A047993 Number of balanced partitions of n: the largest part equals the number of parts.

1, 0, 1, 1, 1, 1, 3, 2, 4, 4, 6, 7, 11, 11, 16, 19, 25, 29, 40, 45, 60, 70, 89, 105, 134, 156, 196, 232, 285, 336, 414, 485, 591, 696, 839, 987, 1187, 1389, 1661, 1946, 2311, 2702, 3201, 3731, 4400, 5126, 6018, 6997, 8195, 9502, 11093, 12849, 14949, 17281, 20062

Offset

1,7

Comments

Useful in the creation of plane partitions with C3 or C3v symmetry.

The function T[m,a,b] used here gives the partitions of m whose Ferrers plot fits within an a X b box.

Central terms of triangle in A063995: a(n) = A063995(n,0). - Reinhard Zumkeller, Jul 24 2013

Sequence enumerates the collection of partitions of size n that are in the semigroup of Dyson rank=0, or balanced partitions, under the binary operation A*B = (a1,a2,...,a[k-1],k)*(b1,...,b[n-1,n) = (a1*b1,...,a1*n,a2*b1,...,a2*n,...,k*b1,...,k*n), where A is a partition with k parts and B is a partition with n parts, and A*B is a partition with k*n parts. Note that the rank of A*B is 0, as required. For example, the product of the rank 0 partitions (1,2,3) of 6 and (1,1,3) of 5 is the rank 0 partition (1,1,2,2,3,3,3,6,9) of 30. There is no rank zero partition of 2, as shown in the sequence. It can be seen that any element of the semigroup that partitions an odd prime p or a composite number of form 2p cannot be a product of smaller nontrivial partitions, whether in this semigroup or not. - Richard Locke Peterson, Jul 15 2018

The "multiplication" given above was noted earlier by Franklin T. Adams-Watters in A122697. - Richard Peterson, Jul 19 2023

The Heinz numbers of these integer partitions are given by A106529. - Gus Wiseman, Mar 09 2019

Links

Vaclav Kotesovec, Table of n, a(n) for n = 1..10000 (terms 1..1000 from T. D. Noe)

Erich Friedman, Illustration of initial terms

Formula

a(n) = p(n-1) - p(n-2) - p(n-5) + p(n-7) + ... + (-1)^k*(p(n-(3*k^2-k)/2) - p(n-(3*k^2+k)/2)) + ..., where p() is A000041(). E.g., A047993 a(20) = p(19) - p(18) - p(15) + p(13) + p(8) - p(5) = 490 - 385 - 176 + 101 + 22 - 7 = 45. - Vladeta Jovovic, Aug 04 2004

G.f.: Sum_{k>=1} (-1)^k * ( x^((3*k^2+k)/2) - x^((3*k^2-k)/2) ) ) / Product_{k>=1} (1-x^k). - Vladeta Jovovic, Aug 05 2004

a(n) ~ exp(Pi*sqrt(2*n/3))*Pi / (48*sqrt(2)*n^(3/2)) ~ p(n) * Pi / (4*sqrt(6*n)), where p(n) is the partition function A000041. - Vaclav Kotesovec, Oct 06 2016

G.f.: Sum_{n>=0} [2n,n]_q q^(2*n), where [m,n]_q are the q-binomial coefficients. - Mamuka Jibladze, Aug 12 2021

G.f.: Sum_{k>=1} x^(2*k-1) * Product_{j=1..k-1} (1-x^(k+j-1)/(1-x^j). - Seiichi Manyama, Jan 24 2022

Example

From Joerg Arndt, Oct 08 2012: (Start)
a(12) = 7 because the partitions of 12 where the largest part equals the number of parts are
   2 + 3 + 3 + 4,
   2 + 2 + 4 + 4,
   1 + 3 + 4 + 4,
   1 + 2 + 2 + 2 + 5,
   1 + 1 + 2 + 3 + 5,
   1 + 1 + 1 + 4 + 5, and
   1 + 1 + 1 + 1 + 2 + 6.
(End)
From Gus Wiseman, Mar 09 2019: (Start)
The a(1) = 1 through a(13) = 11 integer partitions:
  1  21  22  311  321  322   332   333    4222   4322    4332    4333
                       331   4211  4221   4321   4331    4422    4432
                       4111        4311   4411   4421    4431    4441
                                   51111  52111  52211   52221   52222
                                                 53111   53211   53221
                                                 611111  54111   53311
                                                         621111  54211
                                                                 55111
                                                                 622111
                                                                 631111
                                                                 7111111
(End)

Maple

with(combinat): for n from 1 to 36 do P:=partition(n): c:=0: for j from 1 to nops(P) do if P[j][nops(P[j])]=nops(P[j]) then c:=c+1 else c:=c fi od: a[n]:=c: od: seq(a[n], n=1..36); # Emeric Deutsch, Dec 11 2004

Mathematica

Table[ Count[Partitions[n], par_List/; First[par]===Length[par]], {n, 12}] or recur: Sum[T[n-(2m-1), m-1, m-1], {m, Ceiling[Sqrt[n]], Floor[(n+1)/2]}] with T[m_, a_, b_]/; b < a := T[m, b, a]; T[m_, a_, b_]/; m > a*b := 0; T[m_, a_, b_]/; (2m > a*b) := T[a*b-m, a, b]; T[m_, 1, b_] := If[b < m, 0, 1]; T[0, _, _] := 1; T[m_, a_, b_] := T[m, a, b]=Sum[T[m-a*i, a-1, b-i], {i, 0, Floor[m/a]}];
Table[Sum[ -(-1)^k*(p[n-(3*k^2-k)/2] - p[n-(3*k^2+k)/2]), {k, 1, Floor[(1+Sqrt[1+24*n])/6]}] /. p -> PartitionsP, {n, 1, 64}] (* Wouter Meeussen *)
(* also *)
Table[Count[IntegerPartitions[n], q_ /; Max[q] == Length[q]], {n, 24}]
(* Clark Kimberling, Feb 13 2014 *)

Prog

(PARI)
N=66;  q='q + O('q^N );
S=2+2*ceil(sqrt(N));
gf= sum(k=1, S,  (-1)^k * ( q^((3*k^2+k)/2) - q^((3*k^2-k)/2) ) ) / prod(k=1, N, 1-q^k );
/* Joerg Arndt, Oct 08 2012 */
(PARI) my(N=66, x='x+O('x^N)); Vec(sum(k=1, N, x^(2*k-1)*prod(j=1, k-1, (1-x^(k+j-1))/(1-x^j)))) \\ Seiichi Manyama, Jan 24 2022
(Haskell)
a047993 = flip a063995 0  -- Reinhard Zumkeller, Jul 24 2013

Crossrefs

Cf. A000700, A063995, A064173, A064174.

Cf. A003114, A006141, A039900, A090858, A106529, A324516, A324518, A324520.

Column 1 of A350879.

Sequence in context: A327637 A366409 A345082 * A033177 A175512 A240829

Adjacent sequences: A047990 A047991 A047992 * A047994 A047995 A047996

Keyword

nice,nonn

Author

Wouter Meeussen

Status

approved