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

Author

Alois P. Heinz, Jan 26 2011

Keywords

• nonn
• tabf

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.

Examples

			[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];
  ...
		

Links

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

Crossrefs

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

Programs

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]
				

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 *)