A083670 - OEIS

%I #15 Jan 07 2021 12:16:41

%S 1,2,7,44,558,16926,1319358,269695440,146202099255,212360894456310,

%T 834625722216941739,8954592469138636320960,

%U 264305834899495393164591240,21607243912704793462806305720502,4921054357098031770205099867497197328

%N Number of different antisymmetric relations on n unlabeled points.

%H Andrew Howroyd, <a href="/A083670/b083670.txt">Table of n, a(n) for n = 0..50</a>

%H G. Pfeiffer, <a href="http://www.cs.uwaterloo.ca/journals/JIS/VOL7/Pfeiffer/pfeiffer6.html">Counting Transitive Relations</a>, Journal of Integer Sequences, Vol. 7 (2004), Article 04.3.2.

%F Euler transform of A101460. - _Andrew Howroyd_, Oct 24 2019

%t 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];

%t 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]}];

%t a[n_] := Module[{s = 0}, Do[s += permcount[p]*3^edges[p]*2^Length[p], {p, IntegerPartitions[n]}]; s/n!];

%t a /@ Range[0, 14] (* _Jean-François Alcover_, Jan 07 2021, after _Andrew Howroyd_ *)

%o (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;

%o (PARI)

%o 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}

%o edges(v) = {sum(i=2, #v, sum(j=1, i-1, gcd(v[i],v[j]))) + sum(i=1, #v, (v[i]-1)\2)}

%o a(n) = {my(s=0); forpart(p=n, s+=permcount(p)*3^edges(p)*2^#p); s/n!} \\ _Andrew Howroyd_, Oct 24 2019

%Y Cf. A083667 (labeled antisymmetric relations).

%Y Cf. A001174, A101460.

%K easy,nonn

%O 0,2

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