A329812 Number of permutation polynomials (mod n).

1, 2, 6, 8, 120, 12, 5040, 128, 1296, 240, 39916800, 48, 6227020800, 10080, 720, 8192, 355687428096000, 2592, 121645100408832000, 960, 30240, 79833600, 25852016738884976640000, 768, 384000000, 12454041600, 25509168, 40320, 8841761993739701954543616000000, 1440

Offset

1,2

Comments

a(n) is the number of unique bijective functions from Z/nZ to itself induced by polynomials over Z/nZ.

Links

Kenneth G. Hawes, Table of n, a(n) for n = 1..456

Kenneth G. Hawes, SageMath program for generating the sequence

Kenneth G. Hawes, Additional terms including those with more than 1000 digits, n = 1..5000

G. Keller and F. R. Olson, Counting polynomial function (mod p^n), Duke Mathematical Journal, 35 (1968), 835-838.

Formula

a(n) = Product_{i=1..r} a(p_i^k_i) for n having the unique prime factorization n = Product_{i=1..r} p_i^k_i.

a(p^k) = p! if k=1, a(p^k) = p!*(p-1)^p*p^p if k=2, and a(p^k) = p!*(p-1)^p*p^(p+f(p,k)) if k>2, where f(p,k) = Sum_{i=3..k} A002034(p^i).

Example

For n=3, since it is a prime number, a(3) = 3! = 6.
For n=4=2^2, a(4) = 2!*(2-1)^2*2^2 = 8.

Crossrefs

Formula involves the Kempner function A002034.

Sequence in context: A007849 A100621 A002176 * A124675 A279258 A120709

Adjacent sequences: A329809 A329810 A329811 * A329813 A329814 A329815

Keyword

nonn,mult

Author

Kenneth G. Hawes, Nov 21 2019

Status

approved