A362648 Array read by antidiagonals: T(n,k) is the number of nonisomorphic multisets of involutions on an n-set with k involutions.
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, 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, 383, 1532, 3039, 759, 69, 5, 1, 1, 1, 10, 41, 790, 7093, 38800, 28550, 3263, 92, 6, 1
Offset: 0
Examples
Array begins: ============================================================ n/k| 0 1 2 3 4 5 6 7 ... ---+-------------------------------------------------------- 0 | 1 1 1 1 1 1 1 1 ... 1 | 1 1 1 1 1 1 1 1 ... 2 | 1 2 3 4 5 6 7 8 ... 3 | 1 2 4 7 11 16 23 31 ... 4 | 1 3 10 29 74 173 383 790 ... 5 | 1 3 13 63 315 1532 7093 30499 ... 6 | 1 4 27 258 3039 38800 478902 5414462 ... 7 | 1 4 36 759 28550 1203468 46259693 1561933881 ... 8 | 1 5 69 3263 392641 55494682 7010194951 768995611810 ... ...
Links
- Andrew Howroyd, Table of n, a(n) for n = 0..1325 (first 51 antidiagonals).
Crossrefs
Programs
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PARI
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!)) 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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