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%I #21 Jan 24 2019 16:01:11
%S 1,1,1,1,1,1,1,1,2,1,1,1,1,6,1,1,1,2,3,24,1,1,1,1,4,12,120,1,1,1,2,3,
%T 16,60,720,1,1,1,1,6,9,80,270,5040,1,1,1,2,1,24,45,400,1890,40320,1,1,
%U 1,1,6,4,96,225,2800,14280,362880,1,1,1,2,3,24,40,576,1575,22400,128520,3628800,1
%N Number A(n,k) of permutations on [n] that are the k-th power of a permutation; square array A(n,k), n>=0, k>=0, read by antidiagonals.
%C Number of permutations p on [n] such that a permutation q on [n] exists with p=q^k.
%H Alois P. Heinz, <a href="/A247005/b247005.txt">Antidiagonals n = 0..140, flattened</a>
%H H. S. Wilf, <a href="http://www.math.upenn.edu/~wilf/DownldGF.html">Generatingfunctionology</a>, 2nd edn., Academic Press, NY, 1994, Theorem 4.8.2.
%e A(3,0) = 1: (1,2,3).
%e A(3,1) = 6: (1,2,3), (1,3,2), (2,1,3), (2,3,1), (3,1,2), (3,2,1).
%e A(3,2) = 3: (1,2,3), (2,3,1), (3,1,2).
%e A(3,3) = 4: (1,2,3), (1,3,2), (2,1,3), (3,2,1).
%e Square array A(n,k) begins:
%e 1, 1, 1, 1, 1, 1, 1, 1, 1, ...
%e 1, 1, 1, 1, 1, 1, 1, 1, 1, ...
%e 1, 2, 1, 2, 1, 2, 1, 2, 1, ...
%e 1, 6, 3, 4, 3, 6, 1, 6, 3, ...
%e 1, 24, 12, 16, 9, 24, 4, 24, 9, ...
%e 1, 120, 60, 80, 45, 96, 40, 120, 45, ...
%e 1, 720, 270, 400, 225, 576, 190, 720, 225, ...
%e 1, 5040, 1890, 2800, 1575, 4032, 1330, 4320, 1575, ...
%p with(combinat): with(numtheory): with(padic):
%p b:= proc(n, i, k) option remember; `if`(n=0, 1, `if`(i<1, 0, add(
%p `if`(irem(j, mul(p^ordp(k, p), p=factorset(i)))=0, (i-1)!^j*
%p multinomial(n, n-i*j, i$j)/j!*b(n-i*j, i-1, k), 0), j=0..n/i)))
%p end:
%p A:= (n, k)-> `if`(k=0, 1, b(n$2, k)):
%p seq(seq(A(n, d-n), n=0..d), d=0..14);
%t multinomial[n_, k_List] := n!/Times @@ (k!); b[_, 1, _] = 1; b[n_, i_, k_] := b[n, i, k] = If[n == 0, 1, If[i < 1, 0, Sum[If[Mod[j, Product[ p^IntegerExponent[k, p], {p, FactorInteger[i][[All, 1]]}]] == 0, (i - 1)!^j*multinomial[n, Join[{n-i*j}, Array[i&, j]]]/j!*b[n-i*j, i-1, k], 0], {j, 0, n/i}]]]; A[n_, k_] := If[k == 0, 1, b[n, n, k]]; Table[A[n, d - n], {d, 0, 14}, {n, 0, d}] // Flatten (* _Jean-François Alcover_, Jan 14 2017, after _Alois P. Heinz_ *)
%Y Columns k=0-10 give: A000012, A000142, A003483, A103619, A103620, A215716, A215717, A215718, A247006, A247007, A247008.
%Y Main diagonal gives A247009.
%Y Cf. A247026 (the same for endofunctions).
%K nonn,tabl
%O 0,9
%A _Alois P. Heinz_, Sep 09 2014
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