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

%I A181317 #30 Feb 16 2021 08:35:50
%S A181317 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,
%T A181317 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,
%U A181317 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
%N A181317 Triangle in which n-th row lists all partitions of n, in the order of increasing smallest numbers of prime signatures.
%C A181317 The parts of each partition are listed in decreasing order.
%C A181317 Differs from A080577 at a(48) and from A036037 at a(56). A181087 uses the same order for all partitions.
%H A181317 Alois P. Heinz, <a href="/A181317/b181317.txt">Rows n = 1..19, flattened</a>
%e A181317 [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.
%e A181317 Triangle begins:
%e A181317   [1];
%e A181317   [2], [1,1];
%e A181317   [3], [2,1], [1,1,1];
%e A181317   [4], [3,1], [2,2], [2,1,1], [1,1,1,1];
%e A181317   [5], [4,1], [3,2], [3,1,1], [2,2,1], [2,1,1,1], [1,1,1,1,1];
%e A181317   [6], [5,1], [4,2], [3,3], [4,1,1], [3,2,1], [3,1,1,1], [2,2,2];
%e A181317   ...
%p A181317 a:= proc(n) local b, ll;  # gives all parts of partitions of row n
%p A181317   b:= proc(n,i,l)
%p A181317         if n<0 then
%p A181317       elif n=0 then ll:= ll, [mul(ithprime(t)^l[t], t=1..nops(l)), l]
%p A181317       elif i=0 then
%p A181317       else b(n-i, i, [l[], i]), b(n, i-1, l)
%p A181317         fi
%p A181317   end;
%p A181317   ll:= NULL; b(n,n,[]);
%p A181317   map(h-> h[2][], sort([ll], (x,y)-> x[1]<y[1]))[]
%p A181317 end:
%p A181317 seq(a(n), n=1..7);
%t A181317 f[P_] := Times @@ (Prime[Range[Length[P]]]^P);
%t A181317 row[n_] := SortBy[IntegerPartitions[n], f];
%t A181317 Array[row, 7] // Flatten (* _Jean-François Alcover_, Feb 16 2021 *)
%Y A181317 Cf. A036036, A036037, A080576, A080577, A139100, A182937, A181087.
%K A181317 nonn,tabf
%O A181317 1,2
%A A181317 _Alois P. Heinz_, Jan 26 2011