A083670 Number of different antisymmetric relations on n unlabeled points.

1, 2, 7, 44, 558, 16926, 1319358, 269695440, 146202099255, 212360894456310, 834625722216941739, 8954592469138636320960, 264305834899495393164591240, 21607243912704793462806305720502, 4921054357098031770205099867497197328

Offset: 0

Author

Goetz Pfeiffer (Goetz.Pfeiffer(AT)nuigalway.ie), May 02 2003

Keywords

• easy
• nonn

Links

• Andrew Howroyd, Table of n, a(n) for n = 0..50
• G. Pfeiffer, Counting Transitive Relations, Journal of Integer Sequences, Vol. 7 (2004), Article 04.3.2.

Crossrefs

Cf. A083667 (labeled antisymmetric relations).

Cf. A001174, A101460.

Programs

GAP

f := function(n) local s, m, c, t, x, a, j; s := 0; m := [1..n]; c := Combinations(m,2); t := Tuples(m,2); for x in ConjugacyClasses(SymmetricGroup(n)) do a := Representative(x); j := Length(Cycles(a,m)); s := s+Size(x)*2^j*3^(Length(Cycles(a,t,OnPairs))-Length(Cycles(a,c,OnSets))-j); od; return s/Factorial(n); end;

Mathematica

permcount[v_] := Module[{m = 1, s = 0, k = 0, t}, For[i = 1, i <= Length[v], i++, t = v[[i]]; k = If[i > 1 && t == v[[i - 1]], k + 1, 1]; m *= t*k; s += t]; s!/m];
edges[v_] := Sum[Sum[GCD[v[[i]], v[[j]]], {j, 1, i - 1}], {i, 2, Length[v]}] + Sum[Quotient[v[[i]] - 1, 2], {i, 1, Length[v]}];
a[n_] := Module[{s = 0}, Do[s += permcount[p]*3^edges[p]*2^Length[p], {p, IntegerPartitions[n]}]; s/n!];
a /@ Range[0, 14] (* Jean-François Alcover, Jan 07 2021, after Andrew Howroyd *)

PARI

permcount(v) = {my(m=1,s=0,k=0,t); for(i=1,#v,t=v[i]; k=if(i>1&&t==v[i-1],k+1,1); m*=t*k;s+=t); s!/m}
edges(v) = {sum(i=2, #v, sum(j=1, i-1, gcd(v[i],v[j]))) + sum(i=1, #v, (v[i]-1)\2)}
a(n) = {my(s=0); forpart(p=n, s+=permcount(p)*3^edges(p)*2^#p); s/n!} \\ Andrew Howroyd, Oct 24 2019

Formula

Euler transform of A101460. - Andrew Howroyd, Oct 24 2019