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A327903 Number of set-systems covering n vertices where every edge has a different sum.

%I A327903 #8 Oct 03 2019 16:46:19
%S A327903 1,1,5,77,7369,10561753,839653402893,15924566366443524837,
%T A327903 315320784127456186118309342769,
%U A327903 29238175285109256786706269143580213236526609,59347643832090275881798554403880633753161146711444051797893301
%N A327903 Number of set-systems covering n vertices where every edge has a different sum.
%C A327903 A set-system is a set of nonempty sets. It is covering if there are no isolated (uncovered) vertices.
%H A327903 Andrew Howroyd, <a href="/A327903/b327903.txt">Table of n, a(n) for n = 0..20</a>
%H A327903 Gus Wiseman, <a href="/A038041/a038041.txt">Sequences counting and ranking multiset partitions whose part lengths, sums, or averages are constant or strict.</a>
%e A327903 The a(3) = 77 set-systems:
%e A327903   123  1-23    1-2-3      1-2-3-13      1-2-3-13-23     1-2-3-13-23-123
%e A327903        2-13    1-2-13     1-2-3-23      1-2-12-13-23    1-2-12-13-23-123
%e A327903        1-123   1-2-23     1-2-12-13     1-2-3-13-123
%e A327903        12-13   1-3-23     1-2-12-23     1-2-3-23-123
%e A327903        12-23   2-3-13     1-2-13-23     1-2-12-13-123
%e A327903        13-23   1-12-13    1-2-3-123     1-2-12-23-123
%e A327903        2-123   1-12-23    1-3-13-23     1-2-13-23-123
%e A327903        3-123   1-13-23    2-3-13-23     1-3-13-23-123
%e A327903        12-123  1-2-123    1-12-13-23    2-3-13-23-123
%e A327903        13-123  1-3-123    1-2-12-123    1-12-13-23-123
%e A327903        23-123  2-12-13    1-2-13-123    2-12-13-23-123
%e A327903                2-12-23    1-2-23-123
%e A327903                2-13-23    1-3-13-123
%e A327903                2-3-123    1-3-23-123
%e A327903                3-13-23    2-12-13-23
%e A327903                1-12-123   2-3-13-123
%e A327903                1-13-123   2-3-23-123
%e A327903                12-13-23   1-12-13-123
%e A327903                1-23-123   1-12-23-123
%e A327903                2-12-123   1-13-23-123
%e A327903                2-13-123   2-12-13-123
%e A327903                2-23-123   2-12-23-123
%e A327903                3-13-123   2-13-23-123
%e A327903                3-23-123   3-13-23-123
%e A327903                12-13-123  12-13-23-123
%e A327903                12-23-123
%e A327903                13-23-123
%t A327903 stableSets[u_,Q_]:=If[Length[u]==0,{{}},With[{w=First[u]},Join[stableSets[DeleteCases[u,w],Q],Prepend[#,w]&/@stableSets[DeleteCases[u,r_/;r==w||Q[r,w]||Q[w,r]],Q]]]];
%t A327903 qes[n_]:=Select[stableSets[Subsets[Range[n],{1,n}],Total[#1]==Total[#2]&],Union@@#==Range[n]&];
%t A327903 Table[Length[qes[n]],{n,0,4}]
%o A327903 (PARI) \\ by inclusion/exclusion on covered vertices.
%o A327903 C(v)={my(u=Vecrev(-1 + prod(k=1, #v, 1 + x^v[k]))); prod(i=1, #u, 1 + u[i])}
%o A327903 a(n)={my(s=0); forsubset(n, v, s += (-1)^(n-#v)*C(v)); s} \\ _Andrew Howroyd_, Oct 02 2019
%Y A327903 The antichain case is A326572.
%Y A327903 The graphical case is A327904.
%Y A327903 Cf. A003465, A006126, A035470, A275780, A321469, A326030, A326519, A326535, A326565, A326571, A326573.
%K A327903 nonn
%O A327903 0,3
%A A327903 _Gus Wiseman_, Sep 30 2019
%E A327903 Terms a(4) and beyond from _Andrew Howroyd_, Oct 02 2019