A000219 - OEIS

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

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

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

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

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

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.

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

Miklos Bona, editor, Handbook of Enumerative Combinatorics, CRC Press, 2015, page 575.

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).

I. P. Goulden and D. M. Jackson, Combinatorial Enumeration, Wiley, N.Y., 1983, (5.4.5).

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).

Michael Beeler, R. William Gosper and Richard C. Schroeppel, HAKMEM, ITEM 18, Memo AIM-239, Artificial Intelligence Laboratory, Massachusetts Institute of Technology, Cambridge, Mass., 1972.

Steven Finch, Integer Partitions, September 22, 2004. [Cached copy, with permission of the author]

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 = 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).

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

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))

* (1 + c1/n^(2/3) + c2/n^(4/3) + c3/n^2), where

c1 = -0.23994424421250649114273759... = -277/(864*(2*Zeta(3))^(1/3)) - Zeta(3)^(2/3)/(1440*2^(1/3))

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))

c3 = -0.00533195302658826100834286... = -629557/859963392 - 42944125/(7739670528*Zeta(3)) + 14977*Zeta(3)/1114767360 - 22567*Zeta(3)^2/250822656000

and A = A074962 is the Glaisher-Kinkelin constant.

(End)

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

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)

The corresponding partitions of the integer 4 are:

4'''

4''

3'' + 1

2' + 2'

4'

3' + 1

2 + 2'

2' + 1 + 1

4

3 + 1

2 + 2

2 + 1 + 1

1 + 1 + 1 + 1.

(End)

Non-isomorphic representatives of the a(4) = 13 chains of multisets whose dual is also a chain of multisets:

{{1,1,1,1}}

{{1,1,2,2}}

{{1,2,2,2}}

{{1,2,3,3}}

{{1,2,3,4}}

{{1},{1,1,1}}

{{2},{1,2,2}}

{{3},{1,2,3}}

{{1,1},{1,1}}

{{1,2},{1,2}}

{{1},{1},{1,1}}

{{2},{2},{1,2}}

{{1},{1},{1},{1}}

(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 + ...

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

# second Maple program:

a:= proc(n) option remember; `if`(n=0, 1, add(

a(n-j)*numtheory[sigma][2](j), j=1..n)/n)

end:

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

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 *)

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 *)

CoefficientList[Series[Exp[Sum[DivisorSigma[2, n] x^n/n, {n, 50}]], {x, 0, 50}], x] (* Eric W. Weisstein, Feb 01 2018 *)

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

(PARI) A000219(n)=#PlanePartitions(n) \\ See A091298 for PlanePartitions(). For illustrative use: much slower than the above. - M. F. Hasler, Sep 24 2018

(Python)

from sympy import cacheit

from sympy.ntheory import divisor_sigma

@cacheit

if n <= 1:

return 1

return sum(A000219(n - k) * divisor_sigma(k, 2) for k in range(1, n + 1)) // n

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

(Julia)

using Nemo, Memoize

@memoize function a(n)

if n == 0 return 1 end

s = sum(a(n - j) * divisor_sigma(j, 2) for j in 1:n)

return div(s, n)

end

(SageMath) # uses[EulerTransform from A166861]

b = EulerTransform(lambda n: n)

print([b(n) for n in range(37)]) # Peter Luschny, Nov 11 2020