A323712 -id:A323712 - cp's OEIS frontend

A373416 Fixed points of A323712: sizes k of card decks that return to original order after k pile shuffles (but not earlier).

1, 3, 4, 6, 9, 12, 22, 27, 28, 36, 46, 52, 60, 70, 78, 81, 100, 102, 148, 166, 172, 180, 190, 196, 198, 238, 243, 262, 268, 270, 292, 310, 316, 348, 358, 366, 372, 382, 388, 420, 460, 462, 478, 486, 502, 508, 540, 556, 598, 606, 612, 646, 652, 660, 676, 700, 708, 718, 729, 742

Offset: 1

R. J. Mathar, Aug 02 2024

nonn

The pile shuffle of a card stack [1,2,3...,n] puts 1 at the bottom of a first auxiliary stack, 2 at the bottom of a second auxiliary stack, 3 above the 1, 4 above the 2, 5 above the 3, 6 above the 4.. eventually the shuffled stack is the first auxiliary stack on top of the second auxiliary stack.

Sequence A263458 contains sizes 2k of card decks that return to reverse order after k pile shuffles. They return to original order after 2k shuffles (2 reversals), so entries of A263458 are candidates for entries here. The case of 30 cards shows that this is not that simple: 30 cards are in reverse order after 5 shuffles, back to original order after 10 shuffles, in reverse order after 15 shuffles (so 30 is in A263458) etc. Because the return to normal order happens already after 10 shuffles (not only at 30), 30 is not in this sequence here.

Cf. A323712, A163781 (even terms halved ?), A263458.

Equals the ordered set {k: A323712(k) = k}.

A263458 Deal a pack of n cards into two piles and gather them up, n/2 times. All n such that this reverses the order of the deck.

Original entry on oeis.org

4, 6, 12, 22, 28, 30, 36, 46, 52, 60, 70, 78, 100, 102, 108, 126, 148, 150, 156, 166, 172, 180, 190, 196, 198, 222, 228, 238, 262, 268, 270, 276, 292, 310, 316, 348, 358, 366, 372, 382, 388, 396, 420, 430, 438, 460, 462, 478, 486, 502, 508, 540, 556, 598

Offset: 1

Christian Perfect, Oct 19 2015

nonn

This seems to be A003628(n)-1; that is, each element of this sequence is one less than a prime congruent to 5 or 7 modulo 8.

			Take a deck of 52 playing cards. Deal it into two piles, then pick up the first pile and put it on top of the other. Do this 26 times. The order of the deck is reversed, so 52 belongs to this sequence.
6 is in the sequence because the 3 shuffles are [1, 2, 3, 4, 5, 6] -> [5, 3, 1, 6, 4, 2] -> [4, 1, 5, 2, 6, 3] -> [6, 5, 4, 3, 2, 1], original reversed. 8 is not in the sequence because the 4 shuffles are [1, 2, 3, 4, 5, 6, 7, 8] -> [7, 5, 3, 1, 8, 6, 4, 2] -> [4, 8, 3, 7, 2, 6, 1, 5] -> [1, 2, 3, 4, 5, 6, 7, 8] -> [7, 5, 3, 1, 8, 6, 4, 2], not the original reversed. - _R. J. Mathar_, Aug 02 2024

Cf. A003628, A373416.

isA263458 := proc(n)
    local L,itr ;
    L := [seq(i,i=1..n)] ;
    for itr from 1 to n/2 do
        L := pileShuf(L) ; # function code in A323712
    end do:
    for i from 1 to nops(L) do
        if op(-i,L) <> i then
            return false ;
        end if;
    end do:
    true ;
end proc:
n := 1;
for k from 2 do
    if isA263458(k) then
        printf("%d %d\n",n,k) ;
        n := n+1 ;
    end if;
end do: # R. J. Mathar, Aug 02 2024

from itertools import cycle
def into_piles(r,deck):
    packs = [[] for i in range(r)]
    for card, pack in zip(range(1,deck+1),cycle(range(r))):
        packs[pack].insert(0,card)
    out = sum(packs,[])
    return Permutation(out)
def has_reversing_property(deck):
    p = power(into_piles(2,deck), deck/2)
    return p==into_piles(1,deck)
[i for i in range(2,400,2) if has_reversing_property(i)]

cp's OEIS Frontend

A373416 Fixed points of A323712: sizes k of card decks that return to original order after k pile shuffles (but not earlier).

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A263458 Deal a pack of n cards into two piles and gather them up, n/2 times. All n such that this reverses the order of the deck.

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