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.
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
A047993 := proc(n)
a := 0 ;
for p in combinat[partition](n) do
r := max(op(p))-nops(p) ;
if r = 0 then
a := a+1 ;
end if;
end do:
a ;
end proc:
seq(A047993(n), n=1..20) ; # 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 *)
nmax = 100; p = 1; s = 1; Do[p = Normal[Series[p*x^2*(1 - x^(2*k - 1))*(1 + x^k)/(1 - x^k), {x, 0, nmax}]]; s += p; , {k, 1, nmax + 1}]; Take[CoefficientList[s, x], nmax] (* Vaclav Kotesovec, Oct 16 2024 *)
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
KEYWORD
nice,nonn
AUTHOR
STATUS
approved