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A238373 The number of "topped inward" shuffles to reorder a stack of n cards to its original order. 1
3, 4, 2, 4, 7, 12, 9, 8, 21, 12, 15, 15, 30, 16, 30, 40, 35, 60, 21, 57, 24, 24, 90, 63, 27, 28, 12, 12, 31, 220, 33, 63, 180, 420, 37, 225, 39, 24, 182, 99, 60, 40, 306, 264, 195, 48, 49, 60, 51, 144, 306, 84, 462, 60, 264, 265, 180, 240, 35, 35, 63, 144, 612, 544, 67, 1012, 870, 84, 840, 72, 195, 264, 180, 312, 650, 1023, 79, 180, 81, 228, 63, 84, 1740, 783, 87, 88 (list; graph; refs; listen; history; text; internal format)
OFFSET

3,1

COMMENTS

Define a shuffle where the top of the old stack becomes the top of the new stack and is "marked" such that the 2nd, 3rd, 4th card of the old stack are placed right after, just before, right after, just before,... the position of the marked card in the new stack.

This permutes the numbers 1, 2, 3, 4, 5.. ->  1 -> 1,2 -> 3,1,2 -> 3,1,4,2 -> 3,5,1,4,2 etc. Compared to the shuffle in A238371, the odd numbers before the 1 and even numbers after the 1 appear in the opposite order, (old) labels growing inwards towards the (old) top card.

(The "topped inward" classification is a nomenclature invented here; to be replaced by a better name once available.)

The sequence shows how often this shuffle needs to be applied to let the re-shuffled stack return to its original 1, 2, 3,.. order.

Conjecture (equivalent to the Pfister comment in A216066): if the shuffle were defined again in inward order, but with the odd numbers to the left and the even numbers to the right, A003558 becomes the number of shuffles to return to original order.

LINKS

Table of n, a(n) for n=3..88.

EXAMPLE

For n=5, the shuffle of 1,2,3,4,5 is 3,5,1,4,2 (1st shuffle), which becomes 1,2,3,4,5 (2nd shuffle, already original order), so a(5)=2

MAPLE

trackIn := proc(L)

    local ret, po, k ;

    ret := [op(1, L)] ;

    po := 1 ;

    for k from 2 to nops(L) do

        if type(k, 'even') then

            ret := [op(1..po, ret), op(k, L), op(po+1..nops(ret), ret)] ;

        else

            ret := [op(1..po-1, ret), op(k, L), op(po..nops(ret), ret)] ;

            po := po+1 ;

        end if;

    end do:

    ret ;

end proc:

A238373 := proc(n)

    local ca, org, tu ;

    ca := [seq(k, k=1..n)] ;

    org := [seq(k, k=1..n)] ;

    for tu from 1 do

        ca := trackIn(ca) ;

        if ca = org then

            return tu;

        end if:

    end do:

end proc:

seq(A238373(n), n=3..88)

CROSSREFS

Sequence in context: A021296 A323100 A210875 * A078069 A090131 A152833

Adjacent sequences:  A238370 A238371 A238372 * A238374 A238375 A238376

KEYWORD

nonn

AUTHOR

R. J. Mathar, Feb 25 2014

STATUS

approved

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Last modified January 19 09:35 EST 2020. Contains 331048 sequences. (Running on oeis4.)