A301924 - OEIS

%I #12 Aug 27 2019 10:44:34

%S 1,0,1,0,2,1,0,6,3,1,0,21,29,4,1,0,112,2101,150,5,1,0,853,7011181,

%T 7013164,1037,6,1,0,11117,1788775603301,29281354507753847,

%U 1788782615612,12338,7,1,0,261080,53304526022885278403,234431745534048893449761040648508,234431745534048922729326772799024,53304527811667884902,274659,8,1

%N Regular triangle where T(n,k) is the number of unlabeled k-uniform connected hypergraphs spanning n vertices.

%F Column k is the inverse Euler transform of column k of A301922. - _Andrew Howroyd_, Aug 26 2019

%e Triangle begins:

%e    1

%e    0    1

%e    0    2       1

%e    0    6       3       1

%e    0   21      29       4    1

%e    0  112    2101     150    5 1

%e    0  853 7011181 7013164 1037 6 1

%e    ...

%e The T(4,2) = 6 hypergraphs:

%e   {{1,3},{2,4},{3,4}}

%e   {{1,4},{2,4},{3,4}}

%e   {{1,2},{1,3},{2,4},{3,4}}

%e   {{1,4},{2,3},{2,4},{3,4}}

%e   {{1,3},{1,4},{2,3},{2,4},{3,4}}

%e   {{1,2},{1,3},{1,4},{2,3},{2,4},{3,4}}

%o (PARI)

%o InvEulerT(v)={my(p=log(1+x*Ser(v))); dirdiv(vector(#v,n,polcoeff(p,n)), vector(#v,n,1/n))}

%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 rep(typ)={my(L=List(), k=0); for(i=1, #typ, k+=typ[i]; listput(L, k); while(#L<k, listput(L, #L))); Vec(L)}

%o can(v, f)={my(d=1, u=v); while(d>0, u=vecsort(apply(f, u)); d=lex(u, v)); !d}

%o Q(n, k, perm)={my(t=0); forsubset([n, k], v, t += can(Vec(v), t->perm[t])); t}

%o U(n, k)={my(s=0); forpart(p=n, s += permcount(p)*2^Q(n, k, rep(p))); s/n!}

%o A(n)={Mat(vector(n, k, InvEulerT(vector(n,i,U(i,k)-U(i-1,k)))~))}

%o { my(T=A(8)); for(n=1, #T, print(T[n,1..n])) } \\ _Andrew Howroyd_, Aug 26 2019

%Y Row sums are A301920.

%Y Columns k=2..3 are A001349(n > 1), A003190(n > 1).

%Y Cf. A003465, A006129, A038041, A055621, A298422, A299353, A299354, A299471, A301481, A301922, A306017-A306021, A309858.

%K nonn,tabl

%O 1,5

%A _Gus Wiseman_, Jun 19 2018

%E Terms a(16) and beyond from _Andrew Howroyd_, Aug 26 2019