A327904 Number of labeled simple graphs with vertices {1..n} such that every edge has a different sum.

1, 1, 2, 8, 48, 432, 5184, 82944, 1658880, 41472000, 1244160000, 44789760000, 1881169920000, 92177326080000, 5161930260480000, 330363536670720000, 23786174640291840000, 1926680145863639040000, 173401213127727513600000, 17340121312772751360000000

Offset: 0

Author

Gus Wiseman, Sep 30 2019

Keywords

• nonn

Examples

			The graph with edge-set {{1,2},{1,3},{1,4},{2,3}}, which looks like a triangle with a tail, has edges {1,4} and {2,3} with equal sum, so is not counted under a(4).
		

Links

• Andrew Howroyd, Table of n, a(n) for n = 0..100
• Gus Wiseman, Sequences counting and ranking multiset partitions whose part lengths, sums, or averages are constant or strict.
• Gus Wiseman, The a(4) = 48 graphs where every edge has a different sum.

Crossrefs

The generalization to antichains is A326030.

Cf. A006125, A006129, A275780, A321469, A326519, A326535, A327903.

Programs

Maple

a:= proc(n) option remember; `if`(n=0, 1,
      a(n-1)*ceil(n/2)*ceil(n/2+1/4))
    end:
seq(a(n), n=0..20);  # Alois P. Heinz, Oct 03 2019

Mathematica

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]]]];
qes[n_]:=stableSets[Subsets[Range[n],{2}],Total[#1]==Total[#2]&];
Table[Length[qes[n]],{n,0,5}]

PARI

a(n) = {prod(k=1, 2*n+1, ceil(k/4))} \\ Andrew Howroyd, Oct 02 2019

Formula

a(n) = Product_{k=1..2*n+1} ceiling(k/4). - Andrew Howroyd, Oct 02 2019

Extensions

Terms a(8) and beyond from Andrew Howroyd, Oct 02 2019