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A362648 Array read by antidiagonals: T(n,k) is the number of nonisomorphic multisets of involutions on an n-set with k involutions.

%I A362648 #8 May 03 2023 10:52:00
%S A362648 1,1,1,1,1,1,1,1,2,1,1,1,3,2,1,1,1,4,4,3,1,1,1,5,7,10,3,1,1,1,6,11,29,
%T A362648 13,4,1,1,1,7,16,74,63,27,4,1,1,1,8,23,173,315,258,36,5,1,1,1,9,31,
%U A362648 383,1532,3039,759,69,5,1,1,1,10,41,790,7093,38800,28550,3263,92,6,1
%N A362648 Array read by antidiagonals: T(n,k) is the number of nonisomorphic multisets of involutions on an n-set with k involutions.
%C A362648 Isomorphism is up to permutation of the elements of the n-set.
%H A362648 Andrew Howroyd, <a href="/A362648/b362648.txt">Table of n, a(n) for n = 0..1325</a> (first 51 antidiagonals).
%e A362648 Array begins:
%e A362648 ============================================================
%e A362648 n/k| 0 1  2    3      4        5          6            7 ...
%e A362648 ---+--------------------------------------------------------
%e A362648 0  | 1 1  1    1      1        1          1            1 ...
%e A362648 1  | 1 1  1    1      1        1          1            1 ...
%e A362648 2  | 1 2  3    4      5        6          7            8 ...
%e A362648 3  | 1 2  4    7     11       16         23           31 ...
%e A362648 4  | 1 3 10   29     74      173        383          790 ...
%e A362648 5  | 1 3 13   63    315     1532       7093        30499 ...
%e A362648 6  | 1 4 27  258   3039    38800     478902      5414462 ...
%e A362648 7  | 1 4 36  759  28550  1203468   46259693   1561933881 ...
%e A362648 8  | 1 5 69 3263 392641 55494682 7010194951 768995611810 ...
%e A362648   ...
%o A362648 (PARI)
%o A362648 B(c,k)=sum(j=0, c\2, if(k%2, 1, 2^(c-2*j))*k^j*binomial(c, 2*j)*(2*j)!/(2^j*j!))
%o A362648 K(v)=my(S=Set(v)); prod(i=1, #S, my(k=S[i], c=#select(t->t==k, v)); B(c, k))
%o A362648 R(v, m)=concat(vector(#v, i, my(t=v[i], g=gcd(t, m)); vector(g, i, t/g)))
%o A362648 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 A362648 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 A362648 Columns k=0..3 are A000012, A004526, A362649, A362650.
%Y A362648 Main diagonal is A362651.
%Y A362648 Cf. A000085 (involutions), A362644, A362759.
%K A362648 nonn,tabl
%O A362648 0,9
%A A362648 _Andrew Howroyd_, May 01 2023