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A217948 List of numbers 2n for which the riffle permutation permutes all except the first and last of the 2n cards. 2
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,1

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).

FORMULA

Apparently a(n) = A179194(n) - 1. [Joerg Arndt, Dec 15 2012]

a(n) = 2 * A051733(n). [Joerg Arndt, Dec 15 2012]

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 *)

CROSSREFS

Equals twice A051733.

Cf. A002326, A051732.

Sequence in context: A136415 A247456 A266383 * A059891 A020213 A011979

Adjacent sequences:  A217945 A217946 A217947 * A217949 A217950 A217951

KEYWORD

nonn

AUTHOR

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

STATUS

approved

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Last modified February 18 04:10 EST 2018. Contains 299298 sequences. (Running on oeis4.)