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A362763
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Array read by antidiagonals: T(n,k) is the number of nonisomorphic k-sets of permutations of an n-set.
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4
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1, 1, 1, 0, 1, 1, 0, 0, 2, 1, 0, 0, 1, 3, 1, 0, 0, 0, 5, 5, 1, 0, 0, 0, 6, 23, 7, 1, 0, 0, 0, 5, 116, 89, 11, 1, 0, 0, 0, 3, 521, 2494, 484, 15, 1, 0, 0, 0, 1, 1931, 69366, 87984, 2904, 22, 1, 0, 0, 0, 0, 5906, 1592714, 15456557, 4250015, 22002, 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 k-set.
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LINKS
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FORMULA
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T(n,k) = 0 for k > n!.
T(n,k) = T(n, n!-k).
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EXAMPLE
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Array begins:
====================================================================
n/k| 0 1 2 3 4 5 6 ...
---+----------------------------------------------------------------
0 | 1 1 0 0 0 0 0 ...
1 | 1 1 0 0 0 0 0 ...
2 | 1 2 1 0 0 0 0 ...
3 | 1 3 5 6 5 3 1 ...
4 | 1 5 23 116 521 1931 5906 ...
5 | 1 7 89 2494 69366 1592714 30461471 ...
6 | 1 11 484 87984 15456557 2209040882 263190866673 ...
7 | 1 15 2904 4250015 5329123475 5366409944453 4503264576070573 ...
...
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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, k<=1, my(s=0); forpart(q=n, s+=permcount(q) * polcoef(exp(sum(m=1, k, K(R(q, m))*(x^m-x^(2*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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