How to make a permutation of a card problem and compute its order?

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I hope i understood the rule of the game. (If something went wrong on the road, please mention the bad permutation, and the expected one, i will correct the code.)

Here is the sage implementation of the order of the one step permutation implemented by dealing once the set of cards. The lines

The format of the answer

Every line contains - number of cards - number of rounds to get cards into the original order.

are of course a joke. Let us concentrate on the main task, computing the order.

def getOrder(N, pile_len_list, show_one_step_permutation=False):
    """Compute 
    """
    n_piles = len(pile_len_list)

    cards = [1..N]
    piles = [ [] for _ in range(n_piles) ]    # so far, and we prepend
    k_card = -1
    k_pile = -1

    while cards:    # to be dealt
        k_pile += 1
        k_pile = k_pile % n_piles

        pile_cards_number = min( len(cards), pile_len_list[k_pile] )
        # print( f"........ cards = {cards}" )
        pile_cards = cards[:pile_cards_number]
        cards = cards[pile_cards_number:]
        piles[k_pile] = pile_cards[::-1] + piles[k_pile] 
        # print( f"... cards = {cards} k_pile = {k_pile} piles = {piles} "
        #        f"pile_cards_number = {pile_cards_number} pile_cards = {pile_cards}\n" )

    permuted_list = []
    for k in list(range(n_piles))[::-1]:
        permuted_list += piles[k]
    if show_one_step_permutation:
        print( f"ONE STEP PERMUTATION = {permuted_list}" )
    return lcm( [ len(cycle)
                  for cycle in Permutation(permuted_list).to_cycles() ] )

Now, using the sage interpreter with the above defined function, we can ask for...

sage: getOrder(4, [2,1], True)                                                                                                                                                                        
ONE STEP PERMUTATION = [3, 4, 2, 1]
4
sage:

corresponding to the case in the OP. There, instead of $2\clubsuit$, $3\clubsuit$, $4\clubsuit$, $5\clubsuit$, we use a simpler notation and the "new cards" 1, 2, 3, 4. After dealing one step, we get the order $4\clubsuit$, $5\clubsuit$, $3\clubsuit$, $2\clubsuit$, corresponding the "new cards" new order 3, 4, 2, 1. So position 1, goes to position 4, which goes to position 2, which goes to position 3, which goes to position 1.

sage: p = Permutation( [3, 4, 2, 1] ).inverse()                                                                                                                                                       
sage: p(1)                                                                                                                                                                                            
4
sage: p(4)                                                                                                                                                                                            
2
sage: p(3)                                                                                                                                                                                            
1
sage: p(2)                                                                                                                                                                                            
3
sage: p(3)                                                                                                                                                                                            
1

I had to insert the inverse method, convention... but the order does not change. We take the cycles, then compute their lengths, then take their lcm.

We can also ask for:

sage: getOrder( 9, [1, 2, 3], True ) 
....:                                                                                                                                                                                                 
ONE STEP PERMUTATION = [6, 5, 4, 9, 8, 3, 2, 7, 1]
20
sage: p = Permutation( [6, 5, 4, 9, 8, 3, 2, 7, 1] )                                                                                                                                                  
sage: p.inverse().to_cycles()                                                                                                                                                                         
[(1, 9, 4, 3, 6), (2, 7, 8, 5)]

which corresponds to one of the lines in the link in the OP. The one step permutation is a product of two disjoined cycles of lengths $5$ and $4$, so the needed order is $20$.

In more dramatic cases, we get:

sage: getOrder( 77777, [1, 2, 3, 5, 8, 13, 21, 34, 55,] )                                                                                                                                             
359539078526124596396388
sage: _.factor()                                                                                                                                                                                      
2^2 * 3^2 * 7^2 * 13 * 23 * 29 * 41 * 47 * 59 * 73 * 83 * 3593 * 9497