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.

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

Alois P. Heinz, Table of n, a(n) for n = 0..2000

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

Cf. A100471, A100883, A244393, A295261.

Sequence in context: A207890 A008825 A261629 * A271488 A333347 A302592

Adjacent sequences: A244392 A244393 A244394 * A244396 A244397 A244398

Keyword

nonn

Author

David S. Newman, Jul 03 2014

Extensions

More terms from Alois P. Heinz, Jul 03 2014

Status

approved