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A098859 Number of partitions of n into parts each of which is used a different number of times. 16
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,3

COMMENTS

Fill, Janson and Ward refer to these partitions as Wilf partitions. - Peter Luschny, Jun 04 2012

REFERENCES

Stephan Wagner, The Number of Fixed Points of Wilf's Partition Involution, The Electronic Journal of Combinatorics, 20(4) (2013), #P13

LINKS

Maciej Ireneusz Wilczynski, Table of n, a(n) for n = 0..700

James Allen Fill, Svante Janson and Mark Daniel Ward, Partitions with Distinct Multiplicities of Parts: On An "Unsolved Problem" Posed By Herbert 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

Doron Zeilberger, Using generatingfunctionology to enumerate distinct-multiplicity 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}] (* Jean-Fran├žois 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: A026930 A211862 A211863 * A239455 A195012 A238594

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

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Last modified October 1 04:36 EDT 2014. Contains 247503 sequences.