A238371 a(1)=1; for n > 1, a(n) = the number of "topped" Mongean shuffles to reorder a stack of n cards to its original order.

1, 1, 3, 3, 5, 5, 4, 4, 9, 9, 11, 11, 9, 9, 5, 5, 12, 12, 12, 12, 7, 7, 23, 23, 8, 8, 20, 20, 29, 29, 6, 6, 33, 33, 35, 35, 20, 20, 39, 39, 41, 41, 28, 28, 12, 12, 36, 36, 15, 15, 51, 51, 53, 53, 36, 36, 44, 44, 24, 24, 20, 20, 7, 7, 65, 65, 36, 36, 69, 69, 60, 60, 42, 42, 15, 15, 20, 20, 52, 52, 81, 81, 83, 83, 9, 9, 60, 60

Offset: 1

Author

R. J. Mathar, Feb 25 2014

Keywords

• nonn

Comments

In the Mongean shuffle, the top card of the stack becomes the top of the new stack, the second of the old stack goes on top of the new stack, the third to the bottom of the new stack, alternating top and bottom of the new stack.

Here we define a shuffle where the top-bottom placements in the new stack alternate in the same way, but the second card of the old stack moves to the *bottom* of the stack.

A single shuffle is a permutation of 1, 2, 3, 4, 5, 6, ... -> ..., 7, 5, 3, 1, 2, 4, 6, ...

The fixed points, where n=a(n), seem to be in A163778.

(The "topped" classification is a nomenclature invented here, to be replaced if this variant appears elsewhere in the literature.)

Links

• Wikipedia, Mongean shuffle

Crossrefs

Cf. A019567 (Mongean shuffle), A294673 (a bisection).

Programs

Maple

topMong := proc(L)
    ret := [op(1,L)] ;
    for k from 2 to nops(L) do
        if type(k,'even') then
            ret := [op(ret),op(k,L)] ;
        else
            ret := [op(k,L),op(ret)] ;
        end if;
    end do:
    ret ;
end proc:
A238371 := proc(n)
    local ca,org,tu ;
    ca := [seq(k,k=1..n)] ;
    org := [seq(k,k=1..n)] ;
    for tu from 1 do
        ca := topMong(ca) ;
        if ca = org then
            return tu;
        end if:
    end do:
end proc:
seq(A238371(n),n=2..88) ;

Mathematica

topMong[L_] := Module[{ret = {L[[1]]}}, For[k = 2, k <= Length[L], k++, If[ EvenQ[k], ret = Append[ret, L[[k]]], ret = Prepend[ret, L[[k]]]]]; ret];
A238371[n_] := Module[{ca, org, tu}, ca = org = Range[n]; For[tu = 1, True, tu++, ca = topMong[ca]; If[ca == org, Return[tu]]]];
Array[A238371, 88] (* Jean-François Alcover, Jul 03 2018, after R. J. Mathar *)

PARI

apply( A238371(n)=znorder(Mod(bitand(n,2)*2-2,n\2*4+3)), [0..99]) \\ M. F. Hasler, Mar 31 2019

Formula

a(A163778(n)) = A163778(n). - Andrew Howroyd, Nov 11 2017