login
A320810
Number of non-isomorphic multiset partitions of weight n whose part-sizes have a common divisor > 1.
10
0, 2, 3, 12, 7, 84, 15, 410, 354, 3073, 56, 28300, 101, 210036, 126839, 2070047, 297, 25295952, 490, 269662769, 89071291, 3449056162, 1255, 51132696310, 400625539, 713071048480, 145126661415, 11351097702297, 4565, 199926713003444, 6842, 3460838122540969
OFFSET
1,2
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 column sums are not relatively prime.
Also the number of non-isomorphic multiset partitions of weight n in which the multiset union of the parts is periodic, where a multiset is periodic if its multiplicities have a common divisor > 1.
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
a(n) = A007716(n) - A321283(n). - Andrew Howroyd, Jan 17 2023
EXAMPLE
Non-isomorphic representatives of the a(2) = 1 through a(5) = 7 multiset partitions whose part-sizes have a common divisor:
{{1,1}} {{1,1,1}} {{1,1,1,1}} {{1,1,1,1,1}}
{{1,2}} {{1,2,2}} {{1,1,2,2}} {{1,1,2,2,2}}
{{1,2,3}} {{1,2,2,2}} {{1,2,2,2,2}}
{{1,2,3,3}} {{1,2,2,3,3}}
{{1,2,3,4}} {{1,2,3,3,3}}
{{1,1},{1,1}} {{1,2,3,4,4}}
{{1,1},{2,2}} {{1,2,3,4,5}}
{{1,2},{1,2}}
{{1,2},{2,2}}
{{1,2},{3,3}}
{{1,2},{3,4}}
{{1,3},{2,3}}
Non-isomorphic representatives of the a(2) = 1 through a(5) = 7 multiset partitions with periodic multiset union:
{{1,1}} {{1,1,1}} {{1,1,1,1}} {{1,1,1,1,1}}
{{1},{1}} {{1},{1,1}} {{1,1,2,2}} {{1},{1,1,1,1}}
{{1},{1},{1}} {{1},{1,1,1}} {{1,1},{1,1,1}}
{{1,1},{1,1}} {{1},{1},{1,1,1}}
{{1},{1,2,2}} {{1},{1,1},{1,1}}
{{1,1},{2,2}} {{1},{1},{1},{1,1}}
{{1,2},{1,2}} {{1},{1},{1},{1},{1}}
{{1},{1},{1,1}}
{{1},{1},{2,2}}
{{1},{2},{1,2}}
{{1},{1},{1},{1}}
{{1},{1},{2},{2}}
PROG
(PARI) \\ See links in A339645 for combinatorial species functions.
seq(n)={my(A=symGroupSeries(n)); Vec(OgfSeries(sCartProd(sExp(A), -sum(d=2, n, moebius(d) * (-1 + sExp(O(x*x^n) + sum(i=1, n\d, polcoef(A, i*d)*x^(i*d)))) ))), -n)} \\ Andrew Howroyd, Jan 17 2023
KEYWORD
nonn
AUTHOR
Gus Wiseman, Nov 15 2018
EXTENSIONS
Terms a(11) and beyond from Andrew Howroyd, Jan 17 2023
STATUS
approved