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A000219
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Number of planar partitions of n.
(Formerly M2566 N1016)
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58
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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
(list; graph; refs; listen; history; internal format)
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OFFSET
| 0,3
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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 (wouter.meeussen(AT)pandora.be).
Also number of partitions of n objects of 2 colors, each part containing at least one black object. - (Christian G. Bower (bowerc(AT)usa.net), 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 (perry(AT)globalnet.co.uk), May 27 2004
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.) [From Aaron Gable (agable(AT)hmc.edu), May 26 2009]
Contribution from Gary W. Adamson (qntmpkt(AT)yahoo.com), Jun 13 2009: (Start)
(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.
Convolved with the aerated version (1, 0, 1, 0, 3, 0, 6, 0, 13,...) = A026007: (1, 1, 2, 5, 8, 16, 28, 49, 83,...). (End)
Starting with offset 1 = row sums of triangle A162453 [From Gary W. Adamson (qntmpkt(AT)yahoo.com), Jul 03 2009]
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) [From Harry Altman (haltman(AT)umich.edu), Nov 21 2009]
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REFERENCES
| G. Almkvist, The differences of the number of plane partitions, Manuscript, circa 1991.
G. Almkvist, Asymptotic formulas and generalized Dedekind sums, Exper. Math., 7 (No. 4, 1998), pp. 343-359.
G. E. Andrews, The Theory of Partitions, Addison-Wesley, 1976, p. 241.
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. Cambridge, MA: MIT Artificial Intelligence Laboratory, Memo AIM-239, Item 18, Feb. 1972.
Bender, E. A. and Knuth, D. E. ``Enumeration of Plane Partitions.'' J. Combin. Theory A. 13, 40-54, 1972.
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).
D. E. Knuth, A Note on Solid Partitions. Math. Comp. 24, 955-961, 1970.
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.
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).
E. M. Wright, Rotatable partitions, J. London Math. Soc., 43 (1968), 501-505.
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LINKS
| T. D. Noe, Table of n, a(n) for n = 0..400
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 (mathar(AT)strw.leidenuniv.nl), Jan 19 2009]
Beeler, M., Gosper, R. W. and Schroeppel, R., HAKMEM, ITEM 18
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
P. A. MacMahon, Combinatory analysis.
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.
N. J. A. Sloane, Transforms
J. Stienstra, Mahler measure, Eisenstein series and dimers
Eric Weisstein's World of Mathematics, Plane Partition
Index entries for "core" sequences
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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 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).
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 (vladeta(AT)eunet.rs), 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 (vladeta(AT)eunet.rs), Jan 10 2003
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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 (vladeta(AT)eunet.rs), Jan 10 2003
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MAPLE
| series(mul((1-x^k)^(-k), k=1..64), x, 63);
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MATHEMATICA
| Rest@CoefficientList[ Series[ Product[ (1-x^k)^-k, {k, 1, 64} ], {x, 0, 64} ], x ]
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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 */
Contribution from R. J. Mathar (mathar(AT)st.rw.leidenuniv.nl), Oct 18 2009: (Start)
(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)]) (End)
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CROSSREFS
| Cf. A000784, A000785, A000786, A005380, A005987, A048141, A048142, A089300.
Cf. A023871-A023878.
Differences: A191659, A191660, A191661.
Row sums of A089353 and A091438.
Cf. A026007, A001157.
Cf. A162453 [From Gary W. Adamson (qntmpkt(AT)yahoo.com), Jul 03 2009]
Sequence in context: A018081 A001452 A005405 * A191782 A027999 A005196
Adjacent sequences: A000216 A000217 A000218 * A000220 A000221 A000222
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KEYWORD
| nonn,nice,easy,core
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AUTHOR
| N. J. A. Sloane (njas(AT)research.att.com).
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EXTENSIONS
| Corrected Jul 29 2006
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