A245988 Number of pairs of endofunctions f, g on [n] satisfying g^n(f(i)) = f(i) for all i in [n].

1, 1, 10, 141, 9592, 159245, 86252976, 908888155, 1682479423360, 128805405787953, 93998774487116800, 1099662085349496911, 44830846497021739693056, 147548082727234113659293, 3534565745374740945151080448, 1613371163531618738559582856125

Offset: 0

Author

Alois P. Heinz, Aug 08 2014

Keywords

• nonn

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..100

Crossrefs

Main diagonal of A245980.

Cf. A245911.

Programs

Maple

with(numtheory): with(combinat): M:=multinomial:
a:= proc(n) option remember; local l, g; l, g:= sort([divisors(n)[]]),
      proc(k, m, i, t) option remember; local d, j; d:= l[i];
        `if`(i=1, n^m, add(M(k, k-(d-t)*j, (d-t)$j)/j!*
         (d-1)!^j *M(m, m-t*j, t$j) *g(k-(d-t)*j, m-t*j,
        `if`(d-t=1, [i-1, 0], [i, t+1])[]), j=0..min(k/(d-t),
        `if`(t=0, [][], m/t))))
      end; forget(g);
      `if`(n=0, 1, add(g(j, n-j, nops(l), 0)*
      stirling2(n, j)*binomial(n, j)*j!, j=0..n))
    end:
seq(a(n), n=0..20);

Mathematica

multinomial[n_, k_List] := n!/Times @@ (k!); M = multinomial;
b[n_, k0_, p_] := Module[{l, g}, l = Sort[Divisors[p]];
    g[k_, m_, i_, t_] := g[k, m, i, t] = Module[{d, j}, d = l[[i]];
    If[i == 1, n^m, Sum[M[k, Join[{k-(d-t)*j}, Array[(d - t) &, j]]]/j!*
    (d - 1)!^j*M[m, Join[{m - t*j}, Array[t &, j]]]*
    If[d - t == 1, g[k - (d - t)*j, m - t*j, i - 1, 0],
    g[k - (d - t)*j, m - t*j, i, t + 1]], {j, 0, Min[k/(d - t),
    If[t == 0, Infinity, m/t]]}]]]; g[k0, n - k0, Length[l], 0]];
A[n_, k_] := If[k == 0, n^(2*n), Sum[b[n, j, k]*StirlingS2[n,j]*Binomial[n, j]*j!, {j, 0, n}]];
A[0, ] = A[1, ] = 1;
a[n_] := A[n, n];
Table[a[n], {n, 0, 20}] (* Jean-François Alcover, Apr 29 2022, after Alois P. Heinz in A245980 *)

Formula

a(n) = A245980(n,n).