OFFSET

0,3

LINKS

James Enouen, What is the Perfect Shuffle?, arXiv:1911.07426 [math.CO], 2019.

FORMULA

a(n) = Integral_{x=0..oo} (x^4 - 12x^3 + 36x^2 - 24x)^n*exp(-x) dx.

a(n) = 24^n * A321633(n).

Conjecture: Limit_{n->oo} a(n)/(4n)! = 1/e^3. The conjecture is based on the observation of the asymptotic behavior of A007060 and A193624; it seems that it can be generalized in the following way. Let b(n) be the number of ways to shuffle a deck of k*n cards, with k cards in each of n ranks, so that adjacent cards have different ranks. Then, lim_{n->oo} b(n)/(kn)! = 1/e^(k-1); maybe we could prove it with the help of rook polynomials theory or in some other way. - Sergey Kirgizov, Sep 29 2023

EXAMPLE

a(13) = 3668033946384704437729512814619767610579526911188666362431432294400 is the number of ways to shuffle a standard 52-card deck of playing cards so that no two cards of the same rank are adjacent.

MATHEMATICA

Table[Integrate[(x^4 - 12x^3 + 36x^2 - 24x)^n *Exp[-x], {x, 0, Infinity}], {n, 0, 10}] (* Stefano Spezia, Dec 09 2019 *)

CROSSREFS

KEYWORD

nonn

AUTHOR

David Radcliffe, Dec 07 2019

STATUS

approved