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A100471
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Number of integer partitions of n whose sequence of frequencies is strictly increasing.
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17
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1, 1, 2, 2, 4, 4, 7, 8, 11, 13, 18, 20, 27, 32, 40, 44, 60, 67, 82, 93, 114, 129, 161, 175, 209, 239, 285, 315, 372, 416, 484, 545, 631, 698, 811, 890, 1027, 1146, 1304, 1437, 1631, 1805, 2042, 2252, 2539, 2785, 3143, 3439, 3846, 4226, 4722, 5159
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OFFSET
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0,3
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LINKS
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EXAMPLE
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a(4) = 4 because of the 5 unrestricted partitions of 4, only one, 3+1 uses each of its summands just once and 1,1 is not an increasing sequence.
The a(1) = 1 through a(8) = 11 integer partitions:
(1) (2) (3) (4) (5) (6) (7) (8)
(11) (111) (22) (311) (33) (322) (44)
(211) (2111) (222) (511) (422)
(1111) (11111) (411) (4111) (611)
(3111) (22111) (2222)
(21111) (31111) (5111)
(111111) (211111) (41111)
(1111111) (221111)
(311111)
(2111111)
(11111111)
(End)
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MAPLE
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b:= proc(n, i, t) option remember;
if n<0 then 0
elif n=0 then 1
elif i=1 then `if`(n>t, 1, 0)
elif i=0 then 0
else b(n, i-1, t)
+add(b(n-i*j, i-1, j), j=t+1..floor(n/i))
fi
end:
a:= n-> b(n, n, 0):
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MATHEMATICA
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b[n_, i_, t_] := b[n, i, t] = Which[n<0, 0, n==0, 1, i==1, If[n>t, 1, 0], i == 0, 0 , True, b[n, i-1, t] + Sum[b[n-i*j, i-1, j], {j, t+1, Floor[n/i]}]]; a[n_] := b[n, n, 0]; Table[a[n], {n, 0, 60}] (* Jean-François Alcover, Mar 16 2015, after Alois P. Heinz *)
Table[Length[Select[IntegerPartitions[n], OrderedQ@*Split]], {n, 20}] (* Gus Wiseman, Jan 23 2019 *)
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PROG
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(Haskell)
a100471 n = p 0 (n + 1) 1 n where
p m m' k x | x == 0 = if m < m' || m == 0 then 1 else 0
| x < k = 0
| m == 0 = p 1 m' k (x - k) + p 0 m' (k + 1) x
| otherwise = p (m + 1) m' k (x - k) +
if m < m' then p 0 m (k + 1) x else 0
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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