Given an integer \$N\$ consider a permutation \$p=p_1,p_2,p_3,\ldots\$ of \$1,\ldots,N-1\$. Let \$P = p_1 , p_1+p_2 \bmod N, p_1+p_2+p_3 \bmod N, \ldots\$ be its prefix sums modulo \$N\$. Sometimes \$P\$ will be a permutation of \$1,\ldots,N-1\$ itself.

For example, \$N=4: p=3,2,1 \rightarrow P=3,1,2\$

Negative examples: \$p=2,3,1 \rightarrow P=2,1,2\$ is not a permutation ; \$p=3,1,2 \rightarrow P=3,0,2\$ is a permutation but not of \$1,\ldots,3\$

Your task is to write a program or function that takes \$N\$ and returns the number of permutations \$p\$ of \$1,\ldots,N-1\$ such that \$P\$ is also a permutation of \$1,\ldots,N-1\$.


You may return integers or integer-valued numbers.

You may return the \$N\$-th term, the first \$N\$ terms or the entire series.

You may ignore/skip odd \$N\$. If you choose to do so you may take \$N\$ or \$N/2\$ as the input.

Other than that default rules and loopholes for integer sequences apply.

This is code-golf, so shortest code in bytes wins. Different languages compete independently.

First few terms:

\$ \begin{align} 2 &\rightarrow 1 \\ 4 &\rightarrow 2 \\ 6 &\rightarrow 4 \\ 8 &\rightarrow 24 \\ 10 &\rightarrow 288 \end{align}\$

OEIS has more.

  • 1
    \$\begingroup\$ OEIS says the problem is NP hard. I'd be willing to see a solution which isn't brute force \$\endgroup\$ Jun 6 at 7:16
  • 2
    \$\begingroup\$ Given an integer 1,…,N−1 doesn't make any sense. Perhaps you mean: Given an integer N consider 1,…,N−1. Also the next sentence talks about p1, p2 etc without any definition as what they represent. \$\endgroup\$
    – Noodle9
    Jun 6 at 12:39
  • \$\begingroup\$ @Noodle9 Oops, fixed. \$\endgroup\$ Jun 6 at 12:43

12 Answers 12


Wolfram Language (Mathematica), 60 bytes


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J, 24 bytes


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Straightforward brute force.


JavaScript (ES7), 88 bytes


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We compute the bitmask \$z=2^n-2\$ where the bits \$1\$ to \$n-1\$ are set (e.g. \$n=4\$ gives \$z=14=1110_2\$).

We start with \$m=z\$ and \$x=0\$. We recursively clear the bits of \$m\$ in all possible orders while keeping track of the sum of said bit indices in \$p\$ and setting the bits \$p \bmod n\$ in \$x\$. (Note that we do not need to keep track of the permutation itself.)

We have a solution whenever we end up with \$x=z\$, in which case the output value \$o\$ is incremented.


Husk, 13 bytes


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           t   # tail: all except the first element of
            ŀ  # the sequence 0..N-1;
S#o            # now, how many times does this occur among
          P    #  get all permutations of this
   mȯ          #  and for each of them
         ∫     #   get the cumulative sums
      m%¹      #   each modulo the input
     O         #   and sort the results

Alternative, also 13 bytes


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            ←  # decrement the input by 1
           ḣ   # get the sequence 1..N
          P    # and get all permutations of this;
 Sn            # now get all common elements between this and
   m(    )     #  for each permutation
        ∫      #   get the cumulative sums
     †%¹       #   each modulo the input
L              # how long is the resulting list of common elements?

Vyxal, 11 bytes


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ɽ           # exclusive range from 0; range(1, N)
 :          # duplicate top of stack
  Ṗ         # get permutations
   v¦       # vectorized cumulative sum
     ⁰      # push N to top of stack
      %     # modulo (vectorizes)
       vs   # vectorized sort
         ^  # flip stack (so range(1, N) is now on top)
          O # count number of instances

Jelly, 9 bytes


A monadic Link that accepts an integer and yields the count of permutations summing to permutations.

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’Œ!ðÄ%f⁸L - Link: integer, N
’         - decrement -> N-1
 Œ!       - all permutations of [1..N-1]
   ð      - start a new dyadic chain, f(permutations, N)
    Ä     - cumulative sums (of each of the permutations)
     %    - modulo N
      f⁸  - filter keep if in (the permutations)
        L - length

Factor + math.combinatorics math.unicode, 71 bytes

[| n | n [1,b) <permutations> [ dup cum-sum [ n mod ] map ⊂ ] count ]

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  • n [1,b) <permutations> Get all the permutations of [1..n) as a virtual sequence.
  • [ ... ] count Count how many of them...
  • dup cum-sum [ n mod ] map ⊂ ...are supersets of their cumulative sum modulo n.

Burlesque, 30 bytes


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J      # Dup
-.     # Decrement
ro     # Range 1..N-1
r@     # Permutations
J      # Dup
bc     # Infinite cycle
j      # Swap
m{     # Map
 q++   # Sum
 pa    # Partial
x/     # Reorder stack
.%     # Modulo
q~[Z]  # Zip with contained in permutations
++     # Sum (count)

05AB1E, 13 bytes


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L           # Push a list in the range [1, (implicit) input]
 ¨          # Remove the last item to make the range [1,input)
  œ         # Get all its permutations
   ʒ        # Filter the permutations by:
    η       #  Get all prefixes of the current permutation
     O      #  Sum each prefix
      I%    #  Modulo the input
        {   #  Sort it
         ā  #  Push a list in the range [1,length] (without popping)
          Q #  Check if both lists are the same
   }g       # After the filter: pop and push the length
            # (which is output implicitly as result)

K (ngn/k), 23 bytes


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+!x#x All length x combinations of 0 1 ... x-1.
a:?>' The unique results of grading up each combination. This gives all permutation, assign these to a:.
x!+\' Cumulative sum of each permutation, modulo x.
a? Find each row in the result in the list of permutations. This gives nulls for non-permutations.
+/~^ Count the non-null values.


Python, 116 bytes

from itertools import*
f=lambda n:(r:=set(range(1,n)))and sum({sum(x[:i])%n for i in r}==r for x in permutations(r))

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Haskell, 102 bytes

import Data.List
f n=length$filter((`elem`p).tail.map(`mod`n).scanl(+)0)p where p=permutations[1..n-1]

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