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A244395
Number of partitions of n in which the largest summand has frequency 1, the next largest summand has frequency at most 2, the third largest has frequency at most 3, etc.
3
1, 1, 1, 2, 3, 4, 5, 8, 11, 15, 20, 26, 34, 46, 59, 78, 101, 129, 163, 209, 261, 329, 412, 517, 641, 798, 986, 1216, 1493, 1829, 2229, 2721, 3303, 4000, 4841, 5841, 7034, 8458, 10144, 12137, 14512, 17306, 20596, 24483, 29045, 34391, 40680, 48032, 56627, 66666
OFFSET
0,4
LINKS
EXAMPLE
For n=6 the partitions counted are: 6, 51, 42, 411, 321.
MAPLE
b:= proc(n, i, t) option remember; `if`(n=0, 1, `if`(i<1, 0,
b(n, i-1, t) +add(b(n-i*j, i-1, t+1), j=1..min(t, n/i))))
end:
a:= n-> b(n$2, 1):
seq(a(n), n=0..60); # Alois P. Heinz, Jul 03 2014
MATHEMATICA
nend = 20;
For[n = 1, n <= nend, n++,
count[n] = 0;
Ip = IntegerPartitions[n];
For[i = 1, i <= Length[Ip], i++,
m = Max[Ip[[i]]];
condition = True;
Tip = Tally[Ip[[i]]];
For[j = 1, j <= Length[Tip], j++,
condition = condition && (Tip[[j]][[2]] <= j)];
If[condition, count[n]++ (* ; Print[Ip[[i]]] *)]];
]
Table[count[i], {i, 1, nend}]
(* Second program: *)
b[n_, i_, t_] := b[n, i, t] = If[n == 0, 1, If[i < 1, 0,
b[n, i-1, t] + Sum[b[n-i*j, i-1, t+1], {j, 1, Min[t, n/i]}]]];
a[n_] := b[n, n, 1];
a /@ Range[0, 60] (* Jean-François Alcover, Jun 06 2021, after Alois P. Heinz *)
CROSSREFS
KEYWORD
nonn
AUTHOR
David S. Newman, Jul 03 2014
EXTENSIONS
More terms from Alois P. Heinz, Jul 03 2014
STATUS
approved