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A362759
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Array read by antidiagonals: T(n,k) is the number of nonisomorphic multisets of derangements of an n-set with k derangements.
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6
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1, 1, 1, 1, 0, 1, 1, 0, 1, 1, 1, 0, 1, 1, 1, 1, 0, 1, 2, 2, 1, 1, 0, 1, 2, 7, 2, 1, 1, 0, 1, 3, 18, 16, 4, 1, 1, 0, 1, 3, 43, 138, 84, 4, 1, 1, 0, 1, 4, 93, 1559, 4642, 403, 7, 1, 1, 0, 1, 4, 200, 14337, 295058, 211600, 3028, 8, 1, 1, 0, 1, 5, 386, 117053, 15730237, 98019999, 13511246, 25431, 12, 1
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OFFSET
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0,19
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COMMENTS
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Isomorphism is up to permutation of the elements of the n-set. A derangement is a permutation without fixed points. Each derangement can be considered to be a set of disjoint directed cycles excluding singleton loops whose vertices cover the n-set. Permuting the elements of the n-set permutes each of the derangements in the multiset.
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LINKS
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FORMULA
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T(0,k) = T(2,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 0 0 0 0 0 0 ...
2 | 1 1 1 1 1 1 1 ...
3 | 1 1 2 2 3 3 4 ...
4 | 1 2 7 18 43 93 200 ...
5 | 1 2 16 138 1559 14337 117053 ...
6 | 1 4 84 4642 295058 15730237 706921410 ...
7 | 1 4 403 211600 98019999 36414994209 11282515303088 ...
...
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PROG
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(PARI) \\ here B(n, k) gives A320032(n, k).
B(n, k) = sum(j=0, n, (-1)^(n-j)*binomial(n, j)*k^j*j!)
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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