A330054 Number of non-isomorphic set-systems of weight n with no endpoints.

1, 0, 0, 0, 1, 0, 4, 4, 16, 26, 87, 181, 570, 1453, 4464, 13038, 41548, 132217, 442603, 1506803, 5305174, 19092816, 70548770, 266495254, 1029835424, 4063610148, 16366919221, 67217627966, 281326631801, 1199048810660, 5201341196693, 22950740113039, 102957953031700

Offset

0,7

Comments

A set-system is a finite set of finite nonempty set of positive integers. 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.

Links

Andrew Howroyd, Table of n, a(n) for n = 0..50

Wikipedia, Degree (graph theory)

Example

Non-isomorphic representatives of the a(0) = 1 through a(8) = 16 multiset partitions (empty columns not shown):
  0  {1}{2}{12}  {12}{13}{23}    {13}{23}{123}      {12}{134}{234}
                 {1}{23}{123}    {1}{3}{23}{123}    {1}{234}{1234}
                 {1}{2}{13}{23}  {3}{12}{13}{23}    {12}{34}{1234}
                 {1}{2}{3}{123}  {1}{2}{3}{13}{23}  {1}{12}{34}{234}
                                                    {12}{13}{24}{34}
                                                    {1}{2}{134}{234}
                                                    {1}{2}{34}{1234}
                                                    {2}{13}{14}{234}
                                                    {2}{13}{23}{123}
                                                    {3}{13}{23}{123}
                                                    {1}{2}{13}{24}{34}
                                                    {1}{2}{3}{14}{234}
                                                    {1}{2}{3}{23}{123}
                                                    {1}{2}{3}{4}{1234}
                                                    {2}{3}{12}{13}{23}
                                                    {1}{2}{3}{4}{12}{34}

Prog

(PARI)
WeighT(v)={Vec(exp(x*Ser(dirmul(v, vector(#v, n, (-1)^(n-1)/n))))-1, -#v)}
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}
K(q, t, k)={my(g=1+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)}
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)/t, x, x^t) )), n)); s/n!)} \\ Andrew Howroyd, Jan 27 2024

Crossrefs

The complement is counted by A330052.

The multiset partition version is A302545.

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

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

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

Cf. A007716, A055621, A317533, A317794, A319559, A320665, A330055, A330056, A330058.

Sequence in context: A053441 A065732 A092959 * A183433 A322039 A158101

Adjacent sequences: A330051 A330052 A330053 * A330055 A330056 A330057

Keyword

nonn

Author

Gus Wiseman, Nov 30 2019

Extensions

a(11) onwards from Andrew Howroyd, Jan 27 2024

Status

approved