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A088314 Cardinality of set of sets of parts of all partitions of n.

%I A088314 #56 Sep 14 2023 18:06:43
%S A088314 1,1,2,3,5,6,10,12,18,22,30,37,51,61,79,96,124,148,186,222,275,326,
%T A088314 400,473,575,673,811,946,1132,1317,1558,1813,2138,2463,2893,3323,3882,
%U A088314 4461,5177,5917,6847,7818,8994,10251,11766,13334,15281,17309,19732,22307
%N A088314 Cardinality of set of sets of parts of all partitions of n.
%C A088314 Number of different values of A007947(m) when A056239(m) is equal to n.
%C A088314 From _Gus Wiseman_, Sep 11 2023: (Start)
%C A088314 Also the number of finite sets of positive integers that can be linearly combined using all positive coefficients to obtain n. For example, the a(1) = 1 through a(7) = 12 sets are:
%C A088314   {1}  {1}  {1}    {1}    {1}    {1}      {1}
%C A088314        {2}  {3}    {2}    {5}    {2}      {7}
%C A088314             {1,2}  {4}    {1,2}  {3}      {1,2}
%C A088314                    {1,2}  {1,3}  {6}      {1,3}
%C A088314                    {1,3}  {1,4}  {1,2}    {1,4}
%C A088314                           {2,3}  {1,3}    {1,5}
%C A088314                                  {1,4}    {1,6}
%C A088314                                  {1,5}    {2,3}
%C A088314                                  {2,4}    {2,5}
%C A088314                                  {1,2,3}  {3,4}
%C A088314                                           {1,2,3}
%C A088314                                           {1,2,4}
%C A088314 Cf. A365073, A365311, A365312, A365322, A365380.
%C A088314 (End)
%H A088314 Alois P. Heinz, <a href="/A088314/b088314.txt">Table of n, a(n) for n = 0..100</a>
%F A088314 a(n) = 2^(n-1) - A070880(n). - _Alois P. Heinz_, Feb 08 2019
%F A088314 a(n) = A365042(n) + 1. - _Gus Wiseman_, Sep 13 2023
%e A088314 The 7 partitions of 5 and their sets of parts are
%e A088314 [ #]  partition      set of parts
%e A088314 [ 1]  [ 1 1 1 1 1 ]  {1}
%e A088314 [ 2]  [ 2 1 1 1 ]    {1, 2}
%e A088314 [ 3]  [ 2 2 1 ]      {1, 2}  (same as before)
%e A088314 [ 4]  [ 3 1 1 ]      {1, 3}
%e A088314 [ 5]  [ 3 2 ]        {2, 3}
%e A088314 [ 6]  [ 4 1 ]        {1, 4}
%e A088314 [ 7]  [ 5 ]          {5}
%e A088314 so we have a(5) = |{{1}, {1, 2}, {1, 3}, {2, 3}, {1, 4}, {5}}| = 6.
%p A088314 list2set := L -> {op(L)};
%p A088314 a:= N -> list2set(map( list2set, combinat[partition](N) ));
%p A088314 seq(nops(a(n)), n=0..30);
%p A088314 #  Yogy Namara (yogy.namara(AT)gmail.com), Jan 13 2010
%p A088314 b:= proc(n, i) option remember; `if`(n=0, {{}}, `if`(i<1, {},
%p A088314       {b(n, i-1)[], seq(map(x->{x[],i}, b(n-i*j, i-1))[], j=1..n/i)}))
%p A088314     end:
%p A088314 a:= n-> nops(b(n, n)):
%p A088314 seq(a(n), n=0..40);
%p A088314 # _Alois P. Heinz_, Aug 09 2012
%t A088314 Table[Length[Union[Map[Union,IntegerPartitions[n]]]],{n,1,30}] (* _Geoffrey Critzer_, Feb 19 2013 *)
%t A088314 (* Second program: *)
%t A088314 b[n_, i_] := b[n, i] = If[n == 0, {{}}, If[i < 1, {},
%t A088314      Union@Flatten@{b[n, i - 1], Table[If[Head[#] == List,
%t A088314      Append[#, i]]& /@ b[n - i*j, i - 1], {j, 1, n/i}]}]];
%t A088314 a[n_] := Length[b[n, n]];
%t A088314 a /@ Range[0, 40] (* _Jean-François Alcover_, Jun 04 2021, after _Alois P. Heinz_ *)
%t A088314 combp[n_,y_]:=With[{s=Table[{k,i},{k,y}, {i,1,Floor[n/k]}]}, Select[Tuples[s], Total[Times@@@#]==n&]];
%t A088314 Table[Length[Select[Join@@Array[IntegerPartitions,n], UnsameQ@@#&&combp[n,#]!={}&]], {n,0,15}] (* _Gus Wiseman_, Sep 11 2023 *)
%o A088314 (Haskell)
%o A088314 a066186 = sum . concat . ps 1 where
%o A088314    ps _ 0 = [[]]
%o A088314    ps i j = [t:ts | t <- [i..j], ts <- ps t (j - t)]
%o A088314 -- _Reinhard Zumkeller_, Jul 13 2013
%o A088314 (Python)
%o A088314 from sympy.utilities.iterables import partitions
%o A088314 def A088314(n): return len({tuple(sorted(set(p))) for p in partitions(n)}) # _Chai Wah Wu_, Sep 10 2023
%Y A088314 Cf. A182410.
%Y A088314 The complement in subsets of {1..n-1} is A070880(n) = A365045(n) - 1.
%Y A088314 The case of pairs is A365315, see also A365314, A365320, A365321.
%Y A088314 A116861 and A364916 count linear combinations of strict partitions.
%Y A088314 A179822 and A326080 count sum-closed subsets.
%Y A088314 A326083 and A124506 appear to count combination-free subsets.
%Y A088314 A364914 and A365046 count combination-full subsets.
%Y A088314 Cf. A000009, A007865, A088528, A088809, A093971, A326020, A364272, A364534, A365043, A364350.
%K A088314 easy,nonn
%O A088314 0,3
%A A088314 _Naohiro Nomoto_, Nov 05 2003
%E A088314 More terms and clearer definition from _Vladeta Jovovic_, Apr 21 2005