A320664 Number of non-isomorphic multiset partitions of weight n with all parts of odd size.
1, 1, 2, 6, 12, 30, 82, 198, 533, 1459, 4039, 11634, 34095, 102520, 316456, 995709, 3215552, 10591412, 35633438, 122499429, 428988392, 1532929060, 5579867442, 20677066725, 78027003260, 299413756170, 1168536196157, 4635420192861, 18678567555721, 76451691937279, 317625507668759
Offset: 0
Keywords
Examples
Non-isomorphic representatives of the a(1) = 1 through a(4) = 12 multiset partitions with all parts of odd size:
{{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}}
{{1},{1},{1}} {{1},{2,3,3}}
{{1},{2},{2}} {{1},{2,3,4}}
{{1},{2},{3}} {{2},{1,2,2}}
{{3},{1,2,3}}
{{1},{1},{1},{1}}
{{1},{1},{2},{2}}
{{1},{2},{2},{2}}
{{1},{2},{3},{3}}
{{1},{2},{3},{4}}
Links
- Andrew Howroyd, Table of n, a(n) for n = 0..50
Programs
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PARI
\\ See links in A339645 for combinatorial species functions. seq(n)={my(A=symGroupSeries(n)); NumUnlabeledObjsSeq(sCartProd(sExp(A), sExp((A-subst(A,x,-x))/2)))} \\ Andrew Howroyd, Jan 17 2023
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PARI
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} J(q, t, k, y)={1/prod(j=1, #q, my(s=q[j], g=gcd(s,t)); (1 + O(x*x^k) - y^(s/g)*x^(s*t/g))^g)} K(q, t, k) = Vec(J(q,t,k,1)-J(q,t,k,-1), -k)/2 a(n)={my(s=0); forpart(q=n, s+=permcount(q)*polcoef(exp(x*Ser(sum(t=1, n, K(q, t, n)/t))), n)); s/n!} \\ Andrew Howroyd, Jan 17 2023
Extensions
Terms a(11) and beyond from Andrew Howroyd, Jan 16 2023
Comments