0,3

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

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. Roal 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.

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

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.

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

H. Bottomley, Illustration of initial terms

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 ...

N. J. A. Sloane, Transforms

J. Stienstra, Mahler measure, Eisenstein series and dimers

Eric Weisstein's World of Mathematics, Link to a section of The World of Mathematics.

Index entries for "core" sequences

L. Mutafchiev and E. Kamenov, On The Asymptotic Formula for the Number of Plane Partitions...

G.f.: Product_{k >= 1} 1/(1 - x^k)^k. - MacMahon, 1912.

Euler transform of sequence [1, 2, 3, ...].

a(n) ~ (c_2 / n^(25/26)) * exp( c_1 * n^(2/3) ), where c_1 = 2.00945... and c_2 = 0.40099... - Wright, 1931.

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.yu), 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.yu), Jan 10 2003

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.yu), Jan 10 2003

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

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

(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 */

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

Cf. A023871-A023878.

Row sums of A089353 and A091438.

Adjacent sequences: A000216 A000217 A000218 this_sequence A000220 A000221 A000222

Sequence in context: A018081 A001452 A005405 this_sequence A027999 A005196 A032287

nonn,nice,easy,core

njas

Corrected Jul 29 2006