A181317 Triangle in which n-th row lists all partitions of n, in the order of increasing smallest numbers of prime signatures.

1, 2, 1, 1, 3, 2, 1, 1, 1, 1, 4, 3, 1, 2, 2, 2, 1, 1, 1, 1, 1, 1, 5, 4, 1, 3, 2, 3, 1, 1, 2, 2, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 6, 5, 1, 4, 2, 3, 3, 4, 1, 1, 3, 2, 1, 3, 1, 1, 1, 2, 2, 2, 2, 2, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 7, 6, 1, 5, 2, 4, 3, 5, 1, 1, 4, 2, 1, 3, 3, 1, 4, 1, 1, 1, 3, 2, 2, 3, 2, 1, 1, 2

Offset

1,2

Comments

The parts of each partition are listed in decreasing order.

Differs from A080577 at a(48) and from A036037 at a(56). A181087 uses the same order for all partitions.

Links

Alois P. Heinz, Rows n = 1..19, flattened

Example

[3,1,1,1] and [2,2,2] are both partitions of 6, the smallest numbers having these prime signatures are 2^3*3^1*5^1*7^1=840 and 2^2*3^2*5^2=900, thus [3,1,1,1] < [2,2,2] in this order.
Triangle begins:
  [1];
  [2], [1,1];
  [3], [2,1], [1,1,1];
  [4], [3,1], [2,2], [2,1,1], [1,1,1,1];
  [5], [4,1], [3,2], [3,1,1], [2,2,1], [2,1,1,1], [1,1,1,1,1];
  [6], [5,1], [4,2], [3,3], [4,1,1], [3,2,1], [3,1,1,1], [2,2,2];
  ...

Maple

a:= proc(n) local b, ll;  # gives all parts of partitions of row n
  b:= proc(n, i, l)
        if n<0 then
      elif n=0 then ll:= ll, [mul(ithprime(t)^l[t], t=1..nops(l)), l]
      elif i=0 then
      else b(n-i, i, [l[], i]), b(n, i-1, l)
        fi
  end;
  ll:= NULL; b(n, n, []);
  map(h-> h[2][], sort([ll], (x, y)-> x[1]<y[1]))[]
end:
seq(a(n), n=1..7);

Mathematica

f[P_] := Times @@ (Prime[Range[Length[P]]]^P);
row[n_] := SortBy[IntegerPartitions[n], f];
Array[row, 7] // Flatten (* Jean-François Alcover, Feb 16 2021 *)

Crossrefs

Cf. A036036, A036037, A080576, A080577, A139100, A182937, A181087.

Sequence in context: A239512 A334439 A036037 * A330370 A330371 A080577

Adjacent sequences: A181314 A181315 A181316 * A181318 A181319 A181320

Keyword

nonn,tabf

Author

Alois P. Heinz, Jan 26 2011

Status

approved