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

0,3

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.

Formula

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

Example

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).

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}]

Prog

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

Crossrefs

The generalization to antichains is A326030.

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

Sequence in context: A177386 A112541 A052667 * A006925 A185135 A238805

Adjacent sequences: A327901 A327902 A327903 * A327905 A327906 A327907

Keyword

nonn

Author

Gus Wiseman, Sep 30 2019

Extensions

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

Status

approved