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 A098859 Number of partitions of n into parts each of which is used a different number of times. 13

%I

%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 Maciej Ireneusz Wilczynski, <a href="/A098859/b098859.txt">Table of n, a(n) for n = 0..700</a>

%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 Wilf</a>, The Electronic Journal of Combinatorics, Volume 19, Issue 2 (2012)

%H Daniel Kane and Robert C. Rhoades, <a href="http://math.stanford.edu/~rhoades/FILES/wilf.pdf">Asymptotics for Wilf's partitions with distinct multiplicities</a>

%H Doron Zeilberger, <a href="http://www.math.rutgers.edu/~zeilberg/mamarim/mamarimhtml/dmp.html">Using generatingfunctionology to enumerate distinct-multiplicity partitions</a>.

%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.

%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 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

%Y Row sums of A182485.

%Y Cf. A100471, A100881, A105637, A211858, A211859, A211860, A211861, A211862, A211863.

%K nonn,nice,changed

%O 0,3

%A _David S. Newman_, Oct 11 2004

%E Corrected and extended by _Vladeta Jovovic_, Oct 22 2004

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