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
Examples
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, ...
Links
- Alois P. Heinz, Antidiagonals n = 0..80, flattened
Crossrefs
Programs
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Maple
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); -
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, ] = 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 *)