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A362759 Array read by antidiagonals: T(n,k) is the number of nonisomorphic multisets of derangements of an n-set with k derangements.

%I A362759 #10 May 03 2023 21:37:02
%S A362759 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,
%T A362759 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,
%U A362759 200,14337,295058,211600,3028,8,1,1,0,1,5,386,117053,15730237,98019999,13511246,25431,12,1
%N A362759 Array read by antidiagonals: T(n,k) is the number of nonisomorphic multisets of derangements of an n-set with k derangements.
%C A362759 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.
%H A362759 Andrew Howroyd, <a href="/A362759/b362759.txt">Table of n, a(n) for n = 0..1325</a> (first 51 antidiagonals).
%F A362759 T(0,k) = T(2,k) = 1.
%e A362759 Array begins:
%e A362759 ===========================================================
%e A362759 n/k| 0 1   2      3        4           5              6 ...
%e A362759 ---+-------------------------------------------------------
%e A362759 0  | 1 1   1      1        1           1              1 ...
%e A362759 1  | 1 0   0      0        0           0              0 ...
%e A362759 2  | 1 1   1      1        1           1              1 ...
%e A362759 3  | 1 1   2      2        3           3              4 ...
%e A362759 4  | 1 2   7     18       43          93            200 ...
%e A362759 5  | 1 2  16    138     1559       14337         117053 ...
%e A362759 6  | 1 4  84   4642   295058    15730237      706921410 ...
%e A362759 7  | 1 4 403 211600 98019999 36414994209 11282515303088 ...
%e A362759   ...
%o A362759 (PARI) \\ here B(n,k) gives A320032(n,k).
%o A362759 B(n,k) = sum(j=0, n, (-1)^(n-j)*binomial(n,j)*k^j*j!)
%o A362759 K(v)=my(S=Set(v)); prod(i=1, #S, my(k=S[i], c=#select(t->t==k, v)); B(c, k))
%o A362759 R(v, m)=concat(vector(#v, i, my(t=v[i], g=gcd(t, m)); vector(g, i, t/g)))
%o A362759 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}
%o A362759 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!)}
%Y A362759 Columns k=0..3 are A000012, A002865, A362760, A362761.
%Y A362759 Main diagonal is A362762.
%Y A362759 Cf. A000166 (derangements), A320032, A362644, A362648.
%K A362759 nonn,tabl
%O A362759 0,19
%A A362759 _Andrew Howroyd_, May 02 2023