cp's OEIS Frontend

A362899 Array read by antidiagonals: T(n,k) is the number of nonisomorphic multisets of fixed-point-free endofunctions on an n-set with k endofunctions.

%I A362899 #8 May 10 2023 22:39:01
%S A362899 1,1,1,1,0,1,1,0,1,1,1,0,1,2,1,1,0,1,9,6,1,1,0,1,22,162,13,1,1,0,1,63,
%T A362899 3935,4527,40,1,1,0,1,136,81015,1497568,172335,100,1,1,0,1,302,
%U A362899 1369101,384069023,883538845,7861940,291,1,1,0,1,580,19601383,78954264778,3450709120355,725601878962,416446379,797,1
%N A362899 Array read by antidiagonals: T(n,k) is the number of nonisomorphic multisets of fixed-point-free endofunctions on an n-set with k endofunctions.
%C A362899 Isomorphism is up to permutation of the elements of the n-set. Each endofunction can be considered to be a loopless digraph where each node has out-degree 1.
%H A362899 Andrew Howroyd, <a href="/A362899/b362899.txt">Table of n, a(n) for n = 0..1325</a> (first 51 antidiagonals).
%e A362899 Array begins:
%e A362899 ==============================================================
%e A362899 n/k| 0  1      2         3             4                 5 ...
%e A362899 ---+----------------------------------------------------------
%e A362899 0  | 1  1      1         1             1                 1 ...
%e A362899 1  | 1  0      0         0             0                 0 ...
%e A362899 2  | 1  1      1         1             1                 1 ...
%e A362899 3  | 1  2      9        22            63               136 ...
%e A362899 4  | 1  6    162      3935         81015           1369101 ...
%e A362899 5  | 1 13   4527   1497568     384069023       78954264778 ...
%e A362899 6  | 1 40 172335 883538845 3450709120355 10786100835304758 ...
%e A362899   ...
%o A362899 (PARI)
%o A362899 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 A362899 K(v,m) = {prod(i=1, #v, my(g=gcd(v[i],m), e=v[i]/g); (sum(j=1, #v, my(t=v[j]); if(e%(t/gcd(t,m))==0, t)) - 1)^g)}
%o A362899 T(n,k) = {if(n==0, 1, my(s=0); forpart(q=n, s+=permcount(q) * polcoef(exp(sum(m=1, k, K(q,m)*x^m/m, O(x*x^k))), k)); s/n!)}
%Y A362899 Columns k=0..3 are A000012, A001373, A362900, A362901.
%Y A362899 Main diagonal is A362902.
%Y A362899 Cf. A362644, A362759, A362897.
%K A362899 nonn,tabl
%O A362899 0,14
%A A362899 _Andrew Howroyd_, May 10 2023