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%I #88 Mar 28 2021 23:42:35
%S 1,1,2,2,4,5,7,10,13,15,21,28,31,45,55,62,82,105,116,153,172,208,251,
%T 312,341,431,492,588,676,826,905,1120,1249,1475,1676,2003,2187,2625,
%U 2922,3409,3810,4481,4910,5792,6382,7407,8186,9527,10434
%N Number of partitions of n into parts each of which is used a different number of times.
%C Fill, Janson and Ward refer to these partitions as Wilf partitions. - _Peter Luschny_, Jun 04 2012
%H Simon Langowski and Mark Daniel Ward, <a href="/A098859/b098859.txt">Table of n, a(n) for n = 0..2000</a> (terms 0..300 from M. D. Ward, 301..700 from Maciej Ireneusz Wilczynski)
%H James Allen Fill, Svante Janson and Mark Daniel Ward, <a href="http://www.combinatorics.org/ojs/index.php/eljc/article/view/v19i2p18">Partitions with Distinct Multiplicities of Parts: On An "Unsolved Problem" Posed By Herbert S. Wilf</a>, The Electronic Journal of Combinatorics, Volume 19, Issue 2 (2012)
%H Daniel Kane and Robert C. Rhoades, <a href="https://web.archive.org/web/20160809023551/http://math.stanford.edu/~rhoades/FILES/wilf.pdf">Asymptotics for Wilf's partitions with distinct multiplicities</a>
%H Martin Klazar, <a href="http://arxiv.org/abs/1808.08449">What is an answer? — remarks, results and problems on PIO formulas in combinatorial enumeration, part I</a>, arXiv:1808.08449 [math.CO], 2018.
%H Simon Langowski, <a href="https://github.com/SimonLangowski/WilfPartition">Program to compute Wilf Partitions</a>, 2018
%H Stephan Wagner, <a href="http://www.combinatorics.org/ojs/index.php/eljc/article/view/v20i4p13">The Number of Fixed Points of Wilf's Partition Involution</a>, The Electronic Journal of Combinatorics, 20(4) (2013), #P13.
%H Doron Zeilberger, <a href="http://www.math.rutgers.edu/~zeilberg/mamarim/mamarimhtml/dmp.html">Using generatingfunctionology to enumerate distinct-multiplicity partitions</a>; <a href="/A098859/a098859.pdf">Local copy/a> [Pdf file only, no active links]
%F log(a(n)) ~ N*log(N) where N = (6*n)^(1/3) (see Fill, Janson and Ward). - _Peter Luschny_, Jun 04 2012
%e 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.
%e From _Gus Wiseman_, Apr 19 2019: (Start)
%e The a(1) = 1 through a(9) = 15 partitions are the following. The Heinz numbers of these partitions are given by A130091.
%e 1 2 3 4 5 6 7 8 9
%e 11 111 22 221 33 322 44 333
%e 211 311 222 331 332 441
%e 1111 2111 411 511 422 522
%e 11111 3111 2221 611 711
%e 21111 4111 2222 3222
%e 111111 22111 5111 6111
%e 31111 22211 22221
%e 211111 41111 33111
%e 1111111 221111 51111
%e 311111 411111
%e 2111111 2211111
%e 11111111 3111111
%e 21111111
%e 111111111
%e (End)
%t 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 *)
%o (Haskell)
%o a098859 = p 0 [] 1 where
%o p m ms _ 0 = if m `elem` ms then 0 else 1
%o p m ms k x
%o | x < k = 0
%o | m == 0 = p 1 ms k (x - k) + p 0 ms (k + 1) x
%o | m `elem` ms = p (m + 1) ms k (x - k)
%o | otherwise = p (m + 1) ms k (x - k) + p 0 (m : ms) (k + 1) x
%o -- _Reinhard Zumkeller_, Dec 27 2012
%o (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
%Y Row sums of A182485.
%Y Cf. A100471, A100881, A105637, A211858, A211859, A211860, A211861, A211862, A211863, A242882.
%Y Cf. A047966 (each part appears the same number of times), A090858, A116608, A130091, A325242.
%K nonn,nice
%O 0,3
%A _David S. Newman_, Oct 11 2004
%E Corrected and extended by _Vladeta Jovovic_, Oct 22 2004
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