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A362644
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Array read by antidiagonals: T(n,k) is the number of nonisomorphic multisets of permutations of an n-set with k permutations.
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9
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1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 3, 3, 1, 1, 1, 4, 8, 5, 1, 1, 1, 5, 17, 28, 7, 1, 1, 1, 6, 34, 159, 96, 11, 1, 1, 1, 7, 61, 888, 2655, 495, 15, 1, 1, 1, 8, 105, 4521, 76854, 88885, 2919, 22, 1, 1, 1, 9, 170, 20916, 1882581, 15719714, 4255594, 22024, 30, 1
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OFFSET
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0,9
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COMMENTS
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Isomorphism is up to permutation of the elements of the n-set. Each permutation can be considered to be a set of disjoint directed cycles whose vertices cover the n-set. Permuting the elements of the n-set permutes each of the permutations in the multiset.
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LINKS
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FORMULA
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T(0,k) = T(1,k) = 1.
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EXAMPLE
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Array begins:
====================================================================
n/k| 0 1 2 3 4 5 6 ...
---+----------------------------------------------------------------
0 | 1 1 1 1 1 1 1 ...
1 | 1 1 1 1 1 1 1 ...
2 | 1 2 3 4 5 6 7 ...
3 | 1 3 8 17 34 61 105 ...
4 | 1 5 28 159 888 4521 20916 ...
5 | 1 7 96 2655 76854 1882581 39122096 ...
6 | 1 11 495 88885 15719714 2271328951 274390124129 ...
7 | 1 15 2919 4255594 5341866647 5387750530872 4530149870111873 ...
...
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PROG
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(PARI)
B(n, k) = {n!*k^n}
K(v)=my(S=Set(v)); prod(i=1, #S, my(k=S[i], c=#select(t->t==k, v)); B(c, k))
R(v, m)=concat(vector(#v, i, my(t=v[i], g=gcd(t, m)); vector(g, i, t/g)))
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}
T(n, k) = {if(n==0, 1, my(s=0); forpart(q=n, s += permcount(q) * polcoef(exp(sum(m=1, k, K(R(q, m))*x^m/m, O(x*x^k))), k)); s/n!)}
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CROSSREFS
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KEYWORD
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AUTHOR
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STATUS
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approved
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