OFFSET
0,3
COMMENTS
Fill, Janson and Ward refer to these partitions as Wilf partitions. - Peter Luschny, Jun 04 2012
LINKS
Simon Langowski and Mark Daniel Ward, Table of n, a(n) for n = 0..2000 (terms 0..300 from M. D. Ward, 301..700 from Maciej Ireneusz Wilczynski)
James Allen Fill, Svante Janson and Mark Daniel Ward, Partitions with Distinct Multiplicities of Parts: On An "Unsolved Problem" Posed By Herbert S. Wilf, The Electronic Journal of Combinatorics, Volume 19, Issue 2 (2012)
Daniel Kane and Robert C. Rhoades, Asymptotics for Wilf's partitions with distinct multiplicities
Martin Klazar, What is an answer? — remarks, results and problems on PIO formulas in combinatorial enumeration, part I, arXiv:1808.08449 [math.CO], 2018.
Simon Langowski, Program to compute Wilf Partitions, 2018
Stephan Wagner, The Number of Fixed Points of Wilf's Partition Involution, The Electronic Journal of Combinatorics, 20(4) (2013), #P13.
FORMULA
log(a(n)) ~ N*log(N) where N = (6*n)^(1/3) (see Fill, Janson and Ward). - Peter Luschny, Jun 04 2012
EXAMPLE
a(6)=7 because 6= 4+1+1= 3+3= 3+1+1+1= 2+2+2= 2+1+1+1+1= 1+1+1+1+1+1. Four unrestricted partitions of 6 are not counted by a(6): 5+1, 4+2, 3+2+1 because at least two different summands are each used once; 2+2+1+1 because each summand is used twice.
From Gus Wiseman, Apr 19 2019: (Start)
The a(1) = 1 through a(9) = 15 partitions are the following. The Heinz numbers of these partitions are given by A130091.
1 2 3 4 5 6 7 8 9
11 111 22 221 33 322 44 333
211 311 222 331 332 441
1111 2111 411 511 422 522
11111 3111 2221 611 711
21111 4111 2222 3222
111111 22111 5111 6111
31111 22211 22221
211111 41111 33111
1111111 221111 51111
311111 411111
2111111 2211111
11111111 3111111
21111111
111111111
(End)
MATHEMATICA
a[n_] := Length[sp = Split /@ IntegerPartitions[n]] - Count[sp, {___List, b_List, ___List, c_List, ___List} /; Length[b] == Length[c]]; Table[a[n], {n, 0, 48}] (* Jean-François Alcover, Jan 17 2013 *)
PROG
(Haskell)
a098859 = p 0 [] 1 where
p m ms _ 0 = if m `elem` ms then 0 else 1
p m ms k x
| x < k = 0
| m == 0 = p 1 ms k (x - k) + p 0 ms (k + 1) x
| m `elem` ms = p (m + 1) ms k (x - k)
| otherwise = p (m + 1) ms k (x - k) + p 0 (m : ms) (k + 1) x
-- Reinhard Zumkeller, Dec 27 2012
(PARI) a(n)={((r, k, b, w)->if(!k||!r, if(r, 0, 1), sum(m=0, r\k, if(!m || !bittest(b, m), self()(r-k*m, k-1, bitor(b, 1<<m), w+m)))))(n, n, 1, 0)} \\ Andrew Howroyd, Aug 31 2019
CROSSREFS
Row sums of A182485.
KEYWORD
nonn,nice
AUTHOR
David S. Newman, Oct 11 2004
EXTENSIONS
Corrected and extended by Vladeta Jovovic, Oct 22 2004
STATUS
approved