory Analysis. Cambridge Univ. Press, London and New York, Vol. 1, 1915 and Vol. 2, 1916; see vol. 2, p 332.
%D 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
%D N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H Suresh Govindarajan, <a href="/A000219/b000219.txt">Table of n, a(n) for n = 0..6500</a> (first 401 terms from T. D. Noe)
%H G. Almkvist, <a href="https://projecteuclid.org/euclid.em/1047674152">Asymptotic formulas and generalized Dedekind sums</a>, Exper. Math., 7 (No. 4, 1998), pp. 343-359.
%H G. E. Andrews and P. Paule, <a href="http://dx.doi.org/10.1112/jlms/jdm079">MacMahon's partition analysis XII: Plane Partitions</a>, J. Lond. Math. Soc., 76 (2007), 647-666.
%H A. O. L. Atkin, P. Bratley, I. G. McDonald and J. K. S. McKay, <a href="http://dx.doi.org/10.1017/S0305004100042171">Some computations for m-dimensional partitions</a>, Proc. Camb. Phil. Soc., 63 (1967), 1097-1100.
%H A. O. L. Atkin, P. Bratley, I. G. McDonald and J. K. S. McKay, <a href="/A000219/a000219.pdf">Some computations for m-dimensional partitions</a>, Proc. Camb. Phil. Soc., 63 (1967), 1097-1100. [Annotated scanned copy]
%H Michael Beeler, R. William Gosper and Richard C. Schroeppel, <a href="http://www.inwap.com/pdp10/hbaker/hakmem/boolean.html#item18">HAKMEM, ITEM 18</a>, Memo AIM-239, Artificial Intelligence Laboratory, Massachusetts Institute of Technology, Cambridge, Mass., 1972.
%H Edward A. Bender, <a href="http://www.jstor.org/stable/2028691">Asymptotic methods in enumeration</a>, SIAM Review 16 (1974), no. 4, p. 509.
%H E. A. Bender and D. E. Knuth, <a href="http://dx.doi.org/10.1016/0097-3165(72)90007-6">Enumeration of Plane Partitions</a>, J. Combin. Theory A. 13, 40-54, 1972.
%H S. Benvenuti, B. Feng, A. Hanany and Y. H. He, <a href="http://arXiv.org/abs/hep-th/0608050">Counting BPS operators in gauge theories: Quivers, syzygies and plethystics</a>, arXiv:hep-th/0608050, p. 41-42.
%H Henry Bottomley, <a href="/A000219/a000219.gif">Illustration of initial terms</a>
%H D. M. Bressoud and J. Propp, <a href="http://www.ams.org/notices/199906/fea-bressoud.pdf">How the alternating sign matrix conjecture was solved</a>, Notices Amer. Math. Soc., 46 (No. 6, 1999), 637-646.
%H Shouvik Datta, M. R. Gaberdiel, W. Li, and C. Peng, <a href="https://arxiv.org/abs/1606.07070">Twisted sectors from plane partitions</a>, arXiv preprint arXiv:1606.07070 [hep-th], 2016.
%H Wenjie Fang, Hsien-Kuei Hwang, and Mihyun Kang, <a href="https://arxiv.org/abs/2004.08901">Phase transitions from exp(n^(1/2)) to exp(n^(2/3)) in the asymptotics of banded plane partitions</a>, arXiv:2004.08901 [math.CO], 2020.
%H Steven Finch, <a href="/A000219/a000219_1.pdf">Integer Partitions</a>, September 22, 2004. [Cached copy, with permission of the author]
%H P. Flajolet and R. Sedgewick, <a href="http://algo.inria.fr/flajolet/Publications/books.html">Analytic Combinatorics</a>, 2009; see page 580.
%H Bernhard Heim, Markus Neuhauser and Robert Tröger, <a href="https://arxiv.org/abs/2109.15145">Inequalities for Plane Partitions</a>, arXiv:2109.15145 [math.CO], 2021.
%H INRIA Algorithms Project, <a href="http://ecs.inria.fr/services/structure?nbr=141">Encyclopedia of Combinatorial Structures 141</a>
%H Vaclav Kotesovec, <a href="http://arxiv.org/abs/1509.08708">A method of finding the asymptotics of q-series based on the convolution of generating functions</a>, arXiv:1509.08708 [math.CO], 2015-2016, p. 18.
%H Vaclav Kotesovec, <a href="/A000219/a000219.jpg">Graphs - The asymptotic ratio (250000 terms)</a>
%H D. E. Knuth, <a href="http://dx.doi.org/10.1090/S0025-5718-1970-0277401-7">A Note on Solid Partitions</a>, Math. Comp. 24, 955-961, 1970.
%H Oleg Lazarev, Matt Mizuhara and Ben Reid, <a href="http://www.math.oregonstate.edu/~swisherh/LazarevMizuharaReid.pdf">Some Results in Partitions, Plane Partitions, and Multipartitions</a>, 13 August 2010.
%H P. A. MacMahon, <a href="http://www.hti.umich.edu/cgi/t/text/text-idx?c=umhistmath;idno=ABU9009">Combinatory analysis</a>.
%H J. Mangual, <a href="http://arxiv.org/abs/1210.7109">McMahon's Formula via Free Fermions</a>, arXiv preprint arXiv:1210.7109 [math.CO], 2012. - From _N. J. A. Sloane_, Jan 01 2013
%H Ville Mustonen and R. Rajesh, <a href="http://arXiv.org/abs/cond-mat/0303607">Numerical Estimation of the Asymptotic Behaviour of Solid Partitions ...</a>, arXiv:cond-mat/0303607 [cond-mat.stat-mech], 2003.
%H L. Mutafchiev and E. Kamenov, <a href="https://arxiv.org/abs/math/0601253">On The Asymptotic Formula for the Number of Plane Partitions...</a>, arXiv:math/0601253 [math.CO], 2006; C. R. Acad. Bulgare Sci. 59(2006), No. 4, 361-366.
%H Ken Ono, Sudhir Pujahari and Larry Rolen, <a href="https://arxiv.org/abs/2201.01352">Turán inequalities for the plane partition function</a>, arXiv:2201.01352 [math.NT], 2022.
%H I. Pak, <a href="http://dx.doi.org/10.1007/s11139-006-9576-1">Partition bijections, a survey</a>, Ramanujan J. 12 (2006) 5-75.
%H A. Rovenchak, <a href="http://arxiv.org/abs/1401.4367">Enumeration of plane partitions with a restricted number of parts</a>, arXiv preprint arXiv:1401.4367 [math-ph], 2014.
%H Raphael Schumacher, <a href="https://www.fq.math.ca/Papers1/55-2/Schumacher12132016.pdf">The self-counting identity</a>, Fib. Quart., 55 (No. 2 2017), 157-167.
%H N. J. A. Sloane, <a href="/transforms.txt">Transforms</a>
%H J. Stienstra, <a href="https://arxiv.org/abs/math/0502197">Mahler measure, Eisenstein series and dimers</a>, arXiv:math/0502197 [math.NT], 2005.
%H Balázs Szendrői, <a href="http://dx.doi.org/10.2140/gt.2008.12.1171">Non-commutative Donaldson-Thomas invariants and the conifold</a>, Geometry & Topology 12.2 (2008): 1171-1202.
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/PlanePartition.html">Plane Partition</a>
%H E. M. Wright, <a href="https://doi.org/10.1112/jlms/s1-43.1.501">Rotatable partitions</a>, J. London Math. Soc., 43 (1968), 501-505.
%H <a href="/index/Cor#core">Index entries for "core" sequences</a>
%F G.f.: Product_{k >= 1} 1/(1 - x^k)^k. - MacMahon, 1912.
%F Euler transform of sequence [1, 2, 3, ...].
%F 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).
%F The exact value of c_1 is 3*2^(-2/3)*Zeta(3)^(1/3) = 2.0094456608770137530649... - _Vaclav Kotesovec_, Sep 14 2014
%F 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
%F 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
%F From _Vaclav Kotesovec_, Nov 07 2016: (Start)
%F 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))
%F * (1 + c1/n^(2/3) + c2/n^(4/3) + c3/n^2), where
%F c1 = -0.23994424421250649114273759... = -277/(864*(2*Zeta(3))^(1/3)) - Zeta(3)^(2/3)/(1440*2^(1/3))
%F 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))
%F c3 = -0.00533195302658826100834286... = -629557/859963392 - 42944125/(7739670528*Zeta(3)) + 14977*Zeta(3)/1114767360 - 22567*Zeta(3)^2/250822656000
%F and A = A074962 is the Glaisher-Kinkelin constant.
%F (End)
%e A planar partition of 13:
%e 4 3 1 1
%e 2 1
%e 1
%e 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
%e From _David Scambler_ and _Joerg Arndt_, May 01 2013: (Start)
%e There are a(4) = 13 partitions of 4 objects of 2 colors ('b' and 'w'), each part containing at least one black object:
%e 1 black part:
%e [ bwww ]
%e 2 black parts:
%e [ bbww ]
%e [ bww, b ]
%e [ bw, bw ]
%e 3 black parts:
%e [ bbbw ]
%e [ bbw, b ]
%e [ bb, bw ]
%e (but not: [bw, bb ] )
%e [ bw, b, b ]
%e 4 black parts:
%e [ bbbb ]
%e [ bbb, b ]
%e [ bb, bb ]
%e [ bb, b, b ]
%e [ b, b, b, b ]
%e (End)
%e The corresponding partitions of the integer 4 are:
%e 4'''
%e 4''
%e 3'' + 1
%e 2' + 2'
%e 4'
%e 3' + 1
%e 2 + 2'
%e 2' + 1 + 1
%e 4
%e 3 + 1
%e 2 + 2
%e 2 + 1 + 1
%e 1 + 1 + 1 + 1. - _Geoffrey Critzer_, Nov 29 2014
%e From _Gus Wiseman_, Sep 25 2018: (Start)
%e Non-isomorphic representatives of the a(4) = 13 chains of multisets whose dual is also a chain of multisets:
%e {{1,1,1,1}}
%e {{1,1,2,2}}
%e {{1,2,2,2}}
%e {{1,2,3,3}}
%e {{1,2,3,4}}
%e {{1},{1,1,1}}
%e {{2},{1,2,2}}
%e {{3},{1,2,3}}
%e {{1,1},{1,1}}
%e {{1,2},{1,2}}
%e {{1},{1},{1,1}}
%e {{2},{2},{1,2}}
%e {{1},{1},{1},{1}}
%e (End)
%e 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 + ...
%p series(mul((1-x^k)^(-k),k=1..64),x,63);
%p # second Maple program:
%p a:= proc(n) option remember; `if`(n=0, 1, add(
%p a(n-j)*numtheory[sigma][2](j), j=1..n)/n)
%p end:
%p seq(a(n), n=0..50); # _Alois P. Heinz_, Aug 17 2015
%t CoefficientList[Series[Product[(1 - x^k)^-k, {k, 64}], {x, 0, 64}], x]
%t 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 *)
%t 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_ *)
%t CoefficientList[Series[Exp[Sum[DivisorSigma[2, n] x^n/n, {n, 50}]], {x, 0, 50}], x] (* _Eric W. Weisstein_, Feb 01 2018 *)
%o (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 */
%o (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 */
%o (PARI) my(N=66, x='x+O('x^N)); Vec( prod(n=1,N, (1-x^n)^-n) ) \\ _Joerg Arndt_, Mar 25 2014
%o (PARI) A000219(n)=#PlanePartitions(n) \\ See A091298 for PlanePartitions(). For illustrative use: much slower than the above. - _M. F. Hasler_, Sep 24 2018
%o (Python)
%o from sympy import cacheit
%o from sympy.ntheory import divisor_sigma
%o @cacheit
%o def A000219(n):
%o if n <= 1:
%o return 1
%o return sum(A000219(n - k) * divisor_sigma(k, 2) for k in range(1, n + 1)) // n
%o print([A000219(n) for n in range(20)])
%o # _R. J. Mathar_, Oct 18 2009
%o (Julia)
%o using Nemo, Memoize
%o @memoize function a(n)
%o if n == 0 return 1 end
%o s = sum(a(n - j) * divisor_sigma(j, 2) for j in 1:n)
%o return div(s, n)
%o end
%o [a(n) for n in 0:20] # _Peter Luschny_, May 03 2020
%o (SageMath) # uses[EulerTransform from A166861]
%o b = EulerTransform(lambda n: n)
%o print([b(n) for n in range(37)]) # _Peter Luschny_, Nov 11 2020
%Y Cf. A000784, A000785, A000786, A005380, A005987, A048141, A048142, A089300.
%Y Cf. A023871-A023878, A026007, A001157, A162453, A285216.
%Y Differences: A191659, A191660, A191661.
%Y Row sums of A089353 and A091438 and A091298.
%Y Column k=1 of A144048. - _Alois P. Heinz_, Nov 02 2012
%Y 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).
%Y Cf. A249386, A249387.
%Y Cf. A161870, A255610, A255611, A255612, A255613, A255614, A193427.
%K nonn,nice,easy,core
%O 0,3
%A _N. J. A. Sloane_
%E Corrected by _N. J. A. Sloane_, Jul 29 2006
%E Minor edits by _Vaclav Kotesovec_, Oct 27 2014
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