|
|
A098859
|
|
Number of partitions of n into parts each of which is used a different number of times.
|
|
169
|
|
|
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
|
|
|
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.
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
(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
|
|
CROSSREFS
|
|
|
KEYWORD
|
nonn,nice
|
|
AUTHOR
|
|
|
EXTENSIONS
|
|
|
STATUS
|
approved
|
|
|
|