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A000219 Number of planar partitions of n.
(Formerly M2566 N1016)
66
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 (list; graph; refs; listen; history; text; internal format)
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

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

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.

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

Edward A. Bender, Asymptotic methods in enumeration, SIAM Review 16 (1974), no. 4, p. 509.

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

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

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).

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

T. D. Noe and 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, P. Paule, MacMahon's partition analysis XII: Plane Partitions, J. Lond. Math. Soc., 76 (2007), 647-666. [From R. J. Mathar, Jan 19 2009]

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.

Beeler, M., Gosper, R. W. and Schroeppel, R., HAKMEM, ITEM 18

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

H. Bottomley, Illustration of initial terms

P. Flajolet and R. Sedgewick, Analytic Combinatorics, 2009; see page 580

INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 141

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

Oleg Lazarev, Matt Mizuhara, 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, 2012. - From N. J. A. Sloane, Jan 01 2013

Ville Mustonen and R. Rajesh, Numerical Estimation of the Asymptotic Behaviour of Solid Partitions ...

L. Mutafchiev and E. Kamenov, On The Asymptotic Formula for the Number of Plane Partitions..., C. R. Acad. Bulgare Sci. 59(2006), No. 4, 361-366.

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, 2014

N. J. A. Sloane, Transforms

J. Stienstra, Mahler measure, Eisenstein series and dimers

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

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 = 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 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

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)

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);

MATHEMATICA

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

(* Asymptotic formula (after Wright) in the Mathematica format: *) 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) (* Vaclav Kotesovec, Jun 23 2014 *)

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

(Python)

def divisors(n):

...a = {1, n}

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

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

.........a |= {x}

...return a

def sigma(n, k):

...a=0

...for d in divisors(n):

......a += d**k

...return a

def A000219(n):

...if n <=1:

......return 1

...else:

......a=0

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

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

......return a//n

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

#  R. J. Mathar, Oct 18 2009

CROSSREFS

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

Cf. A023871-A023878, A026007, A001157, A091298, A162453.

Differences: A191659, A191660, A191661.

Row sums of A089353 and A091438.

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).

Sequence in context: A225197 A225198 A225199 * A191782 A027999 A005196

Adjacent sequences:  A000216 A000217 A000218 * A000220 A000221 A000222

KEYWORD

nonn,nice,easy,core,changed

AUTHOR

N. J. A. Sloane

EXTENSIONS

Corrected Jul 29 2006

STATUS

approved

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Last modified September 21 06:04 EDT 2014. Contains 247022 sequences.