A330055 - OEIS

%I #12 Jan 27 2024 14:08:53

%S 1,0,0,0,0,0,1,1,3,5,16,24,90,179,567,1475,4623,13650,44475,144110,

%T 492017,1706956,6124330,22442687,84406276,324298231,1273955153,

%U 5106977701,20885538133,87046940269,369534837538,1596793560371,7019424870960,31374394197536,142514998263015

%N Number of non-isomorphic set-systems of weight n with no singletons or endpoints.

%C A set-system is a finite set of finite nonempty set of positive integers. A singleton is an edge of size 1. An endpoint is a vertex appearing only once (degree 1). The weight of a set-system is the sum of sizes of its parts. Weight is generally not the same as number of vertices.

%H Andrew Howroyd, <a href="/A330055/b330055.txt">Table of n, a(n) for n = 0..50</a>

%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Degree_(graph_theory)">Degree (graph theory)</a>

%e Non-isomorphic representatives of the a(7) = 1 through a(10) = 16 set-systems:

%e   {12}{13}{123}  {12}{134}{234}    {12}{134}{1234}    {12}{1345}{2345}

%e                  {12}{34}{1234}    {123}{124}{134}    {123}{124}{1234}

%e                  {12}{13}{24}{34}  {12}{13}{14}{234}  {123}{145}{2345}

%e                                    {12}{13}{23}{123}  {12}{345}{12345}

%e                                    {12}{13}{24}{134}  {12}{13}{124}{134}

%e                                                       {12}{13}{124}{234}

%e                                                       {12}{13}{14}{1234}

%e                                                       {12}{13}{24}{1234}

%e                                                       {12}{13}{245}{345}

%e                                                       {12}{13}{45}{2345}

%e                                                       {12}{34}{123}{124}

%e                                                       {12}{34}{125}{345}

%e                                                       {12}{34}{135}{245}

%e                                                       {13}{24}{123}{124}

%e                                                       {12}{13}{14}{23}{24}

%e                                                       {12}{13}{24}{35}{45}

%o (PARI)

%o WeighT(v)={Vec(exp(x*Ser(dirmul(v, vector(#v, n, (-1)^(n-1)/n))))-1, -#v)}

%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 K(q, t, k)={my(g=x*Ser(WeighT(Vec(sum(j=1, #q, my(g=gcd(t, q[j])); g*x^(q[j]/g)) + O(x*x^k), -k)))); (1-x)*g-subst(g,x,x^2)}

%o S(q, t, k)={(x-x^2)*sum(j=1, #q, if(t%q[j]==0, q[j])) + O(x*x^k)}

%o a(n)={if(n==0, 1, my(s=0); forpart(q=n, s+=permcount(q)*polcoef(exp(sum(t=1, n, subst(K(q, t, n\t)-S(q,t,n\t),x,x^t)/t )), n)); s/n!)} \\ _Andrew Howroyd_, Jan 27 2024

%Y The labeled version is A330056.

%Y The "multi" version is A320665.

%Y Non-isomorphic set-systems with no singletons are A306005.

%Y Non-isomorphic set-systems with no endpoints are A330054.

%Y Non-isomorphic set-systems counted by vertices are A000612.

%Y Non-isomorphic set-systems counted by weight are A283877.

%Y Cf. A007716, A055621, A302545, A317533, A317794, A319559, A321405, A330052, A330058.

%K nonn

%O 0,9

%A _Gus Wiseman_, Nov 30 2019

%E a(11) onwards from _Andrew Howroyd_, Jan 27 2024