A000219 -id:A000219 - cp's OEIS frontend

A052847 G.f.: 1 / Product_{k>=1} (1-x^k)^(k-1).

1, 0, 1, 2, 4, 6, 12, 18, 33, 52, 88, 138, 229, 354, 568, 880, 1378, 2110, 3260, 4942, 7527, 11320, 17031, 25394, 37842, 55956, 82630, 121300, 177677, 258980, 376626, 545352, 787784, 1133764, 1627657, 2329020, 3324559, 4731396, 6717774, 9512060

Offset: 0

encyclopedia(AT)pommard.inria.fr, Jan 25 2000

Euler transform of sequence [0,1,2,3,...]. - Michael Somos, Jul 02 2004
Number of partitions of n objects of 2 colors, where each part must contain at least one of each color. - Franklin T. Adams-Watters, Jan 23 2006
Number of partitions of n without 1s, one kind of 2s, two kinds of 3s, etc. - Joerg Arndt, Jul 31 2011
From Vaclav Kotesovec, Oct 17 2015: (Start)
In general, if v>=0 and g.f. = Product_{k>=1} 1/(1-x^(k+v))^k, then a(n) ~ d1(v) * n^(v^2/6 - 25/36) * exp(-Pi^4 * v^2 / (432*Zeta(3)) + 3*Zeta(3)^(1/3) * n^(2/3)/2^(2/3) - v * Pi^2 * n^(1/3) / (3 * 2^(4/3) * Zeta(3)^(1/3))) / (sqrt(3*Pi) * 2^(v^2/6 + 11/36) * Zeta(3)^(v^2/6 - 7/36)), where Zeta(3) = A002117.
d1(v) = exp(Integral_{x=0..infinity} (1/(x*exp((v-1)*x) * (exp(x)-1)^2) - (6*v^2-1) / (12*x*exp(x)) + v/x^2 - 1/x^3) dx).
d1(v) = (exp(Zeta'(-1) - v*Zeta'(0))) / Product_{j=0..v-1} j!, where Zeta'(0) = -A075700, Zeta'(-1) = A084448 and Product_{j=0..v-1} j! = A000178(v-1).
d1(v) = exp(1/12) * (2*Pi)^(v/2) / (A * G(v+1)), where A = A074962 is the Glaisher-Kinkelin constant and G is the Barnes G-function.
(End)

			1 + x^2 + 2*x^3 + 4*x^4 + 6*x^5 + 12*x^6 + 18*x^7 + 33*x^8 + 52*x^9 + ...
From _Gus Wiseman_, Jan 22 2019: (Start)
The partitions described in Franklin T. Adams-Watters's comment are (n = 2 through 6):
  {{12}}  {{112}}  {{1112}}    {{11112}}    {{111112}}
          {{122}}  {{1122}}    {{11122}}    {{111122}}
                   {{1222}}    {{11222}}    {{111222}}
                   {{12}{12}}  {{12222}}    {{112222}}
                               {{12}{112}}  {{122222}}
                               {{12}{122}}  {{112}{112}}
                                            {{112}{122}}
                                            {{12}{1112}}
                                            {{12}{1122}}
                                            {{12}{1222}}
                                            {{122}{122}}
                                            {{12}{12}{12}}
(End)

Vaclav Kotesovec, Table of n, a(n) for n = 0..1000
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 815
Vaclav Kotesovec, Graph - The asymptotic ratio

Cf. A005380, A000203, A001157, A052812.
Cf. A000219 (v=0), A052847 (v=1), A263358 (v=2), A263359 (v=3), A263360 (v=4), A263361 (v=5), A263362 (v=6), A263363 (v=7), A263364 (v=8).
Cf. A054974, A321407, A321760, A323654, A323655, A323656.

spec := [S,{B=Sequence(Z,1 <= card),C=Prod(B,B),S= Set(C)},unlabeled]: seq(combstruct[count](spec, size=n), n=0..20);
with(numtheory): etr:= proc(p) local b; b:=proc(n) option remember; local d, j; if n=0 then 1 else add(add(d*p(d), d=divisors(j)) *b(n-j), j=1..n)/n fi end end: a:=etr(n-> n-1): seq(a(n), n=0..50); # Vaclav Kotesovec, Mar 04 2015 after Alois P. Heinz

Clear[a]; a[n_]:= a[n] = 1/n*Sum[(DivisorSigma[2,k]-DivisorSigma[1,k])*a[n-k],{k,1,n}]; a[0]=1; Table[a[n],{n,0,100}] (* Vaclav Kotesovec, Mar 04 2015 *)
nmax = 40; CoefficientList[Series[Product[1/(1-x^(k+1))^k, {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Oct 16 2015 *)

{a(n) = if( n<0, 0, polcoeff( 1 / prod(k=1, n, (1 - x^k + x*O(x^n))^(k-1)), n))}

a(n) = 1/n*Sum_{k=1..n} (sigma[2](k)-sigma[1](k))*a(n-k).
G.f.: exp( Sum_{k>0} ( x^k / (1 - x^k) )^2 / k ).
G.f.: exp( sum(n>=0, (sigma[2](n)-sigma[1](n)) *x^n/n ) ). - Joerg Arndt, Jul 31 2011
a(n) ~ 2^(1/36) * Zeta(3)^(1/36) * exp(1/12 - Pi^4/(432*Zeta(3)) - Pi^2 * n^(1/3) / (3 * 2^(4/3) * Zeta(3)^(1/3)) + 3 * Zeta(3)^(1/3) * n^(2/3) / 2^(2/3)) / (A * 3^(1/2) * n^(19/36)), where A = A074962 = 1.2824271291... is the Glaisher-Kinkelin constant. - Vaclav Kotesovec, Mar 07 2015

Edited by Vladeta Jovovic, Sep 10 2002

A055922 Number of partitions of n in which each part occurs an odd number (or zero) times.

Original entry on oeis.org

1, 1, 1, 3, 2, 5, 6, 9, 9, 16, 20, 25, 32, 40, 54, 69, 84, 101, 136, 156, 202, 244, 306, 357, 448, 527, 652, 773, 944, 1103, 1346, 1574, 1885, 2228, 2640, 3106, 3684, 4302, 5052, 5931, 6924, 8079, 9416, 10958, 12718, 14824, 17078, 19820, 22860, 26433

Offset: 0

Christian G. Bower, Jun 23 2000

nonn

			There exist 11 partitions of 6. For six of these partitions, each part occurs an odd number times, they are 6 = 5 + 1 = 4 + 2 = 3 + 2 + 1 = 3 + 1+1+1 = 2+2+2, hence a(6) = 6. The five other partitions are 4 + 1+1 = 3+3 = 2+2 + 1+1 = 2 + 1+1+1+1 = 1+1+1+1+1+1.

Alois P. Heinz, Table of n, a(n) for n = 0..10000
F. C. Auluck, K. S. Singwi and B. K. Agarwala, On a new type of partition, Proc. Nat. Inst. Sci. India 16 (1950) 147-156.
Steven Finch, Integer Partitions, September 22, 2004, page 5. [Cached copy, with permission of the author]
Daniel Herden, Mark R. Sepanski, Jonathan Stanfill, Cordell Hammon, Joel Henningsen, Henry Ickes, and Indalecio Ruiz, Partitions With Designated Summands Not Divisible by 2^L, 2, and 3^L Modulo 2, 4, and 3, arXiv:2101.04058 [math.CO], 2021. See also Integers (2023) Vol. 23, Art. No. A43.
James A. Sellers and Fabrizio Zanello, On the parity of the number of partitions with odd multiplicities, arXiv:2004.06204 [math.CO], 2020.

Cf. A000041, A007690, A055923, A249389.

Column k=0 of A264399.

b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0,
      add(`if`(irem(j, 2)=0, 0, b(n-i*j, i-1)), j=1..n/i)
       +b(n, i-1)))
    end:
a:= n-> b(n$2):
seq(a(n), n=0..60);  # Alois P. Heinz, May 31 2014

b[n_, i_] := b[n, i] = If[n == 0, 1, If[i<1, 0, Sum[If[Mod[j, 2] == 0, 0, b[n-i*j, i-1]], {j, 1, n/i}] + b[n, i-1]]]; a[n_] := b[n, n]; Table[a[n], {n, 0, 60}] (* Jean-François Alcover, Feb 24 2015, after Alois P. Heinz *)

{ my(n=60); Vec(prod(k=1, n, 1 + sum(r=0, n\(2*k), x^(k*(2*r+1))) + O(x*x^n))) } \\ Andrew Howroyd, Dec 22 2017

EULER transform of b where b has g.f. Sum {k>0} c(k)*x^k/(1-x^k) where c is inverse EULER transform of characteristic function of odd numbers.
G.f.: Product_{i>0} (1+x^i-x^(2*i))/(1-x^(2*i)). - Vladeta Jovovic, Feb 03 2004
Asymptotics (Auluck, Singwi, Agarwala, 1950): a(n) ~ B/(2*Pi*n) * exp(2*B*sqrt(n)), where B = sqrt(Pi^2/12 + 2*log(phi)^2), where phi is the golden ratio. - Vaclav Kotesovec, Oct 27 2014

A023871 Expansion of Product_{k>=1} (1 - x^k)^(-k^2).

Original entry on oeis.org

1, 1, 5, 14, 40, 101, 266, 649, 1593, 3765, 8813, 20168, 45649, 101591, 223654, 486046, 1045541, 2225167, 4692421, 9804734, 20318249, 41766843, 85218989, 172628766, 347338117, 694330731, 1379437080, 2724353422, 5350185097, 10449901555, 20304465729, 39254599832

Offset: 0

Olivier Gérard

In general, if g.f. = Product_{k>=1} 1/(1 - x^k)^(c2*k^2 + c1*k + c0) and c2 > 0, then a(n) ~ exp(4*Pi * c2^(1/4) * n^(3/4) / (3*15^(1/4)) + c1*Zeta(3) / Pi^2 * sqrt(15*n/c2) + (Pi * 5^(1/4) * c0 / (2*3^(3/4) * c2^(1/4)) - 15^(5/4) * c1^2 * Zeta(3)^2 / (2*c2^(5/4) * Pi^5)) * n^(1/4) + c1/12 + 75 * c1^3 * Zeta(3)^3 / (c2^2 * Pi^8) - 5*c0 * c1 * Zeta(3) / (4*c2 * Pi^2) - c2*Zeta(3) / (4*Pi^2)) * Pi^(c1/12) * (c2/15)^(1/8 + c0/8 + c1/48) / (A^c1 * 2^((c0 + 3)/2) * n^(5/8 + c0/8 + c1/48)), where A is the Glaisher-Kinkelin constant A074962. - Vaclav Kotesovec, Nov 08 2017
Let A(x) = Product_{k >= 1} (1 - x^k)^(-k^2). The sequence defined by u(n) := [x^n] A(x)^n is conjectured to satisfy the supercongruences u(n*p^r) == u(n*p^(r-1)) (mod p^(3*r)) for all primes p >= 7 and all positive integers n and r. See A380290. - Peter Bala, Feb 02 2025
a(n) is the number of partitions of n where there are k^2 sorts of part k. - Joerg Arndt, Feb 02 2025

Seiichi Manyama, Table of n, a(n) for n = 0..10000 (first 1001 terms from Alois P. Heinz)
G. Almkvist, Asymptotic formulas and generalized Dedekind sums, Exper. Math., 7 (No. 4, 1998), pp. 343-359.
Vaclav Kotesovec, A method of finding the asymptotics of q-series based on the convolution of generating functions, arXiv:1509.08708 [math.CO], Sep 30 2015, p. 21.

Euler transform of squares (A000290).
Cf. A000219, A023872-A023878, A294530, A380290.
Column k=2 of A144048. - Alois P. Heinz, Nov 02 2012

m:=40; R:=PowerSeriesRing(Rationals(), m); Coefficients(R! ( (&*[1/(1-x^k)^k^2: k in [1..m]]) )); // G. C. Greubel, Oct 29 2018

with(numtheory):
a:= proc(n) option remember; `if`(n=0, 1,
      add(add(d*d^2, d=divisors(j)) *a(n-j), j=1..n)/n)
    end:
seq(a(n), n=0..35); # Alois P. Heinz, Nov 02 2012

max = 31; Series[ Product[ 1/(1-x^k)^k^2, {k, 1, max}], {x, 0, max}] // CoefficientList[#, x]& (* Jean-François Alcover, Mar 05 2013 *)

m=40; x='x+O('x^m); Vec(prod(k=1, m, 1/(1-x^k)^k^2)) \\ G. C. Greubel, Oct 29 2018

# uses[EulerTransform from A166861]
b = EulerTransform(lambda n: n^2)
print([b(n) for n in range(32)]) # Peter Luschny, Nov 11 2020

a(n) = 1/n * Sum_{k=1..n} a(n-k)*sigma_3(k), n > 0, a(0)=1, where sigma_3(n) = A001158(n) = sum of cubes of divisors of n. - Vladeta Jovovic, Jan 20 2002
G.f.: Prod_{n>=1} exp(sigma_3(n)*x^n/n), where sigma_3(n) is the sum of cubes of divisors of n (=A001158(n)). - N-E. Fahssi, Mar 28 2010
G.f. (conjectured): 1/Product_{n>=1} E(x^n)^J2(n) where E(x) = Product_{n>=1} 1-x^n and J2(n) = A007434(n) [follows from the identity Sum_{d|n} J2(d) = n^2 - Peter Bala, Feb 02 2025]. - Joerg Arndt, Jan 25 2011
a(n) ~ exp(4 * Pi * n^(3/4) / (3^(5/4) * 5^(1/4)) - Zeta(3) / (4*Pi^2)) / (2^(3/2) * 15^(1/8) * n^(5/8)), where Zeta(3) = A002117 = 1.2020569031595942853997... . - Vaclav Kotesovec, Feb 27 2015

Definition corrected by Franklin T. Adams-Watters and R. J. Mathar, Dec 04 2006

A141199 Number of hierarchical ordered partitions of partitions.

Original entry on oeis.org

1, 1, 3, 7, 17, 38, 87, 191, 421, 911, 1963, 4186, 8885, 18724, 39284, 82005, 170521, 353214, 729290, 1501184, 3081869, 6311404, 12896983, 26301515, 53541702, 108815626, 220824295, 447524559, 905850001, 1831526719

Offset: 0

Thomas Wieder, Jun 13 2008, Jun 29 2008, Jul 28 2008

nonn

Consider the "ordered partitions of partitions" as described in A055887. They are produced by introducing separators (a term used by J. Riordan) between the parts of a partition. If a partition has P parts, then it is possible to introduce 1, 2, ... P-1 separators. Let "|" denote such a separator. We just append 1,2,...,P-1 separators to each integer partition of n and subsequently form all permutation of the resulting list (which is composed of parts and separators).
There are some rules: If we do not append a separator, then we do not perform any permutation. Furthermore, we do not accept permutations which have a dangling separator in front of the integer parts or past them. E.g. the permutations [|,1,2,3] and [1,2,3,|] are forbidden. Furthermore, sequences of separators as "|,|" are forbidden.
Now we impose a further restriction on the permutations. Consider the elements between two separators. We call their number "occupation number". We just request that the occupation number of a ordered partition is monotonically decreasing (if we start from the left to the right of a permutation written in our notation). If we interpret a separator as a level, then we can speak of a hierarchy. E.g. we do not count [1,|,2,3,|,4] as a hierarchy, but we accept [1,2|,3,4] as a hierarchy. We thus speak of "hierarchically ordered partitions of partitions" for this sequence.
With the generating function f := z -> 1/(mul(1-z^i/mul(1-z^j,j=1..i), i=1..25)); we get the asymptotic expansion using the command equivalent (f(z),z,n);
The result is 3.788561346*exp(-n)^(-log(2)) + O(1/n*exp(-n)^(-log(2))). Let fas := n -> 3.788562346*exp(-n)^(-log(2)); then for n=60 we get fas(60)/A141199(60)= .4367915009e19/4344507472742893655 = 1.005387846.
In short, a(n) is the number of finite sequences of integer partitions with weakly decreasing lengths and total sum n. The case of twice-partitions is A358831. A version choosing compositions is A218482. The strictly decreasing case is A358836. For ordered set partitions we have A005651. For weakly decreasing bigomega see A358335. - Gus Wiseman, Dec 05 2022

			n=1:
[1]
-------------------------
n=2:
[1, 1],
[1, "|", 1],
[2]
-------------------------
n=3:
[1, 2],
[1, "|", 1, "|", 1],
[1, 1, 1],
[3],
[2, "|", 1],
[1, 1, "|", 1],
[1, "|", 2]
-------------------------
n=4:
[1, 1, 1, "|", 1],
[1, 1, "|", 1, 1],
[2, 2],
[1, 3],
[1, 1, 1, 1],
[1, 1, 2],
[4],
[1, "|", 1, "|", 1, "|", 1],
[1, 2, "|", 1],
[1, 1, "|", 2],
[1, 1, "|", 1, "|", 1],
[2, "|", 1, "|", 1],
[1, "|", 2, "|", 1],
[1, "|", 1, "|", 2],
[1, "|", 3],
[3, "|", 1],
[2, "|", 2].

Seiichi Manyama, Table of n, a(n) for n = 0..1000
Thomas Wieder, The number of certain rankings and hierarchies formed from labeled or unlabeled elements and sets, Applied Mathematical Sciences, vol. 3, 2009, no. 55, 2707 - 2724. [_Thomas Wieder_, Nov 14 2009]

Cf. A055887, A083355, A140585.

Cf. A000041, A000219, A001970, A005651, A063834, A129519, A218482, A358335, A358831, A358836, A358908.

A Maple program to generate these "hierarchically ordered partitions of partitions" is available on request.
An asymptotic expansion can be found using the generating function given by Vladeta Jovovic. For that purpose we use the Maple program "equivalent" from Bruno Salvy (http://ago.inria.fr/libraries/libraries.html).

my(N=40, x='x+O('x^N)); Vec(1/prod(k=1, N, 1-x^k/prod(j=1, k, 1-x^j))) \\ Seiichi Manyama, Jan 18 2022

G.f.: 1/Product_{i>=1} (1-x^i/Product_{j=1..i} (1-x^j)). - Vladeta Jovovic, Jul 16 2008

More terms from Vladeta Jovovic, Jul 16 2008

a(0)=1 prepended by Seiichi Manyama, Jan 18 2022

A255528 G.f.: Product_{k>=1} 1/(1+x^k)^k.

Original entry on oeis.org

1, -1, -1, -2, 1, 0, 4, 2, 8, -2, 4, -11, -1, -25, -5, -35, 13, -26, 49, -6, 110, 6, 159, -23, 182, -141, 129, -358, 62, -640, 39, -897, 237, -1013, 771, -914, 1793, -664, 3143, -565, 4635, -1157, 5727, -3119, 6121, -7041, 5642, -13088, 5097, -20758, 5879

Offset: 0

Vaclav Kotesovec, Feb 24 2015

sign

In general, if m >= 1 and g.f. = Product_{k>=1} 1/(1 + x^k)^(m*k), then a(n, m) ~ (-1)^n * exp(-m/12 + 3 * 2^(-5/3) * m^(1/3) * Zeta(3)^(1/3) * n^(2/3)) * 2^(m/18 - 5/6) * A^m * m^(1/6 - m/36) * Zeta(3)^(1/6 - m/36) * n^(m/36 - 2/3) / sqrt(3*Pi), where A is the Glaisher-Kinkelin constant A074962. - Vaclav Kotesovec, Apr 13 2017

Seiichi Manyama, Table of n, a(n) for n = 0..10000 (terms 0..5000 from Vaclav Kotesovec)
Vaclav Kotesovec, Graph - the asymptotic ratio (250000 terms)
Vaclav Kotesovec, A method of finding the asymptotics of q-series based on the convolution of generating functions, arXiv:1509.08708 [math.CO], Sep 30 2015, p. 20.

Cf. A000219, A026007, A026011, A073592, A262736.

Cf. A278710 (m=2), A279031 (m=3), A279411 (m=4), A279932 (m=5).

with(numtheory): A000219:=proc(n) option remember; if n = 0 then 1 else add(sigma[2](k)*A000219(n-k), k = 1..n)/n fi: end: A073592:=proc(n) option remember; if n = 0 then 1 else -add(sigma[2](k)*A073592(n-k), k = 1..n)/n fi: end: a:=proc(n); add(A073592(n-2*m)*A000219(m), m = 0..floor(n/2)): end: seq(a(n), n = 0..50); # Vaclav Kotesovec, Mar 09 2015

nmax=100; CoefficientList[Series[Product[1/(1+x^k)^k,{k,1,nmax}],{x,0,nmax}],x]

{a(n) = if(n<0, 0, polcoeff(exp(sum(k=1, n, (-1)^k * x^k / (1-x^k)^2 / k, x*O(x^n))), n))}
for(n=0, 100, print1(a(n), ", "))

a(n) ~ (-1)^n * A * Zeta(3)^(5/36) * exp(3*Zeta(3)^(1/3)*n^(2/3)/2^(5/3) - 1/12) / (2^(7/9) * sqrt(3*Pi) * n^(23/36)), where Zeta(3) = A002117 and A = A074962 is the Glaisher-Kinkelin constant. - Vaclav Kotesovec, Sep 29 2015

a(0) = 1, a(n) = -(1/n)*Sum_{k=1..n} A078306(k)*a(n-k) for n > 0. - Seiichi Manyama, Apr 06 2017

A114736 Number of planar partitions of n where parts strictly decrease along each row and column.

Original entry on oeis.org

1, 1, 1, 3, 4, 6, 10, 15, 22, 33, 49, 70, 102, 146, 205, 290, 405, 561, 779, 1071, 1463, 1999, 2714, 3667, 4946, 6641, 8880, 11848, 15753, 20870, 27586, 36354, 47766, 62621, 81878, 106785, 138975, 180449, 233778, 302270, 390027, 502256, 645603, 828330, 1060851

Offset: 0

Franklin T. Adams-Watters, Mar 16 2006

nonn

If these partitions are "flattened" into a simple partition, the resulting partitions are those for which any part size present with multiplicity k implies the presence of at least k(k-1)/2 larger parts. E.g., [3,1|1] flattens to [3,1^2], 1 has multiplicity 2, so there must be at least 2*1/2 = 1 part larger than 1 - which is the 3.

			For n = 5, we have the 6 partitions [5], [4,1], [4|1], [3,2], [3|2] and [3,1|1].
From _Gus Wiseman_, Nov 15 2018: (Start)
The a(6) = 10 plane partitions:
  6   5 1   4 2   3 2 1
.
  5   4 1   4   3 2   3 1
  1   1     2   1     2
.
  3
  2
  1
(End)

B. Gordon, Multirowed partitions with strict decrease along columns (Notes on plane partitions IV.), Symposia Amer. Math. Soc. 19 (1971) 91-100.

Alois P. Heinz, Table of n, a(n) for n = 0..85

Cf. A000009, A000219, A001970, A007716, A068313, A117433, A120733, A319646, A321645, A321652, A321653, A321655.

prs2mat[prs_]:=Table[Count[prs,{i,j}],{i,Union[First/@prs]},{j,Union[Last/@prs]}];
multsubs[set_,k_]:=If[k==0,{{}},Join@@Table[Prepend[#,set[[i]]]&/@multsubs[Drop[set,i-1],k-1],{i,Length[set]}]];
Table[Length[Select[multsubs[Tuples[Range[n],2],n],And[Union[First/@#]==Range[Max@@First/@#],Union[Last/@#]==Range[Max@@Last/@#],And@@(OrderedQ[#,Greater]&/@prs2mat[#]),And@@(OrderedQ[#,Greater]&/@Transpose[prs2mat[#]])]&]],{n,5}] (* Gus Wiseman, Nov 15 2018 *)

Clarified definition, added 30 terms and reference. - Dennis K Moore, Jan 12 2011

a(40)-a(44) from Alois P. Heinz, Sep 26 2018

A100471 Number of integer partitions of n whose sequence of frequencies is strictly increasing.

Original entry on oeis.org

1, 1, 2, 2, 4, 4, 7, 8, 11, 13, 18, 20, 27, 32, 40, 44, 60, 67, 82, 93, 114, 129, 161, 175, 209, 239, 285, 315, 372, 416, 484, 545, 631, 698, 811, 890, 1027, 1146, 1304, 1437, 1631, 1805, 2042, 2252, 2539, 2785, 3143, 3439, 3846, 4226, 4722, 5159

Offset: 0

David S. Newman, Nov 21 2004

nonn

			a(4) = 4 because of the 5 unrestricted partitions of 4, only one, 3+1 uses each of its summands just once and 1,1 is not an increasing sequence.
From _Gus Wiseman_, Jan 23 2019: (Start)
The a(1) = 1 through a(8) = 11 integer partitions:
  (1)  (2)   (3)    (4)     (5)      (6)       (7)        (8)
       (11)  (111)  (22)    (311)    (33)      (322)      (44)
                    (211)   (2111)   (222)     (511)      (422)
                    (1111)  (11111)  (411)     (4111)     (611)
                                     (3111)    (22111)    (2222)
                                     (21111)   (31111)    (5111)
                                     (111111)  (211111)   (41111)
                                               (1111111)  (221111)
                                                          (311111)
                                                          (2111111)
                                                          (11111111)
(End)

Vaclav Kotesovec, Table of n, a(n) for n = 0..8000 (terms 0..1000 from Alois P. Heinz)

Cf. A000219, A000837 (frequencies are relatively prime), A047966 (frequencies are equal), A098859 (frequencies are distinct), A100881, A100882, A100883, A304686 (Heinz numbers of these partitions).

a100471 n = p 0 (n + 1) 1 n where
   p m m' k x | x == 0    = if m < m' || m == 0 then 1 else 0
              | x < k     = 0
              | m == 0    = p 1 m' k (x - k) + p 0 m' (k + 1) x
              | otherwise = p (m + 1) m' k (x - k) +
                            if m < m' then p 0 m (k + 1) x else 0
-- Reinhard Zumkeller, Dec 27 2012

b:= proc(n,i,t) option remember;
      if n<0 then 0
    elif n=0 then 1
    elif i=1 then `if`(n>t, 1, 0)
    elif i=0 then 0
    else      b(n, i-1, t)
         +add(b(n-i*j, i-1, j), j=t+1..floor(n/i))
      fi
    end:
a:= n-> b(n, n, 0):
seq(a(n), n=0..60);  # Alois P. Heinz, Feb 21 2011

b[n_, i_, t_] := b[n, i, t] = Which[n<0, 0, n==0, 1, i==1, If[n>t, 1, 0], i == 0, 0 , True, b[n, i-1, t] + Sum[b[n-i*j, i-1, j], {j, t+1, Floor[n/i]}]]; a[n_] := b[n, n, 0]; Table[a[n], {n, 0, 60}] (* Jean-François Alcover, Mar 16 2015, after Alois P. Heinz *)
Table[Length[Select[IntegerPartitions[n],OrderedQ@*Split]],{n,20}] (* Gus Wiseman, Jan 23 2019 *)

Corrected and extended by Vladeta Jovovic, Nov 24 2004

Name edited by Gus Wiseman, Jan 23 2019

A117433 Number of planar partitions of n with all part sizes distinct.

Original entry on oeis.org

1, 1, 1, 3, 3, 5, 9, 11, 15, 21, 35, 41, 59, 75, 103, 149, 187, 243, 321, 413, 527, 735, 895, 1165, 1467, 1885, 2335, 2997, 3853, 4765, 5977, 7473, 9269, 11531, 14255, 17537, 22201, 26897, 33233, 40613, 50027, 60637, 74459, 89963, 109751, 134407, 162117, 195859

Offset: 0

Franklin T. Adams-Watters, Mar 16 2006, Apr 01 2008

nonn

Matches A072706 for n < 10, since a unimodal composition into distinct parts can be placed uniquely as a hook. Starting with n = 10, additional partitions are possible (starting with [4,3|2,1] and [4,2|3,1]).

			From _Gus Wiseman_, Nov 15 2018: (Start)
The a(10) = 35 strict plane partitions (A = 10):
  A  64  73  82  532  91  541  631  721  4321
.
  9  54  63  72  432  8  53  71  431  7  43  52  61  421  6  42  51
  1  1   1   1   1    2  2   2   2    3  21  3   3   3    4  31  4
.
  7  6  5  43  42  5  41
  2  3  4  2   3   3  3
  1  1  1  1   1   2  2
.
  4
  3
  2
  1
(End)

Alois P. Heinz, Table of n, a(n) for n = 0..1000 (first 100 terms from Franklin T. Adams-Watters)
OEIS Wiki, Plane partitions

Cf. A000219, A072706, A117434, A000009.

Cf. A001970, A007716, A068313, A114736, A120733, A319646, A321645, A321652, A321653, A321655, A321659, A321660.

b:= proc(n, i) b(n, i):= `if`(n=0, [1], `if`(i<1, [], zip((x, y)
      -> x+y, b(n, i-1), `if`(i>n, [], [0, b(n-i, i-1)[]]), 0)))
    end:
g:= proc(n) g(n):= `if`(n<2, 1, (n-1)*g(n-2) +g(n-1)) end:
a:= proc(n) b(n, n); add(%[i]*g(i-1), i=1..nops(%)) end:
seq(a(n), n=0..60);  # Alois P. Heinz, Nov 18 2012

prs2mat[prs_]:=Table[Count[prs,{i,j}],{i,Union[First/@prs]},{j,Union[Last/@prs]}];
multsubs[set_,k_]:=If[k==0,{{}},Join@@Table[Prepend[#,set[[i]]]&/@multsubs[Drop[set,i-1],k-1],{i,Length[set]}]];
Table[Length[Select[multsubs[Tuples[Range[n],2],n],And[Union[First/@#]==Range[Max@@First/@#],Union[Last/@#]==Range[Max@@Last/@#],UnsameQ@@DeleteCases[Join@@prs2mat[#],0],And@@(OrderedQ[#,Greater]&/@prs2mat[#]),And@@(OrderedQ[#,Greater]&/@Transpose[prs2mat[#]])]&]],{n,5}] (* Gus Wiseman, Nov 15 2018 *)
zip[f_, x_List, y_List, z_] := With[{m = Max[Length[x], Length[y]]}, f[PadRight[x, m, z], PadRight[y, m, z]]];
b[n_, i_] := b[n, i] = If[n == 0, {1}, If[i < 1, {}, zip[Plus, b[n, i - 1], If[i > n, {}, Join[{0}, b[n - i, i - 1]]], 0]]];
g[n_] := g[n] = If[n < 2, 1, (n - 1)*g[n - 2] + g[n - 1]];
a[n_] := With[{bn = b[n, n]}, Sum[bn[[i]]*g[i - 1], {i, 1, Length[bn]}]];
Table[a[n], {n, 0, 60}] (* Jean-François Alcover, Dec 05 2023, after Alois P. Heinz *)

a(n) = Sum_{k=1..floor((sqrt(8*n+1)-1)/2)} A000085(k)*A008289(n,k).

A321405 Number of non-isomorphic self-dual set systems of weight n.

Original entry on oeis.org

1, 1, 1, 2, 2, 3, 6, 9, 16, 28, 47

Offset: 0

Gus Wiseman, Nov 15 2018

Also the number of (0,1) symmetric matrices up to row and column permutations with sum of elements equal to n and no zero rows or columns, in which the rows are all different.
The dual of a multiset partition has, for each vertex, one part consisting of the indices (or positions) of the parts containing that vertex, counted with multiplicity. For example, the dual of {{1,2},{2,2}} is {{1},{1,2,2}}.
The weight of a multiset partition is the sum of sizes of its parts. Weight is generally not the same as number of vertices.

			Non-isomorphic representatives of the a(1) = 1 through a(8) = 16 set systems:
  {{1}}  {{1}{2}}  {{2}{12}}    {{1}{3}{23}}    {{2}{13}{23}}
                   {{1}{2}{3}}  {{1}{2}{3}{4}}  {{1}{2}{4}{34}}
                                                {{1}{2}{3}{4}{5}}
.
  {{12}{13}{23}}        {{13}{23}{123}}          {{1}{13}{14}{234}}
  {{3}{23}{123}}        {{1}{23}{24}{34}}        {{12}{13}{24}{34}}
  {{1}{3}{24}{34}}      {{1}{4}{34}{234}}        {{1}{24}{34}{234}}
  {{2}{4}{12}{34}}      {{2}{13}{24}{34}}        {{2}{14}{34}{234}}
  {{1}{2}{3}{5}{45}}    {{3}{4}{14}{234}}        {{3}{4}{134}{234}}
  {{1}{2}{3}{4}{5}{6}}  {{1}{2}{4}{35}{45}}      {{4}{13}{14}{234}}
                        {{1}{3}{5}{23}{45}}      {{1}{2}{34}{35}{45}}
                        {{1}{2}{3}{4}{6}{56}}    {{1}{2}{5}{45}{345}}
                        {{1}{2}{3}{4}{5}{6}{7}}  {{1}{3}{24}{35}{45}}
                                                 {{1}{4}{5}{25}{345}}
                                                 {{2}{4}{12}{35}{45}}
                                                 {{4}{5}{13}{23}{45}}
                                                 {{1}{2}{3}{5}{46}{56}}
                                                 {{1}{2}{4}{6}{34}{56}}
                                                 {{1}{2}{3}{4}{5}{7}{67}}
                                                 {{1}{2}{3}{4}{5}{6}{7}{8}}

Cf. A000219, A007716, A045778, A049311, A135588, A138178, A283877, A316980, A316983, A319616.

Cf. A320796, A320797, A321401, A321403, A321406.

A098407 Number of different hierarchical orderings that can be formed from n unlabeled elements with no repetition of subhierarchies.

Original entry on oeis.org

1, 1, 2, 6, 13, 33, 78, 186, 436, 1028, 2394, 5566, 12877, 29689, 68198, 156194, 356599, 811959, 1843956, 4177436, 9442166, 21295934, 47932572, 107677140, 241443980, 540441068, 1207689636, 2694452060, 6002389882, 13351958546, 29659179804, 65794744420, 145768641091

Offset: 0

Thomas Wieder, Sep 07 2004; corrected Sep 09 2004

nonn

a(n) is the number of finite sets of compositions with total sum n. The case of constant sums is A358904, cf. A074854. The case of distinct sums is A304961, ordered A336127. The ordered version (sequences of distinct compositions) is A358907. - Gus Wiseman, Dec 12 2022

			Let a pair of parentheses () indicate a subhierarchy and let square brackets [] denote a set of subhierarchies, that is, a hierarchy (also called a society). Let the ranks be ordered from left to right and separated by a colon; e.g., (2:3) is a subhierarchy with three elements ("individuals") on top and two elements on the bottom rank.
Then the hierarchical ordering for n = 4 is composed of the following sets: [(1:1),(2)]; [(1),(3)]; [(1),(1:1:1)]; [(1),(2:1)]; [(1),(1:2)]; [(4)]; [(2:2)]; [(1:3)]; [(3:1)]; [(1:1:2)]; [(1:2:1)]; [(2:1:1)]; [(1:1:1:1)]; thus a(4) = 13.
For example, the following hierarchy is not allowed: [(1),(1),(1),(1)] because of the repetition of (1).

Alois P. Heinz, Table of n, a(n) for n = 0..3217
N. J. A. Sloane and Thomas Wieder, The Number of Hierarchical Orderings, arXiv:math/0307064 [math.CO], 2003; Order 21 (2004), 83-89.

Cf. A011782, A075729.
A034691 counts multisets of compositions, ordered A133494.
A261049 counts sets of partitions, ordered A358906.
Cf. A000009, A000219, A001970, A055887, A075900, A218482, A296122, A304961.

main := proc(n::integer) local a, ListOfPartitions, NumberOfPartitions, APartition, APart, ASet, MultipliticityOfAPart, ndxprttn, ndxprt, Term, Produkt; with(combinat): with(ListTools): a := 0; ListOfPartitions := partition(n); NumberOfPartitions := nops(ListOfPartitions); for ndxprttn from 1 to NumberOfPartitions do APartition := ListOfPartitions[ndxprttn]; ASet := convert(APartition,set); Produkt := 1; for ndxprt from 1 to nops(ASet) do APart := op(ndxprt,ASet); MultipliticityOfAPart := Occurrences(APart, APartition); Term := 2^(APart-1); Term := binomial(Term,MultipliticityOfAPart); Produkt := Produkt * Term; # End of do-loop *** ndxprt ***. end do; a := a + Produkt; # End of do-loop *** ndxprttn ***. end do; print("n, a(n):",n,a); end proc;
PartitionList := proc (n, k) # Authors: # Herbert S. Wilf and Joanna Nordlicht, # Source: # Lecture Notes "East Side West Side,..." # University of Pennsylvania, USA, 2002. # Available from http://www.cis.upenn.edu/~wilf/lecnotes.html # Berechnet die Partitionen von n mit k Summanden. local East, West; if n < 1 or k < 1 or n < k then RETURN([]) elif n = 1 then RETURN([[1]]) else if n < 2 or k < 2 or n < k then West := [] else West := map(proc (x) options operator, arrow; [op(x), 1] end proc, PartitionList(n-1, k-1)) end if; if k <= n-k then East := map(proc(y) options operator, arrow; map(proc (x) options operator, arrow; x+1 end proc, y) end proc, PartitionList(n-k, k)) else East := [] end if; RETURN([op(West), op(East)]) end if end proc;
# second Maple program:
series(exp(add((-1)^(j-1)/j*z^j/(1-2*z^j), j=1..40)), z, 40); # Cf. A102866; Vladeta Jovovic, Feb 19 2008
# alternative Maple program:
b:= proc(n, i) option remember; `if`(n=0 or i=1, `if`(n>1, 0, 1),
      add(b(n-i*j, i-1)*binomial(2^(i-1), j), j=0..n/i))
    end:
a:= n-> b(n$2):
seq(a(n), n=0..32);  # Alois P. Heinz, May 22 2018

terms = 32; CoefficientList[Product[(1 + x^k)^(2^(k-1)), {k, 1, terms+1}] + O[x]^(terms+1), x] // Rest (* Jean-François Alcover, Nov 10 2017, after Vladeta Jovovic *)
nmax = 40; CoefficientList[Series[Exp[Sum[-(-1)^k*x^k/(k*(1 - 2 x^k)), {k, 1, nmax}]], {x, 0, nmax}], x] (* Vaclav Kotesovec, Jun 08 2018 *)

a(n) = Sum_{ partitions n = s_1 + ... + s_n } Product_{ Set{s_i} } C(2^(s_i - 1), m(s_i)), where the sum runs over all partitions of n, the product runs over the set of parts of a given partition, s_i is the i-th part in the set of parts, C(k, l) denotes the binomial coefficient and m(s_i) is the multiplicity of part s_i in the given partition.
G.f.: Product_{k>=1} (1+x^k)^(2^(k-1)). - Vladeta Jovovic, Feb 19 2008
a(n) ~ 2^n * exp(sqrt(2*n) - 1/4 + c) / (sqrt(2*Pi) * 2^(3/4) * n^(3/4)), where c = Sum_{k>=2} -(-1)^k / (k*(2^k-2)) = -0.207530918644117743551169251314627032059... - Vaclav Kotesovec, Jun 08 2018
Weigh transform of A011782. - Alois P. Heinz, Jun 25 2018

More terms from Alois P. Heinz, Apr 21 2012

a(0)=1 prepended by Alois P. Heinz, May 22 2018

cp's OEIS Frontend

A052847 G.f.: 1 / Product_{k>=1} (1-x^k)^(k-1).

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A055922 Number of partitions of n in which each part occurs an odd number (or zero) times.

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A023871 Expansion of Product_{k>=1} (1 - x^k)^(-k^2).

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A141199 Number of hierarchical ordered partitions of partitions.

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A255528 G.f.: Product_{k>=1} 1/(1+x^k)^k.

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A114736 Number of planar partitions of n where parts strictly decrease along each row and column.

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A100471 Number of integer partitions of n whose sequence of frequencies is strictly increasing.