A245988 -id:A245988 - cp's OEIS frontend

A245980 Number A(n,k) of pairs of endofunctions f, g on [n] satisfying g^k(f(i)) = f(i) for all i in [n]; square array A(n,k), n>=0, k>=0, read by antidiagonals.

1, 1, 1, 1, 1, 16, 1, 1, 6, 729, 1, 1, 10, 87, 65536, 1, 1, 6, 213, 2200, 9765625, 1, 1, 10, 141, 8056, 84245, 2176782336, 1, 1, 6, 213, 6184, 465945, 4492656, 678223072849, 1, 1, 10, 87, 9592, 387545, 37823616, 315937195, 281474976710656

Offset: 0

Alois P. Heinz, Aug 08 2014

			Square array A(n,k) begins:
0 :        1,     1,      1,      1,      1,      1, ...
1 :        1,     1,      1,      1,      1,      1, ...
2 :       16,     6,     10,      6,     10,      6, ...
3 :      729,    87,    213,    141,    213,     87, ...
4 :    65536,  2200,   8056,   6184,   9592,   2200, ...
5 :  9765625, 84245, 465945, 387545, 682545, 159245, ...

Alois P. Heinz, Antidiagonals n = 0..80, flattened

Columns k=0-10 give: A062206, A239750, A239771, A241015, A245981, A245982, A245983, A245984, A245985, A245986, A245987.
Main diagonal gives A245988.
Cf. A245910.

with(numtheory): with(combinat): M:=multinomial:
b:= proc(n, k, p) local l, g; l, g:= sort([divisors(p)[]]),
      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; g(k, n-k, nops(l), 0)
    end:
A:= (n, k)-> `if`(k=0, n^(2*n), add(b(n, j, k)*
             stirling2(n, j)*binomial(n, j)*j!, j=0..n)):
seq(seq(A(n, d-n), n=0..d), d=0..12);

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, ] = 1; A[1, ] = 1;
Table[Table[A[n, d - n], {n, 0, d}], {d, 0, 12}] // Flatten (* Jean-François Alcover, Jan 27 2015, after Alois P. Heinz *)

A245911 Number of pairs of endofunctions f, g on [n] satisfying f(g^n(i)) = f(i) for all i in [n].

Original entry on oeis.org

1, 1, 12, 207, 9184, 173225, 46097856, 729481375, 454190410752, 30607186160529, 12762075858688000, 1036636706945881151, 3080713389889966460928, 145084860487902521548921, 124325137916420574135066624, 56537825009822523196823829375

Offset: 0

Alois P. Heinz, Aug 06 2014

nonn

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

Main diagonal of A245910.

Cf. A245988.

with(combinat):
T:= proc(n, j) T(n, j):= binomial(n-1, j-1)*n^(n-j) end:
b:= proc(n, i, k) option remember; `if`(n=0 or i=1, x^n,
      expand(add((i-1)!^j*multinomial(n, n-i*j, i$j)/j!*
      x^(igcd(i, k)*j)*b(n-i*j, i-1, k), j=0..n/i)))
    end:
a:= n-> add((p-> add(n^i*T(n, j)* coeff(p, x, i),
        i=0..degree(p)))(b(j$2, n)), j=0..n):
seq(a(n), n=0..20);

multinomial[n_, k_List] := n!/Times @@ (k!);
b[n_, i_, k_] := b[n, i, k] = Function[{x}, If[n == 0 || i == 1, x^n, Expand[Sum[(i - 1)!^j*multinomial[n, Join[{n - i*j}, Array[i&, j]]]/j!* x^(GCD[i, k]*j)*b[n - i*j, i - 1, k][x], {j, 0, n/i}]]]];
a[n_] := If[n == 0, 1, Sum[Binomial[n - 1, j - 1]*n^(n - j)*b[j, j, n][n], {j, 0, n}]];
a /@ Range[0, 20] (* Jean-François Alcover, Oct 03 2019, after Alois P. Heinz *)

cp's OEIS Frontend

A245980 Number A(n,k) of pairs of endofunctions f, g on [n] satisfying g^k(f(i)) = f(i) for all i in [n]; square array A(n,k), n>=0, k>=0, read by antidiagonals.

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