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There are 3 pending maps:

Identifier

Mp00000: Permutations —topswop⟶ Permutations

Description

Reverse the block of elements at the start of the permutation which has as many elements as the first element of the permutation.

References

[1] Gardner, M. Time travel and other mathematical bewilderments MathSciNet:0905872
[2] Problems in applied mathematics MathSciNet:1086177
[3] Morales, L., Sudborough, H. A quadratic lower bound for Topswops DOI:10.1016/j.tcs.2010.08.011
[4] Pepperdine, A. 73.23 Topswops DOI:10.2307/3619674
[5] Parlett, J. Fixed Point Homing Shuffles DOI:10.46298/dmtcs.14653
[6] Komano, Y., Mizuki, T. Physical Zero-Knowledge Proof Protocols for Topswops and Botdrops DOI:10.1007/s00354-024-00272-3

Sage code

def mapping(pi):
    return pi[:pi[0]][::-1] + pi[pi[0]:]

Created

Apr 20, 2026 at 05:08 by Nathan R. Krause

Updated

Apr 20, 2026 at 05:08 by Nathan R. Krause

1 Comments (hide)

Nathan R. Krause
20 Apr 5:12

I attempted to add the relevant sequences from the OEIS to the references, but the system kept giving me errors. Therefore, I left them out. If they can be added, the sequences are A000375, A000376, A123398, and A174498.

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Identifier

Mp00000: Permutations —topdrop⟶ Permutations

Description

Consider the block of elements at the start of the permutation which has as many elements as the first element of the permutation. Reverse the block and move it to the end.

References

[1] Krause, N. R. The Dynamics and Orbit Structure of the Topdrop Map arXiv:2510.16679

Sage code

def mapping(pi):
    return pi[pi[0]:] + pi[:pi[0]][::-1]

Properties

bijective

Created

Apr 20, 2026 at 04:28 by Nathan R. Krause

Updated

Apr 20, 2026 at 04:28 by Nathan R. Krause

0 Comments

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Identifier

Mp00000: Integer partitions —Decrement the largest parts of a partition⟶ Integer partitions

Description

For partitions with the largest part > 1, we decrement all the largest parts by 1 and increment the other (largest possible) parts to obtain the resulting partition of the same number. The resulting partition may grow in length as needed.

Sage code

def increment_first_part(p,length=None):
    '''
        Increment the part with the smallest index `i` > 1 in partition `p` when such `i` exists.
        When `length` is given it bound the length of the resulting partition. Otherwise, it may become longer than `p`.
    '''
    q = list(p) + [0]*(0 if length is None else length-len(p))
    try:
        # find smallest index i>0 such that q[i] can be incremented
        i = next(i for i in range(1,len(q)) if q[i]<q[i-1])
        q[i] += 1
    except StopIteration:
        if length is None:
            # grow length of the partition
            q.append(1)
    return Partition(q)

def mapping(p):
    if len(p)==0 or p[0]==1: return p
    m0 = max(i+1 for i in range(len(p)) if p[i]==p[0])   # multiplicity of part p[0]
    q = list(p)
    for i in range(m0): q[i] -= 1
    for i in range(m0): q = increment_first_part(q)
    return Partition(q)

Created

Jul 02, 2025 at 19:45 by Max Alekseyev

Updated

Jul 02, 2025 at 19:45 by Max Alekseyev

2 Comments (hide)

Martin Rubey
4 Jul 10:30

Hi Max! Do you have any reference in which context this map appears (possibly implicitly)? Do you know of any desirable properties the map has? I am asking, because each map comes with (a rather high) cost: each map is combined with all the other maps in all possible ways, up to a certain weight. Therefore, I want to make sure that the "best" (or "useful", "naturally occurring", etc) version of the map is added.

Max Alekseyev
7 Jul 2:39

This map participates is the recurrence described in my MO answer: https://mathoverflow.net/q/497101 I'm going to add the corresponding statistics and link it to this map.

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