

A098859


Number of partitions of n into parts each of which is used a different number of times.


27



1, 1, 2, 2, 4, 5, 7, 10, 13, 15, 21, 28, 31, 45, 55, 62, 82, 105, 116, 153, 172, 208, 251, 312, 341, 431, 492, 588, 676, 826, 905, 1120, 1249, 1475, 1676, 2003, 2187, 2625, 2922, 3409, 3810, 4481, 4910, 5792, 6382, 7407, 8186, 9527, 10434
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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.
Doron Zeilberger, Using generatingfunctionology to enumerate distinctmultiplicity partitions.


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.


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}] (* JeanFrançois 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


CROSSREFS

Row sums of A182485.
Cf. A100471, A100881, A105637, A211858, A211859, A211860, A211861, A211862, A211863, A242882.
Sequence in context: A211862 A286948 A211863 * A239455 A195012 A323092
Adjacent sequences: A098856 A098857 A098858 * A098860 A098861 A098862


KEYWORD

nonn,nice


AUTHOR

David S. Newman, Oct 11 2004


EXTENSIONS

Corrected and extended by Vladeta Jovovic, Oct 22 2004


STATUS

approved



