A261430 Number A(n,k) of permutations p of [n] without fixed points such that p^k = Id; square array A(n,k), n>=0, k>=0, read by antidiagonals.
1, 1, 0, 1, 0, 0, 1, 0, 0, 0, 1, 0, 1, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 1, 2, 3, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 0, 9, 0, 15, 0, 0, 1, 0, 0, 2, 0, 0, 40, 0, 0, 0, 1, 0, 1, 0, 3, 24, 105, 0, 105, 0, 0, 1, 0, 0, 0, 0, 20, 0, 0, 0, 0, 0, 0, 1, 0, 1, 2, 9, 0, 175, 0, 2625, 2240, 945, 0, 0
Offset: 0
Examples
Square array A(n,k) begins: 1, 1, 1, 1, 1, 1, 1, 1, 1, ... 0, 0, 0, 0, 0, 0, 0, 0, 0, ... 0, 0, 1, 0, 1, 0, 1, 0, 1, ... 0, 0, 0, 2, 0, 0, 2, 0, 0, ... 0, 0, 3, 0, 9, 0, 3, 0, 9, ... 0, 0, 0, 0, 0, 24, 20, 0, 0, ... 0, 0, 15, 40, 105, 0, 175, 0, 105, ... 0, 0, 0, 0, 0, 0, 210, 720, 0, ... 0, 0, 105, 0, 2625, 0, 4585, 0, 7665, ...
Links
- Alois P. Heinz, Antidiagonals n = 0..140, flattened
Crossrefs
Programs
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Maple
with(numtheory): A:= proc(n, k) option remember; `if`(n<0, 0, `if`(n=0, 1, add(mul(n-i, i=1..j-1)*A(n-j, k), j=divisors(k) minus {1}))) end: seq(seq(A(n, d-n), n=0..d), d=0..14); -
Mathematica
A[0, 0] = A[0, 1] = 1; A[, 0|1] = 0; A[n, k_] := A[n, k] = If[n < 0, 0, If[n == 0, 1, Sum[Product[n - i, {i, 1, j - 1}]*A[n - j, k], {j, Rest @ Divisors[k]}]]]; Table[A[n, d - n], {d, 0, 14}, {n, 0, d}] // Flatten (* Jean-François Alcover, Jan 21 2017, after Alois P. Heinz *)
Formula
E.g.f. of column k: exp(Sum_{d|k, d>1} x^d/d).