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 A098859 Number of partitions of n into parts each of which is used a different number of times. 56
 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 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 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. 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<

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Last modified September 16 20:33 EDT 2019. Contains 327118 sequences. (Running on oeis4.)