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A283877 Number of non-isomorphic set-systems of weight n. 134
1, 1, 2, 4, 9, 18, 44, 98, 244, 605, 1595, 4273, 12048, 34790, 104480, 322954, 1031556, 3389413, 11464454, 39820812, 141962355, 518663683, 1940341269, 7424565391, 29033121685, 115921101414, 472219204088, 1961177127371, 8298334192288, 35751364047676, 156736154469354 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,3

COMMENTS

A set-system is a finite set of finite nonempty sets. The weight of a set-system is the sum of cardinalities of its elements.

LINKS

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

FORMULA

Euler transform of A300913.

EXAMPLE

Non-isomorphic representatives of the a(4)=9 set-systems are:

((1234)),

((1)(234)), ((3)(123)), ((12)(34)), ((13)(23)),

((1)(2)(12)), ((1)(2)(34)), ((1)(3)(23)),

((1)(2)(3)(4)).

PROG

(PARI) \\ SetTypes function referenced by other sequences.

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}

V(n, w)={sumdiv(gcd(n, w), d, moebius(d)*binomial(n/d, w/d))/n}

S(n)={my(v=vector(n)); for(w=0, n, fordiv(gcd(n, w), d, v[n/d] += x^w*V(n/d, w/d))); v}

SetTypes(ptyp, fx)={

my(lim=sum(i=1, #ptyp, ptyp[i]), u=vector(lim, i, O(x*x^(lim\i)))); u[1] += 1;

for(i=1, #ptyp, my(s=S(ptyp[i]), v=vector(#u)); for(j=1, #u, for(k=1, #s, my(g=lcm(j, k)); if(g<=#v, v[g]+=u[j]*s[k]*j*k/g))); u=v);

u[1]-=1; Vec(sum(i=1, #u, subst(fx(u[i]), x, x^i)) + O(x*x^lim), -lim); }

a(n) = {my(s=0); forpart(p=n, s+=permcount(p)*WeighT(SetTypes(p, q->q))[n]); s/n!} \\ Andrew Howroyd, Sep 01 2019

CROSSREFS

Cf. A007716, A034691, A049311, A056156, A089259, A116540, A300913.

Sequence in context: A259803 A032175 A000678 * A081490 A292478 A309267

Adjacent sequences:  A283874 A283875 A283876 * A283878 A283879 A283880

KEYWORD

nonn

AUTHOR

Gus Wiseman, Mar 17 2017

EXTENSIONS

a(0) = 1 prepended and terms a(11) and beyond from Andrew Howroyd, Sep 01 2019

STATUS

approved

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Last modified November 14 10:04 EST 2019. Contains 329111 sequences. (Running on oeis4.)