A217948 List of numbers 2n for which the riffle permutation permutes all except the first and last of the 2n cards.

4, 6, 12, 14, 20, 30, 38, 54, 60, 62, 68, 84, 102, 108, 132, 140, 150, 164, 174, 180, 182, 198, 212, 228, 270, 294, 318, 348, 350, 374, 380, 390, 420, 422, 444, 462, 468, 492, 510, 524, 542, 548, 558, 564, 588, 614, 620, 654, 660, 662, 678, 702, 710, 758, 774, 788, 798

Offset: 1

Author

N. J. A. Sloane, Nov 07 2012, based on an email message from Anthony C Robin

Keywords

• nonn

Comments

With 2n cards, a riffle shuffle can be described as a permutation, where r becomes 2r-1 when r <= n and r becomes 2r-2n when r > n. The first and last cards are always left unaltered. Sequence A002326 describes the lengths of the longest orbits in the permutation. E.g. when 2n=10, the permutation can be described as (2,3,5,9,8,6)(4,7). The present sequence gives the values of 2n for which there is just one orbit on the 2n-2 cards, for example the permutation when 2n=12 is (2,3,5,9,6,11,10,8,4,7) containing all the 10 numbers other than 1 & 12.

Tiago Januario (email, Jan 12 2015; see also reference) conjectures that these terms are always one more than a prime. - N. J. A. Sloane, Mar 02 2015

References

• Tiago Januario and Sebastian Urrutia, An Analytical Study in Connectivity of Neighborhoods for Single Round Robin Tournaments, 14th INFORMS Computing Society Conference, Richmond, Virginia, January 11-13, 2015, pp. 188-199; http://dx.doi.org/10.1287/ics.2015.0014
• Tiago Januario, S Urrutia, D de Werra, Sports scheduling search space connectivity: A riffle shuffle driven approach, Discrete Applied Mathematics, Volume 211, 1 October 2016, Pages 113-120; http://dx.doi.org/10.1016/j.dam.2016.04.018

Links

• Olivier Gérard and Vincenzo Librandi, Table of n, a(n) for n = 1..6000 (first 386 terms from Olivier Gérard).
• Sebastián Urrutia, Dominique de Werra, and Tiago Januario, Recoloring subgraphs of K_(2n) for Sports Scheduling, Theoretical Computer Science (2021) Vol. 877, 36-45.

Crossrefs

Equals twice A051733.

Cf. A002326, A051732.

Programs

Mathematica

(* v8 *)  2*Select[Range[2,1000],Function[n,Sort[First[First[ PermutationCycles@Join[Table[2r-1,{r,1,n}],Table[2r-2n,{r,n+1,2n}]]]]]== Range[2,2n-1]]] (* Olivier Gérard, Nov 08 2012 *)

Formula

From Joerg Arndt, Dec 15 2012: (Start)

Apparently a(n) = A179194(n) - 1.

a(n) = 2 * A051733(n). (End)