|
|
A321283
|
|
Number of non-isomorphic multiset partitions of weight n in which the part sizes are relatively prime.
|
|
11
|
|
|
1, 1, 2, 7, 21, 84, 214, 895, 2607, 9591, 31134, 119313, 400950, 1574123, 5706112, 22572991, 86933012, 356058243, 1427784135, 6044132304, 25342935667, 110414556330, 481712291885, 2166488898387, 9784077216457, 45369658599779, 211869746691055, 1011161497851296, 4871413403219085
(list;
graph;
refs;
listen;
history;
text;
internal format)
|
|
|
OFFSET
|
0,3
|
|
COMMENTS
|
Also the number of nonnegative integer matrices up to row and column permutations with sum of elements equal to n and no zero rows or columns, in which the row sums are relatively prime.
Also the number of non-isomorphic multiset partitions of weight n in which the multiset union of the parts is aperiodic, where a multiset is aperiodic if its multiplicities are relatively prime.
The weight of a multiset partition is the sum of sizes of its parts. Weight is generally not the same as number of vertices.
|
|
LINKS
|
|
|
FORMULA
|
|
|
EXAMPLE
|
Non-isomorphic representatives of the a(1) = 1 through a(4) = 21 multiset partitions with relatively prime part-sizes:
{{1}} {{1},{1}} {{1},{1,1}} {{1},{1,1,1}}
{{1},{2}} {{1},{2,2}} {{1},{1,2,2}}
{{1},{2,3}} {{1},{2,2,2}}
{{2},{1,2}} {{1},{2,3,3}}
{{1},{1},{1}} {{1},{2,3,4}}
{{1},{2},{2}} {{2},{1,2,2}}
{{1},{2},{3}} {{3},{1,2,3}}
{{1},{1},{1,1}}
{{1},{1},{2,2}}
{{1},{1},{2,3}}
{{1},{2},{1,2}}
{{1},{2},{2,2}}
{{1},{2},{3,3}}
{{1},{2},{3,4}}
{{1},{3},{2,3}}
{{2},{2},{1,2}}
{{1},{1},{1},{1}}
{{1},{1},{2},{2}}
{{1},{2},{2},{2}}
{{1},{2},{3},{3}}
{{1},{2},{3},{4}}
Non-isomorphic representatives of the a(1) = 1 through a(4) = 21 multiset partitions with aperiodic multiset union:
{{1}} {{1,2}} {{1,2,2}} {{1,2,2,2}}
{{1},{2}} {{1,2,3}} {{1,2,3,3}}
{{1},{2,2}} {{1,2,3,4}}
{{1},{2,3}} {{1},{2,2,2}}
{{2},{1,2}} {{1,2},{2,2}}
{{1},{2},{2}} {{1},{2,3,3}}
{{1},{2},{3}} {{1,2},{3,3}}
{{1},{2,3,4}}
{{1,2},{3,4}}
{{1,3},{2,3}}
{{2},{1,2,2}}
{{3},{1,2,3}}
{{1},{1},{2,3}}
{{1},{2},{2,2}}
{{1},{2},{3,3}}
{{1},{2},{3,4}}
{{1},{3},{2,3}}
{{2},{2},{1,2}}
{{1},{2},{2},{2}}
{{1},{2},{3},{3}}
{{1},{2},{3},{4}}
|
|
PROG
|
(PARI) \\ See links in A339645 for combinatorial species functions.
seq(n)={my(A=symGroupSeries(n)); NumUnlabeledObjsSeq(sCartProd(sExp(A), 1 + sum(d=1, n, moebius(d) * (-1 + sExp(O(x*x^n) + sum(i=1, n\d, polcoef(A, i*d)*x^(i*d)))) )))} \\ Andrew Howroyd, Jan 17 2023
(PARI) \\ faster self contained program.
EulerT(v)={Vec(exp(x*Ser(dirmul(v, vector(#v, 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)={EulerT(Vec(sum(j=1, #q, my(g=gcd(t, q[j])); g*x^(q[j]/g)) + O(x*x^k), -k))}
a(n)={if(n==0, 1, my(s=0); forpart(q=n, my(u=vector(n, t, K(q, t, n\t))); s+=permcount(q)*polcoef(sum(d=1, n, moebius(d)*exp(sum(t=1, n\d, sum(i=1, n\(t*d), u[t][i*d]*x^(i*d*t))/t, O(x*x^n)) )), n)); s/n!)} \\ Andrew Howroyd, Jan 17 2023
|
|
CROSSREFS
|
Cf. A000740, A000837, A007716, A007916, A100953, A301700, A303386, A303431, A303546, A303547, A320800-A320810.
|
|
KEYWORD
|
nonn
|
|
AUTHOR
|
|
|
EXTENSIONS
|
|
|
STATUS
|
approved
|
|
|
|