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 A000219 Number of planar partitions of n. (Formerly M2566 N1016) 60

%I M2566 N1016

%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 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 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 nxn matrix. (i.e. a 5x5 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 Also number of functions from an n-set to itself, up to permutation of the set (compare to partition function, which is number of conjugacy classes in S_n) [_Harry Altman_, Nov 21 2009]

%D G. Almkvist, The differences of the number of plane partitions, Manuscript, circa 1991.

%D G. Almkvist, Asymptotic formulas and generalized Dedekind sums, Exper. Math., 7 (No. 4, 1998), pp. 343-359.

%D G. E. Andrews, The Theory of Partitions, Addison-Wesley, 1976, p. 241.

%D 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.

%D Beeler, M.; Gosper, R. W.; and Schroeppel, R. HAKMEM. Cambridge, MA: MIT Artificial Intelligence Laboratory, Memo AIM-239, Item 18, Feb. 1972.

%D Bender, E. A. and Knuth, D. E. Enumeration of Plane Partitions.'' J. Combin. Theory A. 13, 40-54, 1972.

%D D. M. Bressoud, Proofs and Confirmations, Camb. Univ. Press, 1999; pp(n) on p. 10.

%D D. M. Bressoud and J. Propp, How the alternating sign matrix conjecture was solved, Notices Amer. Math. Soc., 46 (No. 6, 1999), 637-646.

%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 D. E. Knuth, A Note on Solid Partitions. Math. Comp. 24, 955-961, 1970.

%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 J. Mangual, McMahon's Formula via Free Fermions, arXiv preprint arXiv:1210.7109, 2012. - From _N. J. A. Sloane_, Jan 01 2013

%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).

%D E. M. Wright, Rotatable partitions, J. London Math. Soc., 43 (1968), 501-505.

%H T. D. Noe, <a href="/A000219/b000219.txt">Table of n, a(n) for n = 0..400</a>

%H G. Almkvist, <a href="http://www.emis.de/journals/EM/expmath/volumes/7/7.html">Asymptotic formulas and generalized Dedekind sums</a>, Exper. Math., 7 (No. 4, 1998), pp. 343-359.

%H G. E. Andrews, 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. [From _R. J. Mathar_, Jan 19 2009]

%H Beeler, M., Gosper, R. W. and Schroeppel, R., <a href="http://www.inwap.com/pdp10/hbaker/hakmem/boolean.html#item18">HAKMEM, ITEM 18</a>

%H H. Bottomley, <a href="/A000219/a000219.gif">Illustration of initial terms</a>

%H P. Flajolet and R. Sedgewick, <a href="http://algo.inria.fr/flajolet/Publications/books.html">Analytic Combinatorics</a>, 2009; see page 580

%H INRIA Algorithms Project, <a href="http://algo.inria.fr/ecs/ecs?searchType=1&amp;service=Search&amp;searchTerms=141">Encyclopedia of Combinatorial Structures 141</a>

%H P. A. MacMahon, <a href="http://www.hti.umich.edu/cgi/t/text/text-idx?c=umhistmath;idno=ABU9009">Combinatory analysis</a>.

%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>

%H L. Mutafchiev and E. Kamenov, <a href="http://arXiv.org/abs/math.CO/0601253">On The Asymptotic Formula for the Number of Plane Partitions...</a>, C. R. Acad. Bulgare Sci. 59(2006), No. 4, 361-366.

%H N. J. A. Sloane, <a href="/transforms.txt">Transforms</a>

%H J. Stienstra, <a href="http://arXiv.org/abs/math.NT/0502197">Mahler measure, Eisenstein series and dimers</a>

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/PlanePartition.html">Plane Partition</a>

%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 = 2.00945... and c_2 = 0.23151... - Wright, 1931. Corrected Jun 01 2010 by Rod Canfield - see Mutafchiev and Kamenov. The exact value of c_2 is exp(2c) 2^{-11/36} \zeta(3)^{7/36} (3\pi)^{-1/2}, where c = \int_0^{\infty} \frac{y\log y}{exp(2\pi y)-1}dy = (1/2) \zeta'(-1).

%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

%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)

%p series(mul((1-x^k)^(-k),k=1..64),x,63);

%t Rest@CoefficientList[ Series[ Product[ (1-x^k)^-k, {k, 1, 64} ], {x, 0, 64} ], x ]

%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 (Python)

%o def divisors(n):

%o ...a = {1,n}

%o ...for x in range(2,n//2+1):

%o ......if n % x == 0:

%o .........a |= {x}

%o ...return a

%o def sigma(n,k):

%o ...a=0

%o ...for d in divisors(n):

%o ......a += d**k

%o ...return a

%o def A000219(n):

%o ...if n <=1:

%o ......return 1

%o ...else:

%o ......a=0

%o ......for k in range (1,n+1):

%o .........a += A000219(n-k)*sigma(k,2)

%o ......return a//n

%o print([A000219(n) for n in range(0,20)])

%o # _R. J. Mathar_, Oct 18 2009

%Y Cf. A000784, A000785, A000786, A005380, A005987, A048141, A048142, A089300.

%Y Cf. A023871-A023878.

%Y Differences: A191659, A191660, A191661.

%Y Row sums of A089353 and A091438.

%Y Cf. A026007, A001157.

%Y Cf. A162453. - _Gary W. Adamson_, Jul 03 2009

%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).

%K nonn,nice,easy,core

%O 0,3

%A _N. J. A. Sloane_.

%E Corrected Jul 29 2006

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