A244395 -id:A244395 - cp's OEIS frontend

A244393 Number of partitions of n the largest part of which, call it m, appears once, m-1 appears at most twice, m-2 appears at most thrice, etc.

1, 1, 1, 2, 3, 4, 6, 9, 13, 17, 25, 33, 45, 61, 82, 106, 142, 183, 238, 306, 395, 499, 638, 804, 1014, 1268, 1586, 1967, 2447, 3018, 3721, 4566, 5598, 6827, 8328, 10108, 12257, 14812, 17884, 21508, 25856, 30980, 37076, 44261, 52776, 62768, 74578, 88407, 104681, 123703, 146018, 172019, 202445, 237830, 279087, 326991, 382706

Offset: 0

David S. Newman, Jul 03 2014

nonn

			For n=6 the partitions counted are 6, 51, 42, 411, 321, 3111
The a(9) = 17 such partitions of 9 are:
01:  [ 3 2 2 1 1 ]
02:  [ 4 2 1 1 1 ]
03:  [ 4 2 2 1 ]
04:  [ 4 3 1 1 ]
05:  [ 4 3 2 ]
06:  [ 5 1 1 1 1 ]
07:  [ 5 2 1 1 ]
08:  [ 5 2 2 ]
09:  [ 5 3 1 ]
10:  [ 5 4 ]
11:  [ 6 1 1 1 ]
12:  [ 6 2 1 ]
13:  [ 6 3 ]
14:  [ 7 1 1 ]
15:  [ 7 2 ]
16:  [ 8 1 ]
17:  [ 9 ]

Alois P. Heinz, Table of n, a(n) for n = 0..1000

Cf. A244395.

b:= proc(n, i, t) option remember; `if`(n=0, 1,
      `if`(i<1, 0, b(n, i-1, `if`(t=1, 1, t+1))+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 29 2017

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]]<= m-Tip[[j]][[1]]+1)];
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, If[t == 1, 1, t + 1]] + 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 05 2021, after Alois P. Heinz *)

More terms from Joerg Arndt, Jul 03 2014

A295261 Partitions into parts with frequency less than or equal to their place in the list of summands.

Original entry on oeis.org

1, 1, 1, 2, 2, 4, 4, 6, 8, 11, 12, 18, 22, 28, 34, 44, 54, 69, 82, 102, 125, 154, 185, 226, 271, 327, 393, 474, 562, 673, 797, 947, 1124, 1329, 1563, 1846, 2164, 2541, 2974, 3480, 4062, 4738, 5508, 6403, 7432, 8614, 9966, 11530, 13307, 15345, 17670, 20337

Offset: 0

David S. Newman, Nov 18 2017

nonn

Let the summands of a partition be s(1) < s(2) < ... < s(k) and the frequency of s(i) be f(i). Then we count those partitions for which f(i) <= i.

			The partition 1+1 is not counted because its smallest part, 1, appears twice.
The partition 3+2+2+1 is counted because its smallest part, 1, appears once; its next smallest part, 2 appears twice (and 2 <= 2) and its third part, 3, appears 1 time (and 1 <= 3).

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

Cf. A244395.

b:= proc(n, i, t) option remember; `if`(n=0, 1, `if`(n b(n, 1$2):
seq(a(n), n=0..60);  # Alois P. Heinz, Nov 18 2017

<< Combinatorica`;
nend = 30;
For[n = 1, n <= nend, n++, count[n] = 0;
  part = Partitions[n];
  For[i = 1, i <= Length[part], i++,
   t = Tally[part[[i]]];
   condition = True;
   For[j = 1, j <= Length[t], j++,
    If[t[[-j, 2]] > j, condition = False ]];
   If[condition, count[n]++]]];
Print[Table[count[i], {i, 1, nend}]]

More terms from Alois P. Heinz, Nov 18 2017

cp's OEIS Frontend

A244393 Number of partitions of n the largest part of which, call it m, appears once, m-1 appears at most twice, m-2 appears at most thrice, etc.

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